{"id": "6332.png", "formula": "\\begin{align*} \\cos _ q z & : = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( q ; q ^ 2 ) _ \\infty } ( z ) ^ { 1 / 2 } J ^ { ( 1 ) } _ { - 1 / 2 } ( 2 z ; q ^ 2 ) = \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n \\frac { z ^ { 2 n } } { [ 2 n ] _ q ! } . \\end{align*}"}
{"id": "4101.png", "formula": "\\begin{align*} B { u } _ i ' ( t ) + ( A + B Q ) { u } _ i ( t ) = - H _ i ( u ' ( t ) + Q u ( t ) ) , { u } _ i ( 0 ) = 0 , i = 1 , 2 . \\end{align*}"}
{"id": "1688.png", "formula": "\\begin{align*} T ( b ) - T ( a ) = \\int _ I D T ( p ( t ) ) u \\ d t . \\end{align*}"}
{"id": "5350.png", "formula": "\\begin{align*} B ( u _ f , u _ g ^ * ) & = B ( ( u _ f - f ) + f , ( u _ g ^ * - g ) + g ) \\\\ & = B ( u _ f - f , u _ g ^ * - g ) + B ( u _ f - f , g ) + B ( f , u _ g ^ * - g ) + B ( f , g ) \\\\ & = B ( f , g ) \\end{align*}"}
{"id": "1564.png", "formula": "\\begin{align*} { G } ^ i = \\frac { 1 } { 4 } { g } ^ { i { \\ell } } \\left \\{ \\left [ { F } ^ { 2 } \\right ] _ { x ^ k y ^ { \\ell } } y ^ k - \\left [ { F } ^ { 2 } \\right ] _ { x ^ { \\ell } } \\right \\} . \\end{align*}"}
{"id": "3473.png", "formula": "\\begin{align*} \\mathrm { C M } ( X ^ { n } ) = & ( - 1 ) ^ { n } \\mu ^ { n } + ( - 1 ) ^ { n - 1 } n \\mu ^ { n } + \\sum _ { k = 2 } ^ { n } \\binom { n } { k } ( - \\mu ) ^ { n - k } \\mu ^ { k } \\\\ & + \\sum _ { k = 2 } ^ { n } \\sum _ { i = 2 } ^ { k } \\binom { n } { k } \\binom { k } { i } ( - 1 ) ^ { n - k } \\mu ^ { n - i } \\sigma ^ { i } ( i - 1 ) \\frac { c _ { 1 } } { c _ { ( 1 ) } ^ { \\ast } } L _ { 2 } , ~ n \\geq 2 , \\end{align*}"}
{"id": "5129.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ i } ( \\phi ^ { - 1 } ) _ 0 ( x _ j ) & = \\delta _ { i j } \\\\ \\frac { \\partial } { \\partial x _ i } ( \\phi ^ { - 1 } ) _ 1 ( x _ j ) & = 0 \\end{align*}"}
{"id": "3513.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast \\ast } & = \\frac { \\xi _ { p } \\ln ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) - \\xi _ { q } \\ln ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) } { \\sqrt { 2 \\pi } \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } + \\frac { \\Psi _ { 1 } ^ { \\ast } ( - 1 , \\frac { 3 } { 2 } , 1 ) } { \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) } \\frac { F _ { Y _ { ( 2 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "6377.png", "formula": "\\begin{align*} M : = G - \\frac { g _ { 1 , 1 } + \\cdots + g _ { d , d } } { d } \\ , I _ d , \\end{align*}"}
{"id": "849.png", "formula": "\\begin{align*} \\boldsymbol { \\theta } ^ { i + 1 } = \\boldsymbol { \\theta } ^ { i } + \\tau ^ { i } \\nabla _ { \\boldsymbol { \\theta } } Q ( \\boldsymbol { \\theta } ^ { i } ; \\boldsymbol { \\theta } ^ { i } ) , \\end{align*}"}
{"id": "973.png", "formula": "\\begin{align*} \\phi ( H ) = 0 , \\ \\ \\ H ^ { n } = \\phi _ { 0 } + \\phi _ { 1 } H + \\cdots + \\phi _ { n - 1 } H ^ { n - 1 } . \\end{align*}"}
{"id": "5016.png", "formula": "\\begin{align*} R ^ { n , 4 } : = n ^ { 2 \\alpha + 1 } \\int _ { \\tau _ 1 } ^ { \\tau _ 2 } \\gamma _ s \\left [ \\left ( \\int ^ { s } _ 0 \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) ^ 2 - \\int ^ { s } _ 0 \\psi _ { n , 1 } ^ 2 ( u , s ) d u \\ , \\right ] d s \\rightarrow 0 , \\end{align*}"}
{"id": "8234.png", "formula": "\\begin{align*} | a | = \\sqrt { \\frac { \\sqrt { 1 ^ { 2 } + ( 1 6 m \\beta \\lambda / 3 ) ^ { 2 } } + 1 } { 2 } } \\end{align*}"}
{"id": "189.png", "formula": "\\begin{align*} P _ e ( s ) = \\P ^ s ( ( M ^ 1 , M ^ 2 ) \\neq ( \\hat { M } ^ 1 , \\hat { M } ^ 2 ) ) . \\end{align*}"}
{"id": "7993.png", "formula": "\\begin{align*} h | _ { D ^ \\vee _ 2 } = h _ 1 : D ^ \\vee _ 2 \\rightarrow \\mathbb C \\end{align*}"}
{"id": "7097.png", "formula": "\\begin{align*} \\begin{aligned} ( k , v ) \\cdot _ 0 ( x , 0 ) \\cdot _ 0 ( k , v ) ^ { - 1 } = & ( k , v ) \\cdot _ 0 ( x , 0 ) \\cdot _ 0 ( k ^ { - 1 } , - k \\cdot v ) \\\\ = & ( k \\cdot x , x ^ { - 1 } \\cdot v ) \\cdot _ 0 ( k ^ { - 1 } , - k \\cdot v ) \\\\ = & ( k \\cdot x \\cdot k ^ { - 1 } , k x ^ { - 1 } \\cdot v - k \\cdot v ) . \\end{aligned} \\end{align*}"}
{"id": "7980.png", "formula": "\\begin{align*} E _ + : = \\bigoplus _ { i = 1 } ^ k L _ { i , + } , E _ - : = \\bigoplus _ { i = 1 } ^ k L _ { i , - } . \\end{align*}"}
{"id": "8208.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c } \\lambda _ { + } \\phi ( \\pm a / 2 ) = - \\lambda _ { - } \\phi ( \\pm a / 2 ) \\\\ \\lambda _ { + } \\partial _ { x } \\phi ( \\pm a / 2 ) = - \\lambda _ { - } \\partial _ { x } \\phi ( \\pm a / 2 ) \\end{array} \\right . \\Rightarrow \\left \\lbrace \\begin{array} { c } \\phi ( a / 2 ) = \\phi ( - a / 2 ) = 0 \\\\ \\partial _ { x } \\phi ( a / 2 ) = \\partial _ { x } \\phi ( - a / 2 ) = 0 \\end{array} \\right . , \\end{align*}"}
{"id": "6303.png", "formula": "\\begin{align*} u ( x ) = \\frac { a } { x } + b , \\end{align*}"}
{"id": "1060.png", "formula": "\\begin{align*} \\sum _ { n = \\ell + 1 } ^ \\infty \\| \\widetilde { B _ 0 } S _ n ( t ) \\| _ 1 \\leq \\psi _ 0 * \\theta _ 1 ( t ) , \\end{align*}"}
{"id": "5482.png", "formula": "\\begin{align*} \\mathrm { v o l } ( N _ d ( \\mathrm { C H } ( S ) ) \\cap B ( x , \\rho ) ) & \\leq \\sum _ { r = 0 } ^ { \\lfloor \\rho \\rfloor } \\mathrm { v o l } ( \\mathrm { C H } ( K ) \\cap \\mathrm { A n } ( r ) ) \\\\ & \\lesssim e ^ { a \\rho \\beta } V \\left ( 1 + C '' \\right ) , \\end{align*}"}
{"id": "160.png", "formula": "\\begin{align*} \\forall t > t _ 0 , \\int _ { t _ 0 } ^ t \\dot \\kappa ^ { \\epsilon , L } _ s \\ , d s \\geq \\log t - \\log t _ 0 - \\bar \\chi / t _ 0 = \\log ( t / t _ 0 ) - C ( \\lambda , \\mu , \\bar \\chi ) . \\end{align*}"}
{"id": "4272.png", "formula": "\\begin{align*} M _ b = \\nu _ b . \\end{align*}"}
{"id": "661.png", "formula": "\\begin{align*} q ( A \\times Y ) = \\mu ( A ) , q ( X \\times B ) = \\nu ( B ) , \\end{align*}"}
{"id": "5011.png", "formula": "\\begin{align*} \\gamma _ s : = \\mathbf { 1 } _ { [ 0 , t _ 1 \\wedge t _ 2 ] } ( s ) ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } . \\end{align*}"}
{"id": "928.png", "formula": "\\begin{align*} E _ c ( \\Psi _ { t , \\eta _ i } ^ { q ( t ) } ) ( 0 , h ) = v ' \\Longleftrightarrow \\begin{pmatrix} 0 \\\\ h \\end{pmatrix} = P _ c ( \\Psi _ { t , \\eta _ i } ^ { q ( t ) } ) \\cdot v ' , \\end{align*}"}
{"id": "2885.png", "formula": "\\begin{align*} c ( x ) & = \\{ S ^ i ( x ) : i = 0 , \\dotsc , n - 1 \\} \\\\ & = \\{ ( x _ 1 , x _ 2 , \\dotsc , x _ { n } ) , ( x _ 2 , \\dotsc , x _ { n } , x _ 1 ) , \\dotsc , ( x _ { n } , x _ 1 , \\dotsc , x _ { n - 1 } ) \\} . \\end{align*}"}
{"id": "2194.png", "formula": "\\begin{align*} { \\rm w t } ^ 0 _ { \\rm c l } ( A ) & = { \\rm c l } ( { \\rm w t } ^ 0 ( A ) ) = \\left ( \\sum _ { j \\in \\Z } a _ { 1 j } \\right ) { \\rm c l } ( \\epsilon _ 1 ) + \\dots + \\left ( \\sum _ { j \\in \\Z } a _ { m j } \\right ) { \\rm c l } ( \\epsilon _ m ) \\in P _ { \\rm c l } ^ 0 , \\end{align*}"}
{"id": "7913.png", "formula": "\\begin{align*} D _ + = D { + , 1 } + \\cdots + D _ { + , n _ + } \\subset X _ + D _ - = D _ { - , 1 } + \\cdots + D _ { - , n _ - } \\subset X _ - \\end{align*}"}
{"id": "4157.png", "formula": "\\begin{align*} L _ 0 ^ \\lambda = - \\Delta _ z + \\tfrac 1 4 \\lambda ^ 2 | z | ^ 2 - i \\lambda \\sum _ { j = 1 } ^ n ( a _ j \\partial _ { b _ j } - b _ j \\partial _ { a _ j } ) , \\end{align*}"}
{"id": "7687.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ i ^ k = \\lambda _ { i } ( \\sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \\to \\lambda _ { i } ( - X ( x ^ * ) ) = 0 , \\end{align*}"}
{"id": "6366.png", "formula": "\\begin{align*} Q _ { \\kappa } ( x ) = N _ { \\kappa } \\sum _ { m = 0 } ^ { n - 3 } ( m + 1 ) \\ , c _ { m + 1 } \\ , x ^ { m } + \\frac { N _ { \\kappa } } { 1 - n ^ 2 \\kappa ^ 2 } \\ , x ^ { n - 1 } \\ . \\end{align*}"}
{"id": "2639.png", "formula": "\\begin{align*} ( \\phi ^ { - 1 } ( \\ell ) ) + ( Z _ \\phi ) = ( r _ 1 , r _ 2 , r _ 3 ) + ( Z _ \\phi ) = ( 2 , 2 , 2 ) \\end{align*}"}
{"id": "2909.png", "formula": "\\begin{align*} S ^ m G ( x ) + B ^ { - 1 } d & = S ^ m B ^ { - 1 } F ( A x + e ) = B ^ { - 1 } S ^ { k m } F ( A x + e ) \\\\ & = B ^ { - 1 } F ( S ^ k A x + e ) = B ^ { - 1 } F ( A S x + e ) \\\\ & = G ( S x ) + B ^ { - 1 } d , \\end{align*}"}
{"id": "6202.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + p ( x ) D _ { q ^ { - 1 } } y ( x ) + r ( x ) y ( x ) = 0 , \\end{align*}"}
{"id": "8601.png", "formula": "\\begin{align*} \\begin{bmatrix} P A + A ^ { T } P & P B - A ^ { T } C ^ { T } \\\\ B ^ { T } P - C A & - ( C B + B ^ { T } C ^ { T } ) \\end{bmatrix} = \\begin{bmatrix} - L ^ { T } L & - L ^ { T } W \\\\ - W ^ { T } L & - W ^ { T } W \\end{bmatrix} \\leq 0 . \\end{align*}"}
{"id": "61.png", "formula": "\\begin{align*} \\dot x ( t ) = X _ H ( x ( t ) ) , \\end{align*}"}
{"id": "3526.png", "formula": "\\begin{align*} g _ { 1 } ( u ) = ( 1 + 2 u ) ^ { - t } , \\end{align*}"}
{"id": "4866.png", "formula": "\\begin{align*} \\mathfrak { M } _ { i } ( a _ { i } , q _ { i } ) = \\left \\{ \\alpha _ { i } : \\left | \\alpha _ { i } - \\frac { a _ { i } } { q _ { i } } \\right | \\leq \\frac { 1 } { q _ { i } Q _ { i } } \\right \\} \\end{align*}"}
{"id": "1414.png", "formula": "\\begin{align*} \\textrm { t a n } \\Theta ( Q , \\widetilde Q ) = \\{ \\tan ( \\cos ^ { - 1 } ( \\zeta _ 1 ) ) , \\tan ( \\cos ^ { - 1 } ( \\zeta _ 2 ) ) , \\cdots , \\tan ( \\cos ^ { - 1 } ( \\zeta _ r ) ) \\} . \\end{align*}"}
{"id": "3624.png", "formula": "\\begin{align*} v ( \\tau , w ) \\leq v ( \\tau ' , w ) & v ( \\Delta ( \\tau \\rightarrow \\tau ' ) , w ) = 1 , \\\\ v ( \\tau , w ) > v ( \\tau ' , w ) & v \\left ( { \\sim } \\Delta ( \\tau ' \\rightarrow \\tau ) , w \\right ) = 1 . \\end{align*}"}
{"id": "2777.png", "formula": "\\begin{align*} T _ 2 ( h _ i { } \\geq { } 1 , h _ { i - 1 } { } \\leq { } 1 ) = ( 1 + \\kappa ) { h _ { i - 1 } } \\tfrac { - 1 + h _ { i - 1 } } { 2 - ( 1 + \\kappa ) { h _ { i - 1 } } } { } \\leq { } 0 \\end{align*}"}
{"id": "672.png", "formula": "\\begin{align*} \\int u _ { 1 , s ' } \\ , d \\nu _ { s ' } ^ 1 = \\int u _ { 1 } ^ 0 \\ , d \\nu _ { s } ^ 1 , \\end{align*}"}
{"id": "6436.png", "formula": "\\begin{align*} 0 = \\circ [ \\cdots [ [ [ Q ' , Q ' ] , \\iota _ { \\xi _ { i _ 1 } } ] , \\iota _ { \\xi _ { i _ 2 } } ] , \\ldots , \\iota _ { \\xi _ { i _ k } } ] . \\end{align*}"}
{"id": "542.png", "formula": "\\begin{align*} S _ 2 = O _ { \\alpha , l } \\Big ( \\frac { 1 } { \\sqrt { n } } \\prod \\limits _ { 1 \\leq j \\leq l } \\tau _ 2 ( u _ j / d _ j ) \\Big ) . \\end{align*}"}
{"id": "6129.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\end{gather*}"}
{"id": "2986.png", "formula": "\\begin{align*} H _ 0 ( t ) = 2 \\sqrt { \\pi } t ^ { s } \\frac { \\Gamma ( \\frac { s + 1 } { 2 } ) } { \\Gamma ( \\frac s 2 ) } . \\end{align*}"}
{"id": "7457.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ 0 ^ L \\abs { u ^ n } ^ 2 d x = 0 . \\end{align*}"}
{"id": "8359.png", "formula": "\\begin{align*} \\sum \\limits _ { i } \\alpha _ i = 0 \\end{align*}"}
{"id": "5226.png", "formula": "\\begin{align*} \\alpha = { - ( b _ \\ell + 1 ) ( a _ 1 + 1 ) a _ 1 + ( b _ 1 + 1 ) ( a _ 1 + 1 ) a _ \\ell \\over a _ 1 + b _ 1 + 1 } , \\beta = { ( a _ j + 1 ) ( b _ 1 + 1 ) b _ 1 - ( a _ 1 + 1 ) ( b _ 1 + 1 ) b _ j \\over a _ 1 + b _ 1 + 1 } . \\end{align*}"}
{"id": "3186.png", "formula": "\\begin{align*} \\lim _ { \\gamma _ { + } \\rightarrow 0 ^ { + } } \\mathrm { P } _ { \\Phi + \\Phi ^ { \\mathfrak { b } _ { + } , \\gamma _ { + } } + \\Phi ^ { - \\mathfrak { b } _ { - } , \\gamma _ { - } } } = \\mathrm { P } _ { ( \\Phi + \\Phi ^ { - \\mathfrak { b } _ { - } , \\gamma _ { - } } , \\mathfrak { a } _ { \\mathfrak { b } _ { + } } ) } ^ { \\sharp } \\end{align*}"}
{"id": "7144.png", "formula": "\\begin{align*} g _ { 1 2 } = g _ { 1 3 } & = 0 , \\\\ | g _ { 1 1 } | = 1 , & | g _ { 2 2 } g _ { 3 3 } - g _ { 2 3 } g _ { 3 2 } | = 1 . \\end{align*}"}
{"id": "7414.png", "formula": "\\begin{align*} C ( p / q , f ) = \\sum _ { h = 1 } ^ { q - 1 } | \\hat { f _ q } ( h ) | ^ 2 \\frac { 1 - | \\varphi ( 2 \\pi h p / q ) | ^ 2 } { | 1 - \\varphi ( 2 \\pi h p / q ) | ^ 2 } . \\end{align*}"}
{"id": "938.png", "formula": "\\begin{align*} \\langle \\nabla _ j \\nabla _ i \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { i j } ^ l , \\nabla _ m \\Psi _ t \\rangle = - \\Gamma ^ m _ { i j } + O ( t ) . \\end{align*}"}
{"id": "1549.png", "formula": "\\begin{align*} \\P ( \\widetilde { X } _ { 1 + d } = 1 , \\widetilde { T } _ Y \\leq d ) \\pi ( ( 0 , k ] ) \\leq \\P ( \\widehat { X } _ { 1 + d } = 1 ) \\pi ( ( 0 , k ] ) = \\big ( \\P _ { \\pi } ( X _ 1 = 1 ) \\big ) ^ 2 . \\end{align*}"}
{"id": "7147.png", "formula": "\\begin{align*} | g _ { 2 3 } | = 1 . \\end{align*}"}
{"id": "7125.png", "formula": "\\begin{align*} \\mathbf { u } ' _ 1 + \\cdots + \\mathbf { u } ' _ { \\ell } = \\begin{cases} \\boldsymbol { 0 } , & \\\\ a _ 1 \\mathbf { u } _ 1 + \\cdots + a _ { r } \\mathbf { u } _ { r } , & { } \\end{cases} \\end{align*}"}
{"id": "1028.png", "formula": "\\begin{align*} q _ i = m _ { i + 1 } - m _ i - 1 ( i = 1 , 2 , \\dots , k - 1 ) , \\end{align*}"}
{"id": "8754.png", "formula": "\\begin{align*} \\underline { g } = \\frac { 1 } { p } \\sum _ { g ' \\in \\mathbb { Z } _ p } \\sum _ { k = 0 } ^ { p - 1 } \\xi _ p ^ { ( g - g ' ) k } \\underline { g ' } = \\sum _ { k = 0 } ^ { p - 1 } \\xi _ p ^ { g k } e _ k \\end{align*}"}
{"id": "8288.png", "formula": "\\begin{align*} \\frac { \\sinh ( k ' a ) } { k ' a } = \\mp 1 , \\end{align*}"}
{"id": "6468.png", "formula": "\\begin{align*} [ ( x _ i , X _ \\lambda ) , ( x _ j , X _ \\beta ) ] _ \\mathcal { A } : = \\left ( [ x _ i , x _ j ] _ \\mathfrak { g } , [ X _ \\lambda , \\varrho ( x _ j ) ] - [ X _ \\beta , \\varrho ( x _ i ) ] - \\varphi ( x _ i , x _ j ) + [ X _ \\lambda , X _ \\beta ] \\right ) \\end{align*}"}
{"id": "5814.png", "formula": "\\begin{align*} E _ { 1 } = 1 , E _ { 2 } = 5 , E _ { 4 } = 1 3 8 5 , E _ { n } , \\ldots , \\end{align*}"}
{"id": "8159.png", "formula": "\\begin{align*} \\| \\varphi \\| = \\lim \\limits _ { n \\rightarrow \\infty } \\limsup \\limits _ { t \\rightarrow \\infty } \\| P _ { \\delta } ( W _ { n , m _ 0 ; \\textrm { s u r } } ) e ^ { - i t H _ 0 } \\varphi + P _ { \\delta } ( W _ { n , m _ 0 ; \\textrm { f a r } } ) e ^ { - i t H _ 0 } \\varphi \\| < \\varepsilon \\end{align*}"}
{"id": "4982.png", "formula": "\\begin{align*} N _ t ^ { n , 1 } = n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ t \\psi _ { n , 1 } ( s , t ) d W _ s . \\end{align*}"}
{"id": "4234.png", "formula": "\\begin{align*} 1 - \\frac { \\Psi x _ i } { x _ i } + 1 - \\frac { y _ i } { \\Psi x _ i } - \\left ( 1 - \\frac { y _ i } { x _ i } \\right ) = \\frac { \\Psi ( 1 - \\Psi ) x _ i - ( 1 - \\Psi ) y _ i } { \\Psi x _ i } \\geq \\frac { \\Psi ( 1 - \\Psi ) x _ i - ( 1 - \\Psi ) \\Psi ^ 2 x _ i } { \\Psi x _ i } = ( 1 - \\Psi ) ^ 2 \\geq 0 \\end{align*}"}
{"id": "1148.png", "formula": "\\begin{align*} A = \\sum _ { k = 0 } ^ n A _ k \\end{align*}"}
{"id": "7077.png", "formula": "\\begin{align*} \\varphi _ { i } ( \\gamma _ { i } ) : = a _ { i } \\gamma ^ { b _ { i } } _ { i } + c _ { i } \\gamma _ { i } + d _ { i } , \\forall i \\in \\mathcal { N } , \\end{align*}"}
{"id": "8548.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = A x + B u \\\\ y & = C x + D u , \\end{aligned} \\end{align*}"}
{"id": "6710.png", "formula": "\\begin{align*} ( - 1 ) ^ { ( m - n ) / 2 } \\frac { \\mathbf { c } _ n ( \\lambda ) } { \\mathbf { c } _ m ( \\lambda ) } \\varphi ( \\lambda ) = \\varphi ( - \\lambda ) , \\ ; \\ ; \\lambda \\in \\mathfrak { a } ^ * _ \\mathbb { C } , n , m \\in \\mathbb { Z } . \\end{align*}"}
{"id": "7896.png", "formula": "\\begin{align*} \\mathcal H = \\mathcal H _ + \\oplus \\mathcal H _ - , \\end{align*}"}
{"id": "4727.png", "formula": "\\begin{align*} e _ { ( k ) } H _ { 2 k + 1 } H _ { 2 k } H _ { 2 k + 1 } ^ { - 1 } e _ { ( k ) } = & H _ { 2 k + 1 } e _ { ( k ) } H _ { 2 k } e _ { ( k ) } H _ { 2 k + 1 } ^ { - 1 } \\\\ = & z \\Big ( \\frac { z - z ^ { - 1 } } { q - q ^ { - 1 } } \\Big ) ^ { k - 1 } H _ { 2 k + 1 } e _ { ( k ) } H _ { 2 k + 1 } ^ { - 1 } \\\\ = & e _ { ( k ) } H _ { 2 k } e _ { ( k ) } . \\end{align*}"}
{"id": "6462.png", "formula": "\\begin{align*} \\iota _ { \\nabla _ x ( f e ) } & = [ \\Phi _ 0 ( x ) , \\iota _ { ( f e ) } ] ^ { ( - 1 ) } \\\\ & = \\Phi _ 0 ( x ) [ f ] \\iota _ e + f [ \\Phi _ 0 ( x ) , \\iota _ e ] ^ { ( - 1 ) } \\\\ & = \\iota _ { \\varrho ( x ) [ f ] e + \\nabla _ x e } . \\end{align*}"}
{"id": "7344.png", "formula": "\\begin{align*} f ( w , x , y , z ) = f ( y , z , w , x ) = f ( y , z , x , w ) = f ( x , w , y , z ) . \\end{align*}"}
{"id": "3824.png", "formula": "\\begin{align*} { } _ { 1 ; 2 } ^ { \\kappa } E l l H ^ { \\mu , 4 ; l } _ { k , j , n } ( t _ 1 , t _ 2 ) : = \\sum _ { \\begin{subarray} { c } k _ 1 \\in \\Z _ + , \\mu _ 1 \\in \\{ + , - \\} \\\\ n _ 1 \\in [ - M _ t , - 2 M _ t / 1 5 ] \\cap \\Z , \\\\ \\end{subarray} } \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot \\xi } \\mathcal { F } [ { } _ { } ^ z T _ { k _ 1 , n _ 1 } ^ { \\mu _ 1 ; 2 } ( B ) ( s , \\cdot , V ( s ) ) f ( s , \\cdot , v ) ] ( \\xi ) \\end{align*}"}
{"id": "8871.png", "formula": "\\begin{align*} \\Delta f ( \\lambda ) = - 2 \\sum _ { k = 1 } ^ { N - 1 } \\det \\left ( ( \\lambda I _ N - H ) _ { k k + 1 | k k + 1 } \\right ) . \\end{align*}"}
{"id": "2032.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : p o s t c o l l i s i o n s i g m a } \\left \\{ \\begin{array} { r c l } v ' & = & \\displaystyle { \\frac { v + v _ * } { 2 } + \\frac { | v - v _ * | } { 2 } \\sigma } \\\\ v _ * ' & = & \\displaystyle { \\frac { v + v _ * } { 2 } - \\frac { | v - v _ * | } { 2 } \\sigma } \\end{array} \\right . , \\end{align*}"}
{"id": "4649.png", "formula": "\\begin{align*} 0 \\le \\delta \\leq ( d - \\alpha ) / 2 \\kappa = \\kappa _ \\delta = \\frac { 2 ^ \\alpha \\Gamma ( ( \\delta + \\alpha ) / 2 ) \\Gamma ( ( d - \\delta ) / 2 ) } { \\Gamma ( \\delta / 2 ) \\Gamma ( ( d - \\delta - \\alpha ) / 2 ) } . \\end{align*}"}
{"id": "8064.png", "formula": "\\begin{align*} V ( f ^ { 2 } ) g ( \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) , W ) = W ( f ^ { 2 } ) g ( \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) , V ) . \\end{align*}"}
{"id": "162.png", "formula": "\\begin{align*} C _ { ( m ) } : = \\big ( - \\Delta ^ { \\epsilon } + m ^ 2 \\big ) ^ { - 1 } , \\end{align*}"}
{"id": "4759.png", "formula": "\\begin{align*} J _ { o u t } ^ { ( V ) } \\ ! \\left ( s _ { c h } , J _ { 1 } , \\ldots , J _ { d _ v \\ ! - \\ ! 1 } \\right ) \\ ! = \\ ! J \\ ! \\ ! \\left ( \\ ! \\sqrt { \\sum _ { i = 1 } ^ { d _ v - 1 } \\ ! \\ ! \\left ( J ^ { - 1 } ( J _ { i } ) \\right ) ^ 2 \\ ! + \\ ! s _ { c h } ^ 2 } \\right ) \\ ! , \\end{align*}"}
{"id": "2339.png", "formula": "\\begin{align*} f = f _ 0 + f _ 1 Q + \\ldots + f _ r Q ^ r \\end{align*}"}
{"id": "5081.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { \\tau } ( t - s ) ^ \\alpha \\sigma ' ( X _ s ) \\left ( \\int ^ s _ { \\eta _ n ( s ) } \\left ( s - \\eta _ n ( u ) \\right ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) d s = 0 \\end{align*}"}
{"id": "6925.png", "formula": "\\begin{align*} f - q | _ g = & \\ - L ^ 2 ( x ) p | _ { L ( L ( y ) z ) } + L ( L ( x ) p | _ { L ^ 2 ( y ) z } ) \\\\ \\equiv & \\ - L ^ 2 ( x ) p | _ { L ^ 2 ( y ) z } + L ^ 2 ( x ) p | _ { L ^ 2 ( y ) z } \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "379.png", "formula": "\\begin{align*} \\P _ x [ T _ { A } > t ] = \\frac { \\alpha ( x ) / \\pi ( x ) } { \\| \\alpha / \\pi \\| _ { 2 } ^ 2 } e ^ { - t ( 1 - \\gamma _ 1 ) } + \\sum _ { i = 2 } ^ m c _ { i } f _ i ( x ) e ^ { - t ( 1 - \\gamma _ i ) } . \\end{align*}"}
{"id": "2874.png", "formula": "\\begin{align*} f ( \\mathbf { n } ) = \\sum _ { i = 1 } ^ e a _ i u _ i ( \\mathbf { n } ) , \\end{align*}"}
{"id": "7464.png", "formula": "\\begin{align*} \\mathcal { T } _ { \\left \\{ C _ { i } \\right \\} _ { i = 1 } ^ { r } } : = 2 ^ { - 1 } \\left ( I d + \\mathcal { V } _ { \\left \\{ C _ { i } \\right \\} _ { i = 1 } ^ { r } } \\right ) . \\end{align*}"}
{"id": "8504.png", "formula": "\\begin{align*} t r ( A ^ n ) = t ^ n + \\sum _ { r = 1 } ^ { [ n / 2 ] } ( - 1 ) ^ r \\frac { n } { r } { n - r - 1 \\choose r - 1 } t ^ { n - 2 r } \\delta ^ r , \\end{align*}"}
{"id": "4026.png", "formula": "\\begin{align*} ( Q ^ { - 1 } ) ' ( x ) & = \\frac { 1 } { Q ' ( Q ^ { - 1 } ( x ) ) } = \\frac { - 1 } { h ( x ) } \\\\ ( Q ^ { - 1 } ) '' ( x ) & = - \\frac { Q ^ { - 1 } ( x ) } { h ( x ) ^ 2 } . \\end{align*}"}
{"id": "7987.png", "formula": "\\begin{align*} H _ { ( Y _ + , D _ { Y , + } ) , f _ + , \\vec d _ + } = H _ { ( Y _ - , D _ { Y , - } ) , f _ - , \\vec d _ - } \\end{align*}"}
{"id": "4382.png", "formula": "\\begin{align*} & \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ \\Psi _ 1 \\le 2 \\log r _ 1 \\} } | f | ^ 2 \\\\ \\ge & \\lim _ { j \\rightarrow + \\infty } \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ \\Psi _ 1 \\le 2 \\log r _ 1 \\} \\cap D _ j } | f | ^ 2 \\\\ \\ge & \\lim _ { j \\rightarrow + \\infty } C _ j \\\\ \\ge & G ( 0 ; \\Psi _ 1 , I _ + ( \\Psi _ 1 ) _ o , f ) . \\end{align*}"}
{"id": "505.png", "formula": "\\begin{align*} t _ i & = \\prod _ { \\pi ( x ) = i } t _ x , \\end{align*}"}
{"id": "7101.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { \\sinh ( \\alpha ( t v ) ) } { t } = \\alpha ( v ) . \\end{align*}"}
{"id": "3649.png", "formula": "\\begin{align*} \\limsup _ { R \\to 0 ^ + } \\frac { e ( B _ R ( x _ 0 ) ) } { u ( B _ R ( x _ 0 ) ) } = \\lim _ { \\ell \\to \\infty } \\frac { e ( B _ { \\rho _ \\ell } ( x _ 0 ) ) } { u ( B _ { \\rho _ \\ell } ( x _ 0 ) ) } , \\end{align*}"}
{"id": "6688.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { W } _ t - \\mathcal { W } _ { r r } & \\leq A _ 1 \\mathcal { W } _ r + B ( \\mathcal { W } + m ) + C _ 1 ( \\mathcal { Z } + n ) + E \\\\ & \\leq K _ 1 | \\mathcal { W } _ r | + B \\mathcal { W } + C _ 1 \\mathcal { Z } \\end{aligned} \\end{align*}"}
{"id": "8464.png", "formula": "\\begin{align*} \\frac { C } { 2 } \\log ( M ) + \\frac { C } { 2 } & \\le \\frac { C } { 2 } \\log ( M ) + \\frac { C } { 2 } \\log ( M ) \\\\ & = C \\log ( M ) \\le n . \\end{align*}"}
{"id": "5317.png", "formula": "\\begin{align*} Z ^ { ( n ) } _ t : = n ^ { - 1 } \\sum _ { i = 1 } ^ n \\delta _ { X _ t ^ i } . \\end{align*}"}
{"id": "8511.png", "formula": "\\begin{align*} [ \\textbf { v } \\ , , \\ , \\textbf { w } ] = \\sum _ { j = 1 } ^ n v _ j w _ j , \\end{align*}"}
{"id": "5802.png", "formula": "\\begin{align*} F _ { v - i , i } = \\mathrm { c o n s t . } = C _ v \\end{align*}"}
{"id": "8611.png", "formula": "\\begin{align*} J ( u ) = \\int _ { 0 } ^ { \\infty } \\left \\{ x ^ { T } Q _ c x + 2 x ^ { T } N _ c u + u ^ { T } R _ c u \\right \\} d t . \\end{align*}"}
{"id": "643.png", "formula": "\\begin{align*} H _ n ( s ) = \\xi _ 2 ( s ) + \\Omega ( s ) n ^ { - 2 } + O \\Bigl ( n ^ { - 4 } \\Bigr ) . \\end{align*}"}
{"id": "8424.png", "formula": "\\begin{gather*} L ( f \\otimes D , 1 ) : = { 2 \\pi } \\int _ 0 ^ \\infty ( f \\otimes D ) ( i t ) { \\rm d } t . \\end{gather*}"}
{"id": "6738.png", "formula": "\\begin{align*} \\mu ( \\varepsilon y _ { u + t h } - \\varepsilon y _ u , \\varepsilon v ) + \\nu ( \\mathsf { m } _ \\delta ( \\varepsilon y _ { u + t h } ) - \\mathsf { m } _ \\delta ( \\varepsilon y _ { u } ) , \\varepsilon v ) = ( t h , v ) , \\forall v \\in Y . \\end{align*}"}
{"id": "4742.png", "formula": "\\begin{align*} \\widehat { \\lambda } ( n P ) = n ^ 2 \\widehat { \\lambda } ( P ) + v ( \\psi _ { n } ( P ) ) - \\frac { n ^ 2 - 1 } { 1 2 } v ( \\Delta ) , \\end{align*}"}
{"id": "4775.png", "formula": "\\begin{align*} \\left | S _ { \\kappa , 3 } ^ A \\right | & = \\sum _ { i , j \\in \\mathbb N \\colon \\atop { 3 ( i + j ) \\leq \\kappa } } \\left | \\left \\{ [ n _ 0 , n _ 7 ] \\colon n _ 0 + n _ 7 = \\kappa - 3 ( i + j ) \\right \\} \\right | = \\sum _ { i , j \\in \\mathbb N \\colon \\atop { 3 ( i + j ) \\leq \\kappa } } ( \\kappa - 3 ( i + j ) + 1 ) = a _ \\kappa . \\end{align*}"}
{"id": "1762.png", "formula": "\\begin{align*} \\Omega ( \\alpha , A , z + a f _ k ) = \\{ z + a f _ k \\} \\cup \\bigcup _ { n = 1 } ^ \\infty \\Omega ( \\alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \\end{align*}"}
{"id": "7275.png", "formula": "\\begin{align*} u ( x , t ) & = \\pm { \\sf U } _ \\infty ( x ) + \\Theta _ J ( x , t ) \\\\ & = \\pm { \\sf U } _ \\infty ( x ) + K ( T - t ) ^ { \\frac { \\gamma } { 2 } + J } e _ J ( z ) | x | \\sim \\sqrt { T - t } . \\end{align*}"}
{"id": "2259.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { H _ w ^ 2 } ^ 2 : = \\Vert f '' \\Vert _ { L ^ 2 _ z } ^ 2 + \\Vert ( 1 + z ) f ' \\Vert _ { L ^ 2 _ z } ^ 2 + \\Vert ( 1 + z ) f \\Vert _ { L ^ 2 _ z } ^ 2 . \\end{align*}"}
{"id": "8328.png", "formula": "\\begin{align*} \\delta ( a ) = \\left \\{ \\begin{aligned} & \\left [ \\sum _ { j = 0 } ^ \\infty \\beta _ j P _ { j , j + n } , a \\right ] & & \\textrm { i f } n \\ge 0 \\\\ & \\left [ \\sum _ { j = 0 } ^ \\infty \\beta _ j P _ { j - n , j } , a \\right ] & & \\textrm { i f } n < 0 \\end{aligned} \\right . \\end{align*}"}
{"id": "4173.png", "formula": "\\begin{align*} \\delta _ R ( x , u ) : = ( R x , R ^ 2 u ) . \\end{align*}"}
{"id": "5776.png", "formula": "\\begin{align*} E ( G / H ) = E ( G / ( G ^ 0 H ) ) E ( ( G ^ 0 H ) / H ) = 0 . \\end{align*}"}
{"id": "4457.png", "formula": "\\begin{align*} \\langle f , \\eta ^ { \\odot m } \\rangle = 0 \\mbox { f o r a l l } \\eta \\in A , \\end{align*}"}
{"id": "7822.png", "formula": "\\begin{align*} d \\Theta _ n ( A d _ { \\phi ^ { - 1 } } ( e _ { \\phi ( \\lambda ) } ) ) = \\frac { t ( \\lambda , \\lambda ) } { 2 } e _ { \\theta ( \\lambda ) } ^ \\vee . \\end{align*}"}
{"id": "7232.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left \\{ \\frac { t - x - R } { ( 1 + t - x ) ^ { 1 + b } } \\right \\} ^ { a _ n } \\left \\{ \\frac { ( 2 x ) ^ { 1 - a } } { 1 + t + x } \\right \\} ^ { b _ n } \\quad \\mbox { i n } \\ D \\end{align*}"}
{"id": "6394.png", "formula": "\\begin{align*} V _ 1 = \\{ 4 , 7 \\} , \\ , V _ 2 = \\{ 1 , 6 \\} , \\ , V _ 3 = \\{ 2 , 5 \\} , \\ , V _ 4 = \\{ 3 \\} . \\end{align*}"}
{"id": "8980.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { L ^ 2 _ \\delta ( M , \\partial M ) } = \\| u \\| ^ 2 _ { L ^ 2 _ \\delta ( M ) } + \\| \\gamma u \\| ^ 2 _ { L ^ 2 ( \\partial M ) } . \\end{align*}"}
{"id": "8227.png", "formula": "\\begin{align*} \\phi _ { \\pm } = e ^ { k _ { \\pm } x } , \\end{align*}"}
{"id": "8386.png", "formula": "\\begin{align*} G = \\left [ \\begin{array} { r r r r r r r r r r r r r } 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \\\\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\end{array} \\right ] \\end{align*}"}
{"id": "6864.png", "formula": "\\begin{align*} F ( c ) : = a _ 0 + a _ 1 N ( \\beta ( c ) , c ) - c , c \\in [ a _ 0 , \\infty ) \\end{align*}"}
{"id": "1975.png", "formula": "\\begin{align*} \\bar A ( x ) & : = A ( x _ 0 + \\Theta x ) \\\\ \\bar f ( x ) & : = \\Phi \\Theta ^ { 2 } f ( x _ 0 + \\Theta x ) . \\end{align*}"}
{"id": "7005.png", "formula": "\\begin{align*} X _ { \\Sigma , \\Lambda } = \\bigsqcup _ { i = 1 } ^ s \\widetilde U _ { \\tau _ i } / H , \\end{align*}"}
{"id": "6108.png", "formula": "\\begin{align*} \\mu : = \\max ( \\{ \\abs { y } \\mid y \\in Y \\} \\cup \\{ 2 \\abs { g _ j } + 1 \\mid 1 \\leq j \\leq m \\} ) . \\end{align*}"}
{"id": "2098.png", "formula": "\\begin{align*} ( L x ) ( v ) = \\sum \\limits _ { e \\in E _ v } \\frac { \\delta _ E ( e ) } { \\delta _ V ( v ) } \\frac { 1 } { | e | ^ 2 } \\sum \\limits _ { u \\in e } ( x ( u ) - x ( v ) ) , \\end{align*}"}
{"id": "2566.png", "formula": "\\begin{align*} C _ 2 : = \\dfrac { 1 } { 2 } \\sum _ { j , k = 1 } ^ 3 A _ { j k } A _ { k j } \\ , \\ \\ C _ 3 : = \\sum _ { j , k , l = 1 } ^ 3 A _ { j k } A _ { k l } A _ { l j } \\ , \\end{align*}"}
{"id": "1727.png", "formula": "\\begin{align*} F ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) = \\prod _ { k \\geq 1 } ( 1 - x _ 1 q _ 1 ^ { - k } ) ^ { - 1 } \\cdot \\prod _ { k \\geq 0 } ( 1 - x _ 2 ( q _ 2 \\widetilde { q } _ 2 ) ^ { k / 2 } ) , \\end{align*}"}
{"id": "2247.png", "formula": "\\begin{align*} ( u _ p ^ i , h _ p ^ i ) ( 1 , y ) = ( u _ 0 ^ i , h _ 0 ^ i ) ( y ) , \\ 1 \\leq i \\leq n - 1 . \\end{align*}"}
{"id": "7709.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { g ( n ) } { n } = \\sum _ { n _ 1 , \\ldots , n _ k = 1 } ^ { \\infty } \\frac { ( \\mu * F ) ( n _ 1 , \\ldots , n _ k ) } { n _ 1 \\cdots n _ k } , \\end{align*}"}
{"id": "1966.png", "formula": "\\begin{align*} \\mathbf { d } _ { p + 1 } & = ( - a _ { 1 } ^ { ( p - 1 ) } , \\ldots , - a _ { p - 1 } ^ { ( 1 ) } , z - a _ { p } ^ { ( 0 ) } ) , \\\\ \\mathbf { v } _ { p + 1 } & = ( - a _ { 1 } ^ { ( p ) } A _ { 0 } ^ { ( p + 1 ) } , - \\sum _ { j = p - 1 } ^ { p } a _ { 2 } ^ { ( j ) } A _ { j - p + 1 } ^ { ( p + 1 ) } , \\ldots , - \\sum _ { j = 1 } ^ { p } a _ { p } ^ { ( j ) } A _ { j - 1 } ^ { ( p + 1 ) } ) , \\end{align*}"}
{"id": "3220.png", "formula": "\\begin{align*} \\dim \\mathcal { N } _ { } ( 2 m , \\alpha , L ) = 2 \\dim \\mathcal { M } _ { } ( 2 m , \\alpha , L ) = 2 m ( 2 m + 1 ) ( g - 1 ) + 2 m ^ 2 n . \\end{align*}"}
{"id": "6892.png", "formula": "\\begin{align*} N ^ * = \\bar { \\rho } V _ F N ( g _ 0 + g _ 1 N ^ * , a _ 0 + a _ 1 N ^ * ) \\geq 0 . \\end{align*}"}
{"id": "4599.png", "formula": "\\begin{align*} \\kappa _ t ( \\alpha _ 1 , \\dots , \\alpha _ l ) : = \\kappa ( w _ { \\alpha _ 1 } ( t ) , \\dots , w _ { \\alpha _ l } ( t ) ) \\end{align*}"}
{"id": "5686.png", "formula": "\\begin{align*} x _ 1 ^ { d + 1 } + x _ 2 ^ { d + 1 } + \\cdots + x _ i ^ { d + 1 } & = ( x _ 1 + x _ 2 + \\cdots + x _ i ) x _ { i + 1 } ^ d \\\\ & = ( x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ i ^ 2 ) x _ { i + 1 } ^ { d - 1 } \\\\ & = \\cdots \\\\ & = ( x _ 1 ^ d + x _ 2 ^ d + \\cdots + x _ i ^ d ) x _ { i + 1 } \\end{align*}"}
{"id": "1578.png", "formula": "\\begin{align*} v : = ( v ^ 1 , v ^ 2 , v ^ 3 ) = ( - \\cos x ^ 2 , - \\sin x ^ 2 , f ' ( x ^ 1 ) ) . \\end{align*}"}
{"id": "2599.png", "formula": "\\begin{align*} \\tilde { \\mathbf S } \\psi _ k = \\sum _ { k = 0 } ^ 1 [ \\tilde { \\mathbf S } ] _ { j , k } \\psi _ j \\ , \\end{align*}"}
{"id": "2764.png", "formula": "\\begin{align*} \\lim _ { | x | \\rightarrow \\infty } | x | ^ { s } \\bar { u } _ { j } = 1 , \\quad \\ ; \\ , \\min _ { \\overline { B } _ { 1 } } \\bar { u } _ { j } = \\varepsilon _ { 0 } ^ { s } j \\rightarrow \\infty , \\quad \\end{align*}"}
{"id": "1542.png", "formula": "\\begin{align*} \\Delta _ { r a d } : = \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + \\frac { \\alpha } { 2 } \\frac { 1 } { \\tanh ( \\alpha r ) } \\frac { \\partial } { \\partial r } . \\end{align*}"}
{"id": "3376.png", "formula": "\\begin{align*} \\inf \\left \\{ t \\in [ 0 , 1 ] : \\left ( \\sqrt { x } \\right ) ^ t \\left ( \\frac { x + 1 } { 2 } \\right ) ^ { 1 - t } \\le \\frac { x - 1 } { \\log x } , ( x > 0 ) \\right \\} = \\frac { 2 } { 3 } \\end{align*}"}
{"id": "7456.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\abs { \\lambda ^ n } ^ { - 1 } \\abs { \\Lambda ^ n _ x ( 0 ) } \\abs { u ^ n _ x ( 0 ) } = 0 . \\end{align*}"}
{"id": "1123.png", "formula": "\\begin{align*} a _ i = \\frac { ( \\sigma _ Z ^ 2 \\lambda ^ 2 / 2 + \\rho / 2 ) ( \\omega _ i ^ 2 - \\omega _ { i - 1 } ^ 2 ) } { \\lambda ( \\omega _ i - \\omega _ { i - 1 } ) } = \\left ( \\sigma _ Z ^ 2 \\lambda + \\frac { \\rho } { \\lambda } \\right ) \\cdot \\frac { \\omega _ i + \\omega _ { i - 1 } } { 2 } . \\end{align*}"}
{"id": "7523.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s a _ i ^ j A _ { a _ i } = f _ j | C | - 1 , \\ \\ j = 0 , 1 , \\ldots , s , \\end{align*}"}
{"id": "7631.png", "formula": "\\begin{align*} V _ { \\widehat { a } } \\ = \\ V _ { \\widehat { b m ( a ) } } , \\end{align*}"}
{"id": "2562.png", "formula": "\\begin{align*} \\begin{aligned} Q ( r _ 1 , r _ 2 ; s _ 1 , s _ 2 ) = & \\ , Q ( r _ 1 + r _ 2 , s _ 1 + s _ 2 ) \\oplus \\left ( \\bigoplus _ { k = 1 } ^ { \\min ( r _ 1 , r _ 2 ) } Q ( r _ 1 + r _ 2 - 2 k , s _ 1 + s _ 2 + k ) \\right ) \\\\ & \\ \\ \\oplus \\left ( \\bigoplus _ { k = 1 } ^ { \\min ( s _ 1 , s _ 2 ) } Q ( r _ 1 + r _ 2 + k , s _ 1 + s _ 2 - 2 k ) \\right ) \\ . \\end{aligned} \\end{align*}"}
{"id": "8555.png", "formula": "\\begin{align*} \\Psi ( y ) = \\int _ 0 ^ y \\psi ( v ) \\ d v \\ , y \\geq 0 \\ , \\end{align*}"}
{"id": "3284.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\omega \\cdot A ( x ) e ^ { i x \\cdot \\xi } e ^ { i N ^ { - 1 } _ \\omega ( - \\omega \\cdot A ) ( x ) } d x = \\int _ { \\R ^ 3 } \\omega \\cdot A ( x ) e ^ { i x \\cdot \\xi } d x , \\end{align*}"}
{"id": "1440.png", "formula": "\\begin{align*} c _ { k + 1 } = c _ k \\cdot \\dfrac { A ( k ) } { B ( k + 1 ) } \\ \\ \\ ( k \\ge 0 ) \\enspace . \\end{align*}"}
{"id": "6699.png", "formula": "\\begin{align*} L ( t \\circ \\psi ( \\lambda _ 1 , \\cdot ) ) = l ^ \\tau ( t ) \\circ \\psi ( \\lambda _ 2 , \\cdot ) . \\end{align*}"}
{"id": "7048.png", "formula": "\\begin{align*} \\sum _ { M \\geq 0 } g _ 1 ( M ) \\ , z ^ M & = \\sum _ { k \\geq 1 } \\frac { z ^ { 2 k } } { ( 1 - z ) ^ { k + 1 } } + \\sum _ { k \\geq 0 } \\frac { z ^ { 2 k + 2 } k } { ( 1 - z ) ^ { k + 2 } } \\\\ & = \\sum _ { k \\geq 0 } \\left ( \\frac { z ^ { 2 k + 2 } } { ( 1 - z ) ^ { k + 2 } } + \\frac { z ^ { 2 k + 2 } k } { ( 1 - z ) ^ { k + 2 } } \\right ) \\\\ & = \\sum _ { k \\geq 0 } \\sum _ { j \\geq 0 } z ^ { 2 k + 2 + j } \\binom { k + 1 + j } j ( k + 1 ) . \\end{align*}"}
{"id": "746.png", "formula": "\\begin{align*} \\lim _ { \\zeta _ i \\to z _ i } \\left ( \\frac { e _ n ( z , \\zeta ) } { ( z _ i - \\zeta _ i ) } \\bigg | _ { z _ l = \\zeta _ l , l \\neq i } \\right ) = e _ { n , i } ( z , z ) , \\ ; z \\in \\Omega _ 0 . \\end{align*}"}
{"id": "6955.png", "formula": "\\begin{align*} W ^ { \\{ \\alpha \\} } : = W + \\alpha p p ^ * , \\alpha \\in \\R . \\end{align*}"}
{"id": "7212.png", "formula": "\\begin{align*} \\int _ { \\mu } ^ { \\nu } \\frac { d \\xi } { ( 1 + \\xi ) ^ { 1 + q } } \\le C \\left \\{ \\begin{array} { l l } ( 1 + \\mu ) ^ { - q } & \\mbox { i f } \\ q > 0 , \\\\ \\log ( \\nu + 1 ) & \\mbox { i f } \\ q = 0 , \\\\ ( \\nu + 1 ) ^ { - q } & \\mbox { i f } \\ q < 0 . \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "101.png", "formula": "\\begin{align*} \\dot C _ t = \\frac { 1 } { t ^ 2 } ( A + 1 / t ) ^ { - 2 } = \\frac { C ^ 2 _ t } { t ^ 2 } , \\ddot C _ t & = - \\frac { 2 } { t ^ 3 } A ( A + 1 / t ) ^ { - 3 } \\\\ & = - \\frac { 2 } { t } A C _ t \\dot C _ t . \\end{align*}"}
{"id": "2396.png", "formula": "\\begin{align*} \\dot V ( t ) = - A _ n ^ 2 V ( t ) + A _ n F _ n ( U ( t ) ) , V ( 0 ) = U ( 0 ) , \\end{align*}"}
{"id": "2057.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : n o n l i n e a r j u m p o p } L _ \\mu \\varphi ( x ) : = \\int _ E \\{ \\varphi ( y ) - \\varphi ( x ) \\} P _ \\mu ( x , \\dd y ) . \\end{align*}"}
{"id": "5803.png", "formula": "\\begin{align*} F _ { v - i , i } = 1 . \\end{align*}"}
{"id": "1497.png", "formula": "\\begin{align*} F _ { n , \\lambda } ( x ) = \\sum _ { k = 0 } ^ { n } k ! S _ { 2 , \\lambda } ( n , k ) x ^ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 1 5 ] ) . \\end{align*}"}
{"id": "5010.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) ) ^ 2 \\Psi ^ { n , 6 } _ s d s = 0 , \\end{align*}"}
{"id": "209.png", "formula": "\\begin{align*} \\begin{gathered} R _ { 0 } + \\tilde { R } _ { 0 } > H ( Y _ { 0 } | Y _ { j } Z _ { j } ) , \\\\ R _ { j } + \\tilde { R } _ { j } > H ( Y _ { j } | Y _ { 0 } Z _ { j } ) , \\\\ R _ { 0 } + \\tilde { R } _ { 0 } + R _ { j } + \\tilde { R } _ { j } > H ( Y _ { 0 } Y _ { j } | Z _ { j } ) , \\\\ \\end{gathered} \\end{align*}"}
{"id": "5953.png", "formula": "\\begin{align*} \\Phi ( \\xi x ) ( \\Omega _ \\psi y ) & = L ( \\eta ) ^ \\ast ( \\xi x \\otimes \\Omega _ \\psi y ) \\\\ & = L ( \\eta ) ^ \\ast ( \\xi \\otimes x \\Omega _ \\psi y ) \\\\ & = L ( \\eta ) ^ \\ast ( \\xi \\otimes \\Omega _ \\psi \\sigma ^ \\psi _ { i / 2 } ( x ) y ) \\\\ & = L ( \\eta ) ^ \\ast L ( \\xi \\otimes \\Omega _ \\psi ) \\Omega _ \\psi \\sigma ^ \\psi _ { i / 2 } ( x ) y \\\\ & = \\Phi ( \\xi ) ( x \\Omega _ \\psi y ) \\end{align*}"}
{"id": "4209.png", "formula": "\\begin{align*} \\widehat { A f } ( \\zeta ) = \\hat A \\hat g ( \\zeta ) , \\end{align*}"}
{"id": "5584.png", "formula": "\\begin{align*} \\tau _ { x y } \\eta \\ : \\ z \\in V \\ \\longmapsto \\ \\begin{cases} \\eta ( x ) - 1 & z = x \\ , , \\\\ \\eta ( y ) + 1 & z = y \\ , , \\\\ \\eta ( z ) & \\end{cases} \\end{align*}"}
{"id": "4709.png", "formula": "\\begin{align*} p ( n - k ( 3 k + 1 ) / 2 ) = M _ k ( n ) + \\widetilde { \\mathcal { P } } _ k ( n ) . \\end{align*}"}
{"id": "1955.png", "formula": "\\begin{align*} \\phi _ { 0 } ^ { ( k ) } ( z ) & = \\frac { 1 } { z - a _ { k } ^ { ( 0 ) } - \\sum _ { j = 1 } ^ { p } a _ { k } ^ { ( j ) } \\ , \\phi _ { j - 1 } ^ { ( k + 1 ) } ( z ) } \\\\ \\phi _ { j } ^ { ( k ) } ( z ) & = \\phi _ { 0 } ^ { ( k ) } ( z ) \\ , \\phi ^ { ( k + 1 ) } _ { j - 1 } ( z ) 1 \\leq j \\leq p . \\end{align*}"}
{"id": "276.png", "formula": "\\begin{align*} B Z ^ { \\nu } _ i + Z ^ { \\nu t } _ i B = C _ i ^ { \\nu } . \\end{align*}"}
{"id": "6352.png", "formula": "\\begin{align*} \\mathcal { A } _ { s } ( u , v ) = \\lambda \\int _ \\Omega f ( x , u ) v d x \\end{align*}"}
{"id": "8356.png", "formula": "\\begin{align*} \\sum \\limits _ { i } ^ { N } \\alpha _ i = 0 \\end{align*}"}
{"id": "2925.png", "formula": "\\begin{align*} F A ' A ^ { - 1 } = B ' G A ^ { - 1 } = B ' B ^ { - 1 } F , \\end{align*}"}
{"id": "755.png", "formula": "\\begin{align*} X _ { t , s } ^ \\theta ( x ) = x + \\int _ { t } ^ s \\mu ( X _ { t , r } ^ \\theta ( x ) ) \\ , d r + \\int _ t ^ s \\sigma ( X _ { t , r } ^ \\theta ( x ) ) \\ , d W _ r ^ \\theta , \\end{align*}"}
{"id": "4433.png", "formula": "\\begin{align*} \\sigma u ( e _ 1 , \\cdots , e _ m ) = \\frac { 1 } { m ! } \\sum \\limits _ { \\pi \\in \\Pi _ m } u ( e _ { \\pi ( 1 ) } , \\cdots , e _ { \\pi ( m ) } ) . \\end{align*}"}
{"id": "5826.png", "formula": "\\begin{align*} \\frac { y _ { n + 1 } } { x _ { n + 1 } } = \\prod _ { k \\leq n + 1 } \\frac { p _ k ^ 2 } { p _ k ^ 2 - \\chi ( p _ k ) } = \\left ( \\frac { p _ { n + 1 } ^ 2 } { p _ { n + 1 } ^ 2 - 1 } \\right ) \\left ( \\frac { p _ n ^ 2 } { p _ n ^ 2 - 1 } \\right ) \\prod _ { k \\leq n - 1 } \\frac { p _ k ^ 2 } { p _ k ^ 2 - 1 } . \\end{align*}"}
{"id": "2760.png", "formula": "\\begin{align*} \\partial _ { x _ { k } x _ { l } } ^ { 2 } \\left ( \\frac { 1 } { | x - y | ^ { n + \\sigma p } } \\right ) ( 0 ) = \\frac { ( n + \\sigma p ) [ ( n + \\sigma p + 2 ) y _ { k } y _ { l } - \\delta _ { k l } | y | ^ { 2 } ] } { | y | ^ { n + \\sigma p + 4 } } . \\end{align*}"}
{"id": "8492.png", "formula": "\\begin{align*} & b = z \\sqrt { L / 2 } \\\\ \\iff & z / b = \\sqrt { 2 / L } \\\\ \\iff & ( b - z ) / b = 1 - \\sqrt { 2 / L } \\end{align*}"}
{"id": "8904.png", "formula": "\\begin{align*} \\Phi _ \\rho ( u ) = A ( u ) - \\rho B ( u ) , \\rho \\in I , \\end{align*}"}
{"id": "1665.png", "formula": "\\begin{align*} f _ { i } = { f } _ { i } ^ 0 + \\sum _ { j = 1 } ^ k \\ \\sum _ { \\{ p : j \\in S _ p \\} } ( { f } _ { j } ^ 0 ) | _ { C _ { \\gamma _ p , S _ p } } . \\end{align*}"}
{"id": "5062.png", "formula": "\\begin{align*} L ^ { n , 2 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) ( s - \\eta _ n ( s ) ) ^ \\alpha ( W _ s - W _ { \\eta _ n ( s ) } ) \\left ( \\int _ 0 ^ { \\eta _ n ( s ) } \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) d s , \\end{align*}"}
{"id": "7749.png", "formula": "\\begin{align*} \\mbox { f i n d \\ } u _ { m } ( \\alpha ) \\in V _ { m } ( \\alpha ) \\mbox { s . t . } a ( u _ { m } ( \\alpha ) , v _ { m } ; \\alpha ) = f ( v _ { m } ; \\alpha ) \\forall v _ { m } \\in V _ { m } ( \\alpha ) , \\end{align*}"}
{"id": "833.png", "formula": "\\begin{align*} \\mathbf { H } ^ { r } & = \\mathbf { A } ( \\boldsymbol { \\theta } ) ( \\boldsymbol { x } ^ { r } ) \\mathbf { A } ( \\boldsymbol { \\theta } ) ^ { H } , \\\\ \\mathbf { h } ^ { c } & = \\mathbf { A } ( \\boldsymbol { \\theta } ) \\boldsymbol { x } ^ { c } , \\end{align*}"}
{"id": "2200.png", "formula": "\\begin{align*} \\kappa _ 0 ( A ^ { \\tt s t ' } ) = ( P _ 0 ^ { \\tt s t ' } , Q ^ { \\tt s t } _ 0 , \\rho ) , \\end{align*}"}
{"id": "8450.png", "formula": "\\begin{align*} F & = \\bigcap _ { N = 1 } ^ { \\infty } \\bigcup _ { n = N } ^ { \\infty } \\left \\{ ( x , y ) \\in [ 0 , 1 ] ^ 2 : \\begin{aligned} | T _ { \\beta _ 1 } ^ n x - f _ 1 ( x ) | < \\beta _ 1 ^ { - n \\tau _ 1 ( x ) } \\\\ | T _ { \\beta _ 2 } ^ n y - f _ 2 ( y ) | < \\beta _ 2 ^ { - n \\tau _ 2 ( y ) } \\end{aligned} \\right \\} \\\\ & = \\bigcap _ { N = 1 } ^ { \\infty } \\bigcup _ { n = N } ^ { \\infty } \\bigcup _ { w \\in \\Sigma _ { \\beta _ 1 } ^ n , v \\in \\Sigma _ { \\beta _ 2 } ^ n } J _ { n , \\beta _ 1 } ( w ) \\times J _ { n , \\beta _ 2 } ( v ) . \\end{align*}"}
{"id": "8820.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\psi ( x ) = \\lim _ { x \\to 0 } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha _ { t _ 1 , t _ 2 } ( x ) d t _ 1 d t _ 2 = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha _ { t _ 1 , t _ 2 } ( 0 ) d t _ 1 d t _ 2 = \\psi ( 0 ) \\end{align*}"}
{"id": "16.png", "formula": "\\begin{align*} ( \\alpha _ m , \\beta _ m ) _ { \\rho , E _ { \\rho } ^ m , m } = \\Phi _ { E _ { \\rho } ^ m } ( \\beta _ m ) ( \\xi ) - \\xi = \\Phi _ { E _ { \\rho } ^ n } ( \\operatorname { N } _ { m , n } ( \\beta _ m ) ) ( \\xi ) - \\xi = ( \\alpha , \\beta ) _ { \\rho , E _ { \\rho } ^ n , n } , \\end{align*}"}
{"id": "4608.png", "formula": "\\begin{align*} d _ { 1 , \\mathrm { r e l } } ^ G = \\sum _ { n > 0 } x _ { i , - n } \\frac { \\partial } { \\partial \\varphi _ { i , - n } } - \\frac { 1 } { 2 } \\sum _ { p , q > 0 } ( - 1 ) ^ { \\bar { x } _ i \\bar { x } _ k + \\bar { x } _ i + \\bar { x } _ j } c _ { i , j } ^ k \\varphi _ { k , - p - q } \\frac { \\partial } { \\partial \\varphi _ { i , - p } } \\frac { \\partial } { \\partial \\varphi _ { j , - q } } . \\end{align*}"}
{"id": "5204.png", "formula": "\\begin{align*} \\iota _ { \\Lambda } ^ * \\mathcal { W } = F _ \\Lambda ^ * ( \\mathcal { W } _ { v _ 1 ' } \\boxplus \\mathcal { W } _ { v _ 2 } ) . \\end{align*}"}
{"id": "3771.png", "formula": "\\begin{align*} \\sum _ { i = 0 , 1 } \\big | T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , E ) ( t , x , \\zeta ) + \\hat { \\zeta } \\times T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , B ) ( t , x , \\zeta ) \\big | + \\big | \\widetilde { T } _ { k , j ; n } ^ { b i l ; \\mu , 0 } ( \\mathfrak { m } , E ) ( t , x , \\zeta ) + \\hat { \\zeta } \\times \\widetilde { T } _ { k , j ; n } ^ { b i l ; \\mu , 0 } ( \\mathfrak { m } , B ) ( t , x , \\zeta ) \\big | \\end{align*}"}
{"id": "8895.png", "formula": "\\begin{align*} T ( X , Y ) = \\rho ( Y ) X - \\rho ( X ) Y \\end{align*}"}
{"id": "6905.png", "formula": "\\begin{align*} \\hat { p } ( t , \\xi ) = \\hat { p } ( 0 , e ^ { - t } ( \\xi - \\mu ) + \\mu ) e ^ { - H _ t ( \\xi - \\mu ) } , \\end{align*}"}
{"id": "3265.png", "formula": "\\begin{align*} u ^ s _ 1 : = \\mathcal { M } _ { - A _ 1 , q _ 1 } \\mathcal { S } f _ 1 , u ^ s _ 2 : = \\mathcal { M } _ { A _ 2 , q _ 2 } \\mathcal { S } f _ 2 \\mbox { i n } \\R ^ 3 , \\end{align*}"}
{"id": "2178.png", "formula": "\\begin{align*} \\mathcal { I } = \\left \\{ [ a _ { k } , b _ { k } ] : k \\in K \\right \\} \\end{align*}"}
{"id": "2180.png", "formula": "\\begin{align*} - \\lim _ { \\epsilon \\rightarrow 0 ^ { + } } \\epsilon ^ { - 1 } \\Im \\varphi _ \\mu ( x + i \\epsilon ) = \\int _ \\mathbb { R } \\frac { 1 + s ^ 2 } { ( x - s ) ^ 2 } \\ , d \\sigma _ \\mu ( s ) , x \\in \\mathbb { R } , \\end{align*}"}
{"id": "6725.png", "formula": "\\begin{align*} \\mathsf { F } ' ( E ; H ) = \\dfrac { H } { \\abs { E } } - \\dfrac { E } { \\abs { E } ^ 3 } ( E : H ) . \\end{align*}"}
{"id": "8499.png", "formula": "\\begin{align*} \\rho ( \\varepsilon ) : = \\frac { 2 b d k \\log ( \\frac { 4 b d } { \\varepsilon } ) } { \\varepsilon ^ 2 } + \\frac { 2 k ^ d \\log ( \\frac { 4 k ^ d } { \\varepsilon } ) } { \\varepsilon ^ 2 } + \\frac { \\log ( \\frac { 3 } { \\delta } ) } { 2 \\varepsilon ^ 2 } . \\end{align*}"}
{"id": "1908.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { U } _ { [ n , - 1 ] } } w ( \\gamma ) & = \\sum _ { j = 1 } ^ { p } \\sum _ { k \\in \\mathbb { Z } } a _ { 0 } ^ { ( j ) } A ^ { ( 1 ) } _ { [ k - 1 , j - 1 ] } W _ { [ n - k - 1 , 0 ] } \\\\ & + \\sum _ { j = 2 } ^ { p } \\sum _ { s = 1 } ^ { j - 1 } \\sum _ { \\ell \\in \\mathbb { Z } } \\sum _ { k \\in \\mathbb { Z } } a _ { - s } ^ { ( j ) } \\ , A _ { [ \\ell - 1 , j - s - 1 ] } ^ { ( 1 ) } B _ { [ k - \\ell - 1 , s - 1 ] } ^ { ( 1 ) } W _ { [ n - k - 1 , 0 ] } . \\end{align*}"}
{"id": "6337.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } \\sin ( z ; q ) = \\cos ( q ^ { \\frac { - 1 } { 2 } } z ; q ) . \\end{align*}"}
{"id": "2399.png", "formula": "\\begin{align*} & \\quad \\langle A _ n F _ n ( U ( t ) ) , ( - A _ n ) ^ { - 1 } V ( t ) \\rangle _ { l ^ 2 _ n } = - \\langle F _ n ( U ( t ) ) , V ( t ) \\rangle _ { l ^ 2 _ n } \\\\ & = - \\frac { \\pi } { n } \\sum _ { j = 1 } ^ { n - 1 } \\left [ ( V _ j ( t ) + O _ j ( t ) ) ^ 3 - ( V _ j ( t ) + O _ j ( t ) ) \\right ] V _ j ( t ) \\\\ & \\le - ( 1 - \\epsilon ) \\| V ( t ) \\| _ { l _ n ^ 4 } ^ 4 + C ( \\epsilon ) ( \\| O ( t ) \\| _ { l _ n ^ 4 } ^ 4 + 1 ) . \\end{align*}"}
{"id": "7360.png", "formula": "\\begin{align*} \\max ( x , d ) + \\max ( - x , d ) = d + \\max ( | x | , | d | ) = \\begin{cases} d + | x | & \\ | x | \\ge | d | , \\\\ d + | d | & \\ | x | \\le | d | . \\\\ \\end{cases} \\end{align*}"}
{"id": "3301.png", "formula": "\\begin{align*} D ( s ) \\varphi ( x ) : = \\int _ \\Gamma \\left [ \\partial _ { \\nu _ y } \\mathcal { K } ( | x - y | , s ) \\right ] \\varphi ( y ) d \\Gamma _ y x \\in \\R ^ d \\setminus \\Gamma , \\end{align*}"}
{"id": "1137.png", "formula": "\\begin{align*} e ^ { - a | x | } = \\frac { a } { \\pi } \\int _ { - \\infty } ^ { + \\infty } \\frac { e ^ { - j q x } \\mbox { d } q } { q ^ 2 + a ^ 2 } , ~ ~ ~ ~ a > 0 \\end{align*}"}
{"id": "4632.png", "formula": "\\begin{align*} [ X , X ^ { - 1 } ] = \\sum _ { g \\in T _ 0 ( \\alpha ) } \\epsilon _ g \\{ [ \\alpha ] g [ \\alpha ] ^ { - 1 } g ^ { - 1 } \\} \\end{align*}"}
{"id": "909.png", "formula": "\\begin{align*} A _ 1 = \\frac { 1 } { 3 } ( \\frac { 1 } { 2 } S _ g \\cdot g - \\mathrm { R i c } _ g ) , \\end{align*}"}
{"id": "3426.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & \\cdots & 0 & n \\end{pmatrix} , n > 0 . \\end{align*}"}
{"id": "4016.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\\\ 1 . 0 0 0 0 0 2 5 4 2 6 1 2 6 9 4 & 0 & 0 & 2 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & - 0 . 0 0 0 2 9 4 1 1 7 6 4 7 0 5 9 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & - 0 . 0 0 0 0 0 1 2 7 3 1 1 7 4 7 & 0 \\end{bmatrix} , \\end{align*}"}
{"id": "7331.png", "formula": "\\begin{align*} \\tau _ i = 1 0 ^ { - 3 - \\frac { 2 i } { I } } , ~ i = 1 , \\dots , I . \\end{align*}"}
{"id": "509.png", "formula": "\\begin{align*} \\dim ( M _ i \\cap \\ker A _ i ^ s ) & = \\sum _ { r \\leq s } \\Phi ( x _ i ^ r ) . \\end{align*}"}
{"id": "6562.png", "formula": "\\begin{align*} C : = \\sup \\{ \\| v _ 1 \\| _ { p _ 1 } \\cdots \\| v _ k \\| _ { p _ k } \\colon v = ( v _ 1 , \\ldots , v _ k ) \\in B \\} \\ ; < \\ ; \\infty \\ , . \\end{align*}"}
{"id": "2819.png", "formula": "\\begin{align*} p ( h _ i , \\kappa ) = \\left \\{ \\def \\arraystretch { 2 } \\begin{array} { l l } 2 h _ i - h _ i ^ 2 \\frac { - \\kappa } { 1 - \\kappa } & h _ i \\in \\big ( 0 , 1 \\big ] \\\\ \\frac { h _ i ( 2 - h _ i ) ( 2 - \\kappa h _ i ) } { 2 - ( 1 + \\kappa ) h _ i } & h _ i \\in \\big [ 1 , \\bar { h } ( \\kappa ) \\big ] \\end{array} \\right . \\end{align*}"}
{"id": "2647.png", "formula": "\\begin{align*} \\norm { f } _ { C ^ { 0 , \\beta } [ a , b ] } : = \\sup _ { x \\neq y \\in [ a , b ] } \\frac { | f ( x ) - f ( y ) | } { | x - y | ^ { \\beta } } . \\end{align*}"}
{"id": "3856.png", "formula": "\\begin{align*} \\psi ^ { P , G } [ p ] \\cap M = p , \\psi ^ { Q , G } [ q ] \\cap N = q . \\end{align*}"}
{"id": "1078.png", "formula": "\\begin{align*} \\mathrm { e } ^ { - H _ { 0 } t } f ( \\mathbf { x } ) = \\int _ { \\mathbb { R } ^ { d } } K _ { t } ( \\mathbf { x } , \\mathbf { y } ) f ( \\mathbf { y } ) \\ , \\textnormal { d } \\mathbf { y } . \\end{align*}"}
{"id": "5113.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\limsup _ { n \\rightarrow \\infty } E [ \\Lambda ^ { ( 3 ) } _ { n , \\delta } ] = 0 . \\end{align*}"}
{"id": "3061.png", "formula": "\\begin{align*} c _ { n - 2 , \\xi _ { \\boldsymbol { s } k } , \\xi _ { \\boldsymbol { t } l } } ^ { \\ast \\ast } = \\frac { \\Gamma \\left ( ( n - 2 ) / 2 \\right ) } { ( 2 \\pi ) ^ { ( n - 2 ) / 2 } } \\left [ \\int _ { 0 } ^ { \\infty } x ^ { ( n - 4 ) / 2 } \\overline { \\mathcal { G } } _ { n } \\left ( \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } + \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { t } l } ^ { 2 } + x \\right ) \\mathrm { d } x \\right ] ^ { - 1 } . \\end{align*}"}
{"id": "4146.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } \\chi _ j ( \\lambda ) = 1 \\lambda \\neq 0 , \\end{align*}"}
{"id": "484.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\left ( \\delta ( \\theta _ 1 ) \\cdot \\sqrt { c _ 2 } \\right ) ^ 2 = \\epsilon _ 1 ( \\theta _ 1 , c _ 2 ) = \\epsilon _ 1 ( c _ 1 , c _ 2 ) = \\epsilon _ 1 ( r , g , d , l ) , \\end{align*}"}
{"id": "923.png", "formula": "\\begin{align*} \\Gamma _ c : = \\begin{bmatrix} \\Big ( \\sum _ l \\Gamma _ { i j } ^ l \\Gamma _ { m b } ^ l \\Big ) & \\Big ( \\sum _ l \\Gamma _ { i j } ^ l \\Gamma _ { m m } ^ l \\Big ) \\\\ \\Big ( \\sum _ l \\Gamma _ { m b } ^ l \\Gamma _ { i i } ^ l \\Big ) & \\Big ( \\sum _ l \\Gamma _ { i i } ^ l \\Gamma _ { m m } ^ l \\Big ) \\end{bmatrix} , i \\neq j , \\ , m \\neq b . \\end{align*}"}
{"id": "6442.png", "formula": "\\begin{align*} \\eta ( e _ 1 , e _ 2 ) = - 4 \\ , \\mu \\otimes _ \\mathcal O \\dfrac { \\partial } { \\partial y } . \\end{align*}"}
{"id": "4428.png", "formula": "\\begin{align*} L ( x ) = \\sum _ { j = 1 } ^ n c _ j \\prod _ { i \\neq j } \\frac { x - \\lambda _ i } { \\lambda _ j - \\lambda _ i } \\end{align*}"}
{"id": "1108.png", "formula": "\\begin{align*} - \\frac { 2 } { \\sigma } = k ( \\sigma ) + l ( \\sigma ) \\end{align*}"}
{"id": "6440.png", "formula": "\\begin{align*} \\ell _ 2 \\left ( \\mu \\otimes _ \\mathcal O \\dfrac { \\partial } { \\partial x } , \\mu \\otimes _ \\mathcal O \\dfrac { \\partial } { \\partial y } \\right ) : = \\frac { \\partial \\varphi } { \\partial x } \\ , \\mu \\otimes _ \\mathcal O \\dfrac { \\partial } { \\partial y } - \\frac { \\partial \\varphi } { \\partial y } \\ , \\mu \\otimes _ \\mathcal O \\dfrac { \\partial } { \\partial x } \\end{align*}"}
{"id": "2952.png", "formula": "\\begin{align*} d Y ^ { ( n ) } + b _ { n - 1 } Y ^ { ( n - 1 ) } + \\dots + b _ 1 Y ^ { ( 1 ) } + b _ 0 Y = 0 \\end{align*}"}
{"id": "516.png", "formula": "\\begin{align*} E _ { n s } ( \\dot { x } , x , t ) = - \\left [ 1 + g _ 1 \\dot x L _ { n s } ( \\dot { x } , x , t ) \\right ] L _ { n s } ( \\dot { x } , x , t ) \\ , \\end{align*}"}
{"id": "8257.png", "formula": "\\begin{align*} \\lambda _ { \\pm } = \\frac { \\sqrt { 1 + \\left ( \\frac { 4 \\beta \\hslash ^ { 2 } } { 3 a ^ { 2 } } \\right ) ^ { 2 } } \\pm 1 } { 4 \\beta \\hslash ^ { 2 } / 3 a ^ { 2 } } . \\end{align*}"}
{"id": "2739.png", "formula": "\\begin{align*} \\Phi _ { i } ( x , R \\setminus \\varepsilon ) : = \\Phi _ { i } ( x , R ) - \\Phi _ { i } ( x , \\varepsilon ) , \\quad \\mathcal { M } : = \\| u \\| _ { C ^ { \\sigma p + \\alpha } ( B _ { 2 R } ( 0 ) ) } + 1 , \\end{align*}"}
{"id": "6845.png", "formula": "\\begin{align*} p ( 0 , g ) = p _ { } ( g ) , g \\in \\mathbb { R } , \\end{align*}"}
{"id": "4279.png", "formula": "\\begin{align*} A ( \\rho ) & = \\tilde { \\tau } ^ { - 1 } \\rho ( \\tilde { \\tau } ) \\zeta _ \\phi ^ { p , \\infty } ( \\rho ) \\\\ & = \\tilde { \\tau } _ G ^ { - 1 } \\zeta _ \\phi ^ { p , \\infty } ( \\rho ) \\rho ( \\tilde { \\tau } _ G ) \\zeta _ \\phi ^ { p , \\infty } ( \\rho ) ^ { - 1 } \\zeta _ \\phi ^ { p , \\infty } ( \\rho ) \\\\ & = \\tilde { \\tau } _ G ^ { - 1 } \\zeta _ \\phi ^ { p , \\infty } ( \\rho ) \\rho ( \\tilde { \\tau } _ G ) . \\end{align*}"}
{"id": "6635.png", "formula": "\\begin{align*} Q _ i ^ { - 1 } : = \\{ k a ( t ) n g _ i : \\ , k \\in K , t \\ge 0 , n \\in N _ 0 \\} . \\end{align*}"}
{"id": "2349.png", "formula": "\\begin{align*} \\nu _ Q ( a _ { i j k } Q '^ k ) \\geq \\nu _ Q ( a _ i h ^ j ) = \\nu ( a _ i Q ^ i ) + ( j - i ) \\nu ( Q ) > \\nu ( a _ l ) + ( l + j - i ) \\nu ( Q ) . \\end{align*}"}
{"id": "2139.png", "formula": "\\begin{align*} d \\sigma _ { \\mu } ^ { - } ( s ) = ( q / p ) \\sum \\nolimits _ { k = 1 } ^ { \\infty } \\sigma _ { \\mu } ^ { + } \\big ( [ k ^ m , ( k + 1 ) ^ m ) \\big ) \\ , \\delta _ { - k ^ m } . \\end{align*}"}
{"id": "4760.png", "formula": "\\begin{align*} J _ { o u t } ^ { ( C ) } \\ ! ( J _ { 1 } , \\ldots , J _ { d _ c - 1 } ) \\ ! = \\ ! 1 \\ ! - \\ ! J _ { o u t } ^ { ( V ) } \\ ! \\left ( 0 , 1 \\ ! - \\ ! J _ { 1 } , \\ldots , 1 \\ ! - \\ ! J _ { d _ c \\ ! - \\ ! 1 } \\right ) . \\end{align*}"}
{"id": "641.png", "formula": "\\begin{align*} H _ n ( s ) & : = \\pi ^ { - s } \\Gamma ( s ) \\Bigg ( \\zeta ( \\widetilde { \\Delta } _ n , s ) - V _ \\alpha ( s ) \\widetilde a ( s ) n ^ { 2 - 2 s } \\Bigg ) \\\\ & = \\pi ^ { - s } \\Gamma ( s ) \\Bigg ( \\zeta ( \\Delta , s ) + \\frac { s \\pi ^ 2 } { 3 } \\zeta ( \\Delta , s - 1 ) + O \\Bigl ( n ^ { - 4 } \\Bigr ) \\Bigg ) . \\end{align*}"}
{"id": "3963.png", "formula": "\\begin{align*} \\bigl [ V _ { u } ( f ) h ^ { e _ { n } } _ { u } \\bigr ] ( \\gamma ) = f ( \\gamma ) e _ { n } . \\end{align*}"}
{"id": "142.png", "formula": "\\begin{align*} \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } & \\leq c \\big ( \\lambda \\eta _ t + \\lambda ^ 2 \\gamma _ t + | \\mu - m ^ 2 | \\big ) \\Big ( \\frac { 1 } { m ^ 2 _ t } { \\bf 1 } _ { d = 2 } + \\frac { 1 } { m _ t } { \\bf 1 } _ { d = 3 } + \\frac { 1 } { m ^ 4 _ t } \\Big ) \\\\ & + \\frac { c \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } } { m ^ 2 _ t } \\big ( \\lambda \\eta _ t + \\lambda ^ 2 \\gamma _ t + | \\mu - m ^ 2 | \\big ) + c \\sum _ { i = 1 } ^ 3 \\lambda ^ i R ^ { ( i ) } _ { t } \\big ( \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } \\big ) , \\end{align*}"}
{"id": "5144.png", "formula": "\\begin{align*} \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( 2 a c + 2 ) = \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( 2 a c + 4 ) , \\ ; a = 2 \\bmod 6 \\end{align*}"}
{"id": "4212.png", "formula": "\\begin{align*} ( | \\nu | + | \\nu ' | + n ) \\norm { g } _ { L ^ 2 ( \\R ^ { 2 n } ) } ^ 2 & = \\norm { H ^ { 1 / 2 } g } _ { L ^ 2 ( \\R ^ { 2 n } ) } ^ 2 \\\\ & = \\big ( ( - \\Delta _ z + \\tfrac 1 4 | z | ^ 2 ) g , g \\big ) \\\\ & = \\big \\Vert ( - \\Delta _ z ) ^ { 1 / 2 } g \\big \\Vert _ { L ^ 2 ( \\R ^ { 2 n } ) } ^ 2 + \\tfrac 1 4 \\big \\Vert | z | g \\big \\Vert _ { L ^ 2 ( \\R ^ { 2 n } ) } ^ 2 \\\\ & \\le C \\norm { A ^ { 1 / 2 } g } _ { L ^ 2 ( \\R ^ { 2 n } ) } ^ 2 \\\\ & = C ( 2 | \\nu | + n ) \\norm { g } _ { L ^ 2 ( \\R ^ { 2 n } ) } ^ 2 . \\end{align*}"}
{"id": "1655.png", "formula": "\\begin{align*} a _ { e _ 1 + \\dots + e _ n } \\ , = \\ , \\frac { a _ { e _ 1 + \\dots + e _ { n - 1 } } \\cdot a _ { { e _ 1 } + \\dots + e _ { n - 2 } + e _ n } \\cdot a _ { e _ { n - 1 } + e _ n } \\cdot a _ { { \\bf 0 } } } { a _ { e _ 1 + \\dots + e _ { n - 2 } } \\cdot a _ { e _ { n - 1 } } \\cdot a _ { e _ n } } , \\end{align*}"}
{"id": "827.png", "formula": "\\begin{align*} [ w ] _ { Q B _ { \\beta , p } ( I ) } = [ \\widetilde { w } ] _ { Q B _ { \\beta , p } } . \\end{align*}"}
{"id": "2303.png", "formula": "\\begin{align*} \\sup _ { x \\geq 2 0 } \\left \\| ( u , h ) x ^ { \\frac { 1 } { 4 } } \\right \\| _ { L _ y ^ { \\infty } } \\leq \\sup _ { x \\geq 2 0 } \\left \\| ( u _ x , h _ x ) x ^ { \\frac { 5 } { 4 } } \\right \\| _ { L _ y ^ { \\infty } } \\lesssim \\| ( u , v , h , g ) \\| _ { X _ { 1 } \\cap Y _ { 1 } \\cap Y _ { 2 } } \\end{align*}"}
{"id": "3503.png", "formula": "\\begin{align*} \\overline { G } _ { ( 2 ) } ( u ) = \\ln \\left [ 1 + \\exp ( - u ) \\right ] . \\end{align*}"}
{"id": "1173.png", "formula": "\\begin{align*} \\big ( ( \\hat { \\mu } _ n - \\mu ) * \\gamma _ { \\sigma } \\big ) ( f ) = ( \\hat { \\mu } _ n - \\mu ) ( f * \\phi _ { \\sigma } ) . \\end{align*}"}
{"id": "4695.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ; q ) _ \\infty } \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ { n - 1 } q ^ { n ( n - 1 ) / 2 + n | k | } ( 1 - q ^ n ) . \\end{align*}"}
{"id": "5351.png", "formula": "\\begin{align*} Q ( u , v ) = \\langle \\gamma ( - \\Delta ) ^ { s / 2 } u , ( - \\Delta ) ^ { s / 2 } v \\rangle + \\langle q u , v \\rangle . \\end{align*}"}
{"id": "76.png", "formula": "\\begin{align*} - Y \\circ u + \\nabla _ t w - \\nabla ( F + X ) ( u ) [ w ] = v . \\end{align*}"}
{"id": "360.png", "formula": "\\begin{align*} b ( n ) : = f ( n ) / \\log n , \\end{align*}"}
{"id": "613.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\Big ( \\frac { \\sin ^ 2 ( \\pi x ) } { \\pi ^ 2 } + \\frac { z ^ 2 } { n ^ 2 } \\Big ) ^ { - 1 } d x = \\frac { 1 } { \\frac { z } { n } \\sqrt { ( \\frac { z } { n } ) ^ 2 + \\frac { 1 } { \\pi ^ 2 } } } . \\end{align*}"}
{"id": "1505.png", "formula": "\\begin{align*} e ^ { x } \\phi _ { p , \\lambda } ( x ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( n ) _ { p , \\lambda } } { n ! } x ^ { n } , ( p \\in \\mathbb { Z } ) . \\end{align*}"}
{"id": "8762.png", "formula": "\\begin{align*} q ^ { r - 2 a } [ 2 \\ell - r ] + q ^ { 2 \\ell - a } [ a ] + q ^ { 2 c + r - 2 \\ell - a } & [ r - 2 c - a ] \\frac { q ^ { - 2 r } K ^ { - 2 } - 1 } { q ^ { 4 c - 2 r } K ^ { - 2 } - 1 } \\\\ & + q ^ { 2 c - 2 \\ell } [ 2 c ] \\frac { q ^ { - 2 a } K ^ { - 2 } - 1 } { q ^ { 4 c - 2 r } K ^ { - 2 } - 1 } = [ 2 \\ell ] . \\end{align*}"}
{"id": "1672.png", "formula": "\\begin{align*} q _ { \\vec { \\alpha } } \\Vdash ` ` \\dot { z } _ { \\alpha _ 0 } < \\dots < \\dot { z } _ { \\alpha _ n } \\dot { f } ( \\{ \\dot { z } _ { \\alpha _ 0 } , \\dots , \\dot { z } _ { \\alpha _ n } \\} ) = i ( \\vec { \\alpha } ) \\end{align*}"}
{"id": "7974.png", "formula": "\\begin{align*} X _ + = \\mathbb P _ { \\mathbb P ^ r } ( \\mathcal O ( - 1 ) ^ { r ^ \\prime + 1 } \\oplus \\mathcal O ) X _ - = \\mathbb P _ { \\mathbb P ^ { r ^ \\prime } } ( \\mathcal O ( - 1 ) ^ { r + 1 } \\oplus \\mathcal O ) . \\end{align*}"}
{"id": "1315.png", "formula": "\\begin{align*} f ( t , x , v ) = \\tilde R _ + ( x ) M _ { T _ w } ( x , v ) , \\tilde R _ + ( x ) = \\int _ { \\{ v \\cdot n _ x > 0 \\} } f ( t , x , v ) | v \\cdot n _ x | \\d v , \\end{align*}"}
{"id": "8434.png", "formula": "\\begin{align*} \\varpi _ 1 = 2 \\lambda - 2 - 2 \\beta _ 0 ^ 2 - 4 \\left \\| \\beta \\right \\| ^ 2 \\end{align*}"}
{"id": "6746.png", "formula": "\\begin{align*} \\mu \\int _ { \\Omega } \\varepsilon S ' ( \\overline { u } , h ) : \\varepsilon p + \\nu \\int _ { \\Omega } \\mathsf { m } ' \\left ( \\varepsilon \\overline { y } ; \\varepsilon S ' ( \\overline { u } , h ) \\right ) : \\varepsilon p = - \\alpha \\int _ { \\Omega } h \\cdot \\overline { u } = - J _ u ( \\overline { y } , \\overline { u } ) h . \\end{align*}"}
{"id": "4163.png", "formula": "\\begin{align*} \\Lambda _ k ^ \\lambda g = \\sum _ { | \\nu | _ 1 = k } \\sum _ { \\nu ' \\in \\N ^ n } ( g , \\Phi _ { \\nu , \\nu ' } ^ \\lambda ) \\Phi _ { \\nu , \\nu ' } ^ \\lambda , g \\in L ^ 2 ( \\R ^ n ) , \\end{align*}"}
{"id": "2429.png", "formula": "\\begin{align*} & ( H _ i - q ^ { - 1 } ) ( H _ i + q ) = 0 , \\\\ & H _ i H _ { i + 1 } H _ i = H _ { i + 1 } H _ i H _ { i + 1 } , \\\\ & H _ i H _ j = H _ j H _ i , \\quad | i - j | > 1 . \\end{align*}"}
{"id": "4066.png", "formula": "\\begin{align*} \\psi _ { i _ { \\sigma ( 1 ) } } \\cdots \\psi _ { i _ { \\sigma ( k ) } } = \\psi _ { i _ { \\sigma ' \\tau ( 1 ) } } \\cdots \\psi _ { i _ { \\sigma ' \\tau ( k ) } } = \\psi _ { i _ { \\sigma ' ( 1 ) } } \\cdots \\psi _ { i _ { \\sigma ' ( k ) } } . \\end{align*}"}
{"id": "1154.png", "formula": "\\begin{align*} \\left ( | f | ^ 2 \\right ) ^ 2 = \\left ( \\sum _ { \\mathcal { I } } \\left ( f ^ \\mathcal { I } \\right ) ^ 2 \\right ) ^ 2 = \\sum _ { \\mathcal { I } } \\sum _ \\mathcal { J } \\left ( f ^ \\mathcal { I } \\right ) ^ 2 \\left ( f ^ \\mathcal { J } \\right ) ^ 2 \\end{align*}"}
{"id": "2735.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\sigma } _ { p } u ( x ) = c _ { n , \\sigma p } P . V . \\int _ { \\mathbb { R } ^ { n } } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { n + \\sigma p } } d y , \\end{align*}"}
{"id": "6260.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { x } f ( t ) h ( t / q ) \\left ( \\frac { 1 - t ( 1 + q ) } { q ( 1 - q ) t ^ 2 } u ( t / q ) + \\frac { 1 } { ( 1 - q ) ^ 2 t ^ 2 } \\right ) y ( t ) d _ q t = G ( x ) - \\lim _ { n \\rightarrow \\infty } G ( q ^ n x ) . \\end{align*}"}
{"id": "5539.png", "formula": "\\begin{align*} p _ \\star & = \\sup \\left \\{ p > 0 : \\sum _ { n \\geq 0 } S _ n ^ { ( p ) } < \\infty \\right \\} . \\end{align*}"}
{"id": "2945.png", "formula": "\\begin{align*} \\hat { V } ( \\xi , t ) : = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\int d ^ n x , e ^ { - i x \\cdot \\xi } V ( x , t ) . \\end{align*}"}
{"id": "6673.png", "formula": "\\begin{align*} \\begin{aligned} & \\sup \\limits _ { t \\in [ 0 , T _ 0 ] } \\| u _ 1 ( t , \\cdot ) - u _ 2 ( t , \\cdot ) \\| _ { C ^ 0 ( [ 0 , m ( t ) ] ) } + \\sup \\limits _ { t \\in [ 0 , T _ 0 ] } \\| v _ 1 ( t , \\cdot ) - v _ 2 ( t , \\cdot ) \\| _ { C ^ 0 ( [ 0 , n ( t ) ] ) } + \\| h _ 1 - h _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + \\\\ & + \\| g _ 1 - g _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } \\leq K ( \\| u _ { 0 1 } - u _ { 0 2 } \\| _ { C ^ 2 ( [ 0 , m _ 0 ] ) } + \\| v _ { 0 1 } - v _ { 0 2 } \\| _ { C ^ 2 ( [ 0 , n _ 0 ] ) } + | h _ { 0 1 } - h _ { 0 2 } | + | g _ { 0 1 } - g _ { 0 2 } | ) . \\end{aligned} \\end{align*}"}
{"id": "7093.png", "formula": "\\begin{align*} \\mathcal { G } : = K \\times \\mathfrak { p } \\times [ 0 , 1 ] . \\end{align*}"}
{"id": "1351.png", "formula": "\\begin{align*} \\| u \\| _ { X ^ { s , b } } = \\| \\langle k \\rangle ^ { s } \\langle \\tau + k ^ 2 \\rangle ^ { b } \\widehat { u } \\| _ { L ^ 2 _ \\tau \\ell ^ 2 _ k } = \\| \\langle k \\rangle ^ s \\langle \\tau \\rangle ^ b \\widehat { u } ( k , \\tau - k ^ 2 \\rangle ) \\| _ { \\ell ^ 2 _ k L ^ 2 _ \\tau } = \\| W _ { - t } u \\| _ { H ^ s _ x H ^ b _ t } , \\end{align*}"}
{"id": "3954.png", "formula": "\\begin{align*} \\widehat { \\xi ^ \\mathbf { G } } ( 0 , \\dots , 0 ) = \\int _ { \\R ^ { d ( d - 1 ) } } \\phi ( { y ^ \\star } ) \\prod _ { j = 1 } ^ { d - 1 } g _ { \\tilde { a } ( y ^ \\star ) } ( z ^ { j + 1 } ) d z ^ 2 \\dots d z ^ d = \\phi ( { y ^ \\star } ) \\end{align*}"}
{"id": "6261.png", "formula": "\\begin{align*} f ( q ^ n x ) = q ^ { \\frac { n ( n - 1 ) } { 2 } } x ^ n f ( x ) ( n \\in \\mathbb { N } _ 0 ) . \\end{align*}"}
{"id": "4935.png", "formula": "\\begin{gather*} W ( x , t ) = \\max \\{ h _ { i j } ( x , t ) \\zeta ^ i \\zeta ^ j : g _ { i j } ( x ) \\zeta ^ i \\zeta ^ j = 1 \\} , \\\\ p ( u ) = - \\log ( u - \\frac { 1 } { 2 } \\min u ) , \\end{gather*}"}
{"id": "3115.png", "formula": "\\begin{align*} a ( { w } , \\varphi ) = b ( u - J z _ { \\mathrm { n c } } , \\varphi ) \\varphi \\in V . \\end{align*}"}
{"id": "3073.png", "formula": "\\begin{align*} c _ { n } ^ { \\ast \\ast } & = \\frac { ( m + n - 2 ) ( m + n - 4 ) \\Gamma ( n / 2 ) } { ( 2 \\pi ) ^ { n / 2 } m ^ { 2 } } \\left [ \\int _ { 0 } ^ { \\infty } t ^ { n / 2 - 1 } \\left ( 1 + \\frac { 2 t } { m } \\right ) ^ { - ( m + n - 4 ) / 2 } \\mathrm { d } t \\right ] ^ { - 1 } \\\\ & = \\frac { ( m + n - 2 ) ( m + n - 4 ) \\Gamma ( n / 2 ) } { ( m \\pi ) ^ { n / 2 } m ^ { 2 } B ( \\frac { n } { 2 } , ~ \\frac { m - 4 } { 2 } ) } , ~ i f ~ m > 4 , \\end{align*}"}
{"id": "6122.png", "formula": "\\begin{align*} ( - 1 ) ^ { n - k } a _ k = \\sum _ { 1 \\leq i _ 1 < i _ 2 < \\dots < i _ { n - k } \\leq n } \\lambda _ { i _ 1 } \\lambda _ { i _ 2 } \\cdots \\lambda _ { i _ { n - k } } , \\ , \\ , \\ , k = 0 , 1 , \\dots , n - 1 , \\end{align*}"}
{"id": "7287.png", "formula": "\\begin{align*} { a } _ 0 L _ 1 ^ { p + 1 - q } \\{ \\beta ( \\beta + n - 2 ) - q L _ 1 ^ { q - 1 } \\} & = - L _ 1 ^ p , \\\\ { a } _ 0 L _ 1 ^ { 1 - q } \\{ \\beta ( \\beta + n - 2 ) - q L _ 1 ^ { q - 1 } \\} & = - 1 . \\end{align*}"}
{"id": "752.png", "formula": "\\begin{align*} \\mu _ t = \\mu _ 0 + t _ 1 \\varphi _ 1 + \\cdots + t _ n \\varphi _ n , \\end{align*}"}
{"id": "6836.png", "formula": "\\begin{align*} D ( t ) & = \\int _ { 0 } ^ { t } [ a _ 0 + a _ 1 N ( s ) ] d s - \\frac { ( \\int _ { 0 } ^ { t } e ^ { s - t } [ a _ 0 + a _ 1 N ( s ) ] d s ) ^ 2 } { \\int _ { 0 } ^ { t } e ^ { 2 ( s - t ) } [ a _ 0 + a _ 1 N ( s ) ] d s } . \\end{align*}"}
{"id": "293.png", "formula": "\\begin{align*} \\mu _ x : = e ^ \\gamma \\log x \\prod _ { p < x } \\Big ( 1 - \\frac { 1 } { p } \\Big ) . \\end{align*}"}
{"id": "6236.png", "formula": "\\begin{align*} & \\int \\frac { k ( q x ) } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\Big ( D _ q u ( x ) + u ( x ) u ( q x ) - \\frac { q ^ n x } { ( 1 - q ) } u ( q x ) + \\frac { [ n ] _ q } { ( 1 - q ) } \\Big ) y ( x ) d _ q x = \\\\ & \\frac { k ( x ) } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\Big ( y ( x ) u ( x ) - D _ { q ^ { - 1 } } y ( x ) \\Big ) . \\end{align*}"}
{"id": "1534.png", "formula": "\\begin{align*} C _ { r } : = 2 ( \\pi - \\phi ( r ) ) \\sinh ( \\beta r ) . \\end{align*}"}
{"id": "312.png", "formula": "\\begin{align*} f ( A _ n ) \\le \\frac { \\pi } { 4 } \\frac { M _ q } { m _ q ^ 2 } e ^ \\gamma { \\rm d } ( { \\rm L } _ n ) = \\frac { q } { n } \\ , b _ q f ( q ) \\end{align*}"}
{"id": "5766.png", "formula": "\\begin{align*} G \\mbox { i s t o r s i o n - f r e e } \\ \\Longleftrightarrow \\ | E ( G ) | = 1 \\end{align*}"}
{"id": "4762.png", "formula": "\\begin{align*} \\left | \\mathcal S _ { \\kappa , \\gamma } \\right | = \\binom { \\kappa + 2 ^ \\gamma - 1 } { \\kappa } = \\binom { \\kappa + 2 ^ \\gamma - 1 } { 2 ^ \\gamma - 1 } . \\end{align*}"}
{"id": "5739.png", "formula": "\\begin{align*} \\pi _ b \\cdot \\pi _ { [ a , b ] } = \\pi _ { [ a - 1 , b ] } + ( b - a + 1 ) \\pi _ { [ a , b + 1 ] } . \\end{align*}"}
{"id": "7312.png", "formula": "\\begin{align*} a _ j ( \\tau ) \\lesssim e ^ { - ( \\frac { \\gamma } { 2 } + j ) \\tau } e ^ { ( j - J - { \\sf c } _ 1 ) \\tau } = e ^ { - ( \\frac { \\gamma } { 2 } + J + { \\sf c } _ 1 ) \\tau } = ( T - t ) ^ { \\frac { \\gamma } { 2 } + J + { \\sf c } _ 1 } . \\end{align*}"}
{"id": "5038.png", "formula": "\\begin{align*} Q ^ { n , 6 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( s - \\eta _ n ( s ) ) ^ { 2 \\alpha + 1 } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) d s . \\end{align*}"}
{"id": "7319.png", "formula": "\\begin{align*} | w _ 2 | & \\lesssim T ^ { \\frac { 1 } { 4 } { \\sf d } _ 1 } | x | = 1 , \\\\ w _ 2 | _ { t = 0 } & = 0 | x | > 1 . \\end{align*}"}
{"id": "3869.png", "formula": "\\begin{align*} - \\Delta _ p G = \\delta _ { x _ 0 } \\Omega \\end{align*}"}
{"id": "6521.png", "formula": "\\begin{align*} \\sum \\limits _ { i = m } ^ \\infty \\alpha _ i \\leq C _ { n , r } \\alpha _ m . \\end{align*}"}
{"id": "6408.png", "formula": "\\begin{align*} C _ o & = c _ { 1 , 1 } c _ { 3 , 2 } + c _ { 1 , 3 } c _ { 1 , 2 } + c _ { 1 , 2 } c _ { 1 , 3 } \\\\ & = ( w _ 1 - w _ 1 ^ 2 ) ( - w _ 2 w _ 3 ) + ( - w _ 1 w _ 3 ) ( - w _ 1 w _ 2 ) + ( - w _ 1 w _ 2 ) ( - w _ 1 w _ 3 ) \\\\ & = ( 3 w _ 1 ^ 2 - w _ 1 ) w _ 2 w _ 3 , \\end{align*}"}
{"id": "6796.png", "formula": "\\begin{align*} \\lambda \\frac { \\partial ^ 2 u } { \\partial t ^ 2 } + \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\Biggl ( E I \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } \\Biggr ) = f ( u ) . \\end{align*}"}
{"id": "8865.png", "formula": "\\begin{align*} \\det A ^ { ( k + 1 ) } = a _ { k + 1 k + 1 } \\det A ^ { ( k ) } - a _ { k k - 1 } a _ { k - 1 k } \\det A ^ { ( k - 1 ) } , \\ \\ k = 2 , \\dots , N - 1 , \\end{align*}"}
{"id": "4754.png", "formula": "\\begin{align*} \\theta _ { 0 , 0 } ^ 2 ( 0 , N \\tau ) = - i \\tau \\theta _ { 0 , 0 } ^ 2 ( 0 , \\tau ) . \\end{align*}"}
{"id": "5211.png", "formula": "\\begin{align*} \\int _ { \\Xi _ { \\mu } } x ^ { \\mu ' } e ^ { x ^ r / \\hbar } \\Omega = \\delta _ { \\mu \\mu ' } ; , \\end{align*}"}
{"id": "8279.png", "formula": "\\begin{align*} k \\tanh ( k a / 2 ) = k ' \\tanh ( k ' a / 2 ) , \\end{align*}"}
{"id": "4999.png", "formula": "\\begin{align*} \\Psi ^ { n , 1 } _ s = \\sigma ^ 2 ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) ^ 2 , \\end{align*}"}
{"id": "2732.png", "formula": "\\begin{align*} \\sum _ { u = i } ^ s \\alpha _ u \\alpha ^ * _ u - \\sum _ { v = i } ^ t \\beta _ v \\beta _ v ^ * . \\end{align*}"}
{"id": "3318.png", "formula": "\\begin{align*} \\varphi _ n ^ { h , m } : = \\left \\{ \\begin{array} { c c } \\varphi _ n ^ h & n \\leq m \\\\ 0 & n > m \\end{array} \\right . \\psi _ n ^ { h , m } : = \\left \\{ \\begin{array} { c c } \\psi _ n ^ h & n \\leq m \\\\ 0 & n > m \\end{array} \\right . . \\end{align*}"}
{"id": "256.png", "formula": "\\begin{align*} ( \\delta ^ { ( s ) } ) ^ 2 ( c _ k ) + \\frac { \\sum _ { i j } ( \\sum _ l \\Gamma ^ { k ( s ) } _ { i j , l } c _ l ) ( \\delta ^ { ( s ) } c _ i ) ^ { \\phi ^ s } ( \\delta ^ { ( s ) } c _ j ) ^ { \\phi ^ s } } { ( \\sum _ l r _ l c _ l ) ( \\sum _ l ( \\delta ^ { ( s ) } c _ l ) ^ { \\phi ^ s } ) } = 0 , \\ \\ k \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "3194.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } H = h + E \\\\ h = D + E \\end{array} \\right . . \\end{align*}"}
{"id": "5540.png", "formula": "\\begin{align*} M _ n ( m ) & = 2 \\dbinom { m + n } { n } m > n , \\\\ M _ n ( n ) & = \\dbinom { 2 n } { n } . \\end{align*}"}
{"id": "1139.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { ( a ) _ n t ^ n } { ( q ) _ n } = \\frac { ( a t ) _ { \\infty } } { ( t ) _ { \\infty } } . \\end{align*}"}
{"id": "4533.png", "formula": "\\begin{align*} \\sigma _ { m , 2 m ; n , 2 n } = \\frac { 1 } { 4 } \\left ( 9 \\sigma _ { 3 m - 1 , 3 n - 1 } - 3 \\sigma _ { 3 m - 1 , n - 1 } - 3 \\sigma _ { m - 1 , 3 n - 1 } + \\sigma _ { m - 1 , n - 1 } \\right ) . \\end{align*}"}
{"id": "8119.png", "formula": "\\begin{align*} \\lambda ( ( z _ { k + \\ell + 1 } , z _ { k + 1 } ) ) + \\lambda ( \\{ z _ { k + 1 } \\} ) \\leq \\overline \\lambda ( z _ { k + \\ell + 1 } ) - \\overline \\lambda ( z _ { k + 1 } ) \\leq k + \\ell + 1 - k = \\ell + 1 , \\end{align*}"}
{"id": "228.png", "formula": "\\begin{align*} a ^ 4 & = a ^ 2 \\gamma ^ 2 + 1 = ( a \\gamma + 1 ) \\gamma ^ 2 + 1 = a + \\gamma , \\\\ a ^ 8 & = a ^ 2 + \\gamma ^ 2 = a \\gamma + 1 + \\gamma ^ 2 = a \\gamma + \\gamma . \\end{align*}"}
{"id": "2991.png", "formula": "\\begin{align*} \\partial _ z \\left [ \\sqrt y K _ { i r } ( 2 \\pi | n | y ) e ( n x ) \\right ] = \\pi i n \\sqrt y K _ { i r } ( 2 \\pi | n | y ) e ( n x ) + g ( n , y ) e ( n x ) \\end{align*}"}
{"id": "3843.png", "formula": "\\begin{align*} B _ { k , n } ^ m = ( - 1 ) ^ { m + 1 } ( m + 1 + \\frac { ( 1 - ( - 1 ) ^ m ) } { 2 } ( 2 n + k - 2 + 2 \\gamma _ { k - 1 } ) + 2 \\kappa _ k ) . \\end{align*}"}
{"id": "4176.png", "formula": "\\begin{align*} F _ \\ell ( \\lambda , \\rho ) = F ( \\sqrt \\lambda ) \\chi _ \\ell ( \\lambda / \\rho ) \\quad \\rho \\neq 0 , \\end{align*}"}
{"id": "7067.png", "formula": "\\begin{align*} \\begin{aligned} W _ 2 = & 2 ( m _ 1 - m _ 2 ) / 3 , \\\\ W _ 3 = & \\pi m _ 2 ( 2 l + n _ 1 + n _ 2 ) / 8 , \\\\ W _ 4 = & 4 m _ 2 ( n _ 1 - n _ 2 ) ( 6 l + 4 n _ 1 + 4 n _ 2 - 1 ) / 4 5 . \\end{aligned} \\end{align*}"}
{"id": "2312.png", "formula": "\\begin{align*} \\vec { v } \\left ( { \\bf s ' } _ { \\alpha , \\delta } \\right ) = \\vec { v } \\left ( { \\bf s } _ { \\alpha , \\delta } \\right ) . \\end{align*}"}
{"id": "1662.png", "formula": "\\begin{align*} \\| \\pi ( f ) \\| ^ 2 = \\| \\pi ( f ^ * f ) \\| \\leq \\| f ^ * f \\| _ { \\infty } = \\| f \\| ^ 2 _ { \\infty } . \\end{align*}"}
{"id": "6072.png", "formula": "\\begin{align*} f ^ { ( j ) } ( x ) = ( n ) _ j \\prod _ { i = 1 } ^ { m } \\left ( x - \\lambda _ i \\right ) ^ { \\mu _ { i , j } } \\end{align*}"}
{"id": "6030.png", "formula": "\\begin{align*} f ( x ) & = ( x - 1 ) ^ 4 + b ( x - 1 ) ^ 2 + c \\\\ & = x ^ 4 - 4 x ^ 3 + ( b + 6 ) x ^ 2 - ( 2 b + 4 ) x + ( b + c + 1 ) . \\end{align*}"}
{"id": "5855.png", "formula": "\\begin{align*} \\sum _ { i = - \\infty } ^ k 2 ^ { - i \\frac { r } { p - r } } V _ r ( 0 , x _ i ) ^ { \\frac { p r } { p - r } } \\approx \\sum _ { i = - \\infty } ^ k 2 ^ { - i \\frac { r } { p - r } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \\frac { p r } { p - r } } \\end{align*}"}
{"id": "497.png", "formula": "\\begin{align*} \\left \\{ \\lambda _ { m i n } + \\eta : \\lambda _ { m i n } = \\displaystyle \\sum _ { w s _ i < w } \\omega _ i \\eta \\in \\left ( \\displaystyle \\bigcap _ { j \\in I ' } ( \\alpha _ j ^ \\vee ) ^ \\perp \\right ) \\cap P _ + \\right \\} \\end{align*}"}
{"id": "7156.png", "formula": "\\begin{align*} H ^ * ( X _ { D } ) & = \\mathbb { Z } [ x _ 1 , x _ 2 , x _ 3 ] / \\langle x _ 1 ^ { c } , x _ 2 ( x _ 1 + x _ 2 ) ^ { a - 1 } , x _ 3 ( x _ 1 + x _ 2 + x _ 3 ) ^ { b - 1 } \\rangle , \\\\ H ^ * ( X _ { \\widetilde { D } } ) & = \\mathbb { Z } [ y _ 1 , y _ 2 , y _ 3 ] / \\langle y _ 1 ^ { c } , y _ 2 ( y _ 1 + y _ 2 ) ^ { b - 1 } , y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } \\rangle . \\end{align*}"}
{"id": "8212.png", "formula": "\\begin{align*} k \\cot ( k a / 2 ) - k ' \\coth ( k ' a / 2 ) = 0 \\end{align*}"}
{"id": "5319.png", "formula": "\\begin{align*} X _ k ( t ) = \\begin{cases} X _ k ( t - ) & k \\leq j , \\\\ X _ j ( t - ) & k > j Y _ { i k } = 1 , \\\\ X _ { k - J _ { i k } } ( t - ) & k > j Y _ { i k } = 0 , \\end{cases} \\end{align*}"}
{"id": "5834.png", "formula": "\\begin{align*} u = u _ 2 ^ { q _ 2 } , \\ , \\ , v = v _ 1 ^ { - p _ 2 } v _ 2 ^ { p _ 2 } , \\ , \\ , w = u _ 1 ^ { q _ 1 } , \\end{align*}"}
{"id": "2371.png", "formula": "\\begin{align*} \\partial _ i L ( h _ \\theta ) = b _ { i 0 } + b _ { i 1 } h _ \\theta + \\ldots + b _ { i s } h _ \\theta ^ s , \\end{align*}"}
{"id": "1159.png", "formula": "\\begin{align*} \\delta \\left [ f \\right ] & = \\sum _ { k = 0 } ^ \\infty \\left ( \\sum _ { \\vec { k } } \\delta [ p _ { \\vec { k } } ] a _ { \\vec { k } } \\right ) \\end{align*}"}
{"id": "1248.png", "formula": "\\begin{align*} P ( T , x ) & = \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) - n ( T , l ( T ) - j ) } . \\end{align*}"}
{"id": "4296.png", "formula": "\\begin{align*} 0 = \\mathcal { E } _ 0 \\to \\mathcal { E } _ 1 \\to \\ldots \\to \\mathcal { E } _ N = L , \\end{align*}"}
{"id": "2724.png", "formula": "\\begin{align*} \\sigma _ N ^ 2 = \\begin{cases} \\sqrt { \\mu _ - - \\mu _ + } N \\mbox { i f } \\delta < 1 \\mbox { i n t h e a s s u m p t i o n } T \\sim N ^ { - \\delta } \\\\ N ^ { 1 / 2 } \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "8987.png", "formula": "\\begin{align*} \\langle \\xi _ { \\tilde K } \\nabla w , \\xi _ { \\tilde K } \\nabla w \\rangle = \\frac { ( 1 + \\delta ) ^ 2 } { 1 + 2 \\delta } \\left ( \\langle \\nabla u _ { q , r } , \\nabla \\nu \\rangle - ( 1 + u _ { q , r } ) ^ { 2 \\delta + 1 } \\langle \\nabla u _ { q , r } , \\nabla ( \\xi _ { \\tilde K } ) ^ 2 \\rangle \\right ) . \\end{align*}"}
{"id": "1845.png", "formula": "\\begin{align*} \\mathbf { d } _ { k } ( z ) : = \\begin{cases} ( 0 , \\ldots , 0 , - a _ { 0 } ^ { ( k - 1 ) } , - a _ { 1 } ^ { ( k - 2 ) } , \\ldots , - a _ { k - 2 } ^ { ( 1 ) } , z - a _ { k - 1 } ^ { ( 0 ) } ) , & 1 \\leq k \\leq p , \\\\ [ 0 . 5 e m ] ( - a _ { k - p } ^ { ( p - 1 ) } , - a _ { k - p + 1 } ^ { ( p - 2 ) } , - a _ { k - p + 2 } ^ { ( p - 3 ) } , \\ldots , - a _ { k - 2 } ^ { ( 1 ) } , z - a _ { k - 1 } ^ { ( 0 ) } ) , & k \\geq p + 1 , \\end{cases} \\end{align*}"}
{"id": "1382.png", "formula": "\\begin{align*} \\| \\mathcal { F } ^ { - 1 } ( L _ t [ u ] \\widehat { f } ) \\| _ { X ^ { s , 1 / 2 - } _ T } & \\lesssim \\| \\eta ( t / T ) \\mathcal { F } ^ { - 1 } ( L _ t [ u ] \\widehat { f } ) \\| _ { X ^ { s , 1 / 2 - } } \\\\ & = \\left \\| \\langle k \\rangle ^ s \\| e ^ { - i t k ^ 2 } L _ t [ u ] \\eta ( t / T ) \\| _ { H ^ { 1 / 2 - } _ t } \\widehat { f } \\right \\| _ { \\ell ^ 2 _ k } , \\end{align*}"}
{"id": "8194.png", "formula": "\\begin{align*} u ^ { \\dagger } \\left ( \\begin{array} { l r } I & O \\\\ O & - I \\end{array} \\right ) v = 0 , \\end{align*}"}
{"id": "6046.png", "formula": "\\begin{align*} f _ { r , s } ( \\kappa _ i ) = f \\left ( r \\kappa _ i + s \\right ) = f ( \\lambda _ i ) \\end{align*}"}
{"id": "619.png", "formula": "\\begin{align*} \\Big ( \\sum _ { k = 0 } ^ \\infty a _ k \\Big ) ^ j = \\sum _ { k = 0 } ^ \\infty \\sum _ { k _ 2 = 0 } ^ { k } \\dots \\sum _ { k _ { j } = 0 } ^ { k _ { j - 1 } } a _ { k _ j } \\prod _ { \\ell = 1 } ^ { j - 1 } a _ { k _ { \\ell } - k _ { \\ell + 1 } } . \\end{align*}"}
{"id": "8545.png", "formula": "\\begin{align*} P _ k ^ * ( - x ) & = ( - 1 ) ^ { k - 1 } P _ k ^ * ( x ) \\\\ P _ k ( - x ) & = ( - 1 ) ^ { k } P _ k ( x ) . \\end{align*}"}
{"id": "429.png", "formula": "\\begin{align*} 0 \\le \\frac { d } { d x } H ( x + i f ( x ) ) = H ' ( x + i f ( x ) ) ( 1 + i f ' ( x ) ) . \\end{align*}"}
{"id": "2666.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 2 } p ^ { n - 3 } ( p - 2 ) ( p - 3 ) ( n - k - 2 ) & = p ^ { n - 3 } ( p - 2 ) ( p - 3 ) \\sum _ { k = 1 } ^ { n - 2 } ( n - k - 2 ) \\\\ & = p ^ { n - 3 } ( p - 2 ) ( p - 3 ) \\binom { n - 2 } { 2 } . \\end{align*}"}
{"id": "307.png", "formula": "\\begin{align*} A _ { ( i ) } = \\{ a \\in A : P ( a ) ^ { 1 + v _ { i } } \\le a < P ( a ) ^ { 1 + v _ { i + 1 } } \\} . \\end{align*}"}
{"id": "4129.png", "formula": "\\begin{align*} G ^ { i } | _ { \\ker ( T ^ { i } ) } = 0 , \\end{align*}"}
{"id": "5517.png", "formula": "\\begin{align*} g ' ( z ) = w \\frac { h ' ( w ) } { h ( w ) } . \\end{align*}"}
{"id": "390.png", "formula": "\\begin{align*} d ( o , x ) \\int _ { \\varepsilon D ^ 2 } ^ { b D ^ 2 } \\sqrt { \\frac { d } { e t } } h _ { t / 2 } ( o , o ) d t & \\leq d ( o , x ) ( b - \\varepsilon ) D ^ 2 \\sqrt { \\frac { d } { e \\varepsilon D ^ 2 } } h _ { \\varepsilon D ^ 2 / 2 } ( o , o ) \\\\ & \\leq C ' \\frac { d ( o , x ) } { D } \\frac { D ^ 2 } { n } \\leq \\varepsilon \\frac { D ^ 2 } { n } . \\end{align*}"}
{"id": "3756.png", "formula": "\\begin{align*} K _ k ( y ) : = \\int _ { \\R ^ 3 } e ^ { i y \\cdot \\xi } \\varphi _ k ( \\xi ) d \\xi , \\Longrightarrow | K _ k ( y ) | \\lesssim 2 ^ { 3 k } ( 1 + 2 ^ k | y | ) ^ { - 1 / \\epsilon ^ 3 } . \\end{align*}"}
{"id": "1620.png", "formula": "\\begin{align*} \\gamma _ { i , l } = \\dfrac { P | h [ r ( i , l - 1 ) , r ( i , l ) , l ] | ^ 2 } { \\sigma ^ 2 + \\sum _ { j \\neq i } ^ N P | h [ r ( j , l - 1 ) , r ( i , l ) , l ] | ^ 2 } , \\end{align*}"}
{"id": "6576.png", "formula": "\\begin{align*} d f _ i ( ( x , u _ i ) , ( z , v _ i ) ) = d _ 1 f _ i ( x , u _ i , z ) + d _ 2 f _ i ( x , u _ i , v _ i ) \\end{align*}"}
{"id": "3739.png", "formula": "\\begin{align*} \\big ( ( \\hat { v } + \\omega ) \\times B \\big ) _ a = ( - 1 ) ^ { a - 1 } \\big [ ( \\hat { v } + \\omega ) _ { 3 - a } ( \\omega \\cdot \\mathbf { e } _ 3 ) ( B \\cdot \\omega ) - \\big ( \\sum _ { i = 1 , 2 , 3 } ( \\hat { v } + \\omega ) _ { 3 - a } ( \\omega \\times \\mathbf { e } _ 3 ) \\cdot \\mathbf { e } _ i ( B \\cdot ( \\omega \\times \\mathbf { e } _ i ) ) + ( \\hat { v } + \\omega ) _ 3 B _ 2 \\big ) \\big ] . \\end{align*}"}
{"id": "5181.png", "formula": "\\begin{align*} d ( A ) = { s r ( A ) + r s ( A ) - \\sum _ { i = 1 } ^ l ( s \\alpha _ i + r \\beta _ i + r s ( d _ i - 1 ) ) \\over r s } - 2 . \\end{align*}"}
{"id": "7333.png", "formula": "\\begin{align*} T _ { i , j } = \\begin{cases} - 2 / h ^ 2 , j = i , \\\\ 1 / h ^ 2 , j = i \\pm 1 , \\\\ 0 , \\quad ; \\end{cases} \\end{align*}"}
{"id": "4818.png", "formula": "\\begin{align*} w ( \\alpha ) = ( w _ { p _ { \\sigma ( i ) } } ( \\alpha ) ) _ { 1 \\leq i \\leq \\abs { J } } \\end{align*}"}
{"id": "7366.png", "formula": "\\begin{align*} T _ { i } = \\left ( \\begin{array} { l l } A _ { i } & 0 _ { 3 , N - 3 } \\\\ 0 _ { N - 3 , 3 } & I _ { N - 3 } \\end{array} \\right ) , \\end{align*}"}
{"id": "4259.png", "formula": "\\begin{align*} \\rho \\in \\mathcal D _ k : = \\left [ r _ 1 k \\ln k , r _ 2 k \\ln k \\right ] \\ \\hbox { f o r s o m e } \\ r _ 2 > r _ 1 > 0 . \\end{align*}"}
{"id": "6449.png", "formula": "\\begin{align*} d t _ { i } ( [ X ^ { i + 1 } , \\xi ] ) & = [ d t _ { i } ( X ^ { i + 1 } ) , d t _ { i } ( \\xi ) ] \\\\ & = [ X ^ { i } , d t _ { i } ( \\xi ) ] , \\end{align*}"}
{"id": "2853.png", "formula": "\\begin{align*} \\begin{aligned} h _ * = \\arg \\max _ { 1 { } \\leq { } h { } \\leq { } \\bar { h } ( \\kappa ) } 2 h + \\frac { \\kappa h ^ 3 } { 2 - h \\left ( 1 + \\kappa \\right ) } \\end{aligned} . \\end{align*}"}
{"id": "6157.png", "formula": "\\begin{align*} b _ h ( u ^ { \\Delta t } _ { h } , u ^ { \\Delta t } _ { h } , u ^ { \\Delta t } _ { h } \\phi ) & = - \\left ( u ^ { \\Delta t } _ { h } \\frac 1 2 | u ^ { \\Delta t } _ { h } | ^ 2 , \\nabla \\phi \\right ) . \\end{align*}"}
{"id": "5800.png", "formula": "\\begin{align*} a ( v ) = \\sum _ { i = 1 } ^ { v - 1 } F _ { v - i , i } . \\end{align*}"}
{"id": "6669.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\overline { \\varphi } _ t - A _ 1 \\overline { \\varphi } _ { i i } - ( B _ 1 + h ' C _ 1 + D _ 1 ) \\overline { \\varphi } _ i = a _ { 0 1 } \\chi ^ { p } - \\lambda _ { 0 1 } \\sqrt { A _ 1 } ^ \\alpha | \\varphi _ i | ^ { \\alpha } , & t > 0 , \\ 0 < i < h _ 0 , \\\\ \\overline { \\varphi } _ i ( t , 0 ) = \\overline { \\varphi } ( t , h _ 0 ) = 0 , & t > 0 , \\\\ \\overline { \\varphi } ( 0 , i ) = \\varphi _ 0 ( i ) , & 0 \\leq i \\leq h _ 0 , \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "5292.png", "formula": "\\begin{align*} \\int _ { \\Xi _ d } x ^ { n r + k } e ^ { x ^ r / \\hbar } d x = { } & ( - 1 ) ^ n \\hbar ^ n \\left ( \\prod _ { i = 1 } ^ n ( i - 1 + \\frac { k + 1 } { r } ) \\right ) \\int _ { \\Xi _ d } x ^ { k } e ^ { x ^ r / \\hbar } d x \\\\ = { } & ( - 1 ) ^ n \\hbar ^ n \\frac { \\Gamma \\left ( n + \\frac { k + 1 } { r } \\right ) } { \\Gamma \\left ( \\frac { k + 1 } { r } \\right ) } \\int _ { \\Xi _ d } x ^ { k } e ^ { x ^ r / \\hbar } d x . \\end{align*}"}
{"id": "6846.png", "formula": "\\begin{align*} p ( t , g ) = \\frac { 1 } { \\sqrt { 2 \\pi c ( t ) } } \\exp \\left ( - \\frac { ( g - b ( t ) ) ^ 2 } { 2 c ( t ) } \\right ) , \\end{align*}"}
{"id": "3455.png", "formula": "\\begin{align*} \\mathrm { D T E } _ { ( p , q ) } ( X ) = \\mathrm { E } ( X | x _ { p } < X < x _ { q } ) \\end{align*}"}
{"id": "3852.png", "formula": "\\begin{align*} \\partial _ { v _ { b _ { n } } } & \\partial _ { \\mu } \\bigl [ \\partial _ { c } f ( \\mu , v _ { 1 } , . . . , v _ { p } ) \\cdot e _ { 1 } \\otimes . . . \\otimes e _ { i } \\bigr ] ( v _ { p + 1 } ) \\cdot e _ { i + 1 } \\otimes e _ { { n } } \\\\ & = \\partial _ { \\mu } \\partial _ { v _ { b _ { n } } } \\bigl [ \\partial _ { c } f ( \\mu , v _ { 1 } , \\cdots , v _ { p } ) , e _ { 1 } \\otimes . . . \\otimes e _ { i } \\bigr ] ( v _ { p + 1 } ) \\cdot e _ { { n } } \\otimes e _ { i + 1 } , \\end{align*}"}
{"id": "4670.png", "formula": "\\begin{align*} { \\Gamma } ^ T H { \\Gamma } = I , \\end{align*}"}
{"id": "7276.png", "formula": "\\begin{align*} u ( x , t ) & = \\pm { \\sf U } _ \\infty ( x ) + K { \\sf D } _ J ( T - t ) ^ { \\frac { \\gamma } { 2 } + J } | z | ^ \\gamma ( 1 + o ( 1 ) ) \\\\ & = \\pm { \\sf U } _ \\infty ( x ) + K { \\sf D } _ J ( T - t ) ^ J \\eta ^ \\gamma | \\xi | ^ \\gamma ( 1 + o ( 1 ) ) | z | \\to 0 . \\end{align*}"}
{"id": "4159.png", "formula": "\\begin{align*} \\Phi _ \\nu ^ \\lambda ( \\xi ) = \\lambda ^ { n / 4 } \\prod _ { j = 1 } ^ n h _ { \\nu _ j } ( \\lambda ^ { 1 / 2 } \\xi _ j ) , \\xi \\in \\R ^ n , \\end{align*}"}
{"id": "8735.png", "formula": "\\begin{align*} \\sum \\limits _ { i , j = 1 } ^ { 2 } h _ { \\dot { x } ^ { i } \\dot { x } ^ { j } } y ^ { i } y ^ { j } = \\frac { \\lambda } { ( ( \\dot { x } ^ 1 ) ^ 2 + ( \\dot { x } ^ 2 ) ^ 2 ) ^ { \\frac { 3 } { 2 } } } ( \\dot { x } ^ 2 y ^ 1 - \\dot { x } ^ 1 y ^ 2 ) ^ 2 = \\frac { \\lambda } { a } ( y ^ { 1 } \\cos t + y ^ { 2 } \\sin t ) ^ { 2 } . \\end{align*}"}
{"id": "6448.png", "formula": "\\begin{align*} d s \\left ( \\overleftarrow { [ a , b ] _ { A } } - [ \\overleftarrow { a } , \\overleftarrow { b } ] \\right ) & = \\rho ( [ a , b ] _ { A } ) - [ \\rho ( a ) , \\rho ( b ) ] \\\\ & = 0 , \\end{align*}"}
{"id": "2382.png", "formula": "\\begin{align*} \\overline F _ p : = \\sum _ { i \\in I \\cup \\{ 0 \\} } a _ i Q _ \\theta ^ i . \\end{align*}"}
{"id": "1269.png", "formula": "\\begin{align*} \\Psi ( E _ j ( x , a ) ) & = \\begin{cases} 2 \\sqrt { a } \\left ( \\cot \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) - 1 \\right ) , & 2 \\nmid j , \\\\ 2 \\sqrt { a } \\left ( \\csc \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) - 1 \\right ) , & 2 \\mid j . \\end{cases} \\end{align*}"}
{"id": "5982.png", "formula": "\\begin{align*} \\Pi _ 1 Q ^ { \\beta } \\hat U ( t ) + A \\hat U ( t ) = \\Pi _ 1 f ( t ) + \\hat G ( t ) , t \\in [ 0 , T ] , \\end{align*}"}
{"id": "2583.png", "formula": "\\begin{align*} z \\sim z ' \\iff z = a \\ , z ' \\ , \\ \\ a \\in \\mathbb C ^ \\ast \\ . \\end{align*}"}
{"id": "8435.png", "formula": "\\begin{align*} \\theta ( s ) = 0 , \\ | s | \\le 1 , \\ \\theta ( s ) = 1 , \\ \\ | s | \\ge 2 . \\end{align*}"}
{"id": "844.png", "formula": "\\begin{align*} \\boldsymbol { V } _ { A \\rightarrow B } ^ { e x t } & = \\left ( ( \\overline { \\boldsymbol { V } } _ { A } ^ { p o s t } ) ^ { - 1 } - ( \\boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \\right ) ^ { - 1 } , \\\\ \\boldsymbol { x } _ { A \\rightarrow B } ^ { e x t } & = \\boldsymbol { V } _ { A \\rightarrow B } ^ { e x t } \\left ( ( \\overline { \\boldsymbol { V } } _ { A } ^ { p o s t } ) ^ { - 1 } \\boldsymbol { x } _ { A } ^ { p o s t } - ( \\boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \\boldsymbol { x } _ { A } ^ { p r i } \\right ) , \\end{align*}"}
{"id": "6505.png", "formula": "\\begin{align*} U _ t = \\mathcal { A } _ d U , U ( 0 ) = U _ 0 , \\end{align*}"}
{"id": "6350.png", "formula": "\\begin{align*} | Z _ 1 | = 0 ~ ~ u _ n ( x ) \\longrightarrow u ( x ) ~ ~ x \\in \\Omega \\setminus Z _ 1 . \\end{align*}"}
{"id": "8281.png", "formula": "\\begin{align*} k = k ' = \\frac { 1 } { 2 \\hslash \\sqrt { \\beta / 3 } } . \\end{align*}"}
{"id": "5071.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } L ^ { n , 4 } _ \\tau = \\frac { \\alpha } { 2 ( \\alpha + 1 ) ^ 2 } \\int _ 0 ^ \\tau \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "323.png", "formula": "\\begin{align*} \\langle S \\rangle : = \\{ s \\in S : s \\notin { \\rm L } _ t \\ ; \\forall \\ ; t < s , t \\in S \\} . \\end{align*}"}
{"id": "6196.png", "formula": "\\begin{align*} L _ { M } \\cdot V = \\int _ { V } R _ { h } ^ { T } \\wedge \\eta \\end{align*}"}
{"id": "3058.png", "formula": "\\begin{align*} f _ { \\mathbf { M ^ { \\ast \\ast } } _ { - \\xi _ { \\boldsymbol { s } k } , \\xi _ { \\boldsymbol { t } l } } } ( \\boldsymbol { u } ) = c _ { n - 2 , \\xi _ { \\boldsymbol { s } k } , \\xi _ { \\boldsymbol { t } l } } ^ { \\ast \\ast } \\overline { \\mathcal { G } } _ { n } \\left \\{ \\frac { 1 } { 2 } \\boldsymbol { u } ^ { T } \\boldsymbol { u } + \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } + \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { t } l } ^ { 2 } \\right \\} , ~ \\boldsymbol { u } \\in \\mathbb { R } ^ { n - 2 } , \\end{align*}"}
{"id": "6412.png", "formula": "\\begin{align*} \\nabla _ { [ x , y ] _ \\mathfrak g } - [ \\nabla _ x , \\nabla _ y ] = \\gamma ( x , y ) \\circ \\ell _ 1 - \\ell _ 1 \\circ \\gamma ( x , y ) + \\ell _ 2 ( \\eta ( x , y ) , \\cdot \\ , ) , \\end{align*}"}
{"id": "2612.png", "formula": "\\begin{align*} \\{ e ( ( b + 3 k , b ) ; ( b + k , b + k , b + k ) , J ) : J = k , . . . , b + k \\} \\ . \\end{align*}"}
{"id": "6088.png", "formula": "\\begin{align*} \\sigma \\left ( \\sum _ { i = 1 } ^ { n } c _ { i } x _ { i } \\right ) & \\geq \\rho \\left ( \\varphi \\left ( \\sum _ { i = 1 } ^ { n } c _ { i } x _ { i } \\right ) \\right ) = \\rho \\left ( \\sum _ { i = 1 } ^ { n } c _ { i } \\varphi ( x _ { i } ) \\right ) \\\\ & = \\rho \\left ( \\sum _ { i = 1 } ^ { n } c _ { i } y _ { i } \\right ) \\geq \\delta \\sum _ { i = 1 } ^ { n } \\lvert c _ { i } \\rvert . \\end{align*}"}
{"id": "1779.png", "formula": "\\begin{align*} y ^ * = \\sum _ { w \\in \\mathcal { W } } y ^ * ( w ) \\widetilde { w } , \\end{align*}"}
{"id": "7375.png", "formula": "\\begin{align*} < L _ { n } ( \\phi _ { n } ) , \\ \\psi > = o _ n ( 1 ) \\| \\phi _ n \\| \\| \\psi \\| \\ \\textrm { f o r a n y } \\ \\psi \\in E _ n . \\end{align*}"}
{"id": "8550.png", "formula": "\\begin{align*} \\left ( \\frac q p - 1 \\right ) + \\frac p q \\left ( \\frac q r - 1 \\right ) = \\frac r p - 1 \\ . \\end{align*}"}
{"id": "3263.png", "formula": "\\begin{align*} A ( x ) : = ( A _ 2 - A _ 1 ) ( x ) , q ( x ) : = ( q _ 2 - q _ 1 ) ( x ) , x \\in \\R ^ 3 , \\end{align*}"}
{"id": "7183.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { R e s } _ { z _ 1 } z _ 1 ^ { k + \\frac { j _ 2 } { T } } z _ 0 ^ { - 1 } \\delta ( \\frac { z _ 1 - z _ 2 } { z _ 0 } ) ( \\frac { z _ 1 - z _ 2 } { z _ 0 } ) ^ { \\frac { j _ 1 } { T } } Y _ { M ^ 3 } ( u , z _ 1 ) I ( w _ 1 , z _ 2 ) w _ 2 \\\\ = & \\mbox { R e s } _ { z _ 1 } z _ 1 ^ { k + \\frac { j _ 2 } { T } } z _ 1 ^ { - 1 } \\delta ( \\frac { z _ 2 + z _ 0 } { z _ 1 } ) ( \\frac { z _ 2 + z _ 0 } { z _ 1 } ) ^ { \\frac { j _ 2 } { T } } I ( Y _ { M ^ 1 } ( u , z _ 0 ) w _ 1 , z _ 2 ) w _ 2 . \\end{aligned} \\end{align*}"}
{"id": "2005.png", "formula": "\\begin{align*} n _ 1 = \\displaystyle \\prod _ { p | \\gcd ( n , \\binom { n + s } { s } ) } p ^ { _ p ( n ) } . \\end{align*}"}
{"id": "8650.png", "formula": "\\begin{align*} \\psi ( t ) = \\exp \\left ( - { \\rm i } \\beta ( t ) \\widehat { H } _ 3 \\right ) \\exp \\left ( - { \\rm i } \\alpha ( t ) \\widehat { H } _ 2 \\right ) \\exp \\left ( - { \\rm i } \\tau ( t ) ( \\widehat { H } _ 1 + \\widehat { H } _ 4 ) \\right ) \\psi ' ( 0 ) . \\end{align*}"}
{"id": "8652.png", "formula": "\\begin{align*} \\overline B : = \\sup _ { s > 0 } \\left ( \\int _ 0 ^ s V ( t ) ^ { - \\frac 1 { p - 1 } } \\ , d t \\right ) ^ { p - 1 } \\left ( \\int _ s ^ \\infty W ( t ) \\ , d t \\right ) \\end{align*}"}
{"id": "600.png", "formula": "\\begin{align*} \\zeta _ Q ( s ) : = \\sum _ { k \\in \\Z ^ \\alpha \\setminus \\{ 0 \\} } ( k ^ T Q k ) ^ { - s } , \\textup { R e } ( s ) > \\frac { \\alpha } { 2 } . \\end{align*}"}
{"id": "4138.png", "formula": "\\begin{align*} \\vartheta ( G ) \\vartheta ( \\overline { G } ) = | G | . \\end{align*}"}
{"id": "6840.png", "formula": "\\begin{align*} \\| p _ k ( t , g ) \\| _ { { L ^ 1 ( \\mathbb { R } ) } } = \\| e ^ { i k ( \\frac { 2 \\pi } { V _ F } ) g } ( p _ { t , k } * G _ { t , k } ) ( g ) \\| _ { { L ^ 1 ( \\mathbb { R } ) } } & \\leq \\| p _ { t , k } ( g ) \\| _ { { L ^ 1 ( \\mathbb { R } ) } } \\| G _ { t , k } \\| _ { { L ^ 1 ( \\mathbb { R } ) } } \\\\ & = \\| p _ { 0 , k } ( g ) \\| _ { { L ^ 1 ( \\mathbb { R } ) } } e ^ { - k ^ 2 ( \\frac { 2 \\pi } { V _ F } ) ^ 2 D ( t ) } . \\end{align*}"}
{"id": "6922.png", "formula": "\\begin{align*} f - q | _ g = & \\ L ( p | _ { L ^ 2 ( x y ) } L ( z ) ) + p | _ { L ^ 2 ( x y ) } L ^ 2 ( z ) - L ^ 2 ( p | _ { L ( x L ( y ) ) } z ) - L ^ 2 ( p | _ { x L ^ 2 ( y ) } z ) \\\\ \\equiv & \\ - L ( p | _ { L ( x L ( y ) ) } L ( z ) ) - L ( p | _ { x L ^ 2 ( y ) } L ( z ) ) - p | _ { L ( x L ( y ) ) } L ^ 2 ( z ) - p | _ { x L ^ 2 ( y ) } L ^ 2 ( z ) \\\\ & \\ + L ( p | _ { L ( x L ( y ) ) } L ( z ) ) + p | _ { L ( x L ( y ) ) } L ^ 2 ( z ) + L ( p | _ { x L ^ 2 ( y ) } L ( z ) ) + p | _ { x L ^ 2 ( y ) } L ^ 2 ( z ) \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "4178.png", "formula": "\\begin{align*} H ^ \\mu \\varphi _ k ^ { | \\mu | } = [ k ] | \\mu | \\varphi _ k ^ { | \\mu | } \\end{align*}"}
{"id": "5628.png", "formula": "\\begin{align*} w _ t = \\nabla \\lambda _ t \\cdot \\nabla p + \\nabla \\lambda \\cdot \\nabla p _ t + \\Delta p _ t . \\end{align*}"}
{"id": "8544.png", "formula": "\\begin{align*} y _ { k + 1 } ( x ) = ( A _ k x + B _ k ) y _ k ( x ) - C _ k y _ { k - 1 } ( x ) , k > 0 , \\end{align*}"}
{"id": "1723.png", "formula": "\\begin{align*} \\frac { s ^ r e ^ { z s } } { \\prod _ { i = 1 } ^ { r } ( e ^ { \\omega _ i s } - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } B _ { n , r } ( z \\ , | \\ , \\omega _ 1 , \\cdots , \\omega _ r ) \\frac { s ^ n } { n ! } . \\end{align*}"}
{"id": "2780.png", "formula": "\\begin{align*} \\begin{aligned} T _ 3 ( h _ { N - 1 } { } \\leq { } 1 ) = 0 \\end{aligned} \\end{align*}"}
{"id": "7376.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { { \\C } _ 1 } \\left ( | \\nabla \\phi _ n | ^ 2 + \\phi _ n ^ 2 \\right ) d y = \\frac { 1 } { 4 } . \\end{aligned} \\end{align*}"}
{"id": "1600.png", "formula": "\\begin{align*} E = b ^ 2 \\sum \\limits _ { k = 1 } ^ { 3 } ( - 1 ) ^ { \\gamma + \\tau } z ^ { k } _ { \\tilde { \\gamma } } z ^ { k } _ { \\tilde { \\tau } } z ^ { 3 } _ { \\gamma } z ^ { 3 } _ { \\tau } . \\end{align*}"}
{"id": "4992.png", "formula": "\\begin{align*} \\psi _ { n , 2 } ( u , s ) = ( s - \\eta _ n ( u ) ) ^ { \\alpha } - ( \\eta _ n ( s ) - \\eta _ n ( u ) ) ^ { \\alpha } . \\end{align*}"}
{"id": "6556.png", "formula": "\\begin{align*} d f ( x , y ) : = ( D _ y f ) ( x ) : = \\lim _ { t \\to 0 } \\frac { f ( x + t y ) - f ( x ) } { t } \\end{align*}"}
{"id": "5964.png", "formula": "\\begin{align*} a _ j ( t ) = \\frac 1 { \\Gamma ( 2 - \\beta ) } ( t - t _ j ) ^ { 1 - \\beta } , t \\geq t _ j , \\end{align*}"}
{"id": "2920.png", "formula": "\\begin{align*} F ( A x + e ) = G ( x ) + d \\end{align*}"}
{"id": "4088.png", "formula": "\\begin{align*} p _ n ( i ) - p _ n ( i + 3 ) = a _ i n + b _ i . \\end{align*}"}
{"id": "6245.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n q ^ n \\left ( 1 - q ^ 2 - q ^ n x \\right ) A i _ q ( q ^ n x ) \\\\ & = ( 1 - q ) A i _ q ( \\frac { x } { q } ) - \\frac { x } { q ( 1 + q ) } \\ , _ 1 \\phi _ 1 ( 0 ; - q ^ 2 ; q , - x ) , \\end{align*}"}
{"id": "911.png", "formula": "\\begin{align*} n I _ n - J _ n = \\begin{bmatrix} ( n - 1 ) & - 1 & - 1 & \\dots & - 1 \\\\ - 1 & ( n - 1 ) & - 1 & \\dots & - 1 \\\\ \\cdots \\\\ - 1 & - 1 & - 1 & \\cdots & ( n - 1 ) \\\\ \\end{bmatrix} _ , \\end{align*}"}
{"id": "4680.png", "formula": "\\begin{align*} c _ { v , + } ( h B ^ + / B ^ + ) & = \\sigma _ { v , + } ( h t ^ { - 1 } g ^ { - 1 } \\dot { v } ^ { - 1 } ) \\dot { v } B ^ + / B ^ + = g _ 1 ^ { - 1 } \\dot { v } B ^ + / B ^ + \\\\ & = h _ 1 t _ 1 g ^ { - 1 } B ^ + / B ^ + = h _ 1 B ^ + / B ^ + . \\end{align*}"}
{"id": "6717.png", "formula": "\\begin{align*} \\varphi _ k ( \\lambda ) : = \\sum _ { l = 0 } ^ m h _ l ( \\lambda ^ 2 + k ^ 2 ) \\cdot ( k \\lambda ) ^ l , \\ ; \\ ; \\ ; k = - m , \\dots , m \\end{align*}"}
{"id": "4495.png", "formula": "\\begin{align*} \\varrho _ 1 ( t ) = 2 ^ { \\lfloor \\log _ 2 \\varrho ( t ) \\rfloor } . \\end{align*}"}
{"id": "3863.png", "formula": "\\begin{align*} c \\Big ( \\lfloor T \\rfloor _ i , \\Upsilon , \\lfloor Y \\rfloor _ i \\Big ) = c \\Big ( T , \\Upsilon , Y \\Big ) \\end{align*}"}
{"id": "6972.png", "formula": "\\begin{align*} 1 - \\sum \\limits _ { n \\geq 1 } \\frac { a _ n ^ N } { \\lambda _ { n } ^ 2 } = \\prod \\limits _ { k = 1 } ^ { N } \\left ( \\frac { \\mu _ k ^ 2 } { \\lambda _ k ^ 2 } \\right ) > 0 , \\end{align*}"}
{"id": "7269.png", "formula": "\\begin{align*} \\lim _ { t \\to T } u ( x , t ) = 0 k ^ { - 1 } \\eta ( t ) < | x | < k \\sqrt { T - t } \\end{align*}"}
{"id": "5577.png", "formula": "\\begin{align*} g ( x ) = \\min \\{ 1 , G ( x , x _ 0 ) \\} , x \\in \\Omega , \\end{align*}"}
{"id": "7389.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\int _ { \\mathbb { R } ^ N } \\left ( \\nabla \\bar { \\phi } ( y ) \\cdot \\nabla { \\bar \\psi } ( y ) + \\bar { \\phi } ( y ) { \\bar \\psi } ( y ) - p \\left ( U _ 0 ( y ) \\right ) ^ { p - 1 } \\bar { \\phi } ( y ) { \\bar \\psi } ( y ) \\right ) d y . \\end{aligned} \\end{align*}"}
{"id": "8211.png", "formula": "\\begin{align*} k \\tan ( k a / 2 ) + k ' \\tanh ( k ' a / 2 ) = 0 , \\end{align*}"}
{"id": "1995.png", "formula": "\\begin{align*} A ^ s _ { \\pm } ( x ) = A _ { \\pm } ( x _ 0 + s x ) \\\\ f ^ s _ { \\pm } ( x ) = s f _ { \\pm } ( x _ 0 + s x ) \\\\ Q ^ s ( x ) = Q ( x _ 0 + s x ) . \\end{align*}"}
{"id": "7779.png", "formula": "\\begin{align*} \\lvert \\tilde \\lambda _ i - \\lambda _ i \\rvert \\le \\lVert E \\rVert _ S i = 1 , \\ldots , \\min \\{ N _ h , m p \\} . \\end{align*}"}
{"id": "6220.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } k ( x ) = u ( x ) k ( x ) x \\in I . \\end{align*}"}
{"id": "6105.png", "formula": "\\begin{align*} \\beta \\ , \\| y \\| _ H \\le \\| b ( \\bullet , y ) \\| _ { H ^ * } = \\| x _ h \\| _ H = 1 , \\quad \\| y \\| _ H \\le 1 / \\beta \\end{align*}"}
{"id": "4267.png", "formula": "\\begin{align*} - \\Delta u _ i + V ( x ) u _ i = | u _ i | ^ { p - 1 } u _ i + \\beta | u _ i | ^ { q - 1 } u _ i \\sum \\limits _ { j = 1 \\atop j \\not = i } ^ d | u _ j | ^ { r - 1 } u _ j \\ \\hbox { i n } \\ \\mathbb R ^ n , \\ i = 1 , \\dots , d \\end{align*}"}
{"id": "1386.png", "formula": "\\begin{align*} F ( v , x ) : = \\mathcal { F } ^ { - 1 } _ x ( L _ t [ v ] \\widehat { f } ) . \\end{align*}"}
{"id": "5397.png", "formula": "\\begin{align*} B _ q \\colon H ^ s ( \\R ^ n ) \\times H ^ s ( \\R ^ n ) \\to \\R , B _ q ( u , v ) \\vcentcolon = \\int _ { \\R ^ n } ( - \\Delta ) ^ { s / 2 } u \\ , ( - \\Delta ) ^ { s / 2 } v \\ , d x + \\int _ { \\R ^ n } q u v \\ , d x \\end{align*}"}
{"id": "1717.png", "formula": "\\begin{align*} \\mathbb { S } _ q ( \\ell _ { \\infty } ) ( x _ { \\delta ^ \\vee } ) = \\prod _ { m \\geq 1 } \\prod _ { k = 0 } ^ { m - 1 } \\big ( ( 1 - ( q ^ { \\frac { 1 } { 2 } } ) ^ { 2 - m + 2 k } x _ { m \\delta } ) ( 1 - ( q ^ { \\frac { 1 } { 2 } } ) ^ { - m + 2 k } x _ { m \\delta } ) \\big ) x _ { \\delta ^ \\vee } , \\end{align*}"}
{"id": "7750.png", "formula": "\\begin{align*} A _ m ( \\alpha ) { \\bf u } _ m ( \\alpha ) = { \\bf f } _ m ( \\alpha ) , \\end{align*}"}
{"id": "3482.png", "formula": "\\begin{align*} & \\mathrm { D T V } _ { ( p , q ) } ( X ) = - \\mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \\mu ^ { 2 } + 2 \\mu \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sigma ^ { 2 } \\left ( L _ { 1 } + 1 \\right ) , \\end{align*}"}
{"id": "176.png", "formula": "\\begin{align*} U _ \\beta ^ 0 : = \\frac { 1 } { h ^ N } \\int _ { x _ \\beta + R _ h } u _ 0 ( x ) \\dd x \\quad G _ \\beta ^ j : = \\frac { 1 } { h ^ N \\Delta t _ j } \\int _ { t _ { j - 1 } } ^ { t _ j } \\int _ { x _ \\beta + R _ h } g ( x , \\tau ) \\dd x \\dd \\tau . \\end{align*}"}
{"id": "1970.png", "formula": "\\begin{align*} ( S _ { 0 } ( z ) , \\ldots , S _ { p - 1 } ( z ) ) = \\frac { \\mathbf { 1 } } { ( 0 , \\ldots , 0 , z ) + ( 1 , \\ldots , 1 , - a _ { 0 } ) \\ , ( S _ { 0 } ^ { ( 1 ) } ( z ) , \\ldots , S _ { p - 1 } ^ { ( 1 ) } ( z ) ) } , \\end{align*}"}
{"id": "317.png", "formula": "\\begin{align*} C _ 2 : = \\sum _ { p > 2 3 } b _ p f ( p ) = \\frac { \\pi } { 4 } e ^ \\gamma \\sum _ { p > 2 3 } \\frac { M _ p } { m _ p ^ 2 } { \\rm d } ( { \\rm L } _ p ) \\le \\frac { \\pi } { 4 } \\frac { M e ^ \\gamma } { \\mu _ { 2 3 } ^ 2 } \\prod _ { p \\le 2 3 } \\big ( 1 - \\tfrac { 1 } { p } \\big ) = 0 . 2 5 1 1 3 5 \\cdots , \\end{align*}"}
{"id": "1204.png", "formula": "\\begin{align*} \\big ( \\sqrt { n } T _ n ^ B , \\sqrt { n } T _ n \\big ) & = \\big ( \\sqrt { n } ( \\hat { \\mu } _ n ^ B - \\hat { \\mu } _ n ) * \\gamma _ \\sigma + \\sqrt { n } ( \\hat { \\mu } _ n - \\mu ) * \\gamma _ \\sigma , \\sqrt { n } ( \\hat { \\mu } _ n - \\mu ) * \\gamma _ \\sigma \\big ) \\\\ & \\stackrel { d } { \\to } \\big ( G _ { \\mu } ' + G _ { \\mu } , G _ \\mu \\big ) \\dot { H } ^ { - 1 , p } ( \\mu * \\gamma _ \\sigma ) \\times \\dot { H } ^ { - 1 , p } ( \\mu * \\gamma _ \\sigma ) \\end{align*}"}
{"id": "1813.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\mbox { C o n d a t - V \\ ~ u } & \\sigma = L _ 1 / 2 & \\tau = 1 / ( 2 L _ 1 ) \\\\ \\mbox { P D 3 O } & \\sigma = L _ 2 / 4 & \\tau = 1 / L _ 2 . \\end{array} \\end{align*}"}
{"id": "2235.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ \\infty \\frac { \\beta ( \\sigma , t ) ^ { k } } { k ! } \\right | \\leq \\left | \\frac { \\beta ( \\sigma , t ) } { 1 - \\beta ( \\sigma , t ) } \\right | : = \\beta _ 1 ( \\sigma , t ) . \\end{align*}"}
{"id": "998.png", "formula": "\\begin{align*} \\varphi ( \\alpha , \\beta ) = ( p ^ { 2 } - 1 ) ( q ^ { 2 } - 1 ) . \\end{align*}"}
{"id": "8748.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t ^ 2 u = \\partial _ t \\partial _ x ^ k u + \\partial _ x ^ { 2 k } u \\end{array} \\end{align*}"}
{"id": "8852.png", "formula": "\\begin{align*} \\begin{dcases} & d X ( t ) = d B ( t ) + \\dfrac { \\alpha - 1 } { 2 } \\frac { 1 } { X ( t ) } d t , \\ t < T _ 0 ^ { x , \\alpha } \\\\ & X ( 0 ) = x \\end{dcases} , \\end{align*}"}
{"id": "8004.png", "formula": "\\begin{align*} D _ { Z , + } = ( D _ { 2 , + } + D _ + ) \\cap Z _ + = D _ { 2 , + } \\cap Z _ + + \\sum _ { i \\in \\{ M _ + \\cup M _ - \\} \\setminus S _ + } \\bar D _ i \\cap Z _ + \\end{align*}"}
{"id": "7002.png", "formula": "\\begin{align*} H _ \\sigma = \\mathcal { Z } _ K \\cap \\left ( \\bigcap _ { \\rho _ i \\in \\sigma } \\{ z _ i = 0 \\} \\right ) \\cap \\left ( \\bigcap _ { \\rho _ j \\notin \\sigma } \\{ z _ j \\ne 0 \\} \\right ) \\subset \\C ^ m . \\end{align*}"}
{"id": "4795.png", "formula": "\\begin{align*} { \\rm k } _ { m , n } ( z , w ) = \\sum _ { k = 0 } ^ { n - 1 } P _ { m , k } ( z ) \\overline { P _ { m , k } ( w ) } \\end{align*}"}
{"id": "1993.png", "formula": "\\begin{align*} \\bar A _ { \\pm } ( y ) & = A _ { \\pm } ( x _ 0 + \\rho y ) \\\\ \\bar f _ { \\pm } ( y ) & = \\rho f _ { \\pm } ( x _ 0 + \\rho y ) \\\\ \\bar Q ( y ) & = Q ( x _ 0 + \\rho y ) . \\end{align*}"}
{"id": "5763.png", "formula": "\\begin{align*} G / B = N ' \\rtimes P , \\end{align*}"}
{"id": "3023.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda } + b _ j > \\frac { 2 0 n + 4 } { 2 0 n + 5 } + \\frac { 8 0 n ^ 2 - 4 4 n - 1 5 } { 1 0 ( 2 n + 1 ) ( 4 n + 1 ) } = \\frac { 4 0 n - 7 } { 2 0 n + 1 0 } . \\end{align*}"}
{"id": "5433.png", "formula": "\\begin{align*} M f ( x ) \\vcentcolon = \\sup _ { r > 0 } \\frac { 1 } { | B _ r ( x ) | } \\int _ { B _ r ( x ) } | f ( y ) | \\ , d y . \\end{align*}"}
{"id": "194.png", "formula": "\\begin{align*} \\hat { \\pi } _ t ^ i = \\hat { F } ^ i ( \\hat { \\pi } ^ i _ { t - 1 } , e ^ i _ t , x ^ i _ t ) , \\end{align*}"}
{"id": "249.png", "formula": "\\begin{align*} w ' _ { c } = 0 . \\end{align*}"}
{"id": "2924.png", "formula": "\\begin{align*} g _ b ( x ) = \\sum _ { j = 1 } ^ n b _ j ( g \\circ S ^ { 1 - j } ) ( x ) + d ' . \\end{align*}"}
{"id": "533.png", "formula": "\\begin{align*} \\max _ { 0 \\leq l \\leq n } { n \\choose l } \\alpha ^ l ( 1 - \\alpha ) ^ { n - l } = O _ { \\alpha } \\Big ( \\frac { 1 } { \\sqrt n } \\Big ) , \\end{align*}"}
{"id": "8517.png", "formula": "\\begin{align*} y _ { k + 1 } ( x ) = \\frac { 2 k + 1 } { k + 1 } x y _ k ( x ) - \\frac { k } { k + 1 } y _ { k - 1 } ( x ) , k > 0 , \\end{align*}"}
{"id": "4038.png", "formula": "\\begin{align*} \\frac { A _ 1 + \\sqrt { n P } } { \\sqrt { \\frac { 1 } { n - 1 } \\sum _ { i = 2 } ^ n A _ i ^ 2 } } , \\end{align*}"}
{"id": "4000.png", "formula": "\\begin{align*} \\textstyle C = C _ 0 \\bigcup \\Big ( \\bigcup _ { \\ell = \\lceil \\frac { 1 } { 2 } \\mu _ q ( n ) \\rceil } ^ { \\frac { n - 1 } { 2 } } \\bigcup _ { L \\in \\Omega _ { 2 \\ell } } \\tilde L ^ * \\Big ) ; ~ ~ ~ ~ ~ | \\Omega _ { 2 \\ell } | \\le n ^ { 2 \\ell / \\mu _ q ( n ) } . \\end{align*}"}
{"id": "8449.png", "formula": "\\begin{align*} J _ { n , \\beta _ 1 } ( w ) = \\left \\{ x \\in I _ { n , \\beta _ 1 } ( w ) : | T _ { \\beta _ 1 } ^ n x - f _ 1 ( x ) | < \\beta _ 1 ^ { - n \\tau _ 1 ( x ) } \\right \\} , \\end{align*}"}
{"id": "7981.png", "formula": "\\begin{align*} I _ { ( Y _ + , D _ { Y , + } ) } ( y , z ) = z e ^ { t _ + / z } \\sum _ { d \\in \\mathbb K _ { + } } y ^ { d } \\left ( \\prod _ { i \\in M _ 0 } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\\\ \\left ( \\prod _ { i = 1 } ^ k \\prod _ { 0 < a \\leq v _ i \\cdot d } ( v _ i + a z ) \\right ) \\textbf { 1 } _ { [ - d ] } I _ { D _ { Y , + } , d } , \\end{align*}"}
{"id": "6423.png", "formula": "\\begin{align*} \\vartheta ( x , y , z ) = \\Phi _ 1 \\left ( [ x , y ] _ \\mathfrak { g } , z \\right ) - [ \\Phi _ 0 ( x ) , \\Phi _ 1 ( y , z ) ] + \\circlearrowleft ( x , y , z ) . \\end{align*}"}
{"id": "8479.png", "formula": "\\begin{align*} U & \\le m \\left ( L + \\frac { L ( m - 1 ) } { 2 } \\right ) \\\\ & = \\frac { L m ( m + 1 ) } { 2 } . \\end{align*}"}
{"id": "5491.png", "formula": "\\begin{align*} F ' ( U , V ) ( H , K ) & = \\langle R ( U , V ) , H \\Sigma V ^ { T } + U \\Sigma K ^ { T } - P ( H \\Sigma V ^ { T } + U \\Sigma K ^ { T } ) \\rangle \\\\ & = \\langle R ( U , V ) , H \\Sigma V ^ { T } + U \\Sigma K ^ { T } \\rangle \\\\ & = \\langle R ( U , V ) V \\Sigma ^ { T } , H \\rangle + \\langle \\Sigma ^ { T } U ^ { T } R ( U , V ) , K ^ { T } \\rangle . \\end{align*}"}
{"id": "474.png", "formula": "\\begin{align*} { \\mathcal C } _ X ( A , z ) & = \\sum _ { n = 0 } ^ \\infty z ^ n = \\frac 1 { 1 - z } , \\\\ { \\mathcal C } _ X ( L , z ) & = \\sum _ { n = 1 } ^ \\infty \\frac { z ^ n } n = - \\ln ( 1 - z ) , \\\\ { \\mathcal C } _ X ( { \\bf { N } } _ s , z ) & = \\sum _ { n = 1 } ^ s \\frac { z ^ n } n . \\end{align*}"}
{"id": "2995.png", "formula": "\\begin{align*} ( q - 3 ) h ( \\phi ) \\leq \\sum _ { i = 1 } ^ q N ^ { ( 2 ) } ( D _ i , \\phi ) + 6 \\max \\{ 0 , g ( C ) - 1 \\} . \\end{align*}"}
{"id": "979.png", "formula": "\\begin{align*} t ^ { - 1 } \\left ( x ^ { k } g ( x ) \\right ) = H ^ { k } t ^ { - 1 } ( g ( x ) ) = H ^ { k } \\overline { g } , 0 \\leq k \\leq n - 1 . \\end{align*}"}
{"id": "1037.png", "formula": "\\begin{align*} \\| H ( \\xi _ { 1 } ) - H ( \\xi _ { 0 } ) \\| _ { p } & \\leq \\int _ { 0 } ^ { \\xi _ { 0 } - 2 \\delta } \\big \\| ( F ( \\xi _ { 1 } - s ) - F ( \\xi _ { 0 } - s ) ) G ( s ) \\big \\| _ { p } \\ , \\textnormal { d } s \\\\ & + \\sum _ { j = 0 } ^ { 1 } \\int _ { \\xi _ { 0 } - 2 \\delta } ^ { \\xi _ { j } } \\| F ( \\xi _ { j } - s ) G ( s ) \\| _ { p } \\ , \\textnormal { d } s . \\end{align*}"}
{"id": "1857.png", "formula": "\\begin{align*} A _ { j } ^ { ( q ) } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { A _ { [ n , j ] } ^ { ( q ) } } { z ^ { n + 1 } } = \\sum _ { n \\in \\mathbb { Z } } \\frac { A _ { [ n , j ] } ^ { ( q ) } } { z ^ { n + 1 } } \\\\ B _ { j } ^ { ( q ) } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { B _ { [ n , j ] } ^ { ( q ) } } { z ^ { n + 1 } } = \\sum _ { n \\in \\mathbb { Z } } \\frac { B _ { [ n , j ] } ^ { ( q ) } } { z ^ { n + 1 } } . \\end{align*}"}
{"id": "2721.png", "formula": "\\begin{align*} \\bigg | \\Big \\langle \\mathbb { H } - \\mathbb { H } _ \\mathrm { r i g h t } ^ { ( M _ \\Lambda ) } & - \\mathbb { H } _ \\mathrm { l e f t } ^ { ( M _ \\Lambda ) } - \\mathrm { d } \\Gamma _ \\perp \\left ( Q _ { > M _ \\Lambda } \\left ( h _ { \\mathrm { M F } } - \\mu _ + \\right ) Q _ { > M _ \\Lambda } \\right ) \\Big \\rangle _ \\Phi \\bigg | \\\\ \\le \\ ; & C _ \\Lambda o _ N ( 1 ) + \\frac { C } { \\left ( \\mu _ { 2 M _ \\Lambda + 2 } - \\mu _ + \\right ) ^ { 1 / 2 } } \\end{align*}"}
{"id": "2968.png", "formula": "\\begin{align*} \\{ Q = [ c , | d | , 0 ] : 0 \\leq c < | d | \\} \\{ Q = [ 0 , | d | , c ] : 0 \\leq c < | d | \\} \\end{align*}"}
{"id": "4589.png", "formula": "\\begin{align*} P ( G ) : = \\prod _ { e \\in \\mathcal { E } ( G ) } p ( e ) , p ( e ) : = p _ { i _ 1 i _ 2 } , \\end{align*}"}
{"id": "8029.png", "formula": "\\begin{align*} S : = \\{ a _ 1 , \\ldots , a _ m \\} . \\end{align*}"}
{"id": "8769.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\frac { 1 } { p _ k } > N - 1 , \\end{align*}"}
{"id": "6362.png", "formula": "\\begin{align*} P _ { \\kappa } ( x ) = 1 - \\left [ \\ , R _ { \\kappa } ( x ) + Q _ { \\kappa } ( x ) \\sqrt { 1 + \\kappa ^ 2 x ^ 2 } \\ , \\ , \\right ] \\exp _ { \\kappa } ( - x ) \\ \\ , \\end{align*}"}
{"id": "6524.png", "formula": "\\begin{align*} S \\simeq \\sum \\limits _ { n = 1 } ^ \\infty 2 ^ n \\frac { 1 } { A ( 2 ^ { n } ) } \\simeq \\sum \\limits _ { i = 1 } ^ \\infty \\frac { 1 } { A ( i ) } = \\infty . \\end{align*}"}
{"id": "7851.png", "formula": "\\begin{align*} \\| e _ Q a J b J x J c J d e _ Q \\| _ { \\infty , 1 } = \\sup _ { y , z \\in ( Q ) _ 1 } | \\tau ( a x d y c ^ * b ^ * z ) | \\leq \\| b c \\| _ 2 \\| a x d \\| _ 2 . \\end{align*}"}
{"id": "1095.png", "formula": "\\begin{align*} P _ { f X } Y = f P _ X Y , P _ X f Y = f P _ X Y . \\end{align*}"}
{"id": "7950.png", "formula": "\\begin{align*} D _ + = \\sum _ { i \\in ( M _ + \\cup \\{ j \\} ) } \\bar { D } _ i \\subset X _ + D _ - = \\sum _ { i \\in M _ + } \\bar { D } _ i \\subset X _ - . \\end{align*}"}
{"id": "866.png", "formula": "\\begin{align*} s ^ \\prime x ^ \\prime y ^ \\prime & = s ^ \\prime \\Delta ^ n & ( \\ ; x ^ \\prime y ^ \\prime = \\Delta ^ n ) \\\\ & = \\Delta ^ n t ^ \\prime \\quad \\ ; t ^ \\prime \\in T & ( \\ ; T \\Delta = \\Delta T ) \\\\ & = x ^ \\prime y ^ \\prime t ^ \\prime & ( \\ ; \\Delta ^ n = x ^ \\prime y ^ \\prime ) . \\end{align*}"}
{"id": "1926.png", "formula": "\\begin{align*} \\mathrm { c a r d } ( \\mathcal { S } _ { [ m ( p + 1 ) + j , j ] } ) = \\frac { j + 1 } { p m + j + 1 } \\binom { m ( p + 1 ) + j } { m } . \\end{align*}"}
{"id": "4997.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { s \\in [ 0 , T ] } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ^ 2 ( u , s ) E [ | \\sigma ( X ^ n _ { \\eta _ n ( u ) } ) - \\sigma ( X _ { \\eta _ n ( u ) } ) | ^ 2 ] d u = 0 . \\end{align*}"}
{"id": "8297.png", "formula": "\\begin{align*} \\hat { P } \\phi _ { n } ^ { \\pm } ( x ) = \\hat { T } \\phi _ { n } ^ { \\pm } ( x ) = \\phi _ { n } ^ { \\mp } ( x ) . \\end{align*}"}
{"id": "4622.png", "formula": "\\begin{align*} c _ 1 ( \\alpha , h ) = c _ 2 ( \\alpha , g ) = r \\ , \\ , \\mbox { a n d } \\ , \\ , c _ 1 ( \\alpha , g ) = c _ 2 ( \\alpha , h ) = t , \\end{align*}"}
{"id": "7224.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left \\{ \\frac { ( 2 x ) ^ { 1 - a } } { 1 + t + x } \\log \\left ( \\frac { 1 + t - x } { 1 + R } \\right ) \\right \\} ^ { a _ n } \\quad \\mbox { i n } \\ D \\end{align*}"}
{"id": "6734.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } J ( y _ n , u _ n ) = \\inf _ { ( y , u ) \\in Y \\times U } J ( y , u ) : = \\hat { J } . \\end{align*}"}
{"id": "8883.png", "formula": "\\begin{align*} ( r , s ) \\cdot ( a \\otimes b \\otimes \\dots \\otimes y \\otimes z ) \\cdot ( r ' , s ' ) = ( r \\cdot a \\cdot r ' ) \\otimes b \\otimes \\dots \\otimes y \\otimes ( \\bar s \\cdot z \\cdot \\bar s ' ) \\ , , \\end{align*}"}
{"id": "4850.png", "formula": "\\begin{align*} [ X , X ] _ { \\mathcal { G } _ 1 } ( a , b , c ) = \\frac { 2 } { 3 } ( X ( X ( a , b ) , c ) - X ( a , X ( b , c ) ) + \\\\ X ( X ( b , c ) , a ) - X ( b , X ( c , a ) ) + X ( X ( c , a ) , b ) - X ( c , X ( a , b ) ) \\\\ = \\frac { 4 } { 3 } ( X ( X ( a , b ) , c ) + X ( X ( b , a ) , c ) + X ( X ( c , a ) , b ) ) \\end{align*}"}
{"id": "7054.png", "formula": "\\begin{align*} & \\sum _ { n \\geq 0 } \\sum _ { M \\geq 1 } h _ r ( n , M ) z ^ M q ^ n = \\\\ & \\sum _ { n \\geq 1 } z ^ { n - 1 } ( z q ^ n + z ^ 2 q ^ { 2 n } + \\cdots + z ^ { r - 1 } q ^ { ( r - 1 ) n } ) \\prod _ { j = 1 } ^ { n - 1 } ( 1 + z q ^ j + z ^ 2 q ^ { 2 j } + \\cdots + z ^ { r - 1 } q ^ { ( r - 1 ) j } ) . \\end{align*}"}
{"id": "8033.png", "formula": "\\begin{align*} Z = \\cap _ { i \\in M _ + } \\{ z _ i = 0 \\} \\subset X \\end{align*}"}
{"id": "1107.png", "formula": "\\begin{align*} { \\rm { H o r } } P _ U U = { \\rm { H o r } } P _ V V . \\end{align*}"}
{"id": "8172.png", "formula": "\\begin{align*} h ( 0 ) \\varphi _ 0 = E \\varphi _ 0 \\end{align*}"}
{"id": "2710.png", "formula": "\\begin{align*} \\ell ^ 2 ( \\mathcal { F } _ \\perp ) : = \\bigoplus _ { d \\in \\mathbb { Z } } \\mathcal { F } _ \\perp . \\end{align*}"}
{"id": "545.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { g _ k ( n ) \\tau ^ * ( n ) } { n ^ 2 } = \\prod _ { p } \\Big ( 1 - \\frac { 2 } { p ^ { 2 k } } \\Big ) , \\end{align*}"}
{"id": "8528.png", "formula": "\\begin{align*} \\phi _ { m + 1 } ( t ) = ( t ^ 2 - 1 ) \\phi _ m ( t ) + 4 a _ { m + 1 } \\psi _ m ( t ) + 1 6 \\ : a _ { m + 1 } ^ 2 \\phi _ { m - 1 } ( t ) . \\end{align*}"}
{"id": "5338.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s m - \\frac { ( - \\Delta ) ^ s m _ 1 } { \\gamma _ 1 ^ { 1 / 2 } } m & = 0 \\quad \\Omega , \\\\ m & = m _ 0 \\quad \\Omega _ e . \\end{align*}"}
{"id": "2707.png", "formula": "\\begin{align*} \\mathfrak { H } ^ N = \\ ; & \\Big ( \\mathrm { s p a n } \\{ u _ 1 \\} \\oplus \\mathrm { s p a n } \\{ u _ 2 \\} \\oplus \\bigoplus _ { m \\ge 3 } ^ \\infty \\mathrm { s p a n } \\{ u _ m \\} \\Big ) ^ { \\otimes _ { \\mathrm { s y m } } N } \\end{align*}"}
{"id": "3684.png", "formula": "\\begin{align*} \\begin{aligned} \\beta _ { r r } & = \\varphi '' ( r ) \\big [ \\log \\cos r - f ( s , 0 ) \\big ] - 2 \\varphi ' ( r ) \\tan r - \\varphi ( r ) \\sec ^ 2 r \\\\ & \\leq | \\varphi '' ( r ) | \\big [ - \\log \\cos R + \\bar { f } \\big ] \\end{aligned} \\end{align*}"}
{"id": "3516.png", "formula": "\\begin{align*} \\overline { G } _ { ( 2 ) } ( u ) = ( 3 + 2 u + 3 \\sqrt { 2 u } ) \\exp ( - \\sqrt { 2 u } ) . \\end{align*}"}
{"id": "7315.png", "formula": "\\begin{align*} e _ { j , J } ^ \\bot & = e _ j \\chi _ J - \\sum _ { i = 0 } ^ J ( e _ j \\chi _ J , e _ i ) e _ i \\\\ & = e _ j \\chi _ J - ( e _ j \\chi _ J , e _ j ) e _ j - \\sum _ { i \\not = j } ( e _ j ( \\chi _ J - 1 ) , e _ i ) e _ i \\\\ & = e _ j ( \\chi _ J - 1 ) + ( e _ j ( 1 - \\chi _ J ) , e _ j ) e _ j - \\sum _ { i \\not = j } ( e _ j ( \\chi _ J - 1 ) , e _ i ) e _ i . \\end{align*}"}
{"id": "8704.png", "formula": "\\begin{align*} h ( r , \\dot { r } , t ) = \\frac { 1 } { 2 } ( 1 - b ^ 2 ) ^ \\frac { 3 } { 2 } r ^ 2 + \\lambda ( \\sqrt { r ^ 2 + \\dot { r } ^ 2 } + b ( \\dot { r } \\cos ( \\theta - t ) + r \\sin ( \\theta - t ) ) ) . \\end{align*}"}
{"id": "1676.png", "formula": "\\begin{align*} \\| O _ 1 y _ 1 + O _ 2 y _ 2 \\| = \\| y _ 1 + y _ 2 \\| \\end{align*}"}
{"id": "5850.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p } } B ( x _ k , x _ { k + 1 } ) ^ q a _ k ^ q \\bigg ) ^ { \\frac { 1 } { q } } \\leq C '' \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } a _ k ^ p \\bigg ) ^ { \\frac { 1 } { p } } \\end{align*}"}
{"id": "7763.png", "formula": "\\begin{align*} Q _ j ^ k ( \\alpha _ i ) + \\tilde { \\eta } k = 1 , \\ldots , m , \\ j = 1 , \\ldots , L , \\ i = 1 , \\ldots , p . \\end{align*}"}
{"id": "4756.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 0 } ^ \\infty a _ n q ^ n , \\textrm { w i t h } q = e ^ { 2 \\pi i z } \\end{align*}"}
{"id": "2586.png", "formula": "\\begin{align*} \\mathcal O _ { x , y } \\simeq \\begin{cases} \\mathbb C P ^ 2 \\ , \\ \\ \\mbox { i f } \\ \\ x = 0 \\ \\ \\mbox { o r } \\ \\ y = 0 \\\\ \\mathcal E \\ , \\ \\ \\mbox { i f } \\ \\ x , y > 0 \\end{cases} \\ . \\end{align*}"}
{"id": "5091.png", "formula": "\\begin{align*} P ^ n _ \\tau = P ^ { n , 1 } _ \\tau + P ^ { n , 2 } _ \\tau , \\end{align*}"}
{"id": "8815.png", "formula": "\\begin{align*} 1 _ { L _ 1 ( t ) } ^ * = 1 _ { ( \\phi _ 1 ^ * ) ^ { - 1 } ( \\mathbb { R } _ { > t } ) } = 1 _ { ( - J _ 1 ( t ) , J _ 1 ( t ) ) } , 1 _ { L _ 2 ( t ) } ^ * = 1 _ { ( \\phi _ 2 ^ * ) ^ { - 1 } ( \\mathbb { R } _ { > t } ) } = 1 _ { ( - J _ 2 ( t ) , J _ 2 ( t ) ) } \\end{align*}"}
{"id": "2293.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( \\varphi _ y , \\psi _ y ) ( x , 0 ) = ( \\overline u _ e ^ i , \\overline h _ e ^ i ) ( x ) , ( \\varphi , \\psi ) | _ { y \\rightarrow \\infty } = ( 0 , 0 ) , \\\\ & \\varphi ( 1 , y ) = \\int _ y ^ \\infty u _ 0 ^ i ( \\theta ) { \\rm d } \\theta , \\psi ( 1 , y ) = \\int _ y ^ \\infty h _ 0 ^ i ( \\theta ) { \\rm d } \\theta . \\end{aligned} \\right . \\end{align*}"}
{"id": "3384.png", "formula": "\\begin{align*} v _ t & : = \\left ( ( t + T ) \\wedge \\lambda _ { \\ell , T } \\wedge \\rho _ { r , T } - T \\right ) \\ 1 { T < \\rho _ \\infty } ; \\\\ \\zeta ^ { ( f , \\theta ) } _ t & : = \\left ( f ( \\kappa _ { T + v _ t } ) - \\theta v _ t \\right ) \\ 1 { T < \\rho _ \\infty } . \\end{align*}"}
{"id": "1093.png", "formula": "\\begin{align*} P _ X Y = P _ Y X , \\end{align*}"}
{"id": "8801.png", "formula": "\\begin{align*} 1 _ A ^ { - 1 } ( \\mathbb { R } _ { > t } ) = \\{ g \\in G \\mid 1 _ A ( g ) > t \\} = \\left \\{ \\begin{aligned} & A & & \\ ; t < 1 \\\\ & \\emptyset & & \\ ; t \\geq 1 \\end{aligned} \\right . . \\end{align*}"}
{"id": "7982.png", "formula": "\\begin{align*} I _ { D _ { Y , + } , d } = \\frac { 1 } { \\prod _ { i \\in I _ + , D _ i \\cdot d > 0 } ( \\bar D _ i + ( D _ i \\cdot d ) z ) } [ \\textbf { 1 } ] _ { ( - D _ i \\cdot d ) _ { i \\in I _ + } } ; \\end{align*}"}
{"id": "3142.png", "formula": "\\begin{align*} E _ { 1 } \\doteq \\bigcap \\limits _ { x \\in \\mathbb { Z } ^ { d } , A \\in \\mathcal { U } } \\{ \\rho \\in \\mathcal { U } ^ { \\ast } : \\rho ( \\mathfrak { 1 } ) = 1 , \\ ; \\rho ( | A | ^ { 2 } ) \\geq 0 , \\ ; \\rho = \\rho \\circ \\alpha _ { x } \\} \\ , \\end{align*}"}
{"id": "1033.png", "formula": "\\begin{align*} \\| \\tilde { B } T ( t , A ) - \\tilde { B } T ( t _ { 0 } , A ) \\| _ { q } & = \\| \\tilde { B } T ( t _ { 0 } / 2 , A ) ( T ( t - t _ { 0 } / 2 , A ) - T ( t _ { 0 } / 2 , A ) ) \\| _ { q } \\\\ & \\leq \\| \\tilde { B } T ( t _ { 0 } / 2 , A ) \\| _ { q } \\| T ( t - t _ { 0 } / 2 , A ) - T ( t _ { 0 } / 2 , A ) \\| _ { \\infty } . \\end{align*}"}
{"id": "2962.png", "formula": "\\begin{align*} \\Delta _ k = y ^ 2 \\left ( \\partial _ x ^ 2 + \\partial _ y ^ 2 \\right ) - i k y \\partial _ x \\end{align*}"}
{"id": "1738.png", "formula": "\\begin{align*} \\log F ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) \\sim & \\int _ C \\frac { e ^ { z s } } { e ^ { \\overline \\omega _ 1 s } - 1 } \\sum _ { k \\geq 0 } \\frac { B _ k \\cdot ( \\omega _ 2 ) ^ { k - 1 } ( s ) ^ { k - 2 } } { k ! } \\ , d s \\\\ = & \\sum _ { k \\geq 0 } \\frac { B _ k \\cdot \\omega _ 2 ^ { k - 1 } } { k ! } \\cdot f _ { k - 2 } ( z , \\overline \\omega _ 1 ) . \\end{align*}"}
{"id": "1275.png", "formula": "\\begin{align*} \\Psi ( E _ k ( x , d - 1 ) ) & + \\sum _ { j = 0 } ^ { k - 2 } ( d - 1 ) ^ { k - 1 - j } \\Psi ( E _ { j + 1 } ( x , d - 1 ) ) = \\\\ & = \\sum _ { j = 0 } ^ { k - 1 } ( d - 1 ) ^ { k - 1 - j } \\Psi ( E _ { j + 1 } ( x , d - 1 ) ) , \\end{align*}"}
{"id": "1085.png", "formula": "\\begin{align*} \\eta _ n ( [ 0 , T ] ) : = \\bigcup _ { t \\in [ 0 , T ] } \\eta _ n ( t ) . \\end{align*}"}
{"id": "8976.png", "formula": "\\begin{align*} c _ p ^ 2 \\geq \\frac { 3 L + 3 \\epsilon } { 3 L + \\epsilon } = 1 + \\frac { 2 \\epsilon } { 3 L + \\epsilon } . \\end{align*}"}
{"id": "727.png", "formula": "\\begin{align*} ( \\partial _ s + \\Delta ) \\varphi = - \\rho \\lambda ^ 2 e ^ { \\lambda u } ( 1 + \\rho e ^ { \\lambda u } ) \\varphi | \\nabla u | _ g ^ 2 + \\rho \\lambda e ^ { \\lambda u } \\varphi ( \\partial _ s + \\Delta ) u . \\end{align*}"}
{"id": "5868.png", "formula": "\\begin{align*} C _ { 6 , 1 } & \\approx \\sum _ { k = - \\infty } ^ { M - 1 } \\bigg ( \\int _ { x _ k } ^ { x _ { k + 1 } } u \\bigg ) \\bigg ( \\int _ { x _ k } ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } \\Phi ( 0 , x _ k ) . \\end{align*}"}
{"id": "1597.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { \\partial ^ 2 C ^ 2 } { \\partial z ^ i _ { \\epsilon } \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } v ^ i = f ( x ^ 1 ) \\left [ - f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \\right ] , \\end{align*}"}
{"id": "3124.png", "formula": "\\begin{align*} \\Vert \\widehat { \\lambda } _ h \\widehat { u } _ { \\mathrm { n c } } - \\lambda _ h u _ { \\mathrm { n c } } \\Vert _ { L ^ 2 ( \\Omega ) } \\le & \\lambda _ h \\Vert u - u _ { \\mathrm { n c } } \\Vert _ { L ^ 2 ( \\Omega ) } + \\widehat { \\lambda } _ h \\Vert u - \\widehat { u } _ { \\mathrm { n c } } \\Vert _ { L ^ 2 ( \\Omega ) } + \\vert \\widehat { \\lambda } _ h - \\lambda _ h \\vert . \\end{align*}"}
{"id": "3321.png", "formula": "\\begin{align*} c _ 2 = 0 . 8 9 3 8 1 7 8 5 0 5 2 9 3 1 8 , \\ ; c _ 3 = 0 . 6 8 4 1 5 4 9 0 8 0 2 3 8 3 4 , \\ ; c _ 4 = 0 . 6 2 9 6 4 2 9 9 7 4 6 6 4 2 9 \\end{align*}"}
{"id": "1654.png", "formula": "\\begin{align*} P _ { k \\ell } ( { \\bf u } , { \\bf v } , { \\bf w } ) \\ , \\ , \\ , : = \\ , \\ , \\ , P \\bigl ( \\ , ( { \\bf c } _ k - { \\bf c } _ \\ell ) \\cdot { \\bf u } , \\ , ( { \\bf c } _ k - { \\bf c } _ \\ell ) \\cdot { \\bf v } , \\ , ( { \\bf c } _ k - { \\bf c } _ \\ell ) \\cdot { \\bf w } \\bigr ) , \\end{align*}"}
{"id": "3927.png", "formula": "\\begin{align*} { } | I _ 4 | & \\le c \\frac { \\Delta _ n } { T _ n } \\frac { 1 } { ( \\prod _ { l = 1 } ^ d h _ l ) ^ 2 } \\sum _ { j = j _ D + 1 } ^ { n - 1 } e ^ { - \\rho \\Delta _ n j } \\\\ & \\le c \\frac { \\Delta _ n } { T _ n } \\frac { 1 } { ( \\prod _ { l = 1 } ^ d h _ l ) ^ 2 } e ^ { - \\rho \\Delta _ n ( j _ D + 1 ) } \\\\ & \\le c \\frac { \\Delta _ n } { T _ n } \\frac { 1 } { ( \\prod _ { l = 1 } ^ d h _ l ) ^ 2 } e ^ { - \\rho D } . \\end{align*}"}
{"id": "7465.png", "formula": "\\begin{align*} \\mathrm { F i x } \\left ( \\mathcal { T } _ { \\left \\{ C _ { i } \\right \\} _ { i = 1 } ^ { r } } \\right ) = \\cap _ { i = 1 } ^ { r } C _ { i } . \\end{align*}"}
{"id": "8069.png", "formula": "\\begin{align*} \\operatorname { s g n } ( Z ) = - \\operatorname { s g n } ( g ( \\operatorname { g r a d } ( \\| Z \\| ^ { 2 } ) + 2 \\nabla _ { Z } Z , 2 \\nabla _ { Z } Z ) ) \\end{align*}"}
{"id": "6079.png", "formula": "\\begin{align*} f ( x ) & = x ( x - a ) ( x - b ) ( x - c ) . \\end{align*}"}
{"id": "8707.png", "formula": "\\begin{align*} \\lambda \\left [ 1 + b \\dot { \\theta } \\sin ( \\theta - t ) \\right ] = - a ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } . \\end{align*}"}
{"id": "3349.png", "formula": "\\begin{align*} F _ m ( A ) = \\bigoplus _ { j = 1 } ^ K \\Q ^ { d ( m , j ) } d ( m , j ) = \\begin{cases} 1 & : 1 \\leq m \\leq 2 n _ j - 1 , m , \\\\ 0 & : \\end{cases} \\end{align*}"}
{"id": "3262.png", "formula": "\\begin{align*} \\mathcal { N } _ { A , q } h = ( \\mathcal { M } _ { A , q } \\mathcal { S } h ) | _ { \\partial B } . \\end{align*}"}
{"id": "4995.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { s \\in [ 0 , T ] } n ^ { 2 \\alpha + 1 } E [ | \\widehat \\Theta ^ n _ s - \\Theta ^ n _ s | ^ 2 ] = 0 . \\end{align*}"}
{"id": "6961.png", "formula": "\\begin{align*} \\sigma : = \\sigma ( W ) = \\{ \\lambda _ k ^ 2 \\} _ { k \\ge 1 } \\cup \\{ 0 \\} , \\sigma _ 1 : = \\sigma ( W _ 1 ) = \\{ \\mu _ k ^ 2 \\} _ { k \\ge 1 } \\cup \\{ 0 \\} . \\end{align*}"}
{"id": "2523.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { 1 } { 2 } ( L _ Z g ) ( X , Y ) + g ( X , Y ) d i v ( H ' ) + \\frac { 1 } { \\lambda ^ 2 } R i c ^ N ( \\tilde { X } , \\tilde { Y } ) \\\\ - \\frac { 1 } { 4 } \\lambda ^ 4 g ( X , Y ) | \\nabla _ { \\nu } \\frac { 1 } { \\lambda ^ 2 } | ^ 2 + \\mu g ( X , Y ) + \\frac { n \\lambda ^ 2 } { 2 } g ( X , Y ) \\left ( H ' \\frac { 1 } { \\lambda ^ 2 } \\right ) = 0 . \\end{array} \\end{align*}"}
{"id": "1639.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { F } } = \\{ r _ k ( F ) : F \\in \\mathcal { F } \\mathrm { \\ a n d \\ } k \\le | F | \\} . \\end{align*}"}
{"id": "476.png", "formula": "\\begin{align*} \\epsilon ( g , d ) : = \\min _ { 0 \\leq r \\leq g } \\{ \\epsilon ( r , g , d ) \\} , c ( g , d ) : = \\max _ { 0 \\leq r \\leq g } \\{ c ( r , g , d ) \\} . \\end{align*}"}
{"id": "2578.png", "formula": "\\begin{align*} \\omega _ 1 \\equiv \\dfrac { i } { 2 } \\ , \\lambda _ 3 + \\dfrac { i } { 2 \\sqrt { 3 } } \\ , \\lambda _ 8 = \\dfrac { i } { 3 } \\small { \\begin{pmatrix} 2 & 0 & 0 \\\\ 0 & - 1 & 0 \\\\ 0 & 0 & - 1 \\end{pmatrix} } \\ \\ , \\ \\ \\ \\omega _ 2 = \\dfrac { i } { \\sqrt { 3 } } \\ , \\lambda _ 8 = \\dfrac { i } { 3 } \\small { \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & - 2 \\end{pmatrix} } \\ . \\end{align*}"}
{"id": "5328.png", "formula": "\\begin{align*} N ^ t _ s ( b ) : = \\# \\{ i \\in [ N ^ t _ s ] : | \\pi _ i ^ t ( s ) | \\geq b \\} . \\end{align*}"}
{"id": "3226.png", "formula": "\\begin{align*} t ^ { 2 m } + s _ 2 t ^ { 2 m - 2 } + \\cdots + s _ { 2 m } = 0 . \\end{align*}"}
{"id": "3069.png", "formula": "\\begin{align*} f _ { \\mathbf { M ^ { \\ast } } _ { - \\xi _ { \\boldsymbol { s } k } } } ( \\boldsymbol { w } ) = c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast } \\exp \\left \\{ - \\frac { 1 } { 2 } \\boldsymbol { w } ^ { T } \\boldsymbol { w } - \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } \\right \\} = \\phi _ { n - 1 } ( \\boldsymbol { w } ) , ~ \\boldsymbol { w } \\in \\mathbb { R } ^ { n - 1 } , \\end{align*}"}
{"id": "5733.png", "formula": "\\begin{align*} \\pi _ { [ a , i ] } \\cdot \\pi _ { [ i + 1 , b ] } = \\binom { b - a + 1 } { i - a + 1 } \\pi _ { [ a - 1 , b - 1 ] } + \\binom { b - a } { i - a } \\pi _ { [ a , b - 1 ] } y _ i + \\binom { b - a } { i - a + 1 } \\pi _ { [ a , b - 1 ] } y _ { b } . \\end{align*}"}
{"id": "5858.png", "formula": "\\begin{align*} A _ 2 & \\lesssim C _ 1 + \\sup _ { k \\leq M - 1 } \\sup _ { x \\in ( x _ k , x _ { k + 1 } ) } \\bigg ( \\int _ { x } ^ { \\infty } u \\bigg ) ^ { \\frac { 1 } { q } } \\bigg ( \\int _ { 0 } ^ { x } \\bigg ( \\int _ 0 ^ t w \\bigg ) ^ { - \\frac { p } { p - r } } w ( t ) V _ r ( 0 , t ) ^ { \\frac { p r } { p - r } } d t \\bigg ) ^ { \\frac { p - r } { p r } } \\\\ & = C _ 1 + C _ 3 , \\end{align*}"}
{"id": "6302.png", "formula": "\\begin{align*} & \\int \\frac { x ^ 2 ( x ^ 2 ; q ^ 2 ) _ \\infty } { ( r _ n x ^ 2 ; q ^ 2 ) _ \\infty } p _ n ( x ; - 1 ; q ) d _ q x = \\\\ & \\frac { q ^ { n + 2 } ( \\frac { x ^ 2 } { q ^ 2 } ; q ^ 2 ) _ \\infty } { [ n ] _ q [ n + 1 ] _ q ( \\frac { r _ n x ^ 2 } { q ^ 2 } ; q ^ 2 ) _ \\infty } \\left ( \\frac { q ^ { n + 1 } ( 1 - \\frac { x ^ 2 } { q ^ 2 } ) } { [ n + 1 ] _ q ( 1 - q ^ n ) } \\ , _ 3 \\phi _ 2 ( q ^ { - n } , q ^ { n + 1 } , x ; q ; - q ; q , q ) - x p _ n ( \\frac { x } { q } ; - 1 ; q ) \\right ) . \\end{align*}"}
{"id": "5760.png", "formula": "\\begin{align*} G = N \\rtimes P \\end{align*}"}
{"id": "7936.png", "formula": "\\begin{align*} b _ { m + 1 } = \\sum _ { i \\in M _ + } ( D _ i \\cdot e ) b _ i = \\sum _ { i \\in M _ - } - ( D _ i \\cdot e ) b _ i . \\end{align*}"}
{"id": "4778.png", "formula": "\\begin{align*} | \\mathcal { S } _ { \\kappa , 3 } ^ C | = \\binom { \\kappa + 7 } { 7 } - a _ k - 3 b _ k = 6 c _ k . \\end{align*}"}
{"id": "7349.png", "formula": "\\begin{align*} & [ x \\ast [ y , [ z , u ] ] , v ] + [ v \\ast [ y , [ z , u ] ] , x ] & \\\\ = & ( x [ y , [ z , u ] ] + [ y , [ z , u ] ] x ) v - v ( x [ y , [ z , u ] ] + [ y , [ z , u ] ] x ) & \\\\ & + ( v [ y , [ z , u ] ] + [ y , [ z , u ] ] v ) x - x ( v [ y , [ z , u ] ] + [ y , [ z , u ] ] v ) & \\\\ = & x ( [ y , [ z , u ] ] v ) + [ y , [ z , u ] ] ( x v ) - ( v x ) [ y , [ z , u ] ] - v ( [ y , [ z , u ] ] x ) & \\\\ & + v ( [ y , [ z , u ] ] x ) + [ y , [ z , u ] ] ( v x ) - ( x v ) [ y , [ z , u ] ] - x ( [ y , [ z , u ] ] v ) & \\\\ = & [ y , [ z , u ] ] ( x \\ast v ) - ( x \\ast v ) [ y , [ z , u ] ] & \\\\ = & - [ x \\ast v , [ y , [ z , u ] ] ] . & \\end{align*}"}
{"id": "8049.png", "formula": "\\begin{align*} d H e ^ { \\frac { H } { 2 \\pi i } } \\sum _ { m \\geq 1 } ( - 1 ) ^ m \\frac { ( 2 \\pi i ) e ^ { \\frac { 2 \\pi i m } { d } } } { e ^ { - H } - e ^ { \\frac { 2 \\pi i m } { d } } } \\frac { q ^ { - \\frac m d } } { \\Gamma ( m ) \\Gamma ( 1 + \\frac { d ^ \\prime } { d } m ) \\prod _ { i = 1 } ^ N \\Gamma ( 1 - q _ i m ) } . \\end{align*}"}
{"id": "5644.png", "formula": "\\begin{align*} \\sigma _ k ( \\lambda ( D ^ 2 u ) ) = f \\end{align*}"}
{"id": "1254.png", "formula": "\\begin{align*} P ( T , x ) & = \\dfrac { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { \\displaystyle \\prod _ { j = 0 } ^ { l ( T ) - 1 } W _ j ^ { n ( T , l ( T ) - j ) } } . \\end{align*}"}
{"id": "940.png", "formula": "\\begin{align*} \\langle \\nabla _ i \\nabla _ j \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { i j } ^ l , \\nabla _ m \\nabla _ b \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { m b } ^ l \\rangle = \\frac { 1 } { 2 t } O ( t ) + \\sum _ l \\Gamma _ { i j } ^ l \\Gamma _ { m b } ^ l . \\end{align*}"}
{"id": "5961.png", "formula": "\\begin{align*} \\omega _ j ( t ) & = \\frac 1 { \\Gamma ( 1 - \\beta ) } \\int _ { t _ j } ^ { \\min ( t , t _ { j + 1 } ) } ( t - \\tau ) ^ { - \\beta } \\ , d \\tau \\\\ & = \\frac 1 { \\Gamma ( 2 - \\beta ) } \\left ( ( t - t _ j ) ^ { 1 - \\beta } - ( t - \\min ( t , t _ { j + 1 } ) ) ^ { 1 - \\beta } \\right ) \\end{align*}"}
{"id": "7494.png", "formula": "\\begin{align*} A _ n ( t ) = t ^ { n - 1 } A _ n ( 1 / t ) . \\end{align*}"}
{"id": "1426.png", "formula": "\\begin{align*} \\sum _ { z \\in S ^ 1 _ N } \\exp \\Big ( - \\kappa ( z - w - y ) ^ 2 \\Big ) & \\leq \\sum _ { z = w - 1 } ^ { w + 1 } \\exp \\Big ( - \\kappa ( z - w - y ) ^ 2 \\Big ) \\\\ & + \\int _ { - \\infty } ^ { \\infty } \\exp \\Big ( - \\kappa ( z - w - y ) ^ 2 \\Big ) d z . \\end{align*}"}
{"id": "7502.png", "formula": "\\begin{align*} \\zeta ( t ) = t ^ { \\sigma } , \\ ; \\ ; \\sigma > - 1 . \\end{align*}"}
{"id": "4708.png", "formula": "\\begin{align*} ( - 1 ) ^ k \\sum _ { j = 1 - k } ^ k ( - 1 ) ^ j p ( n - j ( 3 j + 1 ) / 2 ) = M _ k ( n ) . \\end{align*}"}
{"id": "4842.png", "formula": "\\begin{align*} \\tau ^ \\mathrm { t o p } ( M ' / S , \\mathbb { 1 } ) = \\tau ^ \\mathrm { t o p } \\big ( M / S , \\mathrm { I n d } ^ G _ H \\mathbb { 1 } \\big ) \\ ; . \\end{align*}"}
{"id": "1679.png", "formula": "\\begin{align*} \\| O _ 1 y _ 1 + y _ 2 \\| & \\le \\| O _ 1 y _ 1 + z _ 2 \\| + \\| z _ 2 - y _ 2 \\| \\\\ & \\le ( 1 + \\alpha ) \\| y _ 1 + z _ 2 \\| + \\| z _ 2 - y _ 2 \\| \\\\ & \\le ( 1 + \\alpha ) \\| y _ 1 + y _ 2 \\| + ( 2 + \\alpha ) \\| z _ 2 - y _ 2 \\| \\\\ & < ( 1 + \\alpha ) \\| y _ 1 + y _ 2 \\| + ( 2 + \\alpha ) \\alpha \\| y _ 2 \\| \\\\ & \\le ( 1 + \\alpha ( 1 + ( 2 + \\alpha ) A ) ) \\| y _ 1 + y _ 2 \\| . \\end{align*}"}
{"id": "4292.png", "formula": "\\begin{align*} \\rho ^ { \\rm ( s t ) } = \\exp \\biggl ( \\frac 1 2 \\mathfrak { c } ^ { T } M \\mathfrak { c } + s \\biggr ) , \\end{align*}"}
{"id": "6606.png", "formula": "\\begin{align*} N ( t ) = \\Lambda _ \\Omega ( t ) + O _ \\Omega \\big ( t ^ { d - \\delta } \\big ) . \\end{align*}"}
{"id": "4773.png", "formula": "\\begin{align*} | \\mathcal K _ { \\kappa , 2 } | & = \\left | \\mathcal T _ { \\kappa , 2 } \\right | + \\frac 1 2 \\left | \\mathcal S _ { \\kappa , 2 } \\right | - \\frac 1 2 \\left | \\mathcal T _ { \\kappa , 2 } \\right | = \\frac 1 2 \\left ( \\left | \\mathcal S _ { \\kappa , 2 } \\right | + \\left | \\mathcal T _ { \\kappa , 2 } \\right | \\right ) = \\frac 1 2 \\left ( \\sum _ { i = 0 } ^ { \\lfloor \\kappa / 2 \\rfloor } ( \\kappa - 2 i + 1 ) + \\binom { \\kappa + 3 } { 3 } \\right ) . \\end{align*}"}
{"id": "5202.png", "formula": "\\begin{align*} N ( \\pi ) = \\# \\{ ( x _ i , y _ j ) \\ , | \\ , \\hbox { t h e $ s $ - p o i n t $ y _ j $ p r e c e d e s t h e $ r $ - p o i n t $ x _ i $ i n t h e $ \\pi $ - o r d e r } \\} . \\end{align*}"}
{"id": "48.png", "formula": "\\begin{align*} \\Tilde { \\mathcal { N } } ( f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ) \\circ \\rho _ { \\eta } ( X ) = \\Pi _ { w \\in W _ { \\rho } ^ { m _ 0 } } f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ( X + w ) , \\end{align*}"}
{"id": "6839.png", "formula": "\\begin{align*} \\| \\frac { 1 } { V _ F } e ^ { i k \\frac { 2 \\pi } { V _ F } v } p _ k ( t , g ) \\| _ { L ^ 1 ( ( 0 , V _ F ) \\times \\mathbb { R } ) } & \\leq \\int _ { 0 } ^ { V _ F } \\frac { 1 } { V _ F } | e ^ { i k \\frac { 2 \\pi } { V _ F } v } | d v \\int _ { \\mathbb { R } } | p _ k ( t , g ) | d g \\\\ & = \\| p _ k ( t , g ) \\| _ { { L ^ 1 ( \\mathbb { R } ) } } . \\end{align*}"}
{"id": "3602.png", "formula": "\\begin{align*} { \\rm R C } ( I ^ \\vee ) = \\mathbb { R } _ + \\{ e _ 1 , \\ldots , e _ s , ( u _ 1 , 1 ) , \\ldots , ( u _ m , 1 ) \\} . \\end{align*}"}
{"id": "5777.png", "formula": "\\begin{align*} E ( G / H ^ 0 ) = E ( G / H ) E ( H / H ^ 0 ) \\neq 0 \\end{align*}"}
{"id": "8118.png", "formula": "\\begin{align*} \\lambda _ { k + 2 } = \\dots = \\lambda _ { k + \\ell } = \\delta _ { z _ { k + 1 } } , \\lambda _ { k + \\ell + 1 } = \\lambda | _ { ( z _ { k + \\ell + 1 } , z _ { k + 1 } ) } + ( \\lambda ( \\{ z _ { k + 1 } \\} ) - ( \\ell - 1 ) ) \\delta _ { z _ { k + 1 } } , \\end{align*}"}
{"id": "6793.png", "formula": "\\begin{align*} A _ i u : = \\begin{cases} \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\Bigl ( a \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } \\Bigr ) & \\ , \\ , i = 1 , \\\\ a \\frac { \\partial ^ 4 u } { \\partial x ^ 4 } & \\ , \\ , i = 2 , \\end{cases} \\end{align*}"}
{"id": "19.png", "formula": "\\begin{align*} \\rho _ { \\pi ^ { n m _ 0 } } = U _ 1 U _ { n m _ 0 } Q _ { n m _ 0 } Q _ { n m _ 0 - 1 } \\cdots Q _ 1 , \\end{align*}"}
{"id": "8607.png", "formula": "\\begin{align*} A = ( J - R ) Q , , B = - ( J - R ) C ^ T \\end{align*}"}
{"id": "1425.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\hat Z _ { N , 2 , 1 } & \\geq - ( I _ { 1 , 2 , 1 } + I _ { 2 , 2 , 1 } ) \\\\ & \\geq - C \\left [ \\gamma N ^ { 2 } \\big ( ( \\beta ^ { - 1 / 2 } a ^ { - 1 } N \\log N ) \\vee 1 \\big ) + \\beta N ^ 2 a ^ 2 \\right ] . \\end{aligned} \\end{align*}"}
{"id": "5242.png", "formula": "\\begin{align*} \\int _ { \\Xi _ { r ( I ' ) , s ( I ' ) } } e ^ { W ^ { \\nu } / \\hbar } \\Omega = \\int _ { \\Xi _ { r ( I ' ) , s ( I ' ) } } e ^ { W ^ { g ( \\nu ) } / \\hbar } \\Omega , \\end{align*}"}
{"id": "3047.png", "formula": "\\begin{align*} f _ { \\boldsymbol { X } } ( \\boldsymbol { x } ) : = \\frac { c _ { n } } { \\sqrt { | \\boldsymbol { \\Sigma } | } } g _ { n } \\left \\{ \\frac { 1 } { 2 } ( \\boldsymbol { x } - \\boldsymbol { \\mu } ) ^ { T } \\mathbf { \\Sigma } ^ { - 1 } ( \\boldsymbol { x } - \\boldsymbol { \\mu } ) \\right \\} , ~ \\boldsymbol { x } \\in \\mathbb { R } ^ { n } , \\end{align*}"}
{"id": "1712.png", "formula": "\\begin{align*} I ( q ^ { \\frac { k } { 2 } } \\cdot y _ { \\gamma _ e + \\gamma _ m } ) = { \\rm e x p } ( \\pi i \\tau k + 2 \\pi i \\theta ( \\gamma _ e ) ) \\cdot y _ { \\gamma _ m } \\end{align*}"}
{"id": "7395.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\| V _ 1 ( y + h t _ 1 ) U _ { 0 } ( y ) \\right \\| _ { L ^ 2 ( B _ { { \\sigma } h | t _ 1 | } ( 0 ) ) } \\\\ & \\le C _ 1 \\left \\| \\left ( \\frac { 1 } { | y + h t _ 1 | ^ { m } } \\right ) U _ { 0 } ( y ) \\right \\| _ { L ^ 2 ( B _ { { \\sigma } h | t _ 1 | } ( 0 ) ) } \\le \\frac { C _ 2 } { ( 1 - { \\sigma } ) ^ m } \\frac { 1 } { { { h } ^ { m } } } \\end{aligned} \\end{align*}"}
{"id": "7589.png", "formula": "\\begin{align*} x _ { k + 1 } = \\varphi ( Y _ { k + 1 } , x _ k ) \\end{align*}"}
{"id": "2734.png", "formula": "\\begin{align*} \\epsilon ^ { \\bullet } ( k ) = \\epsilon ^ { \\le n } ( k ) \\epsilon ^ { \\le n } ( s ( \\alpha _ u ) ) ^ { - 1 } q _ { \\alpha _ u } \\end{align*}"}
{"id": "4261.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal L ( Y _ \\rho ) = W _ \\rho \\sum _ { i = 2 } ^ d W ^ 2 _ \\rho ( \\hat \\Theta _ i \\cdot ) - \\gamma _ \\rho \\partial _ \\rho W _ \\rho \\ \\hbox { i n } \\ \\mathbb R ^ n \\end{aligned} \\end{align*}"}
{"id": "8097.png", "formula": "\\begin{align*} Z = f X . \\end{align*}"}
{"id": "549.png", "formula": "\\begin{align*} f _ k ( n ) = \\sum _ { w \\mid n } \\frac { 1 } { w } \\sum _ { \\substack { r d = w \\\\ d \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } \\mu ( r ) = \\sum _ { w \\mid n } \\frac { g _ k ( w ) } { w } . \\end{align*}"}
{"id": "5033.png", "formula": "\\begin{align*} E [ | Q ^ { n , 4 } _ \\tau | ^ 2 ] = n ^ { 4 \\alpha + 2 } \\sum _ { j = 0 } ^ { \\lfloor n \\tau \\rfloor } E [ \\xi _ { j , n } ^ 2 ] . \\end{align*}"}
{"id": "8866.png", "formula": "\\begin{align*} f _ { x _ k } ( \\lambda ) = - \\sqrt { 2 } \\det \\left ( ( \\lambda I _ N - H ) _ { k | k } \\right ) , \\ \\ f _ { y _ { k k + 1 } } ( \\lambda ) = 2 \\det \\left ( ( \\lambda I _ N - H ) _ { k | k + 1 } \\right ) , \\end{align*}"}
{"id": "5286.png", "formula": "\\begin{align*} \\mathbb { H } ^ n ( X , ( \\Omega _ X ^ \\bullet , d + \\hbar ^ { - 1 } d W \\wedge - ) ) = \\Omega _ { \\C ^ n } ^ n / \\delta ( \\Omega _ { \\C ^ n } ^ { n - 1 } ) . \\end{align*}"}
{"id": "522.png", "formula": "\\begin{align*} E _ { g n } ( \\dot { x } , x , t ) = - \\left [ \\frac { { \\dot h _ 1 } ( t ) } { h _ 1 ( t ) } - \\frac { \\dot h _ 2 ( t ) } { h _ 2 ( t ) } \\right ] \\Phi _ { g n } ( x , t ) \\end{align*}"}
{"id": "4552.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 7 \\right ) } \\Vert _ { p } = \\mathcal { O } \\left ( 1 \\right ) ( v ( | z _ { 1 } | ) + ( v ( | z _ { 2 } | ) ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "394.png", "formula": "\\begin{align*} \\beta ( \\ell ) = \\min _ { g \\in \\mathcal { \\mathcal { C } _ { \\ell } } } \\left \\{ \\frac { \\langle ( I - P _ { B ( o , \\ell ) } ) g , g \\rangle _ { \\pi _ { \\ell } } } { \\mathbb { E } _ { \\pi _ { \\ell } } [ g ^ 2 ] } \\right \\} . \\end{align*}"}
{"id": "8363.png", "formula": "\\begin{align*} p \\rightarrow { \\textstyle \\frac { 1 } { \\sqrt { 4 \\pi } } } ( p \\cdot v ) ^ { i \\nu - 1 } , \\ , \\ , \\ , \\ , v = ( 1 , 0 , 0 , 0 ) , \\end{align*}"}
{"id": "6927.png", "formula": "\\begin{align*} \\Gamma = \\left ( \\gamma _ { j + k } \\right ) _ { j , k \\ge 0 } \\ , . \\end{align*}"}
{"id": "920.png", "formula": "\\begin{align*} L _ { i j } ( v , v ) \\ , d x ^ i \\otimes d x ^ j : = \\nabla ^ l ( \\nabla _ i v \\ , d x ^ i ) \\cdot \\nabla _ l ( \\nabla _ j v \\ , d x ^ j ) + \\{ \\frac { 1 } { 2 } e \\nabla _ i v \\cdot \\nabla _ j v + \\frac { 1 } { 2 } ( R _ i { } ^ k \\nabla _ j v + R _ j { } ^ k \\nabla _ i v ) \\cdot \\nabla _ k v \\} d x ^ i \\otimes d x ^ j - \\Delta v \\cdot \\nabla _ i ( \\nabla _ j v \\ , d x ^ j ) d x ^ i . \\end{align*}"}
{"id": "7957.png", "formula": "\\begin{align*} t _ - = \\sum _ { a = 1 } ^ \\mathrm { r } \\bar p _ a ^ - \\log \\tilde y _ a , \\tilde y ^ d = \\tilde y _ 1 ^ { p _ 1 ^ - \\cdot d } \\cdots \\tilde y _ \\mathrm { r } ^ { p _ \\mathrm { r } ^ - \\cdot d } . \\end{align*}"}
{"id": "1890.png", "formula": "\\begin{align*} \\mathcal { P } _ { [ n , j , k ] } = \\{ \\gamma \\in \\mathcal { P } _ { [ n , j ] } : \\kappa _ { i } ( \\gamma ) = k \\} i \\leq k \\leq n - j + i . \\end{align*}"}
{"id": "1579.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial z ^ i _ { \\epsilon } } = 2 b ^ 2 \\Big [ z ^ 1 _ { \\tilde { \\epsilon } } ( - 1 ) ^ { { \\tilde { \\epsilon } } + \\tau } z ^ i _ { \\tilde { \\tau } } z ^ 1 _ { \\tau } + \\delta _ { i 1 } \\sum \\limits _ { k } ^ { } ( - 1 ) ^ { \\epsilon + \\tau } z ^ 1 _ { \\tau } z ^ k _ { \\tilde { \\tau } } z ^ k _ { \\tilde { \\epsilon } } \\Big ] , \\end{align*}"}
{"id": "6956.png", "formula": "\\begin{align*} F ^ { \\{ \\alpha \\} } ( z ) : = ( ( W ^ { \\{ \\alpha \\} } - z I ) ^ { - 1 } p , p ) = \\int _ \\R \\frac { d \\rho ^ { \\{ \\alpha \\} } ( s ) } { s - z } \\forall z \\in \\C \\setminus \\R ; \\end{align*}"}
{"id": "3776.png", "formula": "\\begin{align*} H _ { k , j ; n , l , r ; p ; \\star } ^ { \\mu , m , i ; n o n , a } ( t , x , \\zeta ) = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } E B ^ i ( t , s , x - y + ( t - s ) \\omega , \\omega , v ) \\cdot \\nabla _ v \\big ( ( t - s ) \\mathcal { H } ^ { \\star ; \\mu , E , i } _ { k , j ; n , l , r } ( y , \\omega , v , \\zeta ) + \\mathcal { H } ^ { \\star ; e r r ; \\mu , E , i } _ { k , j ; n , l , r } ( y , v , \\omega , \\zeta ) \\big ) \\end{align*}"}
{"id": "5403.png", "formula": "\\begin{align*} \\delta \\psi ( x , y ) \\vcentcolon = \\psi ( x + y ) + \\psi ( x - y ) - 2 \\psi ( x ) \\end{align*}"}
{"id": "3207.png", "formula": "\\begin{align*} r _ { k , t } = B _ k \\log _ 2 \\left ( 1 + \\frac { h _ { k , t } p _ { k , t } } { \\Gamma } \\right ) , \\end{align*}"}
{"id": "7758.png", "formula": "\\begin{align*} { \\bf T } _ j ( \\alpha _ i ) \\cong \\displaystyle \\sum _ { k = 1 } ^ { \\mu _ j } Q _ j ^ k ( \\alpha _ i ) { \\bf r } _ j ^ k j = 1 , \\ldots , L , \\ i = 1 , \\ldots , p . \\end{align*}"}
{"id": "4654.png", "formula": "\\begin{align*} \\rho _ t ( x , y ) = \\frac { t ^ { - d / \\alpha } \\tilde p \\big ( 1 , t ^ { - 1 / \\alpha } x , t ^ { - 1 / \\alpha } y \\big ) } { t ^ { - \\delta / \\alpha } h ( t ^ { - 1 / \\alpha } x ) t ^ { - \\delta / \\alpha } h ( t ^ { - 1 / \\alpha } y ) } = t ^ { \\frac { 2 \\delta - d } { \\alpha } } \\rho _ 1 \\big ( t ^ { - 1 / \\alpha } x , t ^ { - 1 / \\alpha } y \\big ) , \\end{align*}"}
{"id": "8376.png", "formula": "\\begin{align*} a ( \\ell _ 1 - \\ell _ 2 ) = N ( k - 3 ) / 1 6 - ( N - 1 6 ) \\ell _ 2 / 8 . \\end{align*}"}
{"id": "5929.png", "formula": "\\begin{align*} \\left \\langle \\xi \\left ( { { t _ 1 } } \\right ) . . . \\xi \\left ( { { t _ n } } \\right ) \\right \\rangle _ c = { W ^ { \\left ( n \\right ) } } \\left ( { { t _ 1 } - { t _ 2 } , \\ldots , { t _ 1 } - { t _ n } } \\right ) \\end{align*}"}
{"id": "7203.png", "formula": "\\begin{align*} \\langle f , I ^ { \\circ } ( w _ 1 , z _ 2 ) u _ { p + \\frac { j _ 2 } { T } } w _ 2 \\rangle = 0 \\end{align*}"}
{"id": "7199.png", "formula": "\\begin{align*} & z _ 1 ^ { \\frac { j _ 2 } { T } } ( z _ 1 - z _ 2 ) ^ { \\frac { j _ 1 } { T } } Y _ { M ^ 3 } ( u , z _ 1 ) I ^ { \\circ } ( w _ 1 , z _ 2 ) w _ 2 \\\\ - & e ^ { \\frac { j _ 1 } { T } \\pi i } z _ 1 ^ { \\frac { j _ 2 } { T } } ( z _ 2 - z _ 1 ) ^ { \\frac { j _ 1 } { T } } I ^ { \\circ } ( w _ 1 , z _ 2 ) Y _ { M ^ 2 } ( u , z _ 1 ) w _ 2 \\\\ = & \\mbox { R e s } _ { z _ 0 } z _ 0 ^ { \\frac { j _ 1 } { T } } z _ 1 ^ { - 1 } \\delta ( \\frac { z _ 2 + z _ 0 } { z _ 1 } ) ( z _ 2 + z _ 0 ) ^ { \\frac { j _ 2 } { T } } I ^ { \\circ } ( Y _ { M ^ 1 } ( u , z _ 0 ) w _ 1 , z _ 2 ) w _ 2 \\end{align*}"}
{"id": "1083.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\mathrm { e } ^ { - \\Re ( \\lambda _ { k } ) t } \\leq \\frac { 2 | \\Omega | } { ( 4 \\pi t ) ^ { d / 2 } } + 2 M _ { 1 } ^ { 2 } , \\end{align*}"}
{"id": "2708.png", "formula": "\\begin{align*} \\psi _ N = \\ ; & \\sum _ { s = 0 } ^ N \\ ; \\sum _ { d = - N + s , \\ , - N + s + 2 , \\ , \\dots } ^ { \\dots , \\ , N - s - 2 , \\ , N - s } u _ 1 ^ { \\otimes ( N - s + d ) / 2 } \\otimes _ \\mathrm { s y m } u _ 2 ^ { \\otimes ( N - s - d ) / 2 } \\otimes _ \\mathrm { s y m } \\Phi _ { s , d } \\end{align*}"}
{"id": "6510.png", "formula": "\\begin{align*} \\int _ { \\Omega } c \\abs { u } ^ 2 d x = \\int _ { \\Omega } c \\abs { y } ^ 2 d x . \\end{align*}"}
{"id": "8725.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\omega } { d t ^ { 2 } } + \\omega - \\frac { \\mu a ^ { 2 } } { \\lambda } = 0 . \\end{align*}"}
{"id": "5972.png", "formula": "\\begin{align*} C _ { t , \\mu , \\varphi } = \\min _ { r _ 0 \\geq 0 } \\left ( e ^ { r _ 0 t } ( r _ 0 t ) ^ { 1 - \\mu } + \\frac 1 { \\pi } ( \\cos \\varphi ) ^ { \\mu - 1 } \\Gamma ( 1 - \\mu , r _ 0 \\cos \\varphi ) \\right ) . \\end{align*}"}
{"id": "6407.png", "formula": "\\begin{align*} C _ o = \\varphi ( a _ { 1 , 1 } \\ , a _ { 1 , 1 } \\ , a _ { 3 , 3 } \\ , a _ { 2 , 2 } ) , \\ C _ { 1 , 2 } = \\varphi ( a _ { 1 , 2 } \\ , a _ { 2 , 1 } \\ , a _ { 1 , 2 } \\ , a _ { 2 , 1 } ) , \\ C _ { 1 , 3 } = \\varphi ( a _ { 3 , 1 } \\ , a _ { 1 , 3 } ) . \\end{align*}"}
{"id": "2772.png", "formula": "\\begin{align*} \\begin{aligned} \\kappa h _ i ^ 2 - 2 ( 1 + \\kappa ) h _ i + 3 { } \\geq { } 0 , \\\\ \\end{aligned} \\end{align*}"}
{"id": "4119.png", "formula": "\\begin{align*} d _ { V } ^ { i } : = \\left ( \\begin{array} { c c c c c c } d ^ { i , 0 } & - S ^ { i , 1 } & 0 & 0 & \\cdots & 0 \\\\ 0 & d ^ { i , 1 } & - S ^ { i , 2 } & 0 & \\cdots & 0 \\\\ & & \\cdots & \\cdots & & \\\\ 0 & 0 & 0 & 0 & \\cdots & d ^ { i , N } \\\\ \\end{array} \\right ) . \\end{align*}"}
{"id": "7386.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to + \\infty } c _ n = 0 . \\end{align*}"}
{"id": "1581.png", "formula": "\\begin{align*} C ^ 2 = f ^ 2 ( x ^ 1 ) ( 1 + f '^ 2 ( x ^ 1 ) ) , \\end{align*}"}
{"id": "5238.png", "formula": "\\begin{align*} \\psi _ I ( g ( W ) ) = \\widetilde \\psi _ I ( g ) ( \\psi _ I ( W ) ) . \\end{align*}"}
{"id": "7865.png", "formula": "\\begin{align*} \\| P _ j ( a \\otimes 1 ) P _ k ^ \\perp \\| = \\max _ { 0 \\leq i \\leq j } \\| Q _ i a Q _ { k - j + i } ^ \\perp \\| \\leq \\| Q _ j a Q _ { k - j } ^ \\perp \\| < \\varepsilon . \\end{align*}"}
{"id": "6910.png", "formula": "\\begin{align*} W _ i = | \\hat \\theta _ i | - | \\hat \\theta ' _ i | \\ , 1 \\leq i \\leq p \\ . \\end{align*}"}
{"id": "1353.png", "formula": "\\begin{align*} \\left \\| \\prod _ { i = 1 } ^ { \\frac { p + 1 } { 2 } } \\chi ( t / T ) u _ i \\right \\| _ { L ^ 2 _ { x , t } } \\lesssim \\| u _ 1 \\| _ { X ^ { 0 , b ' } _ T } \\prod _ { i = 2 } ^ \\frac { p + 1 } { 2 } N _ i ^ { \\frac { p - 5 } { 2 ( p - 1 ) } + \\varepsilon } \\| u _ i \\| _ { X ^ { 0 , b ' } _ T } , \\end{align*}"}
{"id": "7475.png", "formula": "\\begin{align*} \\mathrm { F i x } ( S _ { n } ) = \\mathrm { F i x } \\left ( \\mathcal { T } _ { \\left \\{ C _ { m , r } \\left ( n \\right ) \\left ( j \\right ) \\right \\} _ { j = 1 } ^ { r } } \\right ) = \\mathrm { F i x } \\left ( \\mathcal { V } _ { \\left \\{ C _ { m , r } \\left ( n \\right ) \\left ( j \\right ) \\right \\} _ { j = 1 } ^ { r } } \\right ) \\supset \\cap _ { i = 1 } ^ { m } C _ { i } . \\end{align*}"}
{"id": "3692.png", "formula": "\\begin{align*} \\Sigma _ { T ^ * M } = \\{ ( x , \\xi ) \\in T ^ * M \\ , | \\ , | \\xi | ^ 2 _ g + V ( x ) = E \\} . \\end{align*}"}
{"id": "1163.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { H } ^ { 1 , p } ( \\rho ) } : = \\| \\nabla f \\| _ { L ^ { p } ( \\rho ; \\R ^ d ) } = \\left ( \\int _ { \\R ^ { d } } | \\nabla f | ^ p d \\rho \\right ) ^ { 1 / p } \\end{align*}"}
{"id": "4064.png", "formula": "\\begin{align*} s _ { ( k ) } = h _ { ( k ) } \\sum _ { t = 0 } ^ { [ k / 2 ] } \\frac { ( - 1 ) ^ t } { 2 ^ t t ! } { \\omega / 2 + k - 2 \\choose t } ^ { - 1 } \\sum _ { 1 \\le a _ i < b _ i \\le k } \\epsilon _ { a _ 1 b _ 1 } \\cdots \\epsilon _ { a _ t b _ t } , \\end{align*}"}
{"id": "2352.png", "formula": "\\begin{align*} f = l ( Q ) = l ( Q _ \\rho + h _ \\rho ) = Q _ \\rho p ( x ) + l ( h _ \\rho ) \\mbox { f o r s o m e } p ( x ) \\in K [ x ] . \\end{align*}"}
{"id": "8427.png", "formula": "\\begin{align*} & u _ 1 ( \\textbf { x } , \\textbf { y } ) = 2 x _ 1 + 2 x _ 2 + 3 y _ 2 , \\\\ & u _ 2 ( \\textbf { x } , \\textbf { y } ) = 2 x _ 1 + y _ 1 + y _ 2 . \\end{align*}"}
{"id": "4833.png", "formula": "\\begin{align*} \\rho = \\frac { 1 } { [ K : \\Q ] } \\sum _ { k = 1 } ^ m \\mathrm { I n d } _ { H _ k } ^ G \\Big ( \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } g . \\varphi _ k \\Big ) \\ ; . \\end{align*}"}
{"id": "6502.png", "formula": "\\begin{align*} \\theta _ 4 ( x ) : = \\left \\{ \\begin{array} { c l c } 1 & & x \\in ( c _ 1 + \\delta , c _ 2 - \\delta ) , \\\\ 0 & & x \\in ( 0 , c _ 1 ) \\cup ( c _ 2 , L ) , \\\\ 0 \\leq \\theta _ 4 \\leq 1 & & . \\end{array} \\right . \\end{align*}"}
{"id": "8095.png", "formula": "\\begin{align*} X = p \\partial _ { 1 } + q \\partial _ { 2 } . \\end{align*}"}
{"id": "7330.png", "formula": "\\begin{align*} { \\sf t } _ 9 & = { \\sf t } _ { 9 , 1 } + { \\sf t } _ { 9 , 2 } + { \\sf t } _ { 9 , 3 } \\\\ & \\lesssim | x | ^ \\frac { 2 ( p - 1 ) } { 1 - q } \\Theta _ { \\sf a } { \\bf 1 } _ { | \\xi | > 2 { \\sf l } _ 2 } { \\bf 1 } _ { | x | < { \\sf r } _ 3 } + { \\sf U } _ \\infty ^ { q - 2 } \\Theta _ { \\sf a } ^ 2 { \\bf 1 } _ { | \\xi | > 2 { \\sf l } _ 2 } { \\bf 1 } _ { | x | < { \\sf r } _ 3 } . \\end{align*}"}
{"id": "5125.png", "formula": "\\begin{align*} k \\cdot ( g ; g _ 1 , \\ldots , g _ n ) = ( g k ^ { - 1 } ; k g _ 1 k ^ { - 1 } , \\ldots , k g _ n k ^ { - 1 } ) \\end{align*}"}
{"id": "2084.png", "formula": "\\begin{align*} u & : = \\ell ( \\gamma _ 2 \\gamma _ 1 \\gamma _ 2 \\gamma _ 1 \\gamma _ 1 \\gamma _ 1 \\gamma _ 2 { \\gamma _ 1 } ^ k \\gamma _ 2 ) \\\\ w _ 1 & : = \\ell ( \\gamma _ 2 \\gamma _ 1 \\gamma _ 1 \\gamma _ 2 \\gamma _ 1 \\gamma _ 1 \\gamma _ 2 { \\gamma _ 1 } ^ k \\gamma _ 2 ) \\\\ w _ 2 & : = \\ell ( \\gamma _ 2 \\gamma _ 1 \\gamma _ 1 \\gamma _ 1 \\gamma _ 2 \\gamma _ 1 \\gamma _ 2 { \\gamma _ 1 } ^ k \\gamma _ 2 ) \\\\ \\end{align*}"}
{"id": "7823.png", "formula": "\\begin{align*} d \\Theta _ n ( e _ \\lambda ^ \\vee ) = d ( \\rho \\circ \\tilde { \\iota } ) ( e _ \\lambda ^ \\vee ) . \\end{align*}"}
{"id": "412.png", "formula": "\\begin{align*} F _ { \\mu } ( H _ { \\mu } ( z ) ) = z , z \\in \\Gamma _ { \\alpha , \\beta } ; \\end{align*}"}
{"id": "7248.png", "formula": "\\begin{align*} b _ { \\xi } ( p , p _ 0 ) = \\lim _ n d ( p _ n , p ) - d ( p _ n , p _ 0 ) . \\end{align*}"}
{"id": "1002.png", "formula": "\\begin{align*} P _ N \\begin{pmatrix} 1 & 0 & 0 \\\\ w & K & u \\\\ q & v ^ { { } } & p \\end{pmatrix} : = \\begin{cases} \\begin{pmatrix} 1 & 0 \\\\ w + \\frac { q } { 1 - p } u & K + \\frac { 1 } { 1 - p } u v ^ { } \\\\ \\end{pmatrix} , & p < 1 , \\\\ & \\\\ \\begin{pmatrix} 1 & 0 \\\\ w + u & K \\\\ \\end{pmatrix} , & p = 1 , \\end{cases} \\end{align*}"}
{"id": "4556.png", "formula": "\\begin{align*} \\sigma _ { m , r m ; n , q n } : = & \\left ( 1 + \\frac { 1 } { r } \\right ) \\left ( 1 + \\frac { 1 } { q } \\right ) \\sigma _ { m ( r + 1 ) - 1 , n ( q + 1 ) - 1 } - \\left ( 1 + \\frac { 1 } { r } \\right ) \\frac { 1 } { q } \\sigma _ { m ( r + 1 ) - 1 , n - 1 } \\\\ & - \\frac { 1 } { r } \\left ( 1 + \\frac { 1 } { q } \\right ) \\sigma _ { m - 1 , n ( q + 1 ) - 1 } + \\frac { 1 } { r q } \\sigma _ { m - 1 , n - 1 } , \\end{align*}"}
{"id": "4214.png", "formula": "\\begin{align*} \\left ( \\mathrm { c u r l } \\ \\boldsymbol { u } , \\mathrm { c u r l } _ w \\ \\boldsymbol { v } _ h \\right ) _ { \\mathcal { T } _ h } + \\left ( \\boldsymbol { u } , \\boldsymbol { v } ^ 0 _ h \\right ) _ { \\mathcal { T } _ h } - \\left ( \\boldsymbol { f } , \\boldsymbol { v } ^ 0 _ h \\right ) _ { \\mathcal { T } _ h } - E ( \\boldsymbol { u } , \\boldsymbol { v } _ h ) = 0 , \\end{align*}"}
{"id": "6934.png", "formula": "\\begin{align*} B A ^ { - 1 } = A ^ { - 1 } B \\end{align*}"}
{"id": "5755.png", "formula": "\\begin{align*} \\pi _ { i _ 1 } \\pi _ J = \\pi _ { i _ 1 } \\cdot \\pi _ { J _ 1 } \\cdots \\pi _ { J _ m } = \\pi _ { J ' } \\end{align*}"}
{"id": "8283.png", "formula": "\\begin{align*} B = \\sinh ( k a / 2 ) \\cosh ( k a / 2 ) \\left [ - \\frac { a } { 2 } + \\sinh ^ { 2 } ( k a / 2 ) \\left ( \\frac { 2 ( a / 2 ) } { 3 } - \\frac { 1 } { 2 ( a / 2 ) k ^ { 2 } } + \\frac { \\sinh ( k a / 2 ) \\cosh ( k a / 2 ) } { 2 ( a / 2 ) ^ { 2 } k ^ { 3 } } \\right ) \\right ] ^ { - 1 / 2 } \\end{align*}"}
{"id": "3423.png", "formula": "\\begin{align*} R = \\ ; & ( n _ 5 ^ { - 1 } n _ 3 ^ { - 1 } ) ^ 3 n _ 2 ^ { - 1 } n _ 3 ^ 2 n _ 4 n _ 2 n _ 1 n _ 3 n _ 5 ^ { - 1 } n _ 1 ^ { - 1 } n _ 4 ^ { - 1 } n _ 5 n _ 2 n _ 4 ^ { - 1 } n _ 5 ( n _ 1 ^ { - 1 } n _ 4 ^ { - 1 } ) ^ 2 \\\\ & n _ 5 n _ 1 ^ { - 1 } n _ 2 ^ { - 1 } n _ 5 n _ 3 n _ 5 n _ 1 n _ 4 n _ 5 ^ { - 1 } n _ 2 ^ { - 1 } n _ 4 n _ 3 ^ { - 1 } n _ 1 n _ 4 n _ 2 n _ 1 . \\end{align*}"}
{"id": "2928.png", "formula": "\\begin{align*} g ( \\pi ) : = \\sum _ { j = 1 } ^ m \\max \\{ F ( x ) : x \\in P ( \\chi _ { H ^ j ( \\pi ) } ) \\} , \\end{align*}"}
{"id": "119.png", "formula": "\\begin{align*} C : = C _ t , S : = S _ t , Q : = Q _ t , E : = S - C . \\end{align*}"}
{"id": "7991.png", "formula": "\\begin{align*} ( X ^ \\vee , h = ( h _ 1 , h _ 2 ) : X ^ \\vee \\rightarrow \\mathbb C ^ 2 ) , \\end{align*}"}
{"id": "5237.png", "formula": "\\begin{align*} \\psi ( x ) = x + v ( x ) + \\cdots = x + ( b + 1 ) g x ^ { a + 1 } y ^ b + \\cdots , \\end{align*}"}
{"id": "5966.png", "formula": "\\begin{align*} 2 ^ { - 1 } \\int _ 0 ^ T g _ { \\beta , T } ( t ) & \\| u ( t ) \\| ^ 2 \\ , d t + \\int _ 0 ^ T | u ( t ) | ^ 2 _ 1 d t \\\\ & \\leq \\int _ 0 ^ T \\langle f ( t ) , u ( t ) \\rangle \\ , d t + \\frac { 1 } { \\Gamma ( 1 - \\beta ) } \\int _ 0 ^ T t ^ { - \\beta } \\langle u ^ 0 , u ( t ) \\rangle \\ , d t , \\end{align*}"}
{"id": "1012.png", "formula": "\\begin{align*} H _ \\mu ( \\P ) = - \\sum _ { A \\in \\P } \\mu ( A ) \\log ( \\mu ( A ) ) . \\end{align*}"}
{"id": "5205.png", "formula": "\\begin{align*} o ' _ { 0 , k _ 1 + 1 , k _ 2 + 1 , 0 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I } } = ( - 1 ) ^ { k _ 1 + k _ 2 + 1 } o _ { 0 , k _ 1 + 1 , k _ 2 + 1 , 0 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I } } , \\end{align*}"}
{"id": "8221.png", "formula": "\\begin{align*} \\phi _ { n } ^ { \\pm } ( x ) = \\frac { e ^ { i \\left [ \\left ( 1 \\pm 2 n \\right ) \\pi x / a \\mp \\theta \\right ] } } { \\sqrt { a } } ; E _ { n } ^ { \\pm } = \\frac { \\hslash ^ { 2 } } { 2 m } \\left ( \\frac { 2 n \\pi \\pm \\pi } { a } \\right ) ^ { 2 } + \\frac { \\beta \\hslash ^ { 4 } } { 3 m } \\left ( \\frac { 2 n \\pi \\pm \\pi } { a } \\right ) ^ { 4 } , \\end{align*}"}
{"id": "8086.png", "formula": "\\begin{align*} \\partial _ { \\theta } f = 0 . \\end{align*}"}
{"id": "8018.png", "formula": "\\begin{align*} \\phi _ D ( n ) = \\langle m _ \\sigma , n \\rangle , \\end{align*}"}
{"id": "7768.png", "formula": "\\begin{align*} Z _ { j , , r s } ^ k = z ^ k _ { j , } ( \\alpha _ r , \\alpha _ s ) = \\exp \\Big [ - \\frac { 1 } { 2 } ( \\alpha _ r - \\alpha _ s ) ^ 2 \\Big ] r , s = 1 , \\ldots , p , \\end{align*}"}
{"id": "4553.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 9 \\right ) } \\Vert _ { p } = \\mathcal { O } \\left ( 1 \\right ) ( v ( | z _ { 1 } | ) + ( v ( | z _ { 2 } | ) ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "2192.png", "formula": "\\begin{align*} R _ i = ( \\underbrace { m _ i , \\dots , m _ i } _ { i } ) ( 1 \\le i \\le l ) , \\end{align*}"}
{"id": "2473.png", "formula": "\\begin{align*} \\overline { X } _ { k } = \\bigcup _ { i = 0 } ^ { k } X _ { i } . \\end{align*}"}
{"id": "6684.png", "formula": "\\begin{align*} \\begin{aligned} \\sup \\limits _ { t \\in [ 0 , T _ 0 ] } & \\big ( \\| w _ 1 ( t , \\cdot ) - w _ 2 ( t , \\cdot ) \\| _ { C ^ { 0 } ( [ 0 , 1 ] } + \\| z _ 1 ( t , \\cdot ) - z _ 2 ( t , \\cdot ) \\| _ { C ^ { 0 } ( [ 0 , 1 ] ) } \\big ) + \\| h _ 1 - h _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + \\| g _ 1 - g _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } \\\\ & \\leq K _ 1 ( \\| w _ { 0 1 } - w _ { 0 2 } \\| _ { C ^ 2 ( [ 0 , 1 ] ) } + \\| z _ { 0 1 } - z _ { 0 2 } \\| _ { C ^ 2 ( [ 0 , 1 ] ) } + | h _ { 0 1 } - h _ { 0 2 } | + | g _ { 0 1 } - g _ { 0 2 } | ) , \\end{aligned} \\end{align*}"}
{"id": "3735.png", "formula": "\\begin{align*} I _ { j , l } ^ { 1 ; 2 } ( t , x ) = \\sum _ { p = 1 , 2 , 3 } \\int _ { \\R ^ 3 } \\int _ { | y - x | \\leq t } \\frac { \\varphi _ { m ; - 1 0 M _ t } ( | y - x | ) } { | y - x | } c _ p T _ p f ( t - | y - x | , y , v ) \\varphi _ { j , l } ( v , \\omega ) d y d v \\end{align*}"}
{"id": "8020.png", "formula": "\\begin{align*} \\hat { y } ^ { \\hat { d } } = y ^ { d _ - } \\cdot ( y ^ \\prime ) ^ { d ^ \\prime } . \\end{align*}"}
{"id": "2278.png", "formula": "\\begin{align*} \\Vert ( w , \\Omega ) \\Vert _ { \\mathcal { Q } ( \\sigma _ 0 ) } : = \\Vert ( w , \\Omega ) \\Vert _ { \\mathcal { Q } ( \\sigma _ 0 , 0 ) } + \\Vert ( w , \\Omega ) \\Vert _ { \\mathcal { Q } ( \\sigma _ 0 , 1 ) } , \\end{align*}"}
{"id": "3986.png", "formula": "\\begin{align*} & \\tilde \\varphi \\big ( \\tilde x \\circ a ( x ) \\big ) = ( a _ { n - 1 } + a _ 0 \\tilde x + \\cdots + a _ { n - 2 } \\tilde x ^ { n - 1 } ) \\varphi \\\\ & = \\big ( \\tilde x a ( \\tilde x ) - a _ { n - 1 } ( \\tilde x ^ n - 1 ) \\big ) \\varphi = \\tilde x a ( \\tilde x ) \\varphi = \\tilde x \\tilde \\varphi \\big ( a ( x ) \\big ) . \\end{align*}"}
{"id": "4356.png", "formula": "\\begin{align*} v _ r \\left ( b _ { 0 , 0 } g ^ { s _ 0 + \\ell _ 0 } \\right ) = v _ r ( f ) + v _ r ( h ) . \\end{align*}"}
{"id": "538.png", "formula": "\\begin{align*} C _ { \\alpha } ( n , s ) : = { n \\choose s } \\alpha ^ s ( 1 - \\alpha ) ^ { n - s } . \\end{align*}"}
{"id": "8997.png", "formula": "\\begin{align*} R ( m , j ) & = \\frac { m ! } { ( m - j ) ! } B _ { m - j } \\left ( H _ m ^ { ( 1 ) } , - H _ m ^ { ( 2 ) } , \\ldots , ( - 1 ) ^ { m - j - 1 } ( m - j - 1 ) ! H _ m ^ { ( m - j ) } \\right ) \\\\ & = \\frac { 1 } { j ! } B _ { j } \\left ( H _ m ^ { ( - 1 ) } , \\ldots , ( - 1 ) ^ { j } ( j - 1 ) ! H _ m ^ { ( - j ) } \\right ) , \\end{align*}"}
{"id": "3092.png", "formula": "\\begin{align*} n \\otimes y \\leftharpoonup a \\# b = \\sum _ { ( y ) } n ( y _ { - 1 } \\rightharpoonup a ) \\otimes y _ 0 b \\end{align*}"}
{"id": "1406.png", "formula": "\\begin{align*} X _ 1 = \\begin{pmatrix} 1 & 1 & 1 \\\\ \\epsilon ^ { \\frac { 1 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 1 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 2 } { 3 } } \\\\ \\epsilon ^ { \\frac { 2 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 2 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 4 } { 3 } } \\\\ 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
{"id": "7411.png", "formula": "\\begin{align*} | \\varphi ( 2 \\pi x ) - \\omega ^ n | & \\le K \\left | x - \\frac { n } { D } \\right | , \\\\ | \\varphi ( 2 \\pi x ) | & \\le 1 - L \\left ( x - \\frac { n } { D } \\right ) ^ 2 , \\end{align*}"}
{"id": "6348.png", "formula": "\\begin{align*} u = 0 ~ ~ \\Omega \\end{align*}"}
{"id": "6281.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } S _ n ( x ; q ) = \\frac { - q } { 1 - q } S _ { n - 1 } ( q x ; q ) , \\end{align*}"}
{"id": "8701.png", "formula": "\\begin{align*} \\gamma ( t ) = ( x ^ { 1 } ( t ) , x ^ { 2 } ( t ) ) = ( r ( t ) \\cos t , r ( t ) \\sin t ) . \\end{align*}"}
{"id": "2348.png", "formula": "\\begin{align*} b _ l = \\sum _ { i - j + k = l } a _ { i j k } . \\end{align*}"}
{"id": "2743.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\mathfrak { G } ( x , R ) = 0 . \\end{align*}"}
{"id": "1883.png", "formula": "\\begin{align*} \\mathcal { L } _ { [ n , j ] } : = \\{ \\gamma \\in \\mathcal { D } _ { [ n , 0 ] } : \\mbox { t h e l a s t s t e p o f } \\ , \\ , \\gamma \\ , \\ , \\mbox { i s } \\ , \\ , ( n - 1 , j ) \\rightarrow ( n , 0 ) \\} , 0 \\leq j \\leq p . \\end{align*}"}
{"id": "5788.png", "formula": "\\begin{align*} ( N S ) ^ g = N ^ g S ^ g = N S ^ x \\end{align*}"}
{"id": "6595.png", "formula": "\\begin{align*} \\psi ( \\mathbb 1 _ { F _ { > N } } ) & = \\sum _ { x \\in F _ { > N } } \\xi ( x ) ^ 2 \\\\ & = \\sum _ { x \\in F _ { > N } } ( \\xi - \\xi _ c ) ( x ) ^ 2 + 2 \\sum _ { x \\in F _ { > N } } \\xi _ c ( x ) ( \\xi - \\xi _ c ) ( x ) + \\sum _ { x \\in F _ { > N } } \\xi _ c ( x ) ^ 2 \\\\ & \\leq \\norm { \\xi - \\xi _ c } _ 2 ^ 2 + 2 \\norm { \\xi _ { c | F _ { > N } } } _ 2 \\cdot \\norm { \\xi - \\xi _ c } _ 2 + \\norm { \\xi _ { c | F _ { > N } } } _ 2 ^ 2 \\\\ & \\leq \\frac 1 h ( 2 d \\sqrt { 2 \\eta } + d \\delta ) + 2 \\sqrt { \\delta } + \\delta . \\end{align*}"}
{"id": "1291.png", "formula": "\\begin{align*} \\mu ( T _ 0 , \\lambda ) \\ge \\sum _ { j = 1 } ^ { k } \\mu ( T _ j , \\lambda ) ( \\forall \\lambda \\in \\mathbb { R } ) . \\end{align*}"}
{"id": "6330.png", "formula": "\\begin{align*} J _ { \\nu } ^ { ( 3 ) } ( z ; q ) : = \\frac { ( q ^ { v + 1 } ; q ) _ \\infty } { ( q ; q ) _ \\infty } \\sum _ { n = 0 } ^ { \\infty } \\dfrac { ( - 1 ) ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( q , q ^ { v + 1 } ; q ) _ n } ( z ) ^ { 2 n + \\nu } , z \\in \\mathbb C . \\end{align*}"}
{"id": "5581.png", "formula": "\\begin{align*} I = \\int _ { B ( x , t ) } ( h _ 1 ( y ) ) ^ q d \\sigma ( y ) \\le C _ 1 ( q , \\kappa ) \\ , [ \\varkappa ( B ( x , 2 \\kappa t ) ) ] ^ { \\frac { q } { 1 - q } } , \\end{align*}"}
{"id": "2680.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\left \\langle \\left ( \\mathcal { N } _ { u _ 1 } - \\frac { N } { 2 } \\right ) ^ 2 \\right \\rangle _ { \\psi _ \\mathrm { g s } } = 0 . \\end{align*}"}
{"id": "2567.png", "formula": "\\begin{align*} A _ { j k } ^ { ( x ) } A _ { l m } ^ { ( y ) } = A _ { l m } ^ { ( y ) } A _ { j k } ^ { ( x ) } \\ . \\end{align*}"}
{"id": "2187.png", "formula": "\\begin{align*} [ K ] & = \\bigsqcup _ { i \\in [ m ] } I _ i = \\bigsqcup _ { j \\in [ n ] } J _ j , \\\\ \\Z & = \\bigsqcup _ { i \\in Z } I _ i = \\bigsqcup _ { j \\in Z } J _ j . \\end{align*}"}
{"id": "8251.png", "formula": "\\begin{align*} a _ { + } = \\left ( \\begin{array} { c } a _ { 1 } \\\\ a _ { 2 } \\\\ a _ { 3 } \\\\ a _ { 4 } \\end{array} \\right ) . \\end{align*}"}
{"id": "626.png", "formula": "\\begin{align*} a _ \\infty ( s ) = \\int _ 1 ^ \\infty z ^ { 2 \\alpha - 2 s - 1 } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\Big ( \\frac { \\sin ^ 2 ( \\pi x ) } { \\pi ^ 2 } + \\frac { \\sin ^ 2 ( \\pi y ) } { \\pi ^ 2 } + z ^ 2 \\Big ) ^ { - \\alpha } d x d y d z . \\end{align*}"}
{"id": "1381.png", "formula": "\\begin{align*} \\| \\mathcal { F } ^ { - 1 } ( L _ t [ u ] \\widehat { f } ) \\| _ { X ^ { s , b - 1 } _ T } & \\lesssim T ^ { 1 / 2 - } \\| \\eta ( t / T ) \\mathcal { F } ^ { - 1 } ( L _ t [ u ] \\widehat { f } ) \\| _ { X ^ { s , 0 } } \\\\ & = T ^ { 1 / 2 - } \\left \\| \\langle k \\rangle ^ s \\| e ^ { - i t k ^ 2 } L _ t [ u ] \\eta ( t / T ) \\| _ { L ^ 2 _ t } \\widehat { f } \\right \\| _ { \\ell ^ 2 _ k } \\\\ & \\lesssim T ^ { 1 - } \\| f \\| _ { H ^ s _ x } . \\end{align*}"}
{"id": "7539.png", "formula": "\\begin{align*} \\widetilde { W } _ { n } ( t ) : = \\frac { 1 } { \\sqrt { n } } \\bigg [ \\sum _ { j = 0 } ^ { [ n t ] - 1 } \\tilde { v } \\circ T ^ j + ( n t - [ n t ] ) \\tilde { v } \\circ T ^ { [ n t ] } \\bigg ] , t \\in [ 0 , 1 ] . \\end{align*}"}
{"id": "74.png", "formula": "\\begin{align*} \\ , ( T _ { u ( t ) } W ^ u ( x ) & , T _ { u ( t ) } W ^ s ( y ) ) \\\\ & = \\ , ( T _ { u ( t ) } W ^ s ( y ) , \\mathcal E ( u ( t ) ) ) + \\dim ( T _ { u ( t ) } W ^ u ( x ) , \\mathcal E ( u ( t ) ) ) \\\\ & = \\mathrm { m } ( x ) - \\mathrm { m } ( y ) . \\end{align*}"}
{"id": "4469.png", "formula": "\\begin{align*} x y '' + y ' - x y = 0 , \\end{align*}"}
{"id": "8703.png", "formula": "\\begin{align*} x ^ { 1 } \\dot { x } ^ 2 - x ^ { 2 } \\dot { x } ^ 1 = | x | ^ 2 = r ^ 2 , ~ ~ | \\dot { x } | ^ { 2 } = r ^ { 2 } + \\dot { r } ^ { 2 } . \\end{align*}"}
{"id": "3178.png", "formula": "\\begin{align*} \\mathrm { P } _ { \\Phi + \\mathcal { K } _ { \\gamma } \\left ( \\Psi , f \\right ) } = - \\inf _ { \\rho \\in E _ { 1 } } \\left \\{ f _ { \\Phi } \\left ( \\rho \\right ) + e _ { \\mathcal { K } _ { \\gamma } \\left ( \\Psi , f \\right ) } \\left ( \\rho \\right ) \\right \\} \\ , \\gamma \\in \\left ( 0 , 1 \\right ) \\ , \\end{align*}"}
{"id": "4440.png", "formula": "\\begin{align*} \\L J _ m ^ k f , g \\R = \\L f , ( J _ m ^ k ) ^ * g \\R \\end{align*}"}
{"id": "4236.png", "formula": "\\begin{align*} \\mathbf { P } ( A _ k \\setminus \\cup _ { i = k _ 1 } ^ { k - 1 } A _ i ) & \\geq \\frac { c } { 1 + \\log \\psi ^ { - 1 } ( a ( k ) ) } \\left ( \\mathbf { P } ( C _ k \\cap G _ k \\cap B _ k \\setminus \\cup _ { i = k _ 1 } ^ { k - 1 } A _ i ) - \\sqrt { e ^ { - a ( k ) + a ( k - 1 ) + 1 } } \\right ) \\vee 0 . \\end{align*}"}
{"id": "6185.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial } { \\partial t } \\varphi = \\log \\frac { ( \\widehat { \\omega } _ { t } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\varphi ) ^ { 2 } \\wedge \\eta _ { 0 } } { \\Omega \\wedge \\eta _ { 0 } } , \\varphi ( 0 ) = 0 . \\end{array} \\end{align*}"}
{"id": "4506.png", "formula": "\\begin{align*} \\nabla _ 1 k _ { 2 3 } - \\nabla _ 2 k _ { 1 3 } = & - \\nabla _ 1 \\frac { \\nabla ^ 2 _ { 2 3 } u } { | \\nabla u | } + \\nabla _ 2 \\frac { \\nabla ^ 2 _ { 1 3 } u } { | \\nabla u | } = R _ { 2 1 3 3 } = 0 . \\end{align*}"}
{"id": "8132.png", "formula": "\\begin{align*} \\psi _ n = \\psi _ { n , \\textrm { b o u n d e d } } + \\psi _ { n , \\textrm { i n } } + \\psi _ { n , \\textrm { o u t } } + o ( 1 ) \\end{align*}"}
{"id": "8549.png", "formula": "\\begin{align*} p ^ { 1 / 2 } s _ { \\sqrt p } ( \\sqrt q ) = \\frac 2 { 1 + \\int \\sqrt { p q } \\ d \\mu } \\left ( \\sqrt { \\frac q p } - \\int \\sqrt { \\frac q p } \\ p \\ d \\mu \\ \\right ) \\end{align*}"}
{"id": "2643.png", "formula": "\\begin{align*} x _ 0 \\ , a _ 1 - x _ 1 \\ , a _ 0 = \\begin{vmatrix} x _ 0 & x _ 1 \\\\ a _ 0 & a _ 1 \\end{vmatrix} \\ , \\ y _ 0 \\ , b _ 1 - y _ 1 \\ , b _ 0 = \\begin{vmatrix} y _ 0 & y _ 1 \\\\ b _ 0 & b _ 1 \\end{vmatrix} \\ , \\end{align*}"}
{"id": "8377.png", "formula": "\\begin{align*} F ( 2 ^ h - 1 , 2 , 2 ) = 2 ^ h . \\end{align*}"}
{"id": "6807.png", "formula": "\\begin{align*} u \\in D ( A _ 1 ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , ( I + A _ 1 ) ( u ) = f . \\end{align*}"}
{"id": "976.png", "formula": "\\begin{align*} H ^ { * } ( \\overline { f } ) & = f _ { 0 } H ^ { * } ( \\overline { e } _ { 1 } ) + f _ { 1 } H ^ { * } ( \\overline { e } _ { 2 } ) + \\cdots + f _ { n - 1 } H ^ { * } ( \\overline { e } _ { n } ) \\\\ & = f _ { 0 } I _ { n } + f _ { 1 } H + \\cdots + f _ { n - 1 } H ^ { n - 1 } = f ( H ) . \\end{align*}"}
{"id": "6696.png", "formula": "\\begin{align*} H ^ { \\sigma , \\lambda } _ { \\infty , ( m ) } : = H ^ { \\sigma } _ { [ \\lambda ] } / m _ \\lambda ^ { m + 1 } H ^ { \\sigma } _ { [ \\lambda ] } , \\end{align*}"}
{"id": "5068.png", "formula": "\\begin{align*} & E [ ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s _ 1 ) } ) \\Lambda ^ n _ { s _ 1 } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s _ 2 ) } ) \\Lambda ^ n _ { s _ 2 } ] \\\\ & = E [ ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s _ 1 ) } ) \\Lambda ^ n _ { s _ 1 } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s _ 2 ) } ) E [ \\Lambda ^ n _ { s _ 2 } | \\mathcal { F } _ { \\eta _ n ( s _ 2 ) } ] ] = 0 . \\end{align*}"}
{"id": "2017.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n v l a s o v - l i m i t } \\dd \\overline { X } _ t = b { \\left ( \\overline { X } _ t , f _ t \\right ) } \\dd t + \\sigma { \\left ( \\overline { X } _ t , f _ t \\right ) } \\dd B _ t . \\end{align*}"}
{"id": "6508.png", "formula": "\\begin{align*} u = 0 \\widetilde { \\Omega } . \\end{align*}"}
{"id": "4786.png", "formula": "\\begin{align*} \\{ B G \\} \\L ^ d = \\{ [ V _ K / G ] \\} - \\sum _ f ( - 1 ) ^ { n _ f } \\{ B N _ G ( f ) \\} \\L ^ { d _ f } , \\end{align*}"}
{"id": "733.png", "formula": "\\begin{align*} { Z ^ { ( 1 ) } } _ { 1 } ^ { n / k } & = Z _ 1 , Z _ { k + 1 } , Z _ { 2 k + 1 } \\dotsc , Z _ { ( n / k - 1 ) + 1 } , \\\\ { Z ^ { ( 2 ) } } _ { 1 } ^ { n / k } & = Z _ 2 , Z _ { k + 2 } , Z _ { 2 k + 2 } \\dotsc , Z _ { ( n / k - 1 ) k + 2 } , \\\\ & \\vdots \\\\ { Z ^ { ( k ) } } _ { 1 } ^ { n / k } & = Z _ { k } , Z _ { 2 k } , Z _ { 3 k } \\dotsc , Z _ { ( n / k ) k } . \\end{align*}"}
{"id": "418.png", "formula": "\\begin{align*} \\Omega _ { \\mu } = \\{ x + i y : x \\in \\mathbb { R } , y > f _ { \\mu } ( x ) \\} , \\end{align*}"}
{"id": "4499.png", "formula": "\\begin{align*} p '' ( \\eta _ 1 ) + p '' ( \\eta _ 3 ) = ~ & - 3 \\eta _ 1 ( 1 + \\eta _ 1 ^ 2 ) ^ { - 5 / 2 } + 3 \\eta _ 3 ( 1 + \\eta _ 3 ^ 2 ) ^ { - 5 / 2 } \\\\ = ~ & \\frac { - 3 \\eta _ 1 ( 1 + \\eta _ 3 ^ 2 ) ^ { 5 / 2 } + 3 \\eta _ 3 ( 1 + \\eta _ 1 ^ 2 ) ^ { 5 / 2 } } { ( 1 + \\eta _ 1 ^ 2 ) ^ { 5 / 2 } ( 1 + \\eta _ 3 ^ 2 ) ^ { 5 / 2 } } , \\end{align*}"}
{"id": "3634.png", "formula": "\\begin{align*} H _ f ( m _ \\ast ) + H _ f ( m ^ \\ast ) = H _ f ( m _ 1 ) + H _ f ( m _ 2 ) . \\end{align*}"}
{"id": "1182.png", "formula": "\\begin{align*} \\partial _ { t } \\mu _ { t } + \\nabla \\cdot ( v _ { t } \\mu _ { t } ) = 0 \\ \\ \\R ^ { d } \\times I \\end{align*}"}
{"id": "3121.png", "formula": "\\begin{align*} \\lambda = \\lambda b ( u , u _ { \\mathrm { n c } } ) + \\lambda b ( u , u - u _ { \\mathrm { n c } } ) = \\lambda b ( u , u _ { \\mathrm { n c } } - J u _ { \\mathrm { n c } } ) + a _ { \\mathrm { p w } } ( u , J u _ { \\mathrm { n c } } ) + \\lambda / 2 \\ \\Vert u - u _ { \\mathrm { n c } } \\Vert ^ 2 _ { L ^ 2 ( \\Omega ) } . \\end{align*}"}
{"id": "8977.png", "formula": "\\begin{align*} E ( k _ u u ) = k _ u ^ 2 E ( u ) . \\end{align*}"}
{"id": "2898.png", "formula": "\\begin{align*} F ( x _ 1 , x _ 2 , \\ldots , x _ n ) = \\left ( f ( x _ 1 , \\ldots , x _ k ) , f ( x _ 2 , \\ldots , x _ { k + 1 } ) , \\ldots , f ( x _ { k } , x _ 1 , \\ldots , x _ { k - 1 } ) \\right ) . \\end{align*}"}
{"id": "4093.png", "formula": "\\begin{align*} \\mathbb { O } _ { k , l } \\left ( A _ { 1 } , A _ { 2 } \\right ) \\doteq \\left \\{ \\begin{array} { c c c } A _ { 1 } A _ { 2 } & & k < l , \\\\ - A _ { 2 } A _ { 1 } & & k > l , \\\\ 0 & & k = l . \\end{array} \\right . \\end{align*}"}
{"id": "2644.png", "formula": "\\begin{gather*} x _ 0 \\ , a _ 1 - x _ 1 \\ , a _ 0 = \\begin{vmatrix} x _ 0 & x _ 1 \\\\ a _ 0 & a _ 1 \\end{vmatrix} \\ , \\\\ y _ 0 \\ , ( x _ 0 \\ , h _ { 0 1 } + x _ 1 h _ { 1 1 } ) - y _ 1 \\ , ( x _ 0 \\ , h _ { 0 0 } + x _ 1 h _ { 1 0 } ) = \\begin{vmatrix} y _ 0 & y _ 1 \\\\ x _ 0 \\ , h _ { 0 0 } + x _ 1 h _ { 1 0 } & x _ 0 \\ , h _ { 0 1 } + x _ 1 h _ { 1 1 } \\end{vmatrix} \\ , \\end{gather*}"}
{"id": "3981.png", "formula": "\\begin{align*} \\textstyle \\lim \\limits _ { t \\to \\infty } \\Pr ( \\mbox { \\rm $ A $ h a s f u l l r a n k } ) \\ge \\lim \\limits _ { t \\to \\infty } \\prod _ { i = 0 } ^ m ( 1 - \\frac { q ^ { d _ i k } } { q ^ { d _ i t } } ) = 1 . \\end{align*}"}
{"id": "259.png", "formula": "\\begin{align*} \\begin{array} { r c l } A _ i ^ { ( s ) } & : = & x ^ { ( p ^ s ) t } \\cdot \\phi ^ { ( s ) } _ { i } ( q ^ { ( s ) } ) \\cdot x ^ { ( p ^ s ) } , \\\\ \\ & \\ & \\ \\\\ B ^ { ( s ) } & : = & ( x ^ t q ^ { ( s ) } x ) ^ { ( p ^ s ) } \\end{array} \\end{align*}"}
{"id": "3510.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast } = \\frac { \\xi _ { p } ^ { 2 } \\phi ( \\xi _ { p } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) - \\xi _ { q } ^ { 2 } \\phi ( \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) } { \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) } , \\end{align*}"}
{"id": "4526.png", "formula": "\\begin{align*} \\tilde { g } ^ { - 1 } = \\begin{bmatrix} - | \\nabla u | ^ { - 2 } & 1 & - a & - b \\\\ 1 & 0 & 0 & 0 \\\\ - a & 0 & 1 & 0 \\\\ - b & 0 & 0 & 1 \\end{bmatrix} . \\end{align*}"}
{"id": "7345.png", "formula": "\\begin{align*} 0 & = 2 f ( w , x , y , z ) & \\\\ & = f ( w , x , y , z ) + f ( x , w , y , z ) & \\\\ & = ( w x , y , z ) + x ( w , y , z ) + ( x , y , z ) w + ( x w , y , z ) + w ( x , y , z ) + ( w , y , z ) x & \\\\ & = ( [ w , x ] , y , z ) + [ w , ( x , y , z ) ] + [ x , ( w , y , z ) ] . & \\end{align*}"}
{"id": "837.png", "formula": "\\begin{align*} p ( \\boldsymbol { s } ^ { c } | \\boldsymbol { s } ) = \\prod _ { m } \\mathop { p ( s _ { m } ^ { c } | s _ { m } ) } = \\prod _ { m } ( 1 - s _ { m } ) \\delta ( s _ { m } ^ { c } ) + s _ { m } \\rho _ { c } ^ { s _ { m } ^ { c } } ( 1 - \\rho _ { c } ) ^ { 1 - s _ { m } ^ { c } } \\end{align*}"}
{"id": "925.png", "formula": "\\begin{align*} P _ c ( \\Psi _ t ) = P ( \\Psi _ t ) - \\frac { 1 } { n } \\begin{bmatrix} 0 & 0 \\\\ 0 & J _ n \\end{bmatrix} P ( \\Psi _ t ) , \\end{align*}"}
{"id": "4888.png", "formula": "\\begin{align*} J & \\leq ( \\log N _ { 1 } \\log N _ { 2 } ) ^ { 2 } \\sum \\limits _ { 1 \\leq m _ { 1 } , \\ldots m _ { 4 } \\leq L } \\prod \\limits _ { i = 1 } ^ { 2 } r _ { i } ( 2 ^ { m _ { 1 } } + 2 ^ { m _ { 2 } } - 2 ^ { m _ { 3 } } - 2 ^ { m _ { 4 } } ) \\\\ & \\leq 3 0 5 . 8 8 6 9 N _ { 1 } N _ { 2 } L ^ { 4 } . \\end{align*}"}
{"id": "6465.png", "formula": "\\begin{align*} \\Psi - \\Phi = \\Bar Q \\circ H + H \\circ Q _ { \\mathfrak { g } } \\end{align*}"}
{"id": "5786.png", "formula": "\\begin{align*} E ( G / H ) = E ( N / N _ N ( S ) ) = \\pm 1 \\end{align*}"}
{"id": "6378.png", "formula": "\\begin{align*} c _ { i , i } = ( d - 1 ) / d ^ 2 \\mbox { f o r $ 1 \\leq i \\leq d $ a n d } c _ { i , j } = c _ { j , i } = - 1 / d ^ 2 \\mbox { f o r $ 1 \\leq i < j \\leq d $ . } \\end{align*}"}
{"id": "1584.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial z ^ i _ { \\epsilon } } v ^ i = 2 b ^ 2 f ( x ^ 1 ) \\cos \\left ( x ^ 2 \\right ) \\left [ - \\delta _ { \\epsilon 1 } f ( x ^ 1 ) f ' ( x ^ 1 ) \\cos \\left ( x ^ 2 \\right ) + \\delta _ { \\epsilon 2 } \\left \\lbrace 1 + f '^ 2 \\left ( x ^ 1 \\right ) \\right \\rbrace \\sin \\left ( x ^ 2 \\right ) \\right ] . \\end{align*}"}
{"id": "2161.png", "formula": "\\begin{align*} \\omega ( s ) = \\sum \\nolimits _ { n = 0 } ^ { \\infty } a _ { n } ( s - s _ { 0 } ) ^ { n } , \\end{align*}"}
{"id": "4597.png", "formula": "\\begin{align*} \\widehat { W } _ T \\stackrel { \\mathrm { d } } { = } \\sqrt { 1 - c T } \\widetilde { W } + \\sqrt { c T } U , \\end{align*}"}
{"id": "400.png", "formula": "\\begin{align*} P ( \\phi , f ) = \\limsup _ { t \\to \\infty } \\frac 1 t \\log \\sum _ { \\tau : p ( \\tau ) \\leq t } e ^ { \\ell _ \\tau ( f ) } . \\end{align*}"}
{"id": "3086.png", "formula": "\\begin{align*} \\mathrm { ( \\widetilde { I I } ) } ~ & \\mathrm { M D T C o v } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { Y } ) = \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } \\left [ \\frac { \\boldsymbol { \\Omega } } { \\overline { F } _ { \\mathbf { Z } } ( \\boldsymbol { \\xi _ { q } } ) } - \\frac { \\boldsymbol { \\delta } \\boldsymbol { \\delta } ^ { T } } { \\overline { F } _ { \\mathbf { Z } } ^ { 2 } ( \\boldsymbol { \\xi _ { q } } ) } \\right ] \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
{"id": "6241.png", "formula": "\\begin{align*} & \\int \\dfrac { x } { ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\frac { q ^ { - 1 } - q ^ { - \\nu } [ \\nu ] _ q ^ 2 } { x } + q x \\right ) J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) d _ q x = \\\\ & \\frac { x } { q ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) - x D _ { q ^ { - 1 } } J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) \\right ) , \\end{align*}"}
{"id": "951.png", "formula": "\\begin{align*} \\eta \\in C _ c ^ { \\infty } ( \\mathbb { R } ) , 0 \\leq \\eta \\leq 1 , \\eta ( s ) = \\begin{cases} 1 \\mbox { i f } | s | \\leq 1 , \\\\ 0 \\mbox { i f } | s | \\geq 2 , \\end{cases} \\zeta \\in C _ c ^ { \\infty } ( \\mathbb { R } ^ 2 ) , 0 \\leq \\zeta \\leq 1 , \\zeta ( s ) = \\begin{cases} 1 \\mbox { i f } | x ' | \\leq 1 , \\\\ 0 \\mbox { i f } | x ' | \\geq 2 . \\end{cases} \\end{align*}"}
{"id": "4926.png", "formula": "\\begin{align*} R ( W , U , Y , Z ) = \\bar { R } ( W , U , Y , Z ) + h ( W , Z ) h ( U , Y ) - h ( W , Y ) h ( U , Z ) \\end{align*}"}
{"id": "8497.png", "formula": "\\begin{align*} \\rho ( \\varepsilon ) = \\frac { 2 b d k \\log ( \\frac { 4 b d k } { \\varepsilon } ) } { \\varepsilon ^ 2 } + \\frac { \\log ( \\frac { 3 } { \\delta } ) } { 2 \\varepsilon ^ 2 } = n . \\end{align*}"}
{"id": "4973.png", "formula": "\\begin{align*} \\widetilde { C } ^ n _ t = n ^ { \\alpha + \\frac 1 2 } \\int ^ t _ 0 ( t - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\left ( X ^ n _ { \\eta _ n ( s ) } - X ^ n _ s \\right ) \\ , d W _ s \\end{align*}"}
{"id": "7791.png", "formula": "\\begin{align*} \\phi = ( - 1 ) ^ k \\gamma _ k \\rho _ { u _ k } \\gamma _ { k - 1 } \\rho _ { u _ { k - 1 } } \\cdots \\gamma _ 1 \\rho _ { u _ 1 } \\gamma _ 0 , \\end{align*}"}
{"id": "443.png", "formula": "\\begin{align*} \\Sigma _ { \\mu \\boxtimes \\nu } ( z ) = \\Sigma _ { \\mu } ( z ) \\Sigma _ { \\nu } ( z ) \\end{align*}"}
{"id": "1999.png", "formula": "\\begin{align*} G ( x ) : = G ( x , n , s ) = \\displaystyle \\sum _ { j = 0 } ^ { n } a _ j b _ j \\frac { x ^ j } { j ! } \\end{align*}"}
{"id": "6495.png", "formula": "\\begin{align*} \\int _ 0 ^ { c _ 1 } \\left ( \\abs { v } ^ 2 + a \\abs { u _ x } ^ 2 \\right ) d x = o ( 1 ) \\int _ { c _ 2 } ^ L \\left ( \\abs { z } ^ 2 + \\abs { y _ x } ^ 2 \\right ) d x = o ( 1 ) . \\end{align*}"}
{"id": "1155.png", "formula": "\\begin{align*} ( f + g ) ^ \\dagger = f ^ \\dagger + g ^ \\dagger , ( \\lambda f ) ^ \\dagger = \\lambda ^ \\dagger f ^ \\dagger = \\lambda f ^ \\dagger , f ^ { \\dagger \\dagger } = f , ( f g ) ^ \\dagger = g ^ \\dagger f ^ \\dagger , \\end{align*}"}
{"id": "4895.png", "formula": "\\begin{align*} D _ { c _ 1 , \\dots , c _ r } ( x _ i , y _ { i j } ) = ( c _ i x _ i , c _ i c _ j y _ { i j } ) , i = 1 , \\dots , r , i < j . \\end{align*}"}
{"id": "6618.png", "formula": "\\begin{align*} S _ { k } ( c ; \\xi ) = & { \\sum _ { a \\ , \\hbox { \\tiny m o d } \\ , k } } ^ { \\ ! \\ ! \\ ! * } \\ ; \\sum _ { z '' \\ , \\hbox { \\tiny m o d } \\ , k } e _ { k \\ell } \\big ( a \\ell F _ h ( z '' ) + c \\cdot ( \\xi + \\ell \\bar \\ell ( z '' - \\xi ) ) \\big ) , \\end{align*}"}
{"id": "6319.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + \\dfrac { 1 - q ^ { \\alpha + 1 } ( 1 + x ) } { q ^ { \\alpha + 1 } x ( 1 + x ) ( 1 - q ) } D _ { q ^ { - 1 } } y ( x ) + \\frac { [ n ] _ q } { x ( 1 - q ) ( 1 + x ) } y ( x ) = 0 . \\end{align*}"}
{"id": "7741.png", "formula": "\\begin{align*} \\lambda _ * H ^ { ( 1 ) } ( \\mu _ N ) - \\lambda _ * \\Psi ( - 1 ) & \\cong \\Psi _ * \\binom { N } { 2 } , \\\\ \\Psi ^ * \\Psi _ * \\binom { N } { 2 } & \\cong * . \\end{align*}"}
{"id": "2746.png", "formula": "\\begin{align*} u _ { i } ^ { p - 1 } ( y ) - \\sum ^ { 3 } _ { j = 1 } | \\mathcal { K } _ { j } | \\leq \\Theta = u _ { i } ^ { p - 1 } ( y ) + \\sum ^ { 3 } _ { j = 1 } \\mathcal { K } _ { j } \\leq u _ { i } ^ { p - 1 } ( y ) + \\sum ^ { 3 } _ { j = 2 } | \\mathcal { K } _ { j } | . \\end{align*}"}
{"id": "8923.png", "formula": "\\begin{align*} v = v ( \\cdot , \\xi ) = \\log ( w _ { i j } \\xi _ i \\xi _ j ) + \\tau | D u | ^ 2 + e ^ { \\kappa \\phi } + \\beta \\log ( u _ 0 - u ) \\end{align*}"}
{"id": "5152.png", "formula": "\\begin{align*} ( D / \\mathcal { W } ) _ { i , j } = \\begin{cases} | W _ j | d _ G ( w _ i , w _ j ) & \\\\ ( | W _ i | - 1 ) & \\\\ 2 ( | W _ i | - 1 ) & \\\\ \\end{cases} \\end{align*}"}
{"id": "7799.png", "formula": "\\begin{align*} \\widetilde { Q } _ { ( X , \\eta , \\omega ) } : = \\{ ( X _ t , \\eta _ t ) \\ : \\ t \\in Q _ { ( X , \\eta , \\omega ) } \\} \\end{align*}"}
{"id": "5223.png", "formula": "\\begin{align*} R _ k : = A [ x _ 1 , \\ldots , x _ n ] / ( x _ 1 , \\ldots , x _ n ) ^ { k + 1 } , \\end{align*}"}
{"id": "8401.png", "formula": "\\begin{gather*} \\# \\Gamma _ 0 ( N ) _ Q = \\begin{cases} 6 & \\\\ 4 & \\\\ 2 & \\end{cases} \\end{gather*}"}
{"id": "5563.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\dbinom { n + 1 } { l } e ^ { - l y } ( 1 - e ^ { - y } ) ^ { n - l + 1 } & = 0 \\\\ \\sup _ { n \\ge 1 } \\dbinom { n + 1 } { l } e ^ { - l y } ( 1 - e ^ { - y } ) ^ { n + 1 - l } & \\le 1 , \\end{align*}"}
{"id": "196.png", "formula": "\\begin{align*} \\tilde { \\pi } _ t = \\tilde { F } ( \\tilde { \\pi } _ { t - 1 } , e _ t , y _ t ) \\end{align*}"}
{"id": "3536.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast } & = \\frac { \\Gamma ( t - 2 ) \\left [ \\left ( 1 + \\xi _ { p } ^ { 2 } \\right ) ^ { - ( t - 2 ) } - \\left ( 1 + \\xi _ { q } ^ { 2 } \\right ) ^ { - ( t - 2 ) } \\right ] } { 4 \\Gamma ( t - \\frac { 1 } { 2 } ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "8305.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { a ( n ) } { n } \\left ( \\{ n x \\} ^ { N } + N ! \\sum _ { k = 0 } ^ { N - 1 } \\frac { ( - 1 ) ^ k \\zeta ( - k ) } { ( N - k ) ! k ! } \\right ) = \\sum _ { n = 1 } ^ { \\infty } a _ n \\sin ( 2 \\pi n x ) + \\sum _ { n = 1 } ^ { \\infty } b _ n \\cos ( 2 \\pi n x ) , \\end{align*}"}
{"id": "6247.png", "formula": "\\begin{align*} p ( x ) = - \\dfrac { 1 + q } { q ( 1 - q ) x } , r ( x ) = \\frac { 1 } { q ( 1 - q ) ^ 2 x } . \\end{align*}"}
{"id": "2478.png", "formula": "\\begin{align*} \\inf _ { 2 \\leq l \\leq s } \\inf _ { ( x , y ) \\in \\bar B } \\sum _ { i = 1 } ^ { n - 2 } ( \\varphi _ i ^ 1 ( x , y ) - \\varphi _ i ^ l ( x , y ) ) ^ 2 > \\varepsilon . \\end{align*}"}
{"id": "7359.png", "formula": "\\begin{align*} | a - b | + | b - c | - | a - c | = 2 \\left ( \\max ( a , b , c ) - \\max ( a , c ) + \\min ( a , c ) - \\min ( a , b , c ) \\right ) . \\end{align*}"}
{"id": "8337.png", "formula": "\\begin{align*} \\alpha ( k ) = \\beta ( k ) \\frac { \\mu ( k + 1 ) } { \\mu ( k ) } \\textrm { w h e r e } \\mu ( 0 ) = 1 . \\end{align*}"}
{"id": "1815.png", "formula": "\\begin{align*} x \\begin{pmatrix} P _ { 0 } ( x ) \\\\ P _ { 1 } ( x ) \\\\ P _ { 2 } ( x ) \\\\ \\vdots \\end{pmatrix} = \\begin{pmatrix} b _ { 0 } & 1 & & & \\\\ a _ { 0 } & b _ { 1 } & 1 & \\\\ & a _ { 1 } & b _ { 2 } & 1 & \\\\ & & \\ddots & \\ddots & \\ddots \\end{pmatrix} \\begin{pmatrix} P _ { 0 } ( x ) \\\\ P _ { 1 } ( x ) \\\\ P _ { 2 } ( x ) \\\\ \\vdots \\end{pmatrix} \\end{align*}"}
{"id": "275.png", "formula": "\\begin{align*} B Z ^ { \\nu } _ i = D ^ { \\nu t } _ i + F _ i ^ { \\nu t } , \\end{align*}"}
{"id": "4990.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) \\Theta ^ n _ s ) ^ 2 d s \\\\ & = \\kappa _ 2 ^ 2 \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "5925.png", "formula": "\\begin{align*} { \\partial _ t } x \\left ( t \\right ) = \\xi \\left ( t \\right ) x \\left ( t \\right ) \\ , \\ \\ x ( 0 ) = 1 \\end{align*}"}
{"id": "2436.png", "formula": "\\begin{align*} \\widetilde { N } _ f : = M _ f S _ \\zeta ; \\widetilde { M } _ f : = \\frac { 1 } { [ | W _ f | ] } M _ f S _ \\zeta , f \\in \\Z ^ { m | n } _ { \\zeta - } . \\end{align*}"}
{"id": "5598.png", "formula": "\\begin{align*} \\Delta \\left ( a \\right ) & = 2 \\ln \\left ( \\frac { \\sqrt { 2 } \\Gamma \\left ( \\frac { 2 \\left | a \\right | + 1 } { 4 } + \\frac { 1 } { 2 } \\right ) } { \\sqrt { \\left | a \\right | } \\Gamma \\left ( \\frac { 2 \\left | a \\right | + 1 } { 4 } \\right ) } \\right ) + \\ln { \\left | a \\right | } = 2 \\ln \\left ( \\frac { \\sqrt { 2 } \\Gamma \\left ( \\frac { \\left | a \\right | } { 2 } + \\frac { 3 } { 4 } \\right ) } { \\Gamma \\left ( \\frac { \\left | a \\right | } { 2 } + \\frac { 1 } { 4 } \\right ) } \\right ) , \\end{align*}"}
{"id": "3405.png", "formula": "\\begin{align*} T = \\{ t _ z : z \\in \\C ^ \\times \\} , t _ z = \\begin{pmatrix} \\frac { 1 } { \\bar z } \\\\ & \\frac { \\bar z } { z } \\\\ & & z \\end{pmatrix} . \\end{align*}"}
{"id": "6474.png", "formula": "\\begin{align*} b ( x ) \\geq b _ 0 > 0 \\ \\ \\ \\ \\omega _ b \\subset \\Omega , c ( x ) \\geq c _ 0 \\neq 0 \\ \\ \\ \\ \\omega _ c \\subset \\Omega c ( x ) = 0 \\ \\ \\ \\ \\Omega \\backslash \\omega _ c \\end{align*}"}
{"id": "7439.png", "formula": "\\begin{align*} \\Re \\left ( \\left < { \\mathcal { A } } _ m U , U \\right > _ { \\mathcal { H } } \\right ) = c ( m - 1 ) \\int _ 0 ^ L \\abs { \\omega _ x } ^ 2 d x + \\frac { c m } { 2 } \\int _ 0 ^ { L } \\int _ 0 ^ \\infty \\sigma ' ( s ) \\abs { \\eta _ x } ^ 2 d s d x \\leq 0 , \\end{align*}"}
{"id": "383.png", "formula": "\\begin{align*} h _ t ( x , y ) = p _ t ( x , y ) - \\frac { 1 } { n } . \\end{align*}"}
{"id": "521.png", "formula": "\\begin{align*} E _ { n u l l } ( \\dot x , x , t ) = - \\frac { \\partial \\Phi _ { n u l l } ( x , t ) } { \\partial t } \\ . \\end{align*}"}
{"id": "785.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ { \\infty } \\prod _ { k = 0 } ^ { n - 2 } \\frac { D } { k + 1 } \\phi _ { n } ( r ) = e ^ { - D } - \\phi _ { 1 } ( r ) , \\end{align*}"}
{"id": "3084.png", "formula": "\\begin{align*} \\lim _ { z _ { k } \\rightarrow + \\infty } H ( \\boldsymbol { z } ) \\overline { \\mathcal { G } } _ { n } \\left ( \\frac { 1 } { 2 } \\boldsymbol { z } ^ { T } \\boldsymbol { z } \\right ) = 0 , ~ k \\in \\{ 1 , 2 , \\cdots , n \\} . \\end{align*}"}
{"id": "8525.png", "formula": "\\begin{align*} \\sum _ { i = m + 1 } ^ { n } z ^ { i } 4 ^ { i - m - 1 } \\int _ { - 1 } ^ 1 \\phi _ m ( t ) \\bigg ( \\sum _ { j \\geq i } c _ j ( t ^ 2 - 1 ) ^ { j - i } \\bigg ) . \\end{align*}"}
{"id": "1905.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { C } _ { [ n , r , s ] } } w ( \\gamma ) = a _ { - s } ^ { ( r + s ) } \\sum _ { k = r } ^ { n - s - 1 } \\sum _ { \\ell = k + s } ^ { n - 1 } A _ { [ k - 1 , r - 1 ] } ^ { ( 1 ) } B _ { [ \\ell - k - 1 , s - 1 ] } ^ { ( 1 ) } W _ { [ n - \\ell - 1 , 0 ] } . \\end{align*}"}
{"id": "1134.png", "formula": "\\begin{align*} A = \\frac { 1 } { n } \\sum _ { t = 1 } ^ n ( w _ t s _ t + \\alpha s _ t ^ 2 ) \\end{align*}"}
{"id": "6316.png", "formula": "\\begin{align*} S _ q ( x ) & = \\frac { b + q [ n ] _ q } { q ( 1 - q ) x ^ 2 } + \\frac { a ( a - [ n ] _ q ) } { x ^ 2 } + \\frac { a b ( 1 + q ) - q b [ n - 1 ] _ q } { q x } \\\\ & + \\frac { a ( 1 - x ) } { ( 1 - q ) x ^ 3 } - \\frac { b } { q ( 1 - q ) x } + \\frac { b ^ 2 } { q } . \\end{align*}"}
{"id": "7100.png", "formula": "\\begin{align*} \\begin{cases} f ( k , v , 0 ) = F ( k , v , 0 ) , \\\\ & \\\\ f ( k \\exp ( t v ) , t ) = F ( k , v , t ) , & t > 0 , \\end{cases} \\end{align*}"}
{"id": "8371.png", "formula": "\\begin{align*} \\| u \\| = \\sqrt { M / N } , \\| u ' \\| = \\sqrt { 1 - M / N } . \\end{align*}"}
{"id": "2454.png", "formula": "\\begin{align*} X _ M : = \\left \\{ x \\in X \\mid M _ { x } \\not = 0 \\right \\} . \\end{align*}"}
{"id": "1782.png", "formula": "\\begin{align*} K = \\bigcap _ { n = 1 } ^ \\infty K ( \\alpha - 1 , A _ n , u _ n + a _ n f _ { q ( n ) } ) . \\end{align*}"}
{"id": "5213.png", "formula": "\\begin{align*} D = \\{ \\mu = ( \\mu _ 1 , \\ldots , \\mu _ n ) \\ , | \\ , \\hbox { $ 0 \\le \\mu _ i \\le r _ i - 2 $ f o r $ 1 \\le i \\le n $ } \\} . \\end{align*}"}
{"id": "5681.png", "formula": "\\begin{align*} x _ j ^ { d + 1 } = x _ j ^ 2 \\cdot x _ j ^ { d - 1 } = \\varpi _ j x _ j ^ { d - 1 } x _ { j + 1 } - \\varpi _ { j - 1 } x _ j ^ d ( 1 \\le j \\le n - 1 ) \\end{align*}"}
{"id": "8101.png", "formula": "\\begin{align*} \\operatorname { d i v } ( Y ) \\mu = L _ { Y } ( \\mu ) = d \\circ i _ { Y } ( \\mu ) , \\end{align*}"}
{"id": "6814.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } ( a u '' v ' ) ( x _ 0 - \\delta ) = \\lim _ { \\delta \\rightarrow 0 } ( a u '' v ' ) ( x _ 0 + \\delta ) . \\end{align*}"}
{"id": "2587.png", "formula": "\\begin{align*} \\alpha : = \\psi _ { x , y } \\circ \\iota \\circ \\psi _ { y , x } ^ { - 1 } : \\mathcal E \\to \\mathcal E \\end{align*}"}
{"id": "4306.png", "formula": "\\begin{align*} \\partial ( \\overline { \\mathcal { M } ( \\alpha , \\beta ) / \\R } ) = \\bigcup _ r \\overline { \\mathcal { M } ( \\alpha , r ) / \\R } \\times \\overline { \\mathcal { M } ( r , \\beta ) / \\R } . \\end{align*}"}
{"id": "2319.png", "formula": "\\begin{align*} R _ { \\widetilde { G } } R _ { \\widetilde { D } } ^ k R _ G = R _ { G } R _ D ^ k R _ { \\widetilde { G } } R _ { \\widetilde { D } } { R _ { \\widetilde { G } } ^ k } R _ D = R _ { D } R _ G ^ k { R _ { \\widetilde { D } } } . \\end{align*}"}
{"id": "2221.png", "formula": "\\begin{align*} \\overline { \\omega } ^ { n + 1 } = r ^ { 2 n + 1 } ( \\Pi ^ { \\ast } \\omega _ { h } ) ^ { n } \\wedge d r \\wedge \\overline { \\eta } , \\end{align*}"}
{"id": "942.png", "formula": "\\begin{align*} \\begin{aligned} P _ c & ( \\Psi _ t ) P _ c ^ T ( \\Psi _ t ) ( x ) \\\\ & = \\begin{bmatrix} I _ n + O ( t ) & \\Upsilon _ c ^ T + O ( t ) \\\\ \\Upsilon _ c + O ( t ) & \\frac { 1 } { 2 t } \\cdot \\begin{pmatrix} \\begin{bmatrix} I _ { \\frac { n ( n - 1 ) } { 2 } } & 0 \\\\ 0 & 3 \\cdot \\Xi ( \\frac { 1 } { 3 } ) - \\frac { n + 2 } { n } \\cdot J _ n \\end{bmatrix} + O ( t ) \\end{pmatrix} + \\Gamma _ c ( x ) \\end{bmatrix} , \\end{aligned} \\end{align*}"}
{"id": "832.png", "formula": "\\begin{align*} \\mathbf { H } ^ { r } & = \\mathbf { A } ( \\boldsymbol { x } ^ { r } ) \\mathbf { A } ^ { H } = \\sum _ { m = 1 } ^ { \\widetilde { M } } x _ { m } ^ { r } \\boldsymbol { a } \\left ( \\overline { \\theta } _ { m } \\right ) \\boldsymbol { a } ^ { H } \\left ( \\overline { \\theta } _ { m } \\right ) , \\\\ \\mathbf { h } ^ { c } & = \\mathbf { A } \\boldsymbol { x } ^ { c } = \\sum _ { m = 1 } ^ { \\widetilde { M } } x _ { m } ^ { c } \\boldsymbol { a } \\left ( \\overline { \\theta } _ { m } \\right ) , \\end{align*}"}
{"id": "8276.png", "formula": "\\begin{align*} k ' = \\frac { 1 } { 2 \\hslash \\sqrt { \\beta / 3 } } \\sqrt { 1 + \\sqrt { 1 - \\frac { 1 6 } { 3 } m \\beta | E | } } . \\end{align*}"}
{"id": "8546.png", "formula": "\\begin{align*} ( 2 p + 1 ) \\frac { 3 ( 2 p + 1 ) } { 3 ( 2 p + 1 ) ^ 2 - 1 } - 1 = \\frac { 1 } { 1 2 ( p ^ 2 + p ) + 2 } . \\end{align*}"}
{"id": "5298.png", "formula": "\\begin{align*} \\mathrm { P a r t } _ h ( \\{ 1 , 1 , 2 , 2 \\} ) = \\bigg \\{ \\{ \\{ 1 \\} , \\{ 1 , 2 , 2 \\} \\} , \\{ \\{ 2 \\} , \\{ 1 , 1 , 2 \\} \\} , \\{ \\{ 1 , 1 \\} , \\{ 2 , 2 \\} \\} , \\{ \\{ 1 , 2 \\} , \\{ 1 , 2 \\} \\} \\bigg \\} . \\end{align*}"}
{"id": "5218.png", "formula": "\\begin{align*} \\sum _ j k _ 1 ( j ) = r ( J ) + n _ 1 r , \\sum _ j k _ 2 ( j ) = s ( J ) + n _ 2 s . \\end{align*}"}
{"id": "1677.png", "formula": "\\begin{align*} \\| y _ 1 + y _ 2 \\| = \\| ( \\| y _ 1 \\| , \\| y _ 2 \\| ) \\| _ Z \\end{align*}"}
{"id": "6906.png", "formula": "\\begin{align*} p ( t , g ) = & \\mathcal { F } ^ { - 1 } _ { \\xi } [ { \\hat { p } ( 0 , e ^ { - t } ( \\xi - \\mu ) + \\mu ) e ^ { - H _ t ( \\xi - \\mu ) } } ] \\\\ & = e ^ { i \\mu g } \\mathcal { F } ^ { - 1 } _ { \\xi } [ { \\hat { p } ( 0 , e ^ { - t } \\xi + \\mu ) e ^ { - H _ t ( \\xi ) } } ] \\\\ & = e ^ { i \\mu g } \\mathcal { F } ^ { - 1 } _ { \\xi } [ \\hat { p } ( 0 , e ^ { - t } \\xi + \\mu ) ] * ( \\frac { 1 } { \\sqrt { 2 \\pi } } \\mathcal { F } ^ { - 1 } _ { \\xi } [ e ^ { - H _ t ( \\xi ) } ] ) . \\end{align*}"}
{"id": "6709.png", "formula": "\\begin{align*} \\varphi ( \\lambda ) = h ( \\lambda ) \\cdot q _ { n , m } ( \\lambda ) , \\ ; \\ ; \\ ; \\lambda \\in \\mathfrak { a } ^ * _ \\mathbb { C } , \\end{align*}"}
{"id": "4733.png", "formula": "\\begin{align*} & \\Delta ( E _ i ) = E _ i \\otimes 1 + K _ i \\otimes E _ i , \\\\ & \\Delta ( F _ i ) = F _ i \\otimes K _ i ^ { - 1 } + 1 \\otimes F _ i , \\\\ & \\Delta ( K _ i ^ { \\pm 1 } ) = K _ i ^ { \\pm 1 } \\otimes K _ i ^ { \\pm 1 } . \\end{align*}"}
{"id": "3578.png", "formula": "\\begin{align*} n _ { \\pi _ 1 } & = \\frac { 1 } { 2 } \\left ( [ 5 ] _ q ! - [ 2 ] _ q ! [ 3 ] _ q ! ( 2 + 2 ( - 1 ) ^ i + 2 ( - 1 ) ^ j + 4 ( - 1 ) ^ { i + j } ) \\right ) \\\\ & = \\frac { 1 } { 2 } ( q + 1 ) ^ 2 ( q ^ 2 + q + 1 ) \\left ( ( q ^ 2 + 1 ) ( q ^ 4 + q ^ 3 + q ^ 2 + q + 1 ) - ( 2 + 2 ( - 1 ) ^ i + 2 ( - 1 ) ^ j + 4 ( - 1 ) ^ { i + j } ) \\right ) \\\\ & = \\frac { 1 } { 2 } ( q + 1 ) ^ 2 ( q ^ 2 + q + 1 ) \\left ( \\{ q ^ 6 + q ^ 5 + 2 ( q ^ 4 + q ^ 3 + q ^ 2 ) \\} + \\{ q - 1 \\} - \\{ 2 ( - 1 ) ^ i + 2 ( - 1 ) ^ j ) + 4 ( - 1 ) ^ { i + j } \\} \\right ) . \\end{align*}"}
{"id": "3275.png", "formula": "\\begin{align*} a _ j ( x ) = ( A _ 2 - A _ 1 ) ( x ) \\cdot e _ j = A ( x ) \\cdot e _ j , j = 1 , 2 , 3 , x \\in \\R ^ 3 , \\end{align*}"}
{"id": "6293.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } p _ n ( x ; - 1 ; q ) = \\frac { - 1 } { 1 - q } \\ , _ 3 \\phi _ 2 ( q ^ { - n } , q ^ { n + 1 } , x ; q ; - q ; q , q ) , \\end{align*}"}
{"id": "5075.png", "formula": "\\begin{align*} M ^ { n , 1 } _ s = \\int ^ s _ { \\eta _ n ( s ) } ( s - \\eta _ n ( u ) ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\end{align*}"}
{"id": "62.png", "formula": "\\begin{align*} \\sum _ { h = 1 } ^ { + \\infty } \\frac { 1 } { h ^ \\alpha ( k + h ) ^ \\beta } & \\leq \\sum _ { h = 1 } ^ { + \\infty } \\frac { 1 } { h ^ { \\alpha + \\beta - \\gamma } ( k + h ) ^ \\gamma } \\\\ & = \\frac { 1 } { k ^ \\gamma } \\sum _ { h = 1 } ^ { + \\infty } \\frac { 1 } { h ^ { \\alpha + \\beta - \\gamma } ( 1 + h / k ) ^ \\gamma } \\\\ & \\leq \\frac { 1 } { k ^ \\gamma } \\sum _ { h = 1 } ^ { + \\infty } \\frac { 1 } { h ^ { \\alpha + \\beta - \\gamma } } \\end{align*}"}
{"id": "5180.png", "formula": "\\begin{align*} E : = s x \\partial _ x + r y \\partial _ y - \\sum _ { \\stackrel { 0 \\le \\alpha \\le r - 2 , \\ 0 \\le \\beta \\le s - 2 , } { d \\ge 0 } } ( s \\beta + r \\alpha + r s ( d - 1 ) ) t _ { \\alpha , \\beta , d } \\partial _ { t _ { \\alpha , \\beta , d } } . \\end{align*}"}
{"id": "6592.png", "formula": "\\begin{align*} A _ 3 & = \\{ x \\in X _ n \\mid ( x , y ) \\in B y \\sim x \\} , \\\\ A _ 1 & = \\{ x \\in X _ n \\setminus A _ 3 \\mid v ( x , y ) \\neq 0 y \\sim x \\} , \\\\ A _ 2 & = X _ n \\setminus ( A _ 1 \\cup A _ 3 ) . \\end{align*}"}
{"id": "8341.png", "formula": "\\begin{align*} \\mathcal { F } f ( p ; \\nu ) = \\int d u \\ , f ( u ) \\ , ( p \\cdot u ) ^ { i \\nu - 1 } , \\end{align*}"}
{"id": "38.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { L , n } = ( \\alpha v _ n \\delta _ n ( \\beta ) , v _ n ) _ { L , n } . \\end{align*}"}
{"id": "5880.png", "formula": "\\begin{align*} e _ n ( { \\mathcal K } ) _ X \\geq { C } \\frac { 1 } { [ \\log _ 2 n ] ^ \\alpha } , n = 1 , 2 , \\ldots , \\hbox { t h e n } d _ n ^ { \\gamma } ( { \\mathcal K } ) _ X \\geq \\frac { C ' } { [ \\log _ 2 n ] ^ { \\alpha } } , \\end{align*}"}
{"id": "5747.png", "formula": "\\begin{align*} ( b - i ) \\Big ( \\pi _ { [ a - 2 , b - 1 ] } + \\pi _ { [ a - 1 , b - 1 ] } y _ { b } \\Big ) = ( b - i ) \\pi _ { [ a - 1 , b ] } \\end{align*}"}
{"id": "6870.png", "formula": "\\begin{align*} H '' ( x ) = \\frac { e ^ { - x ^ 2 } ( 2 \\sqrt { 2 } e ^ { x ^ 2 / 2 } ( x ^ 2 - 3 ) \\int _ 0 ^ { x / \\sqrt { 2 } } e ^ { - t ^ 2 } d t + 2 x ) } { 4 \\pi } . \\end{align*}"}
{"id": "5034.png", "formula": "\\begin{align*} E [ | Q ^ { n , 4 } _ \\tau | ^ 2 ] \\le C \\sum _ { j = 0 } ^ { \\lfloor n \\tau \\rfloor } \\left ( \\int _ { \\frac j n } ^ { \\frac { j + 1 } n \\wedge \\tau } \\gamma _ s d s \\right ) ^ 2 \\le C \\sup _ { 0 \\le j \\le \\lfloor n \\tau \\rfloor } \\left ( \\int _ { \\frac j n } ^ { \\frac { j + 1 } n \\wedge \\tau } \\gamma _ s d s \\right ) \\int _ 0 ^ { \\tau } \\gamma _ s d s , \\end{align*}"}
{"id": "3504.png", "formula": "\\begin{align*} c _ { 1 } = \\frac { 1 } { ( 2 \\pi ) ^ { 1 / 2 } \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) } , \\end{align*}"}
{"id": "6534.png", "formula": "\\begin{align*} \\P \\{ \\ell ^ { ( A ) } _ i \\leq m - i \\} = \\sum \\limits _ { k = 0 } ^ \\infty \\mu ( k ) \\left ( \\P \\left \\{ \\forall t > 0 : t > \\sum \\limits ^ { i + S _ { t } } _ { z = i + 1 } \\frac { 1 } { A ( z ) } S _ { t } < m - i \\right \\} \\right ) ^ k , \\end{align*}"}
{"id": "4060.png", "formula": "\\begin{align*} & q _ { - i } = q _ i ^ { - 1 } , & & q _ { - i , j } = q _ { i j } ^ { - 1 } q _ j ^ 2 . \\end{align*}"}
{"id": "5558.png", "formula": "\\begin{align*} \\int _ { Y _ 1 \\# Y _ 2 } | W _ + | ^ 2 d \\mu _ { g } = \\int _ { Y _ 1 } | W _ + | ^ 2 d \\mu _ { g _ 1 } + \\int _ { Y _ 2 } | W _ + | ^ 2 d \\mu _ { g _ 2 } . \\end{align*}"}
{"id": "7316.png", "formula": "\\begin{align*} | w _ 2 | & \\lesssim \\bar w _ 2 = T ^ { \\frac { 1 } { 2 } { \\sf c } _ 1 } ( t - t ) ^ { \\frac { \\gamma } { 2 } + J + \\frac { 1 } { 2 } { \\sf c } _ 1 } e _ { J + 1 } ( z ) \\\\ & \\lesssim T ^ { \\frac { 1 } { 2 } { \\sf c } _ 1 } ( t - t ) ^ { \\frac { \\gamma } { 2 } + J + \\frac { 1 } { 2 } { \\sf c } _ 1 } | z | ^ \\gamma | z | < { \\sf r } , \\ t \\in ( 0 , T ) . \\end{align*}"}
{"id": "6099.png", "formula": "\\begin{align*} \\theta _ { n + 2 \\ , \\ , n } = b \\ , \\ , \\omega _ { n + 2 \\ , \\ , n + 1 } \\theta _ { n + 2 \\ , \\ , n + 1 } = a \\ , \\ , \\omega _ { n + 2 \\ , \\ , n + 1 } \\end{align*}"}
{"id": "6340.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { N } ^ s _ p u ( x ) = \\int _ { \\Omega } \\dfrac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { N + p s } } d y , ~ ~ \\forall x \\in \\R ^ N \\setminus \\Omega , \\end{aligned} \\end{align*}"}
{"id": "922.png", "formula": "\\begin{align*} \\Upsilon _ c : = \\begin{bmatrix} \\cdots & \\cdots & \\cdots & \\cdots \\\\ \\Gamma _ { i j } ^ 1 & \\Gamma _ { i j } ^ 2 & \\cdots & \\Gamma _ { i j } ^ n \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ \\Gamma _ { i i } ^ 1 & \\Gamma _ { i i } ^ 2 & \\cdots & \\Gamma _ { i i } ^ n \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\end{bmatrix} , i \\neq j , \\end{align*}"}
{"id": "435.png", "formula": "\\begin{align*} K _ { + } = \\{ s + i t \\in K : t \\ge f _ { \\nu } ( s _ { 0 } ) + \\varepsilon \\} \\end{align*}"}
{"id": "2185.png", "formula": "\\begin{align*} A ^ { \\tt s t } = ( A ^ { \\circ } ) ^ { \\circ ' } . \\end{align*}"}
{"id": "7714.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } m ^ t x ^ m = \\frac { x } { ( 1 - x ) ^ { t + 1 } } \\psi _ { t - 1 } ( x ) ( t \\in \\N , | x | < 1 ) , \\end{align*}"}
{"id": "4919.png", "formula": "\\begin{align*} Y _ { ; i j } ^ k - Y _ { ; j i } ^ k = - { R _ { i j m } } ^ k Y ^ m \\end{align*}"}
{"id": "3676.png", "formula": "\\begin{align*} g _ { 0 } = d r ^ 2 + \\sin ^ 2 r \\cdot h _ { \\mathbb { S } ^ { n - 2 } } + \\cos ^ 2 r \\cdot d s ^ 2 \\end{align*}"}
{"id": "6063.png", "formula": "\\begin{align*} b = - 3 = - 1 \\end{align*}"}
{"id": "1482.png", "formula": "\\begin{align*} \\log \\ , D _ { 2 , n } = \\sum _ { p : } \\max _ { 0 \\le k \\le n } \\log \\ , \\left | \\dfrac { k ! } { ( \\beta ) _ k } \\right | _ p & \\le \\log ( | c | + ( n - 1 ) d ) \\pi _ { | c | , d } ( | c | + ( n - 1 ) d ) \\enspace , \\end{align*}"}
{"id": "2733.png", "formula": "\\begin{align*} \\sum _ { u = i } ^ s \\epsilon ^ { \\bullet } ( i ) \\epsilon ^ { \\bullet } ( s ( \\alpha _ i ) ) ^ { - 1 } q _ { \\alpha _ u } \\alpha _ u \\alpha ^ * _ u - \\sum _ { v = i } ^ t \\epsilon ^ { \\bullet } ( t ( \\beta _ i ) ) \\epsilon ^ { \\bullet } ( i ) ^ { - 1 } \\beta _ v \\beta _ v ^ * \\end{align*}"}
{"id": "261.png", "formula": "\\begin{align*} ( \\Lambda _ i ^ { ( s ) } - 1 ) _ { k j } - ( \\Lambda _ j ^ { ( s ) } - 1 ) _ { k i } = \\pi \\cdot L _ { i j } ^ { k ( s ) } ( \\Lambda ^ { ( s ) } ) . \\end{align*}"}
{"id": "5231.png", "formula": "\\begin{align*} \\begin{aligned} x & \\longmapsto x ( 1 + ( b - a ) g x ^ a y ^ b ) ^ { ( b + 1 ) / ( b - a ) } \\\\ y & \\longmapsto y ( 1 + ( b - a ) g x ^ a y ^ b ) ^ { ( a + 1 ) / ( a - b ) } . \\end{aligned} \\end{align*}"}
{"id": "8993.png", "formula": "\\begin{align*} \\Delta _ n = \\frac { n - 2 } { n - 1 } \\Delta _ { n - 2 } - \\frac { 1 } { n - 1 } \\lambda _ { n - 2 } , n \\geq 3 , \\end{align*}"}
{"id": "5553.png", "formula": "\\begin{align*} \\lim \\limits _ { \\ell \\rightarrow \\infty } \\dfrac { \\mathbb { E } [ \\ell , k ] } { \\ell } = \\dfrac { 1 } { 2 } \\end{align*}"}
{"id": "1436.png", "formula": "\\begin{align*} \\sum a _ m x ^ m . v + a _ 0 v + \\sum b _ n y ^ n . v = 0 . \\end{align*}"}
{"id": "668.png", "formula": "\\begin{align*} \\left . \\int u _ { 1 , t } v _ { 2 , t } \\ , d g _ { 1 , t } \\ , \\right | ^ { t = \\bar s } _ { t = s } & = - \\int _ { s } ^ { \\bar s } u _ { 1 , t } \\left ( \\Box ^ * _ { g _ { 1 , t } } v _ { 2 , t } \\right ) \\ , d g _ { 1 , t } , \\end{align*}"}
{"id": "6627.png", "formula": "\\begin{align*} \\sigma _ p ( N , \\xi , h ) = h ^ { - 2 } \\tau _ p ( B _ h ) . \\end{align*}"}
{"id": "1208.png", "formula": "\\begin{align*} \\sqrt { n } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\hat \\theta _ n } ) = \\mathbb { M } _ n ( \\hat t _ n ) + o _ { \\mathbb { P } } ( 1 ) . \\end{align*}"}
{"id": "1097.png", "formula": "\\begin{align*} D _ V X = D _ X V = \\frac { X f } { f } V + P _ X V . \\end{align*}"}
{"id": "1056.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\| \\tilde { B } _ 0 S _ n ( s ) \\| _ { r _ n } \\ , \\mathrm { d } s < \\infty , n = 0 , 1 , \\ldots , \\ell , \\end{align*}"}
{"id": "5017.png", "formula": "\\begin{align*} \\left ( \\int ^ { s } _ 0 \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) ^ 2 - \\int ^ { s } _ 0 \\psi _ { n , 1 } ^ 2 ( u , s ) d u = I _ 2 \\left ( \\psi _ { n , 1 } ^ { \\otimes 2 } ( \\cdot , s ) \\mathbf { 1 } _ { [ 0 , s ] ^ 2 } \\right ) , \\end{align*}"}
{"id": "8216.png", "formula": "\\begin{align*} \\phi ( x ) = A e ^ { i k x } + B e ^ { - i k x } + C e ^ { k ' x } + D e ^ { - k ' x } , \\end{align*}"}
{"id": "996.png", "formula": "\\begin{align*} H ^ { * } ( \\overline { \\alpha } \\otimes \\overline { \\beta } ) = H ^ { * } ( \\overline { \\alpha } ) H ^ { * } ( \\overline { \\beta } ) = \\left ( \\begin{matrix} p q & 0 \\\\ 0 & p q \\end{matrix} \\right ) , \\ \\ L _ { \\alpha , \\beta } = L \\left ( H ^ { * } ( \\overline { \\alpha } \\otimes \\overline { \\beta } ) \\right ) \\end{align*}"}
{"id": "1015.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu ( A \\cap T ^ { - n } B ) = \\mu ( A ) \\mu ( B ) . \\end{align*}"}
{"id": "4557.png", "formula": "\\begin{align*} \\dim ( \\mathcal { F } _ 1 ) = | E ( G ) | - \\# \\{ v \\in V ( G ) : v \\} - 1 \\ , . \\end{align*}"}
{"id": "1331.png", "formula": "\\begin{align*} M _ { T ( x ) } ( v ) \\geq \\frac { 1 } { \\sqrt { 2 \\pi T _ 2 } } e ^ { - \\frac { v ^ 2 } { 2 T _ 1 } } \\geq \\sqrt { \\frac { T _ 1 } { T _ 2 } } M _ { T _ 1 } ( v ) : = \\gamma G ( v ) , \\end{align*}"}
{"id": "2806.png", "formula": "\\begin{align*} \\begin{aligned} c ' ( h ) = - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\left [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\right ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 \\end{aligned} \\end{align*}"}
{"id": "1878.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { D } } _ { [ n , j , k ] } = \\{ \\gamma \\in \\mathcal { D } _ { [ n , j ] } : \\kappa _ { i } ( \\gamma ) = k \\} , i \\leq k \\leq n - j + i . \\end{align*}"}
{"id": "3.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & - 1 & - 1 & - 1 \\\\ 1 & 0 & 1 & - 1 \\\\ 1 & 1 & 1 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "1722.png", "formula": "\\begin{align*} _ r ( z \\ , | \\ , \\omega _ 1 , \\cdots , \\omega _ r ) = \\Gamma _ r ( z \\ , | \\ , \\omega _ 1 , \\cdots , \\omega _ r ) ^ { - 1 } \\ \\Gamma _ r \\big ( \\sum _ { i = 1 } ^ { r } \\omega _ i - z \\ , | \\ , \\omega _ 1 , \\cdots , \\omega _ r \\big ) ^ { ( - 1 ) ^ r } \\ . \\end{align*}"}
{"id": "2398.png", "formula": "\\begin{align*} \\| ( - A _ n ) ^ { \\frac { 1 } { 2 } } a \\| ^ 2 _ { l ^ 2 _ n } = \\langle - A _ n a , a \\rangle _ { l ^ 2 _ n } = \\frac { n } { \\pi } \\sum _ { j = 1 } ^ { n } | a _ { j } - a _ { j - 1 } | ^ 2 . \\end{align*}"}
{"id": "6526.png", "formula": "\\begin{align*} \\left [ \\frac { \\AA ( n + 1 ) } { \\AA ( n ) } \\right ] ^ n = \\left [ 1 + \\frac { A ^ { - 1 } ( n + 1 ) } { \\sum \\limits _ { j = 1 } ^ n A ^ { - 1 } ( j ) } \\right ] ^ n \\leq \\left [ 1 + \\frac { \\varepsilon } { n } \\right ] ^ n \\leq e ^ \\varepsilon . \\end{align*}"}
{"id": "5227.png", "formula": "\\begin{align*} { \\mathcal L } _ v ( d x _ 1 \\wedge \\cdots \\wedge d x _ n ) = d ( \\iota ( v ) ( d x _ 1 \\wedge \\cdots \\wedge d x _ n ) ) . \\end{align*}"}
{"id": "5142.png", "formula": "\\begin{align*} \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( 2 a c ) = \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( 2 a c - 2 ) \\end{align*}"}
{"id": "2127.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow _ { \\sphericalangle } s } ( z - s ) G _ { \\mu } ( z ) = \\mu ( \\{ s \\} ) , s \\in \\mathbb { R } . \\end{align*}"}
{"id": "3057.png", "formula": "\\begin{align*} f _ { \\mathbf { M ^ { \\ast \\ast } } _ { - \\xi _ { \\boldsymbol { s } k } } } ( \\boldsymbol { v } ) = c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast \\ast } \\overline { \\mathcal { G } } _ { n } \\left \\{ \\frac { 1 } { 2 } \\boldsymbol { v } ^ { T } \\boldsymbol { v } + \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } \\right \\} , ~ \\boldsymbol { v } \\in \\mathbb { R } ^ { n - 1 } , \\end{align*}"}
{"id": "6169.png", "formula": "\\begin{align*} \\psi ^ { \\ast } r _ { \\mathbf { S } _ { p } ^ { 1 } } ^ { 2 } = | s | _ { h } ^ { 2 } . \\end{align*}"}
{"id": "4874.png", "formula": "\\begin{align*} \\Xi ( N , k ) = \\{ ( 1 - \\eta ) N \\leq n \\leq N : n = N - 2 ^ { v _ { 1 } } - \\cdots - 2 ^ { v _ { k } } \\} . \\end{align*}"}
{"id": "8911.png", "formula": "\\begin{align*} - v '' + v = v ^ { p - 1 } v \\ge 0 ; \\end{align*}"}
{"id": "4404.png", "formula": "\\begin{align*} M _ p ( g , r ) = \\left ( \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } | g ( r e ^ { i \\varphi } ) | ^ p d \\varphi \\right ) ^ { 1 / p } \\end{align*}"}
{"id": "7087.png", "formula": "\\begin{align*} \\delta ( t _ { k + 1 } ) & = \\delta ( t _ { k } ) + \\frac { ( 2 - \\phi ) \\lambda - \\beta ^ { \\top } u ( t _ { k } ) } { 2 } \\mathsf { T } _ { k } , \\end{align*}"}
{"id": "5821.png", "formula": "\\begin{align*} \\delta ( \\textbf { a } ) = \\lim _ { x \\to \\infty } \\frac { \\# \\{ p \\leq x : p _ n \\equiv a _ n \\bmod q \\} } { x } = \\frac { 1 } { \\varphi ( q ) ^ k } . \\end{align*}"}
{"id": "7609.png", "formula": "\\begin{align*} J ( v , u ) = \\Delta v ( 1 ) ^ 2 + \\int _ 0 ^ 1 \\Delta v ( s ) ^ 2 d s + \\int _ 0 ^ 1 \\Delta u ( s ) ^ 2 d s . \\end{align*}"}
{"id": "3910.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int _ 0 ^ t b ( X _ s ) d s + \\int _ 0 ^ t a ( X _ s ) d W _ s , t \\in [ 0 , T ] , \\end{align*}"}
{"id": "1338.png", "formula": "\\begin{align*} \\Phi ( x ) = \\frac { \\langle x \\rangle ^ \\gamma } { \\gamma } , \\mbox { a n d } \\Psi ( v ) = \\frac { \\langle v \\rangle ^ \\beta } { \\beta } , \\ , \\beta \\geq 2 . \\end{align*}"}
{"id": "1164.png", "formula": "\\begin{align*} \\| \\ell \\| _ { \\dot { H } ^ { - 1 , p } ( \\rho ) } = \\sup \\left \\{ \\ell ( f ) : f \\in \\dot { C } _ { 0 } ^ { \\infty } , \\| f \\| _ { \\dot { H } ^ { 1 , q } ( \\rho ) } \\le 1 \\right \\} . \\end{align*}"}
{"id": "1183.png", "formula": "\\begin{align*} \\int _ { I } \\int _ { \\R ^ { d } } ( \\partial _ { t } \\varphi ( x , t ) + \\langle v _ { t } ( x ) , \\nabla _ { x } \\varphi ( x , t ) \\rangle ) d \\mu _ { t } ( x ) d t = 0 , \\forall \\varphi \\in C _ { 0 } ^ { \\infty } ( \\R ^ { d } \\times I ) . \\end{align*}"}
{"id": "3532.png", "formula": "\\begin{align*} & \\mathrm { D T V } _ { ( p , q ) } ( X ) \\\\ & = - \\mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \\mu ^ { 2 } + 2 \\mu \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sigma ^ { 2 } \\left ( L _ { 1 } + \\frac { 1 } { 2 t - 3 } L _ { 2 } \\right ) , ~ t > \\frac { 3 } { 2 } , \\end{align*}"}
{"id": "8105.png", "formula": "\\begin{align*} { \\mathfrak X } _ { \\omega , c } ( G / H \\times { \\mathbb R } ^ { k } ) ^ { X } : = \\{ Y \\in { \\mathfrak X } _ { \\omega , c } ( G / H \\times { \\mathbb R } ^ { k } ) \\mid [ X , Y ] = 0 \\} . \\end{align*}"}
{"id": "745.png", "formula": "\\begin{align*} \\lim _ { \\zeta _ i \\to z _ i } \\left ( \\frac { f ( z , \\zeta ) } { ( z _ i - \\zeta _ i ) } \\bigg | _ { z _ l = \\zeta _ l , l \\neq i } \\right ) = 0 , \\ ; \\ ; i = 1 , \\ldots , m . \\end{align*}"}
{"id": "1290.png", "formula": "\\begin{align*} \\sigma ( T _ 0 ) \\supseteq \\bigcup _ { j = 1 } ^ { k } \\sigma ( T _ j ) . \\end{align*}"}
{"id": "1671.png", "formula": "\\begin{align*} f _ { j } = { f } _ { j } ^ 0 - \\sum _ { \\{ p : j \\in S _ p \\} } \\rho _ p { f } _ { j } ^ 0 . \\end{align*}"}
{"id": "5528.png", "formula": "\\begin{align*} \\abs { \\gamma ' ( 0 ) } = 1 - \\abs { a } ^ 2 = 1 - \\abs { \\gamma ( 0 ) } ^ 2 . \\end{align*}"}
{"id": "6664.png", "formula": "\\begin{align*} u _ t = ( | D u | ^ { p - 2 } D u ) + | u | ^ { q - 1 } u - \\lambda | D u | ^ { l } \\ \\ \\R ^ { N } \\times ( 0 , T ] , \\end{align*}"}
{"id": "4642.png", "formula": "\\begin{align*} c _ 0 ( \\Z _ + ) \\ni x : = ( x _ k ) _ { k \\in \\Z _ + } \\mapsto B _ w ^ n x & = \\left ( \\left [ \\prod _ { j = 1 } ^ { n } w ^ { - ( k - j ) } \\right ] x _ { k - n } \\right ) _ { k \\in \\Z _ + } \\\\ & = \\left ( w ^ { - n k + \\frac { n ( n + 1 ) } { 2 } } x _ { k - n } \\right ) _ { k \\in \\Z _ + } , \\ n \\in \\N , \\end{align*}"}
{"id": "3725.png", "formula": "\\begin{align*} \\widetilde { M } _ { n } ( t ) = \\sup _ { s \\in [ 0 , t ] } M _ { n } ( s ) + ( 1 + t ) ^ { n ^ 3 } \\lesssim \\big ( \\widetilde { M } _ { n } ( t ) \\big ) ^ { 1 - \\epsilon } + ( 1 + t ) ^ { n ^ 3 } , \\Longrightarrow \\widetilde { M } _ { n } ( t ) \\lesssim ( 1 + t ) ^ { n ^ 3 } . \\end{align*}"}
{"id": "7124.png", "formula": "\\begin{align*} { \\mathcal F } ^ { 4 \\alpha + 1 } ( F _ 6 ) = { \\mathcal F } ^ { 4 \\alpha + 3 } ( F _ 6 ) & = [ 0 , 1 , 1 ] \\cdot [ 0 , 1 , 1 , 0 , 3 , 5 , 0 , 5 , 5 , 0 , 7 , 1 ] \\\\ & = [ 0 , 1 , 1 , 0 , 3 , 5 , 0 , 5 , 5 , 0 , 7 , 1 ] . \\end{align*}"}
{"id": "4117.png", "formula": "\\begin{align*} S ^ { i + 1 , j - 1 } \\circ S ^ { i , j } = 0 , \\forall i , j \\geq 0 . \\end{align*}"}
{"id": "5049.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T } E [ | N ^ { n , 2 } _ \\tau | ^ 2 ] = 0 . \\end{align*}"}
{"id": "4824.png", "formula": "\\begin{align*} F _ \\rho = \\widetilde { M } \\times _ \\rho \\C ^ n \\ ; , \\end{align*}"}
{"id": "3168.png", "formula": "\\begin{align*} \\mathbf { N } _ { \\Lambda } \\doteq \\left \\{ \\begin{array} { l l } \\sum _ { \\mathrm { s } \\in \\mathrm { S } } a _ { x , \\mathrm { s } } ^ { \\ast } a _ { x , \\mathrm { s } } & \\Lambda = \\left \\{ x \\right \\} x \\in \\mathfrak { L } \\\\ 0 & \\end{array} \\right . , \\Lambda \\in \\mathcal { P } _ { \\mathrm { f } } \\ , \\end{align*}"}
{"id": "2029.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : s y m p s i 1 p s i 2 } \\forall ( z _ 1 , z _ 2 ) \\in E ^ 2 , ( \\psi _ 1 , \\psi _ 2 ) ( z _ 1 , z _ 2 , \\cdot ) _ { \\# } \\nu = ( \\psi _ 2 , \\psi _ 1 ) ( z _ 2 , z _ 1 , \\cdot ) _ { \\# } \\nu . \\end{align*}"}
{"id": "5249.png", "formula": "\\begin{align*} \\Lambda _ { Q , j } : = \\Gamma _ { 0 , k _ 1 ( j ) , k _ 2 ( j ) , 1 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I _ j } } \\end{align*}"}
{"id": "2238.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ \\infty \\frac { \\beta ( \\sigma , t ) ^ { k } } { k ! } \\right | \\leq \\left | \\frac { \\beta ( \\sigma , t ) } { 1 - \\beta ( \\sigma , t ) } \\right | : = \\beta _ 1 ( \\sigma , t ) . \\end{align*}"}
{"id": "2774.png", "formula": "\\begin{align*} D = \\max \\left \\{ h _ i + h _ { i - 1 } , \\ , \\tfrac { 2 ( 1 - \\kappa ) \\left ( h _ i + h _ { i - 1 } - ( 1 + \\kappa ) h _ i h _ { i - 1 } \\right ) } { ( 2 - ( 1 + \\kappa ) h _ { i - 1 } ) \\ ( 2 - ( 1 + \\kappa ) h _ i ) } \\right \\} \\end{align*}"}
{"id": "6886.png", "formula": "\\begin{align*} c _ k & : = \\int _ { \\mathbb { R } } | p _ { 0 , k } ( y ) | d y = \\int _ { \\mathbb { R } } d y \\left | e ^ { - i k \\frac { 2 \\pi } { V _ F } g } \\int _ { 0 } ^ { V _ F } p _ { } ( v , g ) e ^ { - i k v \\frac { 2 \\pi } { V _ F } } d v . \\right | \\\\ & \\leq \\int _ { \\mathbb { R } } \\int _ { 0 } ^ { V _ F } | p _ { } ( v , g ) | d v d g = 1 . \\end{align*}"}
{"id": "3599.png", "formula": "\\begin{align*} H { ( a , c ) } : = \\{ x \\in { \\mathbb R } ^ s \\vert \\ , \\langle x , a \\rangle = c \\} \\ \\mbox { a n d } \\ H ^ + { ( a , c ) } : = \\{ x \\in { \\mathbb R } ^ s \\vert \\ , \\langle x , a \\rangle \\geq c \\} . \\end{align*}"}
{"id": "1314.png", "formula": "\\begin{align*} f ( t , x , v ) = f ( t , x , R _ x v ) , R _ x v = v - 2 ( v \\cdot n _ x ) n _ x , \\end{align*}"}
{"id": "2492.png", "formula": "\\begin{align*} T _ U V = g ( U , V ) H ~ ~ T _ U X = - g ( H , X ) U , \\end{align*}"}
{"id": "967.png", "formula": "\\begin{align*} M _ { \\mathbb { R } } ^ { * } = \\left \\{ H ^ { * } ( \\overline { f } ) \\mid \\overline { f } \\in \\mathbb { R } ^ { n } \\right \\} \\ \\ \\ M _ { Q } ^ { * } = \\left \\{ H ^ { * } ( \\overline { f } ) \\mid \\overline { f } \\in Q ^ { n } \\right \\} . \\end{align*}"}
{"id": "7593.png", "formula": "\\begin{align*} \\lambda _ 0 = \\frac { \\beta } { \\delta } , \\lambda _ 1 = - 2 \\log \\delta \\ , , \\lambda _ 2 = - \\frac { \\gamma } { \\delta } . \\end{align*}"}
{"id": "4885.png", "formula": "\\begin{align*} \\begin{aligned} & \\max \\limits _ { \\alpha \\in \\mathfrak { m } _ { i } } | f _ { i } ( \\alpha ) | \\ll N _ { i } ^ { 1 - 1 / 1 8 + \\varepsilon } , \\\\ & \\max \\limits _ { \\alpha \\in \\mathfrak { m } _ { i } } | S _ { i } ( \\alpha ) | \\ll N _ { i } ^ { 1 / 3 - 1 / 4 2 + \\varepsilon } . \\end{aligned} \\end{align*}"}
{"id": "3926.png", "formula": "\\begin{align*} | I _ 3 | \\le \\frac { c } { T _ n } ( \\frac { 1 } { \\prod _ { l \\ge 4 } h _ l } \\frac { 1 } { \\delta _ 2 ^ { \\frac { 1 } { 2 } } } + D ) . \\end{align*}"}
{"id": "3091.png", "formula": "\\begin{align*} ( m \\otimes x ) ( a \\# b ) = \\sum _ { ( b ) } m ( a \\# b _ { - 1 } ) \\otimes x b _ 0 . \\end{align*}"}
{"id": "3154.png", "formula": "\\begin{align*} \\mathfrak { a = } \\underset { } { \\underbrace { \\mathfrak { a } _ { + } } } - \\underset { } { \\underbrace { \\mathfrak { a } _ { - } } } \\end{align*}"}
{"id": "3415.png", "formula": "\\begin{align*} h _ i = w _ i ( g _ 1 , \\ldots , g _ r ) . \\end{align*}"}
{"id": "8433.png", "formula": "\\begin{align*} u \\left ( s \\right ) = \\xi . \\end{align*}"}
{"id": "7491.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\ln ( \\vert \\varphi _ { a , b } ' ( y _ k ) \\vert ) & = \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\ln \\left ( \\left \\vert \\varphi _ { 0 , 1 } ' \\left ( \\frac { y _ k - a } { b } \\right ) \\right \\vert \\right ) = \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\ln \\left ( \\left \\vert \\varphi _ { 0 , 1 } ' \\left ( \\varphi _ { 0 , 1 } ^ { ( k ) } \\left ( \\frac { y _ 0 - a } { b } \\right ) \\right ) \\right \\vert \\right ) \\\\ & \\to \\ln ( 2 ) \\end{align*}"}
{"id": "6269.png", "formula": "\\begin{align*} \\int f ( x ) r ( x ) y ( x ) d _ q x = - f ( x / q ) D _ { q ^ { - 1 } } y ( x ) . \\end{align*}"}
{"id": "5928.png", "formula": "\\begin{align*} \\lim \\limits _ { T \\to \\infty } \\frac 1 T \\ln \\langle { x ^ \\eta } ( T ) \\rangle = w ( \\eta ) \\ , \\end{align*}"}
{"id": "1127.png", "formula": "\\begin{align*} \\dot { C } ( \\lambda W ) + \\frac { \\rho } { \\lambda } \\cdot W = \\zeta \\cdot W , \\end{align*}"}
{"id": "8089.png", "formula": "\\begin{align*} Z = f ( z ) ( y \\partial _ { x } - x \\partial _ { y } ) , \\end{align*}"}
{"id": "5098.png", "formula": "\\begin{align*} \\int ^ { \\eta _ n ( s ) - \\delta } _ 0 \\psi _ { n , 2 } ^ 2 ( u , s ) d u & = \\int ^ { \\eta _ n ( s ) - \\delta } _ 0 [ ( s - \\eta _ n ( u ) ) ^ { \\alpha } - ( \\eta _ n ( s ) - \\eta _ n ( u ) ) ^ \\alpha ] ^ 2 d u \\\\ & \\le C n ^ { - 2 } \\delta ^ { 2 \\alpha - 2 } . \\end{align*}"}
{"id": "5714.png", "formula": "\\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \\ldots , y _ i ) + \\sum _ { \\substack { 0 \\le p \\le i , \\ 1 \\le q \\le n - i \\\\ p + q = k } } ( y _ i - y _ { i + 1 } ) e _ { p } ( y _ 1 , \\ldots , y _ i ) e _ { q } ( y _ { i + 1 } , \\ldots , y _ n ) = 0 . \\end{align*}"}
{"id": "3100.png", "formula": "\\begin{align*} \\boldsymbol { a _ h } ( \\boldsymbol { u _ h } , \\boldsymbol { v _ h } ) = \\lambda _ h \\boldsymbol { b _ h } ( \\boldsymbol { u _ h } , \\boldsymbol { v _ h } ) \\quad \\boldsymbol { v _ h } \\in \\boldsymbol { V _ h } . \\end{align*}"}
{"id": "3667.png", "formula": "\\begin{align*} \\tilde { K } _ { k , l _ 2 } ( y , v , V ( s ) ) = \\int _ { \\R ^ 3 } e ^ { i y \\cdot \\xi } i \\xi | \\xi | ^ { - 2 } \\varphi _ k ( \\xi ) \\psi _ { \\leq l _ 2 } ( \\theta _ { V ( s ) } ( v ) \\cdot \\tilde { \\xi } ) d \\xi . \\end{align*}"}
{"id": "863.png", "formula": "\\begin{align*} u \\left ( \\lambda _ { m } \\right ) = \\frac { \\log \\left ( 1 + \\frac { \\lambda _ { m } } { \\varepsilon } \\right ) } { \\log \\left ( 1 + \\frac { 1 } { \\varepsilon } \\right ) } + o \\left ( \\varepsilon \\right ) , \\end{align*}"}
{"id": "4561.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { T } ( \\Lambda ) & \\to \\ , \\widehat { \\Lambda } \\\\ M ( w ) & \\mapsto M ( \\widehat { w } ) \\end{aligned} \\end{align*}"}
{"id": "9012.png", "formula": "\\begin{align*} \\Delta _ 1 = - \\frac { \\pi \\gamma } { 2 } + 2 \\beta ^ { \\prime } ( 1 ) . \\end{align*}"}
{"id": "3943.png", "formula": "\\begin{align*} \\xi _ { ( w _ i ) _ i , \\phi } ( y _ { \\sigma ( 1 ) } ^ 1 , \\dots , y _ { \\sigma ( d ) } ^ d ) = \\int _ { \\R ^ { d ( d - 1 ) } } \\phi ( y ^ 1 ) \\prod _ { i = 1 } ^ { d - 1 } p _ { w _ { i + 1 } - w _ i } ( y ^ i , y ^ { i + 1 } ) d \\widehat { y } ^ 1 \\dots d \\widehat { y } ^ d , \\end{align*}"}
{"id": "4219.png", "formula": "\\begin{align*} t _ n ^ a = \\inf \\{ m > t _ { n - 1 } ^ a : Z _ m \\leq a Z ^ * _ { m - 1 } \\} \\end{align*}"}
{"id": "2455.png", "formula": "\\begin{align*} M _ { \\operatorname { A d } _ { g } \\left ( x \\right ) } = g M _ { x } , \\end{align*}"}
{"id": "4025.png", "formula": "\\begin{align*} S _ { n , j } = \\sum _ { i = 1 } ^ n A _ { i , j } , j \\in [ d ] , \\end{align*}"}
{"id": "6615.png", "formula": "\\begin{align*} S _ { k } ( a , k , \\xi \\ , ( \\ell ) ) : = \\sum _ { { \\tiny \\begin{tabular} { c } $ z \\ , \\hbox { \\tiny m o d } \\ , k \\ell $ \\\\ $ z = \\xi ( \\ell ) $ \\end{tabular} } } e _ { k \\ell } ( a \\ell N ( z ) + c \\cdot z ) , \\end{align*}"}
{"id": "439.png", "formula": "\\begin{align*} F _ { \\nu _ { n } } ( z ) = H _ { \\nu _ { n } } ^ { - 1 } ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\partial Q } \\frac { \\zeta H _ { \\nu _ { n } } ( \\zeta ) } { H _ { \\nu _ { n } } ( \\zeta ) - z } \\ , d \\zeta \\end{align*}"}
{"id": "2704.png", "formula": "\\begin{align*} \\psi _ \\mathrm { g a u s s } : = \\sum _ { \\substack { - \\sigma _ N ^ 2 \\le d \\le \\sigma _ N ^ 2 \\\\ N + d } } c _ d \\ , u _ 1 ^ { \\otimes ( N + d ) / 2 } \\otimes _ \\mathrm { s y m } u _ 2 ^ { \\otimes ( N - d ) / 2 } , \\end{align*}"}
{"id": "3696.png", "formula": "\\begin{align*} & \\left < \\frac { i } h [ - h ^ 2 \\Delta + V ( x ) - E ( h ) , \\chi _ \\alpha ( x _ n ) h D _ n ] u _ h , \\ , u _ h \\right > _ { L ^ 2 ( \\Omega _ \\Gamma ) } \\\\ & = \\left < O p _ h \\left ( \\{ \\sigma ( - h ^ 2 \\Delta + V ( x ) - E ( h ) ) , \\sigma ( \\chi _ \\alpha ( x _ n ) h D _ n ) \\} \\right ) u _ h , u _ h \\right > _ { L ^ 2 ( \\Omega _ \\Gamma ) } + O ( h ) , \\end{align*}"}
{"id": "2738.png", "formula": "\\begin{align*} \\theta = c _ { n , \\sigma p } \\lim _ { R \\rightarrow \\infty } \\lim _ { i \\rightarrow \\infty } \\int _ { B _ { R } ^ { c } } \\frac { u _ { i } ^ { p - 1 } ( x ) } { | x | ^ { n + \\sigma p } } d x . \\end{align*}"}
{"id": "4685.png", "formula": "\\begin{align*} & ( - 1 ) ^ { k } \\left ( C _ m ( n ) - \\sum _ { j = 1 - k } ^ k ( - 1 ) ^ j u _ m \\big ( n - j ( 3 j - 1 ) / 2 \\big ) \\right ) = \\sum _ { j = 0 } ^ n C _ m ( j ) \\ , M _ k ( n - j ) . \\end{align*}"}
{"id": "348.png", "formula": "\\begin{align*} \\P ( Z _ s = k ) \\to \\exp ( - e ^ { - s } ) \\frac { e ^ { - k s } } { k ! } . \\end{align*}"}
{"id": "8638.png", "formula": "\\begin{align*} \\frac { d U } { d t } ( t ) \\psi = \\sum _ { j = 1 } ^ r \\frac { d f _ j } { d t } ( t ) { \\rm A d } _ { \\prod _ { k = 1 } ^ { j - 1 } \\exp ( { \\rm i } f _ k ( t ) \\widehat H _ k ) } ( { \\rm i } \\widehat H _ j ) U ( t ) \\psi , \\forall \\psi \\in \\mathcal { H } , \\end{align*}"}
{"id": "8109.png", "formula": "\\begin{align*} \\P _ N \\Biggl ( \\prod _ { i = 1 } ^ { N } ( \\Delta \\tau _ i ) ^ { - \\frac d 2 } > C ^ { - N } R \\Biggr ) \\geq C ^ N N ^ { \\frac d 2 N } R ^ { - 1 - \\frac 2 d } \\log ^ { N - 1 } R . \\end{align*}"}
{"id": "1593.png", "formula": "\\begin{align*} E = b ^ 2 f ^ 2 ( x ^ 1 ) \\left [ f '^ 2 ( x ^ 1 ) + \\sin ^ { 2 } \\left ( x ^ 2 - \\theta \\right ) \\right ] . \\end{align*}"}
{"id": "8690.png", "formula": "\\begin{align*} \\bigtriangleup J = \\int \\limits _ { t _ { 0 } } ^ { t _ { 1 } } E ( x ^ { 1 } , x ^ { 2 } , \\dot { x } ^ 1 , \\dot { x } ^ { 2 } , u ^ { 1 } , u ^ { 2 } ) d t , \\end{align*}"}
{"id": "1606.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial z ^ i _ { \\epsilon } } v ^ i = 2 b ^ 2 \\delta _ { \\epsilon 1 } f ^ 2 ( x ^ 1 ) f ' ( x ^ 1 ) . \\end{align*}"}
{"id": "4578.png", "formula": "\\begin{align*} \\left ( \\bigcap _ { e = 1 } ^ \\infty K _ e \\right ) S _ 2 \\subseteq \\left ( \\bigcap _ { e = 1 } ^ n K _ e \\right ) S _ 2 = \\bigcap _ { e = 1 } ^ n ( K _ e S _ 2 ) \\end{align*}"}
{"id": "8074.png", "formula": "\\begin{align*} \\phi ( N \\times \\{ 0 \\} ) = S ^ { 1 } \\cdot x _ { 0 } \\end{align*}"}
{"id": "4493.png", "formula": "\\begin{align*} \\varrho ( t ) : = ( t + 1 ) ^ { - 0 . 4 5 } . \\end{align*}"}
{"id": "6713.png", "formula": "\\begin{align*} \\prescript { } { l } { \\mathcal { F } } _ { n } ( f ) ( \\lambda ) = h _ { \\lambda _ 2 } ( \\lambda _ 1 ) \\cdot q _ { l _ 1 , n _ 1 } = h _ { \\lambda _ 1 } ( \\lambda _ 2 ) \\cdot q _ { l _ 2 , n _ 2 } ( \\lambda _ 2 ) , \\end{align*}"}
{"id": "8461.png", "formula": "\\begin{align*} h _ { d , b , A } \\triangleq \\prod _ { i = 1 } ^ d h _ { 1 , b , A _ i } , \\end{align*}"}
{"id": "8030.png", "formula": "\\begin{align*} H _ { ( X , D ) } ( y ) = e ^ { \\frac { t } { 2 \\pi i } } \\sum _ { \\substack { d \\in \\mathbb K \\\\ ( k _ 1 , \\ldots , k _ m ) \\in ( \\mathbb Z _ { \\geq 0 } ) ^ m } } y ^ d \\left ( \\frac { \\Gamma ( 1 + \\frac { D } { 2 \\pi i } + D \\cdot d ) } { \\prod _ { i = 1 } ^ m \\Gamma ( 1 + \\frac { \\bar D _ i } { 2 \\pi i } + D _ i \\cdot d ) } \\right ) \\textbf { 1 } _ { [ d ] } [ \\textbf { 1 } ] _ { D \\cdot d - \\sum _ { i = 1 } ^ m k _ i a _ i } \\frac { \\prod _ { i = 1 } ^ m x _ i ^ { k _ i } } { \\prod _ { i = 1 } ^ m ( k _ i ! ) } . \\end{align*}"}
{"id": "7725.png", "formula": "\\begin{align*} \\lambda \\leq \\sum _ { t = 1 } ^ k e ^ T A _ { \\bar { S } S } ( k + 1 ) e , \\sum _ { t = 1 } ^ k e ^ T A _ { S \\bar { S } } ( k + 1 ) e \\leq \\Lambda , \\end{align*}"}
{"id": "8353.png", "formula": "\\begin{align*} \\tilde { f } _ \\mu ( p ) = \\sum \\limits _ { i } ^ { N } \\ , \\alpha _ i \\frac { u _ { i \\mu } } { u _ i \\cdot p } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\sum \\limits _ { i } ^ { N } \\alpha _ i = 0 , \\ , \\ , u _ i \\cdot u _ i = 1 , \\ , i = 1 , \\ldots , N . \\end{align*}"}
{"id": "4398.png", "formula": "\\begin{align*} & \\lim _ { j \\rightarrow + \\infty } \\left ( \\sup _ { D _ j } e ^ { - u ( - v _ { t _ 0 , B } ( \\Psi ) ) } \\right ) \\int _ { D _ j } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ { 0 } - B < \\Psi < - t _ { 0 } \\} } | f | ^ 2 e ^ { - \\Psi } \\\\ \\le & \\left ( \\sup _ { D } e ^ { - u ( - v _ { t _ 0 , B } ( \\Psi ) ) } \\right ) \\int _ { D } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ { 0 } - B < \\Psi < - t _ { 0 } \\} } | f | ^ 2 e ^ { - \\Psi } \\\\ < & + \\infty \\end{align*}"}
{"id": "7487.png", "formula": "\\begin{align*} \\varphi _ { a , b } ( x ) : = \\begin{cases} \\frac { b } { 2 } \\left ( \\frac { x - a } { b } - \\frac { 1 } { \\frac { x - a } { b } } \\right ) + a = \\frac { x } { 2 } + \\frac { a } { 2 } - \\frac { \\frac { b ^ 2 } { 2 } } { x - a } & \\\\ a & \\end{cases} \\end{align*}"}
{"id": "4251.png", "formula": "\\begin{align*} \\widetilde { \\Phi } ( x ) = \\Phi ( ( x + M ) ^ 4 ) f ( x ) , \\end{align*}"}
{"id": "8503.png", "formula": "\\begin{align*} p ( x , y ) = \\sum _ { i = 1 } ^ \\infty w _ i f _ i ( x ) g _ i ( y ) \\end{align*}"}
{"id": "3683.png", "formula": "\\begin{align*} \\beta _ r = \\varphi ' ( r ) \\big [ \\log \\cos r - f ( s , 0 ) \\big ] - \\varphi ( r ) \\tan r \\end{align*}"}
{"id": "4096.png", "formula": "\\begin{align*} \\pi _ { \\omega _ { P } } ^ { + } = U _ { + } ^ { * } \\pi _ { P } ^ { + } U _ { + } , \\pi _ { \\omega _ { P } } ^ { - } = U _ { - } ^ { * } \\pi _ { P } ^ { - } U _ { - } , \\end{align*}"}
{"id": "8649.png", "formula": "\\begin{align*} ( k + 2 ) \\frac { d ^ 2 c } { d t ^ 2 } ( t ) = ( k + 3 ) \\frac { ( d c / d t ) ^ 2 ( t ) } { c ( t ) } + \\omega ^ 2 ( t ) ( k + 2 ) ^ 2 c ( t ) . \\end{align*}"}
{"id": "6443.png", "formula": "\\begin{align*} s ^ { - 1 } \\mathcal F = t ^ { - 1 } \\mathcal F = \\Gamma ( \\ker ( \\dd s ) ) + \\Gamma ( \\ker ( \\dd t ) ) . \\end{align*}"}
{"id": "5565.png", "formula": "\\begin{align*} G _ y f ( x ) = \\frac { 1 } { x } ( f ( x e ^ { - y } ) - f ( x ) + y x f ' ( x ) ) \\end{align*}"}
{"id": "2769.png", "formula": "\\begin{align*} \\hat { W } ( y ) : = Z ( y ) + \\tfrac { \\mu } { 2 } \\| y \\| ^ 2 , \\end{align*}"}
{"id": "5415.png", "formula": "\\begin{align*} \\gamma ^ { 1 / 2 } h = ( m + 1 ) h = m h + h = h \\end{align*}"}
{"id": "7832.png", "formula": "\\begin{align*} ( A ^ { M \\sharp M } \\cap A ^ { \\tilde M \\sharp \\tilde M } ) ^ * = ( A ^ { M \\sharp M } ) ^ * \\cap ( A ^ { \\tilde M \\sharp \\tilde M } ) ^ * \\subset M ( A ) ^ { * * } . \\end{align*}"}
{"id": "3478.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast \\ast } = \\frac { c _ { 1 } \\left [ \\xi _ { p } ^ { 3 } \\overline { G } _ { ( 1 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { p } ^ { 2 } \\right ) - \\xi _ { q } ^ { 3 } \\overline { G } _ { ( 1 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { q } ^ { 2 } \\right ) \\right ] } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "1316.png", "formula": "\\begin{align*} M _ { T _ w } ( x , v ) = \\frac { c ( x ) } { ( 2 \\pi T _ w ( x ) ) ^ { d / 2 } } e ^ { \\frac { - | v | ^ 2 } { 2 T _ w ( x ) } } , c ( x ) = \\left ( \\int _ { \\{ v \\cdot n _ x > 0 \\} } \\frac { 1 } { ( 2 \\pi T _ w ( x ) ) ^ { d / 2 } } e ^ { \\frac { - | v | ^ 2 } { 2 T _ w ( x ) } } | v \\cdot n _ x | \\d v \\right ) ^ { - 1 } , \\end{align*}"}
{"id": "1486.png", "formula": "\\begin{align*} ( x ) _ { 0 , \\lambda } = 1 , ( x ) _ { n , \\lambda } = x ( x - \\lambda ) ( x - 2 \\lambda ) \\cdots \\big ( x - ( n - 1 ) \\lambda \\big ) , ( n \\ge 1 ) , \\end{align*}"}
{"id": "1588.png", "formula": "\\begin{align*} f ( x ^ 1 ) f '' ( x ^ 1 ) + f '^ 2 ( x ^ 1 ) + 1 = 0 , \\end{align*}"}
{"id": "4305.png", "formula": "\\begin{align*} \\mathcal { S } = \\int _ 0 ^ 1 d t \\int _ { L _ t } h _ t ( e ^ { - i \\hat { \\theta } } \\Omega ) . \\end{align*}"}
{"id": "1410.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} 1 + \\delta & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & \\frac { 1 } { 2 } & 1 - \\delta _ 1 \\end{matrix} \\right ] , \\Delta A = \\left [ \\begin{matrix} 0 & 0 & 0 \\\\ \\epsilon & 0 & 0 \\\\ 0 & 0 & 0 \\end{matrix} \\right ] , \\end{align*}"}
{"id": "6574.png", "formula": "\\begin{align*} \\psi _ i ^ k ( \\beta _ k ( g , ( \\psi _ j ^ k ) ^ { - 1 } ( x , v ) ) ) = ( \\alpha ( g , x ) , a _ k ( g , x , v ) ) \\end{align*}"}
{"id": "8884.png", "formula": "\\begin{align*} a _ { k + 1 } = \\left ( a _ k ^ 2 + | \\det ( a _ k ^ 2 I _ n - A ^ H A ) | \\left ( \\frac { n - 1 } { \\| A \\| _ F ^ 2 - ( n - 1 ) a _ k ^ 2 } \\right ) ^ { n - 1 } \\right ) ^ { 1 / 2 } , k = 1 , 2 , \\cdots , \\end{align*}"}
{"id": "3867.png", "formula": "\\begin{align*} p _ x [ T ] = p _ 1 [ T ] \\wedge p _ 3 [ T ] , \\quad & p _ y [ T ] = p _ 2 [ T ] \\wedge p _ 4 [ T ] , \\\\ p [ T ] = p _ 5 [ T ] \\wedge p _ 6 [ T ] , \\quad & q [ T ] = q _ 1 [ T ] \\wedge q _ 2 [ T ] . \\end{align*}"}
{"id": "4859.png", "formula": "\\begin{align*} \\begin{cases} N _ { 1 } = p _ { 1 } + p _ { 2 } ^ { 2 } + p _ { 3 } ^ { 2 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k _ { 1 } } } \\\\ N _ { 2 } = p _ { 4 } + p _ { 5 } ^ { 2 } + p _ { 6 } ^ { 2 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k _ { 1 } } } \\end{cases} \\end{align*}"}
{"id": "4307.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 d t \\int _ { L _ t } h _ t ( e ^ { - i \\hat { \\theta } } \\Omega ) = \\int _ { L _ t } f _ { L _ t } ( e ^ { - i \\hat { \\theta } } \\Omega ) | ^ { t = 1 } _ { t = 0 } - \\int _ { \\cup _ t L _ t } \\lambda \\wedge ( e ^ { - i \\hat { \\theta } } \\Omega ) . \\end{align*}"}
{"id": "5660.png", "formula": "\\begin{align*} \\sup _ { \\mathbb { R } ^ n } \\left | u ( x ) - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \\right | \\leq C _ 1 , \\end{align*}"}
{"id": "5356.png", "formula": "\\begin{align*} \\norm { u } _ { L ^ p ( \\Omega ) } ^ p = \\int _ { \\R ^ { n - 1 } } \\norm { u ( x ' , x _ n ) } _ { L ^ p ( a , b ) } ^ p d x ' \\leq C \\int _ { \\R ^ { n - 1 } } \\norm { ( - \\Delta _ { x _ n } ) ^ { s / 2 } u ( x ' , x _ n ) } _ { L ^ p ( \\R ) } ^ p d x ' \\end{align*}"}
{"id": "3274.png", "formula": "\\begin{align*} i \\int _ D A ( x ) \\cdot \\left ( u _ 2 \\nabla u _ 1 - u _ 1 \\nabla u _ 2 \\right ) d x = 2 s \\int _ D \\overline { \\omega } \\cdot A ( x ) e ^ { i x \\cdot \\xi } d x + \\mathcal { R } ( \\xi , s ) , \\end{align*}"}
{"id": "4889.png", "formula": "\\begin{align*} & \\gamma ^ { \\mathcal O \\rtimes E \\mathcal M } \\big ( ( o , u _ \\bullet ) ; ( f ^ { ( 1 ) } , v ^ { ( 1 ) } _ \\bullet ) , \\dots , ( f ^ { ( r ) } , v ^ { ( r ) } _ \\bullet ) \\big ) \\\\ & \\quad { } = \\big ( \\gamma ^ { \\mathcal O } ( o , u _ 1 . f ^ { ( 1 ) } , \\dots , u _ r . f ^ { ( r ) } ) ; \\gamma ^ { E \\mathcal M ^ \\bullet } ( u _ \\bullet ; v _ \\bullet ^ { ( 1 ) } , \\dots , v _ \\bullet ^ { ( r ) } ) \\big ) \\end{align*}"}
{"id": "7373.png", "formula": "\\begin{align*} { \\Delta \\left ( \\frac { \\partial W _ { h } } { \\partial h } \\right ) - \\frac { \\partial W _ { h } } { \\partial h } + p \\left ( \\sum _ { i = 1 } ^ 4 ( U _ { h , i } ) ^ { p - 1 } \\frac { \\partial U _ { h , i } } { \\partial h } \\right ) = 0 . } \\end{align*}"}
{"id": "3980.png", "formula": "\\begin{align*} \\textstyle \\sum _ { j = 1 } ^ n - p _ j ( 0 ) \\log _ q p _ j ( 0 ) \\le - n ( 1 - \\omega ) \\log _ q ( 1 - \\omega ) . \\end{align*}"}
{"id": "4044.png", "formula": "\\begin{align*} E _ { F } \\hat { \\beta } = \\beta ( F ) F \\in \\mathbf { F } _ { 2 } ( \\Sigma ) , \\end{align*}"}
{"id": "4961.png", "formula": "\\begin{align*} \\kappa _ 2 = \\sqrt { \\kappa _ 3 + \\frac { 2 \\alpha + 1 } { 2 ( \\alpha + 1 ) ^ 2 } + \\kappa _ 4 + \\kappa _ 5 } , \\end{align*}"}
{"id": "1973.png", "formula": "\\begin{align*} \\int _ { D } | { v _ k ^ { + } } | ^ p \\ , d x \\ , & \\le \\ , \\int _ { D } | v _ k | ^ p \\ , d x \\\\ \\int _ { D } | \\nabla { v _ k ^ { + } } | ^ p \\ , d x & = \\int _ { D } | \\nabla v _ k | ^ p \\chi _ { \\{ v _ k > 0 \\} } \\ , d x \\ , \\le \\ , \\int _ { D } | \\nabla v _ k | ^ p \\ , d x . \\end{align*}"}
{"id": "3080.png", "formula": "\\begin{align*} \\frac { c _ { n } } { c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast \\ast } } = & \\frac { \\Gamma \\left ( \\frac { n } { 2 } \\right ) 2 ^ { ( n - 3 ) / 2 } } { \\Gamma ( n ) \\Gamma \\left ( \\frac { n - 1 } { 2 } \\right ) \\sqrt { \\pi } } \\bigg \\{ \\int _ { 0 } ^ { \\infty } t ^ { \\frac { n - 3 } { 2 } } \\left [ ( 2 + 3 \\sqrt { 2 } ) \\left ( t + \\frac { \\xi _ { \\boldsymbol { s } , k } ^ { 2 } } { 2 } \\right ) + 3 \\right ] \\exp \\left ( - \\sqrt { 2 t + \\xi _ { \\boldsymbol { s } , k } ^ { 2 } } \\right ) \\mathrm { d } t \\bigg \\} \\end{align*}"}
{"id": "4877.png", "formula": "\\begin{align*} S _ { 1 } ( N , \\alpha ) = \\sum \\limits _ { p \\sim N } ( \\log p ) e ( p \\alpha ) \\end{align*}"}
{"id": "5325.png", "formula": "\\begin{align*} Z _ { i , s } ( t , \\phi ) = \\liminf _ { n \\to \\infty } Z _ { i , s } ^ { ( n ) } ( t , \\phi ) . \\end{align*}"}
{"id": "5414.png", "formula": "\\begin{align*} B _ { \\gamma } ( u , u ) & \\geq \\gamma _ 0 \\| \\nabla ^ s u \\| _ { L ^ 2 ( \\R ^ { 2 n } ) } ^ 2 = \\gamma _ 0 \\| ( - \\Delta ) ^ { s / 2 } u \\| _ { L ^ 2 ( \\R ^ n ) } ^ 2 \\\\ & \\geq \\frac { \\gamma _ 0 } { 2 } ( \\| ( - \\Delta ) ^ { s / 2 } u \\| _ { L ^ 2 ( \\R ^ n ) } ^ 2 + C \\| u \\| _ { L ^ 2 ( \\R ^ n ) } ^ 2 ) \\geq \\frac { \\gamma _ 0 } { 2 } \\min ( 1 , C ) \\| u \\| _ { H ^ s ( \\R ^ n ) } ^ 2 \\end{align*}"}
{"id": "3902.png", "formula": "\\begin{align*} \\| \\mathcal G \\| ^ \\frac { 1 } { p - 1 } _ { q _ 0 , B _ 2 ( 0 ) } = R ^ { \\frac { N ( q _ 0 - 1 ) } { q _ 0 ( p - 1 ) } } \\| f \\| ^ \\frac { 1 } { p - 1 } _ { q _ 0 , B _ { 2 R } ( 0 ) } \\leq M R ^ { \\frac { N ( q _ 0 - 1 ) } { q _ 0 ( p - 1 ) } } , \\end{align*}"}
{"id": "184.png", "formula": "\\begin{align*} \\dot C _ t = ( 1 - A ) C _ t ^ 2 , \\ddot C _ t = 2 ( 1 - A ) C _ t \\dot C _ t . \\end{align*}"}
{"id": "2081.png", "formula": "\\begin{align*} X _ { \\mathcal { F } } = \\{ x \\in \\mathcal { A } ^ { \\Z ^ d } \\mid \\forall p \\in \\mathcal { F } , p \\not \\sqsubset x \\} \\end{align*}"}
{"id": "8294.png", "formula": "\\begin{align*} k \\left [ \\tan ( k a / 2 ) + \\cot ( k a / 2 ) \\right ] = - k ' \\left [ \\tanh ( k ' a / 2 ) - \\coth ( k ' a / 2 ) \\right ] \\end{align*}"}
{"id": "3962.png", "formula": "\\begin{align*} b ^ { * } j ( a ) ( \\gamma \\beta ^ { - 1 } ) c = \\lim _ { n } b ^ { * } j ( f _ { n } ) ( \\gamma \\beta ^ { - 1 } ) c . \\end{align*}"}
{"id": "6444.png", "formula": "\\begin{align*} s ( x , y ) = x \\displaystyle { t ( x , y ) = \\exp _ x \\left ( \\sum _ { i = 1 } ^ n y _ i X _ i \\right ) } = \\varphi _ 1 ^ { \\sum _ { i = 1 } ^ n y _ i X _ i } ( x ) \\end{align*}"}
{"id": "8491.png", "formula": "\\begin{align*} \\int _ a ^ b f ( x ) d x = \\left [ \\frac { 1 } { 2 } L x ^ 2 + c x \\right ] _ { x = a } ^ { x = b } = \\frac { 1 } { 2 } L \\left ( b ^ 2 - a ^ 2 \\right ) + c ( b - a ) = \\left ( L \\frac { a + b } { 2 } + c \\right ) ( b - a ) . \\end{align*}"}
{"id": "2482.png", "formula": "\\begin{align*} K _ r ( x , y , \\varphi ^ l ( x , y ) ) = \\left ( \\sum _ { i = 1 } ^ { n - 2 } \\frac { 3 } { \\varepsilon } ( \\varphi _ i ^ l ( x , y ) - \\Phi _ i ( x , y ) ) ^ 2 \\right ) ^ r > \\left ( \\frac 4 3 \\right ) ^ r . \\end{align*}"}
{"id": "6661.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t - u _ { x x } = u ^ { p } - \\lambda | u _ { x } | ^ { q } , & t > 0 , \\ 0 < x < s ( t ) , \\\\ u _ x ( t , 0 ) = u ( t , x ) = 0 , & t > 0 , \\\\ s ^ { \\prime } ( t ) = - \\mu u _ x ( t , s ( t ) ) , & t > 0 , \\\\ s ( 0 ) = s _ 0 > 0 , \\ u ( 0 , x ) = u _ 0 ( x ) , & 0 \\leq x \\leq s _ 0 , \\end{array} \\right . \\end{align*}"}
{"id": "2289.png", "formula": "\\begin{align*} J _ 1 = 2 \\int _ 0 ^ \\infty ( \\sigma + h _ p ^ 0 ) h _ { p x } ^ 1 u _ { p x } ^ 1 x ^ { 1 - 2 \\sigma _ 1 } { \\rm d } y \\leq \\Vert \\sigma + h _ p ^ 0 \\Vert _ { L ^ \\infty } \\Vert ( u _ { p x } ^ 1 , h _ { p x } ^ 1 ) x ^ { \\frac { 1 } { 2 } - \\sigma _ 1 } \\Vert _ { L _ y ^ 2 } ^ 2 , \\end{align*}"}
{"id": "2445.png", "formula": "\\begin{align*} L _ { f \\cdot \\tau } = M _ { f \\cdot \\tau } + \\sum _ { g \\prec f \\cdot x } \\ell _ { g , f \\cdot \\tau } M _ g . \\end{align*}"}
{"id": "7416.png", "formula": "\\begin{align*} A _ i - A _ i ^ * = \\sum _ { 0 < | h | < H } \\left ( 1 - \\frac { | h | } { H } \\right ) \\hat { f } ( h ) \\left ( e ( h Y _ i \\alpha ) - e ( h ( Y _ i \\alpha - \\delta _ i ) ) \\right ) \\sum _ { k \\in H _ i } e ( h ( S _ k - Y _ i ) \\alpha ) , \\end{align*}"}
{"id": "290.png", "formula": "\\begin{align*} { \\rm L } _ a & : = \\big \\{ b a \\in \\N \\ ; : \\ ; p \\mid b \\implies p \\ge P ( a ) \\big \\} , \\end{align*}"}
{"id": "3622.png", "formula": "\\begin{align*} R ( w ) & = \\{ w ' : w R w ' = 1 \\} , \\\\ R ( w ) & = \\{ w ' : w R w ' \\} . \\end{align*}"}
{"id": "4053.png", "formula": "\\begin{align*} M ^ { ( a ) } = \\sum _ { i , \\alpha } M _ \\alpha ^ i ( 1 \\otimes \\cdots \\otimes E _ i ^ \\alpha \\otimes \\cdots \\otimes 1 ) , \\end{align*}"}
{"id": "2286.png", "formula": "\\begin{align*} p _ p ^ { 1 , a } = \\int _ y ^ \\infty ( R ^ { v , 0 } + R _ E ^ { v , 1 } ) \\end{align*}"}
{"id": "710.png", "formula": "\\begin{align*} { R ^ N _ { \\alpha \\beta \\gamma } } ^ \\delta \\partial _ \\delta = R ^ N ( \\partial _ \\alpha , \\partial _ \\beta ) \\partial _ \\gamma . \\end{align*}"}
{"id": "7356.png", "formula": "\\begin{align*} \\frac { f ( r ) } { f ' ( r ) } = \\frac { k _ { \\phi } ( r ) } { k ' _ { \\phi } ( r ) } = \\frac { r } { \\psi ( - r ) } . \\end{align*}"}
{"id": "4202.png", "formula": "\\begin{align*} \\norm { g _ { \\le \\iota } ^ { ( 1 ) } } _ p ^ p \\lesssim _ \\iota ( \\iota + 2 ) ^ { p - 1 } \\sum _ { \\ell = - 1 } ^ \\iota ( 2 ^ { \\ell ( d _ 1 - d _ 2 ) + 2 \\iota d _ 2 } ) ^ { p / q } \\norm { F ^ { ( \\iota ) } } _ 2 ^ p \\norm { f } _ p ^ p . \\end{align*}"}
{"id": "1132.png", "formula": "\\begin{align*} \\dot { C } ( \\lambda w ) = \\frac { \\delta z _ 0 \\sinh ( z _ 0 \\lambda w ) + ( 1 - \\delta ) \\cdot 2 \\lambda q ^ 2 / ( q ^ 2 - \\lambda ^ 2 w ^ 2 ) ^ 2 } { \\delta \\cosh ( z _ 0 \\lambda w ) + ( 1 - \\delta ) q ^ 2 / ( q ^ 2 - \\lambda ^ 2 w ^ 2 ) } . \\end{align*}"}
{"id": "592.png", "formula": "\\begin{align*} \\sigma ( \\mathcal L _ n ) = \\bigg \\{ \\left . \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi k } { n } \\Big ) \\right | k = 0 \\ , , \\cdots , n - 1 \\bigg \\} . \\end{align*}"}
{"id": "6461.png", "formula": "\\begin{align*} \\langle \\Phi _ 0 ( x ) ^ { ( 0 ) } ( \\alpha ) , e \\rangle = \\varrho ( x ) [ \\langle \\alpha , e \\rangle ] - \\langle \\alpha , \\nabla _ x ( e ) \\rangle , \\ ; \\ ; . \\end{align*}"}
{"id": "2831.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N - 1 } \\frac { p _ i } { S } \\min \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\ , , \\ , \\| \\nabla f ( x _ { i + 1 } ) \\| ^ 2 \\} { } \\leq { } \\sum _ { i = 0 } ^ { N - 1 } \\frac { p _ i } { S } \\frac { f ( x _ i ) - f ( x _ { i + 1 } ) } { p _ i } { } = { } \\frac { f ( x _ 0 ) - f ( x _ { N } ) } { S } . \\end{align*}"}
{"id": "1442.png", "formula": "\\begin{align*} F _ 0 ( z ) = F ( z ) , \\ \\ \\ F _ { s } ( z ) = \\sum _ { k = 0 } ^ { \\infty } ( k + \\gamma _ 1 ) \\cdots ( k + \\gamma _ s ) c _ k z ^ { k + 1 } \\ \\ 1 \\le s \\le r - 1 \\enspace . \\end{align*}"}
{"id": "5427.png", "formula": "\\begin{align*} \\int _ { U } ( - \\Delta ) ^ s m _ 0 g \\ , d x = 0 \\end{align*}"}
{"id": "3005.png", "formula": "\\begin{align*} H = H _ 0 + z H _ 1 \\in \\N _ { A , E } ( a ) \\ , \\end{align*}"}
{"id": "3270.png", "formula": "\\begin{align*} \\rho _ 1 = s \\Big ( i \\omega _ 2 + \\big ( \\frac { \\xi } { 2 s } - \\sqrt { 1 - \\frac { \\vert \\xi \\vert ^ 2 } { 4 s ^ 2 } } \\omega _ 1 \\big ) \\Big ) : = s \\omega ^ { * } _ 1 ( s ) , \\end{align*}"}
{"id": "77.png", "formula": "\\begin{align*} - Y ( t ) + D _ A w ( t ) = v ( t ) , \\forall t \\in \\R , \\end{align*}"}
{"id": "8351.png", "formula": "\\begin{align*} \\tilde { f } _ \\mu = { { w _ 1 } ^ + } _ \\mu \\tilde { f } _ + + { { w _ 1 } ^ - } _ \\mu \\tilde { f } _ - + { w _ { r ^ { - 2 } } } _ \\mu \\tilde { f } _ { 0 + } , \\end{align*}"}
{"id": "4697.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty C _ 1 ( n ) \\ , q ^ n = \\frac { 1 } { ( q ; q ) _ \\infty } \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n q ^ { n ( n + 1 ) / 2 } . \\end{align*}"}
{"id": "6409.png", "formula": "\\begin{align*} C _ { 1 , 2 } = w _ 2 ^ 2 + w _ 1 w _ 2 \\ \\mbox { a n d } \\ C _ { 1 , 3 } = w _ 1 . \\end{align*}"}
{"id": "5262.png", "formula": "\\begin{align*} \\# Z ( H ^ { \\Lambda _ Q } ) = \\# Z ( H ^ { \\Lambda _ { Q , 0 } } ) \\cdot \\# Z ( { \\bf s } ^ { \\Lambda _ { Q , 1 } } _ { t _ i } ) = \\Delta ( \\Lambda _ { Q , 0 } ) Z ^ { \\Lambda _ { Q , 1 } } ( t _ i ^ - ) , \\end{align*}"}
{"id": "7824.png", "formula": "\\begin{align*} \\phi ( \\phi ^ { - 1 } ( \\gamma _ 1 ) \\cup \\phi ^ { - 1 } ( \\gamma _ 2 ) ) = \\gamma _ 1 \\star \\gamma _ 2 . \\end{align*}"}
{"id": "8214.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { r r r r } 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\end{array} \\right ) , \\end{align*}"}
{"id": "6431.png", "formula": "\\begin{align*} \\left ( \\Phi _ 0 ( x ) - \\Psi _ 0 ( x ) - [ Q , \\iota _ { \\varphi ( x ) } ] \\right ) ( f ) & = \\rho ( \\varphi ( x ) ) [ f ] - \\langle Q ( f ) , \\varphi ( x ) \\rangle \\\\ & = 0 . \\end{align*}"}
{"id": "5986.png", "formula": "\\begin{align*} E _ \\beta ( z ) = \\sum _ { m = 0 } ^ \\infty \\frac { z ^ m } { \\Gamma ( \\beta m + 1 ) } \\end{align*}"}
{"id": "2386.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\int _ { - \\infty } ^ \\infty | \\langle f , \\rho [ x , a ] g \\rangle | ^ 2 \\ , \\frac { d x \\ , d a } { a ^ 2 } = \\| f \\| _ 2 ^ { \\ , 2 } \\int _ { - \\infty } ^ \\infty \\frac { | \\widehat { g } ( \\omega ) | ^ 2 } { | \\omega | } \\ , d \\omega , f , g \\in L ^ 2 ( \\R ) . \\end{align*}"}
{"id": "6629.png", "formula": "\\begin{align*} \\sigma _ \\infty ( N , w ) & = \\lim _ { \\epsilon \\to 0 ^ + } ( 2 \\epsilon ) ^ { - 1 } \\int _ { | N ( x ) - 1 | \\le \\epsilon } w ( x ) \\ , d x \\\\ & = \\lim _ { \\epsilon \\to 0 ^ + } ( 2 \\epsilon ) ^ { - 1 } \\int _ { 1 - \\epsilon } ^ { 1 + \\epsilon } \\left ( \\int _ { N ^ { - 1 } ( y ) } w \\ , d \\tau ^ { ( y ) } _ \\infty \\right ) \\ , d y = \\int _ { N ^ { - 1 } ( 1 ) } w \\ , d \\tau ^ { ( 1 ) } _ \\infty , \\end{align*}"}
{"id": "8038.png", "formula": "\\begin{align*} s = - \\frac { H } { z } - \\frac { m } { 3 } , m \\geq 1 \\end{align*}"}
{"id": "3290.png", "formula": "\\begin{align*} \\int _ B \\left ( Q _ { A , q } v _ 1 u _ { - A , q } - Q _ { - A , q } v _ 2 \\right ) d x = \\int _ B \\left ( Q _ { A , q } v _ 1 v _ 2 - Q _ { - A , q } v _ 2 v _ 1 \\right ) d x . \\end{align*}"}
{"id": "8634.png", "formula": "\\begin{align*} \\widehat { H } _ { N H } ( t ) : = \\frac 1 2 \\sum _ { i = 1 } ^ n \\widehat { p } _ i ^ 2 + \\omega ( t ) \\frac 1 2 \\sum _ { i = 1 } ^ n \\widehat { x } ^ 2 _ i + \\tilde { \\omega } ( t ) \\sum _ { i = 1 } ^ n \\widehat { x } _ i ^ \\alpha , \\end{align*}"}
{"id": "2572.png", "formula": "\\begin{align*} \\mathbf S _ { 2 3 } = \\dfrac { 1 } { 2 } ( C _ 3 ^ { 2 1 1 } + 3 C _ 3 ^ { 1 2 2 } + 2 C _ 3 ^ { ( 2 ) } - C _ 2 ^ { 1 2 } ) \\ . \\end{align*}"}
{"id": "1194.png", "formula": "\\begin{align*} & \\varphi ( x ) - \\varphi ( y ) = - \\varphi ( y ) \\\\ & \\le \\langle \\nabla \\varphi ( y ) , x - y \\rangle + \\left ( ( \\| \\varphi \\| _ { \\infty } + \\| \\nabla \\varphi \\| _ { \\infty } ) \\bigvee C _ 1 C _ 4 ^ { 2 - p } \\right ) | x - y | ^ p , \\ \\forall x \\in S ^ c , y \\in S \\end{align*}"}
{"id": "4139.png", "formula": "\\begin{align*} | A | = q ^ { n \\left ( 1 - \\frac { 1 } { k ^ { 2 } } \\right ) } . \\end{align*}"}
{"id": "1171.png", "formula": "\\begin{align*} \\hat { \\rho } _ { n , 1 } ^ B = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { Z _ i ^ B } \\hat { \\rho } _ { n , 2 } ^ B = \\frac { 1 } { n } \\sum _ { i = n + 1 } ^ { 2 n } \\delta _ { Z _ i ^ B } . \\end{align*}"}
{"id": "8843.png", "formula": "\\begin{align*} \\int _ { G } ^ { } f \\circ ( \\phi _ 1 * \\phi _ 2 ) ( g ) d g & = \\lim _ { n \\to \\infty } \\int _ { G } ^ { } f \\circ ( \\phi _ { 1 , n } * \\phi _ { 2 , n } ) ( g ) d g , \\\\ \\int _ { \\mathbb { R } } ^ { } f \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) ( x ) d x & = \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } } ^ { } f \\circ ( \\phi _ { 1 , n } ^ * * \\phi _ { 2 , n } ^ * ) ( x ) d x \\end{align*}"}
{"id": "4264.png", "formula": "\\begin{align*} \\min \\limits _ { i = 2 , \\dots , d \\atop h , \\ell = 1 , \\dots , k } \\left | \\xi _ h - \\eta _ { i \\ell } \\right | = \\left | \\xi _ 1 - \\eta _ { 2 1 } \\right | = 2 \\sin { \\pi \\over d k } . \\end{align*}"}
{"id": "132.png", "formula": "\\begin{align*} C \\star ( { \\bf 1 } ^ \\epsilon _ 0 \\| C ^ 3 \\| _ { L ^ 1 } - { \\bf 1 } ^ \\epsilon _ 0 \\| C _ \\infty ^ 3 \\| _ { L ^ 1 } ) \\star S & = \\big ( \\| C ^ 3 \\| _ { L ^ 1 } - \\| C ^ 3 _ \\infty \\| _ { L ^ 1 } \\big ) C \\star S \\\\ & = : - \\gamma _ t C \\star S , \\end{align*}"}
{"id": "2413.png", "formula": "\\begin{align*} & \\quad \\langle x - y , - ( - A _ n ) ^ { \\frac 1 2 } F _ n ( ( - A _ n ) ^ { \\frac 1 2 } x ) + ( - A _ n ) ^ { \\frac 1 2 } F _ n ( ( - A _ n ) ^ { \\frac 1 2 } y ) \\rangle \\\\ & = - \\langle ( - A _ n ) ^ { \\frac 1 2 } x - ( - A _ n ) ^ { \\frac 1 2 } y , F _ n ( ( - A _ n ) ^ { \\frac 1 2 } x ) - F _ n ( ( - A _ n ) ^ { \\frac 1 2 } y ) \\rangle \\\\ & \\le \\| ( - A _ n ) ^ { \\frac 1 2 } ( x - y ) \\| ^ 2 \\le ( n - 1 ) ^ 2 \\| x - y \\| ^ 2 , \\end{align*}"}
{"id": "1391.png", "formula": "\\begin{align*} ( i \\partial _ t + \\partial _ x ^ 2 + i \\gamma ) G = \\frac { p + 1 } { 2 } \\| v \\| _ { L _ x ^ { p - 1 } } ^ { p - 1 } G + F , \\end{align*}"}
{"id": "8842.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { G } ^ { } f _ { ( n ) } \\circ ( \\phi _ 1 * \\phi _ 2 ) ( g ) d g & = \\int _ { G } ^ { } f \\circ ( \\phi _ 1 * \\phi _ 2 ) ( g ) d g , \\\\ \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } } ^ { } f _ { ( n ) } \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) ( x ) d x & = \\int _ { \\mathbb { R } } ^ { } f \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) ( x ) d x \\end{align*}"}
{"id": "6003.png", "formula": "\\begin{align*} y ' = s y + g y ( 0 ) = 0 . \\end{align*}"}
{"id": "4853.png", "formula": "\\begin{align*} ( X , \\alpha ) = ( - 1 ) ^ p \\int _ M ( \\iota _ X \\alpha ) \\mu . \\end{align*}"}
{"id": "1258.png", "formula": "\\begin{align*} \\sigma ^ { * } ( T ) & = \\bigcup _ { j \\in \\Phi } \\{ x \\in \\mathbb { R } \\colon W _ j ( x ) = 0 \\} . \\end{align*}"}
{"id": "978.png", "formula": "\\begin{align*} x g ( x ) & = g _ { 0 } x + \\cdots + g _ { n - 1 } x ^ { n } \\\\ & = g _ { n - 1 } \\phi _ { 0 } + ( g _ { 0 } + \\phi _ { 1 } g _ { n - 1 } ) x + \\cdots + ( g _ { n - 2 } + \\phi _ { n - 1 } g _ { n - 1 } ) x ^ { n - 1 } . \\end{align*}"}
{"id": "2197.png", "formula": "\\begin{align*} & ( \\underbrace { + \\dots + } _ { a _ { i _ { s } j } } , \\ \\underbrace { - \\dots - } _ { a _ { i _ { s } + 1 \\ , j + 1 } } , \\ \\cdots , \\ \\underbrace { + \\dots + } _ { a _ { u - 1 \\ , j } } , \\ \\underbrace { - \\dots - } _ { a _ { u \\ , j + 1 } } ) . \\end{align*}"}
{"id": "6349.png", "formula": "\\begin{align*} \\int _ { \\R ^ { 2 N } \\setminus ( C \\Omega ) ^ 2 } \\varPhi _ { x , y } \\left ( \\dfrac { | u ( x ) - u ( y ) | } { | x - y | ^ s } \\right ) \\dfrac { d x d y } { | x - y | ^ N } = 0 . \\end{align*}"}
{"id": "1100.png", "formula": "\\begin{align*} \\frac { 1 } { t ^ 2 } \\left \\{ G ( V , V ) G ( U , U ) - G ( U , V ) ^ 2 \\right \\} - G ( \\mbox { H o r } ( P _ V V ) , \\mbox { H o r } ( P _ U U ) ) + G ( \\mbox { H o r } ( P _ V U ) , \\mbox { H o r } ( P _ V U ) ) = 0 . \\end{align*}"}
{"id": "8542.png", "formula": "\\begin{align*} \\mathbf { u } _ { k + 1 } & = R \\mathbf { u } _ k + a _ { k + 1 } \\mathbf { v } _ k + a _ { k + 1 } ^ 2 \\mathbf { u } _ { k - 1 } \\\\ \\mathbf { v } _ { k + 1 } & = ( 2 R + 1 ) \\mathbf { u } _ k + a _ { k + 1 } \\mathbf { v } _ k + a _ { k + 1 } \\mathbf { w } _ k \\\\ \\mathbf { w } _ { k + 1 } & = - \\mathbf { u } _ k - a _ { k + 1 } \\mathbf { w } _ k , \\end{align*}"}
{"id": "6224.png", "formula": "\\begin{align*} & \\int ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty \\left ( ( c q + x ) [ n ] _ q - x \\right ) h _ n ( x ; q ) d _ q x = \\\\ & q ^ { n - 1 } ( 1 - q ) ( x ^ 2 ; q ^ 2 ) _ \\infty \\left ( q h _ n ( \\frac { x } { q } ; q ) - [ n ] _ q ( c q + x ) h _ { n - 1 } ( \\frac { x } { q } ; q ) \\right ) , \\end{align*}"}
{"id": "7422.png", "formula": "\\begin{align*} \\begin{array} { c } v ( 0 , t ) = \\alpha v _ x ( L , t ) - \\gamma \\beta p _ x ( L , t ) = 0 , \\\\ p ( 0 , t ) = \\beta p _ x ( L , t ) - \\gamma \\beta v _ x ( L , t ) = - \\displaystyle { \\frac { V ( t ) } { h } } . \\end{array} \\end{align*}"}
{"id": "5146.png", "formula": "\\begin{align*} \\mathcal { G } ( 1 0 c + 1 ) \\oplus \\mathcal { G } ( 1 ) & = 6 = \\mathcal { G } ( 8 c + 1 ) \\oplus \\mathcal { G } ( 2 c + 1 ) \\\\ \\mathcal { G } ( 1 0 c + 1 ) \\oplus \\mathcal { G } ( 4 c + 1 ) & = 7 = \\mathcal { G } ( 8 c + 1 ) \\oplus \\mathcal { G } ( 6 c + 1 ) \\end{align*}"}
{"id": "6545.png", "formula": "\\begin{align*} r _ n = \\min \\{ x : x \\in \\{ x _ j \\} _ { j \\in \\Z _ + } , x \\geq n \\} . \\end{align*}"}
{"id": "4835.png", "formula": "\\begin{align*} R _ \\mathrm { r a t } ( G ) = R _ \\mathrm { r a t } ( G ' ) \\otimes R _ \\mathrm { r a t } ( G '' ) = \\mathcal { S } ( G ' ) \\otimes \\mathcal { S } ( G '' ) \\subseteq \\mathcal { S } ( G ) \\ ; . \\end{align*}"}
{"id": "3881.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ p \\hat { u } _ n - a _ n | \\hat { u } _ n | ^ { p - 2 } \\hat { u } _ n = \\hat { f } _ n \\qquad & \\Omega \\\\ \\hat { u } _ n = \\hat { g } _ n & \\partial \\Omega \\end{cases} \\end{align*}"}
{"id": "3045.png", "formula": "\\begin{align*} \\Omega _ { \\bullet } ( q ) : = & \\sum _ { \\lambda \\in \\mathcal { P } } q ^ { | \\lambda | ( 1 - g ) } ( q ^ { - n ( \\lambda ) } H _ { \\lambda } ( q ) ) ^ { 2 g + k - 2 } \\prod ^ k _ { j = 1 } s _ { \\lambda } [ \\frac { \\Z _ j } { 1 - q } ] \\\\ \\Omega _ { \\ast } ( q ) : = & \\sum _ { \\lambda \\in \\mathcal { P } } q ^ { | \\lambda | ( 2 - 2 g ) } ( q ^ { - n ( \\lambda ) } H _ { \\lambda } ( q ) ) ^ { 4 g + 2 k - 4 } \\prod ^ k _ { j = 1 } s _ { \\lambda } [ \\frac { \\Z _ j } { 1 - q } ] ^ 2 , \\end{align*}"}
{"id": "6162.png", "formula": "\\begin{align*} K _ { M } ^ { T } = \\pi ^ { \\ast } ( K _ { Z } ^ { o r b } ) = \\pi ^ { \\ast } ( \\varphi ^ { \\ast } ( K _ { Z } + [ \\Delta ] ) ) . \\end{align*}"}
{"id": "7894.png", "formula": "\\begin{align*} \\tilde { T } _ { \\vec s ^ 1 , k _ 1 } \\star \\tilde { T } _ { \\vec s ^ 2 , k _ 2 } = \\sum _ { \\vec s ^ 3 , k _ 3 } \\frac { \\partial ^ 3 \\Phi _ 0 } { \\partial t _ { \\vec s ^ 1 , k _ 1 } \\partial t _ { \\vec s ^ 2 , k _ 2 } \\partial t _ { \\vec s ^ 3 , k _ 3 } } \\tilde { T } _ { - \\vec s ^ 3 } ^ { k _ 3 } . \\end{align*}"}
{"id": "7842.png", "formula": "\\begin{align*} y \\cdot \\overline \\xi \\cdot x = \\overline { x ^ * \\xi y ^ * } , \\end{align*}"}
{"id": "8784.png", "formula": "\\begin{align*} \\| \\phi _ 1 * \\phi _ 2 \\| _ { q ( p _ 1 , p _ 2 ) } \\leq \\| \\phi _ 1 ^ * * \\phi _ 2 ^ * \\| _ { q ( p _ 1 , p _ 2 ) } \\leq C ( p _ 1 , p _ 2 ) \\| \\phi _ 1 ^ * \\| _ { p _ 1 } \\| \\phi _ 2 ^ * \\| _ { p _ 2 } = C ( p _ 1 , p _ 2 ) \\| \\phi _ 1 \\| _ { p _ 1 } \\| \\phi _ 2 \\| _ { p _ 2 } \\end{align*}"}
{"id": "364.png", "formula": "\\begin{align*} \\delta _ { J } \\geq D n ^ { - \\sqrt { b ( n ) } / \\log n } = D e ^ { - \\sqrt { b ( n ) } } . \\end{align*}"}
{"id": "7250.png", "formula": "\\begin{align*} \\int _ { \\Gamma } a _ j ( \\gamma ) d { \\gamma _ j } _ * \\mu ( \\gamma ) = 0 . \\end{align*}"}
{"id": "7243.png", "formula": "\\begin{align*} \\begin{aligned} ( \\rho _ \\alpha - \\alpha ( k + d - 2 ) ) x _ { v } & \\leq ( \\rho _ \\alpha - \\alpha d ( v ) ) x _ { v } = ( 1 - \\alpha ) ( k - 1 ) x _ { u _ 1 } + ( 1 - \\alpha ) \\sum _ { u v \\in E ( H ) } x _ u \\\\ & \\leq ( 1 - \\alpha ) ( k - 1 ) x _ { u _ 1 } + ( 1 - \\alpha ) ( d - 1 ) x _ v , \\end{aligned} \\end{align*}"}
{"id": "2376.png", "formula": "\\begin{align*} \\nu \\left ( \\sum _ { j \\in J } \\partial _ j L ( h _ \\sigma ) ( h _ \\rho - h _ \\sigma ) ^ j \\right ) \\geq \\min _ { j \\in J } \\{ \\beta _ j + j \\gamma _ \\sigma \\} > \\min _ { i \\in I } \\{ \\beta _ i + i \\gamma _ \\rho \\} = \\beta _ b + i \\gamma _ b . \\end{align*}"}
{"id": "8584.png", "formula": "\\begin{align*} g ( x ) & = \\prod _ { j = 1 } ^ n ( 1 + r _ j ( t _ 0 ) \\cos 2 \\pi n _ j x ) \\\\ & = 1 + \\sum _ { A \\subset \\{ 1 , 2 , \\ldots , n \\} } \\left ( \\prod _ { j \\in A } r _ j ( t _ 0 ) \\cos 2 \\pi n _ j x \\right ) \\\\ & = 1 + \\sum _ { A \\subset \\{ 1 , 2 , \\ldots , n \\} } \\left ( w _ A ( t _ 0 ) \\prod _ { j \\in A } \\cos 2 \\pi n _ j x \\right ) , \\end{align*}"}
{"id": "4896.png", "formula": "\\begin{align*} \\sum \\limits _ { l \\ , | \\ , i _ l = i } { t _ i } = 1 , p _ { i j } = \\sum \\limits _ { l < m \\ , | \\ , i _ l = i , i _ m = j } { t _ l t _ m } . \\end{align*}"}
{"id": "3695.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } \\frac { i } h \\int _ { \\Omega _ \\Gamma } [ - h ^ 2 \\Delta + V ( x ) - E ( h ) , \\chi _ \\alpha ( x _ n ) h D _ n ] u _ h \\overline { u _ h } d x = o ( 1 ) . \\end{align*}"}
{"id": "3212.png", "formula": "\\begin{align*} & { \\rm P r } ( s _ { t + 1 } | s _ t , a _ { \\pi } ( s _ t ) ) = \\Big ( \\prod _ { k = 1 } ^ K { \\rm P r } ( h _ { k , t + 1 } ) \\prod _ { i = 1 } ^ n { \\rm P r } ( \\beta _ { k , t + 1 , i } ) \\Big ) \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\times { \\rm P r } ( q ^ { } _ { t + 1 } , \\{ q _ { k , t + 1 } ^ { } , \\forall k \\} | s _ t , a _ { \\pi } ( s _ t ) ) . \\end{align*}"}
{"id": "5481.png", "formula": "\\begin{align*} S _ { R , C } ( x , y ) = \\mathrm { C o n e } \\left ( x , \\{ z \\in \\partial _ \\infty M : \\mathrm { d i s t } ( x , [ y , z ] ) \\geq R \\} \\right ) \\cap B ( x , R + C ) \\setminus B ( x , R ) . \\end{align*}"}
{"id": "861.png", "formula": "\\begin{align*} \\nu _ { f _ { m } ^ { r } \\rightarrow x _ { m } ^ { r } } ( s _ { m } ^ { r } ) = & \\underset { s _ { m } ^ { r } } { \\sum } \\nu _ { \\eta _ { m } ^ { r } \\rightarrow s _ { m } ^ { r } } ( s _ { m } ^ { r } ) \\times f _ { m } ^ { r } ( x _ { m } ^ { r } , s _ { m } ^ { r } ) \\\\ = & \\pi _ { s ^ { r } , m } ^ { o u t } \\mathcal { C N } ( x _ { m } ^ { r } ; 0 , ( \\sigma _ { m } ^ { r } ) ^ { 2 } ) + ( 1 - \\pi _ { s ^ { r } , m } ^ { o u t } ) \\delta ( x _ { m } ^ { r } ) . \\end{align*}"}
{"id": "8087.png", "formula": "\\begin{align*} \\nabla _ { Z } Z = f ^ { 2 } \\nabla _ { \\partial _ { \\theta } } \\partial _ { \\theta } = - f ^ { 2 } c _ { 1 } \\partial _ r c _ { 1 } \\partial _ { 2 } \\end{align*}"}
{"id": "6406.png", "formula": "\\begin{align*} \\Bigl ( \\ , \\sum _ { j = 1 } ^ d \\ w _ j ^ { | R _ 1 \\cap B _ { \\pi } | } \\ , \\Bigr ) \\ , \\cdots \\ , \\Bigl ( \\ , \\sum _ { j = 1 } ^ d w _ j ^ { | R _ p \\cap B _ { \\pi } | } \\ , \\Bigr ) . \\end{align*}"}
{"id": "158.png", "formula": "\\begin{align*} \\chi ^ { \\epsilon , L } _ t \\leq \\frac { 1 } { 1 + \\frac { 1 } { t } } + p _ { d , t , \\mu , m ^ 2 = 1 } ( \\lambda ) , \\int _ { 0 } ^ { t _ 0 } \\frac { p _ { d , t , \\mu , m ^ 2 = 1 } ( \\lambda ) } { t ^ 2 } \\ , d t \\leq c _ d ( \\lambda , \\mu , m ^ 2 = 1 ) , \\end{align*}"}
{"id": "2788.png", "formula": "\\begin{align*} P : = \\left [ \\begin{array} { l l l l l } g _ 0 & g _ 1 & \\dots & g _ N & x _ 0 \\end{array} \\right ] \\end{align*}"}
{"id": "1721.png", "formula": "\\begin{align*} B _ 0 ( t , q ^ { \\frac { 1 } { 2 } } ) \\cdot B _ 0 ( - t , q ^ { \\frac { 1 } { 2 } } ) = \\prod _ { n \\geq 0 } \\big ( 1 - x y ^ { n } \\big ) \\cdot \\prod _ { n \\geq 1 } \\big ( 1 - x ^ { - 1 } y ^ { n } ) ^ { - 1 } , \\end{align*}"}
{"id": "2313.png", "formula": "\\begin{align*} \\vec { v } \\left ( { \\bf s ' } _ { \\alpha , 0 } \\right ) = \\left ( 1 - \\alpha , \\alpha , 1 \\right ) \\vec { v } \\left ( { \\bf s } _ { \\alpha , 0 } \\right ) = \\left ( 1 - \\alpha , \\alpha , 0 \\right ) . \\end{align*}"}
{"id": "5261.png", "formula": "\\begin{align*} Q = \\{ ( I _ 0 , k _ 1 ( 0 ) + 1 , k _ 2 ( 0 ) + 1 ) , ( I _ 1 , k _ 1 ( 1 ) , k _ 2 ( 1 ) ) \\} \\end{align*}"}
{"id": "4631.png", "formula": "\\begin{align*} \\phi \\tilde c _ 1 ( \\alpha , g ) = \\tilde c _ 1 ( \\alpha , h ) \\ ; \\ ; \\ ; \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ ; \\ ; \\ ; \\phi \\tilde c _ 2 ( \\alpha , g ) = \\tilde c _ 2 ( \\alpha , h ) . \\end{align*}"}
{"id": "1267.png", "formula": "\\begin{align*} f _ j & = \\begin{cases} 2 \\csc \\left ( \\dfrac { \\pi } { 2 j + 4 } \\right ) - 2 \\cot \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) , & 2 \\nmid j , \\\\ 2 \\cot \\left ( \\dfrac { \\pi } { 2 j + 4 } \\right ) - 2 \\csc \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) , & 2 \\mid j . \\end{cases} \\end{align*}"}
{"id": "5678.png", "formula": "\\begin{align*} ( x _ j - x _ { j + 1 } ) ( \\varpi _ { j - 1 } + x _ j ) = 0 ( 1 \\le j \\le n - 1 ) . \\end{align*}"}
{"id": "5447.png", "formula": "\\begin{align*} m = l ( m ) r ( m ) = l ( m ' ) r ( m ' ) = m ' . \\end{align*}"}
{"id": "5078.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int ^ { \\tau _ 2 } _ { \\tau _ 1 } \\gamma _ s M ^ { n , 1 } _ s M ^ { n , 2 } _ s d s = 0 , \\end{align*}"}
{"id": "5313.png", "formula": "\\begin{align*} \\psi _ \\Lambda ( u ) = \\Lambda ( \\{ 0 \\} ) u ^ 2 + \\int _ { ( 0 , 1 ] } \\left ( e ^ { - u x } - 1 + u x \\right ) x ^ { - 2 } \\Lambda ( d x ) . \\end{align*}"}
{"id": "5693.png", "formula": "\\begin{align*} \\begin{cases} m _ { d + 1 , 1 ^ { k - 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } + m _ { d , 1 ^ k } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d \\ge 2 ) , \\\\ \\\\ m _ { 2 , 1 ^ { k - 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } + ( k + 1 ) m _ { 1 ^ { k + 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d = 1 ) . \\end{cases} \\end{align*}"}
{"id": "7911.png", "formula": "\\begin{align*} f _ + ^ * ( K _ { X _ + } ) = f _ - ^ * ( K _ { X _ - } ) . \\end{align*}"}
{"id": "3248.png", "formula": "\\begin{align*} \\partial _ { x _ i } \\partial _ { x _ j } \\bigl ( a ^ { i j } \\varrho \\bigr ) - \\partial _ { x _ i } \\bigl ( b ^ i \\varrho \\bigr ) + c \\varrho = 0 \\end{align*}"}
{"id": "2621.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ p a _ j ( j + 1 ) = 1 \\ . \\end{align*}"}
{"id": "7338.png", "formula": "\\begin{align*} [ x y , z ] = - [ y z , x ] - [ z x , y ] \\mbox { i f } \\mathsf { c h a r } ( F ) = 3 ; \\end{align*}"}
{"id": "2325.png", "formula": "\\begin{align*} R = \\begin{pmatrix} A & B & 0 \\\\ C & D & 0 \\\\ E & F & 1 \\end{pmatrix} \\end{align*}"}
{"id": "4068.png", "formula": "\\begin{align*} c _ 1 ( F ^ i / F ^ { i + 1 } ) = ( 0 , a - i , 0 ) + i \\cdot ( 1 , - 1 , 0 ) = ( i , a - 2 i , 0 ) . \\end{align*}"}
{"id": "5329.png", "formula": "\\begin{align*} E = \\{ Z ( t , B _ \\epsilon ) \\geq b \\cdot p ( t , \\epsilon ) \\} . \\end{align*}"}
{"id": "2095.png", "formula": "\\begin{align*} ( a v g ( x ) , \\beta ) _ E & = \\sum \\limits _ { e \\in E } \\delta _ E ( e ) ( a v g ( x ) ) ( e ) \\beta ( e ) \\\\ & = \\sum \\limits _ { e \\in E } \\delta _ E ( e ) \\beta ( e ) \\frac { \\sum \\limits _ { v \\in e } x ( v ) } { | e | } \\\\ & = \\sum \\limits _ { v \\in V } \\delta _ V ( v ) x ( v ) \\sum \\limits _ { e \\in E _ v } \\frac { \\beta ( e ) } { | e | } \\frac { \\delta _ E ( e ) } { \\delta _ V ( v ) } \\\\ & = ( x , a v g ^ * ( \\beta ) ) _ V . \\end{align*}"}
{"id": "4494.png", "formula": "\\begin{align*} e ^ { i \\tau \\Phi ( \\xi , \\eta _ 1 , \\eta _ 2 ) } = \\frac { 1 } { i \\Phi ( \\xi , \\eta _ 1 , \\eta _ 2 ) } \\left [ \\partial _ \\tau e ^ { i \\tau \\Phi ( \\xi , \\eta _ 1 , \\eta _ 2 ) } \\right ] , \\end{align*}"}
{"id": "3372.png", "formula": "\\begin{align*} C = C ^ { ( < j ) } \\oplus C ^ { ( j ) } \\oplus C ^ { ( > j ) } . \\end{align*}"}
{"id": "8591.png", "formula": "\\begin{align*} & \\# ( z ) \\le l - 1 , \\\\ & u \\wedge v = u \\wedge z = v \\wedge z = 0 , \\\\ & 0 \\le \\# ( u ) , \\# ( v ) \\le l - \\# ( z ) , u + v > 0 . \\end{align*}"}
{"id": "5663.png", "formula": "\\begin{align*} ( h ( 1 ) , h ( 2 ) , h ( 3 ) , h ( 4 ) , h ( 5 ) ) = ( 3 , 3 , 4 , 5 , 5 ) , \\end{align*}"}
{"id": "7178.png", "formula": "\\begin{align*} \\langle u ( n ) f , a \\rangle = \\langle f , \\theta ( u ( n ) ) a \\rangle . \\end{align*}"}
{"id": "8708.png", "formula": "\\begin{align*} \\lambda = - \\frac { a ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } } { 1 + b \\dot { \\theta } \\sin ( \\theta - t ) } . \\end{align*}"}
{"id": "5387.png", "formula": "\\begin{align*} B _ q ( v , v ) & \\geq \\norm { ( - \\Delta ) ^ { s / 2 } v } _ { L ^ 2 ( \\R ^ n ) } ^ 2 + \\langle q _ { s , 1 } , v ^ 2 \\rangle + \\langle q _ { s , 2 } , v ^ 2 \\rangle \\\\ & \\geq \\norm { ( - \\Delta ) ^ { s / 2 } v } _ { L ^ 2 ( \\R ^ n ) } ^ 2 + \\langle q _ { s , 1 } , v ^ 2 \\rangle - \\abs { \\langle q _ { s , 2 } , v ^ 2 \\rangle } \\\\ & \\geq \\left ( \\delta ( \\Omega ) - \\norm { q _ { s , 2 } } _ { s , - s } \\right ) \\norm { v } _ { H ^ s ( \\R ^ n ) } ^ 2 + \\langle q _ { s , 1 } , v ^ 2 \\rangle \\end{align*}"}
{"id": "4343.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ k \\sum _ { j = 0 } ^ { k - l } ( - 1 ) ^ \\dagger m _ { k + 1 - l } ( p _ k , \\ldots , p _ { j + l + 1 } , m _ l ( p _ { j + l } , \\ldots p _ { j + 1 } ) , p _ j , \\ldots p _ 1 ) = 0 . \\end{align*}"}
{"id": "4037.png", "formula": "\\begin{align*} \\mathrm { A r e a } ( \\mathbb { S } ^ { n - 1 } ) = \\frac { 2 \\pi ^ { \\frac { n } { 2 } } } { \\Gamma ( \\frac { n } { 2 } ) } , \\end{align*}"}
{"id": "5463.png", "formula": "\\begin{align*} m A = l ( m ) r ( m ) A & \\subseteq l ( m ) A r ( m ) \\\\ & = A r ( m ) = ( A l ( m ) ^ { - 1 } ) l ( m ) r ( m ) = A m . \\end{align*}"}
{"id": "6125.png", "formula": "\\begin{align*} \\left | \\sum _ { 1 \\leq i _ 1 < i _ 2 < \\cdots < i _ { n - m } \\leq n - 1 } \\omega _ { i _ 1 } \\omega _ { i _ 2 } \\cdots \\omega _ { i _ { n - m } } \\right | = \\left | \\omega _ m \\omega _ { m + 1 } \\cdots \\omega _ { n - 1 } \\right | \\geq \\left | \\omega _ m \\right | ^ { n - m } . \\end{align*}"}
{"id": "6001.png", "formula": "\\begin{align*} u ( t ) = \\frac { d ^ m } { d t ^ m } \\int _ 0 ^ t k _ m ( t - \\tau ) g ( \\tau ) d \\tau , \\end{align*}"}
{"id": "8230.png", "formula": "\\begin{align*} k _ { \\pm } ^ { ' 2 } = \\frac { 1 - \\sqrt { 1 \\pm 1 6 i m \\beta \\lambda / 3 } } { 4 \\beta \\hslash ^ { 2 } / 3 } . \\end{align*}"}
{"id": "7134.png", "formula": "\\begin{align*} q _ c + q _ { c + 1 } + \\cdots + q _ { d - 1 } = \\sum _ { \\{ i , c \\} \\in E ( \\mathsf { P } _ { n + 2 } ) \\atop b \\leq i < c } x _ { i c } - \\sum _ { \\{ j , d \\} \\in E ( \\mathsf { P } _ { n + 2 } ) \\atop c \\leq j < d } x _ { j d } \\end{align*}"}
{"id": "6022.png", "formula": "\\begin{align*} f ( x ) = x ^ 3 - \\frac { 3 } { 2 } x ^ 2 = ( x - 1 ) ^ 3 + \\frac { 3 } { 2 } ( x - 1 ) ^ 2 - \\frac { 1 } { 2 } \\end{align*}"}
{"id": "127.png", "formula": "\\begin{align*} \\| T _ { \\lambda } \\| _ { L ^ 1 \\cap L ^ \\infty } \\leq c \\lambda \\big ( \\eta _ t + \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } ) \\Big [ \\frac { \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } } { m ^ 2 _ t } + \\frac { 1 } { m ^ 4 _ t } + \\frac { 1 } { m ^ 2 _ t } { \\bf 1 } _ { d = 2 } + \\frac { 1 } { m _ t } { \\bf 1 } _ { d = 3 } \\Big ] . \\end{align*}"}
{"id": "860.png", "formula": "\\begin{align*} \\nu _ { \\eta _ { m } ^ { r } \\rightarrow s _ { m } ^ { r } } ( s _ { m } ^ { r } ) \\propto & \\underset { s _ { m } } { \\sum } \\eta _ { m } ^ { r } ( s _ { m } ^ { r } , s _ { m } ) \\times \\nu _ { s _ { m } \\rightarrow \\eta _ { m } ^ { r } } ( s _ { m } ) \\\\ = & \\pi _ { s ^ { r } , m } ^ { o u t } \\delta ( s _ { m } ^ { r } - 1 ) + ( 1 - \\pi _ { s ^ { r } , m } ^ { o u t } ) \\delta ( s _ { m } ^ { r } ) , \\end{align*}"}
{"id": "3481.png", "formula": "\\begin{align*} c _ { 1 } = c _ { ( 1 ) } ^ { \\ast } = c _ { ( 2 ) } ^ { \\ast } = ( 2 \\pi ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "2653.png", "formula": "\\begin{align*} g _ { n } ( p ^ { n + 1 } ) & = \\frac { 1 } { 2 ( p - 1 ) ^ 2 ( p + 1 ) } \\bigg ( 2 p ^ { 2 n - 3 } + ( n ^ 2 - n ) p ^ { n + 1 } \\\\ & - ( n ^ 2 - n ) p ^ n - ( n ^ 2 - n + 2 ) p ^ { n - 1 } + ( n ^ 2 - n - 2 ) p ^ { n - 2 } + 2 \\bigg ) \\end{align*}"}
{"id": "3255.png", "formula": "\\begin{align*} \\mathcal { H } _ { A + \\nabla \\varphi , q } \\tilde { u } : = - ( \\nabla + i ( A + \\nabla \\varphi ) ) ^ 2 \\tilde { u } + q ( x ) \\tilde { u } = e ^ { - i \\varphi ( x ) } \\mathcal { H } _ { A , q } u . \\end{align*}"}
{"id": "1974.png", "formula": "\\begin{align*} { v ' } ^ + = \\phi ^ + \\mbox { o n $ \\partial D $ } \\end{align*}"}
{"id": "4933.png", "formula": "\\begin{align*} u ^ * ( \\xi , t ) = \\frac { 1 } { \\rho ( \\xi , t ) } . \\end{align*}"}
{"id": "5559.png", "formula": "\\begin{align*} \\delta W _ + : = - \\nabla \\cdot W _ + = 0 . \\end{align*}"}
{"id": "5012.png", "formula": "\\begin{align*} R ^ { n , 1 } _ \\tau & : = n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) ) ^ 2 \\Psi ^ { n , 1 } _ s d s \\\\ & = n ^ { 2 \\alpha + 1 } \\int _ 0 ^ \\tau \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) ^ 2 d s \\\\ & = : R ^ { n , 2 } _ \\tau + R ^ { n , 3 } _ \\tau , \\end{align*}"}
{"id": "3392.png", "formula": "\\begin{align*} \\mu ( z ) & : = \\frac { 1 } { 2 } \\Bigl [ b ( g ( z ) ) \\Delta _ { \\Sigma } g ( z ) + b ' ( g ( z ) ) \\| \\Sigma ^ { 1 / 2 } ( z ) \\nabla g ( z ) \\| _ { d + 1 } ^ 2 \\Bigr ] , \\end{align*}"}
{"id": "3682.png", "formula": "\\begin{align*} \\beta ( s , r ) = \\varphi ( r ) \\log \\cos r + ( 1 - \\varphi ( r ) ) f ( s , 0 ) , \\end{align*}"}
{"id": "5782.png", "formula": "\\begin{align*} G = \\left \\{ \\begin{pmatrix} a & x \\\\ 0 & 1 \\end{pmatrix} : x \\in \\R , a = \\pm 1 \\right \\} \\end{align*}"}
{"id": "4013.png", "formula": "\\begin{align*} \\delta x = \\frac { \\partial f } { \\partial x } \\bigg | _ { ( x _ 0 , u _ 0 ) } \\delta x + \\frac { \\partial f } { \\partial u } \\bigg | _ { ( x _ 0 , u _ 0 ) } \\delta u . \\end{align*}"}
{"id": "4301.png", "formula": "\\begin{align*} ( e ^ { - i \\hat { \\theta } } ( \\omega _ { X ^ \\vee } + \\sqrt { - 1 } \\alpha _ \\phi ) ^ n ) = 0 , [ \\alpha _ \\phi = \\alpha + \\sqrt { - 1 } \\partial \\bar { \\partial } \\phi ] \\in - c _ 1 ( E ) . \\end{align*}"}
{"id": "987.png", "formula": "\\begin{align*} \\tau ( A ) = L \\left ( H ^ { * } ( \\overline { \\alpha } ) \\right ) + L \\left ( H ^ { * } ( \\overline { \\beta } ) \\right ) . \\end{align*}"}
{"id": "3999.png", "formula": "\\begin{align*} \\tilde L = C _ 0 \\oplus L , ~ ~ ~ ~ ~ \\tilde L ^ * = \\{ c \\in \\tilde L \\ , | \\ , \\ell _ c = \\ell \\} . \\end{align*}"}
{"id": "8814.png", "formula": "\\begin{align*} J _ 1 ( t ) : = \\frac { \\mu ( L _ 1 ( t ) ) } { 2 } , J _ 2 ( t ) : = \\frac { \\mu ( L _ 2 ( t ) ) } { 2 } \\end{align*}"}
{"id": "555.png", "formula": "\\begin{align*} \\mathbb { E } ( \\overline { S } _ n ) = \\frac { 1 } { n } \\sum _ { 1 \\leq i \\leq n } \\mathbb { E } ( X _ i ) . \\end{align*}"}
{"id": "3784.png", "formula": "\\begin{align*} \\lesssim \\| \\mathfrak { m } ( \\cdot , \\zeta ) \\| _ { \\mathcal { S } ^ \\infty } 2 ^ { m / 4 + k / 2 + n + \\max \\{ - j , p , n + \\epsilon M _ t \\} / 2 - p / 2 + \\max \\{ l , p \\} + 8 \\epsilon M _ t } \\big ( 2 ^ { \\min \\{ n , p \\} + j } \\mathbf { 1 } _ { l = - j , l > n + \\epsilon M _ t } \\end{align*}"}
{"id": "3736.png", "formula": "\\begin{align*} c _ p : = ( \\hat { v } _ 2 \\delta _ { 3 p } - \\hat { v } _ 3 \\delta _ { 2 p } ) - \\frac { \\hat { v } _ 2 \\omega _ 3 - \\hat { v } _ 3 \\omega _ 2 } { 1 + \\hat { v } \\cdot \\omega } \\hat { v } _ p . \\end{align*}"}
{"id": "8243.png", "formula": "\\begin{align*} k ' _ { \\pm } = \\frac { 1 } { \\sqrt { 4 \\beta \\hslash ^ { 2 } / 3 } } \\left ( \\sqrt { \\frac { \\sqrt { ( 1 - | a | ) ^ { 2 } + | b | ^ { 2 } } + ( 1 - | a | ) } { 2 } } \\mp i \\sqrt { \\frac { \\sqrt { ( 1 - | a | ) ^ { 2 } + | b | ^ { 2 } } - ( 1 - | a | ) } { 2 } } \\right ) . \\end{align*}"}
{"id": "3107.png", "formula": "\\begin{align*} a _ { \\mathrm { p w } } ( z _ { \\mathrm { n c } } , v _ { \\mathrm { n c } } ) = ( \\lambda u , v _ { \\mathrm { n c } } ) _ { 1 + \\delta } \\quad v _ { \\mathrm { n c } } \\in V ( { \\mathcal { T } } ) . \\end{align*}"}
{"id": "3747.png", "formula": "\\begin{align*} I _ { j ; l ; 2 } ^ { m ; p , q } ( t , x ) : = \\int _ { 0 } ^ { t } \\int _ { \\R ^ 3 } \\int _ { 0 } ^ { 2 \\pi } \\int _ 0 ^ \\pi 2 ^ { m - j - l + \\max \\{ l , p \\} } \\sum _ { \\begin{subarray} { c } \\tilde { j } \\in [ 0 , ( 1 + 2 \\epsilon ) M _ t ) \\cap \\Z _ + , \\tilde { l } \\in [ - \\tilde { j } , 2 ] \\cap \\Z \\\\ \\tilde { m } \\in [ - 1 0 M _ t , \\epsilon M _ t ] \\cap \\Z \\end{subarray} } | B _ { S ; \\tilde { j } , \\tilde { l } } ^ { \\tilde { m } ; 1 } ( s , x + ( t - s ) \\omega ) | f ( s , x + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "7149.png", "formula": "\\begin{align*} a g _ { 1 1 } g _ { 1 2 } ^ { a - 1 } & = ( a - 1 ) g _ { 1 2 } ^ a , \\\\ a g _ { 1 1 } ^ { a - 1 } g _ { 1 2 } & = g _ { 1 2 } ^ a . \\end{align*}"}
{"id": "580.png", "formula": "\\begin{align*} M _ 4 = \\sum _ { \\substack { 0 \\leq m _ 2 \\leq j - i - 1 \\\\ \\gcd ( l + m _ 1 + m _ 2 , j ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } \\\\ \\gcd ( l + m _ 1 + m _ 2 + 1 , j + 1 ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } C _ { \\alpha } ( j - i - 1 , m _ 2 ) . \\end{align*}"}
{"id": "5176.png", "formula": "\\begin{align*} & \\prod _ { i = - 2 } ^ 2 b _ { j _ \\sigma - i } \\times P _ 2 \\Big ( I _ 5 , ( 1 / b _ { j _ \\sigma - 2 } , 1 / b _ { j _ \\sigma - 1 } , 1 / a _ m , 1 / b _ { j _ \\sigma + 1 } , 1 / b _ { j _ \\sigma + 2 } ) \\Big ) \\\\ & = \\prod _ { i = 0 } ^ 4 a _ { m - i } \\times P _ 2 \\Big ( I _ 5 , ( 1 / b _ { j _ \\sigma - 2 } , 1 / b _ { j _ \\sigma - 1 } , 1 / a _ m , 1 / b _ { j _ \\sigma + 1 } , 1 / b _ { j _ \\sigma + 2 } ) \\Big ) . \\end{align*}"}
{"id": "1996.png", "formula": "\\begin{align*} \\Delta _ n ^ { ( \\alpha ) } = \\displaystyle \\prod _ { j = 1 } ^ { n } j ^ j ( \\alpha + j ) ^ { j - 1 } \\end{align*}"}
{"id": "3131.png", "formula": "\\begin{align*} \\frac { 1 - C _ a } { 1 + C _ b } = \\frac { 1 - \\big ( ( 1 + \\alpha ) \\varkappa ^ 2 + ( 1 + 1 / \\alpha ) \\Lambda _ 1 ^ 2 ( \\epsilon _ 3 + \\widehat \\Lambda _ 3 \\varkappa ^ 2 ) \\big ) } { 1 + ( 1 + 1 / \\alpha ) \\Lambda _ 1 ^ 2 \\Lambda _ 3 } = \\theta _ 0 ( \\varkappa , \\alpha ) < 1 . \\end{align*}"}
{"id": "6745.png", "formula": "\\begin{align*} \\mu \\int _ { \\Omega } \\varepsilon S ' ( \\overline { u } , h ) : \\varepsilon p + \\nu \\int _ { \\Omega } \\mathsf { m } ' \\left ( \\varepsilon \\overline { y } ; \\varepsilon S ' ( \\overline { u } , h ) \\right ) : \\varepsilon p = \\int _ { \\Omega } h \\cdot p . \\end{align*}"}
{"id": "7767.png", "formula": "\\begin{align*} m _ { j , , r } ^ k = q ^ k _ { j , } ( \\alpha _ r ) = \\frac { 1 } { 4 } \\alpha _ r ^ 2 r = 1 , \\ldots , p , \\end{align*}"}
{"id": "4555.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 1 2 \\right ) } \\Vert _ { p } = \\mathcal { O } \\left ( \\frac { 1 } { m } \\right ) ( v ( | z _ { 1 } | ) + ( v ( | z _ { 2 } | ) ) \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } . \\end{align*}"}
{"id": "7549.png", "formula": "\\begin{align*} \\bar s ^ 0 = \\min _ { k \\in \\mathcal K } \\ \\max _ { \\Vert \\vec p _ c \\Vert ^ 2 \\le P } \\ | { \\vec { h } } _ { k } ^ { H } \\vec { p } _ { c } | ^ 2 = \\min _ { k \\in \\mathcal K } P \\Vert \\vec h _ k \\Vert ^ 2 . \\end{align*}"}
{"id": "6827.png", "formula": "\\begin{align*} p ( 0 , v , g ) = p _ { } ( v , g ) , v \\in ( 0 , V _ F ) , g \\in \\mathbb { R } . \\end{align*}"}
{"id": "8430.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { \\varepsilon \\to \\varepsilon _ 0 } \\mathop { \\sup } \\limits _ { ( x , j ) \\in K \\times S } P \\left ( { \\| { y ^ \\varepsilon \\left ( { t , s , \\xi , j } \\right ) - y ^ { \\varepsilon _ 0 } \\left ( { t , s , \\xi , j } \\right ) } \\| _ H \\ge \\eta } \\right ) = 0 . \\end{align*}"}
{"id": "2353.png", "formula": "\\begin{align*} l ( h _ \\rho ) = a _ { \\rho 0 } ( f ) + b _ 1 Q _ \\rho + \\ldots + b _ l Q _ \\rho ^ l \\end{align*}"}
{"id": "2163.png", "formula": "\\begin{align*} x + i f ( x ) = \\omega \\left ( H ( x + i f ( x ) ) \\right ) = \\omega \\left ( h ( x ) \\right ) , \\end{align*}"}
{"id": "2726.png", "formula": "\\begin{align*} \\psi _ \\mathrm { t r i a l } : = \\sum _ { s = 0 } ^ N \\sum _ { | d | \\le \\sigma _ N ^ 2 } c _ { d , s } u _ 1 ^ { \\otimes ( N - s + d ) / 2 } \\otimes _ { \\mathrm { s y m } } u _ 2 ^ { \\otimes ( N - s - d ) / 2 } \\otimes _ { \\mathrm { s y m } } \\Phi _ { \\mathrm { t r i a l } , s } . \\end{align*}"}
{"id": "6214.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { a } D _ q f ( t ) d _ q t = f ( a ) - \\lim _ { n \\rightarrow \\infty } f ( a q ^ n ) . \\end{align*}"}
{"id": "4827.png", "formula": "\\begin{align*} \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } \\tau ^ \\mathrm { a n } ( M / S , g . \\rho ) = \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } \\tau ^ \\mathrm { t o p } ( M / S , g . \\rho ) \\ ; . \\end{align*}"}
{"id": "1027.png", "formula": "\\begin{align*} n _ i = i + \\sum _ { j = 1 } ^ { i - 1 } p _ j \\end{align*}"}
{"id": "8858.png", "formula": "\\begin{align*} \\nabla g : = \\left ( \\frac { \\partial g } { \\partial x _ { 1 } } , \\frac { \\partial g } { \\partial x _ { 2 } } , \\dots , \\frac { \\partial g } { \\partial x _ { N } } , \\frac { \\partial g } { \\partial y _ { 1 2 } } , \\dots , \\frac { \\partial g } { \\partial y _ { N - 1 N } } \\right ) , \\ \\ \\Delta g : = \\sum _ { k = 1 } ^ N \\frac { \\partial ^ 2 g } { \\partial x _ { k } ^ 2 } + \\sum _ { k = 1 } ^ { N - 1 } + \\frac { \\partial ^ 2 g } { \\partial y _ { k k + 1 } ^ 2 } . \\end{align*}"}
{"id": "3878.png", "formula": "\\begin{align*} \\int _ { \\Omega _ \\epsilon } \\eta ^ p \\langle | \\nabla G _ \\epsilon | ^ { p - 2 } \\nabla G _ \\epsilon , \\nabla ( G _ \\epsilon - g ) \\rangle = - p \\int _ { \\Omega _ \\epsilon } \\eta ^ { p - 1 } ( G _ \\epsilon - g ) \\langle | \\nabla G _ \\epsilon | ^ { p - 2 } \\nabla G _ \\epsilon , \\nabla \\eta \\rangle \\leq C \\end{align*}"}
{"id": "5455.png", "formula": "\\begin{align*} b = l ^ C _ A ( b ) r ^ C _ A ( b ) . \\end{align*}"}
{"id": "6141.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\big [ \\left ( u , \\partial _ t \\phi \\right ) - \\nu \\left ( \\nabla u , \\nabla \\phi \\right ) - \\left ( ( u \\cdot \\nabla ) \\ , u , \\phi \\right ) \\big ] \\ , d t + \\left ( u _ 0 , \\phi ( 0 ) \\right ) = 0 , \\end{align*}"}
{"id": "4690.png", "formula": "\\begin{align*} u _ m \\big ( n \\big ) = \\sum _ { j = 0 } ^ { \\infty } C _ m \\big ( n + k ( 3 k + 1 ) / 2 - j \\big ) \\ , \\big ( M _ k ( j ) + \\widetilde { P } _ k ( j ) \\big ) . \\end{align*}"}
{"id": "5035.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } E \\left [ | Q ^ { n , 4 } _ \\tau | ^ 2 \\right ] = 0 . \\end{align*}"}
{"id": "2301.png", "formula": "\\begin{align*} \\rho _ { k } ( x ) = \\left \\{ \\begin{array} { l l } 0 & 1 \\leq x \\leq 5 0 + 5 0 ( k - 2 ) , \\\\ 1 & x \\geq 6 0 + 5 0 ( k - 2 ) , \\end{array} \\right . \\end{align*}"}
{"id": "13.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { \\rho , L , n } = T _ { L | K } ( \\lambda _ { \\rho } ( \\alpha ) \\psi _ { L , n } ( \\beta ) ) \\cdot _ { \\rho } v _ n \\end{align*}"}
{"id": "5138.png", "formula": "\\begin{align*} \\mathcal { G } ( O _ n ) & = \\mathcal { G } ( h _ 0 ) \\oplus \\mathcal { G } ( h _ 1 ) \\oplus \\dots \\oplus \\mathcal { G } ( h _ d ) \\\\ & = ( k _ 0 \\oplus k _ 1 \\oplus \\dots \\oplus k _ d ) s + [ \\mathcal { G } ( r _ 0 ) \\oplus \\mathcal { G } ( r _ 1 ) \\oplus \\dots \\oplus \\mathcal { G } ( r _ d ) ] & \\mathcal { G } ( r _ i ) < s = 2 ^ t \\\\ & = ( k _ 0 \\oplus k _ 1 \\oplus \\dots \\oplus k _ d ) s + \\mathcal { G } ( ( r _ 0 , r _ 1 , \\dots , r _ d ) ) \\end{align*}"}
{"id": "1472.png", "formula": "\\begin{align*} \\theta ^ { \\ell _ i - i - 1 } _ t & = \\bigcirc _ { \\ell = 0 } ^ { i } ( \\theta _ t + \\gamma _ { r - i + \\ell } ) ^ { - 1 } \\circ ( \\theta _ t + \\xi _ { i , 1 } ) \\circ \\cdots \\circ ( \\theta _ t + \\xi _ { i , \\ell _ i } ) \\\\ & = \\bigcirc _ { \\ell = 0 } ^ { j } ( \\theta _ t + \\gamma _ { r - j + \\ell } ) ^ { - 1 } \\circ ( \\theta _ t + \\xi _ { j , 1 } ) \\circ \\cdots \\circ ( \\theta _ t + \\xi _ { j , \\ell _ j } ) = \\theta ^ { \\ell _ j - j - 1 } _ t \\enspace . \\end{align*}"}
{"id": "9015.png", "formula": "\\begin{align*} - 2 \\ln \\left ( \\frac { 2 \\Gamma \\left ( \\frac { 3 } { 4 } \\right ) } { \\Gamma \\left ( \\frac { 1 } { 4 } \\right ) } \\right ) = \\frac { \\gamma } { 2 } + \\frac { 1 } { 2 } \\ln \\left ( \\frac { \\pi } { 2 } \\right ) - \\frac { 1 } { \\pi } \\left ( \\Delta _ 1 + \\frac { \\pi \\gamma } { 2 } \\right ) . \\end{align*}"}
{"id": "8848.png", "formula": "\\begin{align*} d \\lambda _ i ( t ) = \\sqrt { \\frac { 2 } { \\beta } } d B _ i ( t ) + \\sum _ { j ( \\neq i ) } \\frac { 1 } { \\lambda _ i ( t ) - \\lambda _ j ( t ) } d t - \\frac { \\lambda _ i ( t ) } { 2 } d t , \\ \\ i = 1 , \\dots , N , \\end{align*}"}
{"id": "822.png", "formula": "\\begin{align*} [ w ] _ { Q B _ { \\beta , \\infty } } : = \\inf \\left \\{ [ w ] _ { Q B _ { \\beta , p } } : w \\in Q B _ { \\beta , p } , ~ p > 0 \\right \\} . \\end{align*}"}
{"id": "3893.png", "formula": "\\begin{align*} \\| H _ { \\lambda } \\| _ \\infty < + \\infty \\Rightarrow | \\nabla H _ { \\lambda } ( x ) | = o ( | \\nabla \\Gamma ( x ) | ) \\hbox { a s } x \\to 0 . \\end{align*}"}
{"id": "1567.png", "formula": "\\begin{align*} R ^ i _ j = K ( x , y ) \\left \\lbrace F ^ 2 \\delta ^ i _ j - F F _ { y ^ j } y ^ i \\right \\rbrace . \\end{align*}"}
{"id": "8494.png", "formula": "\\begin{align*} M & \\le \\left ( \\frac { 4 b d } { \\varepsilon } \\right ) ^ { b d k } \\left ( \\frac { 4 k } { \\varepsilon } \\right ) ^ k \\\\ & \\le \\left ( \\frac { 4 b d k } { \\varepsilon } \\right ) ^ { b d k } \\left ( \\frac { 4 b d k } { \\varepsilon } \\right ) ^ { b d k } \\\\ & = \\left ( \\frac { 4 b d k } { \\varepsilon } \\right ) ^ { 2 b d k } . \\end{align*}"}
{"id": "6931.png", "formula": "\\begin{align*} | \\Gamma | ^ 2 - | \\Gamma _ 1 | ^ 2 = \\Gamma ^ * \\Gamma - \\Gamma ^ * S S ^ * \\Gamma = \\Gamma ^ * ( I - S S ^ * ) \\Gamma = \\Gamma ^ * ( e _ 0 e _ 0 ^ * ) \\Gamma , \\end{align*}"}
{"id": "3356.png", "formula": "\\begin{align*} F _ { p } ( s , f ) ( z ) = f ( e ^ { 2 \\pi i s d _ { p } } z ^ { b _ { p } } ) \\end{align*}"}
{"id": "4112.png", "formula": "\\begin{align*} W | _ K ^ { - 1 } T W | _ K P _ K ( S ^ n e _ j ) & = P _ K ( S ^ { n + 1 } e _ j ) = ( P _ K S ) ( S ^ { n } e _ j ) \\\\ & = ( P _ K S ) ( P _ K S ^ { n } e _ j ) + ( P _ K S ) ( P _ { K ^ \\bot } S ^ { n } e _ j ) \\\\ & \\stackrel { ( 1 ) } { = } ( P _ K S ) ( P _ K S ^ { n } e _ j ) , \\end{align*}"}
{"id": "2541.png", "formula": "\\begin{align*} \\begin{array} { l l } \\nabla _ { e _ 1 } e _ 1 = \\nabla _ { X } X = e _ 1 , \\\\ \\nabla _ { e _ 2 } e _ 2 = \\nabla _ { U } U = 0 , \\\\ \\nabla _ { e _ 1 } e _ 2 = \\nabla _ { X } U = 0 , \\\\ \\nabla _ { e _ 2 } e _ 1 = \\nabla _ { U } X = 0 . \\end{array} \\end{align*}"}
{"id": "4080.png", "formula": "\\begin{align*} T r ( \\boldsymbol { X } ) = E ( \\boldsymbol { X } , \\boldsymbol { X } ) - ( \\boldsymbol { M } , \\boldsymbol { M } ) . \\end{align*}"}
{"id": "1237.png", "formula": "\\begin{align*} P ( T , x ) & = \\prod _ { j = 1 } ^ { 8 } \\mathcal { G } ( u _ j , x ) \\\\ & = x ^ 5 \\ \\frac { x ^ 2 - 2 } { x } \\ \\frac { x ^ 2 - 3 } { x } \\ \\frac { x ^ 5 - 7 x ^ 3 + 1 1 x } { ( x ^ 2 - 2 ) ( x ^ 2 - 3 ) } \\\\ & = x ^ 4 ( x ^ 4 - 7 x ^ 2 + 1 1 ) . \\end{align*}"}
{"id": "531.png", "formula": "\\begin{align*} L ^ { \\prime } _ { n s } [ \\dot { x } ^ { \\prime } ( t ^ { \\prime } ) , x ^ { \\prime } ( t ^ { \\prime } ) , t ^ { \\prime } ] = \\frac { 1 } { C ^ { \\prime } _ 1 f ^ { \\prime \\ 2 } ( t ^ { \\prime } ) [ f ^ { \\prime } ( t ^ { \\prime } ) \\dot { x } ^ { \\prime } ( t ) - a ^ { \\prime } _ o x ^ { \\prime } ( t ^ { \\prime } ) + C ^ { \\prime } _ 2 + v ^ { \\prime } _ o V _ 0 ] } \\ . \\end{align*}"}
{"id": "859.png", "formula": "\\begin{align*} \\nu _ { s _ { m } \\rightarrow \\eta _ { m } ^ { r } } ( s _ { m } ) \\propto & \\nu _ { h _ { m } \\rightarrow s _ { m } } ( s _ { m } ) \\times \\nu _ { h _ { m + 1 } \\rightarrow s _ { m } } ( s _ { m } ) \\times \\nu _ { \\eta _ { m } ^ { c } \\rightarrow s _ { m } } ( s _ { m } ) \\\\ = & \\pi _ { s , m } ^ { r , o u t } \\delta ( s _ { m } - 1 ) + ( 1 - \\pi _ { s , m } ^ { r , o u t } ) \\delta ( s _ { m } ) , \\end{align*}"}
{"id": "2311.png", "formula": "\\begin{align*} \\left \\| \\frac { f } { y } x ^ { \\alpha } \\right \\| _ { L ^ { 2 } } = \\left \\| \\left \\| \\frac { f } { y } \\right \\| _ { L _ { y } ^ { 2 } } x ^ { \\alpha } \\right \\| _ { L _ { x } ^ { 2 } } \\leq \\left \\| f _ { y } x ^ { \\alpha } \\right \\| _ { L ^ { 2 } } \\end{align*}"}
{"id": "6816.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } u '''' v \\ , d x = [ u ''' v ] ^ { x = 1 } _ { x = 0 } - [ u '' v ' ] ^ { x = 1 } _ { x = 0 } + \\int _ { 0 } ^ { 1 } u '' v '' d x . \\end{align*}"}
{"id": "7835.png", "formula": "\\begin{align*} \\| \\mu \\| _ { C o n } = \\max \\{ \\| \\mu \\| _ { M \\otimes _ { e h } N } , \\| \\mu ^ * \\| _ { M \\otimes _ { e h } N } \\} . \\end{align*}"}
{"id": "7273.png", "formula": "\\begin{align*} { \\sf U } ( \\xi ) = { \\sf U } _ \\infty ( \\xi ) + { \\sf B } _ 1 | \\xi | ^ \\gamma + O ( | \\xi | ^ { \\gamma - { \\sf k } _ 1 } ) | \\xi | \\to \\infty , \\end{align*}"}
{"id": "2823.png", "formula": "\\begin{align*} \\begin{aligned} - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\big [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\big ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 = 0 . \\end{aligned} \\end{align*}"}
{"id": "5886.png", "formula": "\\begin{align*} \\mathbb { V } _ { n , k } = \\{ V \\in \\mathbb { R } ^ { k \\times n } \\colon V V ^ T = I _ k \\} , \\end{align*}"}
{"id": "5631.png", "formula": "\\begin{align*} 0 \\leq A : = \\frac { \\varrho _ 1 - \\varrho _ 2 } { ( \\varrho _ 1 - \\varrho _ 2 ) + ( p _ 1 - p _ 2 ) } \\leq 1 0 \\leq B : = \\frac { p _ 1 - p _ 2 } { ( \\varrho _ 1 - \\varrho _ 2 ) + ( p _ 1 - p _ 2 ) } \\leq 1 . \\end{align*}"}
{"id": "3931.png", "formula": "\\begin{align*} V a r ( \\hat { \\pi } _ { h , n } ( x ) ) & \\le \\frac { c } { T _ n } \\frac { \\sum _ { l = 1 } ^ d | \\log ( h ^ * _ l ) | } { \\prod _ { l \\ge 3 } h ^ * _ l } + \\frac { c } { T _ n } \\frac { 1 } { \\prod _ { l = 1 } ^ d h _ l ^ * } \\Delta _ n \\\\ & \\le \\frac { c } { T _ n } \\frac { \\sum _ { l = 1 } ^ d | \\log ( h ^ * _ l ) | } { \\prod _ { l \\ge 3 } h ^ * _ l } . \\end{align*}"}
{"id": "3296.png", "formula": "\\begin{align*} u ^ s _ { - A , q } ( y , x ) & = \\underset { ( \\ell _ 2 , m _ 2 ) \\in \\Gamma } \\sum \\beta _ { \\ell _ 2 m _ 2 } ( x ) h ^ { ( 1 ) } _ { \\ell _ 2 } ( k \\vert y \\vert ) Y ^ { m _ 2 } _ { \\ell _ 2 } \\left ( \\widehat { y } \\right ) . \\end{align*}"}
{"id": "235.png", "formula": "\\begin{align*} a ( P ) = ( P ^ { ( s ) } \\{ a \\} ) ( P _ 0 ) , \\ \\ a \\in R . \\end{align*}"}
{"id": "8716.png", "formula": "\\begin{align*} \\frac { \\partial h } { \\partial \\dot { x } ^ { 2 } } = \\frac { ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } } { 2 } x ^ { 1 } + \\lambda \\left ( \\frac { \\dot { x } ^ { 2 } } { \\sqrt { ( \\dot { x } ^ { 1 } ) ^ { 2 } + ( \\dot { x } ^ { 2 } ) ^ { 2 } } } + b \\sin ( t + c ) \\right ) . \\end{align*}"}
{"id": "3976.png", "formula": "\\begin{align*} \\textstyle \\big ( \\sum _ { x \\in G } a _ x x \\big ) \\cdot \\big ( \\sum _ { y \\in G } b _ y y \\big ) = \\sum _ { z \\in G } \\big ( \\sum _ { x y = z } a _ x b _ y \\big ) z . \\end{align*}"}
{"id": "1124.png", "formula": "\\begin{align*} \\omega _ i = \\frac { \\lambda ( a _ i + a _ { i + 1 } ) } { 2 [ ( \\sigma _ N ^ 2 + \\sigma _ Z ^ 2 ) \\lambda ^ 2 + \\rho ) ] } . \\end{align*}"}
{"id": "2467.png", "formula": "\\begin{align*} \\omega \\left ( y , z \\right ) : = \\left ( x , \\left [ y , z \\right ] \\right ) . \\end{align*}"}
{"id": "2199.png", "formula": "\\begin{align*} ( \\underbrace { + } _ { a _ { u j } } , \\ , \\underbrace { - \\dots - } _ { a _ { u + 1 \\ , j + 1 } } , \\ , \\cdots , \\ , \\underbrace { + \\dots + } _ { a _ { u ' - 1 \\ , j } } , \\ , \\underbrace { - \\dots - } _ { a _ { u ' \\ , j + 1 } } ) . \\end{align*}"}
{"id": "2144.png", "formula": "\\begin{align*} \\int _ { [ - n ^ m , - 1 ] } \\frac { n s ^ { 2 } \\ , d \\sigma _ { \\mu } ^ { - } ( s ) } { ( \\widetilde { x } _ n - s ) ^ { 2 } } & = \\frac { n q } { p } \\sum _ { k = 1 } ^ n \\frac { k ^ { 2 m } \\sigma _ { \\mu } ^ { + } ( [ k ^ m , ( k + 1 ) ^ m ) ) } { ( \\widetilde { x } _ n + k ^ m ) ^ 2 } \\\\ & \\leq \\frac { 4 q c } { m \\cdot n ^ { 2 m - 3 } } \\sum _ { k = 1 } ^ n k ^ { ( 2 - a ) m - 1 } \\\\ & \\leq \\frac { 4 q c } { m \\cdot n ^ { 2 m - 3 } } \\int _ 1 ^ { n + 1 } s ^ { ( 2 - a ) m - 1 } \\ ; d \\lambda ( s ) \\\\ & \\leq \\frac { 2 ^ { ( 2 - a ) m + 2 } q c } { ( 2 - a ) m ^ 2 } \\cdot \\frac { 1 } { n ^ { a m - 3 } } . \\end{align*}"}
{"id": "6992.png", "formula": "\\begin{align*} \\int _ \\Delta \\omega = \\int _ { S ^ { n - 1 } } f ^ * \\alpha , \\end{align*}"}
{"id": "8509.png", "formula": "\\begin{align*} Q _ 0 ( x ) : = - 1 6 x + 1 0 4 x ^ 2 - 3 5 2 x ^ 3 + 6 6 0 x ^ 4 - 6 7 2 x ^ 5 + 3 3 6 x ^ 6 - 6 4 x ^ 7 \\in S ^ { 1 6 } , \\ ; \\forall x \\in R . \\end{align*}"}
{"id": "6042.png", "formula": "\\begin{align*} \\Lambda _ { r , s } = ( \\kappa _ 1 , \\kappa _ 2 , \\ldots , \\kappa _ { m } ) \\end{align*}"}
{"id": "2007.png", "formula": "\\begin{align*} H _ { s , c } = \\{ n \\in \\mathbb { N } , n > 1 2 7 \\ p | n , p ^ { \\nu _ p ( n ) } \\leq s \\ \\ p > s c ^ { - 1 } \\ \\ d + \\left [ \\frac { s } { p } \\right ] \\geq p \\} \\end{align*}"}
{"id": "1274.png", "formula": "\\begin{align*} \\mathcal { E } ( \\mathcal { B } _ { d , k } ) & = \\Psi ( E _ k ( x , d - 1 ) ) + \\sum _ { j = 1 } ^ { k - 1 } ( d - 2 ) ( d - 1 ) ^ { k - 1 - j } \\Psi ( E _ j ( x , d - 1 ) ) . \\end{align*}"}
{"id": "6152.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\| v ^ { \\Delta t } _ { h } - u ^ { \\Delta t } _ { h } \\| _ { 2 } ^ 2 \\ , d t \\leq \\frac { \\Delta t } { 1 2 } \\sum _ { m = 1 } ^ N \\| u _ h ^ m - u _ h ^ { m - 1 } \\| _ { 2 } ^ 2 \\leq C \\ , \\bigg ( ( \\Delta t ) ^ { 2 } + \\frac { \\Delta t } { h ^ { 1 / 2 } } \\bigg ) , \\end{align*}"}
{"id": "8693.png", "formula": "\\begin{align*} L = \\int \\limits _ { t _ { 0 } } ^ { t _ { 1 } } \\left ( \\sqrt { ( { \\dot { x } ^ { 1 } } ) ^ 2 + { ( \\dot { x } ^ { 2 } } ) ^ 2 } + b \\left ( \\cos \\theta ( x ^ 1 , x ^ 2 ) \\dot { x } ^ { 1 } + \\sin \\theta ( x ^ 1 , x ^ 2 ) \\dot { x } ^ { 2 } \\right ) \\right ) d t . \\end{align*}"}
{"id": "887.png", "formula": "\\begin{align*} f = f _ 0 \\oplus f _ 1 \\oplus f _ 2 \\oplus \\cdots \\end{align*}"}
{"id": "7361.png", "formula": "\\begin{align*} \\Delta v = \\lambda ^ 2 \\Big ( \\hat { V } ( | y | ) v - | v | ^ { p - 1 } v \\Big ) \\mathbb { R } ^ N , \\end{align*}"}
{"id": "2254.png", "formula": "\\begin{align*} \\begin{cases} u _ p ^ 0 ( x , \\eta ) + \\delta = \\phi _ * ( z ) + w ( x , \\eta ) , \\\\ h _ p ^ 0 ( x , \\eta ) + \\sigma = \\psi _ * ( z ) + \\Omega ( x , \\eta ) . \\end{cases} \\end{align*}"}
{"id": "4502.png", "formula": "\\begin{align*} \\nabla ^ 2 u = & - k | \\nabla u | , \\\\ \\mu | \\nabla u | = & - \\langle J , \\nabla u \\rangle . \\end{align*}"}
{"id": "8114.png", "formula": "\\begin{align*} c _ { N , i } = \\frac { N ^ i \\Gamma ( N - i + \\beta ) } { i ! ( N - i - 1 ) ! ( \\alpha + 1 ) ^ { N - i + \\beta } } . \\end{align*}"}
{"id": "3813.png", "formula": "\\begin{align*} a = a _ 1 \\mathbf { e } _ 1 + a _ 2 \\mathbf { e } _ 2 , c = \\cos \\theta \\cos \\phi \\mathbf { e } _ 1 + \\cos \\theta \\sin \\phi \\mathbf { e } _ 2 + \\sin \\theta \\mathbf { e } _ 3 , ( a - c ) \\times b = - ( a _ 2 - \\cos \\theta \\sin \\phi ) \\mathbf { e } _ 3 + \\sin \\theta \\mathbf { e } _ 2 \\end{align*}"}
{"id": "8426.png", "formula": "\\begin{align*} & F _ 1 ( y _ 1 , y _ 2 ) = \\bigg \\{ \\frac { 2 } { \\sqrt { { y _ 1 } ^ 2 + { y _ 2 } ^ 2 } } ( y _ 1 , y _ 2 ) + ( v _ 1 , v _ 2 ) \\bigg | ~ ( v _ 1 , v _ 2 ) \\in [ 0 , 1 ] \\times [ 0 , 1 ] \\bigg \\} , \\\\ & F _ 2 ( x _ 1 , x _ 2 ) = \\bigg \\{ \\frac { \\sqrt 2 } { \\sqrt { { x _ 1 } ^ 2 + { x _ 2 } ^ 2 } } ( x _ 1 , x _ 2 ) + ( v _ 1 , v _ 2 ) \\bigg | ~ ( v _ 1 , v _ 2 ) \\in [ 0 , 1 ] \\times [ 0 , 1 ] \\bigg \\} . \\end{align*}"}
{"id": "3276.png", "formula": "\\begin{align*} H = A + \\nabla \\vartheta . \\end{align*}"}
{"id": "1917.png", "formula": "\\begin{align*} \\mathrm { c a r d } ( \\mathcal { R } _ { [ n , j ] } ) = \\binom { m ( p + 1 ) + j } { m } , \\mathrm { c a r d } ( \\mathcal { S } _ { [ n , j ] } ) = \\frac { j + 1 } { p m + j + 1 } \\binom { m ( p + 1 ) + j } { m } . \\end{align*}"}
{"id": "5965.png", "formula": "\\begin{align*} t _ j = T ( j / N ) ^ k k \\geq \\frac { 2 - \\beta } { \\beta } , j = 0 , \\dots , N . \\end{align*}"}
{"id": "5857.png", "formula": "\\begin{align*} \\sum _ { i = - \\infty } ^ { k - 1 } 2 ^ { - i \\frac { r } { p - r } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \\frac { p r } { p - r } } & \\approx \\sum _ { i = - \\infty } ^ { k - 1 } \\int _ { x _ i } ^ { x _ { i + 1 } } \\bigg ( \\int _ 0 ^ t w \\bigg ) ^ { - \\frac { p } { p - r } } w ( t ) d t V _ r ( x _ { i - 1 } , x _ i ) ^ { \\frac { p r } { p - r } } \\\\ & \\le \\int _ { 0 } ^ { x _ { k } } \\bigg ( \\int _ 0 ^ t w \\bigg ) ^ { - \\frac { p } { p - r } } w ( t ) V _ r ( 0 , t ) ^ { \\frac { p r } { p - r } } d t . \\end{align*}"}
{"id": "8830.png", "formula": "\\begin{align*} \\| f _ t \\circ ( \\phi _ 1 * \\phi _ 2 ) \\| = 0 \\leq \\| f _ t \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) \\| . \\end{align*}"}
{"id": "7244.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { v \\in V ( G ) } x _ v & \\leq \\frac { 2 e ( L ) + 2 \\epsilon e ( S ) + ( 1 + \\epsilon ) e ( L , S ) } { \\rho _ \\alpha } \\\\ & \\leq \\frac { \\epsilon n + \\epsilon C _ 1 n + ( 1 + \\epsilon ) ( r - 2 + \\epsilon ) n } { \\rho _ \\alpha } \\\\ & = \\frac { ( ( 1 + C _ 1 ) \\epsilon + ( 1 + \\epsilon ) ( r - 2 + \\epsilon ) ) n } { \\rho _ \\alpha } . \\end{aligned} \\end{align*}"}
{"id": "6572.png", "formula": "\\begin{align*} g _ { x , m } ( z , y ) = m ^ k h ( z , ( 1 / m ) y ) = m ^ k f ( z , ( 1 / m ) y ) = f ( z , y ) = n ^ k f ( t , ( 1 / n ) y ) = g _ { x , n } ( z , y ) , \\end{align*}"}
{"id": "7924.png", "formula": "\\begin{align*} S : = \\{ i \\in \\{ 1 , \\ldots , m \\} | \\overline { \\{ i \\} } \\not \\in \\mathcal A _ \\omega \\} . \\end{align*}"}
{"id": "7391.png", "formula": "\\begin{align*} \\int _ { B _ R ( h _ n t _ 1 ) } \\phi _ n ^ 2 d y = o _ n ( 1 ) . \\end{align*}"}
{"id": "1219.png", "formula": "\\begin{align*} \\mathcal { H } ( v , x ) & = x - d ( v ) - \\sum _ { w \\in c ( v ) } \\dfrac { 1 } { \\mathcal { H } ( w , x ) } ( \\forall v \\in V ( T ) ) , \\end{align*}"}
{"id": "7806.png", "formula": "\\begin{align*} ( h ' _ { X _ 1 } \\otimes 1 ) + ( 1 \\otimes h ' _ { X _ 2 } ) = ( h ' _ { X _ 1 } \\otimes 1 ) + ( 1 \\otimes A d _ { \\psi _ q } ( h ' _ { X _ 1 } ) ) \\end{align*}"}
{"id": "2798.png", "formula": "\\begin{align*} \\begin{aligned} x _ i { } & = { } \\frac { U } { L } + \\frac { U } { L } \\sum \\limits _ { j = i } ^ { N - 1 } h _ j \\\\ f _ i { } & = { } \\Delta - \\frac { U ^ 2 } { L } \\sum \\limits _ { j = 0 } ^ { i - 1 } h _ j \\ , \\left ( 1 - \\frac { h _ j } { 2 } \\frac { - \\kappa } { 1 - \\kappa } \\right ) \\\\ g _ i { } & = { } U \\end{aligned} \\end{align*}"}
{"id": "6187.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\mathrm { d i s t } ^ { 2 } ( p , S _ { 0 } ^ { 1 } ) = | z _ { 1 } | ^ { 2 } + | z _ { 2 } | ^ { 2 } : = r _ { S ^ { 1 } } ^ { 2 } . \\end{array} \\end{align*}"}
{"id": "4441.png", "formula": "\\begin{align*} \\delta ^ { k + 1 } N _ m ^ k f = 0 . \\end{align*}"}
{"id": "2911.png", "formula": "\\begin{align*} S ^ { j \\ell } F ( A x + e ) = F ( A S x + e ) = B G ( S x ) + d = S ^ { k m } B G ( x ) + d = S ^ { k m } F ( A x + e ) , \\end{align*}"}
{"id": "7195.png", "formula": "\\begin{align*} \\langle f , \\theta ( u ( n ) ) I ^ { \\circ } ( w _ 1 , z ) w _ 2 \\rangle = 0 . \\end{align*}"}
{"id": "3249.png", "formula": "\\begin{align*} \\partial _ { y _ k } \\partial _ { y _ m } \\bigl ( q ^ { k m } \\sigma \\bigr ) - \\partial _ { y _ k } \\bigl ( h ^ k \\sigma \\bigr ) = 0 , \\end{align*}"}
{"id": "6923.png", "formula": "\\begin{align*} f - q | _ g = & \\ L ( x L ( p | _ { L ^ 2 ( y z ) } ) ) + x L ^ 2 ( p | _ { L ^ 2 ( y z ) } ) - L ^ 2 ( x p | _ { L ( y L ( z ) ) } ) - L ^ 2 ( x p | _ { y L ^ 2 ( z ) } ) \\\\ \\equiv & \\ - L ( x L ( p | _ { L ( y L ( z ) ) } ) ) - L ( x L ( p | _ { y L ^ 2 ( z ) } ) ) - x L ^ 2 ( p | _ { L ( y L ( z ) ) } ) - x L ^ 2 ( p | _ { y L ^ 2 ( z ) } ) \\\\ & \\ + L ( x L ( p | _ { L ( y L ( z ) ) } ) ) + x L ^ 2 ( p | _ { L ( y L ( z ) ) } ) + L ( x L ( p | _ { y L ^ 2 ( z ) } ) ) + x L ^ 2 ( p | _ { y L ^ 2 ( z ) } ) \\\\ \\equiv & \\ 0 , \\end{align*}"}
{"id": "7545.png", "formula": "\\begin{align*} \\sin ( \\alpha _ k ^ * ) \\Re \\{ e _ k \\} = \\cos ( \\alpha _ k ^ * ) \\Im \\{ e _ k \\} \\end{align*}"}
{"id": "6327.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + \\frac { q x + x - 1 } { q x ( q x - 1 ) } D _ { q ^ { - 1 } } y ( x ) + \\frac { [ n ] _ q [ n + 1 ] _ q } { q ^ n x ( 1 - q x ) } y ( x ) = 0 . \\end{align*}"}
{"id": "8807.png", "formula": "\\begin{align*} \\phi _ 1 * \\phi _ 2 ( g ) = \\int _ { G } ^ { } \\phi _ 1 ( g ' ) \\phi _ 2 ( g '^ { - 1 } g ) d g ' . \\end{align*}"}
{"id": "1273.png", "formula": "\\begin{align*} ( \\zeta - 1 ) \\left ( \\frac { 1 } { \\zeta } - 1 \\right ) = 2 - \\left ( \\zeta + \\frac { 1 } { \\zeta } \\right ) = 2 - 2 \\cos \\left ( \\frac { 1 } { j + 1 } \\pi \\right ) , \\end{align*}"}
{"id": "5173.png", "formula": "\\begin{align*} & \\Delta ^ { \\sigma ( 2 + k ) } _ { h } \\cdots \\Delta ^ { \\sigma ( m - 1 + k ) } _ { h } H ( x _ 0 , \\ldots , x _ { m - 1 } ) \\\\ & = \\Delta ^ { \\sigma ( 2 + k ) } _ { h } \\cdots \\Delta ^ { \\sigma ( m - 1 + k ) } _ { h } \\Bigg [ \\sum \\limits _ { s = 0 } ^ { m - 1 } \\prod \\limits _ { i = 0 } ^ { m - 3 } x _ { i + s } - \\sum \\limits _ { s = 0 } ^ { m - 1 } \\prod \\limits _ { i = 0 } ^ { m - 3 } x _ { \\sigma ( i + s ) } \\Bigg ] . \\end{align*}"}
{"id": "3013.png", "formula": "\\begin{align*} S _ { S _ n , \\frac { 1 } { 2 } W } ( R _ i ) = \\frac { 4 n ^ 2 + 3 n + 1 } { 4 n ( 2 n + 1 ) } . \\end{align*}"}
{"id": "4403.png", "formula": "\\begin{align*} S ( A ) = \\{ g : \\mathbb { D } \\rightarrow \\mathbb { C } \\mbox { h o l o m o r p h i c : t h e r e i s } f \\in A \\mbox { w i t h } | \\hat { g } ( k ) | \\leq | \\hat { f } ( k ) | \\mbox { f o r a l l } k \\} . \\end{align*}"}
{"id": "4955.png", "formula": "\\begin{align*} X ^ n _ t = X _ 0 + \\int ^ t _ 0 ( t - \\eta _ n ( s ) ) ^ { \\alpha } \\sigma ( X ^ n _ { \\eta _ { n } ( s ) } ) \\ , d W _ s , \\end{align*}"}
{"id": "1090.png", "formula": "\\begin{align*} \\tilde \\Lambda _ l \\coloneqq \\sum _ { j = 0 } ^ { \\min ( l , k ) } \\frac { \\gamma ^ { 2 k + l - 2 j } \\kappa _ { k , j , l } } { ( l - j ) ! j ! ( ( k - j ) ! ) ^ 2 } . \\end{align*}"}
{"id": "6928.png", "formula": "\\begin{align*} S ( x _ 0 , x _ 1 , x _ 2 , \\ldots ) & = ( 0 , x _ 0 , x _ 1 , x _ 2 , \\ldots ) \\ , , \\intertext { a n d b y $ S ^ * $ i t s a d j o i n t ( t h e \\emph { b a c k w a r d s h i f t } ) , } S ^ * ( x _ 0 , x _ 1 , x _ 2 , \\ldots ) & = ( x _ 1 , x _ 2 , x _ 3 , \\ldots ) , \\end{align*}"}
{"id": "2620.png", "formula": "\\begin{align*} k _ n = c _ n \\sqrt { \\dfrac { 2 ( n + 1 ) ^ 3 } { ( p + 1 ) ( p + 2 ) } } \\ . \\end{align*}"}
{"id": "1170.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\Psi ( t _ n h _ n ) - \\Psi ( 0 ) } { t _ n } = h ( g ) . \\end{align*}"}
{"id": "1810.png", "formula": "\\begin{align*} f ( x ) = \\delta _ H ( x ) , g ( y ) = \\lambda \\| y \\| _ 1 , g ^ \\ast ( z ) = \\begin{cases} 0 & \\| z \\| _ \\infty \\leq \\lambda \\\\ + \\infty & \\mbox { o t h e r w i s e , } \\end{cases} h ( x ) = \\frac { 1 } { 2 } \\| C x - b \\| ^ 2 , \\end{align*}"}
{"id": "1742.png", "formula": "\\begin{align*} - \\int _ C \\frac { e ^ { \\omega s } \\cdot s ^ { 1 - d } } { ( e ^ { \\omega s } - 1 ) ^ 2 } \\ , d s = \\frac { ( d - 1 ) \\zeta ( d ) } { 2 \\pi i } \\cdot \\big ( \\frac { \\omega } { 2 \\pi i } \\big ) ^ { d - 2 } , \\end{align*}"}
{"id": "1106.png", "formula": "\\begin{align*} \\frac { \\| \\mbox { g r a d } b \\| ^ 2 } { b ^ 2 } \\left \\{ G ( V , V ) G ( U , U ) - G ( U , V ) ^ 2 \\right \\} - G ( \\mbox { H o r } ( P _ V V ) , \\mbox { H o r } ( P _ U U ) ) + G ( \\mbox { H o r } ( P _ V U ) , \\mbox { H o r } ( P _ V U ) ) = 0 , \\end{align*}"}
{"id": "634.png", "formula": "\\begin{align*} & \\zeta ( \\Delta _ n , s ) = V _ \\alpha ( s ) \\Bigg ( a ( s ) \\ , n ^ { \\alpha - 2 s } + \\sum _ { m = 0 } ^ { M - 1 } b _ m ( s ) \\ , n ^ { - 2 m } \\Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) , \\\\ & \\zeta ( \\widetilde { \\Delta } _ n , s ) = V _ \\alpha ( s ) \\Bigg ( \\widetilde { a } ( s ) \\ , n ^ { \\alpha - 2 s } + \\sum _ { m = 0 } ^ { M - 1 } \\widetilde { b } _ m ( s ) \\ , n ^ { - 2 m } \\Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) . \\end{align*}"}
{"id": "7932.png", "formula": "\\begin{align*} \\Sigma _ \\pm = \\{ \\sigma _ I : \\bar { I } \\in \\mathcal A _ \\pm \\} . \\end{align*}"}
{"id": "1437.png", "formula": "\\begin{align*} & | p | _ v = p ^ { - \\tfrac { [ K _ v : \\Q _ p ] } { [ K : \\Q ] } } \\ \\ v \\in { { \\mathfrak { M } } } ^ { f } _ K \\ \\ v \\mid p \\enspace , \\\\ & | x | _ v = | \\iota _ v x | ^ { \\tfrac { [ K _ v : \\R ] } { [ K : \\Q ] } } \\ \\ v \\in { { \\mathfrak { M } } } ^ { \\infty } _ K \\enspace , \\end{align*}"}
{"id": "7279.png", "formula": "\\begin{align*} \\lambda ^ { - \\frac { n - 2 } { 2 } } \\sigma { \\sf A } _ 1 & = - \\eta ^ \\frac { 2 } { 1 - q } | y | \\to \\infty , \\ | \\xi | \\to 0 , \\\\ - \\eta ^ \\frac { 2 } { 1 - q } { \\sf B } _ 1 & = K { \\sf D } _ J ( T - t ) ^ J \\eta ^ \\gamma | \\xi | \\to \\infty , \\ | z | \\to 0 . \\end{align*}"}
{"id": "5731.png", "formula": "\\begin{align*} \\pi _ { [ a , i - 1 ] } \\pi _ { [ i , b - 1 ] } y _ i = \\binom { b - a } { i - a } \\pi _ { [ a , b - 1 ] } y _ i \\end{align*}"}
{"id": "1533.png", "formula": "\\begin{align*} \\P ( T _ { d e t } \\geq \\ss ) = \\exp \\Big ( - \\int _ { B _ O ( R ) } \\P _ { x _ 0 } ( T _ { d e t } \\leq \\ss ) d \\mu ( x _ 0 ) \\Big ) \\end{align*}"}
{"id": "4718.png", "formula": "\\begin{align*} e ( H _ { 2 } ^ { - 1 } H _ { 3 } ^ { - 1 } H _ { 1 } H _ { 2 } ) e = e ( H _ { 2 } ^ { - 1 } H _ { 3 } ^ { - 1 } H _ { 1 } H _ { 2 } ) e H _ { 2 } ^ { - 1 } H _ { 3 } ^ { - 1 } H _ { 1 } H _ { 2 } , \\end{align*}"}
{"id": "1842.png", "formula": "\\begin{align*} \\frac { ( f _ { 1 } , \\ldots , f _ { p } ) } { ( g _ { 1 } , \\ldots , g _ { p } ) } : = \\left ( \\frac { f _ { 1 } } { g _ { p } } , \\frac { f _ { 2 } \\ , g _ { 1 } } { g _ { p } } , \\frac { f _ { 3 } \\ , g _ { 2 } } { g _ { p } } , \\ldots , \\frac { f _ { p } \\ , g _ { p - 1 } } { g _ { p } } \\right ) . \\end{align*}"}
{"id": "8051.png", "formula": "\\begin{align*} g ( \\nabla _ { V } X , W ) = - g ( \\nabla _ { W } X , V ) \\end{align*}"}
{"id": "4294.png", "formula": "\\begin{align*} \\omega | _ L = 0 , ( e ^ { - i \\hat { \\theta } } \\Omega | _ L ) = 0 . \\end{align*}"}
{"id": "2096.png", "formula": "\\begin{align*} \\mathcal Q ( x ) ( v ) = ( a v g ^ * \\circ a v g ) ( x ) ( v ) = \\sum \\limits _ { e \\in E _ v } \\frac { ( a v g ( x ) ) ( e ) } { | e | } \\frac { \\delta _ E ( e ) } { \\delta _ V ( v ) } , \\end{align*}"}
{"id": "888.png", "formula": "\\begin{align*} g _ l = \\sum _ 0 ^ l h _ k l = 0 , 1 , 2 , \\ldots \\end{align*}"}
{"id": "7336.png", "formula": "\\begin{align*} ( x , y , z ) = - [ x , y ] z + x [ y , z ] + [ x z , y ] ; \\end{align*}"}
{"id": "5601.png", "formula": "\\begin{align*} x y z = a , x + y + z = a n . \\end{align*}"}
{"id": "8975.png", "formula": "\\begin{align*} E ( c _ p u _ p ) = c _ p ^ 2 E ( u _ p ) \\geq L + \\epsilon , \\end{align*}"}
{"id": "6974.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 1 } ^ { \\infty } \\frac { \\mu _ { k } ^ { 2 } } { \\lambda _ { k } ^ { 2 } } = 1 - \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { a _ { k } } { \\lambda _ k ^ 2 } ; \\end{align*}"}
{"id": "5829.png", "formula": "\\begin{align*} \\chi ( p _ n ) = 1 , \\chi ( p _ { n + 1 } ) = 1 , \\chi ( p _ { n + 2 } ) = 1 , \\end{align*}"}
{"id": "5099.png", "formula": "\\begin{align*} E [ | B ^ { n , 4 } _ \\tau | ^ 2 ] = n ^ { 2 \\alpha + 1 } \\int _ { [ ( \\delta + \\frac 1 n ) \\wedge \\tau , \\tau ] ^ 2 } ( t - s _ 1 ) ^ { \\alpha } ( t - s _ 2 ) ^ { \\alpha } E \\left [ \\sigma ' ( X _ { \\eta _ n ( s _ 1 ) - \\delta } ) M _ { s _ 1 } ^ { n , 3 } \\sigma ' ( X _ { \\eta _ n ( s _ 2 ) - \\delta } ) M _ { s _ 2 } ^ { n , 3 } \\right ] d s _ 1 d s _ 2 , \\end{align*}"}
{"id": "6044.png", "formula": "\\begin{align*} f _ { r , s } ( x ) = f ( r x + s ) \\in K [ x ] . \\end{align*}"}
{"id": "7486.png", "formula": "\\begin{align*} x _ { n + 1 } & = x _ n - \\frac { f ( x _ n ) } { f ' ( x _ n ) } = x _ n - \\frac { ( x _ n - a ) ^ 2 + b ^ 2 } { 2 ( x _ n - a ) } = \\frac { b } { 2 } \\left ( \\frac { x _ n - a } { b } - \\frac { 1 } { \\frac { x _ n - a } { b } } \\right ) + a \\end{align*}"}
{"id": "8999.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { n - 1 } ( k + x - z ) : = \\sum _ { k = 0 } ^ { n - 1 } P _ { k , n } ( z ) x ^ k , \\Re \\ , z > 0 . \\end{align*}"}
{"id": "7509.png", "formula": "\\begin{align*} P ( S _ i ) = \\frac { { m _ x \\choose i } \\sum _ { j = 0 } ^ { i } ( - 1 ) ^ { j } { { i } \\choose { j } } ( i - j ) ^ { \\gamma _ x k _ x } } { m _ x ^ { \\gamma _ x k _ x } } . \\end{align*}"}
{"id": "3253.png", "formula": "\\begin{align*} u _ { A , q } ( \\cdot , d ) = u ^ i ( \\cdot , d ) + u _ { A , q } ^ s ( \\cdot , d ) \\mathrm { i n } \\ ; \\R ^ 3 , \\end{align*}"}
{"id": "699.png", "formula": "\\begin{align*} & \\partial _ s u + \\Delta u = \\frac { f ' ( u ) } { f ( u ) } \\left ( m + \\frac { | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 } { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 } \\right ) + \\mathcal { H } \\ , \\frac { f ( u ) } { \\sqrt { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 } } , \\\\ & u ( 0 , \\cdot ) = u _ 0 , \\end{align*}"}
{"id": "5978.png", "formula": "\\begin{align*} \\hat w ( s ) = ( s ^ \\beta { + } A ) ^ { - 1 } \\left ( \\hat f ( s ) - \\frac 1 { s } A u _ 0 \\right ) , \\end{align*}"}
{"id": "4721.png", "formula": "\\begin{align*} e _ { ( l + 1 ) } = \\Big ( \\frac { q - q ^ { - 1 } } { z - z ^ { - 1 } } \\Big ) ^ { l - 1 } e _ { ( l ) } H _ { 2 l } H _ { 2 l + 1 } H _ { 2 l - 1 } ^ { - 1 } H _ { 2 l } ^ { - 1 } e _ { ( l ) } \\end{align*}"}
{"id": "4377.png", "formula": "\\begin{align*} e ^ { t _ 0 } \\lim _ { j \\rightarrow + \\infty } \\frac { G ( t _ 0 ) - G ( t _ 0 + B _ j ) } { B _ j } & = \\lim _ { j \\to + \\infty } \\frac { G ( t _ 0 ) - G ( t _ 0 + B _ j ) } { e ^ { - t _ 0 } - e ^ { - t _ 0 - B _ j } } \\\\ & = \\liminf _ { B \\to 0 + 0 } \\frac { G ( t _ 0 ) - G ( t _ 0 + B ) } { e ^ { - t _ 0 } - e ^ { - t _ 0 - B } } \\end{align*}"}
{"id": "4582.png", "formula": "\\begin{align*} f ^ h = t ^ { \\deg f } f \\left ( \\frac { x _ 1 } { t } , \\ldots , \\frac { x _ n } { t } \\right ) . \\end{align*}"}
{"id": "856.png", "formula": "\\begin{align*} \\nu _ { \\eta _ { m } ^ { c } \\rightarrow s _ { m } } ( s _ { m } ) = \\pi _ { s , m } ^ { c , i n } \\delta ( s _ { m } - 1 ) + ( 1 - \\pi _ { s , m } ^ { c , i n } ) \\delta ( s _ { m } ) , \\end{align*}"}
{"id": "5473.png", "formula": "\\begin{align*} \\norm { \\nabla _ x \\tilde { f } } \\lesssim L , \\\\ \\norm { H ( \\tilde { f } ) _ x } \\leq \\sum _ { i = 0 } ^ { N _ 0 } P ^ { \\circ i } ( L ) , \\end{align*}"}
{"id": "6924.png", "formula": "\\begin{align*} f - q | _ g = & \\ - L ^ 2 ( p | _ { L ( L ( x ) y ) } ) z + L ( L ( p | _ { L ^ 2 ( x ) y } ) z ) \\\\ \\equiv & \\ - L ^ 2 ( p | _ { L ^ 2 ( x ) y } ) z + L ^ 2 ( p | _ { L ^ 2 ( x ) y } ) z \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "7716.png", "formula": "\\begin{align*} V = \\sum _ { d _ 1 , \\ldots , d _ k \\le x } h _ { f , k } ( d _ 1 , \\ldots , d _ k ) \\Big ( \\Big ( \\frac { x } { d _ 1 \\cdots d _ k } \\Big ) ^ { r + 1 } T _ { k - 1 } \\Big ( \\log \\frac { x } { d _ 1 \\cdots d _ k } \\Big ) + O \\Big ( \\Big ( \\frac { x } { d _ 1 \\cdots d _ k } \\Big ) ^ { r + \\theta _ { k } + \\varepsilon } \\Big ) \\Big ) , \\end{align*}"}
{"id": "8519.png", "formula": "\\begin{align*} u _ k = p u _ { k - 1 } + a _ k u _ { k - 2 } , k \\geq 1 , \\end{align*}"}
{"id": "1898.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { U } _ { [ n , j , k ] } } w ( \\gamma ) = a _ { - j } ^ { ( j ) } \\ , B _ { [ k - 1 , j - 1 ] } ^ { ( 1 ) } W _ { [ n - k - 1 , 0 ] } , \\end{align*}"}
{"id": "8394.png", "formula": "\\begin{align*} P _ a = \\begin{cases} P ^ { - 1 } , & a \\in P ^ { - 1 } \\\\ P \\cup \\{ b \\in A \\mid b < f ( a ) ^ { - 1 } \\} & a \\in P . \\end{cases} \\end{align*}"}
{"id": "4755.png", "formula": "\\begin{align*} \\begin{aligned} \\theta _ { 0 0 , 0 0 } ^ 2 ( 0 , M _ 1 \\tau ) & = - \\tau _ { 1 , 1 } \\theta _ { 0 1 , 0 0 } ^ 2 ( 0 , \\tau ) , \\\\ \\theta _ { 0 0 , 0 0 } ^ 2 ( 0 , M _ 2 \\tau ) & = - \\tau _ { 2 , 2 } \\theta _ { 1 0 , 0 0 } ^ 2 ( 0 , \\tau ) , \\quad \\\\ \\theta _ { 0 0 , 0 0 } ^ 2 ( 0 , N _ { 1 , 2 } \\tau ) & = ( \\tau _ { 1 , 2 } ^ 2 - \\tau _ { 1 , 1 } \\tau _ { 2 , 2 } ) \\theta _ { 0 0 , 0 0 } ^ 2 ( 0 , \\tau ) . \\end{aligned} \\end{align*}"}
{"id": "6694.png", "formula": "\\begin{align*} f = \\{ ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) , \\dots , ( x _ n , y _ n ) \\} , \\end{align*}"}
{"id": "3457.png", "formula": "\\begin{align*} \\mathrm { D T M } _ { ( p , q ) } ( X ^ { n } ) = \\mathrm { E } \\left [ ( X - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { n } | x _ { p } < X < x _ { q } \\right ] , \\end{align*}"}
{"id": "2812.png", "formula": "\\begin{align*} x _ { i + 1 } = x _ i - \\tfrac { h _ i } { L } g _ i , \\forall i = 0 , \\dots , N - 1 . \\end{align*}"}
{"id": "6702.png", "formula": "\\begin{align*} J _ { w , \\tau , \\lambda } \\psi ( \\lambda , \\cdot ) = \\mathbf { c } _ { w ^ { - 1 } , \\tau } ( - w \\lambda ) \\psi ( w \\lambda , \\cdot ) , \\ ; \\ ; \\ ; \\lambda \\in \\mathfrak { a } ^ * _ \\mathbb { C } , w \\in W _ A . \\end{align*}"}
{"id": "7161.png", "formula": "\\begin{align*} y _ 3 \\{ g _ { 1 1 } y _ 1 + g _ { 1 2 } y _ 2 + ( g _ { 1 3 } + g _ { 2 3 } ) y _ 3 \\} ^ { a - 1 } = ( g _ { 1 3 } + g _ { 2 3 } ) ^ { a - 1 } y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } , \\end{align*}"}
{"id": "984.png", "formula": "\\begin{align*} \\operatorname { t r } ( \\alpha ) = \\operatorname { T r } \\left ( H ^ { * } ( \\overline { \\alpha } ) \\right ) , \\ \\ \\ N ( \\alpha ) = \\operatorname { d e t } \\left ( H ^ { * } ( \\overline { \\alpha } ) \\right ) . \\end{align*}"}
{"id": "7119.png", "formula": "\\begin{align*} 0 = \\frac { \\partial \\tau ( t , x ) } { \\partial t } + \\frac { \\partial \\tau ( t , x ) } { \\partial x } \\frac { \\partial p ( t ; t , x ) } { \\partial \\tau } \\ , . \\end{align*}"}
{"id": "768.png", "formula": "\\begin{align*} P _ { \\rm F A } ^ { } ( \\lambda ) = Q \\left ( \\frac { { ( 1 - \\alpha ) \\left ( { \\lambda } - 1 \\right ) \\bar { N } { M } - { M } } } { \\sqrt { 2 { M } + \\left ( 2 - \\alpha - { \\lambda } \\right ) ^ 2 \\bar { N } { M } } } \\right ) \\end{align*}"}
{"id": "4179.png", "formula": "\\begin{align*} | \\mu | ^ { - n } \\norm { \\varphi _ k ^ { | \\mu | } } _ 2 ^ 2 = | \\{ \\nu \\in \\N ^ { n } : | \\nu | _ 1 = k \\} | = \\binom { k + n - 1 } { k } \\sim ( k + 1 ) ^ { n - 1 } . \\end{align*}"}
{"id": "1476.png", "formula": "\\begin{align*} \\psi _ s : K [ t ] \\longrightarrow K ; \\ t ^ k \\mapsto \\dfrac { 1 } { ( k + \\gamma _ { r - s } ) \\cdots ( k + \\gamma _ r ) } = \\dfrac { 1 } { ( k + \\zeta _ 1 ) \\cdots ( k + \\zeta _ { s + 1 } ) } \\enspace , \\end{align*}"}
{"id": "1405.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} B & { \\bf 0 } \\\\ { \\bf 0 } & C \\end{matrix} \\right ] , B = \\left [ \\begin{matrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ \\epsilon & 0 & 1 \\end{matrix} \\right ] , \\end{align*}"}
{"id": "8312.png", "formula": "\\begin{align*} \\sum _ { j = m } ^ { \\infty } | c _ j | = O \\left ( v \\left ( \\frac { 1 } { m } \\right ) \\right ) , \\end{align*}"}
{"id": "6514.png", "formula": "\\begin{align*} \\sum _ { { z = 1 } } ^ \\infty \\frac { 1 } { A ( z ) } = \\infty \\end{align*}"}
{"id": "4845.png", "formula": "\\begin{align*} & \\left . \\frac { d } { d t } ( X , \\delta _ X D f \\wedge \\delta _ X D g ) \\right | _ { t = 0 } = ( \\dot X , \\delta _ X D f \\wedge \\delta _ X D g ) + ( X , \\dot { \\delta _ X } D f \\wedge \\delta _ X D g ) \\\\ & + ( X , \\delta _ X \\dot { D f } \\wedge \\delta _ X D g ) + ( X , \\delta _ X { D f } \\wedge \\dot { \\delta _ X } D g ) + ( X , \\delta _ X { D f } \\wedge \\delta _ X \\dot { D g } ) . \\end{align*}"}
{"id": "3039.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ g [ \\alpha _ i , \\beta _ i ] \\prod _ { j = 1 } ^ k \\gamma _ j = 1 , \\end{align*}"}
{"id": "2147.png", "formula": "\\begin{align*} d \\nu ( s ) = \\frac { 1 8 } { \\pi ^ { 4 } } \\sum \\nolimits _ { m \\in \\mathbb { Z } } \\frac { 1 } { m ^ { 2 } } I _ { [ m , m + 1 ] } ( s ) \\left \\{ \\sum \\nolimits _ { n \\in \\mathbb { N } } \\frac { 1 } { 2 ^ { n } n ^ { 2 } } \\sum \\nolimits _ { j = 1 } ^ { 2 ^ { n } } d \\tau \\left ( s - m - j 2 ^ { - n } \\right ) \\right \\} , \\end{align*}"}
{"id": "5546.png", "formula": "\\begin{align*} M _ { n , k } ( k ) = \\dfrac { k - n + 1 } { k + 1 } \\dbinom { k + n } { n } = \\dbinom { k + n } { n } - \\dbinom { k + n } { n - 1 } . \\end{align*}"}
{"id": "2544.png", "formula": "\\begin{align*} R i c ( X _ 1 , Y _ 1 ) = a _ 1 a _ 3 ( 1 - 2 e ^ { 2 x _ 1 } ) ( e ^ { 2 x _ 1 } - 1 ) . \\end{align*}"}
{"id": "2701.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\ , \\left \\langle \\left ( \\mathcal { N } _ 1 - \\frac { N } { 2 } \\right ) ^ 2 \\right \\rangle _ { \\psi _ \\mathrm { g s } } = 0 , \\end{align*}"}
{"id": "2193.png", "formula": "\\begin{align*} Q & = \\left ( \\tau ^ { \\rho ^ { ( l ) } + \\eta ^ { ( l ) } _ { \\tt r e v } } ( Q ^ { ( l ) } _ 0 ) , \\dots , \\tau ^ { \\rho ^ { ( 1 ) } + \\eta ^ { ( 1 ) } _ { \\tt r e v } } ( Q ^ { ( 1 ) } _ 0 ) \\right ) = \\left ( Q ^ { ( l ) } , \\dots , Q ^ { ( 1 ) } \\right ) . \\end{align*}"}
{"id": "3775.png", "formula": "\\begin{align*} H _ { k , j ; n , l , r ; \\star } ^ { \\mu , m , i ; l i n } ( t , x , \\zeta ) = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big [ ( t - s ) \\mathcal { K } _ { k , n , l , r } ^ { \\star ; \\mu , m } ( y , v , \\omega , \\zeta ) + \\mathcal { K } _ { k , n , l , r } ^ { \\star ; e r r , \\mu , m } ( y , v , \\omega , \\zeta ) \\big ] f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "8150.png", "formula": "\\begin{align*} \\varphi _ n = \\varphi _ { n ; \\textrm { o u t } } + \\varphi _ { n ; \\textrm { i n } } + \\varphi _ { n ; \\textrm { s u r } } \\end{align*}"}
{"id": "8590.png", "formula": "\\begin{align*} S _ 2 = \\sum _ { u , v , z } b _ { u \\vee z } b _ { v \\vee z } \\int _ { E ^ c } e ^ { 2 \\pi i ( u - v ) x } d x , \\end{align*}"}
{"id": "88.png", "formula": "\\begin{align*} a ^ \\epsilon ( \\lambda , m ^ 2 ) = \\begin{cases} - c _ 1 \\lambda \\log ( \\epsilon ^ { - 2 } ) + O _ { m ^ 2 , \\lambda } ( 1 ) & ( d = 2 ) , \\\\ - c _ 1 \\lambda \\epsilon ^ { - 1 } + c _ 2 \\lambda ^ 2 \\log ( \\epsilon ^ { - 2 } ) + O _ { m ^ 2 , \\lambda } ( 1 ) & ( d = 3 ) . \\end{cases} \\end{align*}"}
{"id": "3933.png", "formula": "\\begin{align*} \\alpha : = \\frac { \\bar { \\beta } _ 3 } { 2 \\bar { \\beta } _ 3 + d - 2 } ( \\frac { 1 } { \\beta _ 1 } + \\frac { 1 } { \\beta _ 2 } ) = \\frac { \\bar { \\beta } _ 3 } { 2 \\bar { \\beta } _ 3 + d - 2 } ( \\frac { d } { \\bar { \\beta } } - \\frac { d - 2 } { \\bar { \\beta _ 3 } } ) = \\frac { \\bar { \\beta } _ 3 d - ( d - 2 ) \\bar { \\beta } } { \\bar { \\beta } ( 2 \\bar { \\beta } _ 3 + d - 2 ) } . \\end{align*}"}
{"id": "8331.png", "formula": "\\begin{align*} b _ { i - l + k - n , i } ( k , l ) P _ { i - l + k - n , j } + b _ { l , l + j - i + n } ( i , j ) P _ { k , l + j - i + n } = \\chi _ { l , i } \\delta ( P _ { k , j } ) \\ , . \\end{align*}"}
{"id": "6497.png", "formula": "\\begin{align*} \\theta _ 1 ( x ) = \\left \\{ \\begin{array} { c l c } 1 & & x \\in ( b _ 1 + \\varepsilon , b _ 2 - \\varepsilon ) , \\\\ 0 & & x \\in ( 0 , b _ 1 ) \\cup ( b _ 2 , L ) , \\\\ 0 \\leq \\theta _ 1 \\leq 1 & & . \\end{array} \\right . \\end{align*}"}
{"id": "3077.png", "formula": "\\begin{align*} & c _ { n - 2 , \\xi _ { \\boldsymbol { s } k } , \\xi _ { \\boldsymbol { t } l } } ^ { \\ast \\ast } = \\frac { \\Gamma ( ( n - 2 ) / 2 ) } { ( 2 \\pi ) ^ { ( n - 2 ) / 2 } } \\left \\{ \\int _ { 0 } ^ { \\infty } t ^ { ( n - 4 ) / 2 } \\ln \\left [ 1 + \\exp \\left ( - \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } , k } ^ { 2 } - \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { t } , l } ^ { 2 } \\right ) \\exp ( - t ) \\right ] \\mathrm { d } t \\right \\} ^ { - 1 } . \\end{align*}"}
{"id": "7740.png", "formula": "\\begin{align*} Q ( a ) ^ k = b ( a \\otimes a ) ^ { \\binom { k } { 2 } } Q ( a ^ k ) . \\end{align*}"}
{"id": "4161.png", "formula": "\\begin{align*} \\Phi _ { \\nu , \\nu ' } ^ \\lambda ( z ) : = ( 2 \\pi ) ^ { - n / 2 } \\lambda ^ { n / 2 } ( \\pi _ \\lambda ( z , 0 ) \\Phi _ \\nu ^ \\lambda , \\Phi _ { \\nu ' } ^ \\lambda ) , z \\in \\R ^ { 2 n } \\end{align*}"}
{"id": "73.png", "formula": "\\begin{align*} W _ A ^ u = T _ { u ( 0 ) } W ^ u ( x ) , W ^ s _ A = T _ { u ( 0 ) } W ^ s ( y ) , \\end{align*}"}
{"id": "6356.png", "formula": "\\begin{align*} \\lambda _ * = \\dfrac { \\rho ^ { \\varphi ^ + - { q ^ \\pm } } } { 2 c _ 2 c ^ { q ^ \\pm } } . \\end{align*}"}
{"id": "14.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { \\rho , L , n } = \\frac { 1 } { \\eta ^ n } T _ { L | K } ( \\lambda _ { \\rho } ( \\alpha ) \\delta _ n ( \\beta ) ) \\cdot _ { \\rho } v _ n , \\end{align*}"}
{"id": "7290.png", "formula": "\\begin{align*} \\theta _ 1 ( x ) & = { \\sf c } _ 1 | x | ^ 2 { \\sf U } _ \\infty ^ { p - 1 } \\theta _ 0 + { \\sf h } _ 1 ( x ) = { a } _ 1 | x | ^ \\frac { 2 ( p - q ) } { 1 - q } \\theta _ 0 + { \\sf h } _ 1 ( x ) ( { a } _ 1 \\not = 0 ) , \\\\ { \\sf h } _ 1 ( x ) & = | x | ^ \\frac { 2 ( p - q ) } { 1 - q } \\theta _ 0 \\sum _ { i = 1 } ^ { N - 1 } b _ i | x | ^ \\frac { 2 ( p - q ) i } { 1 - q } . \\end{align*}"}
{"id": "3353.png", "formula": "\\begin{align*} \\varphi ( f ) ( t ) = u ( t ) \\begin{pmatrix} f ( \\lambda _ 1 ( t ) ) & 0 & \\ldots & 0 & 0 \\\\ 0 & f ( \\lambda _ 2 ( t ) ) & \\ldots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\ldots & f ( \\lambda _ a ( t ) ) & 0 \\\\ 0 & 0 & \\ldots & 0 & 0 _ { m - n a } \\end{pmatrix} u ( t ) ^ { \\ast } . \\end{align*}"}
{"id": "2446.png", "formula": "\\begin{align*} L _ { f \\cdot \\tau } S _ \\zeta = q ^ { - \\ell ( \\tau ) } \\widetilde { N } _ { f \\cdot \\tau } + \\sum _ { g \\prec f \\cdot \\tau } \\ell _ { g , f \\cdot \\tau } M _ g S _ \\zeta , \\end{align*}"}
{"id": "2172.png", "formula": "\\begin{align*} p _ { \\mu \\boxplus \\nu } ( h ( x ) ) = \\frac { f ( x ) } { \\pi } \\int _ { \\mathbb { R } } \\frac { d \\nu ( s ) } { ( x - s ) ^ { 2 } + f ( x ) ^ { 2 } } , x \\in V , \\end{align*}"}
{"id": "6056.png", "formula": "\\begin{align*} \\Lambda _ 0 = ( 0 , 1 ) \\end{align*}"}
{"id": "1755.png", "formula": "\\begin{align*} Z ( \\overline \\delta , \\overline \\mu , \\beta ) = \\exp ( F ^ { P } \\big ( g _ s , Q , \\beta ) + F ^ { N P } ( g _ s , Q , \\beta ) \\big ) . \\end{align*}"}
{"id": "7175.png", "formula": "\\begin{align*} \\langle Y _ M ' ( u , z ) f , a \\rangle = \\langle f , Y _ { M } ( e ^ { z L ( 1 ) } ( - z ^ { - 2 } ) ^ { L ( 0 ) } u , z ^ { - 1 } ) a \\rangle \\end{align*}"}
{"id": "4131.png", "formula": "\\begin{align*} B ^ { i } \\phi : = P _ { \\Upsilon ^ { i } } ( I - d _ { V } ^ { i - 1 } G ^ { i } ) \\phi , \\end{align*}"}
{"id": "6703.png", "formula": "\\begin{align*} \\varphi ( \\lambda ) \\gamma ( m _ w ^ { - 1 } ) \\mathbf { c } _ { w ^ { - 1 } , \\gamma } ( - w \\lambda ) = \\tau ( m _ w ^ { - 1 } ) \\mathbf { c } _ { w ^ { - 1 } , \\tau } ( - w \\lambda ) \\circ \\varphi ( w \\lambda ) , \\ ; \\ ; \\ ; \\lambda \\in \\mathfrak { a } ^ * _ \\mathbb { C } , w \\in W _ A . \\end{align*}"}
{"id": "596.png", "formula": "\\begin{align*} a ^ 2 + b ^ 2 - \\frac { 2 } { 3 } a ^ 2 b ^ 2 \\geq a ^ 2 + b ^ 2 - \\frac { 2 } { 3 } a ^ 2 = \\frac { 1 } { 3 } a ^ 2 + b ^ 2 \\geq 0 . \\end{align*}"}
{"id": "79.png", "formula": "\\begin{align*} \\| Y ( t ) - u ( t ) \\| & = \\| Y _ i - \\tilde v ( t ) \\| \\leq \\| Y _ i - \\tilde v ( \\sigma _ i ) \\| + \\| \\tilde v ( \\sigma _ i ) - \\tilde v ( t ) \\| < \\frac { \\epsilon } { 2 } , \\end{align*}"}
{"id": "2158.png", "formula": "\\begin{align*} \\left | 1 - M ( x + i f ( x ) ) \\right | = \\frac { 2 } { \\sqrt { x ^ { 2 } + [ f ( x ) + 1 ] ^ { 2 } } } \\leq \\frac { 2 } { | x | } \\rightarrow 0 \\quad ( | x | \\rightarrow \\infty ) . \\end{align*}"}
{"id": "6401.png", "formula": "\\begin{align*} V _ 1 = \\{ 4 , 7 \\} , \\ , V _ 2 = \\{ 1 , 6 \\} , \\ , V _ 3 = \\{ 2 , 5 \\} , \\ , V _ 4 = \\{ 3 \\} . \\end{align*}"}
{"id": "8733.png", "formula": "\\begin{align*} \\sum \\limits _ { i , j = 1 } ^ { 2 } h _ { \\dot { x } ^ { i } \\dot { x } ^ { j } } y ^ { i } y ^ { j } \\leq 0 , \\end{align*}"}
{"id": "5047.png", "formula": "\\begin{align*} N ^ n _ \\tau = N ^ { n , 1 } _ \\tau + N ^ { n , 2 } _ \\tau , \\end{align*}"}
{"id": "7989.png", "formula": "\\begin{align*} \\hat { h } _ i : = ( h _ 1 , \\ldots , h _ { i - 1 } , h _ { i + 1 } , \\ldots , h _ n ) \\end{align*}"}
{"id": "1122.png", "formula": "\\begin{align*} g ^ { - 1 } ( s | \\rho , \\lambda ) = \\frac { \\lambda s } { ( \\sigma _ N ^ 2 + \\sigma _ Z ^ 2 ) \\lambda ^ 2 + \\rho } . \\end{align*}"}
{"id": "4144.png", "formula": "\\begin{align*} \\hat f ( \\xi ) = \\int _ { \\R ^ n } f ( x ) e ^ { - i \\xi x } \\ , d x , \\xi \\in \\R ^ n , \\end{align*}"}
{"id": "7200.png", "formula": "\\begin{align*} \\langle f , \\sum _ { i \\geq 0 } \\binom { \\frac { j _ 1 } { T } } { i } ( - 1 ) ^ i z _ 2 ^ { \\frac { j _ 1 } { T } - i } I ^ { \\circ } ( w _ 1 , z _ 2 ) u _ { p + \\frac { j _ 2 } { T } + i } w _ 2 \\rangle = 0 , \\end{align*}"}
{"id": "4677.png", "formula": "\\begin{align*} h _ 1 b _ 1 = b _ 2 h _ 2 , h _ 2 \\in U ^ { - } _ { u ^ { - 1 } , > 0 } , b _ 2 \\in B ^ + _ { \\ge 0 } . \\end{align*}"}
{"id": "1233.png", "formula": "\\begin{align*} [ Z _ j ] _ { \\alpha , \\alpha } & = \\sum _ { i = 1 } ^ { n ( T , j + 1 ) } [ D _ { j + 1 } ] _ { i , \\alpha } \\ [ D _ { j + 1 } ] _ { i , \\alpha } \\ [ R _ { j + 1 } ^ { - 1 } ] _ { i , i } \\\\ & = \\sum _ { i = 1 } ^ { n ( T , j + 1 ) } [ D _ { j + 1 } ] _ { i , \\alpha } \\ [ R _ { j + 1 } ^ { - 1 } ] _ { i , i } \\ , . \\end{align*}"}
{"id": "8724.png", "formula": "\\begin{align*} \\Psi ( \\omega ) = h _ { 2 } \\omega - \\frac { d } { d t } ( h _ { 1 } \\omega ' ) . \\end{align*}"}
{"id": "4195.png", "formula": "\\begin{align*} A _ \\ell ( y ) : = \\{ ( x , u ) \\in B _ { 3 R } ^ { d _ { \\mathrm { C C } } } ( 0 ) : | x - y | \\ge 2 ^ { \\gamma \\iota } C R _ \\ell \\} \\end{align*}"}
{"id": "2377.png", "formula": "\\begin{align*} F _ p ( x ) = L ( h _ \\theta ) + \\sum _ { i \\in I } \\partial _ { i } L ( h _ \\theta ) ( Q - h _ \\theta ) ^ { i } = : L _ p ( Q ) . \\end{align*}"}
{"id": "1116.png", "formula": "\\begin{align*} C ( v ) = \\ln \\cosh ( z _ 0 v ) . \\end{align*}"}
{"id": "5930.png", "formula": "\\begin{align*} W \\left [ { \\eta \\left ( t \\right ) } \\right ] = \\ln \\left \\langle { e ^ { \\mathop \\int d t \\eta ( t ) \\xi ( t ) } } \\right \\rangle \\end{align*}"}
{"id": "8957.png", "formula": "\\begin{align*} M _ { 2 , 2 } ( \\R ^ d ) = L ^ 2 ( \\R ^ d ) . \\end{align*}"}
{"id": "6228.png", "formula": "\\begin{align*} p ( x ) = - \\dfrac { x } { 1 - q } , r ( x ) = \\frac { q ^ { 1 - n } [ n ] _ q } { 1 - q } . \\end{align*}"}
{"id": "1916.png", "formula": "\\begin{align*} S ^ { ( q ) } _ { j } ( z ) & : = \\sum _ { m = 0 } ^ { \\infty } \\frac { S ^ { ( q ) } _ { [ m ( p + 1 ) + j , j ] } } { z ^ { m ( p + 1 ) + j + 1 } } , \\\\ T ^ { ( q ) } _ { j } ( z ) & : = \\sum _ { m = 0 } ^ { \\infty } \\frac { T ^ { ( q ) } _ { [ m ( p + 1 ) + j , j ] } } { z ^ { m ( p + 1 ) + j + 1 } } . \\end{align*}"}
{"id": "1005.png", "formula": "\\begin{align*} \\{ X _ N ( 0 ) = a _ 0 , \\ldots X _ N ( \\ell ) = a _ { \\ell } \\} \\end{align*}"}
{"id": "2120.png", "formula": "\\begin{align*} x _ { n + 1 } = f ( x _ { n } ) + \\epsilon ( L _ G ( g ( x _ n ) ) ) , \\end{align*}"}
{"id": "6552.png", "formula": "\\begin{align*} E _ 2 ( i , m ) = \\underbrace { \\frac { 1 } { e \\sqrt { i + 1 } } } _ { \\geq \\frac { 1 } { e \\sqrt { m + 1 } } } \\underbrace { \\Big ( \\frac { e ^ { 1 - \\frac { 1 } { A ( m + 1 ) } } } { 2 A ( m + 1 ) } \\Big ) ^ { i + 1 } } _ { \\geq \\Big ( \\frac { 1 } { 2 A ( m + 1 ) } \\Big ) ^ { i + 1 } } \\geq \\Big ( \\frac { 1 } { 2 A ( m + 1 ) } \\Big ) ^ { i + 2 } \\end{align*}"}
{"id": "114.png", "formula": "\\begin{align*} \\eta _ t = C _ \\infty ( 0 ) - C _ t ( 0 ) \\geq 0 , \\gamma _ t = \\| C _ \\infty ^ 3 \\| _ { L ^ 1 } - \\| C _ t ^ 3 \\| _ { L ^ 1 } \\geq 0 . \\end{align*}"}
{"id": "801.png", "formula": "\\begin{align*} | g ( z ^ m ) | + \\sum _ { k = N } ^ { \\infty } | b _ k | | z | ^ k \\leq d ( 0 , \\partial { \\Omega } ) \\end{align*}"}
{"id": "1683.png", "formula": "\\begin{align*} \\| T x \\| = \\| x \\| \\end{align*}"}
{"id": "2965.png", "formula": "\\begin{align*} ( T _ p \\varphi ) ( z ) = 2 \\sqrt y \\sum _ { n \\neq 0 } \\left ( a _ \\varphi ( p n ) + p ^ { - 1 } a _ \\varphi ( n / p ) \\right ) K _ { i r } ( 2 \\pi | n | y ) e ( n x ) . \\end{align*}"}
{"id": "4777.png", "formula": "\\begin{align*} | \\mathcal { S } _ { \\kappa , 3 } ^ B | = 3 \\left ( \\sum _ { i , j \\in \\mathbb N \\colon \\atop { 2 ( i + j ) \\leq \\kappa } } \\binom { \\kappa - 2 ( i + j ) + 3 } { 3 } - A _ \\kappa \\right ) = 3 b _ \\kappa . \\end{align*}"}
{"id": "8674.png", "formula": "\\begin{align*} W ( \\sigma ) = R _ { \\varphi ( \\sigma ) } ( u ) \\wedge v + u \\wedge R _ { \\varphi ( \\sigma ) } ( v ) . \\end{align*}"}
{"id": "948.png", "formula": "\\begin{align*} \\int \\limits _ { V } \\left ( \\overline { \\rho _ R \\log \\rho _ R } - \\rho _ R \\log \\rho _ R \\right ) ( \\tau , \\cdot ) \\ d x = \\frac { 1 } { 2 \\mu + \\lambda } \\int \\limits _ 0 ^ { \\tau } \\int \\limits _ { V } \\left ( \\overline { p ( \\rho _ R ) } \\rho _ R - \\overline { p ( \\rho _ R ) \\rho _ R } \\right ) \\ d x \\ d t . \\end{align*}"}
{"id": "2650.png", "formula": "\\begin{align*} \\zeta _ { \\Z ^ n } ^ R ( s ) & = \\sum _ { \\substack { S \\\\ \\Z ^ n } } [ \\Z ^ n : S ] ^ { - s } \\\\ & = \\sum _ { k \\ge 1 } f _ n ( k ) k ^ { - s } . \\end{align*}"}
{"id": "4629.png", "formula": "\\begin{align*} \\epsilon _ g = s i g n ( b _ g \\tilde c _ 1 ( \\alpha , g ) , \\tilde c _ 2 ( \\alpha , g ) ) \\end{align*}"}
{"id": "5198.png", "formula": "\\begin{align*} k _ 1 ( \\Gamma ) = r ( J ) + p r k _ 2 ( \\Gamma ) = s ( J ) + ( \\ell _ 1 + \\ell _ 2 - | J | + 1 + \\sum _ { j \\in J } d _ j - p ) s , \\end{align*}"}
{"id": "5677.png", "formula": "\\begin{align*} \\alpha _ j \\varpi _ j = 0 ( 1 \\le j \\le n - 1 ) , \\end{align*}"}
{"id": "5813.png", "formula": "\\begin{align*} \\beta ( 2 n + 1 ) = \\frac { \\pi ^ { 2 n + 1 } E _ { n } } { 4 ^ { n + 1 } ( 2 n ) ! } \\end{align*}"}
{"id": "7579.png", "formula": "\\begin{align*} E _ \\mathcal { N } = | | x _ \\mathcal { N } - x ( b ) | | , \\end{align*}"}
{"id": "1247.png", "formula": "\\begin{align*} W _ 0 & = 1 , \\\\ W _ 1 & = x , \\\\ W _ j & = x W _ { j - 1 } - c _ { l ( T ) + 1 - j } W _ { j - 2 } ( \\forall j \\in \\overline { 2 , l ( T ) } ) . \\end{align*}"}
{"id": "1363.png", "formula": "\\begin{align*} \\widehat { \\left ( | u | ^ { p - 1 } u \\right ) } _ k = 2 \\pi \\sum _ { k = k _ 1 - k _ 2 + \\cdots + k _ { p } } \\widehat { u } _ { k _ 1 } \\overline { \\widehat { u } } _ { k _ 2 } \\cdots \\widehat { u } _ { k _ { p - 2 } } \\overline { \\widehat { u } } _ { k _ { p - 1 } } \\widehat { u } _ { k _ p } = \\frac { 1 } { L _ t [ u ] } \\widehat { \\left ( | v | ^ { p - 1 } v \\right ) } _ k . \\end{align*}"}
{"id": "6171.png", "formula": "\\begin{align*} r & = a b _ { 1 } - a _ { 1 } , \\\\ a & = a _ { 1 } b _ { 2 } - a _ { 2 } , \\\\ a _ { i } & = a _ { i + 1 } b _ { i + 2 } - a _ { i + 2 } , \\\\ & \\dots . \\end{align*}"}
{"id": "3267.png", "formula": "\\begin{align*} N ^ { - 1 } _ \\omega ( g ) ( x ) = \\frac { 1 } { ( 2 \\pi ) ^ 3 } \\int _ { \\R ^ 3 } e ^ { - i x \\cdot \\xi } \\Big ( \\frac { \\hat { g } ( \\xi ) } { \\omega \\cdot \\xi } \\Big ) d \\xi , \\end{align*}"}
{"id": "4024.png", "formula": "\\begin{align*} \\lambda _ n ( x ) \\triangleq \\sum _ { i = 0 } ^ \\infty a _ i x ^ i \\end{align*}"}
{"id": "3723.png", "formula": "\\begin{align*} t ( v ) = t ' ( h _ k v ) \\end{align*}"}
{"id": "2297.png", "formula": "\\begin{align*} \\mathcal { E } _ u ^ { ( n ) } = & ( \\chi - 1 ) f _ u ^ { ( n ) } + \\mathcal { \\tilde E } _ u ^ { ( n ) } \\end{align*}"}
{"id": "7888.png", "formula": "\\begin{align*} \\mathfrak H : = \\bigoplus _ { \\vec s \\in \\mathbb Z ^ n } \\mathfrak H _ { \\vec s } , \\end{align*}"}
{"id": "6466.png", "formula": "\\begin{align*} \\Phi _ b - \\Phi _ a & = \\int _ a ^ b \\frac { \\dd J _ t } { \\dd t } \\ , d t \\\\ \\Psi - \\Phi & = \\int _ a ^ b \\left ( \\Bar Q \\circ H _ t + H _ t \\circ Q _ { \\mathfrak { g } } \\right ) d t \\\\ & = \\Bar Q \\circ \\left ( \\int _ a ^ b H _ t \\ , d t \\right ) + \\left ( \\int _ a ^ b H _ t \\ , d t \\right ) \\circ Q _ { \\mathfrak { g } } . \\end{align*}"}
{"id": "939.png", "formula": "\\begin{align*} \\begin{aligned} & \\langle \\nabla _ i \\nabla _ j \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { i j } ^ l , \\nabla _ m \\nabla _ m \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { m m } ^ l - \\frac { \\sum _ k ( \\nabla _ k \\nabla _ k \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { k k } ^ l ) } { n } \\rangle \\\\ & = \\frac { 1 } { 2 t } O ( t ) + \\sum _ l \\Gamma _ { i j } ^ l ( \\Gamma _ { m m } ^ l - \\frac { \\sum _ k \\Gamma ^ l _ { k k } } { n } ) . \\end{aligned} \\end{align*}"}
{"id": "6308.png", "formula": "\\begin{align*} & \\int x ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty h _ n ( x ; q ) d _ q x = \\frac { ( 1 - q ) x ( x ^ 2 ; q ^ 2 ) _ \\infty } { [ n ] _ q - 1 } \\left ( \\frac { q ^ n } { x } h _ n ( \\frac { x } { q } ; q ) - q ^ { n - 1 } [ n ] _ q h _ { n - 1 } ( \\frac { x } { q } ; q ) \\right ) , \\end{align*}"}
{"id": "3832.png", "formula": "\\begin{align*} E r r U ^ 4 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\sum _ { \\begin{subarray} { c } k _ 2 \\in \\Z _ + , n _ 2 \\in [ - M _ t , 2 ] \\cap \\Z \\\\ \\mu _ 2 \\in \\{ + , - \\} , i _ 2 \\in \\{ 0 , 1 , 2 , 3 \\} \\end{subarray} } \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { ( \\R ^ 3 ) ^ 3 } e ^ { i X ( s ) \\cdot \\xi + i s \\mu _ 1 | \\xi - \\eta | } \\mathcal { F } \\big [ \\big ( \\sum _ { l _ 2 \\in [ - M _ t , 2 ] \\cap \\Z } I n i _ { k _ 2 , j _ 2 , n _ 2 } ^ { \\mu _ 2 , i _ 2 } ( s , X ( s ) , V ( s ) ) \\end{align*}"}
{"id": "1379.png", "formula": "\\begin{align*} i u _ t + \\bigtriangleup u + i \\gamma u = 0 , \\end{align*}"}
{"id": "5312.png", "formula": "\\begin{align*} | \\pi _ i ( t ) | : = \\lim _ { n \\to \\infty } n ^ { - 1 } \\# ( \\pi _ i ( t ) \\cap [ n ] ) . \\end{align*}"}
{"id": "7938.png", "formula": "\\begin{align*} f _ + ^ * K _ { X _ + } = f _ - ^ * K _ { X _ - } + \\left ( \\sum _ { i = 1 } ^ m D _ i \\cdot e \\right ) E . \\end{align*}"}
{"id": "2651.png", "formula": "\\begin{align*} a _ n ( p ^ e ) = \\binom { n - 1 + e } { e } _ p , \\end{align*}"}
{"id": "3395.png", "formula": "\\begin{align*} T _ n ( f ) : = n \\wedge S _ n ( f ) , f \\in C . \\end{align*}"}
{"id": "7538.png", "formula": "\\begin{align*} \\mathcal { W } _ { \\frac { p } { 2 } } ( g \\circ W _ { n } , g \\circ W ) & = \\mathcal { W } _ { \\frac { p } { 2 } } ( g \\circ W _ { n } \\circ \\pi _ \\Delta , W ) \\\\ & \\le \\mathcal { W } _ { \\frac { p } { 2 } } ( g \\circ W _ { n } \\circ \\pi _ \\Delta , \\sigma X _ n ) + \\mathcal { W } _ { \\frac { p } { 2 } } ( \\sigma X _ n , \\sigma B ) \\\\ & \\le C n ^ { - \\frac { 1 } { 4 } + \\frac { 1 } { 4 ( p - 1 ) } } + C n ^ { - \\frac { 1 } { 4 } + \\delta } \\le C n ^ { - \\frac { 1 } { 4 } + \\frac { 1 } { 4 ( p - 1 ) } } , \\end{align*}"}
{"id": "8818.png", "formula": "\\begin{align*} \\lim _ { x ' \\to x } \\psi ( x ' ) = \\lim _ { x ' \\to x } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha _ { t _ 1 , t _ 2 } ( x ' ) d t _ 1 d t _ 2 = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha _ { t _ 1 , t _ 2 } ( x ) d t _ 1 d t _ 2 = \\psi ( x ) \\end{align*}"}
{"id": "2466.png", "formula": "\\begin{align*} \\dim \\left [ \\gg , x \\right ] = | \\Delta _ { 1 } \\backslash { A } ^ { \\perp } | + 2 k . \\end{align*}"}
{"id": "7065.png", "formula": "\\begin{align*} \\Psi ( x , y ) = \\begin{cases} \\Psi ^ + ( x , y ) , & y > 0 , \\\\ \\Psi ^ - ( x , y ) , & y < 0 , \\end{cases} \\end{align*}"}
{"id": "2011.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : N p a r t i c l e m e a n f i e l d g e n e r a t o r } \\mathcal { L } _ N \\varphi _ N ( \\mathbf { x } ^ N ) = \\sum _ { i = 1 } ^ N L _ { \\mu _ { \\mathbf { x } ^ N } } \\diamond _ i \\varphi _ N ( \\mathbf { x } ^ N ) , \\end{align*}"}
{"id": "5105.png", "formula": "\\begin{align*} \\Lambda _ n = \\Lambda ^ { ( 1 ) } _ { n , \\delta } + \\Lambda ^ { ( 2 ) } _ { n , \\delta } , \\end{align*}"}
{"id": "7859.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\| e _ i ^ { n , g } J e _ i ^ { n , g } J ( K _ { g , n } ^ i - [ T _ g ^ i , J x _ n J ] ) J e _ i ^ { n , g } J e _ i ^ { n , g } \\| = 0 . \\end{align*}"}
{"id": "2066.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : T i n f t y t p h i k M c K e a n } T _ { \\infty , t } \\Phi _ k ( \\nu ) = \\prod _ { i = 1 } ^ k \\big \\langle \\nu , T ^ { \\nu } _ t \\varphi ^ i \\big \\rangle , \\end{align*}"}
{"id": "1169.png", "formula": "\\begin{align*} \\Lambda _ { \\mu } : = D _ { \\mu } \\cap \\big \\{ h = ( \\rho - \\mu ) * \\gamma _ { \\sigma } : \\big \\} . \\end{align*}"}
{"id": "1190.png", "formula": "\\begin{align*} \\| E ( h ) \\| _ { L ^ p ( \\rho ; \\R ^ d ) } = \\sup \\left \\{ \\int _ { \\R ^ d } \\langle \\nabla \\varphi , E ( h ) \\rangle d \\rho : \\varphi \\in C _ 0 ^ \\infty , \\| \\nabla \\varphi \\| _ { L ^ q ( \\rho ; \\R ^ d ) } \\le 1 \\right \\} = \\| h \\| _ { \\dot { H } ^ { - 1 , p } ( \\rho ) } . \\end{align*}"}
{"id": "3529.png", "formula": "\\begin{align*} c _ { 1 } = \\frac { \\Gamma \\left ( t \\right ) } { \\Gamma ( t - 1 / 2 ) \\pi ^ { \\frac { 1 } { 2 } } } , ~ t > \\frac { 1 } { 2 } , \\end{align*}"}
{"id": "6735.png", "formula": "\\begin{align*} J ( \\overline { y } , \\overline { u } ) \\leq \\liminf _ { n } J ( y _ n , u _ n ) = \\hat { J } . \\end{align*}"}
{"id": "5134.png", "formula": "\\begin{align*} \\omega = \\sum _ { i = 1 } ^ g [ \\alpha _ i , \\alpha ^ \\# _ i ] \\in \\pi _ { 2 n } ( Z _ { g , 1 } ) , \\end{align*}"}
{"id": "8695.png", "formula": "\\begin{align*} A _ { H T } = \\frac { 1 } { 2 } \\int \\limits _ { t _ { 0 } } ^ { t _ { 1 } } ( x ^ { 1 } \\dot { x } ^ { 2 } - x ^ { 2 } \\dot { x } ^ { 1 } ) d t . \\end{align*}"}
{"id": "3558.png", "formula": "\\begin{align*} J M & = \\sum _ { i = 1 } ^ { n } \\binom { n - 1 } { i - 1 } ( 1 - y ) ^ i ( 1 + y ) ^ { n - i } i - 1 = \\alpha \\\\ & = \\sum _ { \\alpha = 0 } ^ { n - 1 } \\binom { n - 1 } { \\alpha } ( 1 - y ) ^ { \\alpha + 1 } ( 1 + y ) ^ { n - 1 - \\alpha } \\\\ & = ( 1 - y ) \\sum _ { \\alpha = 0 } ^ { n - 1 } \\binom { n - 1 } { \\alpha } ( 1 - y ) ^ { \\alpha } ( 1 + y ) ^ { n - 1 - \\alpha } \\\\ & = ( 1 - y ) ( 1 - y + 1 + y ) ^ { n - 1 } \\\\ & = 2 ^ { n - 1 } ( 1 - y ) \\\\ & = [ 2 ^ { n - 1 } , - 2 ^ { n - 1 } , 0 , \\ldots , 0 ] . \\end{align*}"}
{"id": "7921.png", "formula": "\\begin{align*} C _ \\omega : = \\bigcap _ { I \\in \\mathcal A _ \\omega } \\angle _ I \\subset \\mathbb L ^ \\vee \\otimes \\mathbb R . \\end{align*}"}
{"id": "7963.png", "formula": "\\begin{align*} H _ { ( X _ + , D _ + ) } ( y ) = e ^ { \\frac { t _ + } { 2 \\pi i } } \\sum _ { d \\in \\mathbb K _ { + } } y ^ d \\left ( \\frac { 1 } { \\prod _ { i \\in M _ 0 } \\Gamma ( 1 + \\frac { \\bar D _ i } { 2 \\pi i } + D _ i \\cdot d ) } \\right ) \\textbf { 1 } _ { [ d ] } [ \\textbf { 1 } ] _ { ( D _ i \\cdot d ) _ { i \\in I _ + } } . \\end{align*}"}
{"id": "1207.png", "formula": "\\begin{align*} \\inf _ { \\theta \\in N _ 1 \\cap \\Theta _ n ^ c } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\theta } ) & > { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\mu ) + o _ { \\mathbb P } ( n ^ { - 1 / 2 } ) \\\\ & \\geq \\inf _ { \\theta \\in N _ 1 \\cap \\Theta _ n } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ \\theta ) + o _ { \\mathbb P } ( n ^ { - 1 / 2 } ) . \\end{align*}"}
{"id": "3770.png", "formula": "\\begin{align*} { H } ^ { m , i ; p , q } _ { k , j ; n , l , r } ( t , x , \\zeta ) : = \\int _ { 0 } ^ { t } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ 0 ^ { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\| \\mathfrak { m } ( \\cdot , \\zeta ) \\| _ { \\mathcal { S } ^ \\infty } 2 ^ { 3 k + 2 n } 2 ^ { - j - r + 2 \\epsilon M _ t } | \\hat { v } + \\omega | | B ( s , x - y + ( t - s ) \\omega ) | f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "8665.png", "formula": "\\begin{align*} Q ( x ) \\leq \\begin{cases} \\frac { ( d - 2 ) ^ 2 } { 4 | x | ^ 2 } & \\ d \\neq 2 \\ , , \\\\ \\frac { 1 } { 4 | x | ^ 2 ( \\ln | x | ) ^ 2 } & \\ d = 2 \\ , , \\end{cases} \\qquad \\ | x | \\geq R \\ , . \\end{align*}"}
{"id": "2129.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { | z - s | ^ 2 } - \\frac { 1 } { s ^ 2 } \\right | & = \\frac { 1 } { s ^ 2 } \\frac { | 2 ( s - x ) x + x ^ 2 - y ^ 2 | } { ( x - s ) ^ 2 + y ^ 2 } \\\\ & \\leq \\frac { 1 } { s ^ 2 } \\left ( \\frac { 2 \\delta | x - s | y } { ( x - s ) ^ 2 + y ^ 2 } + \\frac { x ^ 2 + y ^ 2 } { y ^ 2 } \\right ) \\leq \\frac { 1 } { s ^ 2 } \\left ( \\delta + \\delta ^ 2 + 1 \\right ) \\in L ^ { 1 } ( \\nu ) . \\end{align*}"}
{"id": "7363.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { y } _ j & = r \\ , \\left ( \\sqrt { 1 - h ^ 2 } \\cos \\left ( \\frac { 2 ( j - 1 ) \\pi } { k } \\right ) , \\sqrt { 1 - h ^ 2 } \\sin \\left ( \\frac { 2 ( j - 1 ) \\pi } { k } \\right ) , h \\right ) \\\\ \\underline { y } _ j & = r \\ , \\left ( \\sqrt { 1 - h ^ 2 } \\cos \\left ( \\frac { 2 ( j - 1 ) \\pi } { k } \\right ) , \\sqrt { 1 - h ^ 2 } \\sin \\left ( \\frac { 2 ( j - 1 ) \\pi } { k } \\right ) , - h \\right ) \\end{aligned} \\end{align*}"}
{"id": "255.png", "formula": "\\begin{align*} \\delta ^ { ( s ) } c _ k = v _ k , \\ \\ k \\in \\{ 1 , \\ldots , n \\} \\end{align*}"}
{"id": "3561.png", "formula": "\\begin{align*} c _ i ^ 2 N _ i ' ( N _ i ' - 1 ) + c _ i ^ 2 \\cdot 2 N _ i ' N _ i '' + c _ i ^ 2 { N _ i '' } ^ 2 & = c _ i ^ 2 \\left ( { N _ i ' } ^ 2 + 2 N _ i ' N _ i '' + { N _ i '' } ^ 2 - N _ i ' \\right ) \\\\ & = c _ i ^ 2 \\left ( ( N _ i ' + N _ i '' ) ^ 2 - N _ i ' \\right ) \\\\ & = c _ i ^ 2 \\left ( N _ i ^ 2 - N _ i ' \\right ) , \\ , \\ , ( N _ i ' + N _ i '' = N _ i ) \\\\ & = c _ i \\left ( N _ i - N _ i ' \\right ) \\pmod 2 \\\\ & = c _ i N _ i '' \\pmod 2 . \\\\ \\end{align*}"}
{"id": "292.png", "formula": "\\begin{align*} f ( a ) = \\frac { 1 } { a \\log a } \\ \\le \\ \\frac { 1 } { a \\log 2 P ( a ) } \\ < \\ \\frac { e ^ \\gamma } { a } \\prod _ { p < P ( a ) } \\Big ( 1 - \\frac { 1 } { p } \\Big ) \\ = \\ e ^ \\gamma \\ , { \\rm d } ( { \\rm L } _ a ) . \\end{align*}"}
{"id": "7587.png", "formula": "\\begin{align*} \\frac { d x } { d t } = \\sum _ { \\alpha = 1 } ^ r b _ { \\alpha } ( t ) X _ { \\alpha } ( x ) , x ( a ) = x _ 0 , \\end{align*}"}
{"id": "4689.png", "formula": "\\begin{align*} C _ m ( n ) = \\sum _ { j = - \\infty } ^ \\infty ( - 1 ) ^ j u _ m \\big ( n - j ( 3 j - 1 ) / 2 \\big ) . \\end{align*}"}
{"id": "8123.png", "formula": "\\begin{align*} \\tilde \\lambda ( h \\iota ) = \\lambda ( h ) , h \\in H \\end{align*}"}
{"id": "3302.png", "formula": "\\begin{align*} S ( \\partial _ t ) \\varphi ( x , t ) : = \\int _ 0 ^ t \\int _ \\Gamma k ( | x - y | , t - \\tau ) \\varphi ( y , \\tau ) d \\Gamma _ y d \\tau x \\in \\R ^ d \\setminus \\Gamma \\end{align*}"}
{"id": "8947.png", "formula": "\\begin{align*} D w ( x _ 0 ) = 0 , { \\rm a n d } \\ \\ L w ( x _ 0 ) \\le 0 , \\end{align*}"}
{"id": "2590.png", "formula": "\\begin{align*} [ E _ { j k } , E _ { l m } ] = \\delta _ { l , k } E _ { j m } - \\delta _ { j , m } E _ { l k } . \\end{align*}"}
{"id": "654.png", "formula": "\\begin{align*} D L _ { ( g , f ) } ( 0 , h ) = & \\ 2 \\Delta ^ f _ g \\left ( 2 \\Delta _ g ^ f h + h \\right ) - ( 4 \\pi ) ^ { - \\frac { n } { 2 } } \\int _ { M } h e ^ { - f } d v _ g \\\\ & - 2 \\left \\langle \\nabla _ g h , \\nabla _ g \\left ( 2 \\Delta _ g f + f - | \\nabla f | ^ 2 + R _ g \\right ) \\right \\rangle . \\end{align*}"}
{"id": "7706.png", "formula": "\\begin{align*} \\sum _ { n _ 1 \\cdots n _ k \\le x } f ( [ n _ 1 , \\ldots , n _ k ] ) = x ^ { r + 1 } Q _ { f , k - 1 } ( \\log x ) + O \\big ( x ^ { r + \\theta _ k + \\varepsilon } \\big ) , \\end{align*}"}
{"id": "7798.png", "formula": "\\begin{align*} P ( X _ t , \\eta _ t ) = t , \\end{align*}"}
{"id": "6321.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) - \\dfrac { x } { 1 - q } D _ { q ^ { - 1 } } y ( x ) + \\frac { q ^ { 1 - n } [ n ] _ q } { 1 - q } y ( x ) = 0 . \\end{align*}"}
{"id": "899.png", "formula": "\\begin{align*} \\norm { h } ^ 2 = u ( a ) . \\end{align*}"}
{"id": "5689.png", "formula": "\\begin{align*} m _ { d + 1 , 1 ^ k } ( x _ 1 , x _ 2 , \\ldots , x _ i ) = \\begin{cases} m _ { d , 1 ^ k } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d \\ge 2 ) , \\\\ \\\\ ( k + 1 ) m _ { 1 ^ { k + 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d = 1 ) . \\end{cases} \\end{align*}"}
{"id": "8443.png", "formula": "\\begin{align*} \\dim _ { \\rm H } D _ { \\beta } ( \\varphi , x _ 0 ) = \\frac { 1 } { 1 + \\alpha } \\alpha = \\liminf _ { n \\to \\infty } \\frac { \\log _ { \\beta } \\varphi ( n ) ^ { - 1 } } { n } , \\end{align*}"}
{"id": "5919.png", "formula": "\\begin{align*} { \\partial _ t } { x _ k } \\left ( t \\right ) = { A _ { k p } } \\left ( t \\right ) { x _ p } \\left ( t \\right ) , \\ \\ k = 1 \\dots d \\end{align*}"}
{"id": "8789.png", "formula": "\\begin{align*} q ( P ) = \\left ( 1 - \\frac { p ' } { 2 } + \\frac { p ' } { 2 } \\cdot \\frac { 1 } { p } \\right ) ^ { - 1 } = 2 \\end{align*}"}
{"id": "431.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\min _ { 1 \\le i \\le k _ { n } } \\mu _ { n , i } \\left ( ( - \\varepsilon , \\varepsilon ) \\right ) = 1 \\end{align*}"}
{"id": "5398.png", "formula": "\\begin{align*} B _ { \\gamma } ( u , \\phi ) = F ( \\phi ) \\quad u - f \\in \\Tilde { H } ^ s ( \\Omega ) \\end{align*}"}
{"id": "1407.png", "formula": "\\begin{align*} s e p ( R _ { X _ 1 } \\Lambda _ 1 R _ { X _ 1 } ^ { - 1 } , R _ { V _ 2 } \\Lambda _ 2 R _ { V _ 2 } ^ { - 1 } ) \\geq \\frac { s e p ( \\Lambda _ 1 , \\Lambda _ 2 ) } { \\kappa _ 2 ( R _ { X _ 1 } ) \\kappa _ 2 ( R _ { V _ 2 } ) } = \\frac { \\delta _ \\lambda } { \\kappa _ 2 ( X _ 1 ) } , \\end{align*}"}
{"id": "4476.png", "formula": "\\begin{align*} A _ { \\mu } ( \\zeta ) = \\sum \\limits _ { k = 1 } ^ \\infty \\frac 1 { k ! } K _ 0 ^ { ( k ) } ( \\zeta ) | \\zeta | ^ { k - 2 \\mu } \\left ( \\sum \\limits _ { | \\vec { i } | = k , \\sum i _ l l = \\mu } ^ \\infty \\left ( \\begin{array} { c } k \\\\ \\vec { i } \\end{array} \\right ) \\left [ \\prod \\limits _ { l = 1 } ^ \\infty \\left ( \\begin{array} { c } \\frac 1 2 \\\\ l \\end{array} \\right ) ^ { i _ l } \\right ] \\right ) . \\end{align*}"}
{"id": "6911.png", "formula": "\\begin{align*} W _ i = \\mathsf { s g n } ( | \\hat \\theta _ i | - | \\hat \\theta ' _ i | ) \\ , \\max ( | \\hat \\theta _ i | , | \\hat \\theta ' _ i | ) \\ . \\end{align*}"}
{"id": "2778.png", "formula": "\\begin{align*} T _ 2 ( h _ i { } \\geq { } 1 , h _ { i - 1 } { } \\geq { } 1 ) = 0 \\end{align*}"}
{"id": "5706.png", "formula": "\\begin{align*} \\pi _ { \\emptyset } = 1 \\in M . \\end{align*}"}
{"id": "7118.png", "formula": "\\begin{align*} \\tau ( t , x ) = \\tau ( \\sigma , p ( \\sigma ; t , x ) ) \\ , . \\end{align*}"}
{"id": "5690.png", "formula": "\\begin{align*} m _ { d + 1 , 1 ^ 0 } ( x _ 1 , x _ 2 , \\ldots , x _ i ) = \\begin{cases} m _ { d , 1 ^ 0 } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d \\ge 2 ) , \\\\ \\\\ m _ { 1 ^ { 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d = 1 ) \\end{cases} \\end{align*}"}
{"id": "1680.png", "formula": "\\begin{align*} \\| O _ 1 y _ 1 + y _ 2 \\| & \\ge \\| O _ 1 y _ 1 + z _ 2 \\| - \\| z _ 2 - y _ 2 \\| \\\\ & \\ge ( 1 - \\alpha ) \\| y _ 1 + z _ 2 \\| - \\| z _ 2 - y _ 2 \\| \\\\ & \\ge ( 1 - \\alpha ) \\| y _ 1 + y _ 2 \\| - ( 2 - \\alpha ) \\| z _ 2 - y _ 2 \\| \\\\ & > ( 1 - \\alpha ) \\| y _ 1 + y _ 2 \\| - ( 2 - \\alpha ) \\alpha \\| y _ 2 \\| \\\\ & \\ge ( 1 - \\alpha ( 1 + ( 2 - \\alpha ) A ) ) \\| y _ 1 + y _ 2 \\| . \\end{align*}"}
{"id": "2417.png", "formula": "\\begin{align*} \\mathbb P \\big ( \\sup _ { t , x } | u ^ n ( t , x ) - u ( t , x ) | ^ { 2 p } \\ge 1 \\big ) \\le \\sum _ { i = 1 } ^ 3 \\mathbb P ( R _ i ^ { ( n ) } \\ge 3 ^ { - 2 p } ) \\end{align*}"}
{"id": "1490.png", "formula": "\\begin{align*} ( x ) _ { n } = \\sum _ { k = 0 } ^ { n } S _ { 1 , \\lambda } ( n , k ) ( x ) _ { k , \\lambda } , ( n \\ge 0 ) , \\end{align*}"}
{"id": "3627.png", "formula": "\\begin{align*} \\dfrac { w \\ ! : \\ ! 1 \\ ! : \\ ! \\lozenge \\phi \\ ! = \\ ! \\frac { r } { m + 1 } } { w \\mathsf { R } w '' ; w '' \\ ! : \\ ! 1 \\ ! : \\ ! \\phi \\ ! = \\ ! \\frac { r } { m + 1 } } & & \\dfrac { w \\ ! : \\ ! 1 \\ ! : \\ ! \\lozenge \\phi \\ ! = \\ ! \\frac { r } { m + 1 } ; w \\mathsf { R } w ' } { w ' \\ ! : \\ ! 1 \\ ! : \\ ! \\phi \\ ! = \\ ! 0 \\mid \\ldots \\mid w ' \\ ! : \\ ! 1 \\ ! : \\ ! \\phi \\ ! = \\ ! \\frac { r - 1 } { m + 1 } } \\end{align*}"}
{"id": "2915.png", "formula": "\\begin{align*} x ^ n - 1 = \\prod _ { i = 1 } ^ t m ^ { ( ( q ^ m - 1 ) s _ i / n ) } ( x ) , \\end{align*}"}
{"id": "8750.png", "formula": "\\begin{align*} \\begin{array} { l l } A _ n ( t ) = \\Pi _ { k = 1 } ^ n \\left [ ( \\kappa { \\mathcal U } _ k ( s , t ) + I ) \\partial _ t { \\rm L o g } ( { \\mathcal U _ k } ( t , s ) + \\kappa I ) \\right ] , \\end{array} \\end{align*}"}
{"id": "910.png", "formula": "\\begin{align*} f ^ * g _ N - \\frac { \\mathrm { t r } _ { g _ M } f ^ * g _ N } { m } g _ M = 0 , \\end{align*}"}
{"id": "3339.png", "formula": "\\begin{align*} g _ 4 ( K ( 3 , n \\tilde { q } , n \\tilde { p } ) ) = d - 1 \\end{align*}"}
{"id": "6628.png", "formula": "\\begin{align*} ( \\Phi _ z ) _ * \\big ( \\tau _ p ^ { ( y ) } \\big ) = | z | _ p ^ { 2 } \\ , \\tau _ p ^ { ( z ^ { - 2 } y ) } . \\end{align*}"}
{"id": "1162.png", "formula": "\\begin{align*} g ^ c ( y ) = \\inf _ { x \\in \\R ^ d } \\big [ c ( x , y ) - g ( x ) \\big ] , y \\in \\R ^ d . \\end{align*}"}
{"id": "1941.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { B } _ { q } e _ { 0 } = \\sum _ { k = 0 } ^ { p } a _ { - k - q } ^ { ( k ) } \\ , e _ { k } , \\\\ \\mathcal { B } _ { q } e _ { n } = e _ { n - 1 } + \\sum _ { k = 0 } ^ { p } a _ { - k - q - n } ^ { ( k ) } \\ , e _ { n + k } , n \\geq 1 . \\end{cases} \\end{align*}"}
{"id": "47.png", "formula": "\\begin{align*} \\Tilde { \\mathcal { N } } ( f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ) = f ^ { \\tilde { \\phi } ^ { - n } } . \\end{align*}"}
{"id": "2019.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : g r a d i e n t m c k e a n } b ( x , \\mu ) = - \\nabla V ( x ) - \\int _ { \\R ^ d } \\nabla W ( x - y ) \\mu ( \\dd y ) , \\end{align*}"}
{"id": "7140.png", "formula": "\\begin{align*} m _ 2 p q ^ { m _ 2 - 1 } & = q ^ { m _ 2 } ( m _ 2 - 1 ) , \\\\ { m _ 2 \\choose 2 } p ^ 2 q ^ { m _ 2 - 2 } & = q ^ { m _ 2 } { m _ 2 - 1 \\choose 2 } \\end{align*}"}
{"id": "5897.png", "formula": "\\begin{align*} \\mu _ { V X ^ { ( n ) } } ( A ) = \\mathbb { P } ( V X ^ { ( n ) } \\in A ) = \\mathbb { P } \\Big ( \\sum _ { j = 1 } ^ n n ^ { 1 / p } \\frac { Z _ j } { ( \\lVert Z ^ { ( n ) } \\rVert _ p ^ p + W _ n ) ^ { 1 / p } } V _ { \\bullet , j } \\in A \\Big ) , \\end{align*}"}
{"id": "3448.png", "formula": "\\begin{align*} \\ker ( \\kappa _ f ) \\cong & \\langle \\ , x _ i \\frac { \\partial f } { \\partial x _ i } , i = 1 , . . . , n \\\\ & \\ , \\ , \\ , w _ { - \\alpha } ( f ) , \\alpha \\in R ( N , \\Sigma _ { \\Delta } ) \\ , \\rangle . \\end{align*}"}
{"id": "369.png", "formula": "\\begin{align*} \\frac { \\tilde p ^ \\# _ m ( x , y ) } { \\tilde \\pi ( y ) } = 1 + \\sum _ { j = 2 } ^ { n } f _ j ( x ) f _ j ( y ) \\lambda _ j ^ m , \\end{align*}"}
{"id": "8196.png", "formula": "\\begin{align*} A = ( D P ) ^ { \\dagger } \\left ( \\begin{array} { c c } I & 0 \\\\ 0 & - I \\end{array} \\right ) ( D P ) , \\end{align*}"}
{"id": "8124.png", "formula": "\\begin{align*} F ( t ) & = P f ( \\cdot , t ) , \\\\ F _ n ( t ) & = P _ n f ( \\cdot , t ) . \\end{align*}"}
{"id": "6764.png", "formula": "\\begin{align*} \\nu _ p \\left ( \\binom { \\alpha \\beta } { p ^ y } \\right ) = x , \\end{align*}"}
{"id": "5701.png", "formula": "\\begin{align*} k ! ( x _ i - x _ { i + 1 } ) e _ k ( x _ 1 , \\ldots , x _ i ) = \\alpha _ i \\varpi _ { i - k + 1 } \\varpi _ { i - k + 2 } \\cdots \\varpi _ { i } . \\end{align*}"}
{"id": "2686.png", "formula": "\\begin{align*} u _ 1 : = \\frac { u _ + + u _ - } { \\sqrt { 2 } } , u _ 2 : = \\frac { u _ + - u _ - } { \\sqrt { 2 } } \\end{align*}"}
{"id": "4646.png", "formula": "\\begin{align*} { \\hat { A } } ^ n x & = J ^ { - 1 } A ^ n J = \\left ( y _ k + y _ 0 \\right ) _ { k \\in \\N } \\\\ & = \\left ( \\left [ \\prod _ { j = k } ^ { k + n - 1 } w ^ j \\right ] \\left ( x _ { k + n } - l ( x ) \\right ) + \\left [ \\prod _ { j = 0 } ^ { n - 1 } w ^ j \\right ] \\left ( x _ { n } - l ( x ) \\right ) \\right ) _ { k \\in \\N } \\in c ( \\N ) . \\end{align*}"}
{"id": "6150.png", "formula": "\\begin{align*} u ^ { \\Delta t } _ { h } - v ^ { \\Delta t } _ { h } & = \\frac { 1 } { 2 } \\ , u _ h ^ m + ( 1 - \\frac { 1 } { 2 } ) \\ , u _ h ^ { m - 1 } - u _ { h } ^ { m - 1 } - \\frac { t - t _ { m - 1 } } { \\Delta t } ( u _ { h } ^ { m } - u _ { h } ^ { m - 1 } ) \\\\ & = \\left ( \\frac { 1 } { 2 } - \\frac { t - t _ { m - 1 } } { \\Delta t } \\right ) \\left ( u _ h ^ { m } - u _ { h } ^ { m - 1 } \\right ) . \\end{align*}"}
{"id": "1444.png", "formula": "\\begin{align*} ( \\theta _ t - k ) ^ n \\circ [ t ^ k ] ( t ^ m ) = ( k + m - k ) ^ n t ^ { m + k } = m ^ n t ^ { m + k } \\enspace . \\end{align*}"}
{"id": "632.png", "formula": "\\begin{align*} & \\left | \\ E ^ { \\frac { n } { 2 } } _ { y , M } \\circ E ^ { \\frac { n } { 2 } } _ { x , M } \\frac { G ' \\cdot ( x ^ 2 + y ^ 2 ) ^ { N ' } } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { \\alpha } } \\ \\right | \\leqslant C \\log ( x ^ 2 + y ^ 2 + z ^ 2 ) \\Big | _ { x = 0 } ^ { x = \\frac { n } { 2 } } \\Big | _ { y = 0 } ^ { y = \\frac { n } { 2 } } \\leqslant C \\log n . \\end{align*}"}
{"id": "3662.png", "formula": "\\begin{align*} \\mathcal { F } ( f ) ( \\xi ) = \\int e ^ { - i x \\cdot \\xi } f ( x ) d x . \\end{align*}"}
{"id": "2411.png", "formula": "\\begin{align*} \\| L _ { \\mu } ( s ) \\| ^ 2 _ { \\mathrm F } & \\le C \\sum _ { l = 1 } ^ { n - 1 } ( - \\lambda _ { l , n } ) ^ { 2 \\mu } e ^ { - \\lambda ^ 2 _ { l , n } ( t - s ) } \\| \\mathbb U ( s ) - U ( s ) \\| ^ 2 \\\\ & \\le C _ \\epsilon ( t - s ) ^ { - \\frac { 1 } { 4 } - \\mu - \\epsilon } \\| \\mathbb U ( s ) - U ( s ) \\| ^ 2 \\\\ & \\le C _ \\epsilon ( t - s ) ^ { - \\frac { 1 } { 4 } - \\mu - \\epsilon } \\| \\mathbb U ( s ) - \\tilde U ( s ) \\| ^ 2 + C _ \\epsilon ( t - s ) ^ { - \\frac { 1 } { 4 } - \\mu - \\epsilon } \\| E ( s ) \\| ^ 2 , \\end{align*}"}
{"id": "8361.png", "formula": "\\begin{align*} u \\times v \\mapsto \\langle u | v \\rangle _ t = e ^ { t ( ( \\tilde { f } ^ { | u \\rangle } , \\tilde { f } ^ { | v \\rangle } ) _ { { } _ { \\mathfrak { J } } } + 4 \\pi ) } = e ^ { - t 4 \\pi ( \\lambda \\textrm { c o t h } \\lambda - 1 ) } \\end{align*}"}
{"id": "621.png", "formula": "\\begin{align*} F ' _ { k , j } ( x , y ) = { \\alpha + j - 1 \\choose j } \\sum _ { k _ 2 = 0 } ^ { k } \\dots \\sum _ { k _ { j } = 0 } ^ { k _ { j - 1 } } d _ { k _ j } & \\prod _ { \\ell = 1 } ^ { j - 1 } d _ { k _ { \\ell } - k _ { \\ell + 1 } } \\\\ ( x ^ { 2 k _ j + 4 } + y ^ { 2 k _ j + 4 } ) & \\prod _ { \\ell = 1 } ^ { j - 1 } ( x ^ { 2 ( k _ { \\ell } - k _ { \\ell + 1 } ) + 4 } + y ^ { 2 ( k _ { \\ell } - k _ { \\ell + 1 } ) + 4 } ) . \\end{align*}"}
{"id": "4078.png", "formula": "\\begin{align*} \\boldsymbol { M } = \\sum _ { j = 0 } ^ 6 \\mathbf { s } ( j ) P ( \\boldsymbol { X } = \\mathbf { s } ( j ) ) . \\end{align*}"}
{"id": "1726.png", "formula": "\\begin{align*} \\frac { F ( z + \\overline \\omega _ 1 \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) } { F ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) } = \\frac { 1 } { 1 - x _ 2 } , \\frac { F ( z + \\omega _ 2 \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) } { F ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) } = \\frac { 1 } { 1 - x _ 1 } . \\end{align*}"}
{"id": "8343.png", "formula": "\\begin{align*} \\widetilde { f } _ \\mu ( p ) = \\sum \\limits _ { i } ^ { N } \\ , \\alpha _ i { \\textstyle \\frac { u _ { i \\mu } } { u _ i \\cdot p } } , \\ , \\ , \\ , \\sum \\limits _ { i } ^ { N } \\alpha _ i = 0 , \\end{align*}"}
{"id": "2217.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } H = \\frac { | \\bigtriangledown ^ { T } u | ^ { 2 } } { u + A } + t r _ { \\omega } \\widehat { \\omega } _ { \\infty } . \\end{array} \\end{align*}"}
{"id": "2329.png", "formula": "\\begin{align*} \\widetilde A \\widetilde D - \\widetilde B \\widetilde C = 1 , \\widetilde { E } < \\widetilde { A } + \\widetilde { C } , \\widetilde { F } < \\widetilde { B } + \\widetilde { D } , - \\widetilde C \\leq \\widetilde C \\widetilde F - \\widetilde D \\widetilde E < \\widetilde D . \\end{align*}"}
{"id": "5131.png", "formula": "\\begin{align*} ( a b + c d ) ^ 2 = \\Big ( ( a + d ) ( b + c ) \\Big ) \\cdot ( a b + c d ) - \\Big ( ( a + d ) ^ 2 b c + a d ( b + c ) ^ 2 - 4 ( a d ) ( b c ) \\Big ) \\cdot 1 . \\end{align*}"}
{"id": "7761.png", "formula": "\\begin{align*} u _ { { \\rm H i P O D } } ^ { L , M _ L } ( \\alpha ^ * ) = \\displaystyle \\sum _ { k = 1 } ^ { m } \\Big [ \\sum _ { j = 1 } ^ { N _ h } { u } _ { { \\rm P O D } , k , j } ^ { \\alpha ^ * } \\ , \\vartheta _ j ( x ) \\Big ] \\varphi _ k ( \\psi _ x ( { \\bf y } ) ) , \\end{align*}"}
{"id": "3997.png", "formula": "\\begin{align*} \\alpha _ i '^ { - 1 } \\alpha _ i c _ i = c _ i \\beta _ i ' \\beta _ i ^ { - 1 } , ~ ~ ~ ~ ~ ~ i = 0 , 1 , \\cdots , r . \\end{align*}"}
{"id": "8418.png", "formula": "\\begin{gather*} t ^ s _ { d } ( \\tau , z ) : = \\overline { \\theta _ { d ^ 2 , s d ^ 2 } ^ 0 ( \\tau ) } \\theta _ { 1 , s d } ( \\tau , z ) \\end{gather*}"}
{"id": "3584.png", "formula": "\\begin{align*} \\mu ( y , x ) & = \\begin{cases} 1 & y = x , \\\\ - \\sum \\limits _ { y \\leq t < x } \\mu ( y , t ) & y < x , \\\\ 0 & y > x . \\end{cases} \\end{align*}"}
{"id": "4671.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { m - 1 } \\sum \\limits _ { j = 1 } ^ { m - 1 } \\sum \\limits _ { k = 1 } ^ { m - 1 } l _ { j k } \\tau _ { i j } \\tau _ { i k } = t r ( \\ { \\Gamma } ^ T L { \\Gamma } ) = t r ( L \\ { \\Gamma } { \\Gamma } ^ T ) \\\\ = \\sum \\limits _ { i = 1 } ^ { m - 1 } \\sum \\limits _ { j = 1 } ^ { m - 1 } l _ { i j } h ^ { j i } = \\sum \\limits _ { i = 1 } ^ { m - 1 } \\sum \\limits _ { j = 1 } ^ { m - 1 } l _ { i j } h ^ { i j } . \\end{align*}"}
{"id": "422.png", "formula": "\\begin{align*} \\Re H ' _ { \\mu } ( z ) > 0 z = s + i t , \\ , t \\ge f _ { \\mu } ( s ) . \\end{align*}"}
{"id": "2078.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m a r t i n g a l e P o i s s o n m e a s u r e } M _ t = \\int _ 0 ^ t \\int _ \\Theta a ( s , \\theta ) \\mathcal { N } ( \\dd s , \\dd \\theta ) - \\int _ 0 ^ t \\int _ \\Theta a ( s , \\theta ) \\nu ( \\dd \\theta ) \\dd s , \\end{align*}"}
{"id": "8888.png", "formula": "\\begin{align*} \\sigma = \\left ( \\sigma ^ 2 - | \\det ( \\sigma ^ 2 I _ n - A ^ H A ) | \\left ( \\frac { n - 1 } { ( n + 1 ) \\sigma ^ 2 - \\| A \\| _ F ^ 2 } \\right ) ^ { n - 1 } \\right ) ^ { 1 / 2 } \\end{align*}"}
{"id": "204.png", "formula": "\\begin{align*} \\hat { H } _ { 1 } = g ^ { n } _ { 1 } ( Z ^ { n } _ { 1 } , M _ 0 , M _ 1 ) . \\end{align*}"}
{"id": "2321.png", "formula": "\\begin{align*} \\mathcal { R } ( \\psi ) = \\left ( \\begin{array} { l l l } m _ { 0 0 } & m _ { 0 1 } & 0 \\\\ m _ { 1 0 } & m _ { 1 1 } & 0 \\\\ E & F & 1 \\end{array} \\right ) , \\ \\left ( \\begin{array} { l l } m _ { 0 0 } & m _ { 0 1 } \\\\ m _ { 1 0 } & m _ { 1 1 } \\end{array} \\right ) = M _ \\psi \\ \\ \\ \\ E , F \\in \\mathbb { N } . \\end{align*}"}
{"id": "1849.png", "formula": "\\begin{align*} \\gamma = e _ { 1 } e _ { 2 } \\cdots e _ { k } , \\end{align*}"}
{"id": "6697.png", "formula": "\\begin{align*} \\mathcal { F } _ \\tau f ( \\lambda , k ) = \\int _ G e ^ \\tau _ { \\lambda , k } ( g ) f ( g ) \\ ; d g = \\int _ { G / K } e ^ \\tau _ { \\lambda , k } ( g ) f ( g ) \\ ; d g , \\end{align*}"}
{"id": "1627.png", "formula": "\\begin{align*} \\mathcal { A D } _ k = \\{ r _ k ( M ) : M \\in \\mathcal { D } \\} , \\end{align*}"}
{"id": "8164.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } \\| \\chi _ { S _ { v t } ^ c } e ^ { - i t H } \\psi \\| \\leq \\lim \\limits _ { t \\rightarrow \\infty } \\sup _ { \\tau \\geq 0 } \\| \\chi _ { S _ { v t } ^ c } e ^ { - i \\tau H } \\psi \\| = \\lim \\limits _ { R \\rightarrow \\infty } \\sup _ { \\tau \\geq 0 } \\| \\chi _ { S _ { R } ^ c } e ^ { - i \\tau H } \\psi \\| \\end{align*}"}
{"id": "1061.png", "formula": "\\begin{align*} \\mathrm { e } ^ { - t \\operatorname { S p e c } ( A _ j ) } = \\operatorname { S p e c } ( T ( t , A _ j ) ) \\end{align*}"}
{"id": "6759.png", "formula": "\\begin{align*} \\log \\left ( 1 + \\frac { \\log ( 1 + x ) } { u } \\right ) = \\sum _ { n = 1 } ^ \\infty \\ell _ n ( u ) x ^ n , \\end{align*}"}
{"id": "6060.png", "formula": "\\begin{align*} f ^ { ( j ) } ( x ) = \\sum _ { ( k _ 1 , k _ 2 , \\ldots , k _ { m } ) } \\binom { j } { k _ 1 , k _ 2 , \\ldots , k _ { m } } \\prod _ { i = 1 } ^ { m } h _ i ^ { ( k _ i ) } ( x ) \\end{align*}"}
{"id": "5444.png", "formula": "\\begin{align*} \\emph { \\textsf { K e r } } ( l ) = B \\emph { \\textsf { K e r } } ( r ) = A . \\end{align*}"}
{"id": "4540.png", "formula": "\\begin{align*} J _ { 2 2 } = \\frac { 4 } { m n \\pi ^ { 2 } } \\int _ { h _ { 1 } } ^ { \\pi } \\int _ { 0 } ^ { h _ { 2 } } H _ { x , z _ { 1 } , y , z _ { 2 } } ( t _ { 1 } , t _ { 2 } ) \\frac { S ( t _ { 1 } , t _ { 2 } ) } { t _ { 1 } ^ { 2 } \\left ( 2 \\sin \\frac { t _ { 2 } } { 2 } \\right ) ^ { 2 } } d t _ { 1 } d t _ { 2 } \\end{align*}"}
{"id": "3329.png", "formula": "\\begin{align*} S = \\{ \\beta \\in B ( 3 ) \\slash \\exists n \\in \\mathbb { N } , \\exists i , j \\in \\{ 1 , 2 \\} \\ \\ \\mbox { s u c h t h a t } \\ \\ \\ s i g n \\big ( ( \\beta \\sigma _ i \\beta ^ { - 1 } \\sigma _ j ) ^ n \\big ) \\neq 0 \\} \\end{align*}"}
{"id": "5656.png", "formula": "\\begin{align*} \\sigma _ k ( \\lambda ( D ^ 2 \\underline { u } ) ) & = ( \\underline { u } ' ) ^ k + \\underline { u } '' ( \\underline { u } ' ) ^ { k - 1 } \\Sigma _ { i = 1 } ^ n \\sigma _ { k - 1 ; i } ( a ) ( a _ i x _ i ^ 2 ) \\\\ & \\geq ( \\underline { u } ' ) ^ k + 2 h _ k ( a ) \\underline { u } '' ( \\underline { u } ' ) ^ { k - 1 } \\tau \\\\ & = \\overline { f } \\quad \\ \\mathbb { R } ^ n \\setminus \\overline { D } _ { s _ 0 } . \\end{align*}"}
{"id": "487.png", "formula": "\\begin{align*} | P - Q | \\leq 2 \\sqrt { \\epsilon _ 0 } + 2 \\sqrt { \\epsilon _ 0 } = 4 \\sqrt { \\epsilon _ 0 } . \\end{align*}"}
{"id": "4098.png", "formula": "\\begin{align*} { B } \\overline { u } ^ \\prime ( t ) + A \\overline { u } ( t ) + B Q \\overline { u } ( t ) = - H ( { u } ^ \\prime ( t ) + Q u ( t ) ) , \\ t \\in \\ , [ 0 , T ] , \\quad \\overline { u } ( 0 ) = 0 . \\end{align*}"}
{"id": "5526.png", "formula": "\\begin{align*} \\Phi ^ * = \\Phi \\setminus \\mathcal { H } \\subseteq \\Phi \\setminus \\bigcup _ { v \\in \\overline { \\Phi } \\cap \\mathbb { S } ^ 1 } H ^ v \\end{align*}"}
{"id": "7723.png", "formula": "\\begin{align*} \\pi _ { S } ^ T ( k + 1 ) A _ { S \\bar { S } } ( k ) e = \\left ( \\pi _ { \\bar { S } } ^ T ( k ) - \\pi _ { \\bar { S } } ^ T ( k + 1 ) \\right ) e + \\pi _ { \\bar { S } } ^ T ( k + 1 ) A _ { \\bar { S } S } ( k ) e \\end{align*}"}
{"id": "5305.png", "formula": "\\begin{align*} \\int _ { \\Psi _ { j + 1 } - \\Psi _ j } x ^ k e ^ { x ^ r / \\hbar } d x = \\frac { 1 } { r } e ^ { \\pi i \\frac { k + 1 } { r } + 2 \\pi i j \\frac { k + 1 } { r } } \\hbar ^ { ( k + 1 ) / r } \\Gamma \\left ( \\frac { k + 1 } { r } \\right ) \\left ( e ^ { 2 \\pi i ( k + 1 ) / r } - 1 \\right ) = \\frac { 1 } { r } C _ k \\zeta ^ { j ( k + 1 ) } \\end{align*}"}
{"id": "6873.png", "formula": "\\begin{align*} p _ { t , 0 } ( y ) = e ^ { t } p _ { } ( e ^ t y ) , p ( 0 , g ) = p _ { } ( g ) . \\end{align*}"}
{"id": "7577.png", "formula": "\\begin{align*} \\frac { x _ { k + 1 } - x _ k } { h } = f _ h ( t _ k , x _ k , x _ { k + 1 } ) , \\end{align*}"}
{"id": "4282.png", "formula": "\\begin{align*} \\phi ( q _ \\rho ) = g _ \\rho \\rtimes \\rho , \\qquad \\phi _ 0 ( q _ \\rho ) = e \\phi ( q _ \\rho ) = e _ \\rho g _ \\rho \\rtimes \\rho \\end{align*}"}
{"id": "2219.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } R ^ { T } ( t ) = - \\Delta _ { B } u - t r _ { \\omega } \\widehat { \\omega } _ { \\infty } . \\end{array} \\end{align*}"}
{"id": "3811.png", "formula": "\\begin{align*} | E l l ^ { a } _ { k , j , n } ( s , x , \\zeta ) | \\lesssim \\sum _ { l \\in [ - M _ t , n + \\epsilon M _ t ] \\cap \\Z } | E l l ^ { a ; l } _ { k , j , n } ( s , x , \\zeta ) | , E l l ^ { a ; l } _ { k , j , n } ( s , x , \\zeta ) : = 2 ^ { 3 k + 4 n + 3 \\epsilon M _ t } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } f ( s , x - y , v ) \\end{align*}"}
{"id": "4418.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ d S _ { d , i } = [ 1 , 4 k d ] \\cup [ 4 k N + 4 e + 1 , 4 k N + 4 e + d ] \\cup [ ( 4 k + 2 ) N - d + 1 , ( 4 k + 2 ) N ] ; \\end{align*}"}
{"id": "7302.png", "formula": "\\begin{align*} \\epsilon _ 0 = \\alpha _ 1 \\psi _ 1 \\alpha _ 1 = - \\int _ 0 ^ \\infty e ^ { \\mu _ 1 s _ 1 } ( \\mathcal { R } _ 1 + \\mathcal { R } _ 2 , \\psi _ 1 ) _ { L _ y ^ 2 ( B _ { 4 { \\sf R } _ { \\sf i n } } ) } d s _ 1 . \\end{align*}"}
{"id": "4156.png", "formula": "\\begin{align*} L ^ \\mu ( g \\circ T _ { \\bar \\mu } ^ { - 1 } ) ( T _ { \\bar \\mu } z ) = L _ 0 ^ { | \\mu | } g ( z ) , \\end{align*}"}
{"id": "653.png", "formula": "\\begin{align*} & \\widetilde \\xi ( s ) = ( s - 1 ) \\Gamma \\left ( \\frac s 2 + 1 \\right ) \\pi ^ { s / 2 } \\zeta _ R ( s ) , \\\\ & \\xi ( s , \\chi ) = \\left ( \\frac \\pi 4 \\right ) ^ { - s / 2 } \\Gamma \\left ( \\frac s 2 \\right ) \\beta ( s ) , \\end{align*}"}
{"id": "7.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 1 \\\\ 0 & 0 & 1 & 1 \\\\ \\end{pmatrix} \\end{align*}"}
{"id": "8562.png", "formula": "\\begin{align*} \\left ( \\frac { t - 1 } t , 1 \\right ) = \\frac { t - 1 } t ( 1 , 0 ) + \\frac 1 t ( 0 , t ) \\end{align*}"}
{"id": "3951.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\left ( \\prod _ { i = 1 } ^ d \\abs { K _ { h _ { \\sigma ( i ) } } ( x _ { \\sigma ( i ) } - y _ { \\sigma ( i ) } ^ i ) } \\right ) \\abs { d _ { ( w _ i ) _ i , \\phi } ^ { ( l ) } ( y ^ 1 _ { \\sigma ( 1 ) } , \\dots , y ^ d _ { \\sigma ( d ) } ) } d y _ { \\sigma ( 1 ) } ^ 1 \\dots d y _ { \\sigma ( d ) } ^ d \\le C \\sqrt { \\Delta _ n } . \\end{align*}"}
{"id": "8413.png", "formula": "\\begin{gather*} ( W , W ' ) : = ( \\phi ^ W , \\phi ^ { W ' } ) \\end{gather*}"}
{"id": "6075.png", "formula": "\\begin{align*} f _ 1 ( x ) & = x ( x - 1 ) ( x - 2 ) ( x - 3 ) ^ 5 \\\\ & = x ^ 8 - 1 8 x ^ 7 + 1 3 7 x ^ 6 - 5 7 0 x ^ 5 + 1 3 9 5 x ^ 4 - 1 9 9 8 x ^ 3 + 1 5 3 9 x ^ 2 - 4 8 6 x . \\end{align*}"}
{"id": "8668.png", "formula": "\\begin{align*} d _ \\sigma A ( \\sigma ) = A ( \\sigma ) \\ne 0 \\end{align*}"}
{"id": "8946.png", "formula": "\\begin{align*} w = e ^ { - \\kappa \\phi } ( u - u _ 0 ) , \\end{align*}"}
{"id": "781.png", "formula": "\\begin{align*} & \\beta ( 1 + D r ^ m ) ( 1 + E r ^ m ) ^ { \\frac { D - 2 E } { E } } + ( 1 - \\beta ) r ^ m ( 1 + E r ^ m ) ^ { \\frac { D - E } { E } } \\\\ & + \\sum _ { n = 2 } ^ { \\infty } \\prod _ { k = 0 } ^ { n - 2 } \\frac { | E - D + E k | } { k + 1 } \\phi _ { n } ( r ) = ( 1 - E ) ^ { \\frac { D - E } { E } } - \\phi _ { 1 } ( r ) , \\end{align*}"}
{"id": "6656.png", "formula": "\\begin{align*} [ \\operatorname { s o c } _ { G } H ^ { 1 } ( \\lambda ) : L ( \\sigma ) ] = [ \\operatorname { s o c } _ { L _ { \\alpha } } R ^ { 1 } \\operatorname { i n d } _ { B } ^ { P _ { \\alpha } } L _ { \\mathfrak f } ( \\lambda ) : L _ { P _ { \\alpha } } ( \\bar { \\sigma } ) ] . \\end{align*}"}
{"id": "4210.png", "formula": "\\begin{align*} \\hat A : = | \\zeta | ^ 2 - \\tfrac 1 4 \\Delta _ \\zeta + i \\sum _ { j = 1 } ^ n ( \\partial _ { \\alpha _ j } \\beta _ j - \\partial _ { \\beta _ j } \\alpha _ j ) \\end{align*}"}
{"id": "2998.png", "formula": "\\begin{align*} \\deg _ { \\vec { s } } \\vec { v } = \\max _ k \\{ \\deg v _ k + s _ k \\} \\ . \\end{align*}"}
{"id": "3435.png", "formula": "\\begin{align*} \\ker ( \\kappa _ f ) \\cong & \\langle \\ , x _ i \\frac { \\partial f } { \\partial x _ i } , i = 1 , . . . , n \\\\ & \\ , \\ , \\ , w _ { - \\alpha } ( f ) , \\alpha \\in R ( N , \\Sigma _ { \\Delta } ) \\ , \\rangle . \\end{align*}"}
{"id": "8790.png", "formula": "\\begin{align*} C ( P ) = \\frac { B ( p ) ^ { p ' / 2 } } { B ( 2 ) } = B ( p ) ^ { p ' / 2 } \\end{align*}"}
{"id": "7020.png", "formula": "\\begin{align*} \\small { \\begin{aligned} \\dot { \\hat { \\tau } } & = \\dot { \\hat { \\tau } } ^ * \\left ( K _ 0 ( \\hat { \\tau } ^ * ) - K _ { \\bar { \\tau } } ( \\hat { \\tau } ^ * ) \\right ) + \\hat { \\tau } ^ * ( \\dot { K } _ 0 ( \\hat { \\tau } ^ * ) - \\dot { K } _ { \\bar { \\tau } } ( \\hat { \\tau } ^ * ) ) + \\bar \\tau \\dot { K } _ { \\bar \\tau } ( \\hat \\tau ^ * ) , \\end{aligned} } \\normalsize \\end{align*}"}
{"id": "1522.png", "formula": "\\begin{align*} f ( r , \\theta ) & : = \\begin{cases} \\displaystyle \\frac { \\alpha \\sinh ( \\alpha r ) } { 2 \\pi ( \\cosh ( \\alpha R ) - 1 ) } , & , \\\\ [ 2 e x ] 0 , & \\end{cases} \\end{align*}"}
{"id": "4844.png", "formula": "\\begin{align*} & \\tau ^ \\mathrm { a n } ( M / S , \\rho ) - \\tau ^ \\mathrm { t o p } ( M / S , \\rho ) \\\\ & = \\frac { 1 } { [ K : \\Q ] } \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } \\Big ( \\tau ^ \\mathrm { a n } ( M / S , g . \\rho ) - \\tau ^ \\mathrm { t o p } ( M / S , g . \\rho ) \\Big ) = 0 \\ ; . \\end{align*}"}
{"id": "3665.png", "formula": "\\begin{align*} \\gamma _ 2 : = \\inf \\{ k : k \\in \\R _ + , | V ( x , v , \\tau _ { \\ast } , 0 ) | | \\leq 2 ^ { k M _ t } \\} . \\end{align*}"}
{"id": "7785.png", "formula": "\\begin{align*} \\sin \\theta \\Big ( R ( \\tilde \\Xi _ 1 ) , R ( \\Xi _ 1 ) \\Big ) \\le \\sin \\theta ( \\tilde { \\boldsymbol { \\xi } } _ 1 , \\boldsymbol { \\xi } _ 1 ) = { \\mathcal P } _ { B 2 } , \\end{align*}"}
{"id": "5859.png", "formula": "\\begin{align*} C _ { 4 , 1 } & \\lesssim A _ 3 ^ { \\frac { p - q } { p q } } + \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p - q } } \\int _ { x _ k } ^ { x _ { k + 1 } } \\bigg ( \\int _ t ^ { x _ { k + 1 } } u \\bigg ) ^ { \\frac { q } { p - q } } u ( t ) V _ r ( x _ k , t ) ^ { \\frac { p q } { p - q } } d t . \\end{align*}"}
{"id": "1370.png", "formula": "\\begin{align*} \\varepsilon = s - \\frac { p - 5 } { 2 ( p - 1 ) } . \\end{align*}"}
{"id": "4070.png", "formula": "\\begin{align*} B = \\bigcup \\limits _ { l = 0 } ^ m D ' _ l . \\end{align*}"}
{"id": "7693.png", "formula": "\\begin{align*} d _ { M K } ( \\mu , \\nu ) = \\sup \\left \\{ \\left | \\int _ { X } f \\mathrm { d } \\mu - \\int _ { X } f \\mathrm { d } \\nu \\right | : f \\in \\operatorname { L i p } _ { 1 } ( X , \\mathbb { R } ) \\right \\} , \\end{align*}"}
{"id": "5543.png", "formula": "\\begin{align*} \\lim \\limits _ { \\ell \\rightarrow \\infty } \\dfrac { \\mathbb { E } [ \\ell ] } { \\ell } = \\dfrac { 1 } { 2 } \\end{align*}"}
{"id": "5365.png", "formula": "\\begin{align*} C ( s , \\Omega ) = C ( s , A \\Omega ) \\leq C ( s , \\Omega _ { \\infty } ) \\leq C ( s , \\omega ) \\leq \\frac { ( \\omega _ k \\abs { \\omega } ) ^ { s / k } } { ( 2 \\pi ) ^ s } \\sqrt { \\frac { \\delta ^ { 1 + \\frac { 2 s } { k } } } { \\delta - 1 } } = \\vcentcolon c ( s , \\Omega ) , \\delta = 1 + \\frac { k } { 2 s } , \\end{align*}"}
{"id": "4175.png", "formula": "\\begin{align*} U : = ( - ( U _ 1 ^ 2 + \\dots + U _ { d _ 2 } ^ 2 ) ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "3728.png", "formula": "\\begin{align*} \\mathcal { F } ( f ) ( \\xi ) = \\int e ^ { - i x \\cdot \\xi } f ( x ) d x . \\end{align*}"}
{"id": "4916.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial X } { \\partial t } ( x , t ) = \\frac { \\phi ( t ) \\psi u ^ \\alpha \\rho ^ \\delta } { K ^ \\frac { \\beta } { n } } \\nu - X , \\\\ [ 4 p t ] & X ( x , 0 ) = X _ 0 ( x ) , \\end{cases} \\end{align*}"}
{"id": "5854.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow - \\infty } V _ r ( 0 , x _ k ) = 0 . \\end{align*}"}
{"id": "36.png", "formula": "\\begin{align*} h _ n ( X ) P _ { n m _ 0 - 1 } ( X ) = P _ { n m _ 0 } ( X ) \\end{align*}"}
{"id": "1826.png", "formula": "\\begin{align*} A _ { n } = \\sum _ { \\gamma \\in \\mathcal { M } _ { n } } w ( \\gamma ) . \\end{align*}"}
{"id": "718.png", "formula": "\\begin{align*} & \\Psi ^ * \\circ L \\circ ( \\Psi ^ * ) ^ { - 1 } = - a ^ { i j } ( s , x ) \\partial _ { x _ i } \\partial _ { x _ j } + b ^ { j } ( s , x ) \\partial _ { x _ j } + c ( s , x ) , \\\\ & \\textup { w h e r e } \\Lambda ^ { - 1 } \\| \\xi \\| ^ 2 \\leq a ^ { i j } ( s , x ) \\xi _ i \\xi _ j \\leq \\Lambda \\| \\xi \\| ^ 2 , \\\\ & \\textup { a n d } \\| b ( s , x ) \\| \\leq \\Lambda ^ { - 1 } , \\ 0 \\leq c ( s , x ) \\leq \\Lambda ^ { - 1 } , \\end{align*}"}
{"id": "4217.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\frac { 1 } { u ( 1 \\vee \\log \\Phi ^ { - 1 } ( u ) ) } \\mathrm { d } u = \\infty \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( 1 \\vee \\log \\phi ^ { - 1 } ( n ) ) } = \\infty . \\end{align*}"}
{"id": "5667.png", "formula": "\\begin{align*} h _ J ( j ) = \\begin{cases} j + 1 & , \\\\ j & . \\end{cases} \\end{align*}"}
{"id": "5025.png", "formula": "\\begin{align*} R ^ { n , 3 } _ \\tau = \\frac 1 n \\sum _ { j = 0 } ^ { \\lfloor n \\tau \\rfloor } \\int _ { j } ^ { ( j + 1 ) \\wedge n \\tau } \\gamma _ { \\frac x n } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\frac x n } ) \\sum _ { i = 0 } ^ { \\lfloor x \\rfloor } \\int _ { i } ^ { ( i + 1 ) \\wedge x } [ ( x - i ) ^ \\alpha - ( x - v ) ^ \\alpha ] ^ 2 d v d x . \\end{align*}"}
{"id": "2652.png", "formula": "\\begin{align*} \\zeta _ { \\Z ^ 2 } ^ R ( s ) & = \\zeta ( s ) \\\\ \\zeta _ { \\Z ^ 3 } ^ R ( s ) & = \\frac { \\zeta ( 3 s - 1 ) \\zeta ( s ) ^ 3 } { \\zeta ( 2 s ) ^ s } \\\\ \\zeta _ { \\Z ^ 4 } ^ R ( s ) & = \\prod _ p \\frac { 1 } { ( 1 - p ^ { - s } ) ^ 2 ( 1 - p ^ { 2 - 4 s } ) ( 1 - p ^ { 3 - 6 s } ) } \\bigg ( 1 + 4 p ^ { - s } \\\\ & + 2 p ^ { - 2 s } + ( 4 p - 3 ) p ^ { - 3 s } + ( 5 p - 1 ) p ^ { - 4 s } + ( p ^ 2 - 5 p ) p ^ { - 5 s } \\\\ & + ( 3 p ^ 2 - 4 p ) p ^ { - 6 s } - 2 p ^ { 2 - 7 s } - 4 p ^ { 2 - 8 s } - p ^ { 2 - 9 s } \\bigg ) . \\end{align*}"}
{"id": "2985.png", "formula": "\\begin{align*} H _ m ( t ) = \\frac { 2 \\sqrt \\pi \\Gamma ( \\frac { s + 1 } { 2 } ) t ^ { 1 / 2 } } { \\Gamma ( \\frac s 2 ) } J _ { s - \\frac 1 2 } ( 2 \\pi | m | t ) \\end{align*}"}
{"id": "1050.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\mathrm { e } ^ { - \\omega s } \\psi ^ { [ n * ] } _ 1 ( s ) \\ , \\textnormal { d } s = \\left ( \\int _ 0 ^ \\infty \\mathrm { e } ^ { - \\omega s } \\psi _ 1 ( s ) \\ , \\textnormal { d } s \\right ) ^ n , \\end{align*}"}
{"id": "1942.png", "formula": "\\begin{align*} \\phi _ { j } ^ { ( q ) } ( z ) & : = \\langle ( z I - \\mathcal { H } _ { q } ) ^ { - 1 } e _ { j } , e _ { 0 } \\rangle = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\langle \\mathcal { H } _ { q } ^ { n } \\ , e _ { j } , e _ { 0 } \\rangle } { z ^ { n + 1 } } , 0 \\leq j \\leq p , q \\geq 0 , \\\\ \\beta _ { j } ^ { ( q ) } ( z ) & : = \\langle ( z I - \\mathcal { B } _ { q } ) ^ { - 1 } e _ { j } , e _ { 0 } \\rangle = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\langle \\mathcal { B } _ { q } ^ { n } \\ , e _ { j } , e _ { 0 } \\rangle } { z ^ { n + 1 } } , 0 \\leq j \\leq p , q \\geq 0 , \\end{align*}"}
{"id": "2049.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n v l a s o v - l i m i t s y n c h r o n o u s c o u p l i n g } \\dd \\overline { X } { } ^ i _ t = b \\big ( \\overline { X } { } ^ i _ t , f _ t \\big ) \\dd t + \\sigma \\big ( \\overline { X } { } ^ i _ t , f _ t \\big ) \\dd B ^ i _ t , \\end{align*}"}
{"id": "8023.png", "formula": "\\begin{align*} \\tilde { D } _ { m + 1 } = ( - D , 1 ) , \\ , \\tilde { D } _ { m + 2 } = ( 0 , 1 ) , D = \\sum _ { i \\in M _ + \\cup M _ - } D _ i . \\end{align*}"}
{"id": "1648.png", "formula": "\\begin{align*} \\theta _ { \\mathcal { C } } ( { \\bf z + h } ) = a _ { 0 0 0 } & + a _ { 1 0 0 } \\exp [ h _ 1 ] \\exp [ z _ 1 ] + a _ { 0 1 0 } \\exp [ h _ 2 ] \\exp [ z _ 2 ] + a _ { 0 0 1 } \\exp [ h _ 3 ] \\exp [ z _ 3 ] \\\\ & + a _ { 1 1 0 } \\exp [ h _ 1 + h _ 2 ] \\exp [ z _ 1 + z _ 2 ] + a _ { 1 0 1 } \\exp [ h _ 1 + h _ 3 ] \\exp [ z _ 1 + z _ 3 ] \\\\ & + a _ { 0 1 1 } \\exp [ h _ 2 + h _ 3 ] \\exp [ z _ 2 + z _ 3 ] + a _ { 1 1 1 } \\exp [ h _ 1 + h _ 2 + h _ 3 ] \\exp [ z _ 1 + z _ 2 + z _ 3 ] . \\end{align*}"}
{"id": "1988.png", "formula": "\\begin{align*} \\mathbf { S ^ 3 _ { k } } : = \\int _ { B _ { 1 / 2 } } \\eta _ r ( 2 - \\eta _ r ) | \\nabla ( U _ k - U _ 0 ) | ^ 2 \\ , d x \\end{align*}"}
{"id": "3439.png", "formula": "\\begin{align*} H ^ k ( \\mathbb { P } , \\Omega _ { \\mathbb { P } } ^ { n - 1 } \\otimes \\mathcal { O } ( - m Y - m K _ { \\mathbb { P } } ) ) \\cong & H ^ { n - k } ( \\mathbb { P } , \\Omega _ { \\mathbb { P } } ^ 1 \\otimes \\mathcal { O } ( m Y + m K _ { \\mathbb { P } } ) ) = 0 \\end{align*}"}
{"id": "5963.png", "formula": "\\begin{align*} \\omega _ { n , j } & : = \\omega _ j ( t _ { n + 1 } ) \\\\ & = \\frac 1 { \\Gamma ( 2 - \\beta ) } \\left ( ( t _ { n + 1 } - t _ j ) ^ { 1 - \\beta } - ( t _ { n + 1 } - t _ { j + 1 } ) ^ { 1 - \\beta } \\right ) , \\end{align*}"}
{"id": "6149.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\| u ^ { \\Delta t } _ { h } - v ^ { \\Delta t } _ { h } \\| _ 2 ^ 2 \\ , d t \\leq \\frac { \\Delta t } { 1 2 } \\sum _ { m = 1 } ^ N \\| u _ h ^ m - u _ h ^ { m - 1 } \\| _ { 2 } ^ 2 . \\end{align*}"}
{"id": "5450.png", "formula": "\\begin{align*} q [ q ( r ^ B _ A ( m ) ) ^ { - 1 } r ^ B _ A ( m ) ] \\stackrel { ( L 2 ) } = q ( r ^ B _ A ( m ) ) ^ { - 1 } q ( r ^ B _ A ( m ) ) = \\textsf { 1 } _ A , \\end{align*}"}
{"id": "262.png", "formula": "\\begin{align*} \\begin{array} { r c l } L _ { i j k } ^ { ( s ) } & : = & \\sum _ m L _ { i j } ^ { m ( s ) } ( q _ { m k } ^ { ( s ) } ) ^ { p ^ s } , \\\\ \\ & \\ & \\ \\\\ \\Gamma _ { i j k } ^ { ( s ) } & : = & \\sum _ m \\Gamma _ { i j } ^ { m ( s ) } ( q _ { m k } ^ { ( s ) } ) ^ { p ^ s } . \\end{array} \\end{align*}"}
{"id": "2750.png", "formula": "\\begin{align*} v _ { j } ( x ) : = j ^ { - s } w _ { j } ( R _ { j } ^ { - 1 } x ) , w _ { j } ( x ) : = \\begin{cases} j ^ { s } + j ^ { t } \\phi ( x ) , \\quad \\mathrm { i n } \\ ; B _ { 6 } , \\\\ ( 1 - \\psi ( x ) ) ( j ^ { s } + j ^ { t } ) , \\quad \\mathrm { i n } \\ ; B _ { 6 } ^ { c } , \\end{cases} \\end{align*}"}
{"id": "2559.png", "formula": "\\begin{align*} Q ( p , q ) ^ \\ast = Q ( q , p ) \\ . \\end{align*}"}
{"id": "736.png", "formula": "\\begin{align*} G _ 0 ( z , w ) ^ 2 \\partial \\bar { \\partial } \\log G _ 0 ( z , w ) & = G _ 0 ( z , w ) \\partial \\bar { \\partial } G _ 0 ( z , w ) - \\partial G _ 0 ( z , w ) \\bar { \\partial } G _ 0 ( z , w ) \\\\ & = ( 8 + 8 z \\bar { w } - ( z \\bar { w } ) ^ 2 ) ( 8 - 4 z \\bar { w } ) - ( 8 z - 2 z ^ 2 \\bar { w } ) ( 8 \\bar { w } - 2 z \\bar { w } ^ 2 ) \\\\ & = 6 4 - 3 2 z \\bar { w } - 8 ( z \\bar { w } ) ^ 2 , \\end{align*}"}
{"id": "3367.png", "formula": "\\begin{align*} F _ m ( \\psi _ n ) ( x , y ) = ( 2 x , y ) . \\end{align*}"}
{"id": "2340.png", "formula": "\\begin{align*} S _ Q ( f ) = \\{ i , 0 \\leq i \\leq r \\mid \\nu _ Q ( f ) = \\nu ( f _ i Q ^ i ) \\} \\mbox { a n d } \\delta _ Q ( f ) = \\max S _ Q ( f ) . \\end{align*}"}
{"id": "2684.png", "formula": "\\begin{align*} h _ \\mathrm { M F } : = - \\Delta + V _ { \\mathrm { D W } } + \\lambda w * | u _ + | ^ 2 . \\end{align*}"}
{"id": "5254.png", "formula": "\\begin{align*} A ^ 1 _ Q = \\varnothing ; A ^ 2 _ Q = \\bigcup _ { S _ 0 = 0 } ^ { s - 1 } A ^ 2 _ { Q , j , S _ 0 } . \\end{align*}"}
{"id": "6916.png", "formula": "\\begin{align*} \\Big ( \\gamma _ j - \\gamma ' _ j \\big | \\boldsymbol { \\gamma } _ { - j } , \\boldsymbol { \\gamma } ' _ { - j } \\Big ) \\overset { d } { = } \\big ( \\gamma _ j - \\gamma _ j ' \\big ) \\ , \\end{align*}"}
{"id": "8455.png", "formula": "\\begin{align*} s = \\min _ { A \\in \\mathcal { A } } s _ A = \\min \\left \\{ \\frac { 2 + \\theta _ { 1 } } { 1 + \\theta _ { 1 } + \\epsilon } , \\frac { 2 + \\theta _ { 2 } - ( \\theta _ { 1 } + 2 \\epsilon ) \\log _ { \\beta _ 2 } \\beta _ 1 } { 1 + \\theta _ { 2 } + \\epsilon } \\right \\} . \\end{align*}"}
{"id": "7066.png", "formula": "\\begin{align*} H _ j ( x , y ) = ( f _ { j , 1 } ( x , y ) ) ^ { \\lambda _ { j , 1 } } ( f _ { j , 2 } ( x , y ) ) ^ { \\lambda _ { j , 2 } } ( f _ { j , 3 } ( x , y ) ) ^ { \\lambda _ { j , 3 } } . \\end{align*}"}
{"id": "1989.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } \\int _ { B _ { 1 / 2 } } \\big ( | \\nabla U _ k | ^ 2 - | \\nabla V _ { k , r } | ^ 2 \\big ) \\ , d x & = \\liminf _ { k \\to \\infty } \\Big ( \\int _ { B _ { 1 / 2 } } \\big ( | \\nabla U _ 0 | ^ 2 - | \\nabla V | ^ 2 \\big ) \\ , d x + \\mathbf { T ^ 1 _ { k } } + \\mathbf { T _ { k } ^ 2 } + \\mathbf { T ^ 3 _ { k } } \\Big ) \\\\ & \\ge \\int _ { B _ { 1 / 2 } } ( U _ 0 - V ) \\cdot \\nabla ( U _ 0 + V ) \\ , d x = \\int _ { B _ { 1 / 2 } } ( | \\nabla U _ 0 | ^ 2 - | \\nabla V | ^ 2 ) \\ , d x . \\end{align*}"}
{"id": "6756.png", "formula": "\\begin{align*} h ^ * ( X , G ) = \\inf _ { k \\in \\mathbb { N } } \\ \\sup _ { m \\in \\mathbb { N } } h _ G ( \\P _ m | \\P _ k ) . \\end{align*}"}
{"id": "2691.png", "formula": "\\begin{align*} E _ { 2 - \\mathrm { m o d e } } : = \\inf \\Big \\{ \\left \\langle \\psi _ N , H _ N \\psi _ N \\right \\rangle , \\ ; | \\ ; \\psi _ N \\in P ^ { \\otimes N } \\mathcal { H } _ N , \\ ; \\int _ { \\mathbb { R } ^ { d N } } | \\psi _ N | ^ 2 = 1 \\Big \\} . \\end{align*}"}
{"id": "7047.png", "formula": "\\begin{align*} \\sum _ { M \\geq 0 } h ( M ) \\ , z ^ M & = \\sum _ { M \\geq 0 } z ^ M \\sum _ { 0 \\leq 2 j \\leq M - 1 } ( j + 1 ) \\binom { M - j - 1 } j \\\\ & = \\sum _ { M , j \\geq 0 } z ^ { M + 2 j + 1 } ( j + 1 ) \\binom { M + j } j = \\sum _ { j \\geq 0 } { z ^ { 2 j + 1 } ( j + 1 ) } { ( 1 - z ) ^ { - ( j + 1 ) } } \\\\ & = \\sum _ { j \\geq 1 } { z ^ { 2 j - 1 } j } { ( 1 - z ) ^ { - j } } = z ^ { - 1 } \\frac { \\frac { z ^ 2 } { 1 - z } } { \\left ( 1 - \\frac { z ^ 2 } { 1 - z } \\right ) ^ 2 } = \\frac { z - z ^ 2 } { ( 1 - z - z ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "3264.png", "formula": "\\begin{align*} \\mathcal { P } _ { ( A _ 1 , A _ 2 , q ) } v : = i \\textrm { d i v } ( A v ) + i A \\cdot \\nabla v + ( \\abs { A _ 2 } ^ 2 - \\abs { A _ 1 } ^ 2 + q ) v , v \\in H ^ 1 ( \\R ^ 3 ) , \\end{align*}"}
{"id": "7061.png", "formula": "\\begin{align*} Z _ j = \\begin{cases} \\dot { x } = - y + d _ j x + l _ j \\ , x ^ 2 + m _ j \\ , x \\ , y + n _ j \\ , y ^ 2 , \\\\ \\dot { y } = x ( 1 - y ) , \\end{cases} \\ \\ ( x , y ) \\in \\Sigma ^ { \\mathcal { H } } _ j , \\end{align*}"}
{"id": "4269.png", "formula": "\\begin{align*} \\lim \\limits _ { | \\zeta | \\to \\infty } \\partial _ { \\zeta _ i } f ( \\zeta ) = 0 . \\end{align*}"}
{"id": "6113.png", "formula": "\\begin{align*} \\iota ( o _ 1 h o _ 2 ) = g _ j ^ { - 1 } p _ 1 g _ j \\ , \\iota ( h ) \\ , g _ k ^ { - 1 } p _ 2 g _ k \\end{align*}"}
{"id": "4978.png", "formula": "\\begin{align*} \\widetilde { Y } ^ n _ t = \\widetilde { A } ^ n _ t + \\widetilde { C } ^ n _ t + \\int _ 0 ^ t ( t - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\widetilde { Y } ^ n _ s \\ , d W _ s . \\end{align*}"}
{"id": "8535.png", "formula": "\\begin{align*} S _ n & = \\sum _ { p = 1 } ^ { n - 1 } A _ p + b _ p - \\epsilon _ p \\\\ & = \\int _ 1 ^ n \\log x \\ , d x + \\frac { 1 } { 2 } \\log n - \\sum _ { p = 1 } ^ { n - 1 } \\epsilon _ p \\\\ & = \\big ( n + \\frac { 1 } { 2 } \\big ) \\log n - n + 1 - \\sum _ { p = 1 } ^ { n - 1 } \\epsilon _ p . \\end{align*}"}
{"id": "2143.png", "formula": "\\begin{align*} \\sigma _ { \\mu } ^ { + } ( [ k ^ m , ( k + 1 ) ^ m ) ) & = p c m \\int _ k ^ { k + 1 } s ^ { - ( a m + 1 ) } \\ ; d \\lambda ( s ) \\\\ & \\leq p c m k ^ { - ( a m + 1 ) } , \\ ; \\ ; \\ ; \\ ; \\ ; k \\in \\mathbb { N } , \\end{align*}"}
{"id": "673.png", "formula": "\\begin{align*} g ^ { \\lambda } _ t : = \\lambda \\cdot \\psi _ { s _ + } ^ * \\big [ g _ { t / \\lambda } \\big ] , t \\leq 0 . \\end{align*}"}
{"id": "6262.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } A _ q ( x ) = \\frac { q } { 1 - q } A _ q ( q x ) , \\end{align*}"}
{"id": "7055.png", "formula": "\\begin{align*} \\sum _ { M \\geq 1 } h _ r ( M ) z ^ M & = \\sum _ { n \\geq 1 } z ^ { n - 1 } \\cdot \\frac { z - z ^ { r } } { 1 - z } \\cdot \\left ( \\frac { 1 - z ^ { r } } { 1 - z } \\right ) ^ { n - 1 } \\\\ & = \\frac { z - z ^ { r } } { 1 - z } \\cdot \\frac { 1 } { 1 - \\frac { z ( 1 - z ^ r ) } { 1 - z } } \\\\ & = \\frac { z - z ^ r } { 1 - 2 z + z ^ { r + 1 } } . \\end{align*}"}
{"id": "7447.png", "formula": "\\begin{align*} c ( 1 - m ) \\int _ 0 ^ L \\abs { \\omega _ x } ^ 2 d x - \\frac { c m } { 2 } \\int _ 0 ^ L \\int _ 0 ^ { \\infty } \\sigma ' ( s ) \\abs { \\eta _ x } ^ 2 d s d x = - \\Re \\left ( \\left < { \\mathcal { A } } _ m U ^ n , U ^ n \\right > _ { \\mathcal { H } } \\right ) \\leq \\| F _ n \\| _ { \\mathcal { H } } \\| U \\| _ { \\mathcal { H } } . \\end{align*}"}
{"id": "8145.png", "formula": "\\begin{align*} \\| \\chi _ { S _ { r _ 0 } } e ^ { - i t H _ 0 } P _ { \\delta } ( W _ { n ; \\textrm { o u t } } ) \\| _ { \\textrm { o p } } = \\| e ^ { - i t H _ 0 ^ \\parallel } \\otimes ( \\chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \\perp } P ^ \\perp _ \\delta ( W _ { n ; \\textrm { o u t } } ^ \\perp ) ) \\| _ \\textrm { o p } = \\| \\chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \\perp } P ^ \\perp _ \\delta ( W _ { n ; \\textrm { o u t } } ^ \\perp ) \\| _ \\textrm { o p } \\end{align*}"}
{"id": "42.png", "formula": "\\begin{align*} \\dfrac { ( \\Tilde { \\mathcal { N } } ( f ) \\circ \\rho _ { \\eta } ( X ) ) ' } { \\Tilde { \\mathcal { N } } ( f ) \\circ \\rho _ { \\eta } ( X ) } = \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { ( f ( X + w ) ) ' } { f ( X + w ) } = \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { f ' ( X + w ) } { f ( X + w ) } . \\end{align*}"}
{"id": "8032.png", "formula": "\\begin{align*} X : = X _ - , \\tilde X : = X _ + , \\quad , D : = D _ - , \\tilde D : = D _ + . \\end{align*}"}
{"id": "6704.png", "formula": "\\begin{align*} \\int _ { K / M } e _ { w \\lambda , k } ^ \\tau ( g ) \\beta ( w \\lambda , k ) d k = \\int _ { K / M } e _ { - \\lambda , k } ^ \\tau ( g ) \\beta ( \\lambda , k ) d k , \\ ; \\ ; \\ ; \\ ; \\ ; w \\in W _ A . \\end{align*}"}
{"id": "3453.png", "formula": "\\begin{align*} \\mathrm { T C K } _ { q } ( X ) = \\frac { \\mathrm { E } \\left [ ( X - \\mathrm { T C E } _ { q } ( X ) ) ^ { 4 } | X > x _ { q } \\right ] } { \\mathrm { T V } _ { q } ^ { 2 } ( X ) } - 3 , \\end{align*}"}
{"id": "7507.png", "formula": "\\begin{align*} \\mathcal { Q } u = \\mathbf { S } _ { d , 0 } ( t ) u _ 0 + \\int _ { 0 } ^ t \\mathbf { S } _ { d , - \\alpha } ( t - s ) | u ( s ) | ^ p d s + \\int _ { 0 } ^ t \\mathbf { S } _ { d , 0 } ( t - s ) \\zeta ( s ) \\mathbf { w } d s \\end{align*}"}
{"id": "5737.png", "formula": "\\begin{align*} m _ { 2 , 1 ^ { k - 1 } } ( y _ 1 , \\ldots , y _ { b } ) = k m _ { 1 ^ { k } } ( y _ 1 , \\ldots , y _ { b } ) y _ { b + 1 } = k e _ { k } ( y _ 1 , \\ldots , y _ { b } ) y _ { b + 1 } . \\end{align*}"}
{"id": "8342.png", "formula": "\\begin{align*} K ( \\nu ; z ) = - { \\textstyle \\frac { 4 \\pi } { \\nu } } z ^ 2 e ^ { z } \\sum \\limits _ { n = - \\infty } ^ { + \\infty } { \\textstyle \\frac { [ \\nu + i ( 2 n + 1 - z ) ] ^ { n - 1 } } { [ \\nu + i ( 2 n + 1 + z ) ] ^ { n + 2 } } } \\end{align*}"}
{"id": "2973.png", "formula": "\\begin{align*} \\Phi ( m , n ; s ) = \\sum _ { c > 0 } \\frac { K ( m , n , c ) } { c } \\begin{cases} I _ { 2 s - 1 } ( 4 \\pi \\sqrt { | m n | } \\ , c ^ { - 1 } ) & m n < 0 , \\\\ J _ { 2 s - 1 } ( 4 \\pi \\sqrt { | m n | } \\ , c ^ { - 1 } ) & m n > 0 \\end{cases} \\end{align*}"}
{"id": "6945.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} Q _ { 0 } & 0 \\\\ 0 & I \\end{pmatrix} , \\end{align*}"}
{"id": "6546.png", "formula": "\\begin{align*} t _ { n + 1 } - t _ n \\leq \\sum \\limits _ { j = x _ n + 1 } ^ { x _ { n + 1 } } \\frac { 1 } { A ( j ) } ; \\end{align*}"}
{"id": "8586.png", "formula": "\\begin{align*} & n _ k ( m ) = [ 1 0 ^ { m l } \\lambda _ l ^ { k } ] + 3 ^ { k } , k = 1 , 2 , \\ldots , l - 1 , \\\\ & n _ l ( m ) = m + n _ { l - 1 } ( m ) + \\ldots + n _ 1 ( m ) . \\end{align*}"}
{"id": "8845.png", "formula": "\\begin{align*} y = \\frac { t ( t ' - y ) + t ' ( y - t ) } { t ' - t } = \\frac { t ( t ' - y ) + t ' f _ t ( y ) } { t ' - t } . \\end{align*}"}
{"id": "6600.png", "formula": "\\begin{align*} \\norm { \\pi ( \\beta ) } ^ 2 = \\norm { \\pi ( \\beta ^ \\ast \\ast \\beta ) } = \\abs { \\frac { 1 } { m _ S ( B ) } \\int _ { B } \\omega _ \\pi ( g ) \\ , d m _ S ( g ) } ^ 2 \\le \\frac { 1 } { m _ S ( B ) } \\int _ { B } \\abs { \\omega _ \\pi ( g ) } ^ 2 \\ , d m _ S ( g ) , \\end{align*}"}
{"id": "2317.png", "formula": "\\begin{align*} G D ^ k \\widetilde { G } = \\widetilde { G } \\widetilde { D } ^ k G D G ^ k \\widetilde { D } = \\widetilde { D } \\widetilde { G } ^ k D . \\end{align*}"}
{"id": "1618.png", "formula": "\\begin{align*} y [ j , l ] = \\sum _ { i = 1 } ^ { N } \\sqrt { P } h [ r ( i , l - 1 ) , j , l ] \\ ; x [ r ( i , l - 1 ) , l - 1 ] + n [ j , l ] \\end{align*}"}
{"id": "7272.png", "formula": "\\begin{align*} L _ 1 ^ { q - 1 } = \\beta _ 0 ( \\beta _ 0 + n - 2 ) , \\beta _ 0 = \\tfrac { 2 } { 1 - q } . \\end{align*}"}
{"id": "6254.png", "formula": "\\begin{align*} h ( q ^ n x ) = ( - q ) ^ n ( \\frac { - x } { q ^ 2 } ; q ) _ n h ( x ) ( n \\in \\mathbb { N } _ 0 ) . \\end{align*}"}
{"id": "3034.png", "formula": "\\begin{align*} \\tilde { K } _ { \\lambda , \\tau } ( q ) = q ^ { n ( \\tau ) } K _ { \\lambda , \\tau } ( q ^ { - 1 } ) . \\end{align*}"}
{"id": "6565.png", "formula": "\\begin{align*} \\sum _ { \\nu = 1 } ^ k \\beta \\Big ( f _ 1 ( x + t _ n y ) , \\ldots , f _ { \\nu - 1 } ( x + t _ n y ) , \\frac { f _ \\nu ( x + t _ n y ) - f _ \\nu ( x ) } { t _ n } , f _ { \\nu + 1 } ( x ) , \\ldots , f _ k ( x ) \\Big ) , \\end{align*}"}
{"id": "511.png", "formula": "\\begin{align*} \\dim ( ( \\mathbb { C } \\phi ^ 1 ) _ 2 \\cap \\ker A _ 2 ) + \\dim ( ( \\mathbb { C } \\phi ^ 2 ) _ 2 \\cap \\ker A _ 2 ) + \\dim ( ( \\mathbb { C } \\phi ^ 3 ) _ 2 \\cap \\ker A _ 2 ) & = 1 + 1 + 1 \\\\ & = \\dim ( M _ 2 \\cap \\ker A _ i ) . \\end{align*}"}
{"id": "4271.png", "formula": "\\begin{align*} \\partial _ { \\zeta _ i } h ( \\zeta ) = s { \\zeta _ i \\over | \\zeta | } h ( \\zeta ) ( 1 + o ( 1 ) ) , \\end{align*}"}
{"id": "5021.png", "formula": "\\begin{align*} R ^ { n , M , 1 } _ \\tau = n ^ { 2 \\alpha + 1 } \\sum _ { i = 0 } ^ { M - 1 } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\tau _ i } ) \\int _ { \\tau _ i } ^ { \\tau _ { i + 1 } } \\gamma _ s \\Xi ^ { n , 1 } _ s d s \\end{align*}"}
{"id": "1702.png", "formula": "\\begin{align*} | x - y | = & | x - 2 ^ { | j | + 2 N } e _ 1 + 2 ^ { | j | + 2 N } e _ 1 - 2 ^ { | k | + 2 N } e _ 1 + 2 ^ { | k | + 2 N } e _ 1 - y | \\\\ \\ge & | 2 ^ { | j | + 2 N } e _ 1 - 2 ^ { | k | + 2 N } e _ 1 | - | y - 2 ^ { | k | + 2 N } e _ 1 | - 2 ^ { - j } \\ge C 2 ^ { 2 N } . \\end{align*}"}
{"id": "100.png", "formula": "\\begin{align*} \\sup _ { \\| Y \\| _ 2 = 1 } Y ^ T \\Sigma _ t ( \\varphi ) Y = Y ^ T _ 0 \\Sigma _ t ( \\varphi ) Y _ 0 \\leq Y ^ T _ 0 \\Sigma _ t ( 0 ) Y _ 0 \\leq \\| \\Sigma _ t ( 0 ) \\| = \\sup _ { \\| Y \\| _ 2 = 1 } Y ^ T \\Sigma _ t ( 0 ) Y . \\end{align*}"}
{"id": "8487.png", "formula": "\\begin{align*} f ( x ) & = f ( x ) - f ( c ) + f ( c ) \\\\ & = \\int _ c ^ x \\partial _ y f ( y ) d y + \\int _ 0 ^ c \\partial _ y f ( y ) d y + f ( 0 ) \\\\ \\Rightarrow \\left | f ( x ) \\right | & \\le \\int _ 0 ^ 1 \\left | \\partial _ y f ( y ) \\right | d y + \\left | f ( 0 ) \\right | . \\end{align*}"}
{"id": "1870.png", "formula": "\\begin{align*} S _ { j } ( z ) = \\sum _ { m = 0 } ^ { \\infty } \\frac { S _ { [ m ( p + 1 ) + j , j ] } } { z ^ { m ( p + 1 ) + j + 1 } } , 0 \\leq j \\leq p - 1 . \\end{align*}"}
{"id": "7727.png", "formula": "\\begin{align*} A _ S ( t _ 1 : t _ 0 ) & = A _ S ( t _ 1 : \\tau ) A _ S ( \\tau : t _ 0 ) + A _ { S \\bar S } ( t _ 1 : \\tau ) A _ { \\bar S S } ( \\tau : t _ 0 ) \\cr A _ { \\bar S S } ( t _ 1 : t _ 0 ) & = A _ { \\bar S S } ( t _ 1 : \\tau ) A _ { S } ( \\tau : t _ 0 ) + A _ { \\bar S } ( t _ 1 : \\tau ) A _ { \\bar S S } ( \\tau : t _ 0 ) . \\end{align*}"}
{"id": "6032.png", "formula": "\\begin{align*} ( x ) _ j = x ( x - 1 ) \\cdots ( x - j + 1 ) . \\end{align*}"}
{"id": "3277.png", "formula": "\\begin{align*} \\mathcal { H } _ { - \\tilde { A } _ 1 , q _ 1 } \\tilde { u } _ 1 & = e ^ { - i \\varphi / 2 } \\mathcal { H } _ { - A _ 1 , q _ 1 } u _ 1 = k ^ 2 \\tilde { u } _ 1 , \\cr \\mathcal { H } _ { \\tilde { A } _ 2 , q _ 2 } \\tilde { u } _ 2 & = e ^ { - i \\varphi / 2 } \\mathcal { H } _ { A _ 2 , q _ 2 } u _ 1 = k ^ 2 \\tilde { u } _ 2 , \\textrm { i n } \\ , B . \\end{align*}"}
{"id": "8610.png", "formula": "\\begin{align*} & \\min _ { \\delta , P } \\delta ^ 2 \\\\ & \\ \\ \\\\ & \\begin{bmatrix} - P A - A ^ { T } P & - P B + A ^ { T } C ^ { T } \\\\ - B ^ { T } P + C A & ( C B + B ^ { T } C ^ { T } ) \\end{bmatrix} + \\delta I _ { n + m } \\geq 0 , \\end{align*}"}
{"id": "7605.png", "formula": "\\begin{align*} x _ { k + 1 } = \\varphi ( \\tilde Y _ { k + 1 } , x _ k ) , \\end{align*}"}
{"id": "322.png", "formula": "\\begin{align*} \\sup _ { A \\textnormal { L - p r i m i t i v e } } f ( A ) \\ & = \\ \\sum _ p \\max \\{ f ( p ) , e ^ \\gamma { \\rm d } ( { \\rm L } _ p ) \\} , \\lim _ { x \\to \\infty } \\sup _ { \\substack { A \\subset [ x , \\infty ) \\\\ A \\textnormal { L - p r i m i t i v e } } } f ( A ) \\ = \\ e ^ \\gamma . \\end{align*}"}
{"id": "5327.png", "formula": "\\begin{align*} N _ t ( b ) = \\# \\{ i \\in [ N _ t ] : | \\pi _ i ( t ) | \\geq b \\} . \\end{align*}"}
{"id": "6363.png", "formula": "\\begin{align*} A + B \\ , \\sqrt { 1 + \\kappa ^ 2 x ^ 2 } = 0 \\ \\ . \\end{align*}"}
{"id": "6026.png", "formula": "\\begin{align*} a & = b - 3 \\\\ 0 & = 3 - 2 b \\\\ 0 & = b + c - 1 \\end{align*}"}
{"id": "8267.png", "formula": "\\begin{align*} k \\cot ( k a / 2 ) - k ' \\coth ( k ' a / 2 ) = 0 \\end{align*}"}
{"id": "2427.png", "formula": "\\begin{align*} M _ f : = v _ { f ( 1 ) } \\otimes v _ { f ( 2 ) } \\otimes \\cdots \\otimes v _ { f ( m ) } \\otimes w _ { f ( m + 1 ) } \\otimes \\cdots \\otimes w _ { f ( m + n ) } . \\end{align*}"}
{"id": "4625.png", "formula": "\\begin{align*} \\theta = \\omega ^ i \\ , \\ , \\mbox { f o r s o m e } i \\in \\Z , i \\neq 0 \\end{align*}"}
{"id": "4953.png", "formula": "\\begin{align*} | F ( x n _ { k + 1 } ) - F ( x n _ k ) | & = | F ^ \\prime ( \\xi ) | | x n _ { k + 1 } - x n _ k | \\\\ & \\leq \\frac { \\xi + 2 } { \\xi ^ 2 } | x | ( n _ { k + 1 } - n _ k ) \\\\ & \\leq \\frac { x n _ { k + 1 } + 2 } { x ^ 2 n _ k ^ 2 } | x | ( n _ { k + 1 } - n _ k ) \\\\ & = \\frac { 2 n _ { k + 1 } } { n _ k ^ 2 } ( n _ { k + 1 } - n _ k ) . \\end{align*}"}
{"id": "4675.png", "formula": "\\begin{align*} x _ i ( a ) y _ j ( \\pm b ) & = y _ j ( \\pm b ) x _ i ( a ) , \\\\ x _ i ( a ) y _ i ( b + c ) & = y _ i ( \\frac { b + c } { a ( b + c ) + 1 } ) \\alpha _ i ^ \\vee ( a ( b + c ) + 1 ) x _ i ( \\frac { a } { a ( b + c ) + 1 } ) , \\\\ x _ i ( \\frac { a } { a ( b + c ) + 1 } ) y _ i ( - c ) & = y _ i ( \\frac { - c ( a ( b + c ) + 1 ) } { a b + 1 } ) \\alpha _ i ^ \\vee ( \\frac { a ( b ) + 1 } { a ( b + c ) + 1 } ) x _ i ( \\frac { a } { a b + 1 } ) . \\end{align*}"}
{"id": "2195.png", "formula": "\\begin{align*} r = \\theta - \\eta . \\end{align*}"}
{"id": "2789.png", "formula": "\\begin{align*} x _ i = x _ 0 + \\tfrac { 1 } { L } \\sum \\limits _ { k = 0 } ^ { i - 1 } h _ k g _ k \\end{align*}"}
{"id": "1411.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} 1 + \\delta & \\ & \\ & \\ \\\\ \\ & 1 & \\ & \\ \\\\ \\ & \\frac { 1 } { 2 } & 1 - \\delta _ 1 & \\ \\\\ \\ & \\ & \\ & ( 1 - 2 \\delta _ 1 ) I _ { n - 3 } \\end{matrix} \\right ] , \\Delta A = \\left [ \\begin{matrix} 0 & \\cdots & 0 \\\\ \\epsilon & \\vdots & \\vdots \\\\ 0 & \\vdots & \\vdots \\\\ \\vdots & \\vdots & \\vdots \\\\ 0 & \\cdots & 0 \\end{matrix} \\right ] . \\end{align*}"}
{"id": "7688.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ i ^ k = \\lambda _ { i } ^ k ( \\sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \\geq \\sigma _ { k - 1 } \\lambda _ { 1 } ( Z _ { k - 1 } ) + \\lambda _ { i } ( - X ( x _ k ) ) . \\end{align*}"}
{"id": "1539.png", "formula": "\\begin{align*} \\Delta _ { r a d } : = \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + \\frac { \\alpha } { 2 } \\frac { 1 } { \\tanh ( \\alpha r ) } \\frac { \\partial } { \\partial r } . \\end{align*}"}
{"id": "2960.png", "formula": "\\begin{align*} L ( s , \\varphi \\times \\chi _ d ) = \\sum _ { n = 1 } ^ \\infty \\frac { a ( n ) \\chi _ d ( n ) } { n ^ s } . \\end{align*}"}
{"id": "6405.png", "formula": "\\begin{align*} f \\circ \\varphi = \\psi \\circ f \\ \\mbox { ( e q u a l i t y o f m a p s f r o m $ B _ { \\pi } $ t o $ \\{ 1 , \\ldots , \\ell + 1 \\} $ ) , } \\end{align*}"}
{"id": "3476.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast } = \\frac { c _ { 1 } \\left [ \\xi _ { p } ^ { 2 } \\overline { G } _ { ( 1 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { p } ^ { 2 } \\right ) - \\xi _ { q } ^ { 2 } \\overline { G } _ { ( 1 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { q } ^ { 2 } \\right ) \\right ] } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "3802.png", "formula": "\\begin{align*} E r r ^ 0 _ { i , i _ 1 , i _ 2 , i _ 3 } ( t _ 1 , t _ 2 ) = \\sum _ { b = 0 , 1 } \\int _ { ( \\R ^ 3 ) ^ 4 } e ^ { i t _ b ( \\mu _ 3 | \\kappa | + \\mu _ 2 | \\sigma | + \\mu | \\xi | + \\mu _ 1 | \\eta | ) } \\clubsuit K _ { k _ 3 , j _ 3 ; n _ 3 } ^ { \\mu _ 3 , i _ 3 } ( t _ b , \\kappa , V ( t _ b ) ) \\cdot { } ^ 3 \\clubsuit K ( t _ b , \\xi , \\eta , \\sigma , X ( t _ b ) , V ( t _ b ) ) \\end{align*}"}
{"id": "4673.png", "formula": "\\begin{align*} L k ( \\overline { { } ^ J U ^ - _ { v , w , > 0 } } ) = \\overline { { } ^ J U ^ - _ { v , w , > 0 } } \\bigcap \\{ | | x | | = 1 \\vert x \\in L \\} \\end{align*}"}
{"id": "7024.png", "formula": "\\begin{align*} \\omega _ i ^ * = \\min \\left ( \\max ( \\omega _ i ^ 0 , - \\bar { \\omega } ) , \\bar { \\omega } \\right ) , \\end{align*}"}
{"id": "3436.png", "formula": "\\begin{align*} \\kappa ( Y ) = \\min ( n - 1 , \\dim \\ , F ( \\Delta ) ) . \\end{align*}"}
{"id": "1525.png", "formula": "\\begin{align*} \\P ( T _ { d e t } \\geq \\ss ) = \\exp \\big ( { - } \\Theta ( \\ss ^ { \\frac { 1 } { 2 \\alpha } } ) \\big ) . \\end{align*}"}
{"id": "4638.png", "formula": "\\begin{align*} A B x = x , \\ x \\in Y . \\end{align*}"}
{"id": "8623.png", "formula": "\\begin{align*} \\langle \\vartheta _ { p _ 1 } , \\ldots , \\vartheta _ { p _ r } \\rangle ^ { n a i v e } : = \\sum _ { \\textbf { A } \\in N E ( \\tilde { X } ) } t ^ { \\pi _ * ( \\textbf { A } ) } N _ \\textbf { A } ^ { n a i v e } ( \\textbf { p } ) \\in \\mathbb { Z } [ N E ( X ) ] . \\end{align*}"}
{"id": "3731.png", "formula": "\\begin{align*} U ^ m _ { S ; j , l } ( t , x ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } ( t - s ) \\big ( E ( s , x + ( t - s ) \\omega ) + \\hat { v } \\times B ( s , x + ( t - s ) \\omega ) \\big ) \\cdot \\nabla _ v \\big ( \\frac { m _ U ( v , \\omega ) \\varphi _ { j , l } ( v , \\omega ) } { 1 + \\hat { v } \\cdot \\omega } \\big ) \\end{align*}"}
{"id": "1786.png", "formula": "\\begin{align*} z _ n ^ * = y _ n ^ * - \\sum _ { m = 1 } ^ \\infty a _ m f _ { q ( m ) } ( y _ n ^ * ) \\widetilde { u } _ m . \\end{align*}"}
{"id": "3600.png", "formula": "\\begin{align*} { \\rm R C } ( I ) : = \\mathbb { R } _ + \\{ e _ 1 , \\ldots , e _ s , ( v _ 1 , 1 ) , \\ldots , ( v _ q , 1 ) \\} . \\end{align*}"}
{"id": "7872.png", "formula": "\\begin{align*} \\| V _ \\alpha ( \\hat { x } ) \\| ^ 2 = \\tau ( \\rho _ \\alpha ( x ) x ^ * ) + \\frac { 1 } { \\alpha } \\tau ( \\Delta \\circ \\rho _ \\alpha ( x ) x ^ * ) = \\| x \\| _ 2 ^ 2 , \\end{align*}"}
{"id": "9009.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { k + z } = - \\frac { 1 } { 2 } \\left ( \\psi \\left ( \\frac { z } { 2 } \\right ) - \\psi \\left ( \\frac { z + 1 } { 2 } \\right ) \\right ) , z \\in \\mathbb { C } \\setminus - \\mathbb { N } _ 0 . \\end{align*}"}
{"id": "2645.png", "formula": "\\begin{gather*} \\begin{pmatrix} x _ 0 \\ , g _ { 0 0 } + x _ 1 g _ { 1 0 } & x _ 0 \\ , g _ { 0 1 } + x _ 1 g _ { 1 1 } \\\\ x _ 0 \\ , h _ { 0 0 } + x _ 1 h _ { 1 0 } & x _ 0 \\ , h _ { 0 1 } + x _ 1 h _ { 1 1 } \\end{pmatrix} \\begin{pmatrix} y _ 0 \\\\ y _ 1 \\end{pmatrix} \\xmapsto { t _ j \\mapsto f _ j } \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} \\end{gather*}"}
{"id": "751.png", "formula": "\\begin{align*} a _ { \\lambda } ( m ' _ i ) = \\sum ^ { n } _ { j = 1 } h ' _ { j } ( a _ { \\lambda } ( m ' _ i ) ) h _ j , 1 \\leq i \\leq r . \\end{align*}"}
{"id": "3494.png", "formula": "\\begin{align*} \\mathrm { D T E } _ { ( p , q ) } ( X ) = \\mu + \\sigma \\frac { \\Gamma \\left ( ( m + 1 ) / 2 \\right ) \\sqrt { m } \\left [ \\left ( 1 + \\frac { \\xi _ { p } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } - \\left ( 1 + \\frac { \\xi _ { q } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } \\right ] } { \\Gamma ( m / 2 ) ( m - 1 ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "2250.png", "formula": "\\begin{align*} ( 1 + u _ p ^ 0 ) - ( \\sigma + h _ p ^ 0 ) \\geq 1 - \\sigma - \\mathcal { O } ( \\delta , \\sigma ) : = c _ 1 > 0 . \\end{align*}"}
{"id": "4954.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int ^ t _ 0 ( t - s ) ^ { \\alpha } \\sigma ( X _ s ) \\ , d W _ s , t \\in [ 0 , T ] , \\end{align*}"}
{"id": "3447.png", "formula": "\\begin{align*} & x _ i \\frac { \\partial } { \\partial x _ i } : x ^ m \\mapsto m _ i \\cdot x ^ m , i = 1 , . . . , n , \\\\ & \\frac { \\partial e _ { - \\alpha } ^ { \\lambda } } { \\partial \\lambda _ i } _ { \\vert { \\lambda = 0 } } : x ^ m \\mapsto h t _ { - \\alpha } ( m ) \\cdot x ^ { m - \\alpha } , \\alpha \\in R ( N , \\Sigma _ { C ( \\Delta ) } ) . \\end{align*}"}
{"id": "4454.png", "formula": "\\begin{align*} N _ 0 ( ( R f ^ { i _ 1 \\dots i _ k } ) _ { p _ 1 q _ 1 \\dots p _ { m - k } q _ { m - k } } ) = 0 \\mbox { i n } U . \\end{align*}"}
{"id": "1260.png", "formula": "\\begin{align*} E _ 0 ( x , a ) & = 1 , \\\\ E _ 1 ( x , a ) & = x , \\\\ E _ j ( x , a ) & = x E _ { j - 1 } ( x , a ) - a E _ { j - 2 } ( x , a ) ( \\forall j \\ge 2 ) , \\end{align*}"}
{"id": "5600.png", "formula": "\\begin{align*} \\frac { x } { y } + \\frac { y } { z } + \\frac { z } { x } = n \\end{align*}"}
{"id": "4321.png", "formula": "\\begin{align*} \\int _ { A _ m } \\Omega = \\int _ { L _ m } \\Omega > 0 , \\sum _ 1 ^ m \\int _ { A _ i } \\Omega \\geq \\sum _ 1 ^ m \\ \\int _ { L _ i } \\Omega , \\int _ L \\Omega = \\sum _ 1 ^ N \\int _ { L _ i } \\Omega . \\end{align*}"}
{"id": "4971.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\in [ 0 , T ] } E [ | Z ^ n _ t - \\widetilde { Z } ^ n _ t | ^ 2 ] = 0 , \\end{align*}"}
{"id": "7181.png", "formula": "\\begin{align*} & I ( L ( - 1 ) w _ 1 , z ) = \\frac { d } { d z } I ( w _ 1 , z ) . \\end{align*}"}
{"id": "6163.png", "formula": "\\begin{align*} \\lbrack \\omega _ { 0 } ] _ { B } - T c _ { 1 } ^ { B } ( M ) = [ \\psi ^ { \\ast } \\omega _ { N } ] _ { B } . \\end{align*}"}
{"id": "8233.png", "formula": "\\begin{align*} z ^ { 2 } = a ^ { 2 } - b ^ { 2 } + 2 i a b = 1 \\pm 1 6 i m \\beta \\lambda / 3 \\Rightarrow \\left \\lbrace \\begin{array} { l } a ^ { 2 } - b ^ { 2 } = 1 \\\\ 2 a b = \\pm 1 6 m \\beta \\lambda / 3 \\end{array} \\right . . \\end{align*}"}
{"id": "7660.png", "formula": "\\begin{align*} c & = \\deg ( N _ { s - 1 } ) = \\prod _ { i = 0 } ^ 1 \\frac { ( n + i + m ) ! \\ , i ! } { ( s + i - 1 ) ! \\ , ( n + m - s + i + 1 ) ! } \\\\ & = \\frac { 1 } { s } \\ , \\binom { 2 s - m - 2 } { s - 1 } \\ , \\binom { 2 s - m - 3 } { s - 1 } . \\end{align*}"}
{"id": "7751.png", "formula": "\\begin{align*} { \\bf u } _ m ( \\alpha ) = \\big [ { \\tilde u } _ { 1 , 1 } ^ \\alpha , \\ldots , { \\tilde u } _ { 1 , N _ h } ^ \\alpha , { \\tilde u } _ { 2 , 1 } ^ \\alpha , \\ldots , { \\tilde u } _ { 2 , N _ h } ^ \\alpha , \\ldots , { \\tilde u } _ { m , 1 } ^ \\alpha , \\ldots , { \\tilde u } _ { m , N _ h } ^ \\alpha \\big ] ^ T \\in \\mathbb { R } ^ { m N _ h } \\end{align*}"}
{"id": "7264.png", "formula": "\\begin{align*} \\psi _ 3 ^ { ( R ) } ( r ) & = k \\mu _ 3 ^ { ( R ) } Z _ 2 ( r ) \\int _ 0 ^ r \\psi _ 3 ^ { ( R ) } Z _ 1 r _ 1 ^ { n - 1 } d r _ 1 + k \\mu _ 3 ^ { ( R ) } Z _ 1 ( r ) \\int _ r ^ R \\psi _ 3 ^ { ( R ) } Z _ 2 r _ 1 ^ { n - 1 } d r _ 1 \\\\ & - k \\mu _ 3 ^ { ( R ) } \\frac { Z _ 2 ( R ) } { Z _ 1 ( R ) } Z _ 1 ( r ) \\int _ 0 ^ R \\psi _ 3 ^ { ( R ) } Z _ 1 r ^ { n - 1 } d r . \\end{align*}"}
{"id": "7744.png", "formula": "\\begin{align*} b ( \\alpha , \\lambda ) = \\langle \\check { \\alpha } , \\lambda \\rangle Q ( \\alpha ) , ( \\alpha \\in \\Delta , \\lambda \\in \\Lambda ) . \\end{align*}"}
{"id": "4427.png", "formula": "\\begin{align*} 4 k d ' + ( 4 k - 1 ) e ' + j ' \\leq 4 k d ' + 4 k e ' < 4 k d ' + ( 8 k - 1 ) e ' + \\ell ' \\leq 4 k d ' + 8 k e ' = 4 k N < 4 k N + 4 e + i , \\end{align*}"}
{"id": "8220.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c } \\phi ( + a / 2 ) = - \\phi ( - a / 2 ) \\\\ \\partial _ { x } \\phi ( + a / 2 ) = - \\partial _ { x } \\phi ( - a / 2 ) \\\\ \\partial _ { x } ^ { 2 } \\phi ( + a / 2 ) = - \\partial _ { x } ^ { 2 } \\phi ( - a / 2 ) \\\\ \\partial _ { x } ^ { 3 } \\phi ( + a / 2 ) = - \\partial _ { x } ^ { 3 } \\phi ( - a / 2 ) \\end{array} \\right . , \\end{align*}"}
{"id": "5502.png", "formula": "\\begin{align*} \\lambda = \\sqrt { \\sum _ { i } \\lambda _ { i } ^ 2 } . \\end{align*}"}
{"id": "8530.png", "formula": "\\begin{align*} u _ { k + 1 } = B _ k u _ k + A _ k u _ { k - 1 } , \\end{align*}"}
{"id": "3765.png", "formula": "\\begin{align*} \\widetilde { H } ^ { S ; U ; m , i } _ { j , n , l , r } ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big ( 1 + ( t - s ) \\omega \\cdot \\nabla \\big ) \\big [ \\big ( E ( s , x + ( t - s ) \\omega ) + \\hat { v } \\times B ( s , x + ( t - s ) \\omega ) \\big ) \\cdot \\nabla _ v \\big ( \\frac { m _ { U } ( v , \\omega ) \\varphi _ { j , n } ^ { i ; r } ( v , \\zeta ) \\varphi _ { l ; r } ( \\tilde { v } + \\omega ) } { 1 + \\hat { v } \\cdot \\omega } \\big ) \\end{align*}"}
{"id": "3169.png", "formula": "\\begin{align*} \\mathbf { C } _ { \\Lambda } \\doteq \\left \\{ \\begin{array} { l l } a _ { x , \\uparrow } a _ { x , \\downarrow } & \\Lambda = \\left \\{ x \\right \\} x \\in \\mathfrak { L } \\\\ 0 & \\end{array} \\right . , \\Lambda \\in \\mathcal { P } _ { \\mathrm { f } } \\ . \\end{align*}"}
{"id": "6792.png", "formula": "\\begin{align*} \\partial ^ \\alpha _ x K _ j ( x - y ) = ( - 1 ) ^ { | \\alpha | } \\partial ^ \\alpha _ y K _ j ( x - y ) \\end{align*}"}
{"id": "3759.png", "formula": "\\begin{align*} T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , U ) ( t , x , \\zeta ) : = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i x \\cdot \\xi + i \\mu ( t - s ) | \\xi | - i s \\hat { v } \\cdot \\xi } | \\xi | ^ { - 1 } \\mathfrak { m } ( \\xi , \\zeta ) m ^ { 0 } _ U ( \\xi , v ) \\hat { g } ( s , \\xi , v ) \\varphi _ k ( \\xi ) \\varphi _ { n ; - M _ t } \\big ( \\tilde { \\xi } + \\mu \\tilde { \\zeta } \\big ) \\varphi _ { j , n } ^ i ( v , \\zeta ) \\end{align*}"}
{"id": "6094.png", "formula": "\\begin{align*} \\Psi _ { A B } = \\theta _ { A \\ , N + 2 } \\wedge \\theta _ { N + 2 \\ , B } \\end{align*}"}
{"id": "3208.png", "formula": "\\begin{align*} E _ { k , t } ^ { } = \\tau p _ { k , t } , ~ \\forall k \\in { \\cal K } . \\end{align*}"}
{"id": "6016.png", "formula": "\\begin{align*} f ^ { ( j + 1 ) } ( x ) & = { \\mu _ j } ( x - \\lambda ) ^ { { \\mu _ j } - 1 } g _ j ( x ) + ( x - \\lambda ) ^ { \\mu _ j } g ' _ j ( x ) \\\\ & = ( x - \\lambda ) ^ { { \\mu _ j } - 1 } g _ { j + 1 } ( x ) \\end{align*}"}
{"id": "5128.png", "formula": "\\begin{align*} \\phi \\frac { \\partial } { \\partial x _ i } \\phi ^ { - 1 } ( x _ j ) = \\phi \\frac { \\partial } { \\partial x _ i } ( \\phi ^ { - 1 } ) _ 0 ( x _ j ) + \\phi \\frac { \\partial } { \\partial x _ i } ( \\phi ^ { - 1 } ) _ 1 ( x _ j ) \\end{align*}"}
{"id": "2581.png", "formula": "\\begin{align*} H : = \\left \\{ \\small { \\begin{pmatrix} \\det ( U ) ^ { - 1 } & 0 \\\\ 0 & U \\end{pmatrix} } \\ , \\ \\ U \\in U ( 2 ) \\right \\} \\simeq S \\big ( U ( 2 ) \\times U ( 1 ) \\big ) \\simeq U ( 2 ) \\ , \\end{align*}"}
{"id": "4826.png", "formula": "\\begin{align*} g . \\rho : \\pi _ 1 ( M ) & \\rightarrow \\mathrm { G L } _ n ( K ) \\\\ \\gamma & \\mapsto \\big ( g \\big ( { \\rho ( \\gamma ) } _ { i , j } \\big ) \\big ) _ { 1 \\leqslant i , j \\leqslant n } \\ ; , \\end{align*}"}
{"id": "59.png", "formula": "\\begin{align*} - j r ^ { - 1 } ( ( r ( \\dfrac { \\zeta { \\pi ' _ n } ^ j } { 1 - \\zeta { \\pi ' _ n } ^ j } \\alpha ) , \\pi ' _ n ) _ { \\rho ' , L , n } ) = - j r ^ { - 1 } ( [ r ( \\dfrac { \\zeta { \\pi ' _ n } ^ j } { 1 - \\zeta { \\pi ' _ n } ^ j } \\alpha ) , \\pi ' _ n ] _ { \\rho ' , L , n } ) . \\end{align*}"}
{"id": "211.png", "formula": "\\begin{align*} \\begin{gathered} R _ { 0 } + \\tilde { R } _ { 0 } > H ( Y _ { 0 } | Y _ { j } Z _ { j } ) , \\\\ R _ { j } + \\tilde { R } _ { j } > H ( Y _ { j } | Y _ { 0 } Z _ { j } ) , \\\\ R _ { 0 } + \\tilde { R } _ { 0 } + R _ { j } + \\tilde { R } _ { j } > H ( Y _ { 0 } Y _ { j } | Z _ { j } ) . \\\\ \\end{gathered} \\end{align*}"}
{"id": "4040.png", "formula": "\\begin{align*} Y = X \\beta + e \\end{align*}"}
{"id": "506.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\in \\overline { E } \\\\ \\mathrm { h e a d } ( a ) = i } } \\epsilon ( a ) a a ^ * = 0 . \\end{align*}"}
{"id": "2100.png", "formula": "\\begin{align*} & = \\sum \\limits _ { v \\in e } \\frac { \\delta _ E ( e ) } { | e | } [ ( a v g ( x ) ) ( e ) - x ( v ) ) ] x ( v ) \\\\ & = \\sum \\limits _ { i = 1 } ^ k \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\left [ \\left \\{ \\sum \\limits _ { j = 1 } ^ k x ( v _ j ) \\right \\} - | e | x ( v _ i ) ) \\right ] x ( v _ i ) \\\\ & = - \\frac { 1 } { 2 } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum _ { i , j = 1 } ^ k ( x ( v _ i ) - x ( v _ j ) ) ^ 2 \\le 0 . \\end{align*}"}
{"id": "7855.png", "formula": "\\begin{align*} \\varphi ( a ^ * J b ^ * J T J b J a ) \\leq \\| T \\| \\varphi ( a ^ * J b ^ * J e _ A J b J a ) = 0 \\end{align*}"}
{"id": "660.png", "formula": "\\begin{align*} \\left ( I + E ( g ) \\right ) ^ { - 1 } = \\sum _ { k = 0 } ^ { \\infty } ( - 1 ) ^ k \\left ( E ( g ) \\right ) ^ { k } \\left \\| \\left ( I + E ( g ) \\right ) ^ { - 1 } \\right \\| _ { \\mathcal B _ 3 } \\le \\frac { 1 } { 1 - \\left \\| E ( g ) \\right \\| _ { \\mathcal B _ 3 } } . \\end{align*}"}
{"id": "2656.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ { n - 1 } ( n - k ) p ^ { n - 3 } ( p - 1 ) & = ( p - 1 ) p ^ { n - 3 } \\sum _ { k = 2 } ^ { n - 1 } ( n - k ) \\\\ & = ( p - 1 ) p ^ { n - 3 } \\binom { n - 1 } { 2 } . \\end{align*}"}
{"id": "2449.png", "formula": "\\begin{align*} & { \\bf T } _ { w _ 0 ^ W } P ( \\mu ) = T ^ r ( \\mu ' ) , \\end{align*}"}
{"id": "3471.png", "formula": "\\begin{align*} \\mathrm { D T M } _ { ( p , q ) } ( X ^ { n } ) & = \\mathrm { E } [ ( X - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { n } | x _ { p } < X < x _ { q } ] \\\\ & = \\mathrm { E } \\left [ \\sum _ { k = 0 } ^ { n } \\binom { n } { k } X ^ { k } ( - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { n - k } | x _ { p } < X < x _ { q } \\right ] \\\\ & = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { n - k } \\mathrm { E } [ X ^ { k } | x _ { p } < X < x _ { q } ] . \\end{align*}"}
{"id": "445.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( z ) \\Sigma _ { \\mu } ( \\eta _ { \\mu } ( z ) ) = z \\end{align*}"}
{"id": "5375.png", "formula": "\\begin{align*} B _ P ( v , w ) \\vcentcolon = \\langle ( - \\Delta ) ^ { s / 2 } v , ( - \\Delta ) ^ { s / 2 } w \\rangle + \\sum _ { | \\alpha | \\leq m } \\langle a _ \\alpha , ( D ^ \\alpha v ) w \\rangle \\end{align*}"}
{"id": "4375.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t \\} } | f _ t | ^ 2 + \\int _ { \\{ \\Psi < - t \\} } | \\hat { f } - f _ t | ^ 2 = \\int _ { \\{ \\Psi < - t \\} } | \\hat { f } | ^ 2 . \\end{align*}"}
{"id": "7325.png", "formula": "\\begin{align*} | x | ^ \\frac { 2 p } { 1 - q } & = | x | ^ \\frac { 2 ( p - q + 2 ) } { 1 - q } \\eta ^ { - \\frac { 4 } { 1 - q } } | \\xi | ^ { - 2 \\gamma } \\cdot | x | ^ \\frac { 2 ( q - 2 ) } { 1 - q } \\eta ^ \\frac { 4 } { 1 - q } | \\xi | ^ { 2 \\gamma } \\\\ & = \\eta ^ \\frac { 2 ( p - q ) } { 1 - q } | \\xi | ^ { \\frac { 2 ( p - q + 2 ) } { 1 - q } - 2 \\gamma } \\cdot | x | ^ \\frac { 2 ( q - 2 ) } { 1 - q } \\eta ^ \\frac { 4 } { 1 - q } | \\xi | ^ { 2 \\gamma } , \\end{align*}"}
{"id": "7973.png", "formula": "\\begin{align*} H _ { ( \\mathbb P ^ 2 , H + H ) , ( d , d ) } ( y ) = H _ { ( F _ 1 , D _ 2 + D _ 3 + D _ 4 ) , ( d , 0 , 0 ) } ( 0 , y ) . \\end{align*}"}
{"id": "2845.png", "formula": "\\begin{align*} U : = \\sqrt { \\frac { 2 L \\ \\Delta } { \\sum \\limits _ { i = 0 } ^ { N - 1 } h _ i \\big ( 2 - h _ i \\frac { - \\kappa } { 1 - \\kappa } \\big ) } } . \\end{align*}"}
{"id": "3136.png", "formula": "\\begin{align*} P ^ { \\mathrm { M F } } \\left ( \\eta \\right ) = \\inf _ { c \\in \\mathbb { C } } \\left \\{ \\eta \\left \\vert c \\right \\vert ^ { 2 } + P \\left ( c , \\eta \\right ) \\right \\} \\ , \\eta \\in \\mathbb { R } _ { 0 } ^ { + } \\ , \\end{align*}"}
{"id": "4486.png", "formula": "\\begin{align*} \\upsilon _ \\pm ( \\eta ) = \\left \\{ \\begin{aligned} & 1 , { } \\pm \\eta \\geq 0 , \\\\ & 0 , { } \\pm \\eta < 0 , \\end{aligned} \\right . \\end{align*}"}
{"id": "4789.png", "formula": "\\begin{align*} \\Psi _ N = \\frac { W _ { N _ 0 } T _ \\nu e ^ { A _ \\nu } \\Omega } { \\| e ^ { A _ { \\nu } } \\Omega \\| ^ 2 } . \\end{align*}"}
{"id": "8838.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } ^ { } f \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) ( x ) d x = f _ + ' ( 0 ) \\| \\phi _ 1 ^ * * \\phi _ 2 ^ * \\| + \\int _ { 0 } ^ { \\infty } \\| f _ t \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) \\| d \\nu ( t ) . \\end{align*}"}
{"id": "1464.png", "formula": "\\begin{align*} C _ { u , m } = \\prod _ { i = 1 } ^ m \\alpha _ i ^ { r ( u + 1 ) + r ^ 2 n + { r \\choose 2 } } \\left ( \\Psi _ { \\boldsymbol { 1 } } \\left ( Q _ u \\right ) \\right ) \\enspace . \\end{align*}"}
{"id": "6485.png", "formula": "\\begin{align*} u _ x = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) y _ x = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) . \\end{align*}"}
{"id": "3193.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\rightarrow 0 ^ { + } } \\mathfrak { F } _ { \\gamma } \\left ( \\Phi , f \\right ) = 0 \\ , \\Phi \\in \\mathcal { W } _ { 0 } \\subseteq \\mathcal { W } _ { 1 } \\ , \\end{align*}"}
{"id": "4579.png", "formula": "\\begin{align*} C _ { R } ( J ) R [ x ] = C _ { R [ x ] } ( J ' ) \\textrm { a n d } C _ { R [ x ] } ( J ' ) \\cap R = C _ R ( J ) . \\end{align*}"}
{"id": "3521.png", "formula": "\\begin{align*} L _ { 1 } = \\frac { \\xi _ { p } ( 1 + | \\xi _ { p } | ) \\exp ( - | \\xi _ { p } | ) - \\xi _ { q } ( 1 + | \\xi _ { q } | ) \\exp ( - | \\xi _ { q } | ) } { 2 F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , ~ L _ { 2 } & = \\frac { F _ { Y _ { ( 1 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "1008.png", "formula": "\\begin{align*} \\mathsf { F r } _ x ( B ) = \\lim _ { n \\to \\infty } \\frac 1 n \\# \\big \\{ i \\in \\{ 1 , 2 , \\dots , n \\} : ( x _ i , x _ { i + 1 } , \\dots , x _ { i + k - 1 } ) = B \\big \\} , \\end{align*}"}
{"id": "365.png", "formula": "\\begin{align*} q _ A = k \\frac { \\displaystyle \\sum _ { m = 0 } ^ \\infty [ p ^ \\# _ m ( o , o ) - \\pi ( o ) ] } { \\displaystyle \\sum _ { m = 0 } ^ \\infty [ \\tilde p ^ \\# _ m ( A , A ) - \\pi ( A ) ] } \\end{align*}"}
{"id": "349.png", "formula": "\\begin{align*} S ( \\delta ) = o ( 1 ) \\end{align*}"}
{"id": "486.png", "formula": "\\begin{align*} ( 1 + 4 \\sqrt { c _ 1 } ) ^ \\rho \\cdot r N M = ( 1 + 4 \\sqrt { c _ 1 } ) ^ \\rho \\cdot r \\cdot c ( r - 1 , g , d ^ { 2 g } ) ^ { \\rho + 1 } \\cdot M ( c _ 1 ) \\leq c _ 3 ^ { 1 + \\rho } \\end{align*}"}
{"id": "6583.png", "formula": "\\begin{align*} \\{ . , . \\} \\ , = \\ , \\beta _ * \\circ C ^ \\infty ( U , [ . , . ] ) \\circ ( D \\times D ) \\end{align*}"}
{"id": "6234.png", "formula": "\\begin{align*} & \\int \\frac { ( - q ^ { n + 3 } x ^ 2 ; q ^ 2 ) _ \\infty } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\widetilde { h } _ n ( x ; q ) d _ q x = \\\\ & \\frac { ( - q ^ { n + 1 } x ^ 2 ; q ^ 2 ) _ \\infty } { [ n + 1 ] _ q ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( q ^ n x \\widetilde { h } _ n ( x ; q ) - q ^ { 1 - n } ( 1 - q ^ n ) \\widetilde { h } _ { n - 1 } ( x ; q ) \\right ) , \\end{align*}"}
{"id": "7324.png", "formula": "\\begin{align*} N _ 3 [ \\epsilon , v , w ] & = n _ 3 ( 1 - \\chi _ 2 ) \\chi _ 4 \\\\ & = \\underbrace { n _ 3 ( \\chi _ { 2 , { \\sf a } } - \\chi _ 2 ) } _ { = { \\sf t } _ 7 } + \\underbrace { n _ 3 ( \\chi _ 4 - \\chi _ { 3 , { \\sf e } } ) } _ { = { \\sf t } _ 8 } + \\underbrace { n _ 3 ( 1 - \\chi _ { 2 , { \\sf a } } ) \\chi _ { 3 , { \\sf e } } } _ { = { \\sf t } _ 9 } . \\end{align*}"}
{"id": "6034.png", "formula": "\\begin{align*} \\Lambda _ { \\sigma } = ( \\kappa _ 1 , \\kappa _ 2 , \\ldots , \\kappa _ { m } ) \\end{align*}"}
{"id": "4313.png", "formula": "\\begin{align*} \\dim ( \\ker \\bar { \\partial } \\subset W ^ { 1 , 2 ; \\mu } ) = \\deg q - \\sum _ 1 ^ k \\deg p _ i + \\frac { n \\mu } { \\pi } . \\end{align*}"}
{"id": "1984.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / 2 } } Q ( 0 ) \\lambda ( U _ 0 ) \\ , d x & \\le \\int _ { B _ { 1 / 2 } } Q ( 0 ) \\liminf _ { k \\to \\infty } \\lambda ( U _ k ) \\ , d x \\le \\liminf _ { k \\to \\infty } \\int _ { B _ { 1 / 2 } } Q ( 0 ) \\lambda ( U _ k ) \\ , d x . \\\\ \\end{align*}"}
{"id": "4145.png", "formula": "\\begin{align*} \\check f ( x ) = ( 2 \\pi ) ^ { - n } \\int _ { \\R ^ n } f ( \\xi ) e ^ { i x \\xi } \\ , d \\xi , x \\in \\R ^ n . \\end{align*}"}
{"id": "877.png", "formula": "\\begin{align*} s _ { b _ 1 } ^ { \\beta _ 1 } s _ { b _ 2 } ^ { \\beta _ 2 } \\cdots s _ { b _ q } ^ { \\beta _ q } f _ 1 ( t , s ) ^ { \\mu _ 1 } f _ 2 ( t , s ) ^ { \\mu _ 2 } \\cdots f _ n ( t , s ) ^ { \\mu _ n } s _ { w _ 1 } ^ { \\partial _ 1 } s _ { w _ 2 } ^ { \\partial _ 2 } \\cdots s _ { w _ l } ^ { \\partial _ l } = s _ { a _ 1 } ^ { \\alpha _ 1 } s _ { a _ 2 } ^ { \\alpha _ 2 } \\cdots s _ { a _ p } ^ { \\alpha _ p } . \\end{align*}"}
{"id": "4867.png", "formula": "\\begin{align*} \\mathfrak { M } = \\mathfrak { M } _ { 1 } \\times \\mathfrak { M } _ { 2 } = \\{ ( \\alpha _ { 1 } , \\alpha _ { 2 } ) : \\alpha _ { 1 } \\in \\mathfrak { M } _ { 1 } , \\alpha _ { 2 } \\in \\mathfrak { M } _ { 2 } \\} , \\ \\mathfrak { m } = \\left [ \\frac { 1 } { Q _ { i } } , 1 + \\frac { 1 } { Q _ { i } } \\right ] ^ 2 \\setminus \\mathfrak { M } . \\end{align*}"}
{"id": "2753.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\sigma } _ { p } u _ { j } ( x ) = K _ { j } ( x ) u _ { j } ^ { q ( p - 1 ) } ( x ) , \\ ; \\ , x \\in \\mathbb { R } ^ { n } , \\quad | x | ^ { s } u _ { j } ( x ) \\rightarrow 1 , \\quad , \\end{align*}"}
{"id": "4092.png", "formula": "\\begin{align*} E ( \\mathbf { R } _ n ) _ 3 = \\varepsilon n \\left [ \\frac { n - 1 } { 2 } a _ 3 + b _ 3 \\right ] . \\end{align*}"}
{"id": "3234.png", "formula": "\\begin{align*} \\dim \\mathcal { A } _ { 2 m } = h ^ 0 ( K _ { X } ^ { 2 m } ( D ^ { 2 m - 1 } ) ) = ( 4 m - 1 ) ( g - 1 ) + ( 2 m - 1 ) n . \\end{align*}"}
{"id": "4353.png", "formula": "\\begin{align*} v _ r \\left ( a _ s g ^ s \\right ) = v _ r ( f ) < v _ r ( f - h ) \\le v _ r \\left ( ( a _ s - a ' _ s ) g ^ s \\right ) . \\end{align*}"}
{"id": "3110.png", "formula": "\\begin{align*} v _ { \\mathrm { n c } } : = \\sum _ { { \\ell } = 1 } ^ { j } \\xi _ { \\ell } \\phi _ { \\mathrm { n c } } ( { \\ell } ) , v _ { \\mathrm { p w } } : = \\sum _ { { \\ell } = 1 } ^ { j } \\xi _ { \\ell } \\phi _ { \\mathrm { p w } } ( { \\ell } ) , w _ { \\mathrm { p w } } : = \\sum _ { { \\ell } = 1 } ^ { j } \\xi _ { \\ell } \\lambda _ h ( \\ell ) \\phi _ { \\mathrm { p w } } ( { \\ell } ) . \\end{align*}"}
{"id": "6430.png", "formula": "\\begin{align*} \\frac { \\dd \\Theta _ t ^ { ( 1 ) } } { \\dd t } & = \\Bar { Q } ^ { ( 0 ) } \\circ H _ t ^ { ( 1 ) } \\\\ & = [ Q , H ^ { ( 1 ) } ] = \\Bar { \\Psi } ^ { ( 1 ) } - \\Xi _ 1 ^ { ( 1 ) } . \\end{align*}"}
{"id": "8905.png", "formula": "\\begin{align*} c _ \\rho : = \\inf _ { \\gamma \\in \\Gamma } \\ \\max _ { t \\in [ 0 , 1 ] } \\Phi _ \\rho ( \\gamma ( t ) ) > \\max \\{ \\Phi _ \\rho ( w _ 1 ) , \\Phi _ \\rho ( w _ 2 ) \\} , \\rho \\in I . \\end{align*}"}
{"id": "332.png", "formula": "\\begin{align*} \\sum _ { n \\in L ^ x \\cap { \\rm L } _ { A ^ y } } \\frac { 1 } { n } & = \\sum _ { a \\in A ^ y } \\frac { 1 } { a } \\prod _ { P ( a ) \\le p \\le x } ( 1 - \\tfrac { 1 } { p } ) ^ { - 1 } = \\sum _ { a \\in A ^ y } \\frac { 1 } { a } \\prod _ { p < P ( a ) } ( 1 - \\tfrac { 1 } { p } ) \\prod _ { p \\le x } ( 1 - \\tfrac { 1 } { p } ) ^ { - 1 } \\\\ & = { \\rm d } ( { \\rm L } _ { A ^ y } ) \\prod _ { p \\le x } ( 1 - \\tfrac { 1 } { p } ) ^ { - 1 } . \\end{align*}"}
{"id": "1660.png", "formula": "\\begin{align*} f ( \\alpha ) & = \\sum _ { \\{ B \\in { H } : \\alpha \\in B \\} } a _ B \\\\ & = \\sum _ { \\{ D _ B \\in F _ 1 : \\alpha \\in D _ B \\} } a _ B + \\sum _ { B \\in S _ \\gamma } a _ B \\\\ & = \\sum _ { \\{ D _ B \\in F _ 1 : \\alpha \\in D _ B \\} } a _ B \\end{align*}"}
{"id": "8277.png", "formula": "\\begin{align*} \\psi _ { A _ { k } } = A _ { k } \\left [ \\frac { \\cosh ( k x ) } { \\cosh ( k a / 2 ) } - \\frac { \\cosh ( k ' x ) } { \\cosh ( k ' a / 2 ) } \\right ] \\end{align*}"}
{"id": "4180.png", "formula": "\\begin{align*} F ^ { ( \\iota ) } : = ( \\hat F \\chi _ \\iota ) ^ \\vee \\iota \\in \\N , \\end{align*}"}
{"id": "2627.png", "formula": "\\begin{align*} \\frac { d f } { d t } = [ f , h ] _ { \\star } \\ , \\end{align*}"}
{"id": "5899.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { V \\in \\mathbb { V } _ { n , k } } \\rho _ \\mathrm { L P } ( \\mu _ { V X ^ { ( n ) } } , \\tilde { \\mu } _ { V X ^ { ( n ) } } ) = 0 . \\end{align*}"}
{"id": "453.png", "formula": "\\begin{align*} \\Phi _ { n } ( \\eta _ { \\nu _ { n } } ( z ) ) = z \\end{align*}"}
{"id": "5505.png", "formula": "\\begin{align*} \\d u ( t ) = \\left [ ( 1 + i \\alpha ) \\Delta u ( t ) - ( \\gamma ( t ) + i \\beta ) | u ( t ) | ^ { 2 } u ( t ) + f ( t , u ( t ) ) \\right ] \\d t + g ( t , u ( t ) ) \\d W ( t ) , ~ t \\in \\R , \\end{align*}"}
{"id": "682.png", "formula": "\\begin{align*} - \\Delta _ { g _ o } ^ { f _ o } \\varphi _ i = \\lambda _ i \\varphi _ i , \\int _ M \\varphi _ i \\varphi _ j d \\nu _ * = \\delta _ { i j } . \\end{align*}"}
{"id": "7369.png", "formula": "\\begin{align*} { \\Delta W _ h - W _ h + \\sum _ { i = 1 } ^ 4 U _ { h , i } ^ p = 0 . } \\end{align*}"}
{"id": "2997.png", "formula": "\\begin{align*} e = e _ 1 e _ 2 e _ 3 & \\le 2 \\log _ { q } ( 5 g + 1 + n ) + e _ 1 e _ 2 \\\\ & \\le 2 \\log _ { q } ( 5 g + 1 + n ) + 2 \\log _ { q } \\max \\{ n + 1 , 2 g + 1 \\} + e _ 1 \\\\ & \\le 2 \\log _ { q } ( 5 g + 1 + n ) + 2 \\log _ { q } \\max \\{ n + 1 , 2 g + 1 \\} + \\log _ q ( n ) + 2 . \\end{align*}"}
{"id": "3403.png", "formula": "\\begin{align*} D _ { n , v } = \\begin{cases} \\displaystyle { \\sum _ { w \\in \\Z [ \\zeta ] ^ 3 + v \\ ; : \\ ; \\langle w , w \\rangle = n } w ^ \\perp } & \\\\ [ 8 m m ] \\displaystyle { \\frac { 1 } { 2 } \\sum _ { w \\in \\Z [ \\zeta ] ^ 3 + v \\ ; : \\ ; \\langle w , w \\rangle = n } w ^ \\perp } & \\end{cases} \\end{align*}"}
{"id": "5324.png", "formula": "\\begin{align*} Z _ { i , s } ^ { ( n ) } ( t , \\phi ) = n ^ { - 1 } \\sum _ { j = 1 } ^ n 1 ( j \\in \\pi ^ t _ i ( s ) ) \\phi ( X _ j ( t ) ) \\end{align*}"}
{"id": "6651.png", "formula": "\\begin{align*} \\log n ! = n \\log n - n \\log e + o ( n ) \\ ; . \\ , \\end{align*}"}
{"id": "1466.png", "formula": "\\begin{align*} f _ { n , u } ( \\alpha , \\beta , \\underline { X } , \\underline { Y } ) = f ( \\alpha , \\beta , \\underline { X } , \\underline { Y } ) & = \\displaystyle c ( t _ { i , s } ) \\prod _ { k \\geq 3 } \\prod _ { s { { = 0 } } } ^ { { r - 1 } } \\prod _ { 1 \\leq i , j \\leq r } ( t _ { k , s } - X _ i ) ( t _ { k , s } - Y _ j ) \\\\ & \\cdot \\displaystyle \\prod _ { s = { { 0 } } } ^ { { r - 1 } } ( X _ s Y _ s ) ^ u [ ( X _ s - \\alpha ) ( X _ s - \\beta ) ( Y _ s - \\alpha ) ( Y _ s - \\beta ) ] ^ { r n } \\enspace , \\end{align*}"}
{"id": "2821.png", "formula": "\\begin{align*} \\| \\nabla f ( x _ N ) \\| ^ 2 { } \\leq { } \\frac { L \\big [ f ( x _ 0 ) - f ( x _ N ) \\big ] } { \\sum \\limits _ { i = 0 } ^ { N - 1 } h _ i } \\| \\nabla f ( x _ N ) \\| ^ 2 { } \\leq { } \\frac { L \\big [ f ( x _ 0 ) - f _ * \\big ] } { \\tfrac { 1 } { 2 } + \\sum \\limits _ { i = 0 } ^ { N - 1 } h _ i } \\end{align*}"}
{"id": "804.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ { \\infty } | b _ k | | z | ^ k & = M _ { r } ^ { N } ( g ) \\leq M _ { r } ^ { N } ( f ) \\\\ & = \\int _ { 0 } ^ { r } \\frac { M _ { t } ^ { N } ( G ) M _ { t } ^ { N } ( \\psi \\circ \\omega ) } { t } d t \\\\ & \\leq \\int _ { 0 } ^ { r } \\frac { M _ { t } ^ { N } ( h _ { \\psi } ) M _ { t } ^ { N } ( \\psi ) } { t } d t = : R ^ N ( r ) , \\end{align*}"}
{"id": "4444.png", "formula": "\\begin{align*} \\L I _ m ^ * g , f \\R = \\L g , I _ m f \\R \\end{align*}"}
{"id": "5280.png", "formula": "\\begin{align*} \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { r + k _ 1 ( 0 ) } a ^ { \\widehat { P } ^ { C o n t , + r } } & + \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { s + k _ 2 ( 0 ) } b ^ { \\widehat { P } ^ { C o n t , + s } } \\\\ & = 1 + \\sum _ { \\substack { j = 1 \\\\ 2 \\notin I _ j } } ^ h \\left ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { P } ^ { C o n t } _ j } + \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { P } ^ { C o n t } _ j } \\right ) . \\end{align*}"}
{"id": "8559.png", "formula": "\\begin{align*} 1 \\geq \\int \\cosh _ 2 f \\ d \\mu \\geq \\frac 1 { ( 2 k ) ! } \\int f ^ { 2 k } \\ d \\mu \\end{align*}"}
{"id": "6867.png", "formula": "\\begin{align*} \\begin{aligned} A _ 1 & : = \\int _ { - \\infty } ^ { \\lambda } \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp ( - g ^ 2 / 2 ) d g , \\\\ A _ 2 & : = \\frac { \\frac { 1 } { 2 \\sqrt { 2 \\pi } } \\exp ( - \\lambda ^ 2 / 2 ) } { \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp ( - \\lambda ^ 2 / 2 ) + \\lambda \\int _ { - \\lambda } ^ { + \\infty } \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp ( - g ^ 2 / 2 ) d g } . \\end{aligned} \\end{align*}"}
{"id": "3827.png", "formula": "\\begin{align*} E r r H ^ 0 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\sum _ { a = 1 , 2 } ( - 1 ) ^ a \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( t _ a ) \\cdot \\xi - i t _ a \\hat { v } \\cdot \\eta + i t _ a \\mu _ 1 | \\xi - \\eta | } ( \\hat { V } ( t _ a ) \\cdot \\xi - \\hat { v } \\cdot \\eta + \\mu _ 1 | \\xi - \\eta | ) ^ { - 1 } \\hat { g } ( t _ a , \\eta , v ) \\end{align*}"}
{"id": "8084.png", "formula": "\\begin{align*} Z = f ( r ) \\partial _ { \\theta } . \\end{align*}"}
{"id": "826.png", "formula": "\\begin{align*} \\widetilde { w } ( t ) = \\chi _ { ( 0 , s ) } ( t ) t ^ \\alpha . \\end{align*}"}
{"id": "5823.png", "formula": "\\begin{align*} \\frac { y _ { n } } { x _ { n } } = \\prod _ { k \\leq n } \\left ( 1 - \\frac { \\chi ( p _ k ) } { p _ k ^ 2 } \\right ) ^ { - 1 } = \\prod _ { k \\leq n } \\frac { p _ k ^ 2 } { p _ k ^ 2 - \\chi ( p _ k ) } . \\end{align*}"}
{"id": "1342.png", "formula": "\\begin{align*} \\beta = \\frac { \\chi } { 1 + \\chi } , \\gamma \\leq \\min \\left \\{ \\frac { \\tilde { \\lambda } \\chi ( 1 - \\chi ) \\xi } { 8 ( 1 + \\chi ) } , \\frac { 1 + \\chi } { 2 ( 2 + \\chi ) R _ 0 \\| \\nabla _ x M \\| _ \\infty } \\right \\} , \\mbox { w i t h } \\xi : = \\begin{cases} \\tilde C ^ { k - 2 } , & \\mbox { i f } k < 2 , \\\\ 1 , & \\mbox { i f } k = 2 , \\\\ \\| \\nabla _ x M \\| _ \\infty ^ { k - 2 } , & \\mbox { i f } k > 2 . \\end{cases} \\end{align*}"}
{"id": "2412.png", "formula": "\\begin{align*} \\mathrm { d } _ { \\mathrm { T V } } ( X _ n , X _ \\infty ) = \\| p _ { X _ n } - p _ { X _ \\infty } \\| _ { L ^ 1 ( \\R ) } . \\end{align*}"}
{"id": "2725.png", "formula": "\\begin{align*} \\Phi _ { \\mathrm { t r i a l } , s } : = \\frac { \\big ( \\mathbb { U } _ \\mathrm { l e f t } \\mathbb { U } _ \\mathrm { r i g h t } \\Omega \\big ) _ { s } } { \\sqrt { \\sum _ { s = 0 } ^ N \\left \\| \\big ( \\mathbb { U } _ \\mathrm { l e f t } \\mathbb { U } _ \\mathrm { r i g h t } \\Omega \\big ) _ { s } \\right \\| ^ 2 } } \\in \\mathcal { F } _ \\perp . \\end{align*}"}
{"id": "2261.png", "formula": "\\begin{align*} \\Phi = ( \\phi , \\psi ) \\in H _ 0 ^ 1 ( 0 , r ) \\times H _ 0 ^ 1 ( 0 , r ) \\triangleq X , ~ { \\rm { a n d } } ~ F = ( f _ 1 , f _ 2 ) \\in H ^ { - 1 } \\times H ^ { - 1 } \\triangleq X ^ { - 1 } . \\end{align*}"}
{"id": "5175.png", "formula": "\\begin{align*} f _ 1 ( x _ { - 2 } , x _ { - 1 } , x _ 1 , x _ 2 ) & = x _ { - 1 } ( a _ m - x _ 1 ) \\le 0 \\\\ f _ 2 ( x _ { - 2 } , x _ { - 1 } , x _ 1 , x _ 2 ) & = x _ { - 2 } ( a _ m - x _ 1 ) + x _ 1 ( a _ m - x _ { - 2 } ) \\le 0 \\\\ f _ 3 ( x _ { - 2 } , x _ { - 1 } , x _ 1 , x _ 2 ) & = x _ { - 1 } ( a _ m - x _ { 2 } ) + x _ 2 ( a _ m - x _ { - 1 } ) \\le 0 \\\\ f _ 4 ( x _ { - 2 } , x _ { - 1 } , x _ 1 , x _ 2 ) & = x _ 1 ( a _ m - x _ { - 1 } ) \\le 0 . \\end{align*}"}
{"id": "6513.png", "formula": "\\begin{align*} u = 0 y = 0 \\omega _ c . \\end{align*}"}
{"id": "3998.png", "formula": "\\begin{align*} \\Omega _ { 2 \\ell } = \\{ L \\ , | \\ , \\mbox { $ L $ i s a d i r e c t s u m o f s o m e o f $ C _ 1 , \\cdots , C _ r $ , \\ , $ \\dim L = 2 \\ell $ } \\} ; \\end{align*}"}
{"id": "5185.png", "formula": "\\begin{align*} S = L \\otimes \\O \\left ( - \\sum _ { i \\in I _ 0 } [ z _ i ] \\right ) , \\end{align*}"}
{"id": "7044.png", "formula": "\\begin{align*} g ( M ) = \\sum _ { 0 \\leq 2 n \\leq M - 1 } ( M - 2 n ) \\binom { M - n - 1 } n . \\end{align*}"}
{"id": "3870.png", "formula": "\\begin{align*} - \\Delta _ p ( \\Gamma + H _ \\lambda ) + \\Delta _ p \\Gamma = \\lambda G _ \\lambda ^ { p - 1 } \\Omega \\setminus \\{ x _ 0 \\} \\end{align*}"}
{"id": "3703.png", "formula": "\\begin{align*} \\| \\lambda _ j ^ { - 1 } \\partial _ \\nu u _ { \\lambda _ j } \\| _ { L ^ 2 ( \\Gamma ) } = o ( 1 ) . \\end{align*}"}
{"id": "1652.png", "formula": "\\begin{align*} a _ { 0 0 0 } a _ { 1 1 0 } a _ { 1 0 1 } a _ { 0 1 1 } \\ , = \\ , a _ { 1 0 0 } a _ { 0 1 0 } a _ { 0 0 1 } a _ { 1 1 1 } \\end{align*}"}
{"id": "5582.png", "formula": "\\begin{align*} \\widetilde { G } ( x , y ) = \\frac { G ( x , y ) } { g ( x ) g ( y ) } \\end{align*}"}
{"id": "2408.png", "formula": "\\begin{align*} \\| ( - A _ n ) ^ { \\frac { 1 } { 2 } } ( F _ n ( a ) - F _ n ( b ) ) \\| ^ 2 _ { l _ n ^ 2 } = \\frac { n } { \\pi } \\sum _ { j = 1 } ^ n | f ( a _ j ) - f ( b _ j ) - f ( a _ { j - 1 } ) + f ( b _ { j - 1 } ) | ^ 2 \\\\ \\le \\frac { 2 n } { \\pi } \\sum _ { j = 1 } ^ n | a _ j ^ 3 - b _ j ^ 3 - a _ { j - 1 } ^ 3 + b _ { j - 1 } ^ 3 | ^ 2 + 2 \\| ( - A _ n ) ^ { \\frac { 1 } { 2 } } ( a - b ) \\| ^ 2 _ { l _ n ^ 2 } . \\end{align*}"}
{"id": "8702.png", "formula": "\\begin{align*} \\dot { \\gamma } ( t ) = ( \\dot { x } ^ { 1 } ( t ) , \\dot { x } ^ { 2 } ( t ) ) = ( \\dot { r } ( t ) \\cos t - r ( t ) \\sin t , \\dot { r } ( t ) \\sin t + r ( t ) \\cos t ) . \\end{align*}"}
{"id": "6901.png", "formula": "\\begin{align*} n ( \\tau , v ) = \\rho _ { } ( v - \\tau ) , \\end{align*}"}
{"id": "5680.png", "formula": "\\begin{align*} x _ 1 ^ { \\ell + 1 } + x _ 2 ^ { \\ell + 1 } + \\cdots + x _ i ^ { \\ell + 1 } = ( x _ 1 ^ { \\ell } + x _ 2 ^ { \\ell } + \\cdots + x _ i ^ { \\ell } ) x _ { i + 1 } ( 1 \\le \\ell \\le d - 1 ) , \\end{align*}"}
{"id": "7241.png", "formula": "\\begin{align*} & \\dot { x } ( t , s ) = \\lambda x ( t , s ) + \\partial _ s ^ 2 x ( t , s ) , s \\in [ 0 , 1 ] , \\\\ & \\dot { \\hat x } ( t , s ) = \\lambda \\hat x ( t , s ) + \\partial _ s ^ 2 \\hat x ( t , s ) \\\\ & \\qquad + \\int _ 0 ^ 1 l ( s ) \\left ( \\partial _ s ^ 2 x ( t , \\theta ) - \\partial _ s ^ 2 \\hat x ( t , \\theta ) \\right ) d \\theta , \\\\ & x ( t , 0 ) = 0 , x ( t , 1 ) = 0 , \\\\ & \\hat x ( t , 0 ) = 0 , \\hat x ( t , 1 ) = 0 . \\end{align*}"}
{"id": "3445.png", "formula": "\\begin{align*} & x _ i \\frac { \\partial } { \\partial x _ i } : x ^ m \\mapsto m _ i \\cdot x ^ m , i = 1 , . . . , n , \\\\ & \\frac { \\partial e _ { - \\alpha } ^ { \\lambda } } { \\partial \\lambda } _ { \\vert { \\lambda = 0 } } : x ^ m \\mapsto h t _ { - \\alpha } ( m ) \\cdot x ^ { m - \\alpha } , \\alpha \\in R ( N , \\Sigma _ { C ( \\Delta ) } ) . \\end{align*}"}
{"id": "8691.png", "formula": "\\begin{align*} E ( x ^ { 1 } , x ^ { 2 } , \\dot { x } ^ 1 , \\dot { x } ^ { 2 } , u ^ { 1 } , u ^ { 2 } ) = h ( x ^ { 1 } , x ^ { 2 } , u ^ { 1 } , u ^ { 2 } ) - h ( x ^ { 1 } , x ^ { 2 } , \\dot { x } ^ 1 , \\dot { x } ^ { 2 } ) - \\sum \\limits _ { j = 1 } ^ { 2 } ( u ^ { j } - \\dot { x } ^ { j } ) \\frac { \\partial h ( x ^ { 1 } , x ^ { 2 } , \\dot { x } ^ 1 , \\dot { x } ^ { 2 } ) } { \\partial \\dot { x } ^ j } . \\end{align*}"}
{"id": "6832.png", "formula": "\\begin{align*} \\begin{aligned} B ( t ) = \\int _ { 0 } ^ { t } e ^ { - ( t - s ) } g _ { } ( s ) d s = \\int _ { 0 } ^ { t } e ^ { - ( t - s ) } ( g _ 0 + g _ 1 N ( s ) ) d s . \\\\ C ( t ) = 2 \\int _ { 0 } ^ { t } e ^ { - 2 ( t - s ) } a ( s ) d s = 2 \\int _ { 0 } ^ { t } e ^ { - 2 ( t - s ) } ( a _ 0 + a _ 1 N ( s ) ) d s . \\end{aligned} \\end{align*}"}
{"id": "4887.png", "formula": "\\begin{align*} t _ { i } ( n ) = \\sum \\limits _ { \\omega N _ { i } < p _ { 1 } , p _ { 2 } \\leq N _ { i } \\atop p _ { 1 } - p _ { 2 } = n } \\log p _ { 1 } \\log p _ { 2 } . \\end{align*}"}
{"id": "7552.png", "formula": "\\begin{align*} \\chi _ 1 ( x ) & = { \\hat { T } } ( s _ 1 ( x ) ) - s _ 1 ( T ( x ) ) \\\\ & = { \\hat { T } } ( s _ 2 ( x ) + \\gamma ( x ) ) - ( s _ 2 ( T ( x ) ) + \\gamma ( T ( x ) ) ) \\\\ & = ( { \\hat { T } } ( s _ 2 ( x ) ) - s _ 2 ( T ( x ) ) ) + { \\hat { T } } ( \\gamma ( x ) ) - \\gamma ( T ( x ) ) \\\\ & = \\chi _ 2 ( x ) + T _ M ( \\gamma ( x ) ) - \\gamma ( T ( x ) ) \\\\ & = \\chi _ 2 ( x ) - \\Phi ^ 1 ( \\gamma ) ( x ) . \\end{align*}"}
{"id": "2694.png", "formula": "\\begin{align*} \\mathrm { T r } _ \\perp ( A ) : = \\sum _ { m \\ge 1 } \\langle u _ m , A u _ m \\rangle . \\end{align*}"}
{"id": "1580.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 1 + ( f ' ( x ^ 1 ) ) ^ 2 & 0 \\\\ 0 & ( f ( x ^ 1 ) ) ^ 2 \\\\ \\end{pmatrix} . \\end{align*}"}
{"id": "5478.png", "formula": "\\begin{align*} - \\Delta \\int _ M \\chi ( y ) G ( x , y ) d \\mathrm { v o l } ( y ) = \\chi ( x ) \\geq \\left \\lbrace \\begin{matrix} 1 & N _ { C + 1 } ( K ) , \\\\ 0 & M . \\end{matrix} \\right . \\end{align*}"}
{"id": "5488.png", "formula": "\\begin{align*} \\iota ( \\mathbb { H } ^ 2 ) = \\bigcup _ { z \\in S ^ 1 } \\iota ( [ - z , z ] ) . \\end{align*}"}
{"id": "8827.png", "formula": "\\begin{align*} \\phi _ 1 ^ * * \\phi _ 2 ^ * ( x ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha ( t _ 1 , t _ 2 ) d t _ 1 d t _ 2 . \\end{align*}"}
{"id": "3619.png", "formula": "\\begin{align*} V : = \\{ V _ x \\} _ { x \\in b \\Z ^ d } : = \\{ ( V _ { x , i } , U _ { x , i } ) _ { i \\ge 1 } \\} _ { x \\in b \\Z ^ d } \\end{align*}"}
{"id": "2264.png", "formula": "\\begin{align*} I _ 1 & = \\int _ 0 ^ r \\left [ \\frac { z } { 2 } ( \\bar \\phi + e _ \\delta - \\delta ) ( \\bar \\phi ' , \\bar \\psi ' ) \\cdot ( \\phi , \\psi ) \\right ] { \\rm { d } } z \\\\ & \\leq \\Vert ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert ( \\bar \\phi ' , \\bar \\psi ' ) \\Vert _ { L ^ 2 } \\cdot \\frac { 1 } { 2 } \\left [ \\Vert z \\bar \\phi \\Vert _ { L ^ \\infty } + \\Vert z ( e _ \\delta - \\delta ) \\Vert _ { L ^ \\infty } \\right ] , \\end{align*}"}
{"id": "7404.png", "formula": "\\begin{align*} \\frac { \\sum _ { k = 1 } ^ N f ( S _ k \\alpha ) - E ( f ) N } { \\sqrt { N } } \\overset { d } { \\to } \\mathcal { N } ( 0 , \\sigma ^ 2 ) \\end{align*}"}
{"id": "8941.png", "formula": "\\begin{align*} G ( x , u , p ) = \\phi ^ * \\circ Y ( x , u , p ) , \\end{align*}"}
{"id": "2323.png", "formula": "\\begin{align*} \\mathcal { E } = \\{ R \\in S l ( \\mathbb { Z } , 3 ) : R C _ 1 \\subset C _ 1 , \\ R ^ { - 1 } C _ 2 \\subset C _ 2 , \\ R C _ 3 \\subset C _ 3 \\} , \\end{align*}"}
{"id": "3742.png", "formula": "\\begin{align*} J ^ { e s s ; 1 } _ { k , m ' } ( t , x ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } e ^ { i x \\cdot \\xi + i ( t - s ) \\omega \\cdot \\xi } \\mathcal { F } [ ( E + \\hat { v } \\times B ) f ] \\cdot \\nabla _ v \\big [ \\frac { \\varphi ^ { e s s } ( \\omega , \\xi ) } { i ( \\omega + \\hat { v } ) \\cdot \\xi } \\varphi _ k ( \\xi ) \\varphi _ j ( v ) \\big ] \\end{align*}"}
{"id": "4785.png", "formula": "\\begin{align*} \\mathcal R = \\{ ( 1 ^ 4 ) , ( 2 , 1 ^ 2 ) , ( 3 , 1 ) , ( 2 ^ 2 ) , ( 4 ) \\} \\end{align*}"}
{"id": "1508.png", "formula": "\\begin{align*} y _ { p , \\lambda } ( x ) = e ^ { x } \\int _ { 0 } ^ { x } \\phi _ { p , \\lambda } ( t ) d t . \\end{align*}"}
{"id": "5477.png", "formula": "\\begin{align*} \\Phi ( x ) = \\sum _ { n = \\lceil C \\rceil } ^ \\infty e ^ { - a n } \\phi _ n - \\int _ { M } \\chi ( y ) G ( x , y ) d \\mathrm { v o l } ( y ) . \\end{align*}"}
{"id": "7400.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } c _ n = 0 , \\ \\ \\int _ { \\mathbb { R } ^ N } \\sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \\frac { \\partial U _ { h _ n , i } } { \\partial h _ n } \\left ( \\psi _ n + c _ n \\sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \\frac { \\partial U _ { h _ n , i } } { \\partial h _ n } \\right ) ~ d y = 0 . \\end{align*}"}
{"id": "8245.png", "formula": "\\begin{align*} b _ { \\pm } = \\frac { 1 \\pm | a | } { 4 } = \\frac { 1 \\pm \\frac { \\sqrt { 1 + \\sqrt { 1 + \\left ( \\frac { 1 6 \\beta m \\lambda } { 3 } \\right ) ^ { 2 } } } } { \\sqrt { 2 } } } { 4 } , \\end{align*}"}
{"id": "1653.png", "formula": "\\begin{align*} A \\ , \\ , = \\ , \\ , \\begin{small} \\begin{pmatrix} \\lambda _ 1 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & \\lambda _ 2 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & \\lambda _ 3 & 1 \\end{pmatrix} . \\end{small} \\end{align*}"}
{"id": "309.png", "formula": "\\begin{align*} \\sum _ { 0 \\le i \\le j } { \\rm d } ( { \\rm L } _ { A _ { ( i ) } } ) = { \\rm d } ( { \\rm L } _ { A ^ { ( j ) } } ) \\le \\sqrt { v _ { j + 1 } } \\ , r _ q \\ , { \\rm d } ( { \\rm L } _ n ) . \\end{align*}"}
{"id": "302.png", "formula": "\\begin{align*} p _ j = P ( a ' ) > P ( ( a ' ) ^ * ) ^ { 1 / v } \\ge P ( a ^ * ) ^ { 1 / v } . \\end{align*}"}
{"id": "7476.png", "formula": "\\begin{align*} \\cap _ { i = 1 } ^ { m } C _ { i } \\subset \\cap _ { n = 0 } ^ { \\infty } \\mathrm { F i x } ( S _ { n } ) \\subset \\cap _ { n = 0 } ^ { j _ { f } } \\mathrm { F i x } ( S _ { n } ) . \\end{align*}"}
{"id": "3469.png", "formula": "\\begin{align*} & \\mathrm { E } [ X ^ { n } | x _ { p } < X < x _ { q } ] \\\\ & = \\mu ^ { n } + n \\mu ^ { n - 1 } \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sum _ { i = 2 } ^ { n } \\binom { n } { i } \\mu ^ { n - i } \\sigma ^ { i } \\left [ L _ { 1 } + ( i - 1 ) \\frac { c _ { 1 } } { c _ { ( 1 ) } ^ { \\ast } } L _ { 2 } \\right ] , \\end{align*}"}
{"id": "5380.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u _ 1 + \\sum _ { | \\alpha | \\leq m } a _ { 1 , \\alpha } D ^ \\alpha u _ 1 & = 0 \\mbox { i n } \\ ; \\ ; \\Omega , u _ 1 - f _ 1 \\in \\widetilde H ^ s ( \\Omega ) \\end{align*}"}
{"id": "1641.png", "formula": "\\begin{align*} \\mathcal { X } \\cap [ \\emptyset , N ] = \\bigcup _ { j \\in J } [ B _ j , N ] ^ * , \\end{align*}"}
{"id": "778.png", "formula": "\\begin{align*} & \\beta | f ' ( z ^ m ) | + ( 1 - \\beta ) | f ( z ^ m ) | + \\sum _ { n = 1 } ^ { \\infty } | a _ n | \\phi _ { n } ( r ) \\\\ \\leq & \\beta | f ' _ { 0 } ( z ^ m ) | + ( 1 - \\beta ) | f _ { 0 } ( z ^ m ) | + \\phi _ { 1 } ( r ) + \\sum _ { n = 2 } ^ { \\infty } \\left | \\frac { f ^ { ( n ) } _ { 0 } ( 0 ) } { n ! } \\right | \\phi _ { n } ( r ) \\\\ \\leq & - f _ { 0 } ( - 1 ) \\\\ \\leq & d ( 0 , \\partial { \\Omega } ) , \\end{align*}"}
{"id": "3668.png", "formula": "\\begin{align*} \\mathfrak { H } ^ { j ; 2 } _ { k , \\tilde { k } } ( t _ 1 , t _ 2 ) = \\sum _ { \\tilde { j } \\in [ 0 , j + 2 ] \\cap \\Z } \\mathfrak { H } ^ { \\tilde { j } , j ; 2 } _ { k , \\tilde { k } } ( t _ 1 , t _ 2 ) , \\mathfrak { H } ^ { \\tilde { j } , j ; 2 } _ { k , \\tilde { k } } ( t _ 1 , t _ 2 ) : = \\sum _ { a = 1 , 2 } \\int _ 0 ^ { t _ a } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } ( - 1 ) ^ { a - 1 } V _ 3 ( t _ a ) \\big ( ( t _ a - \\tau ) \\mathfrak { K } ^ { j ; 1 } _ { k , \\tilde { k } } ( y , v , V ( t _ a ) , \\omega ) \\end{align*}"}
{"id": "2759.png", "formula": "\\begin{align*} | J _ { 3 } | \\leq & 2 ^ { ( \\sigma + 1 ) p + n - 2 } \\int _ { B _ { R } ^ { c } } \\frac { ( 1 + j ^ { 1 - q } ) ^ { p - 1 } + j ^ { - q ( p - 1 ) } | y | ^ { - s ( p - 1 ) } } { | y | ^ { n + \\sigma p } } d y \\\\ = & 2 ^ { ( \\sigma + 1 ) p + n - 2 } \\gamma ( 1 + j ^ { 1 - q } ) ^ { p - 1 } R ^ { - \\sigma p } + 2 ^ { ( \\sigma + 1 ) p + n - 2 } \\tau j ^ { - q ( p - 1 ) } R ^ { - \\sigma p - s ( p - 1 ) } . \\end{align*}"}
{"id": "8855.png", "formula": "\\begin{align*} f _ { \\lambda } ( \\lambda _ i ) = \\prod _ { j ( \\neq i ) } ( \\lambda _ i - \\lambda _ j ) \\neq 0 , \\end{align*}"}
{"id": "1211.png", "formula": "\\begin{align*} \\| v _ { t } \\| _ { L ^ { p } ( \\mu _ { t } ; \\R ^ d ) } ^ { p } = \\int _ { \\R ^ { d } } \\frac { | E | ^ { p } } { f _ { t } ^ { p - 1 } } d \\rho \\le c ^ { - p + 1 } t ^ { - p + 1 } \\| E \\| _ { L ^ { p } ( \\rho ; \\R ^ d ) } ^ { p } , \\end{align*}"}
{"id": "8034.png", "formula": "\\begin{align*} \\pi : E : = \\mathbb P ( N _ { Z / X } ) \\rightarrow Z . \\end{align*}"}
{"id": "7353.png", "formula": "\\begin{align*} g ' ( z ) = \\Phi ( z ) f ' ( z ) + \\Phi ' ( z ) f ( z ) . \\end{align*}"}
{"id": "937.png", "formula": "\\begin{align*} \\langle \\nabla _ i \\nabla _ i \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { i i } ^ l - \\frac { \\sum _ k ( \\nabla _ k \\nabla _ k \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { k k } ^ l ) } { n } , \\nabla _ m \\Psi _ t \\rangle = - \\Gamma ^ m _ { i i } + \\frac { \\sum _ k \\Gamma ^ k _ { i i } } { n } + O ( t ) . \\end{align*}"}
{"id": "6522.png", "formula": "\\begin{align*} \\sum \\limits _ { i = n } ^ \\infty \\alpha _ i \\leq \\sum \\limits _ { i = n } ^ \\infty r ^ n \\alpha _ n = \\frac { \\alpha _ n } { 1 - r } . \\end{align*}"}
{"id": "8882.png", "formula": "\\begin{align*} f _ { k } ( \\lambda ) = ( \\lambda - a _ k ) f _ { k - 1 } ( \\lambda ) - b _ { k - 1 } ^ 2 f _ { k - 2 } ( \\lambda ) , \\ \\ 2 \\le k \\le n , \\end{align*}"}
{"id": "6194.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial } { \\partial t } \\varphi = \\log \\frac { ( \\widehat { \\omega } _ { t , N } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\varphi ) ^ { 2 } \\wedge \\eta _ { N } } { \\Omega _ { N } \\wedge \\eta _ { N } } , \\varphi ( T _ { 0 } ) = \\phi _ { T _ { 0 } } , t \\in ( T _ { 0 } , T ^ { \\prime } ] \\end{array} \\end{align*}"}
{"id": "1223.png", "formula": "\\begin{align*} \\det ( x I - B _ 1 ( T ) ) = \\det ( x I - B _ 2 ( T ) ) & = \\prod _ { v \\in V ( T ) } \\mathcal { F } ( v , x ) . \\end{align*}"}
{"id": "5712.png", "formula": "\\begin{align*} & ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \\ldots , y _ n ) = 0 . \\end{align*}"}
{"id": "7126.png", "formula": "\\begin{align*} E ( \\mathsf { P } ^ { ( 1 ) } ) & = \\{ \\{ 0 , 1 \\} , \\dots , \\{ i _ 1 - 1 , i _ 1 \\} , \\{ i _ 1 , j _ 1 \\} , \\{ j _ 1 , j _ 1 + 1 \\} , \\dots , \\{ n , n + 1 \\} \\} , \\\\ E ( \\mathsf { P } ^ { ( 2 ) } ) & = \\{ \\{ i _ 1 , i _ 1 + 1 \\} , \\dots , \\{ j _ 1 - 1 , j _ 1 \\} \\} . \\end{align*}"}
{"id": "6114.png", "formula": "\\begin{align*} \\abs { h } _ { Y \\cup \\mathcal { O } } \\leq \\abs { \\iota ( h ) } \\leq \\lambda \\abs { h } _ { X \\cup \\mathcal { P } } + \\lambda c = \\lambda \\abs { g _ j ^ { - 1 } p g _ k } _ { X \\cup \\mathcal { P } } + \\lambda c \\leq \\lambda ( \\mu + c ) \\end{align*}"}
{"id": "4491.png", "formula": "\\begin{align*} ( \\partial _ { \\eta _ 1 } - \\partial _ { \\eta _ 2 } ) e ^ { i t \\Phi ( \\eta _ 1 , \\eta _ 2 , \\xi ) } = ( p ' ( \\eta _ 1 ) - p ' ( \\eta _ 2 ) ) i t e ^ { i t \\Phi ( \\eta _ 1 , \\eta _ 2 , \\xi ) } , \\end{align*}"}
{"id": "313.png", "formula": "\\begin{align*} b _ q = \\frac { \\pi } { 4 } \\frac { M _ q } { m _ q ^ 2 } \\mu _ q \\ > \\ \\frac { \\log q } { \\log ( 2 q ) } \\quad \\qquad q \\le 2 3 . \\end{align*}"}
{"id": "7568.png", "formula": "\\begin{align*} \\frac { d x } { d t } = X ( t , x ) , x \\in N , X \\in \\mathfrak { X } _ t ( N ) , \\end{align*}"}
{"id": "8616.png", "formula": "\\begin{align*} N I L Q G ( s ) = \\frac { 1 3 . 7 5 s ^ 2 + 6 . 7 7 s + 1 3 2 . 5 } { s ^ 4 + 3 . 8 4 7 s ^ 3 + 2 6 . 6 6 s ^ 2 + 4 6 . 8 6 s + 1 2 5 . 1 } \\end{align*}"}
{"id": "2619.png", "formula": "\\begin{align*} X ^ 0 _ { ( 0 , 0 , 0 ) , 0 } \\equiv 1 \\end{align*}"}
{"id": "1853.png", "formula": "\\begin{align*} A _ { [ n , j ] } ^ { ( q ) } & : = \\sum _ { \\gamma \\in \\mathcal { D } _ { [ n , j ] } } w ( \\gamma + q ) , \\\\ B _ { [ n , j ] } ^ { ( q ) } & : = \\sum _ { \\gamma \\in \\widehat { \\mathcal { D } } _ { [ n , j ] } } w ( \\gamma - q ) . \\end{align*}"}
{"id": "8411.png", "formula": "\\begin{gather*} V = \\bigoplus _ { i \\in I } V _ i \\end{gather*}"}
{"id": "4864.png", "formula": "\\begin{align*} 1 \\leq a _ { i } \\leq q _ { i } \\leq P _ { i } , \\ \\ ( a _ { i } , q _ { i } ) = 1 , \\end{align*}"}
{"id": "691.png", "formula": "\\begin{align*} u \\geq e ^ { - C _ 1 s _ 0 ^ { - \\theta / 2 } ( s - s _ 0 ) } \\hat { u } - C s _ 0 ^ { - \\theta / 4 } & \\geq e ^ { - C _ 1 s _ 0 ^ { - \\theta / 2 } \\log A } \\hat { u } - C s _ 0 ^ { - \\theta / 4 } \\\\ & \\geq \\left ( 1 - 2 C _ 1 s _ 0 ^ { - \\theta / 2 } \\log A \\right ) \\hat { u } - C s _ 0 ^ { - \\theta / 4 } . \\end{align*}"}
{"id": "4636.png", "formula": "\\begin{align*} D ( A ^ { 3 } ) & = \\left \\{ x : = \\left ( x _ k \\right ) _ { k \\in \\N } \\in D ( A ^ 2 ) \\ , \\middle | \\ , A ^ 2 x \\in D ( A ) \\right \\} \\\\ & = \\left \\{ x : = \\left ( x _ k \\right ) _ { k \\in \\N } \\in c _ 0 \\ , \\middle | \\ , \\left ( w ^ { k } w ^ { k + 1 } w ^ { k + 2 } x _ { k + 3 } \\right ) _ { k \\in \\N } \\in c _ 0 \\right \\} \\end{align*}"}
{"id": "7053.png", "formula": "\\begin{align*} & \\sum _ { n \\geq 0 } ( M - 3 n - 1 ) \\binom { M - n - 1 } n \\\\ = & \\sum _ { n \\geq 0 } ( M - 2 n ) \\binom { M - n - 1 } n - \\sum _ { n \\geq 0 } ( n + 1 ) \\binom { M - n - 1 } n \\\\ = & g ( M ) - h ( M ) . \\end{align*}"}
{"id": "7786.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) : = \\int _ X \\alpha ^ \\vee \\cup \\beta . \\end{align*}"}
{"id": "485.png", "formula": "\\begin{align*} \\Xi : = \\{ P _ j , P _ { j + 1 } , . . . , P _ { j + N } \\} , \\end{align*}"}
{"id": "6596.png", "formula": "\\begin{align*} \\langle \\Delta _ G v , v \\rangle & \\leq \\langle \\Delta _ { F _ { \\leq k } } v , v \\rangle \\\\ & = \\langle P _ { F _ { \\leq k - 1 } } \\Delta _ X v , v \\rangle + \\langle P _ { F _ k } \\Delta _ { F _ { \\leq k } } P _ { F _ { k - 1 } \\cup F _ k } v , v \\rangle \\\\ & = \\eta \\langle P _ { F _ { \\leq k - 1 } } v , v \\rangle + \\norm { P _ { F _ k } \\Delta _ { F _ { \\leq k } } } \\cdot \\norm { P _ { F _ { k - 1 } \\cup F _ k } v } \\cdot \\norm v \\\\ & \\leq 2 ^ \\frac 3 2 d h ^ { - 1 } \\eta ^ \\frac 3 2 + 2 d \\delta . \\end{align*}"}
{"id": "477.png", "formula": "\\begin{align*} \\frac { \\langle v _ 1 , v _ 2 \\rangle } { | v _ 1 | | v _ 2 | } = \\frac { | v _ 1 | ^ 2 } { | v _ 1 | | v _ 2 | } + \\frac { \\langle v _ 1 , v _ 2 - v _ 1 \\rangle } { | v _ 1 | | v _ 2 | } \\geq 1 - \\frac { | v _ 1 | | v _ 2 - v _ 1 | } { | v _ 1 | | v _ 2 | } \\geq 1 - \\frac { \\epsilon \\cdot c } { c - \\epsilon \\cdot c } = \\cos \\theta . \\end{align*}"}
{"id": "7448.png", "formula": "\\begin{align*} c ( 1 - m ) \\int _ 0 ^ L \\abs { \\omega _ x } ^ 2 d x + \\frac { c m d _ { \\sigma } } { 2 } \\int _ 0 ^ L \\int _ 0 ^ { \\infty } \\sigma ( s ) \\abs { \\eta _ x } ^ 2 d s d x = - \\Re \\left ( \\left < { \\mathcal { A } } _ m U , U \\right > _ { \\mathcal { H } } \\right ) \\leq \\| F \\| _ { \\mathcal { H } } \\| U \\| _ { \\mathcal { H } } , \\end{align*}"}
{"id": "7070.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { F } _ 1 = & \\{ d _ 1 = d _ 2 = m _ 1 = m _ 2 = 0 , l _ 1 = - 1 3 / 4 , l _ 2 = - 3 / 2 , n _ 2 = - 1 / 2 \\} , \\\\ \\mathcal { F } _ 2 = & \\{ d _ 1 = d _ 2 = m _ 1 = m _ 2 = 0 , l _ 1 = 9 / 4 , l _ 2 = 1 / 2 , n _ 2 = 3 / 2 \\} , \\\\ \\mathcal { F } _ 3 ^ \\pm = & \\{ d _ 1 = d _ 2 = m _ 1 = m _ 2 = 0 , \\\\ & l _ 1 = ( \\pm 1 + 6 l _ 2 + f ( l _ 2 ) ) / 4 , n _ 2 = 1 / 2 \\pm f ( l _ 2 ) / 5 , l _ 2 \\not \\in \\mathcal { L } \\} , \\end{aligned} \\end{align*}"}
{"id": "3551.png", "formula": "\\begin{align*} \\pi \\mid _ { C _ 2 ^ n } = \\left ( m _ 0 + \\sum _ { i = 1 } ^ n m _ i \\right ) \\mathbb { 1 } \\bigoplus _ { i = 1 } ^ n r _ { i } \\rho _ i , \\end{align*}"}
{"id": "1023.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { C _ N } { D _ N } = \\lambda ( [ 1 , 1 ] ) . \\end{align*}"}
{"id": "6517.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\prod _ { i = 1 } ^ m \\mu ( [ 0 , A ( m ) ^ { { \\rho } i } ] ) < \\infty . \\end{align*}"}
{"id": "6312.png", "formula": "\\begin{align*} & \\int \\frac { x ^ { n - 2 } } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\widetilde { h } _ n ( x ; q ) d _ q x = \\dfrac { x ^ n } { [ n - 1 ] _ q ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\frac { \\widetilde { h } _ n ( x ; q ) } { x } - \\widetilde { h } _ { n - 1 } ( x ; q ) \\right ) . \\end{align*}"}
{"id": "1103.png", "formula": "\\begin{align*} G _ { ( t , \\rho ) } \\left ( \\frac { \\partial } { \\partial t } , \\frac { \\partial } { \\partial t } \\right ) = 1 . \\end{align*}"}
{"id": "4674.png", "formula": "\\begin{align*} \\pi ( z ) & = \\lim _ { n \\to \\infty } \\pi ( h _ { 1 , n } h _ { 2 , n } h _ { 3 , n } B ^ + / B ^ + ) \\\\ & = \\lim _ { n \\to \\infty } h _ { 1 , n } B ^ + / B ^ + \\in U ^ - _ { J , \\ge 0 } B ^ + / B ^ + \\bigcap U ^ + \\dot { u } _ J \\cdot B ^ + / B ^ + . \\end{align*}"}
{"id": "5572.png", "formula": "\\begin{align*} \\frac { ( k + n - j ) ! } { ( n - j ) ! } = \\mathrm O ( n ^ k ) \\end{align*}"}
{"id": "7567.png", "formula": "\\begin{align*} x ( t ) = \\frac { x _ { ( 2 ) } ( t ) ( x _ { ( 3 ) } ( t ) - x _ { ( 1 ) } ( t ) ) + \\rho x _ { ( 3 ) } ( t ) ( x _ { ( 1 ) } ( t ) - x _ { ( 2 ) } ( t ) ) } { ( x _ { ( 3 ) } ( t ) - x _ { ( 1 ) } ( t ) ) + \\rho ( x _ { ( 1 ) } ( t ) - x _ { ( 2 ) } ( t ) ) } , \\end{align*}"}
{"id": "7432.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\left ( \\mu \\int _ 0 ^ L \\abs { y _ t } ^ 2 d x + \\beta \\int _ 0 ^ L \\abs { y _ x } ^ 2 d x \\right ) - \\gamma \\beta \\Re \\left ( \\int _ 0 ^ L u _ x \\overline { y _ { x t } } \\right ) = 0 . \\end{align*}"}
{"id": "4270.png", "formula": "\\begin{align*} \\partial _ { \\zeta _ i } f = g \\partial _ { \\zeta _ i } h + h \\partial _ { \\zeta _ i } g \\end{align*}"}
{"id": "4415.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { i = 1 } ^ { d } S _ { d , i } \\right ) \\cup \\left ( \\bigcup _ { i = 1 } ^ { e } T _ { e , i } \\right ) = [ 1 , 4 k N ] , \\end{align*}"}
{"id": "5094.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , t ] } E [ | P ^ { n , 2 } _ \\tau | ^ 2 ] = 0 . \\end{align*}"}
{"id": "3903.png", "formula": "\\begin{align*} \\omega ( \\frac { R _ 0 } { S ^ j } ) \\leq \\theta ^ j [ \\omega ( R _ 0 ) + \\tau R _ 0 ^ \\sigma \\sum _ { k = 0 } ^ { j - 1 } ( \\theta S ^ \\sigma ) ^ { - k } ] . \\end{align*}"}
{"id": "4716.png", "formula": "\\begin{align*} B _ { k , n } ^ { * } : = \\{ \\omega \\in B _ { k } ^ { * } \\ : | \\ : d ^ { * } = \\omega e _ { ( k ) } \\in \\mathcal { D } _ { k , n } ^ { * } \\hbox { a n d } \\ell ( d ^ { * } ) = \\ell ( \\omega ) \\} , \\end{align*}"}
{"id": "8409.png", "formula": "\\begin{gather*} ( f | _ { k } ( \\gamma , \\upsilon ) ) ( \\tau ) : = f \\left ( \\frac { a \\tau + b } { c \\tau + d } \\right ) \\frac 1 { \\upsilon ( \\tau ) ^ { 2 k } } \\end{gather*}"}
{"id": "8252.png", "formula": "\\begin{align*} - i \\frac { 3 a ^ { 2 } } { 2 \\beta \\hslash ^ { 2 } } \\lambda a _ { 2 } + a _ { 1 } = \\lambda ^ { 2 } a _ { 1 } \\ , \\Rightarrow \\ , a _ { 2 } = i \\frac { 2 \\beta \\hslash ^ { 2 } } { 3 a ^ { 2 } } . \\frac { \\lambda ^ { 2 } - 1 } { \\lambda } a _ { 1 } \\end{align*}"}
{"id": "4607.png", "formula": "\\begin{align*} \\Delta _ F ( X _ { n } ) = n , \\Delta _ F ( \\varphi _ { i , n } ) = | n | , \\Delta _ F ( \\varphi ^ * _ { i , n } ) = - | n | , \\Delta _ F | _ { M _ \\Delta } = - \\Delta \\end{align*}"}
{"id": "959.png", "formula": "\\begin{align*} \\alpha = \\sum _ { i = 0 } ^ { n - 1 } a _ { i } \\theta ^ { i } \\in E \\stackrel { \\tau } { \\longrightarrow } \\overline { \\alpha } = \\begin{pmatrix} a _ 0 \\\\ a _ 1 \\\\ \\vdots \\\\ a _ { n - 1 } \\end{pmatrix} \\in Q ^ { n } \\end{align*}"}
{"id": "148.png", "formula": "\\begin{align*} \\limsup _ { t \\downarrow 0 } \\frac { c \\eta _ t } { m _ t } \\leq 0 { \\bf 1 } _ { d = 2 } + c _ 0 { \\bf 1 } _ { d = 3 } , \\lim _ { t \\downarrow 0 } \\frac { \\gamma _ t } { m _ t } = 0 . \\end{align*}"}
{"id": "4930.png", "formula": "\\begin{align*} \\omega ^ { - 1 } h _ { i j } = - \\rho _ { ; i j } + \\frac { 1 } { \\rho } g _ { i j } - \\frac { 1 } { \\rho } \\rho _ { i } \\rho _ { j } . \\end{align*}"}
{"id": "2354.png", "formula": "\\begin{align*} l ( h _ \\rho ) = a _ 0 + a _ 1 h _ \\rho + \\ldots + a _ r h _ \\rho ^ r . \\end{align*}"}
{"id": "3868.png", "formula": "\\begin{align*} \\Big | \\bigotimes _ { i = 1 } ^ n x _ i - \\bigotimes _ { i = 1 } ^ n \\hat { x } _ i \\Big | \\leq \\sum _ { k = 1 } ^ n \\bigg ( \\prod _ { i = 1 } ^ { k - 1 } | x _ i | \\bigg ) \\cdot \\big | x _ k - \\hat { x } _ k \\big | \\cdot \\bigg ( \\prod _ { i = k + 1 } ^ n \\big | \\hat { x } _ i \\big | \\bigg ) \\end{align*}"}
{"id": "8589.png", "formula": "\\begin{align*} t \\wedge s = \\tau ( A \\cap B ) , t \\vee s = \\tau ( A \\cup B ) , \\end{align*}"}
{"id": "8956.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } i \\partial _ t u + \\Delta u & = \\pm | u | ^ 2 u , ( t , x ) \\in \\R \\times \\R ^ 2 , \\\\ u ( 0 ) & = u _ 0 \\in M ^ s _ { 4 , 2 } ( \\R ^ 2 ) + L ^ 2 ( \\R ^ 2 ) \\end{array} \\right . \\end{align*}"}
{"id": "7623.png", "formula": "\\begin{align*} | V ( R _ 1 ) | \\ge \\sum _ { i \\in [ 3 ] } \\deg ( v _ i , V ( R _ 1 ) ) & \\geq 3 \\big ( ( 2 / 3 + 8 \\gamma ) n - | V ( R _ 2 ) | \\big ) \\\\ & > 2 n - ( n + | V ( R _ 2 ) | ) \\\\ & = n - | V ( R _ 2 ) | , \\end{align*}"}
{"id": "1950.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { S } _ { 1 } ( n + 1 , r ) } w ( \\gamma ) & = \\sum _ { \\gamma \\in \\mathcal { S } _ { 1 } ( n , r - 1 ) } w ( \\gamma ) + \\sum _ { k = 0 } ^ { p } a _ { r + q } ^ { ( k ) } \\Big ( \\sum _ { \\gamma \\in \\mathcal { S } _ { 1 } ( n , r + k ) } w ( \\gamma ) \\Big ) \\\\ & = \\langle \\mathcal { H } _ { q } ^ { n } e _ { r - 1 } , e _ { 0 } \\rangle + \\sum _ { k = 0 } ^ { p } a _ { r + q } ^ { ( k ) } \\langle \\mathcal { H } _ { q } ^ { n } e _ { r + k } , e _ { 0 } \\rangle \\\\ & = \\langle \\mathcal { H } _ { q } ^ { n + 1 } e _ { r } , e _ { 0 } \\rangle \\end{align*}"}
{"id": "5258.png", "formula": "\\begin{align*} \\Delta ( \\Gamma ) : = \\# Z ( H ^ { \\Gamma } ) . \\end{align*}"}
{"id": "6555.png", "formula": "\\begin{align*} f ^ \\wedge \\colon U \\times E \\to F \\ , , f ^ \\wedge ( x , v ) \\ , : = \\ , f ( x ) ( v ) \\ , ? \\end{align*}"}
{"id": "8983.png", "formula": "\\begin{align*} \\| u _ { q , r } \\| ^ 2 _ { W ^ { 1 , 2 } _ { \\delta ^ \\ast } ( M ) } = \\| \\nabla u _ { q , r } \\| ^ 2 _ { L ^ 2 ( M ) } + \\| u _ { q , r } \\| ^ 2 _ { L ^ 2 _ { \\delta ^ \\ast } ( M ) } < C . \\end{align*}"}
{"id": "7145.png", "formula": "\\begin{align*} ( g _ { 3 1 } y _ 1 + g _ { 3 2 } y _ 2 + g _ { 3 3 } y _ 3 ) \\{ ( g _ { 1 1 } + g _ { 2 1 } + g _ { 3 1 } ) y _ 1 + ( g _ { 2 2 } + g _ { 3 2 } ) y _ 2 + ( g _ { 2 3 } + g _ { 3 3 } ) y _ 3 \\} ^ { a - 1 } \\\\ = g _ { 3 1 } ( g _ { 1 1 } + g _ { 2 1 } + g _ { 3 1 } ) ^ { a - 1 } y _ 1 ^ a + g _ { 3 2 } ( g _ { 2 2 } + g _ { 3 2 } ) ^ { a - 1 } y _ 2 ( y _ 1 + y _ 2 ) ^ { a - 1 } \\end{align*}"}
{"id": "5650.png", "formula": "\\begin{align*} | \\hat { h } ( y ) | \\leq \\begin{cases} C | y | ^ { 2 - \\delta } , & \\ \\delta \\neq n , \\\\ C | y | ^ { 2 - n } \\ln | y | , & \\ \\delta = n . \\end{cases} \\end{align*}"}
{"id": "4928.png", "formula": "\\begin{align*} \\bar { R } _ { \\alpha \\beta \\gamma \\delta } = 0 . \\end{align*}"}
{"id": "6877.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d B ( t ) } { d t } & = g _ 0 + g _ 1 N ( t ) - B ( t ) , \\\\ \\frac { d C ( t ) } { d t } & = 2 ( a _ 0 + a _ 1 N ( t ) - C ( t ) ) . \\end{aligned} \\end{align*}"}
{"id": "2699.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\ , \\left \\langle \\left ( \\mathcal { N } _ 1 - \\mathcal { N } _ 2 \\right ) ^ 2 \\right \\rangle _ { \\psi _ \\mathrm { g s } } = 0 \\end{align*}"}
{"id": "8705.png", "formula": "\\begin{align*} \\frac { \\partial h } { \\partial r } - \\frac { d } { d t } \\frac { \\partial h } { \\partial \\dot { r } } = 0 . \\end{align*}"}
{"id": "3331.png", "formula": "\\begin{align*} s _ 1 = { \\mathcal B } _ { - 1 } ( \\sigma _ 1 ) = \\begin{pmatrix} 1 & 0 \\\\ - 1 & 1 \\end{pmatrix} \\ \\ \\ \\ s _ 2 = { \\mathcal B } _ { - 1 } ( \\sigma _ 2 ) = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "1650.png", "formula": "\\begin{align*} \\tilde { \\tau } _ { \\mathcal { C } } ( x , y , t ) \\ , = \\ , \\frac { 1 } { E _ 2 E _ 4 \\cdots E _ { 2 g } } \\biggl ( \\displaystyle \\sum _ { I } \\displaystyle \\prod _ { i < j \\in I } ( \\kappa _ i - \\kappa _ j ) \\displaystyle \\prod _ { \\{ i \\ , : \\ , c _ i = 1 \\} } \\lambda _ i \\prod _ { i \\in I } E _ i \\biggr ) , \\end{align*}"}
{"id": "4389.png", "formula": "\\begin{align*} & \\int _ { D \\backslash \\{ F = 0 \\} } | F _ 1 - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f F ^ { 1 + \\delta } | ^ 2 e ^ { - \\varphi + v _ { t _ 0 , B } ( \\Psi ) - \\Psi } \\\\ \\le & \\left ( \\frac { 1 } { \\delta } + 1 - e ^ { - t _ 0 - B } \\right ) \\int _ D \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f | ^ 2 e ^ { - \\Psi } . \\end{align*}"}
{"id": "8133.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\liminf _ { t \\rightarrow \\infty } \\| \\chi _ { B _ n } \\psi _ t \\| = \\| \\psi \\| \\end{align*}"}
{"id": "2159.png", "formula": "\\begin{align*} 0 < \\Re \\left [ \\frac { H ( z _ { 1 } ) - H ( z _ { 2 } ) } { z _ { 1 } - z _ { 2 } } \\right ] = \\frac { s _ { 1 } - s _ { 2 } } { | \\omega ( s _ { 1 } ) - \\omega ( s _ { 2 } ) | ^ { 2 } } \\left [ h ^ { - 1 } ( s _ { 1 } ) - h ^ { - 1 } ( s _ { 2 } ) \\right ] , \\end{align*}"}
{"id": "5007.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) ) ^ 2 \\Psi ^ { n , 3 } _ s d s \\\\ & = \\kappa _ 4 \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "3051.png", "formula": "\\begin{align*} \\overline { \\mathcal { G } } _ { n } ( u ) = \\int _ { u } ^ { \\infty } { G } _ { n } ( v ) \\mathrm { d } v , \\end{align*}"}
{"id": "514.png", "formula": "\\begin{align*} p ' ( \\bar { x } , \\bar { y } ) & = m \\big ( \\ , w _ 1 ( \\bar { x } , \\bar { y } ) , \\dots , w _ n ( \\bar { x } , \\bar { y } ) \\ , \\big ) \\\\ & = m _ 1 \\big ( \\ , w _ { i _ 1 } ( \\bar { x } , \\bar { y } ) , \\dots , w _ { i _ p } ( \\bar { x } , \\bar { y } ) \\ , \\big ) \\cdot m _ 2 \\big ( \\ , w _ { j _ 1 } ( \\bar { x } , \\bar { y } ) , \\dots , w _ { j _ q } ( \\bar { x } , \\bar { y } ) \\ , \\big ) \\end{align*}"}
{"id": "3173.png", "formula": "\\begin{align*} \\int _ { \\Theta _ { d } } f ( \\theta ) g ( \\theta ) \\mathrm { d } ( \\mathrm { F } _ { \\ast } \\mathrm { P } ) \\left ( \\theta \\right ) = \\int _ { \\Theta _ { d } } f ( \\theta ) \\mathrm { d } ( \\mathrm { F } _ { \\ast } \\mathrm { P } ) \\left ( \\theta \\right ) \\int _ { \\Theta _ { d } } g ( \\theta ) \\mathrm { d } ( \\mathrm { F } _ { \\ast } \\mathrm { P } ) \\left ( \\theta \\right ) \\ . \\end{align*}"}
{"id": "2230.png", "formula": "\\begin{align*} a _ i ( t , s ) = { \\rm L o g } ( U _ i ( t , s ) + \\kappa I ) \\end{align*}"}
{"id": "8285.png", "formula": "\\begin{align*} \\psi _ { B } = B \\left [ \\frac { \\sinh ( k ' x ) } { \\sinh ( k ' a / 2 ) } - \\frac { x } { a / 2 } \\frac { \\cosh ( k ' x ) } { \\cosh ( k ' a / 2 ) } \\right ] , \\end{align*}"}
{"id": "6274.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } p _ n ( x ; a , b ; q ) = \\dfrac { q ^ { 1 - n } [ n ] _ q } { ( 1 - a q ) ( 1 - b q ) } p _ { n - 1 } ( x ; a q , b q ; q ) , \\end{align*}"}
{"id": "3579.png", "formula": "\\begin{align*} w _ 4 ( \\pi ) = \\begin{cases} \\sum \\limits _ { i = 1 } ^ 4 t _ i ^ 2 , & , \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "5765.png", "formula": "\\begin{align*} E ( G / H ) = 0 . \\end{align*}"}
{"id": "7760.png", "formula": "\\begin{align*} { \\bf U } _ { { \\rm H i P O D } } ^ k ( \\alpha ^ * ) = [ u _ { { \\rm P O D } , k , 1 } ^ { \\alpha ^ * } , \\ldots , u _ { { \\rm P O D } , k , N _ h } ^ { \\alpha ^ * } ] ^ T = \\displaystyle \\sum _ { j = 1 } ^ { L } T _ j ^ k ( \\alpha ^ * ) { \\boldsymbol \\xi } _ j k = 1 , \\ldots , m \\end{align*}"}
{"id": "3202.png", "formula": "\\begin{align*} \\sum _ { i = n _ k } ^ { n _ { k + 1 } } ( 2 ^ i ) ^ { \\frac { p } { p - 1 } } \\mu ^ { \\frac { 1 } { 1 - p } } ( A _ { 2 ^ i } ) > 2 ^ k 2 ^ { i _ k } \\mu ^ { - 1 } ( A _ { 2 ^ { i _ k } } ) > 2 ^ k . \\end{align*}"}
{"id": "8732.png", "formula": "\\begin{align*} \\det ( P ) = \\frac { a ^ 2 } { \\lambda } U [ ( t _ 1 - t _ 0 ) \\sin ( t _ 1 - t _ 0 ) + 2 \\cos ( t _ 1 - t _ 0 ) - 2 ] . \\end{align*}"}
{"id": "4014.png", "formula": "\\begin{align*} \\dot { x } = A x + B u . \\end{align*}"}
{"id": "3768.png", "formula": "\\begin{align*} \\widetilde { T } _ { k , j ; n , l , r } ^ { T ; \\mu , m , i } ( \\mathfrak { m } , U ) ( t , x , \\zeta ) = \\sum _ { \\tilde { j } \\in [ j - 2 , j + 2 ] } \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big [ i \\mu \\omega ^ { m ; U } _ { j , l , r } ( t - s , v , \\omega ) K _ { k ; n } ^ { } ( \\mathfrak { m } ) ( y , \\zeta ) + c ^ { m , U } _ { j , l , r ; q } ( t - s , v , \\omega ) K _ { k ; n } ^ { q } ( \\mathfrak { m } ) ( y , \\zeta ) \\end{align*}"}
{"id": "4878.png", "formula": "\\begin{align*} S _ { k } ( N , \\alpha ) = \\sum \\limits _ { p \\sim N } ( \\log p ) e ( p ^ { k } \\alpha ) \\end{align*}"}
{"id": "1737.png", "formula": "\\begin{align*} \\frac { 1 } { e ^ { \\omega _ 2 s } - 1 } = \\sum _ { k \\geq 0 } \\frac { B _ k \\cdot ( \\omega _ 2 s ) ^ { k - 1 } } { k ! } , \\end{align*}"}
{"id": "4554.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 1 1 \\right ) } \\Vert _ { p } = \\mathcal { O } \\left ( \\frac { 1 } { m } \\right ) ( v ( | z _ { 1 } | ) + ( v ( | z _ { 2 } | ) ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "7650.png", "formula": "\\begin{align*} \\pi _ { v , w } = \\langle M _ v ^ t \\cdot \\alpha _ w , M _ w ^ t \\cdot \\alpha _ v \\rangle \\subset V _ { n + m } . \\end{align*}"}
{"id": "1079.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } | V ( \\mathbf { x } ) \\mathrm { e } ^ { - H _ { 0 } t } f ( \\mathbf { x } ) | ^ { 2 } \\ , \\textnormal { d } \\mathbf { x } = \\int _ { \\mathbb { R } ^ { d } } | V ( \\mathbf { x } ) | ^ { 2 } \\bigg | \\int _ { \\mathbb { R } ^ { d } } K _ { t } ( \\mathbf { x } , \\mathbf { y } ) f ( \\mathbf { y } ) \\ , \\textnormal { d } \\mathbf { y } \\bigg | ^ { 2 } \\ , \\textnormal { d } \\mathbf { x } . \\end{align*}"}
{"id": "8521.png", "formula": "\\begin{align*} \\Delta _ 1 ( k , m ) = z \\Delta _ 1 ( k , m - 1 ) + a _ { m } \\Delta _ 2 ( k , m - 1 ) + a _ { m } ^ 2 \\Delta _ 1 ( k , m - 2 ) . \\end{align*}"}
{"id": "3820.png", "formula": "\\begin{align*} { } ^ 1 _ 3 L a s t E l l ^ i ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta ) + i s \\mu _ 1 | \\eta | - i s \\hat { v } \\cdot \\xi } K ( s , X ( s ) , V ( s ) ) \\cdot \\nabla _ \\zeta \\big [ i ( \\hat { \\zeta } \\cdot ( \\xi + \\eta ) + \\mu _ 1 | \\eta | - \\hat { v } \\cdot \\xi ) ^ { - 1 } \\end{align*}"}
{"id": "8473.png", "formula": "\\begin{align*} y ^ * _ i = \\left ( ( i - n + m ) L + U / m - L ( m + 1 ) / 2 \\right ) _ + , \\end{align*}"}
{"id": "4722.png", "formula": "\\begin{align*} e _ { ( l + 1 ) } = \\Big ( \\frac { q - q ^ { - 1 } } { z - z ^ { - 1 } } \\Big ) ^ { l - 2 } e _ { ( l - 1 ) } H _ { 2 l - 2 } H _ { 2 l - 1 } H _ { 2 l } H _ { 2 l + 1 } H _ { 2 l - 3 } ^ { - 1 } H _ { 2 l - 2 } ^ { - 1 } H _ { 2 l - 1 } ^ { - 1 } H _ { 2 l } ^ { - 1 } e _ { ( l ) } . \\end{align*}"}
{"id": "4424.png", "formula": "\\begin{align*} 4 k N + 4 e + i = ( 4 k + 2 ) N - 2 d + i \\leq ( 4 k + 2 ) N - d < ( 4 k + 2 ) N - d ' \\leq ( 4 k + 2 ) N - i ' < ( 4 k + 2 ) N - i ' + 1 . \\end{align*}"}
{"id": "2706.png", "formula": "\\begin{align*} \\sigma _ N ^ 2 : = \\begin{cases} \\sqrt { \\mu _ - - \\mu _ + } \\ , N \\qquad & \\delta > 2 \\\\ C \\qquad & . \\end{cases} \\end{align*}"}
{"id": "3753.png", "formula": "\\begin{align*} \\times \\varphi _ k ( \\xi ) d { \\omega } d v d s \\big | \\lesssim \\sup _ { s \\in [ 0 , t ] } \\sup _ { g \\in L ^ 2 , \\| g \\| _ { L ^ 2 } = 1 } 2 ^ { 2 m + k + l } 2 ^ { 2 \\tilde { c } ( m , k , l ) } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\big | \\hat { f } ( s , \\xi , v ) \\big | \\varphi _ k ( \\xi ) \\psi _ { [ l - 2 , l + 2 ] } ( \\tilde { v } \\times \\tilde { \\xi } ) \\big | \\overline { \\hat { g } ( \\xi ) } \\big | \\varphi _ j ( v ) d v d \\xi \\end{align*}"}
{"id": "5715.png", "formula": "\\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \\ldots , y _ i ) = 0 . \\end{align*}"}
{"id": "2080.png", "formula": "\\begin{align*} \\| u \\| _ E : = \\left ( \\| \\partial _ x u \\| _ { L ^ 2 } ^ 2 + \\| | D _ y | ^ { \\frac { 1 } { 2 } } u \\| _ { L ^ 2 } ^ 2 + \\| u \\| _ { L ^ 2 } ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "2655.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ { n - 1 } ( n - k ) p ^ { 2 n - k - 4 } & = \\frac { p ^ { n - 3 } ( ( n - 2 ) p ^ { n - 1 } - ( n - 1 ) p ^ { n - 2 } + 1 ) ) } { ( p - 1 ) ^ 2 } . \\end{align*}"}
{"id": "2569.png", "formula": "\\begin{align*} C _ 3 ^ { x y y } = \\sum _ { j , k , l } A _ { j k } ^ { ( x ) } A _ { k l } ^ { ( y ) } A _ { l j } ^ { ( y ) } = \\sum _ { j , k , l } A _ { k l } ^ { ( y ) } A _ { l j } ^ { ( y ) } A _ { j k } ^ { ( x ) } = C _ 3 ^ { y y x } \\ , \\end{align*}"}
{"id": "4945.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": "944.png", "formula": "\\begin{align*} H ( \\rho ) = \\rho \\int _ 1 ^ { \\rho } \\frac { p ( s ) } { s ^ 2 } d s , \\end{align*}"}
{"id": "4883.png", "formula": "\\begin{align*} r _ { i } ( n ) = \\# \\{ \\omega N < p _ { i } \\leq N _ { i } : n = p _ { 1 } - p _ { 2 } \\} . \\end{align*}"}
{"id": "6645.png", "formula": "\\begin{align*} \\log L & \\geq n \\log \\frac { A } { \\sqrt { a } } - \\frac { 1 } { 4 } ( 1 + b ) \\ , n \\log n + \\frac { 1 } { 2 } n \\log n - n \\log e + o ( n ) \\\\ & = \\left ( \\frac { 1 - b } { 4 } \\right ) \\ , n \\log n + n ( \\log \\frac { A } { e \\sqrt { a } } ) + o ( n ) \\ ; , \\ , \\end{align*}"}
{"id": "1903.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { V } _ { [ n , \\ell ] } } w ( \\gamma ) = \\sum _ { k = \\ell } ^ { n - 1 } \\sum _ { \\gamma \\in \\mathcal { V } _ { [ n , \\ell , k ] } } w ( \\gamma ) = \\sum _ { k = \\ell } ^ { n - 1 } a _ { 0 } ^ { ( \\ell ) } A ^ { ( 1 ) } _ { [ k - 1 , \\ell - 1 ] } W _ { [ n - k - 1 , 0 ] } . \\end{align*}"}
{"id": "8699.png", "formula": "\\begin{align*} h = f + \\lambda g , \\end{align*}"}
{"id": "393.png", "formula": "\\begin{align*} G ( o , o ) = \\int _ 0 ^ { \\infty } h _ t ( o , o ) d t \\lesssim _ d 1 + \\frac { 1 } { R } + \\frac { \\log ( D / R ) } { R } + \\frac { D ^ 2 } { n } \\lesssim _ { d , C } \\frac { \\log n } { R } = \\frac { 1 } { n } D ^ 2 \\log n , \\end{align*}"}
{"id": "3187.png", "formula": "\\begin{align*} \\xi _ { \\mathcal { I } } \\left ( z \\right ) \\doteq \\left ( y _ { 1 } , \\dots , y _ { d } \\right ) , z = \\left ( z _ { 1 } , \\dots , z _ { n } \\right ) \\in \\mathbb { Z } ^ { n } \\ , \\end{align*}"}
{"id": "6379.png", "formula": "\\begin{align*} c _ { i , i } = w _ i - w _ i ^ 2 \\mbox { f o r $ 1 \\leq i \\leq d $ , a n d } c _ { i , j } = c _ { j , i } = - w _ i w _ j \\mbox { f o r $ 1 \\leq i < j \\leq d $ . } \\end{align*}"}
{"id": "6947.png", "formula": "\\begin{align*} Q ^ * Q R x = Q ^ * Q R Q ^ * Q x = R x - R ^ { - 1 / 2 } p ( R ^ { - 1 / 2 } p ) ^ * x = R x ; \\end{align*}"}
{"id": "5511.png", "formula": "\\begin{align*} \\mathcal { F } _ = \\{ f \\in \\mathcal { F } : f \\circ \\gamma = \\gamma \\circ f \\gamma \\in ( \\Omega ) \\} . \\end{align*}"}
{"id": "8240.png", "formula": "\\begin{align*} k ' _ { \\pm } = \\sqrt { \\frac { 1 - ( | a | \\pm i | b | ) } { 4 \\beta \\hslash ^ { 2 } / 3 } } = \\sqrt { \\frac { ( 1 - | a | ) \\mp i | b | } { 4 \\beta \\hslash ^ { 2 } / 3 } } . \\end{align*}"}
{"id": "6470.png", "formula": "\\begin{align*} u ( 0 , t ) = u ( L , t ) = y ( 0 , t ) = y ( L , t ) = 0 , \\ t \\in ( 0 , \\infty ) , \\end{align*}"}
{"id": "6256.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } q ^ { \\frac { n ( n - 3 ) } { 2 } } x ^ n \\left ( 1 - q + q ^ { n + 2 } x \\right ) A _ q ( q ^ n x ) = ( 1 - q ) A _ q ( \\frac { x } { q } ) - x A _ q ( q x ) , \\end{align*}"}
{"id": "3127.png", "formula": "\\begin{align*} \\sum _ { k = \\ell } ^ { \\ell + m } \\delta ^ 2 ( \\mathcal { T } _ k , \\mathcal { T } _ { k + 1 } ) \\leq \\Lambda _ { \\mathrm { q o } } \\eta ^ 2 _ \\ell \\quad \\ell , m \\in \\mathbb { N } _ 0 . \\end{align*}"}
{"id": "4806.png", "formula": "\\begin{align*} \\max _ { 1 \\le i \\le n } \\bigg | \\sum _ { k = 1 } ^ i \\widehat \\eta _ { n , k } \\bigg | = o _ P ( n ^ { - 1 / 2 + p / 4 } ) \\quad \\mbox { a s \\ } n \\to \\infty \\end{align*}"}
{"id": "54.png", "formula": "\\begin{align*} \\Phi _ K ( u ) ( \\omega ) = \\rho _ { u ^ { - 1 } } ( \\omega ) \\end{align*}"}
{"id": "6570.png", "formula": "\\begin{align*} f _ x | _ { U \\cap V _ x \\cap V _ y } \\ ; = \\ ; f | _ { U \\cap V _ x \\cap V _ y } \\ ; = \\ ; f _ y | _ { U \\cap V _ x \\cap V _ y } \\ , , \\end{align*}"}
{"id": "3205.png", "formula": "\\begin{align*} q ^ { } _ { k , t + 1 } = \\Big [ q ^ { } _ { k , t } - \\sum _ { i = 1 } ^ n f _ { k , t , i } ^ { } \\tau \\Big ] ^ + + \\sum _ { i = 1 } ^ n \\beta _ { k , t , i } W , \\end{align*}"}
{"id": "5385.png", "formula": "\\begin{align*} q = q _ s + q _ 0 , q _ s \\in M _ \\Omega ( H ^ s \\to H ^ { - s } ) , q _ 0 \\in M _ { \\geq 0 } ( H ^ s \\to H ^ { - s } ) . \\end{align*}"}
{"id": "391.png", "formula": "\\begin{align*} 0 = \\sum _ { x \\in \\Gamma } G ( o , x ) & = \\sum _ { \\underset { G ( o , x ) \\geq 0 } { x \\in \\Gamma } } G ( o , x ) + \\sum _ { \\underset { 0 > G ( o , x ) > - \\tfrac { C _ 1 C _ 3 } { 2 } \\tfrac { D ^ 2 } { n } } { x \\in \\Gamma } } G ( o , x ) + \\sum _ { \\underset { - \\tfrac { C _ 1 C _ 3 } { 2 } \\tfrac { D ^ 2 } { n } \\geq G ( o , x ) } { x \\in \\Gamma } } G ( o , x ) \\\\ & = : A _ 1 + A _ 2 + A _ 3 . \\end{align*}"}
{"id": "7231.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left \\{ \\log \\left ( \\frac { 1 + t - x } { 1 + R } \\right ) \\right \\} ^ { a _ n } \\left \\{ \\frac { ( 2 x ) ^ { 1 - a } } { 1 + t + x } \\right \\} ^ { b _ n } \\quad \\mbox { i n } \\ D \\end{align*}"}
{"id": "7515.png", "formula": "\\begin{align*} P ( E _ { x \\rightarrow y } ) = \\sum _ { i = 1 } ^ { m i n ( m _ x , k _ x \\gamma _ x ) } P ( E _ { x \\rightarrow y } | S _ i ) P ( S _ i ) , \\end{align*}"}
{"id": "6037.png", "formula": "\\begin{align*} f _ { \\sigma } ( x ) = \\sum _ { k = 0 } ^ n \\sigma ( c _ k ) x ^ k \\in K [ x ] . \\end{align*}"}
{"id": "413.png", "formula": "\\begin{align*} \\varphi _ { \\mu } ( z ) = H _ { \\mu } ( z ) - z \\end{align*}"}
{"id": "8444.png", "formula": "\\begin{align*} x = \\frac { \\epsilon _ 1 ( x , \\beta ) } { \\beta } + \\frac { \\epsilon _ 2 ( x , \\beta ) } { \\beta ^ { 2 } } + \\cdots + \\frac { \\epsilon _ n ( x , \\beta ) } { \\beta ^ n } + \\cdots , \\end{align*}"}
{"id": "762.png", "formula": "\\begin{align*} T _ m \\triangleq \\frac { 1 } { \\bar { N } \\sigma ^ 2 } \\sum _ { i = 0 } ^ { \\bar { N } - 1 } | y _ m [ i ] | ^ 2 , m = 1 , 2 , \\ldots , M . \\end{align*}"}
{"id": "298.png", "formula": "\\begin{align*} f ( A ) = \\sum _ { a \\in A } \\frac { 1 } { a \\log a } \\ \\le \\ \\frac { e ^ \\gamma } { m _ q } \\frac { 1 } { 1 + v } \\sum _ { a \\in A } { \\rm d } ( { \\rm L } _ a ) = \\frac { e ^ \\gamma } { m _ q } \\frac { { \\rm d } ( { \\rm L } _ A ) } { 1 + v } . \\end{align*}"}
{"id": "6532.png", "formula": "\\begin{align*} \\ 1 _ { \\{ \\sigma ^ r _ \\infty = \\infty \\} } = \\ 1 _ { \\{ \\theta _ \\infty = \\infty \\} } . \\end{align*}"}
{"id": "7746.png", "formula": "\\begin{align*} \\mbox { f i n d \\ } u ( \\alpha ) \\in V \\mbox { s . t . } a ( u ( \\alpha ) , v ; \\alpha ) = f ( v ; \\alpha ) \\forall v \\in V , \\end{align*}"}
{"id": "1038.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\sum _ { j = 0 } ^ { 1 } & \\int _ { \\xi _ { 0 } - 2 \\delta } ^ { \\xi _ { j } } \\| F ( \\xi _ { j } - s ) G ( s ) \\| _ { p } \\ , \\textnormal { d } s \\\\ & \\leq 2 \\lim _ { \\delta \\to 0 } \\left ( g ( \\xi _ { 0 } - 2 \\delta , \\xi _ { 0 } + \\delta ) \\int _ { 0 } ^ { 3 \\delta } \\| F ( s ) \\| _ { q } \\ , \\textnormal { d } s \\right ) = 0 . \\end{align*}"}
{"id": "5233.png", "formula": "\\begin{align*} \\psi ( x ) = { } & x \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } f _ { x , n } ( x ^ a y ^ b ) ^ n \\right ) = x f _ x \\\\ \\psi ( y ) = { } & y \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } f _ { y , n } ( x ^ a y ^ b ) ^ n \\right ) = y f _ y \\end{align*}"}
{"id": "8507.png", "formula": "\\begin{align*} t r ( A ( t , \\delta ) ^ { 1 3 } ) = t ^ { 1 3 } - 1 3 [ t ^ { 1 1 } \\delta - 5 t ^ { 9 } \\delta ^ 2 + 1 2 t ^ { 7 } \\delta ^ 3 - 1 4 t ^ 5 \\delta ^ 4 + 7 t ^ 3 \\delta ^ 5 - t \\delta ^ 6 ] \\in S ^ { 1 3 } , \\ ; \\forall t , \\delta \\in R . \\end{align*}"}
{"id": "7610.png", "formula": "\\begin{align*} \\frac { d P } { d t } = P B R ^ { - 1 } B ^ \\textsf { T } P - P A - A ^ \\textsf { T } P - Q = P ^ 2 + 4 P - 1 , \\end{align*}"}
{"id": "730.png", "formula": "\\begin{align*} I _ 1 \\leq \\ & - \\rho \\lambda e ^ { \\lambda u } \\Bigl ( \\lambda - 2 c \\Bigr ) | \\nabla u | _ g ^ 2 \\varphi v - 3 \\left ( \\frac { f ' ( u ) } { f ( u ) } \\right ) ^ 2 | \\nabla u | _ g ^ 2 \\varphi v \\\\ & + 2 c \\left | \\frac { f ' ( u ) } { f ( u ) } \\right | \\varepsilon ^ { - 1 } \\varphi v ^ 3 + \\frac { \\varepsilon ' } { ( 1 - \\varepsilon ' ) } \\rho ^ 2 \\lambda ^ 2 e ^ { 2 \\lambda u } | \\nabla u | ^ 2 _ g \\varphi v . \\end{align*}"}
{"id": "3505.png", "formula": "\\begin{align*} c _ { ( 1 ) } ^ { \\ast } = \\frac { 1 } { ( 2 \\pi ) ^ { 1 / 2 } \\Psi _ { 1 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) } \\end{align*}"}
{"id": "4996.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { s \\in [ 0 , T ] } n ^ { 2 \\alpha + 1 } \\int _ { \\eta _ n ( s ) } ^ s ( s - \\eta _ n ( u ) ) ^ { 2 \\alpha } E [ | \\sigma ( X ^ n _ { \\eta _ n ( u ) } ) - \\sigma ( X _ { \\eta _ n ( u ) } ) | ^ 2 ] d u = 0 \\end{align*}"}
{"id": "1691.png", "formula": "\\begin{align*} H _ { \\psi , r } = \\frac { k - 1 } { r } + \\delta ' ( r ) , \\end{align*}"}
{"id": "7572.png", "formula": "\\begin{align*} \\frac { d g } { d t } = \\sum _ { \\alpha = 1 } ^ { r } b _ \\alpha ( t ) X _ \\alpha ^ \\mathrm { R } ( g ) = \\widehat { X } ^ G _ R ( t , g ) , \\ , \\end{align*}"}
{"id": "316.png", "formula": "\\begin{align*} f ( A ) & \\le f ( 2 ^ K ) + \\sum _ { \\substack { p > 2 \\\\ p \\in A } } f ( p ) + ( 2 - 2 ^ { 1 - K } ) \\sum _ { \\substack { 2 < p \\le 2 3 \\\\ p \\notin A } } b _ p f ( p ) + 2 \\sum _ { \\substack { p > 2 3 \\\\ p \\notin A } } b _ p f ( p ) \\\\ & \\le f ( 2 ^ K ) + ( 2 - 2 ^ { 1 - K } ) \\sum _ { 2 < p \\le 2 3 } b _ p f ( p ) + 2 \\sum _ { p > 2 3 } b _ p f ( p ) \\\\ & = : \\ f ( 2 ^ K ) + ( 2 - 2 ^ { 1 - K } ) C _ 1 + 2 C _ 2 . \\end{align*}"}
{"id": "901.png", "formula": "\\begin{align*} \\tilde { h } '' - \\dfrac { \\tilde { h } ^ { 2 } - 1 } { r ^ { 2 } } \\tilde { h } = \\left ( g '^ { 2 } \\sigma ^ { 2 } - B ^ { 2 } \\right ) \\tilde { h } \\end{align*}"}
{"id": "2923.png", "formula": "\\begin{align*} f \\circ S ^ { ( 1 - i ) k } ( A x + e ) = f ( A S ^ { ( 1 - i ) k } x + e ) \\quad g \\circ S ^ { ( 1 - i ) k } ( x ) + d ' , \\end{align*}"}
{"id": "5626.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } X ( t , x _ 0 ) = - \\vec { b } ( X ( t , x _ 0 ) , t ) ( 0 , \\infty ) \\\\ X ( 0 , x _ 0 ) = x _ 0 \\end{cases} . \\end{align*}"}
{"id": "6237.png", "formula": "\\begin{align*} D _ q u ( x ) + u ( x ) u ( q x ) = 0 . \\end{align*}"}
{"id": "6751.png", "formula": "\\begin{align*} 2 K _ X & \\equiv f ^ * \\left ( 2 E + \\sum \\limits _ { i = 1 } ^ { n } { \\left ( F _ { i } + F _ { i } ^ { ' } \\right ) } \\right ) . \\end{align*}"}
{"id": "6276.png", "formula": "\\begin{align*} \\int \\frac { \\widetilde { h } _ n ( x ; q ) } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } d _ q x = - \\dfrac { q ^ { 1 - n } ( 1 - q ) } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\widetilde { h } _ { n - 1 } ( x ; q ) . \\end{align*}"}
{"id": "962.png", "formula": "\\begin{align*} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ E = Q ( \\theta ) \\ \\ \\ \\ \\ \\ \\ R = \\mathbb { Z } [ \\theta ] , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{align*}"}
{"id": "2953.png", "formula": "\\begin{align*} y t Z ^ { \\prime \\prime } - ( y + 1 ) Z ^ \\prime = 0 \\end{align*}"}
{"id": "8059.png", "formula": "\\begin{align*} \\{ x \\in M \\mid \\| X _ { 2 } \\| = 0 \\} \\end{align*}"}
{"id": "1548.png", "formula": "\\begin{align*} \\P ( Z \\ge 1 2 \\rho L _ { } \\tau ) & \\le e ^ { - 1 2 h \\rho L _ { } \\tau } 4 ^ { \\tau } = \\big ( 4 / e ^ { 3 } \\big ) ^ { \\tau } \\leq e ^ { - \\tau } . \\end{align*}"}
{"id": "6195.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial } { \\partial t } \\omega = - \\mathrm { R i c } ^ { T } ( \\omega ) \\mathrm { w i t h } \\omega ( T _ { 0 } ) = \\omega ^ { \\prime } , \\end{array} \\end{align*}"}
{"id": "7490.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\ln ( \\vert \\varphi ' _ { 0 , 1 } ( x _ k ) \\vert ) \\to \\ln ( 2 ) \\end{align*}"}
{"id": "5229.png", "formula": "\\begin{align*} f \\mapsto \\sum _ { n = 0 } ^ { \\infty } { v ^ n ( f ) \\over n ! } , \\end{align*}"}
{"id": "7473.png", "formula": "\\begin{align*} \\cap _ { i = 1 } ^ { m } C _ { i } \\subset \\cap _ { n = 0 } ^ { \\infty } \\mathrm { F i x } ( S _ { n } ) \\subset \\cap _ { n = 0 } ^ { j _ { f } } \\mathrm { F i x } ( S _ { n } ) , \\end{align*}"}
{"id": "558.png", "formula": "\\begin{align*} \\mathbb { E } ( \\overline { S } _ n ) = \\frac { 1 } { n } \\sum _ { 1 \\leq i \\leq n } f _ k ( i ) + O _ \\alpha ( n ^ { - 1 / 2 + \\varepsilon } ) \\end{align*}"}
{"id": "8289.png", "formula": "\\begin{align*} k = \\frac { 1 } { 2 \\hslash \\sqrt { \\beta / 3 } } \\sqrt { \\frac { \\sqrt { \\frac { 1 6 } { 3 } m \\beta | E | } - 1 } { 2 } } \\end{align*}"}
{"id": "3341.png", "formula": "\\begin{align*} Q _ { 3 , \\tilde { q } , \\tilde { p } } = \\sigma _ 2 ^ { \\lambda ( 1 ) } \\sigma _ 1 ^ { \\lambda ( 2 ) } \\sigma _ 2 ^ { \\lambda ( 3 ) } \\sigma _ 1 ^ { \\lambda ( 4 ) } . . . \\sigma _ 2 ^ { \\lambda ( \\tilde { q } - 2 ) } \\sigma _ 1 ^ { \\lambda ( \\tilde { q } - 1 ) } \\end{align*}"}
{"id": "5234.png", "formula": "\\begin{align*} d x \\wedge d y = \\psi ^ * ( d x \\wedge d y ) = ( f ^ { b - a } + ( a - b ) x ^ a y ^ b f ' f ^ { b - a - 1 } ) d x \\wedge d y . \\end{align*}"}
{"id": "8396.png", "formula": "\\begin{gather*} y ^ 2 = x ^ 3 - 1 2 9 8 7 D ^ 2 x - 2 6 3 4 6 6 D ^ 3 . \\end{gather*}"}
{"id": "8670.png", "formula": "\\begin{align*} x _ 1 V _ 1 ( x ) + x _ 2 V _ 2 ( x ) + x _ 3 V _ 3 ( x ) = 0 \\end{align*}"}
{"id": "3085.png", "formula": "\\begin{align*} \\mathrm { ( \\widetilde { I } ) } ~ \\mathrm { M T E } _ { \\boldsymbol { q } } ( \\mathbf { Y } ) & = \\boldsymbol { \\mu } + \\frac { \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } \\boldsymbol { \\delta } } { \\overline { F } _ { \\mathbf { Z } } ( \\boldsymbol { \\xi _ { q } } ) } , \\end{align*}"}
{"id": "7670.png", "formula": "\\begin{align*} E ( S _ { 5 , 6 , 1 } ; u , v ) = 1 + 2 u v + 2 u ^ 2 v ^ 2 + u ^ 3 v ^ 3 - ( 2 9 u ^ 3 + 5 2 0 u ^ 2 v + 5 2 0 u v ^ 2 + 2 9 v ^ 3 ) . \\end{align*}"}
{"id": "5171.png", "formula": "\\begin{align*} \\sum \\limits _ { r = 0 } ^ { m - 2 } \\sum \\limits _ { k = 0 } ^ { m - 1 } \\prod \\limits _ { i = 0 } ^ { r } \\rho _ { i + k } = \\sum \\limits _ { r = 0 } ^ { m - 2 } \\sum \\limits _ { k = 0 } ^ { m - 1 } \\prod \\limits _ { i = 0 } ^ { r } \\rho _ { \\sigma ( i + k ) } \\end{align*}"}
{"id": "1409.png", "formula": "\\begin{align*} A = I _ { r + 1 } - \\left [ \\begin{matrix} 0 & & & & & \\\\ - 1 & \\delta & & & & \\\\ & - 1 & 2 \\delta & & & \\\\ & & \\ddots & \\ddots & & \\\\ & & & - 1 & ( r - 1 ) \\delta & \\\\ & & & & 0 & r \\delta \\\\ \\end{matrix} \\right ] , \\Delta A = \\left [ \\begin{matrix} 0 & \\cdots & \\cdots & \\cdots & 0 \\\\ \\vdots & \\cdots & \\cdots & \\cdots & \\vdots \\\\ 0 & \\cdots & 0 & \\epsilon & 0 \\end{matrix} \\right ] . \\end{align*}"}
{"id": "3338.png", "formula": "\\begin{align*} B ( N , \\tilde { q } , \\tilde { p } ) = Q _ { N , \\tilde { q } , \\tilde { p } } \\sigma _ 2 ^ { \\epsilon ( 2 ) } Q _ { N , \\tilde { q } , \\tilde { p } } ^ { - 1 } \\sigma _ 1 ^ { \\epsilon ( 1 ) } \\end{align*}"}
{"id": "1176.png", "formula": "\\begin{align*} \\max _ { \\sum _ { k = 1 } ^ { r - 1 } N _ { k } + 1 \\le j \\le \\sum _ { k = 1 } ^ { r } N _ { j } } M _ j \\lesssim \\sigma ^ { - \\lfloor d / 2 \\rfloor } \\exp \\left ( \\frac { ( p - 1 ) r ^ 2 \\delta ^ 2 } { 2 \\sigma ^ 2 ( 1 - \\eta ) } \\right ) \\end{align*}"}
{"id": "4949.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty | A _ { n _ k } f ( x ) - A _ { n _ { k - 1 } } f ( x ) | & \\leq \\sum _ { k = 1 } ^ \\infty \\sum _ { j \\in J ( k ) } | A _ { m _ j } f ( x ) - A _ { m _ { j - 1 } } f ( x ) | . \\\\ & = \\sum _ { j = 1 } ^ \\infty | A _ { m _ j } f ( x ) - A _ { m _ { j - 1 } } f ( x ) | . \\end{align*}"}
{"id": "3303.png", "formula": "\\begin{align*} D ( \\partial _ t ) \\psi ( x , t ) : = \\int _ 0 ^ t \\int _ \\Gamma \\left [ \\partial _ { \\nu _ y } k ( | x - y | , t - \\tau ) \\right ] \\psi ( y , \\tau ) d \\Gamma _ y d \\tau x \\in \\R ^ d \\setminus \\Gamma , \\end{align*}"}
{"id": "3569.png", "formula": "\\begin{align*} \\kappa ( c _ 1 ( \\chi ^ j \\circ \\det \\mid _ M ) ) & = \\kappa ( ( q + 1 ) j u ) \\\\ & = ( q + 1 ) j \\kappa ( u ) \\\\ & = 0 . \\end{align*}"}
{"id": "8900.png", "formula": "\\begin{align*} - u '' + \\lambda u = | u | ^ { p - 2 } u \\end{align*}"}
{"id": "2943.png", "formula": "\\begin{align*} & \\partial _ t \\langle \\tilde { B } : \\phi ( t ) \\rangle _ t = \\pm \\langle \\phi ( t ) , C ^ * C \\phi ( t ) \\rangle + g ( t ) \\\\ & g ( t ) \\in L ^ 1 ( d t ) , C ^ * C \\geq 0 . \\end{align*}"}
{"id": "5692.png", "formula": "\\begin{align*} m _ { d + 1 , 1 ^ { k - 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) = \\begin{cases} m _ { d , 1 ^ { k - 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d \\ge 2 ) , \\\\ \\\\ k m _ { 1 ^ { k } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d = 1 ) . \\end{cases} \\end{align*}"}
{"id": "6823.png", "formula": "\\begin{align*} J _ v ( 0 , g ) p ( t , 0 , g ) = J _ v ( V _ F , g ) p ( t , V _ F , g ) , g > g _ F , t > 0 . \\end{align*}"}
{"id": "7120.png", "formula": "\\begin{align*} \\frac { d \\tau ( t , x ) } { d t } = \\frac { \\partial \\tau ( t , x ) } { \\partial x } g ( t , x ) ( u - \\mu ( t , x ) ) \\ , . \\end{align*}"}
{"id": "6370.png", "formula": "\\begin{align*} P _ { \\kappa } ( x ) \\ ! = \\ ! \\frac { \\int _ { 0 } ^ { x } \\ ! h ( t ) \\exp _ { \\kappa } [ - f ( t ) ] \\ , d t } { \\int _ { 0 } ^ { x } \\ ! h ( t ) \\exp _ { \\kappa } [ - f ( t ) ] \\ , d t \\ ! + \\ ! \\lambda \\int _ { x } ^ { \\infty } \\ ! h ( t ) \\exp _ { \\kappa } [ - f ( t ) ] \\ , d t } \\ , \\end{align*}"}
{"id": "1082.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Re ( \\lambda _ { n } ) = \\infty . \\end{align*}"}
{"id": "1574.png", "formula": "\\begin{align*} E = b ^ 2 \\sum \\limits _ { k = 1 } ^ { 3 } ( - 1 ) ^ { \\gamma + \\tau } z ^ { k } _ { \\tilde { \\gamma } } z ^ { k } _ { \\tilde { \\tau } } z ^ { 1 } _ { \\gamma } z ^ { 1 } _ { \\tau } , \\end{align*}"}
{"id": "8154.png", "formula": "\\begin{align*} & \\limsup \\limits _ { t \\rightarrow \\infty } \\| P _ \\delta ( W _ { n , m ; \\textrm { f a r } } ) e ^ { - i t H } \\psi - P _ \\delta ( W _ { n , m ; \\textrm { f a r } } ) e ^ { - i t H _ 0 } \\varphi \\| \\\\ & \\leq \\limsup \\limits _ { t \\rightarrow \\infty } \\| P _ \\delta ( W _ { n , m ; \\textrm { f a r } } ) \\| _ { \\textrm { o p } } \\| e ^ { - i t H } \\psi - e ^ { - i t H _ 0 } \\varphi \\| = 0 \\end{align*}"}
{"id": "1373.png", "formula": "\\begin{align*} \\left ( \\frac { p + 1 } { 2 } \\right ) ^ 2 \\sum _ { \\substack { k = k _ 1 - k _ 2 + \\cdots - k _ { p - 1 } + k _ { 2 p - 1 } \\\\ | k _ 1 | \\gg k _ 2 ^ * \\\\ | \\Phi ( k _ 1 - k _ 2 + \\cdots + k _ p , k _ 1 \\cdots , k _ p ) | \\gtrsim | k _ 1 | } } \\frac { \\widehat { ( W _ t u _ 0 ) } _ { k _ 1 } } { \\Phi ( k _ 1 - k _ 2 + \\cdots + k _ p , k _ 1 \\cdots , k _ p ) } \\prod _ { \\substack { j = 3 \\\\ o d d } } ^ { 2 p - 1 } \\widehat { v } _ { k _ i } \\prod _ { \\substack { l = 2 \\\\ e v e n } } ^ { 2 p - 1 } \\overline { \\widehat { v } _ { k _ l } } , \\end{align*}"}
{"id": "5436.png", "formula": "\\begin{align*} \\begin{cases} f ^ k \\to f F ^ s _ { p _ 1 , q } , \\ , & \\| f ^ k \\| _ { L ^ { \\infty } ( \\R ^ n ) } \\leq C \\\\ g ^ k \\to g F ^ s _ { p _ 1 , q } , \\ , & \\| g ^ k \\| _ { L ^ { \\infty } ( \\R ^ n ) } \\leq C \\end{cases} \\end{align*}"}
{"id": "2927.png", "formula": "\\begin{align*} K ( x _ 1 ; x _ 2 ) : = v ( x _ 1 - x _ 2 ) \\varphi ( x _ 1 ) \\varphi ( x _ 2 ) \\ , , \\end{align*}"}
{"id": "7421.png", "formula": "\\begin{align*} \\alpha = \\alpha _ 1 + \\gamma ^ 2 \\beta . \\end{align*}"}
{"id": "8421.png", "formula": "\\begin{gather*} A \\otimes D : = A _ F , \\end{gather*}"}
{"id": "4912.png", "formula": "\\begin{align*} \\begin{cases} \\varphi ( t ) = e ^ { \\eta t } & \\alpha = 1 - \\delta - \\beta , \\\\ \\varphi ( t ) = ( 1 + ( 1 - \\beta - \\delta - \\alpha ) \\eta t ) ^ { \\frac { 1 } { 1 - \\beta - \\delta - \\alpha } } & \\alpha \\not = 1 - \\delta - \\beta , \\end{cases} \\end{align*}"}
{"id": "4987.png", "formula": "\\begin{align*} \\langle M ^ { ( n ) } \\rangle _ \\tau = n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau } \\left | \\sum _ { i = 1 } ^ Q \\rho _ i \\mathbf { 1 } _ { [ 0 , t _ i ] } ( \\tau ) ( t _ i - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\Theta ^ n _ s \\right | ^ 2 d s , \\end{align*}"}
{"id": "8162.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } \\| \\chi _ { S _ { 2 v t } } e ^ { - i t H } \\psi \\| = \\| \\psi \\| \\end{align*}"}
{"id": "2359.png", "formula": "\\begin{align*} \\nu _ \\sigma ( f ) = \\nu ( l ( h _ \\sigma ) ) \\mbox { f o r e v e r y } \\sigma , \\rho < \\sigma < \\lambda . \\end{align*}"}
{"id": "2963.png", "formula": "\\begin{align*} \\lambda = \\tfrac 1 4 + r ^ 2 r \\geq 0 . \\end{align*}"}
{"id": "4921.png", "formula": "\\begin{align*} \\bar \\nabla _ Z Y = \\nabla _ Z Y - h ( Z , Y ) \\nu , \\end{align*}"}
{"id": "4023.png", "formula": "\\begin{align*} \\kappa ^ { ( X + Y ) } ( z ) = \\kappa ^ { ( X ) } ( z ) + \\kappa ^ { ( Y ) } ( z ) . \\end{align*}"}
{"id": "246.png", "formula": "\\begin{align*} \\theta ( \\lambda ) = ( c ^ * ( \\theta ) ) ( P _ 0 ) \\ \\ \\ \\textup { f o r } \\ \\ c \\in R ^ n , \\ \\ \\lambda = c ( P _ 0 ) , \\ \\theta \\in R _ { \\pi } . \\end{align*}"}
{"id": "125.png", "formula": "\\begin{align*} T _ { \\lambda } : = 3 \\lambda \\big [ S ( 0 ) - C ( 0 ) + C ( 0 ) - C _ \\infty ( 0 ) \\big ] S \\star C = 3 \\lambda \\big [ S ( 0 ) - C ( 0 ) - \\eta _ t \\big ] S \\star C , \\end{align*}"}
{"id": "2886.png", "formula": "\\begin{align*} 0 = ( 0 , 0 , \\dotsc , 0 ) 1 = ( 1 , 1 , \\dotsc , 1 ) , \\end{align*}"}
{"id": "8412.png", "formula": "\\begin{gather*} \\phi ^ W = \\sum _ { i , \\chi } m _ { i , \\chi } \\varphi _ i \\otimes \\chi \\end{gather*}"}
{"id": "8805.png", "formula": "\\begin{align*} \\mu ( 1 _ { \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) } ^ { - 1 } ( \\mathbb { R } _ { > t ' } ) ) = \\left \\{ \\begin{aligned} & \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) & & \\ ; t ' < 1 \\\\ & 0 & & \\ ; t ' \\geq 1 \\end{aligned} \\right . \\end{align*}"}
{"id": "2531.png", "formula": "\\begin{align*} \\begin{array} { l l } \\nabla _ { e _ 1 } e _ 1 = \\nabla _ { X } X = e ^ { - 2 x _ 2 } e _ 2 , \\\\ \\nabla _ { e _ 2 } e _ 2 = \\nabla _ { U } U = 0 , \\\\ \\nabla _ { e _ 1 } e _ 2 = \\nabla _ { X } U = - e _ 1 , \\\\ \\nabla _ { e _ 2 } e _ 1 = \\nabla _ { U } X = - e _ 1 . \\end{array} \\end{align*}"}
{"id": "520.png", "formula": "\\begin{align*} L _ { g n } ( \\dot { x } , x , t ) = { \\frac { h _ 1 ( t ) [ h _ 2 ( t ) \\dot { x } + { \\dot h _ 2 } ( t ) x ] + { \\dot h _ 4 } ( t ) } { h _ 2 ( t ) [ h _ 2 ( t ) x + h _ 4 ] } } \\end{align*}"}
{"id": "6989.png", "formula": "\\begin{align*} I _ Q ( f ) : = \\sum W ( U , f ) \\int _ U Q \\omega . \\end{align*}"}
{"id": "6258.png", "formula": "\\begin{align*} p ( x ) = \\dfrac { 1 - q x } { q ( 1 - q ) x ^ 2 } , r ( x ) = \\frac { 1 } { ( 1 - q ) ^ 2 x ^ 2 } . \\end{align*}"}
{"id": "904.png", "formula": "\\begin{align*} \\psi ( r ) = C r ^ { 2 } + O \\left ( r ^ { 2 + 2 \\alpha } \\right ) \\ , ( r \\rightarrow 0 ^ { + } ) . \\end{align*}"}
{"id": "3167.png", "formula": "\\begin{align*} d \\doteq d _ { - } + \\mathrm { r } _ { + } ( d _ { - } ) = e _ { ( \\cdot ) } ( \\hat { \\omega } ) \\ . \\end{align*}"}
{"id": "1087.png", "formula": "\\begin{align*} B _ x = Z _ x Y _ x , \\end{align*}"}
{"id": "7729.png", "formula": "\\begin{align*} \\dot x ( t ) = A ( t ) x ( t ) t \\geq 0 , \\end{align*}"}
{"id": "1840.png", "formula": "\\begin{align*} q _ { n } ( z ) \\ , \\phi _ { k } ( z ) - q _ { n , k + 1 } ( z ) = O ( z ^ { - n _ { k } - 1 } ) , z \\rightarrow \\infty , 0 \\leq k \\leq p - 1 , \\end{align*}"}
{"id": "615.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\Big ( \\frac { \\sin ^ 2 ( \\pi x ) } { 3 \\pi ^ 2 } + \\frac { z ^ 2 } { n ^ 2 } \\Big ) ^ { - 1 } d x = \\frac { \\sqrt { 3 } } { \\frac { z } { n } \\sqrt { 3 ( \\frac { z } { n } ) ^ 2 + \\frac { 1 } { \\pi ^ 2 } } } , \\end{align*}"}
{"id": "5209.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { v - \\sigma } ( k _ { 1 2 } ( \\Xi _ j ) + 2 p _ j ) = - \\beta + 1 . \\end{align*}"}
{"id": "1994.png", "formula": "\\begin{align*} & \\tilde A ( y , s ) = A ( R _ 0 ^ { k _ 0 } y , s ) \\\\ & \\tilde f ( y , s ) = R _ 0 ^ { ( 2 k _ 0 ( 1 - \\alpha ) + k _ 0 \\alpha ) } f ( R _ 0 ^ { k _ 0 } y , s ) \\\\ & \\tilde q ( y ) = R _ 0 ^ { 2 k _ 0 ( 1 - \\alpha ) } q ( R _ 0 ^ { k _ 0 } y ) \\end{align*}"}
{"id": "6266.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n q ^ { 2 n } A i _ q ( q ^ n x ) = \\frac { q ( 1 - q ^ 2 ) + x } { q x ( 1 + q ) } \\ , _ 1 \\phi _ 1 ( 0 ; - q ^ 2 ; q , - x ) - \\frac { ( 1 - q ) } { x } A i _ q ( \\frac { x } { q } ) , \\\\ & \\sum _ { n = 0 } ^ { \\infty } q ^ { \\frac { n ( n - 1 ) } { 2 } } x ^ n A _ q ( q ^ n x ) = \\frac { 1 - q } { q ^ 2 x } A _ q ( \\frac { x } { q } ) + \\frac { 1 - q - x } { q ^ 2 x } A _ q ( q x ) . \\end{align*}"}
{"id": "2000.png", "formula": "\\begin{align*} g _ 1 ( x ) : = n ! g ( x ) = n ! \\displaystyle \\sum _ { j = 0 } ^ { n } \\binom { n + s - j } { n - j } \\frac { x ^ j } { j ! } \\end{align*}"}
{"id": "5460.png", "formula": "\\begin{align*} k a = a _ 1 k _ 1 \\end{align*}"}
{"id": "4610.png", "formula": "\\begin{align*} & B _ i = ( \\bar { \\alpha } + k _ { i - 1 } ) ( \\bar { x } _ i + 1 ) , \\\\ & B _ { i , j } = 1 + \\bar { \\alpha } + ( k _ { i - 1 } + 1 ) ( \\bar { x } _ i + 1 ) + ( k _ { j - 1 } + \\bar { x } _ i + 1 ) ( \\bar { x } _ j + 1 ) . \\end{align*}"}
{"id": "2428.png", "formula": "\\begin{align*} ^ j ( f ) : = \\sum _ { j \\le i } ( - 1 ) ^ { b _ i } \\varepsilon _ { f ( i ) } \\in \\texttt { P } , ( f ) : = ^ { 1 } ( f ) \\in \\texttt { P } . \\end{align*}"}
{"id": "705.png", "formula": "\\begin{align*} v = \\frac { f ( u ) } { \\sqrt { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | ^ 2 } } . \\end{align*}"}
{"id": "3022.png", "formula": "\\begin{align*} 0 \\leq \\hat { \\Lambda } \\cdot \\hat { R } _ j = \\frac { 3 } { 4 ( 4 n + 1 ) } - \\frac { b } { 4 n + 1 } + \\frac { b _ j ( 2 n + 1 ) } { 2 ( 4 n + 1 ) } - \\frac { b _ i n } { 4 n + 1 } - \\frac { \\mu } { 4 n + 1 } \\end{align*}"}
{"id": "7098.png", "formula": "\\begin{align*} \\left ( \\pi _ z ( x , v ) s \\right ) ( u ) = e ^ { i \\langle \\mathrm { A d } _ u z , v \\rangle } \\cdot s ( x ^ { - 1 } u ) , u \\in K , \\end{align*}"}
{"id": "2640.png", "formula": "\\begin{align*} Q = ( \\alpha _ 0 : \\alpha _ 1 ) \\times ( \\beta _ 0 : \\beta _ 1 ) \\times ( \\gamma _ 0 : \\gamma _ 1 ) \\in X \\end{align*}"}
{"id": "8950.png", "formula": "\\begin{align*} \\begin{array} { l l } \\ ! \\ ! & \\ ! \\ ! \\log B _ 0 ( x _ 0 , u ( x _ 0 ) , D u ( x _ 0 ) ) - \\log B _ 0 ( x _ 0 , u _ 0 ( x _ 0 ) , D u _ 0 ( x _ 0 ) ) \\\\ = \\ ! \\ ! & \\ ! \\ ! D _ u ( \\log B _ 0 ) ( x _ 0 , \\tilde u ( x _ 0 ) , D u ( x _ 0 ) ) ( u - u _ 0 ) ( x _ 0 ) \\\\ \\ ! \\ ! & \\ ! \\ ! + D _ { p _ k } ( \\log B _ 0 ) ( x _ 0 , u _ 0 ( x _ 0 ) , \\tilde p ( x _ 0 ) ) D _ k ( u - u _ 0 ) ( x _ 0 ) , \\end{array} \\end{align*}"}
{"id": "7667.png", "formula": "\\begin{align*} E ( S _ 4 ; u , v ) = 1 + 4 u ^ 2 + 4 5 u v + 4 v ^ 2 + u ^ 2 v ^ 2 , \\end{align*}"}
{"id": "6342.png", "formula": "\\begin{align*} a ( x , y , t ) = a ( y , x , t ) ~ ~ \\forall ( x , y , t ) \\in \\overline { \\Omega } \\times \\overline { \\Omega } \\times \\R , \\end{align*}"}
{"id": "315.png", "formula": "\\begin{align*} \\sum _ { \\substack { p > 2 \\\\ p \\notin A } } \\sum _ { k = 1 } ^ { K - 1 } f ( ( A ^ k ) _ { 2 ^ k p } ) & \\le ( 1 - 2 ^ { 1 - K } ) \\sum _ { \\substack { 2 < p \\le 2 3 \\\\ p \\notin A } } b _ p f ( p ) + \\sum _ { \\substack { p > 2 3 \\\\ p \\notin A } } f ( p ) \\Big ( ( 1 - 2 ^ { 1 - J } ) b _ p + 2 ^ { - J } \\frac { \\log p } { \\log ( 2 p ) } \\Big ) \\\\ & \\le ( 1 - 2 ^ { 1 - K } ) \\sum _ { \\substack { 2 < p \\le 2 3 \\\\ p \\notin A } } b _ p f ( p ) + \\sum _ { \\substack { p > 2 3 \\\\ p \\notin A } } b _ p f ( p ) , \\end{align*}"}
{"id": "4118.png", "formula": "\\begin{align*} Z ^ i : = Z ^ { i , 0 } \\oplus Z ^ { i , 1 } \\oplus \\dots \\oplus Z ^ { i , N } . \\end{align*}"}
{"id": "1196.png", "formula": "\\begin{align*} \\frac { d ( \\mu * \\gamma _ { \\sigma } + t _ n \\tilde { h } _ i ) } { d ( \\mu * \\gamma _ { \\sigma } ) } \\ge ( 1 - c t _ n ) \\ge \\frac { 1 } { 2 } , \\ i = 1 , 2 . \\end{align*}"}
{"id": "4728.png", "formula": "\\begin{align*} & e _ { ( k ) } H _ { 2 k } ^ { - 1 } H _ { 2 k + 1 } ^ { - 1 } H _ { 2 k - 1 } H _ { 2 k } e _ { ( k ) } \\\\ = & e _ { ( k ) } H _ { 2 k } ^ { - 1 } H _ { 2 k + 1 } H _ { 2 k - 1 } ^ { - 1 } H _ { 2 k } e _ { ( k ) } + ( q - q ^ { - 1 } ) e _ { ( k ) } ( H _ { 2 k } - H _ { 2 k } ^ { - 1 } ) e _ { ( k ) } - ( q - q ^ { - 1 } ) ^ { 2 } e _ { ( k ) } ^ { 2 } \\\\ = & e _ { ( k ) } H _ { 2 k } ^ { - 1 } H _ { 2 k + 1 } H _ { 2 k - 1 } ^ { - 1 } H _ { 2 k } e _ { ( k ) } . \\end{align*}"}
{"id": "8582.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ s \\cos ( 2 \\pi n _ { k _ j } x ) = 2 ^ { - s } \\sum _ { \\varepsilon _ i = \\pm 1 } \\cos 2 \\pi ( \\varepsilon _ 1 n _ { k _ 1 } + \\varepsilon _ { 2 } n _ { k _ 2 } + \\ldots + \\varepsilon _ s n _ { k _ s } ) x , \\end{align*}"}
{"id": "7397.png", "formula": "\\begin{align*} K _ h & : = 2 \\int _ { { \\C } _ 1 } \\big ( \\sum _ { i = 2 } ^ { 4 } ( U _ { h , i } ) ^ { p } \\big ) { W } _ { h } - \\sum _ { i = 2 } ^ { 4 } ( U _ { h , i } ) ^ { p + 1 } d y \\\\ & - 4 \\int _ { { \\C } _ 1 } \\frac { p } { 2 } \\big ( U _ { h , 1 } + \\lambda \\sum _ { i = 2 } ^ 4 U _ { h , i } \\big ) ^ { p - 1 } \\big ( \\sum _ { i = 2 } ^ { 4 } U _ { h , i } \\big ) ^ 2 - \\frac { 1 } { p + 1 } \\sum _ { i = 2 } ^ { 4 } ( U _ { h , i } ) ^ { p + 1 } \\ : d y , \\end{align*}"}
{"id": "1528.png", "formula": "\\begin{align*} \\P ( T _ { d e t } \\geq t ) \\ , = \\ , e ^ { - \\Theta ( t ^ { \\frac { 1 } { 2 \\alpha } } ) } . \\end{align*}"}
{"id": "6130.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\end{gather*}"}
{"id": "5355.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s / 2 } ( u \\circ A ) ( x ) = ( ( - \\Delta ) ^ { s / 2 } u \\circ A ) ( x ) \\end{align*}"}
{"id": "7261.png", "formula": "\\begin{align*} u ( x , t ) = \\begin{cases} \\lambda ( t ) ^ { - \\frac { 2 } { p - 1 } } { \\sf Q } ( y ) & , \\\\ & | y | \\to \\infty , \\ | x | \\ll 1 . \\end{cases} \\end{align*}"}
{"id": "756.png", "formula": "\\begin{align*} \\mu _ i ( x ) = r _ i x _ i \\left ( 1 - \\sum _ { j = 1 } ^ { d } a _ { i j } x _ j ^ + \\right ) , \\sigma ( x ) = \\mathrm { d i a g } \\left ( \\frac { x _ 1 } { 1 + \\lVert x \\rVert ^ 2 } , \\frac { x _ 2 } { 1 + \\lVert x \\rVert ^ 2 } , \\ldots , \\frac { x _ d } { 1 + \\lVert x \\rVert ^ 2 } \\right ) , \\end{align*}"}
{"id": "8902.png", "formula": "\\begin{align*} \\mu _ 1 : = \\ell \\ , \\Big ( \\frac { \\lambda _ 2 ( \\mathcal { G } ) } { p - 2 } \\Big ) ^ { \\frac { 2 } { p - 2 } } , \\end{align*}"}
{"id": "3585.png", "formula": "\\begin{align*} { \\sim } \\left ( \\sum \\limits _ { i = 1 } ^ { n } a _ i \\cdot \\mathtt { w } ^ + ( \\phi _ i ) \\geqslant c \\right ) \\dashv \\vdash _ { \\mathtt { w } \\mathsf { B D } } \\sum \\limits _ { i = 1 } ^ { n } - a _ i \\cdot \\mathtt { w } ^ + ( \\phi _ i ) > - c \\end{align*}"}
{"id": "9001.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ m ( - 1 ) ^ k P _ k \\left ( m , x \\right ) \\left ( j + x \\right ) ^ { m - k } = \\sum _ { k = 0 } ^ m ( - 1 ) ^ { m - k } P _ { m - k } ( m , x ) ( j + x ) ^ k . \\end{align*}"}
{"id": "6642.png", "formula": "\\begin{align*} L ( n , R ) = 2 ^ { ( n \\log n ) R + n ( \\log \\frac { A } { e \\sqrt { a } } ) + o ( n ) } \\ ; . \\ , \\end{align*}"}
{"id": "1092.png", "formula": "\\begin{align*} 2 g ( \\nabla _ X Y , Z ) = X g ( Y , Z ) + Y g ( X , Z ) - Z g ( X , Y ) + g ( [ X , Y ] , Z ) - g ( Y , \\nabla ^ * _ X Z - \\nabla _ Z X ) - g ( X , [ Y , Z ] ) . \\end{align*}"}
{"id": "3531.png", "formula": "\\begin{align*} c _ { ( 2 ) } ^ { \\ast } = \\frac { 4 ( t - 1 ) ( t - 2 ) } { B ( \\frac { 1 } { 2 } , t - \\frac { 5 } { 2 } ) } , ~ t > \\frac { 5 } { 2 } . \\end{align*}"}
{"id": "2589.png", "formula": "\\begin{align*} \\int _ { S U ( 3 ) } \\tilde f ( g ) \\ , d g = \\int _ { \\mathcal O _ { x , y } } f ( x ) \\ , d x \\end{align*}"}
{"id": "4958.png", "formula": "\\begin{align*} E [ Z _ { t _ 1 } Z _ { t _ 2 } ] = \\mathbf { 1 } _ { \\{ t _ 1 = t _ 2 \\} } . \\end{align*}"}
{"id": "8664.png", "formula": "\\begin{align*} \\int _ \\epsilon ^ M | u ' ( r ) - r ^ { - 1 } u ( r ) | ^ 2 \\ , d r & = \\int _ \\epsilon ^ M | u ' ( r ) | ^ 2 \\ , d r - M ^ { - 1 } | u ( M ) | ^ 2 + \\epsilon ^ { - 1 } | u ( \\epsilon ) | ^ 2 \\\\ & \\leq \\int _ \\epsilon ^ M | u ' ( r ) | ^ 2 \\ , d r + \\epsilon ^ { - 1 } | u ( \\epsilon ) | ^ 2 \\ , . \\end{align*}"}
{"id": "6344.png", "formula": "\\begin{align*} | | u | | _ { s , \\varPhi _ { x , y } } = | | u | | _ { \\widehat { \\varPhi } _ x } + [ u ] _ { s , \\varPhi _ { x , y } } , \\end{align*}"}
{"id": "2004.png", "formula": "\\begin{align*} n = n _ 0 \\cdot n _ 1 \\ { \\rm w i t h } \\ \\gcd ( n _ 0 , n _ 1 ) = 1 \\end{align*}"}
{"id": "408.png", "formula": "\\begin{align*} Y _ { n } = \\frac { X _ { 1 } + \\cdots + X _ { n } } { \\sqrt { n } } \\end{align*}"}
{"id": "7697.png", "formula": "\\begin{align*} d \\left ( y _ { k } , p \\right ) \\leq \\frac { \\alpha ^ { k } } { 1 - \\alpha } \\ ; d \\left ( x _ { 0 } , T x _ { 0 } \\right ) + \\sum _ { i = 1 } ^ { k } \\alpha ^ { k - i } \\ ; \\varepsilon _ { k } , \\ ; k \\in \\mathbb { N } , \\end{align*}"}
{"id": "6000.png", "formula": "\\begin{align*} u ( t ) = \\int _ 0 ^ t k ( t - \\tau ) g ( \\tau ) d \\tau , \\end{align*}"}
{"id": "4804.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , 1 ] } | W _ t - W _ { \\lfloor n t \\rfloor / n } | = O _ P \\big ( n ^ { - 1 / 2 } ( \\ln n ) ^ { 1 / 2 } \\big ) , \\end{align*}"}
{"id": "3568.png", "formula": "\\begin{align*} \\chi _ { \\pi } ( h _ 1 ) & = 2 \\chi _ j ( \\det ( h _ 1 ) ) \\\\ & = 2 \\chi _ j ( - 1 ) \\\\ & = 2 ( - 1 ) ^ j . \\end{align*}"}
{"id": "6978.png", "formula": "\\begin{align*} - \\frac { 1 } { z } \\prod \\limits _ { k = 1 } ^ { \\infty } \\Bigg ( \\frac { z - \\mu _ { k } ^ { 2 } } { z - \\lambda _ { k } ^ { 2 } } \\Bigg ) = \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { a _ k } { \\lambda _ k ^ 2 ( \\lambda _ { k } ^ { 2 } - z ) } . \\end{align*}"}
{"id": "1628.png", "formula": "\\begin{align*} \\mathcal { A D } = \\bigcup _ { k = 1 } ^ { \\infty } \\mathcal { A D } _ k , \\end{align*}"}
{"id": "3160.png", "formula": "\\begin{align*} f _ { \\mathfrak { m } } ^ { \\sharp } = \\underset { ^ { \\ast } } { \\underbrace { \\Delta _ { \\mathfrak { a } _ { + } } } } + \\underset { ^ { \\ast } } { \\underbrace { \\left ( - \\Delta _ { \\mathfrak { a } _ { - } } + e _ { \\Phi } - \\beta ^ { - 1 } s \\right ) } } \\ . \\end{align*}"}
{"id": "1289.png", "formula": "\\begin{align*} \\sigma ^ { * } ( \\mathcal { A } _ 1 ) & = \\{ 0 \\} . \\end{align*}"}
{"id": "6795.png", "formula": "\\begin{align*} u ( 0 ) = u ' ( 0 ) = u '' ( 1 ) = u ''' ( 1 ) = 0 , \\end{align*}"}
{"id": "3560.png", "formula": "\\begin{align*} 2 \\binom { n - 2 } { i - 2 } \\binom { c _ i } { 2 } + c _ i ^ 2 N _ i ' ( N _ i ' - 1 ) + c _ i ^ 2 \\cdot 2 N _ i ' N _ i '' + c _ i ^ 2 { N _ i '' } ^ 2 \\end{align*}"}
{"id": "8968.png", "formula": "\\begin{align*} \\left \\vert \\int _ \\Omega 2 R \\xi _ E \\xi _ { \\tilde K } u ^ 2 d V _ g \\right \\vert = \\left \\vert \\int _ \\Omega \\tilde R u ^ 2 d V _ g \\right \\vert \\leq \\epsilon \\| \\nabla u \\| ^ 2 _ { L ^ 2 ( \\Omega \\cap \\tilde K ) } + C ' _ { \\epsilon } \\| u \\| ^ 2 _ { L ^ 2 ( \\Omega \\cap \\tilde K ) } . \\end{align*}"}
{"id": "1335.png", "formula": "\\begin{align*} \\phi ( x , v ) = H ( x , v ) + \\alpha \\frac { x \\cdot v } { \\langle x \\rangle } + \\beta \\langle x \\rangle , \\mbox { a n d } V ( s ) = 1 + s ^ { \\frac { \\xi } { 1 + \\xi } } . \\end{align*}"}
{"id": "5253.png", "formula": "\\begin{align*} A ^ 1 _ Q = \\bigcup _ { R _ 0 = 0 } ^ { r - 1 } A ^ 1 _ { Q , j , R _ 0 } ; A ^ 2 _ Q = \\varnothing . \\end{align*}"}
{"id": "6943.png", "formula": "\\begin{align*} T K = K A . \\end{align*}"}
{"id": "4079.png", "formula": "\\begin{align*} E ( \\boldsymbol { X } , \\boldsymbol { M } ) = ( \\boldsymbol { M } , \\boldsymbol { M } ) . \\end{align*}"}
{"id": "8317.png", "formula": "\\begin{align*} G ( t , x ) \\left ( e ^ { t } - 1 \\right ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\bar { P } _ n ( x ) } { n ! } t ^ n , \\end{align*}"}
{"id": "1596.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } = \\frac { f ' ( x ^ 1 ) } { \\sqrt { 1 + f '^ 2 ( x ^ 1 ) } } \\delta _ { \\epsilon 1 } \\left [ f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \\right ] , \\end{align*}"}
{"id": "4703.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ; q ) _ \\infty } \\sum _ { n = - k } ^ { k } ( - 1 ) ^ { n } \\ , q ^ { n ( 3 n - 1 ) / 2 } = 1 + ( - 1 ) ^ k \\frac { q ^ { k ( k + 1 ) / 2 } } { ( q ; q ) _ k } \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { ( n + k + 1 ) ( k + 1 ) } } { ( q ^ { n + k + 1 } ; q ) _ \\infty } . \\end{align*}"}
{"id": "6749.png", "formula": "\\begin{align*} L _ { \\chi } + L _ { \\chi ' } \\equiv L _ { \\chi \\chi ' } + \\sum \\limits _ { \\chi \\left ( \\sigma \\right ) = \\chi ' \\left ( \\sigma \\right ) = - 1 } { D _ { \\sigma } } \\end{align*}"}
{"id": "5864.png", "formula": "\\begin{align*} C _ 6 ^ { \\frac { p q } { p - q } } & \\lesssim \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k } \\sup _ { i \\leq k } 2 ^ { i } \\bigg ( \\int _ { x _ i } ^ { x _ { i + 1 } } u ( t ) \\bigg ( \\int _ t ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } d t \\bigg ) \\Phi ( 0 , x _ i ) \\\\ & + \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k } \\sup _ { i \\leq k } 2 ^ { i } \\sup _ { y \\in ( x _ i , x _ { i + 1 } ) } \\bigg ( \\int _ y ^ { x _ { i + 1 } } u ( t ) \\bigg ( \\int _ t ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } d t \\bigg ) \\Phi ( x _ i , y ) . \\end{align*}"}
{"id": "4073.png", "formula": "\\begin{align*} l e n g t h ( X ) \\geqslant n X = D [ i _ 0 , i _ 1 , \\ldots , i _ n , i _ { n + 1 } , \\ldots , i _ m ] i _ { n + 1 } , \\ldots , i _ m . \\end{align*}"}
{"id": "7374.png", "formula": "\\begin{align*} \\| \\phi _ n \\| = 1 . \\end{align*}"}
{"id": "3996.png", "formula": "\\begin{align*} \\textstyle \\omega = \\{ i \\ , | \\ , 1 \\le i \\le r , \\ , c _ i \\ne 0 \\} , ~ ~ ~ \\ell _ c = \\sum _ { i \\in \\omega } k _ i , ~ ~ ~ \\omega ' = \\{ 1 , \\cdots , r \\} - \\omega . \\end{align*}"}
{"id": "8698.png", "formula": "\\begin{align*} A _ { B H } = \\iint \\limits _ { R } ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } d x ^ 1 d x ^ 2 , \\end{align*}"}
{"id": "6447.png", "formula": "\\begin{align*} [ \\widetilde { X } , s ^ { - 1 } ( \\mathcal { F } ) ] & = [ \\widetilde { X } , \\Gamma ( \\ker \\dd s ) + \\Gamma ( \\ker \\dd t ) ] \\\\ & = [ \\widetilde { X } , \\Gamma ( \\ker ( \\dd s ) ] + [ \\widetilde { X } , \\Gamma ( \\ker \\dd t ) ] \\\\ & \\subset \\Gamma ( \\ker \\dd s ) + \\Gamma ( \\ker \\dd t ) = s ^ { - 1 } ( \\mathcal { F } ) . \\end{align*}"}
{"id": "3236.png", "formula": "\\begin{align*} \\lim _ { \\kappa \\to 0 } \\kappa ^ p \\nu _ p ( \\{ m _ f > \\kappa \\} ) = c _ { d , p } \\ , \\| \\nabla f \\| _ { L ^ p ( \\R ^ d ) } ^ p \\end{align*}"}
{"id": "1747.png", "formula": "\\begin{align*} B _ 0 ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) = & B _ 0 ( v , w , t ) = F ^ * ( v \\ , | \\ , w , - t ) , \\\\ D _ 0 ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) = & G ^ * ( v \\ , | \\ , w - t \\tau / 2 , w + t \\tau / 2 , - t ) , \\\\ B _ n ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) = & B _ 0 ( v + n w , w , t ) , \\\\ D _ n ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) = & D _ 0 ( v + n w - n t \\tau / 2 , w , t , q ^ { \\frac { 1 } { 2 } } ) \\cdot \\prod _ { k = 0 } ^ { n - 1 } B _ 0 ( v + n w + ( 1 - n + 2 k ) t \\tau / 2 , w + t \\tau / 2 , t ) , \\\\ & q ^ { \\frac { 1 } { 2 } } = ( \\pi i \\tau ) . \\end{align*}"}
{"id": "1901.png", "formula": "\\begin{align*} \\mathcal { V } _ { [ n , \\ell ] } = \\bigcup _ { k = \\ell } ^ { n - 1 } \\mathcal { V } _ { [ n , \\ell , k ] } \\end{align*}"}
{"id": "8926.png", "formula": "\\begin{align*} \\omega [ u ] ( \\Omega ^ \\prime , \\Omega ) = \\inf _ { x _ 0 \\in \\Omega ^ \\prime , R < R _ 0 } \\omega [ u ] ( x _ 0 , R ) \\end{align*}"}
{"id": "776.png", "formula": "\\begin{align*} \\beta | f ' ( z ^ m ) | & + ( 1 - \\beta ) | f ( z ^ m ) | + \\sum _ { n = 1 } ^ { \\infty } | a _ n | \\phi _ { n } ( r ) \\\\ & \\leq \\beta f ' _ { 0 } ( r ^ m ) + ( 1 - \\beta ) f _ { 0 } ( r ^ m ) + \\sum _ { n = 1 } ^ { \\infty } M ( n ) \\phi _ { n } ( r ) \\\\ & \\leq d ( 0 , \\partial { \\Omega } ) , \\end{align*}"}
{"id": "4868.png", "formula": "\\begin{align*} U _ { i } = \\left ( \\frac { N _ { i } } { 1 6 ( 1 + \\delta ) } \\right ) ^ { 1 / 3 } , V _ { i } = U _ { i } ^ { 5 / 6 } , L = \\log _ { 2 } \\left ( \\frac { N _ { 1 } } { \\log N _ { 1 } } \\right ) \\end{align*}"}
{"id": "7288.png", "formula": "\\begin{align*} - { a } _ 0 ^ { - 1 } = L _ 1 ^ { 1 - q } \\beta ( \\beta + n - 2 ) - q = \\tfrac { \\beta ( \\beta + n - 2 ) } { \\beta _ 0 ( \\beta _ 0 + n - 2 ) } - q > 1 - q . \\end{align*}"}
{"id": "8940.png", "formula": "\\begin{align*} g ( x , y , g ^ * ( x , y , u ) ) = u , \\end{align*}"}
{"id": "6801.png", "formula": "\\begin{align*} | | | u | | | ^ 2 : = \\| u \\| ^ 2 _ { L ^ 2 ( 0 , 1 ) } + \\| \\sqrt { a } u ^ { ( i ) } \\| ^ 2 _ { L ^ 2 ( 0 , 1 ) } , \\end{align*}"}
{"id": "5867.png", "formula": "\\begin{align*} C _ { 6 , 1 } & \\approx \\sum _ { k = - \\infty } ^ { M - 1 } \\bigg ( \\int _ { x _ k } ^ { \\infty } u \\bigg ) ^ { \\frac { p } { p - q } } \\big [ \\Phi ( 0 , x _ k ) - \\Phi ( 0 , x _ { k - 1 } ) \\big ] . \\end{align*}"}
{"id": "1519.png", "formula": "\\begin{align*} \\Big ( x \\frac { d } { d x } \\Big ) _ { p , \\lambda } \\Big ( \\frac { 1 } { 1 - x } \\Big ) ^ { r + 1 } & = \\sum _ { n = 0 } ^ { \\infty } \\binom { n + r } { n } \\Big ( x \\frac { d } { d x } \\Big ) _ { p , \\lambda } x ^ { n } = \\sum _ { n = 0 } ^ { \\infty } \\binom { n + r } { n } ( n ) _ { p , \\lambda } x ^ { n } . \\\\ \\end{align*}"}
{"id": "3027.png", "formula": "\\begin{align*} N _ { \\lambda } ( u , z , w ) = \\prod _ { x \\in \\lambda } ( z ^ { a ( x ) + 1 } - u w ^ { l ( x ) } ) ( z ^ { a ( x ) } - u ^ { - 1 } w ^ { l ( x ) + 1 } ) . \\end{align*}"}
{"id": "7789.png", "formula": "\\begin{align*} \\phi ( \\gamma _ 1 \\cup \\gamma _ 2 ) = \\phi ( \\gamma _ 1 ) \\star \\phi ( \\gamma _ 2 ) , \\end{align*}"}
{"id": "2604.png", "formula": "\\begin{align*} H = \\sum _ { i = 1 } ^ 3 \\left ( \\dfrac { 1 } { 2 m } P _ i ^ 2 + \\dfrac { 1 } { 2 } m \\omega ^ 2 X _ i ^ 2 \\right ) \\ , \\end{align*}"}
{"id": "665.png", "formula": "\\begin{align*} \\left . \\int u _ { 1 , t } d \\nu ^ 2 _ { t } \\ , \\right | _ { t = \\bar s } ^ { t = s ' } & = \\int _ { \\bar s } ^ { s ' } \\int _ M \\left ( \\Box _ { g _ { 2 , t } } u _ { 1 , t } \\right ) d \\nu ^ 2 _ { t } d t . \\end{align*}"}
{"id": "8416.png", "formula": "\\begin{gather*} W = \\bigoplus _ D W _ D \\end{gather*}"}
{"id": "3109.png", "formula": "\\begin{align*} a _ { \\mathrm { p w } } ( \\phi _ { \\mathrm { n c } } ( { j } ) , v _ { \\mathrm { n c } } ) = \\lambda _ h ( j ) b ( \\phi _ { \\mathrm { p w } } ( { j } ) , v _ { \\mathrm { n c } } ) \\phi _ { \\mathrm { p w } } ( { j } ) - \\phi _ { \\mathrm { n c } } ( { j } ) = \\lambda _ h ( { j } ) \\kappa _ { m } ^ 2 h _ { \\mathcal { T } } ^ { 2 m } \\phi _ { \\mathrm { p w } } ( { j } ) . \\end{align*}"}
{"id": "5700.png", "formula": "\\begin{align*} k ! e _ k ( x _ 1 , \\ldots , x _ i ) = \\varpi _ { i - k + 1 } \\varpi _ { i - k + 2 } \\cdots \\varpi _ { i } . \\end{align*}"}
{"id": "797.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ { \\infty } | b _ k | r ^ k & \\leq \\sum _ { n = N } ^ { \\infty } \\frac { | \\tilde { b } _ n | } { n } r ^ n = \\int _ { 0 } ^ { r } \\frac { M _ { t } ^ { N } ( \\tilde { G } ) } { t } d t \\\\ & \\leq \\int _ { 0 } ^ { r } \\frac { M _ { t } ^ { N } ( G ) . M _ { t } ^ { N } ( \\psi \\circ \\omega ) } { t } d t \\\\ & \\leq \\int _ { 0 } ^ { r } \\frac { M _ { t } ^ { N } ( \\psi ) t ^ { 2 N } } { t ^ 2 ( 1 - t ^ 2 ) } d t : = R ^ N ( r ) . \\end{align*}"}
{"id": "5389.png", "formula": "\\begin{align*} P = \\sum _ { \\abs { \\alpha } \\leq m } ( a _ \\alpha + b _ \\alpha ) D ^ \\alpha , a _ { \\alpha } \\in M _ 0 ( H ^ { s - | \\alpha | } \\rightarrow H ^ { - s } ) , P _ s \\vcentcolon = \\sum _ { \\abs { \\alpha } \\leq m } b _ \\alpha D ^ \\alpha \\in \\mathcal { P } _ { m , s } ( \\Omega ) . \\end{align*}"}
{"id": "964.png", "formula": "\\begin{align*} V = V _ { \\phi } = \\left [ \\theta _ { j } ^ { i } \\right ] _ { 0 \\leq i , j \\leq n - 1 } , \\ \\ \\ \\ \\Delta = ( V _ { \\phi } ) \\neq 0 . \\end{align*}"}
{"id": "8531.png", "formula": "\\begin{align*} \\Delta ( 2 i , m ) & = ( 4 i - 1 ) ( 4 z + 1 ) \\Delta ( 2 i - 1 , m ) - ( 2 i - 1 ) ^ 2 \\Delta ( 2 i - 2 , m ) , \\\\ \\Delta ( 2 i + 1 , m ) & = ( 4 i + 1 ) \\Delta ( 2 i , m ) - ( 2 i ) ^ 2 \\Delta ( 2 i - 1 , m ) . \\end{align*}"}
{"id": "1475.png", "formula": "\\begin{align*} { { \\mathcal { L } } } _ m ( { { \\mathfrak { A } } } ( \\boldsymbol { t } ) ) = E \\cdot { \\rm { d e t } } \\left ( \\varphi _ { \\zeta _ j , s _ j } ( t ^ { u + \\ell } ( t - 1 ) ^ { r n } ) \\right ) _ { \\substack { 0 \\le \\ell \\le r - 1 \\\\ 1 \\le j \\le d , 1 \\le s _ j \\le r _ j } } \\enspace . \\end{align*}"}
{"id": "3651.png", "formula": "\\begin{gather*} u ( \\{ x _ 0 \\} ) = \\lim _ { \\ell \\to \\infty } \\int _ { B _ { r _ \\ell } ( x _ 0 ) } u _ { n _ \\ell } , e ( \\{ x _ 0 \\} ) = \\lim _ { \\ell \\to \\infty } e _ { n _ \\ell } ( B _ { R _ \\ell } ( x _ 0 ) ) , \\\\ \\shortintertext { a n d } \\limsup _ { \\ell \\to \\infty } \\frac { e ( B _ { R _ \\ell } ( x _ 0 ) ) } { u ( B _ { r _ \\ell } ( x _ 0 ) ) } = \\lim _ { \\ell \\to \\infty } \\frac { e _ { n _ \\ell } ( B _ { R _ \\ell } ( x _ 0 ) ) } { \\int _ { B _ { r _ \\ell } ( x _ 0 ) } u _ { n _ \\ell } } . \\end{gather*}"}
{"id": "5264.png", "formula": "\\begin{align*} \\left \\langle \\tau _ 0 ^ { ( a , b ) } \\prod _ { i \\in I \\setminus B } \\tau _ { d _ i } ^ { ( a _ i , b _ i ) } \\right \\rangle ^ { } \\not = 0 . \\end{align*}"}
{"id": "1193.png", "formula": "\\begin{align*} \\varphi ( x ) - \\varphi ( y ) = \\varphi ( x ) \\le ( \\| \\varphi \\| _ { \\infty } \\vee C _ 1 C _ 3 ^ { 2 - p } ) | x - y | ^ p , \\ \\forall x \\in S , y \\in S ^ c \\end{align*}"}
{"id": "2208.png", "formula": "\\begin{align*} \\varrho ( r ) = \\int _ 0 ^ r \\frac { 1 } { \\omega ( \\rho ) } d \\rho . \\end{align*}"}
{"id": "8462.png", "formula": "\\begin{align*} \\Lambda _ { d , b , A } \\triangleq \\prod _ { i = 1 } ^ d \\left [ \\frac { A _ i - 1 } { b } , \\frac { A _ i } { b } \\right ) . \\end{align*}"}
{"id": "3388.png", "formula": "\\begin{align*} \\partial _ x g ( x , y ) = 1 - \\gamma \\frac { b ' ( x ) } { b ( x ) ^ 2 } \\| y \\| _ d ^ 2 , \\nabla _ y g ( x , y ) = 2 \\gamma \\frac { y } { b ( x ) } , x > 1 . \\end{align*}"}
{"id": "1053.png", "formula": "\\begin{align*} \\tilde { B } _ 0 T ( t , A + B ) = \\widetilde { B _ 0 } S _ 0 ( t ) + \\cdots + \\widetilde { B _ 0 } S _ { \\ell } ( t ) + W ( t ) , \\end{align*}"}
{"id": "4193.png", "formula": "\\begin{align*} g _ { \\le \\iota } = \\sum _ { \\ell = - 1 } ^ \\iota \\sum _ { m = 1 } ^ { M _ { \\ell } } \\chi _ { m } ^ { ( \\ell ) } g _ { m } ^ { ( \\ell ) } + \\sum _ { \\ell = - 1 } ^ \\iota \\sum _ { m = 1 } ^ { M _ { \\ell } } ( 1 - \\chi _ { m } ^ { ( \\ell ) } ) g _ { m } ^ { ( \\ell ) } = : g _ { \\le \\iota } ^ { ( 1 ) } + g _ { \\le \\iota } ^ { ( 2 ) } . \\end{align*}"}
{"id": "1608.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } = 2 b ^ 2 \\delta _ { \\epsilon 1 } f ( x ^ 1 ) f ' ( x ^ 1 ) . \\end{align*}"}
{"id": "8579.png", "formula": "\\begin{align*} m = n _ { j _ 1 } + \\ldots + n _ { j _ s } - n _ { k _ 1 } - \\ldots - n _ { k _ t } , \\end{align*}"}
{"id": "994.png", "formula": "\\begin{align*} S _ { \\alpha , \\beta } = \\left \\{ \\left ( x _ { 1 } , x _ { 2 } , \\cdots , x _ { n } \\right ) \\in \\mathbb { Z } ^ { n } \\mid x _ { i } \\in \\mathbb { Z } \\ \\ \\ \\ 0 \\leq x _ { i } < b _ { i } \\right \\} . \\end{align*}"}
{"id": "1609.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { \\partial ^ 2 C ^ 2 } { \\partial z ^ i _ { \\epsilon } \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } v ^ i = f ( x ^ 1 ) \\left [ - f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \\right ] , \\end{align*}"}
{"id": "6084.png", "formula": "\\begin{align*} f ( x ) = ( x - 1 ) ^ 4 + c ( x - 1 ) ^ 2 + d ( x - 1 ) \\end{align*}"}
{"id": "3227.png", "formula": "\\begin{align*} \\dim ( h ^ { - 1 } ( \\mathcal { D } _ i ) \\setminus ( T ^ * \\mathcal { U } \\cap h ^ { - 1 } ( \\mathcal { D } _ i ) ) ) & = \\dim ( ( \\mathcal { Z } \\cap h ^ { - 1 } ( \\mathcal { D } _ i ) ) \\cup ( T ^ * \\mathcal { W } \\cap h ^ { - 1 } ( \\mathcal { D } _ i ) ) ) \\\\ & \\leq 2 N - 3 \\\\ & = \\dim h ^ { - 1 } ( \\mathcal { D } _ i ) - 2 \\end{align*}"}
{"id": "4906.png", "formula": "\\begin{align*} \\mathrm { q a d j } ( \\lambda _ i ( A ) E _ n - A ) = \\prod _ { k = 1 ; k \\ne i } ^ n ( \\lambda _ i ( A ) - \\lambda _ k ( A ) ) v _ i v _ i ^ * . \\end{align*}"}
{"id": "4743.png", "formula": "\\begin{align*} \\widehat { \\lambda } ( R + Q ) + \\widehat { \\lambda } ( R - Q ) = 2 \\widehat { \\lambda } ( R ) + 2 \\widehat { \\lambda } ( Q ) + v ( x ( R ) - x ( Q ) ) - v ( \\Delta ) / 6 \\end{align*}"}
{"id": "1423.png", "formula": "\\begin{align*} x _ i = ( \\phi _ { 0 } ) ^ { d - 1 } \\sum _ { j \\in S _ N ^ 1 \\setminus \\{ 0 \\} } \\phi _ { j } ( x _ i ) \\alpha _ { j } ^ { ( i ) } , \\end{align*}"}
{"id": "6761.png", "formula": "\\begin{align*} k \\cdot \\gcd \\left ( C , C _ 1 , C _ 2 , \\dots , C _ m \\right ) & = \\gcd \\left ( k C , k C _ 1 , k C _ 2 , \\dots , k C _ m \\right ) \\\\ & = \\gcd \\left ( k C , j _ 1 C , j _ 2 C , \\dots , j _ m C \\right ) \\\\ & = C \\gcd ( k , j _ 1 , j _ 2 , \\dots , j _ m ) . \\end{align*}"}
{"id": "7612.png", "formula": "\\begin{align*} \\overline { p } ( n ) = \\frac { 1 } { 2 \\pi } \\underset { 2 \\nmid k } { \\sum _ { k = 1 } ^ { N } } \\sqrt { k } \\underset { ( h , k ) = 1 } { \\sum _ { h = 0 } ^ { k - 1 } } \\dfrac { \\omega ( h , k ) ^ 2 } { \\omega ( 2 h , k ) } e ^ { - \\frac { 2 \\pi i n h } { k } } \\dfrac { d } { d n } \\biggl ( \\dfrac { \\sinh \\frac { \\pi \\sqrt { n } } { k } } { \\sqrt { n } } \\biggr ) + R _ { 2 } ( n , N ) , \\end{align*}"}
{"id": "6215.png", "formula": "\\begin{align*} D _ q h ( x ) = u ( x ) h ( x ) ( x \\in I ) . \\end{align*}"}
{"id": "7933.png", "formula": "\\begin{align*} S _ \\pm = \\left \\{ i \\in \\{ 1 , \\ldots , m \\} | \\overline { \\{ i \\} } \\not \\in \\mathcal A _ { \\pm } \\} \\right \\} . \\end{align*}"}
{"id": "7677.png", "formula": "\\begin{align*} \\Phi ( x , y , Z ) : = r _ { \\mathrm { V } } ( x ) + \\kappa r _ { \\mathrm { O } } ( x , y , Z ) , \\Psi ( x , y , Z ) : = \\kappa r _ { \\mathrm { V } } ( x ) + r _ { \\mathrm { O } } ( x , y , Z ) , \\end{align*}"}
{"id": "6976.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } \\frac { a _ k } { \\lambda _ k ^ 4 } = \\infty . \\end{align*}"}
{"id": "7808.png", "formula": "\\begin{align*} \\widetilde { H } ( \\varphi ^ { [ n ] } ) = \\det ( \\varphi ) ^ { n + 1 } \\tilde { \\iota } _ \\varphi . \\end{align*}"}
{"id": "7074.png", "formula": "\\begin{align*} W _ 7 ( v _ 7 ) = - \\frac { 8 0 3 3 9 } { 3 1 1 0 4 } v _ 7 ^ 3 + O ( v _ 7 ^ 4 ) . \\end{align*}"}
{"id": "2770.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { W } ( x _ i ) = f _ i \\nabla \\hat { W } ( x _ i ) = g _ i . \\end{aligned} \\end{align*}"}
{"id": "8831.png", "formula": "\\begin{align*} \\| f _ t \\circ ( \\phi _ 1 * \\phi _ 2 ) \\| \\leq ( \\| \\phi _ 1 \\| - t ) ( \\| \\phi _ 2 \\| - t ) = \\| f _ t \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) \\| \\end{align*}"}
{"id": "4603.png", "formula": "\\begin{align*} D ^ k _ { X ^ \\pm } ( n , m ) = \\begin{cases} C ^ k _ { X ^ \\pm } ( n , m ) , & \\ X = A , C , \\\\ C ^ k _ { X ^ \\pm } ( n , m ) ^ { \\Z _ 2 } , & \\ X = B , D , O . \\end{cases} \\end{align*}"}
{"id": "6177.png", "formula": "\\begin{align*} i _ { \\frac { \\partial } { \\partial r } } \\overline { \\omega } ^ { n + 1 } = ( \\Pi ^ { \\ast } \\omega _ { h } ) ^ { n } \\wedge \\eta . \\end{align*}"}
{"id": "1066.png", "formula": "\\begin{align*} B R ( w , A ) = B R ( z , A ) ( I + ( z - w ) R ( w , A ) ) , \\end{align*}"}
{"id": "6853.png", "formula": "\\begin{align*} N ( b , c ) = \\frac { 1 } { V _ F } \\int _ { 0 } ^ { + \\infty } g \\frac { 1 } { \\sqrt { 2 \\pi c } } \\exp \\left ( - \\frac { ( g - b ) ^ 2 } { 2 c } \\right ) d g \\geq 0 , b \\in \\mathbb { R } , c > 0 . \\end{align*}"}
{"id": "6222.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } u ( x ) + \\frac { 1 } { q } u ( x ) u ( x / q ) + A ( x ) u ( x / q ) + r ( x ) = 0 , \\end{align*}"}
{"id": "7903.png", "formula": "\\begin{align*} \\nabla _ { z \\partial z } L ( t , z ) \\alpha = L ( t , z ) ( \\mu \\alpha - \\frac { \\rho } { z } \\alpha ) , \\end{align*}"}
{"id": "7057.png", "formula": "\\begin{align*} \\sum _ { M \\geq 1 } g _ r ^ { ( d ) } ( M ) z ^ M & = z ^ d \\sum _ { s \\geq 0 } z ^ { s } \\sum _ { n \\geq 0 } \\binom { n + s } { n } ( z ^ r ) ^ n = z ^ d \\sum _ { s \\geq 0 } \\frac { z ^ s } { ( 1 - z ^ r ) ^ { s + 1 } } \\\\ & = \\frac { z ^ d } { 1 - z ^ r } \\cdot \\frac { 1 } { 1 - \\frac { z } { 1 - z ^ r } } = \\frac { z ^ d } { 1 - z - z ^ r } . \\end{align*}"}
{"id": "855.png", "formula": "\\begin{align*} \\nu _ { \\eta _ { m } ^ { r } \\rightarrow s _ { m } } ( s _ { m } ) = & \\underset { \\boldsymbol { s ^ { r } } } { \\sum } \\eta _ { m } ^ { r } ( s _ { m } ^ { r } , s _ { m } ) \\times \\nu _ { s _ { m } ^ { r } \\rightarrow \\eta _ { m } ^ { r } } ( s _ { m } ^ { r } ) \\\\ = & \\pi _ { s , m } ^ { r , i n } \\delta ( s _ { m } - 1 ) + ( 1 - \\pi _ { s , m } ^ { r , i n } ) \\delta ( s _ { m } ) , \\end{align*}"}
{"id": "2236.png", "formula": "\\begin{align*} z _ s = e ^ { - \\pi t + \\pi \\sigma i } , \\cos \\frac { \\pi s } { 2 } = \\frac { e ^ { \\frac { \\pi } { 2 } \\left ( t - \\sigma i \\right ) } } { 2 } \\left ( 1 + z _ s \\right ) , \\quad \\sin \\frac { \\pi s } { 2 } = \\frac { e ^ { \\frac { \\pi } { 2 } \\left ( t - \\sigma i \\right ) } i } { 2 } \\left ( 1 - z _ s \\right ) . \\end{align*}"}
{"id": "2628.png", "formula": "\\begin{align*} \\max \\{ | S | : S \\in \\bigcap _ { i = 1 } ^ m \\mathcal { F } _ i \\} & = \\min \\{ r _ 1 ( X ) + r _ 2 ( X _ 1 \\backslash X ) + \\dots + r _ { m - 1 } ( X _ { m - 2 } \\backslash X _ { m - 3 } ) + r _ m ( E \\backslash X _ { m - 2 } ) : \\\\ & X \\subseteq X _ 1 \\subseteq X _ 2 \\subseteq \\dots \\subseteq X _ { m - 2 } \\subseteq E \\} \\end{align*}"}
{"id": "7633.png", "formula": "\\begin{align*} { \\sigma _ { i } ^ 2 = ( u - u ^ { - 1 } ) \\sigma _ { i } + 1 } . \\end{align*}"}
{"id": "4572.png", "formula": "\\begin{align*} \\phi \\cdot \\psi = \\phi \\circ F _ * ^ e \\psi . \\end{align*}"}
{"id": "6908.png", "formula": "\\begin{align*} e ^ { - H _ t ( \\xi ) } = A ( t ) e ^ { - \\frac { 1 } { 2 } C ( t ) \\xi ^ 2 - \\mu ( 2 C _ 0 ( t ) + i B ( t ) ) \\xi } , \\end{align*}"}
{"id": "6011.png", "formula": "\\begin{align*} M _ f ( \\lambda ) = ( \\mu _ 0 , \\mu _ 1 , \\ldots , \\mu _ n ) \\end{align*}"}
{"id": "5516.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ \\gamma \\frac { h ' ( z ) } { h ( z ) } d z & = \\int _ 0 ^ 1 g ' ( t + i R ) d t = 1 . \\end{align*}"}
{"id": "4043.png", "formula": "\\begin{align*} \\Sigma = I _ { n } . \\end{align*}"}
{"id": "6963.png", "formula": "\\begin{align*} \\Phi ( z ) : = \\prod _ { k \\ge 0 } \\left ( \\frac { z - \\mu _ k ^ 2 } { z - \\lambda _ k ^ 2 } \\right ) \\ , . \\end{align*}"}
{"id": "3390.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\sup _ { y : \\| y \\| _ d \\leq b ( x ) } \\bigl | b ( x ) G ( x , y ) - \\gamma \\sigma ^ 2 \\bigr | = 0 . \\end{align*}"}
{"id": "2214.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial } { \\partial t } \\omega ( t ) = - \\mathrm { R i c } _ { \\omega ( t ) } ^ { T } , \\omega ( 0 ) = \\omega _ { 0 } . \\end{array} \\end{align*}"}
{"id": "2346.png", "formula": "\\begin{align*} a _ i Q ^ i = \\sum _ { j = 0 } ^ i { i \\choose j } a _ i h ^ j Q '^ { i - j } . \\end{align*}"}
{"id": "6126.png", "formula": "\\begin{align*} | H _ i \\cap H _ j | & \\le r _ i + r _ j - r \\ \\tag * { ( H 1 ) } \\\\ | S - H _ i | + r _ i & \\ge r \\ \\tag * { ( H 2 ) } \\end{align*}"}
{"id": "8797.png", "formula": "\\begin{align*} 1 _ { \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) } ^ * = 1 _ { ( \\phi ^ * ) ^ { - 1 } ( \\mathbb { R } _ { > t } ) } = 1 _ { ( - \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) / 2 , \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) / 2 ) } \\end{align*}"}
{"id": "7556.png", "formula": "\\begin{align*} \\| S ^ m ( f _ 1 , \\dots , f _ m ) \\| _ { L ^ p ( \\R ) } \\leq C \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { p _ i } ( \\R ) } \\end{align*}"}
{"id": "7928.png", "formula": "\\begin{align*} \\mathcal A _ 0 : = \\mathcal A _ { \\omega _ 0 } = \\{ I \\subset \\{ 1 , \\ldots , m \\} : \\omega _ 0 \\in \\angle _ I \\} . \\end{align*}"}
{"id": "7652.png", "formula": "\\begin{align*} M _ { u _ 1 } ^ t \\cdot a _ 2 = M _ { u _ 2 } ^ t \\cdot a _ 1 = 0 \\ ; . \\end{align*}"}
{"id": "318.png", "formula": "\\begin{align*} C _ 1 : = \\sum _ { 2 < p \\le 2 3 } b _ p f ( p ) = \\sum _ { 2 < p \\le 2 3 } \\frac { \\pi } { 4 } \\frac { M } { m _ p ^ 2 } e ^ \\gamma { \\rm d } ( { \\rm L } _ p ) = \\frac { \\pi } { 4 } \\ , M e ^ \\gamma \\cdot 0 . 3 9 0 1 2 \\cdots = 0 . 5 4 6 3 \\cdots \\end{align*}"}
{"id": "4792.png", "formula": "\\begin{align*} \\partial _ { t } \\mathcal { P } m - \\bar { \\mu } \\Delta \\mathcal { P } m = \\mathcal { P } g . \\end{align*}"}
{"id": "8067.png", "formula": "\\begin{align*} g ( \\operatorname { g r a d } ( \\| Z \\| ^ { 2 } ) + 2 \\nabla _ { Z } Z , 2 \\nabla _ { Z } Z ) = - f ^ { 2 } \\| X \\| ^ { 2 } g ( \\operatorname { g r a d } ( f ^ { 2 } ) , \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) ) . \\end{align*}"}
{"id": "3686.png", "formula": "\\begin{align*} g = \\frac { d r ^ 2 } { \\alpha ( r ) } + r ^ 2 \\cdot h _ { \\mathbb { S } ^ { n - 2 } } + e ^ { 2 \\beta ( s , r ) } \\cdot d s ^ 2 . \\end{align*}"}
{"id": "4734.png", "formula": "\\begin{align*} E _ { i } \\cdot v _ r & = \\delta _ { r , i + 1 } v _ { r - 1 } , \\\\ F _ { i } \\cdot v _ r & = \\delta _ { r , i } v _ { r + 1 } , \\\\ K _ i \\cdot v _ r & = \\begin{cases} q v _ { i } & \\hbox { i f } r = i , \\\\ q ^ { - 1 } v _ { i + 1 } & \\hbox { i f } r = i + 1 , \\\\ v _ { r } & \\hbox { e l s e } . \\end{cases} \\end{align*}"}
{"id": "8554.png", "formula": "\\begin{align*} \\Phi ( x ) = \\int _ 0 ^ x \\phi ( u ) \\ d u \\ , x \\geq 0 \\ , \\end{align*}"}
{"id": "6692.png", "formula": "\\begin{align*} S = \\{ A > 0 : T ( A w _ 0 ) = \\infty , \\ h _ { \\infty } ( A w _ 0 ) + g _ { \\infty } ( A w _ 0 ) < \\infty \\ \\ u ( A w _ 0 , \\cdot ) , v ( ( A w _ 0 , \\cdot ) \\ \\} . \\end{align*}"}
{"id": "8366.png", "formula": "\\begin{align*} d \\mu ( \\nu ) = { \\textstyle \\frac { 1 } { 2 \\pi ^ 2 } } \\nu ^ 2 \\ , K ( \\nu ; z ) d \\nu \\end{align*}"}
{"id": "8057.png", "formula": "\\begin{align*} 0 = \\operatorname { d i v } ( X _ { 1 } ) = \\operatorname { d i v } ( h X _ { 2 } ) , \\end{align*}"}
{"id": "4986.png", "formula": "\\begin{align*} \\kappa _ 2 ^ 2 \\sum _ { i = 1 } ^ Q \\rho _ i \\int ^ { \\tau \\wedge t _ i } _ 0 ( t _ i - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ( X _ s ) d B _ s , \\end{align*}"}
{"id": "8768.png", "formula": "\\begin{align*} Y _ O ( P , G ) \\leq C ( P ) , Y _ R ( P , G ) \\geq C ( P ) , C ( P ) : = \\frac { B ( p _ 1 ) B ( p _ 2 ) } { B \\circ q ( p _ 1 , p _ 2 ) } . \\end{align*}"}
{"id": "8139.png", "formula": "\\begin{align*} \\| V e ^ { - i t H _ 0 } \\psi \\| & = \\| V \\chi _ { S _ { r _ 0 } } e ^ { - i t H _ 0 } \\psi \\| \\leq M \\| \\chi _ { S _ { r _ 0 } } e ^ { - i t H _ 0 } \\psi \\| = M \\| e ^ { - i t H _ 0 ^ \\parallel } \\psi ^ \\parallel \\| \\| \\chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \\perp } \\psi ^ \\perp \\| \\\\ & = M \\| \\psi ^ \\parallel \\| \\| \\chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \\perp } \\psi ^ \\perp \\| \\end{align*}"}
{"id": "3911.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int _ 0 ^ t b ( X _ s ) d s + \\int _ 0 ^ t a ( X _ s ) d W _ s , t \\in [ 0 , T ] , \\end{align*}"}
{"id": "8972.png", "formula": "\\begin{align*} f _ b ( x ) = a x ^ q + b x ^ r , \\end{align*}"}
{"id": "3629.png", "formula": "\\begin{align*} f ' _ - ( x _ 0 , 0 ^ \\pm , 0 ) \\coloneqq \\displaystyle \\liminf _ { ( x , u , \\xi ) \\to ( x _ 0 , 0 ^ \\pm , 0 ) } \\frac { f ( x , u , \\xi ) } { \\abs { u } } \\geq \\limsup _ { u \\to 0 ^ \\pm } \\sup _ { \\abs { \\xi } = 1 } \\frac { f ( x _ 0 , u , \\rho ( \\abs { u } ) \\xi ) } { \\abs { u } } , \\end{align*}"}
{"id": "2563.png", "formula": "\\begin{align*} \\begin{aligned} Q ( p , q ) \\otimes Q ( q , p ) = & \\ , \\bigoplus _ { n = 0 } ^ { p } \\bigoplus _ { m = 0 } ^ { q } \\Bigg \\{ Q ( p + q - n - m , p + q - n - m ) \\\\ & \\ \\ \\oplus \\Bigg [ \\bigoplus _ { k = 1 } ^ { \\min ( p - n , q - m ) } \\Bigg ( Q ( p + q - n - m - 2 k , p + q - n - m + k ) \\\\ & \\ \\ \\oplus Q ( p + q - n - m + k , p + q - n - m - 2 k ) \\Bigg ) \\Bigg ] \\Bigg \\} \\ . \\end{aligned} \\end{align*}"}
{"id": "8945.png", "formula": "\\begin{align*} F [ u ] : = \\log \\det [ D ^ 2 u - A ( \\cdot , u , D u ) ] = [ \\tau ( 1 - t ) + \\epsilon ] ( u - u _ 0 ) + \\log B _ t ( \\cdot , u , D u ) , \\end{align*}"}
{"id": "6156.png", "formula": "\\begin{align*} \\begin{aligned} | R _ { n l } | & \\leq \\| n l _ h ( u ^ { \\Delta t } _ { h } , u ^ { \\Delta t } _ { h } ) \\| _ { H ^ { - 1 } } \\| P _ h ( u ^ { \\Delta t } _ { h } \\phi ) - u ^ { \\Delta t } _ { h } \\phi \\| _ { H ^ 1 } \\leq c \\ , \\sqrt { h } \\| u ^ { \\Delta t } _ { h } \\| _ { 2 } \\ , \\| u ^ { \\Delta t } _ { h } \\| _ { H ^ 1 } ^ { 2 } , \\end{aligned} \\end{align*}"}
{"id": "2179.png", "formula": "\\begin{align*} H ( z ) = z + ( \\varphi _ { \\mu } \\circ F _ { \\nu } ) ( z ) = z + \\gamma _ { \\mu } + \\int _ { \\mathbb { R } } \\frac { 1 + F _ { \\nu } ( z ) s } { F _ { \\nu } ( z ) - s } \\ , d \\sigma _ { \\mu } ( s ) \\end{align*}"}
{"id": "7112.png", "formula": "\\begin{align*} \\boldsymbol { R } ( \\tau ; t , x ) \\triangleq \\begin{cases} \\boldsymbol { R } _ 1 ( \\tau ; t , x ) & h ( \\tau , p ( \\tau ; t , x ) ) > 0 \\\\ \\tau & h ( \\tau , p ( \\tau ; t , x ) ) \\leq 0 \\end{cases} , \\end{align*}"}
{"id": "5951.png", "formula": "\\begin{align*} { \\rm { \\Delta } } _ { i j . . . k p } ^ { \\left ( n \\right ) } = w _ { i j . . . k p } ^ { \\left ( n \\right ) } \\delta \\left ( { { t _ 1 } - { t _ 2 } } \\right ) . . . \\delta \\left ( { { t _ 1 } - { t _ n } } \\right ) \\ , \\end{align*}"}
{"id": "5029.png", "formula": "\\begin{align*} Q ^ { n . 4 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( s - \\eta _ n ( s ) ) ^ { 2 \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) \\ , \\left [ \\left ( W _ s - W _ { \\eta _ n ( s ) } \\right ) ^ 2 - ( s - \\eta _ n ( s ) ) \\right ] \\ , d s . \\end{align*}"}
{"id": "2265.png", "formula": "\\begin{align*} I _ 2 = \\int _ 0 ^ r \\left [ - \\frac { z } { 2 } ( \\bar \\psi \\bar \\psi ' \\cdot \\phi + \\bar \\psi \\bar \\phi ' \\cdot \\psi ) \\right ] { \\rm { d } } z \\leq \\Vert ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert ( \\bar \\phi ' , \\bar \\psi ' ) \\Vert _ { L ^ 2 } \\cdot \\frac { 1 } { 2 } \\Vert z \\bar \\psi \\Vert _ { L ^ \\infty } , \\end{align*}"}
{"id": "29.png", "formula": "\\begin{align*} \\psi _ { L , n } ( \\beta ) = \\eta ^ n y \\mod \\eta ^ n \\mathfrak { X } _ L . \\end{align*}"}
{"id": "3180.png", "formula": "\\begin{align*} \\mathrm { P } _ { \\Phi + \\Phi ^ { \\mathfrak { b } , \\gamma } } = - \\inf _ { \\rho \\in E _ { 1 } } \\left \\{ f _ { \\Phi } \\left ( \\rho \\right ) + e _ { \\Phi ^ { \\mathfrak { b } , \\gamma } } \\left ( \\rho \\right ) \\right \\} \\ , \\end{align*}"}
{"id": "1431.png", "formula": "\\begin{align*} \\varphi _ { \\beta , a } ( P ) = \\frac { 1 } { P ^ { a + \\beta } } \\int _ { 0 } ^ { P } u ^ \\beta \\kappa ( u ) d u , \\end{align*}"}
{"id": "5892.png", "formula": "\\begin{align*} \\nu = \\mathcal { D } \\Big ( \\Big ( \\frac { 1 } { 1 + \\alpha } \\Big ) ^ { 1 / p } \\sum _ { j = 1 } ^ \\infty A _ { \\bullet , j } Z _ j + \\sigma _ { p , \\alpha } ( I _ k - A A ^ T ) ^ { 1 / 2 } N _ k \\Big ) \\end{align*}"}
{"id": "1020.png", "formula": "\\begin{align*} \\underline d ( S ) & = \\liminf _ { n \\to \\infty } \\frac { \\# ( S \\cap \\{ 1 , \\dots , n \\} ) } { n } , \\\\ \\overline d ( S ) & = \\limsup _ { n \\to \\infty } \\frac { \\# ( S \\cap \\{ 1 , \\dots , n \\} ) } { n } . \\end{align*}"}
{"id": "468.png", "formula": "\\begin{align*} \\bar F ( x ) = P ( B > x ) = L ( x ) x ^ { - \\alpha } . \\end{align*}"}
{"id": "4560.png", "formula": "\\begin{align*} \\mathcal { T } ( \\Lambda ( G ) ) : = \\ , \\Lambda \\cup \\{ P _ i [ 1 ] : i \\in Q _ 0 \\} \\ , . \\end{align*}"}
{"id": "2505.png", "formula": "\\begin{align*} A _ Y X = - A _ X Y - \\lambda ^ 2 g ( X , Y ) \\left ( \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
{"id": "5359.png", "formula": "\\begin{align*} s _ { \\theta } = ( 1 - \\theta ) s _ 0 + \\theta s _ 1 = \\frac { t - r } { t - z } z + \\frac { r - z } { t - z } t = r . \\end{align*}"}
{"id": "6668.png", "formula": "\\begin{align*} \\dfrac { \\partial i } { \\partial r } = \\dfrac { 1 } { 1 + \\xi _ 2 ' ( i ) ( g ( t ) - g _ 0 ) } : = \\sqrt { A _ 2 ( g ( t ) , i ) } , \\\\ - \\dfrac { 1 } { g ' ( t ) } \\dfrac { \\partial i } { \\partial t } = \\dfrac { \\xi _ 2 ( i ) } { 1 + \\xi _ 2 ' ( i ) ( g ( t ) - g _ 0 ) } : = C _ 2 ( g ( t ) , i ) , \\\\ \\dfrac { \\partial ^ 2 i } { \\partial r ^ 2 } = - \\dfrac { \\xi _ 2 '' ( i ) ( g ( t ) - g _ 0 ) } { [ 1 + \\xi _ 2 ' ( i ) ( g ( t ) - g _ 0 ) ] ^ 3 } : = B _ 2 ( g ( t ) , i ) , \\end{align*}"}
{"id": "1696.png", "formula": "\\begin{align*} z ( x y ) = ( - 1 ) ^ { \\vert y \\vert ( \\vert x \\vert + \\vert z \\vert ) } ( y z ) x . \\end{align*}"}
{"id": "5849.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } \\bigg ( \\sum _ { i = - \\infty } ^ { k } 2 ^ { - i \\frac { r } { p } } V _ r ( x _ { i - 1 } , x _ i ) ^ r a _ i ^ r \\bigg ) ^ { \\frac { q } { r } } \\int _ { x _ k } ^ { x _ { k + 1 } } u \\bigg ) ^ { \\frac { 1 } { q } } \\leq C ' \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } a _ k ^ p \\bigg ) ^ { \\frac { 1 } { p } } \\end{align*}"}
{"id": "4520.png", "formula": "\\begin{align*} \\partial _ t g _ t ( \\partial _ i , \\partial _ j ) = & 2 \\bar { \\nabla } ^ 2 _ { i j } t , \\\\ \\bar { g } | _ { t = 0 } = & g , \\\\ \\partial _ t ( \\bar { \\nabla } ^ 2 _ { i j } t ) - ( \\bar { \\nabla } ^ 2 t ) ^ 2 _ { i j } = & 0 , \\\\ \\bar { \\nabla } ^ 2 _ { i j } t | _ { t = 0 } = & k _ { i j } \\end{align*}"}
{"id": "1281.png", "formula": "\\begin{align*} H _ 0 ( x ) & = 1 , \\\\ H _ 1 ( x ) & = 2 x , \\\\ H _ j ( x ) & = 2 x H _ { j - 1 } ( x ) - 2 ( j - 1 ) H _ { j - 2 } ( x ) ( \\forall j \\ge 2 ) . \\end{align*}"}
{"id": "4122.png", "formula": "\\begin{align*} d K ^ { \\ell + 1 } - K ^ { \\ell + 1 } d & = d K ^ { \\ell } K - K ^ { \\ell + 1 } d = K ^ \\ell d K + \\ell S K ^ \\ell - K ^ { \\ell + 1 } d \\\\ & = K ^ { \\ell } S + \\ell S K ^ { \\ell } = ( \\ell + 1 ) S K ^ \\ell , \\end{align*}"}
{"id": "2384.png", "formula": "\\begin{align*} \\pi ( x ) C _ \\pi \\pi ( x ) ^ * = \\Delta _ G ( x ) ^ { 1 / 2 } C _ \\pi , x \\in G , \\end{align*}"}
{"id": "2249.png", "formula": "\\begin{align*} & ( u , v , h , g ) | _ { y = 0 } = ( u , v , h , g ) | _ { y \\rightarrow \\infty } = ( 0 , 0 , 0 , 0 ) , \\\\ & ( u , v , h , g ) | _ { x = 1 } = ( u , v , h , g ) | _ { x \\rightarrow \\infty } = ( 0 , 0 , 0 , 0 ) . \\end{align*}"}
{"id": "8354.png", "formula": "\\begin{align*} \\widetilde { f } ^ { | u \\rangle } _ { \\mu } ( p ) = \\frac { u _ \\mu } { u \\cdot p } \\end{align*}"}
{"id": "3006.png", "formula": "\\begin{align*} \\delta ( f ^ t Q ^ { ( t ) } ) = \\delta _ { t G } ( f ^ t ) + \\delta _ { - t G } ( Q ^ { ( t ) } ) \\le \\delta _ { - t G } ( Q ^ { ( t ) } ) \\leq \\delta _ G ( Q ) \\ , \\end{align*}"}
{"id": "7138.png", "formula": "\\begin{align*} ( p x _ 1 + q x _ 2 + r x _ 3 ) ^ { m } = \\alpha x _ 2 ( x _ 1 + x _ 2 ) ^ { { m } _ 2 - 1 } + \\beta x _ 3 ( x _ 1 + x _ 2 + x _ 3 ) ^ { { m } _ 3 - 1 } \\end{align*}"}
{"id": "3231.png", "formula": "\\begin{align*} \\mathcal { A } _ { 2 m } = H ^ 0 ( X , K ^ { 2 m } ( D ^ { 2 m - 1 } ) ) \\subset \\mathcal { A } . \\end{align*}"}
{"id": "7620.png", "formula": "\\begin{align*} t _ e = t ' _ e - \\omega ( e ) z = t ' _ e - \\omega ( e ) \\lfloor \\tfrac { n } { 4 m } \\rfloor \\ , , \\end{align*}"}
{"id": "4543.png", "formula": "\\begin{align*} \\Vert J _ { 2 2 } \\Vert _ { p } = \\mathcal { O } \\left ( \\left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \\right ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) \\right ) . \\end{align*}"}
{"id": "7187.png", "formula": "\\begin{align*} & n = j \\end{align*}"}
{"id": "7975.png", "formula": "\\begin{align*} D _ + = \\underbrace { h _ + + \\cdots + h _ + } _ { r - r ^ \\prime } , D _ - = \\underbrace { ( \\xi _ - - h _ - ) + \\cdots + ( \\xi _ - - h _ - ) } _ { r - r ^ \\prime } . \\end{align*}"}
{"id": "8423.png", "formula": "\\begin{gather*} ( f \\otimes D ) ( \\tau ) = \\sum _ n c _ f ( n ) \\left ( \\frac { D } { n } \\right ) q ^ n \\end{gather*}"}
{"id": "7300.png", "formula": "\\begin{align*} - H _ y | y | ^ { - 2 } & > - \\tfrac { 1 } { 2 } \\Delta _ y | y | ^ { - 2 } = ( n - 4 ) | y | ^ { - 4 } | y | > { \\sf m } _ 1 , \\\\ - H _ y | y | ^ { - ( n - 4 ) } & > - \\tfrac { 1 } { 2 } \\Delta _ y | y | ^ { - ( n - 4 ) } = ( n - 4 ) | y | ^ { - ( n - 2 ) } | y | > { \\sf m } _ 1 . \\end{align*}"}
{"id": "1462.png", "formula": "\\begin{align*} \\Theta = \\dfrac { \\prod _ { i = 1 } ^ m \\alpha ^ r _ i \\prod _ { s = 0 } ^ { r - 1 } a _ { 0 , s } ^ m } { ( n - 1 ) ! ^ { r ^ 2 m } } \\Psi ( { \\hat { P } } _ { n , n } ) \\enspace . \\end{align*}"}
{"id": "7901.png", "formula": "\\begin{align*} \\mu ( \\tilde { T } _ { \\vec s , k } ) = \\left ( \\frac 1 2 \\deg ( \\tilde { T } _ { \\vec s , k } ) + \\# \\{ i : s _ i < 0 \\} - \\frac 1 2 \\dim _ { \\mathbb C } X \\right ) \\tilde { T } _ { \\vec s , k } . \\end{align*}"}
{"id": "5725.png", "formula": "\\begin{align*} \\pi _ { [ a , i ] } \\cdot \\pi _ { [ i + 1 , b ] } = \\binom { b - a + 1 } { i - a + 1 } \\pi _ { [ a , b ] } \\end{align*}"}
{"id": "6136.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\end{gather*}"}
{"id": "6296.png", "formula": "\\begin{align*} u ( x ) = \\frac { - 1 } { f ( x ) } \\int _ { 0 } ^ { q x } f ( t ) r ( t ) d _ q t \\end{align*}"}
{"id": "6311.png", "formula": "\\begin{align*} & \\int \\frac { x } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\widetilde { h } _ n ( x ; q ) d _ q x = \\dfrac { ( 1 - q ) x } { [ n - 1 ] _ q ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\frac { \\widetilde { h } _ n ( x ; q ) } { q x } - q ^ { - n } [ n ] _ q \\widetilde { h } _ { n - 1 } ( x ; q ) \\right ) , \\end{align*}"}
{"id": "4534.png", "formula": "\\begin{align*} F _ { m , n } ( t _ { 1 } , t _ { 2 } ) : = \\frac { 1 } { ( m + 1 ) ( n + 1 ) } \\frac { ( 1 - \\cos ( m + 1 ) t _ { 1 } ) ( 1 - \\cos ( n + 1 ) t _ { 2 } ) } { \\left ( 4 \\sin \\frac { t _ { 1 } } { 2 } \\sin \\frac { t _ { 2 } } { 2 } \\right ) ^ { 2 } } , \\end{align*}"}
{"id": "8060.png", "formula": "\\begin{align*} V ( h ) = 0 \\end{align*}"}
{"id": "5547.png", "formula": "\\begin{align*} \\mathbb { E } [ \\ell , k ] = \\left . \\left [ \\dfrac { \\ell + 1 } { 2 } \\dbinom { \\ell } { k } + \\sum \\limits _ { i = 0 } ^ { k / 2 - 1 } \\dbinom { \\ell } { k - 2 i - 2 } \\right ] \\middle / \\dbinom { \\ell } { k } \\right . \\end{align*}"}
{"id": "7352.png", "formula": "\\begin{align*} ( 1 - r ^ 2 ) m ( r ) - 2 r = 0 . \\end{align*}"}
{"id": "8453.png", "formula": "\\begin{align*} W ( \\mathbf { t } ) = \\bigcap _ { N = 1 } ^ { \\infty } \\bigcup _ { n = N } ^ { \\infty } \\bigcup _ { w \\in \\Sigma _ { \\beta _ 1 } ^ n , v \\in \\Sigma _ { \\beta _ 2 } ^ n } B \\left ( \\overline { x } _ { n , w } , \\beta _ 2 ^ { - n ( 1 + \\theta _ { 1 } + \\epsilon ) \\log _ { \\beta _ 2 } \\beta _ 1 } \\right ) \\times B \\left ( \\overline { y } _ { n , v } , \\beta _ 2 ^ { - n ( 1 + \\theta _ { 2 } + \\epsilon ) } \\right ) , \\end{align*}"}
{"id": "6822.png", "formula": "\\begin{align*} J _ v ( v , g ) = - g _ L v + g ( V _ E - v ) , \\end{align*}"}
{"id": "5135.png", "formula": "\\begin{align*} \\Lambda = \\big ( \\Lambda ( x , y , z , u , w ) , d \\big ) , \\end{align*}"}
{"id": "3841.png", "formula": "\\begin{align*} A _ m = 1 + m + ( 1 - ( - 1 ) ^ m ) \\kappa _ 1 + 2 \\kappa _ 2 . \\end{align*}"}
{"id": "5222.png", "formula": "\\begin{align*} \\int _ { \\Xi } e ^ { W ' / \\hbar } d x _ 1 \\wedge \\cdots \\wedge d x _ n = \\int _ { \\Xi } \\psi ^ * \\left ( e ^ { W ' / \\hbar } d x _ 1 \\wedge \\cdots \\wedge d x _ n \\right ) = \\int _ { \\Xi } e ^ { \\psi ( W ' ) / \\hbar } d x _ 1 \\wedge \\cdots \\wedge d x _ n . \\end{align*}"}
{"id": "7780.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\min \\{ N _ h , m p \\} } ( \\tilde \\lambda _ i - \\lambda _ i ) ^ 2 \\le \\lVert E \\rVert _ F . \\end{align*}"}
{"id": "3512.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast \\ast } = \\frac { \\xi _ { p } ^ { 3 } \\phi ( \\xi _ { p } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) - \\xi _ { q } ^ { 3 } \\phi ( \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) } { \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) } , \\end{align*}"}
{"id": "1007.png", "formula": "\\begin{align*} A _ n ( x ) = \\frac 1 n \\sum _ { i = 0 } ^ { n - 1 } \\delta _ { T ^ i ( x ) } \\end{align*}"}
{"id": "7908.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial t _ { \\vec s , k } } ( L ( t , - z ) \\alpha , L ( t , z ) \\beta ) \\\\ = & \\frac { 1 } { z } ( \\tilde { T } _ { \\vec s , k } \\star _ t L ( t , - z ) \\alpha , L ( t , z ) \\beta ) - \\frac { 1 } { z } ( L ( t , - z ) \\alpha , \\tilde { T } _ { \\vec s , k } \\star _ t L ( t , z ) \\beta ) \\\\ = & 0 \\end{align*}"}
{"id": "8239.png", "formula": "\\begin{align*} k _ { \\pm } = \\sqrt { \\frac { 1 + ( | a | \\pm i | b | ) } { 4 \\beta \\hslash ^ { 2 } / 3 } } = \\sqrt { \\frac { ( 1 + | a | ) \\pm i | b | } { 4 \\beta \\hslash ^ { 2 } / 3 } } \\end{align*}"}
{"id": "7998.png", "formula": "\\begin{align*} H _ { ( \\mathbb P ^ 3 , K 3 ) , 4 d } ( y ) = H _ { ( Q _ 4 , K 3 ) , d } ( y ) . \\end{align*}"}
{"id": "4528.png", "formula": "\\begin{align*} 0 = \\tilde { \\nabla } ^ 2 _ { i j } u | _ { T M ^ 3 } = \\nabla ^ 2 _ { i j } u + _ { i j } \\hat { N } ( u ) = ( _ { i j } - k _ { i j } ) | \\nabla u | \\end{align*}"}
{"id": "1356.png", "formula": "\\begin{align*} \\Gamma ^ { k , \\tau } _ p & : = \\{ \\tau = \\tau _ 1 - \\tau _ 2 + \\cdots + \\tau _ p , \\ , \\ , k = k _ 1 - k _ 2 + \\cdots k _ p \\} , \\end{align*}"}
{"id": "4143.png", "formula": "\\begin{align*} f _ 1 * f _ 2 ( g ) = \\int _ { G } f _ 1 ( h ) f _ 2 ( h ^ { - 1 } g ) \\ , d h , g \\in G , \\end{align*}"}
{"id": "7953.png", "formula": "\\begin{align*} I _ { X } ( y , z ) = z e ^ { t / z } \\sum _ { d \\in \\mathbb K } y ^ { d } \\left ( \\prod _ { i = 0 } ^ { m } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\textbf { 1 } _ { [ - d ] } , \\end{align*}"}
{"id": "2252.png", "formula": "\\begin{align*} ( \\hat { u _ p ^ 0 } , \\hat { h _ p ^ 0 } ) ( x , 0 ) = ( 0 , 0 ) , ( \\hat { u _ p ^ 0 } , \\hat { h _ p ^ 0 } ) ( x , \\infty ) = ( \\delta , \\sigma ) . \\end{align*}"}
{"id": "8303.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c } \\phi ^ { * } ( + a / 2 ) = e ^ { - i \\varphi } . \\phi ^ { * } ( - a / 2 ) \\\\ \\partial _ { x } \\phi ^ { * } ( + a / 2 ) = e ^ { - i \\varphi } . \\partial _ { x } \\phi ^ { * } ( - a / 2 ) \\\\ \\partial _ { x } ^ { 2 } \\phi ^ { * } ( + a / 2 ) = e ^ { - i \\varphi } . \\partial _ { x } ^ { 2 } \\phi ^ { * } ( - a / 2 ) \\\\ \\partial _ { x } ^ { 3 } \\phi ^ { * } ( + a / 2 ) = e ^ { - i \\varphi } . \\partial _ { x } ^ { 3 } \\phi ^ { * } ( - a / 2 ) \\end{array} \\right . , \\end{align*}"}
{"id": "2288.png", "formula": "\\begin{align*} I _ 4 = & \\int _ 0 ^ \\infty ( f _ u ^ { ( 1 ) } , f _ h ^ { ( 1 ) } ) \\cdot ( u _ p ^ 1 , h _ p ^ 1 ) x ^ { - 2 \\sigma _ 1 } { \\rm d } y \\\\ \\leq & \\Vert ( f _ u ^ { ( 1 ) } , f _ h ^ { ( 1 ) } ) x ^ { \\frac { 1 } { 2 } - \\sigma _ 1 } \\Vert _ { L _ y ^ 2 } \\Vert ( u _ p ^ 1 , h _ p ^ 1 ) x ^ { - \\frac { 1 } { 2 } - \\sigma _ 1 } \\Vert _ { L _ y ^ 2 } \\\\ \\leq & C x ^ { - \\frac { 3 } { 2 } } + \\delta _ 0 \\Vert ( u _ p ^ 1 , h _ p ^ 1 ) x ^ { - \\frac { 1 } { 2 } - \\sigma _ 1 } \\Vert _ { L _ y ^ 2 } ^ 2 . \\end{align*}"}
{"id": "6566.png", "formula": "\\begin{align*} \\sum _ { \\nu = 1 } ^ k \\beta ( f _ 1 ( x ) , \\ldots , f _ { \\nu - 1 } ( x ) , d f _ \\nu ( x , y ) , f _ { \\nu + 1 } ( x ) , \\ldots , f _ k ( x ) ) = d ( \\beta \\circ f ) ( x , y ) \\end{align*}"}
{"id": "7619.png", "formula": "\\begin{align*} n ^ { r - 1 } \\Biggl [ \\dfrac { ( r - 1 ) ! } { 2 ^ r } - \\dfrac { C _ 2 ( r ) + 1 } { \\sqrt { n } } \\Biggr ] = n ^ { r - 1 } \\Biggl [ \\dfrac { ( r - 1 ) ! } { 2 ^ { r + 1 } } + \\dfrac { ( r - 1 ) ! } { 2 ^ { r + 1 } } - \\dfrac { C _ 2 ( r ) + 1 } { \\sqrt { n } } \\Biggr ] > n ^ { r - 1 } \\dfrac { ( r - 1 ) ! } { 2 ^ { r + 1 } } . \\end{align*}"}
{"id": "2063.png", "formula": "\\begin{align*} T _ { N , t } \\boldsymbol { \\mu } _ N ^ \\mathrm { T } - \\boldsymbol { \\mu } _ N ^ \\mathrm { T } T _ { \\infty , t } & = - \\int _ 0 ^ t \\frac { \\dd } { \\dd s } { \\left [ T _ { N , t - s } \\boldsymbol { \\mu } _ N ^ \\mathrm { T } T _ { \\infty , s } \\right ] } \\dd s \\\\ & = \\int _ 0 ^ t T _ { N , t - s } { \\left [ \\mathcal { L } _ N \\boldsymbol { \\mu } _ N ^ \\mathrm { T } - \\boldsymbol { \\mu } _ N ^ \\mathrm { T } \\mathcal { L } _ { \\infty } \\right ] } T _ { \\infty , s } \\dd s , \\end{align*}"}
{"id": "8304.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\sum _ { n = 1 } ^ { \\infty } \\frac { A ( n ) } { n } \\sin ( 2 \\pi n x ) = - \\sum _ { n = 1 } ^ { \\infty } \\frac { a ( n ) } { n } \\left ( \\{ n x \\} - \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
{"id": "5721.png", "formula": "\\begin{align*} ( y _ i - y _ { i + 1 } ) \\pi _ { [ a , i ] } = ( y _ i - y _ { i + 1 } ) e _ { i - a + 1 } ( y _ 1 , \\ldots , y _ { i } ) = 0 \\end{align*}"}
{"id": "3122.png", "formula": "\\begin{align*} \\textup { L H S } = & ( 1 + \\lambda _ h ) \\Vert u _ { \\mathrm { n c } } \\Vert _ { \\delta } ^ 2 + \\lambda \\Vert u - u _ { \\mathrm { n c } } \\Vert ^ 2 _ { L ^ 2 ( \\Omega ) } + 2 \\lambda b ( u , u _ { \\mathrm { n c } } - J u _ { \\mathrm { n c } } ) + 2 a _ { \\mathrm { p w } } ( u , J u _ { \\mathrm { n c } } - u _ { \\mathrm { n c } } ) . \\end{align*}"}
{"id": "1603.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial z ^ i _ { \\epsilon } } = 2 b ^ 2 \\Big [ z ^ 3 _ { \\tilde { \\epsilon } } \\sum \\limits _ { \\tau } ^ { } ( - 1 ) ^ { { \\tilde { \\epsilon } } + \\tau } z ^ i _ { \\tilde { \\tau } } z ^ 3 _ { \\tau } + \\delta _ { i 3 } \\sum \\limits _ { \\tau , k } ^ { } ( - 1 ) ^ { \\epsilon + \\tau } z ^ 3 _ { \\tau } z ^ k _ { \\tilde { \\tau } } z ^ k _ { \\tilde { \\epsilon } } \\Big ] , \\end{align*}"}
{"id": "4416.png", "formula": "\\begin{align*} S _ { d , i , 1 } = \\{ i , d + i , 2 d + i , \\ldots , ( 4 k - 1 ) d + i \\} \\mbox { a n d } S _ { d , i , 2 } = \\{ 4 k N + 4 e + i , ( 4 k + 2 ) N - i + 1 \\} , \\end{align*}"}
{"id": "6270.png", "formula": "\\begin{align*} \\int F ( x ) r ( x ) y ( x ) d _ q x = - F ( x ) D _ { q ^ { - 1 } } y ( x ) . \\end{align*}"}
{"id": "2709.png", "formula": "\\begin{align*} \\mathcal { F } _ \\perp : = \\mathbb { C } \\oplus \\bigoplus _ { s = 1 } ^ \\infty \\Big ( \\mathrm { s p a n } \\{ u _ 1 , u _ 2 \\} ^ \\perp \\Big ) ^ { \\otimes _ { \\mathrm { s y m } } s } . \\end{align*}"}
{"id": "4429.png", "formula": "\\begin{align*} ( A ) = \\sum \\limits _ { j = 1 } ^ n ( \\lambda _ j ) \\prod \\limits _ { i \\neq j } \\dfrac { ( A - \\lambda _ i ) } { ( \\lambda _ j - \\lambda _ i ) } . \\end{align*}"}
{"id": "6662.png", "formula": "\\begin{align*} u _ t = \\Delta u + a ( x ) u ^ p \\ \\ \\R ^ { N } , t > 0 \\end{align*}"}
{"id": "8334.png", "formula": "\\begin{align*} g ( z ) = \\frac { \\alpha z + \\beta } { \\bar \\beta z + \\bar \\alpha } \\end{align*}"}
{"id": "1632.png", "formula": "\\begin{align*} [ B , M ] ^ * = \\{ N \\in \\mathcal { D } : \\max ( B ) \\sqsubseteq \\max ( r _ m ( N ) ) \\mathrm { \\ a n d \\ } N \\le M \\} . \\end{align*}"}
{"id": "5753.png", "formula": "\\begin{align*} f = \\pi _ { i _ 1 } \\sum _ { J \\subseteq [ n - 1 ] } c _ { J } \\pi _ J = \\sum _ { J \\subseteq [ n - 1 ] } c _ { J } \\pi _ { i _ 1 } \\pi _ J . \\end{align*}"}
{"id": "143.png", "formula": "\\begin{align*} \\forall t > 0 , \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } \\leq c ( \\mu ) \\lambda , \\lim _ { \\mu \\rightarrow \\infty } c ( \\mu ) = 0 . \\end{align*}"}
{"id": "8486.png", "formula": "\\begin{align*} & \\int _ 0 ^ x \\partial _ y f ( y ) d y = f ( x ) - f ( 0 ) \\\\ \\Rightarrow & f ( x ) = \\int _ 0 ^ x \\partial _ y f ( y ) d y + f ( 0 ) \\\\ \\Rightarrow & \\left | f ( x ) \\right | \\le \\int _ 0 ^ x \\left | \\partial _ y f ( y ) \\right | d y + \\left | f ( 0 ) \\right | \\le \\int _ 0 ^ 1 \\left | \\partial _ y f ( y ) \\right | d y + \\left | f ( 0 ) \\right | . \\end{align*}"}
{"id": "8146.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int \\limits _ 0 ^ \\infty \\| \\chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \\perp } P _ { \\delta } ^ \\perp ( W _ { n ; \\textrm { o u t } } ^ \\perp ) \\| _ { \\textrm { o p } } \\ , d t = 0 \\end{align*}"}
{"id": "3053.png", "formula": "\\begin{align*} c _ { n } ^ { \\ast \\ast } = \\frac { \\Gamma ( n / 2 ) } { ( 2 \\pi ) ^ { n / 2 } } \\left [ \\int _ { 0 } ^ { \\infty } s ^ { n / 2 - 1 } \\overline { \\mathcal { G } } _ { n } ( s ) \\mathrm { d } s \\right ] ^ { - 1 } . \\end{align*}"}
{"id": "3437.png", "formula": "\\begin{align*} & \\Omega _ X ^ p : = \\iota _ * \\Omega _ { U } ^ p 1 \\leq p \\leq n , \\\\ & T _ X : = ( \\Omega _ X ^ 1 ) ^ * , \\end{align*}"}
{"id": "4265.png", "formula": "\\begin{align*} x \\in \\widetilde \\Sigma _ i \\ \\Rightarrow \\ | x - \\rho \\tilde \\eta _ { \\ell } | \\ge \\frac \\rho 2 | \\tilde \\eta _ { \\ell } - \\tilde \\eta _ { i } | \\ \\hbox { i f } \\ \\ell \\not = i , \\end{align*}"}
{"id": "4430.png", "formula": "\\begin{align*} P ( x ) = \\sum \\limits _ { i = 1 } ^ k \\left ( \\sum \\limits _ { p = 0 } ^ { m _ i - 1 } \\dfrac { c _ i ^ p } { p ! } ( x - \\lambda _ i ) ^ p \\right ) L _ i ( x ) . \\end{align*}"}
{"id": "3487.png", "formula": "\\begin{align*} g _ { 1 } ( u ) = \\left ( 1 + \\frac { 2 u } { m } \\right ) ^ { - ( m + 1 ) / 2 } , \\end{align*}"}
{"id": "5489.png", "formula": "\\begin{align*} S = \\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix} \\quad C = \\begin{pmatrix} 1 & 1 & 0 \\\\ 1 & 1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "8984.png", "formula": "\\begin{align*} \\begin{cases} w = ( 1 + u _ { q , r } ) ^ { 1 + \\delta } \\\\ \\nu = \\xi _ { \\tilde K } ^ 2 ( 1 + u _ { q , r } ) ^ { 1 + 2 \\delta } \\end{cases} \\end{align*}"}
{"id": "7466.png", "formula": "\\begin{align*} C _ { m , r } \\left ( n \\right ) \\left ( j \\right ) : = C _ { f \\left ( \\left ( r - 1 \\right ) n + j - 1 \\right ) } \\end{align*}"}
{"id": "761.png", "formula": "\\begin{align*} T _ { } \\triangleq \\sum _ { m = 1 } ^ { M } w _ m T _ m \\overset { \\mathcal { H } _ 1 } { \\underset { \\mathcal { H } _ 0 } { \\gtrless } } \\lambda \\end{align*}"}
{"id": "3533.png", "formula": "\\begin{align*} \\mathrm { D T E } _ { ( p , q ) } ( X ) = \\mu + \\sigma \\frac { \\Gamma ( t - 1 ) \\left [ \\left ( 1 + \\xi _ { p } ^ { 2 } \\right ) ^ { - ( t - 1 ) } - \\left ( 1 + \\xi _ { q } ^ { 2 } \\right ) ^ { - ( t - 1 ) } \\right ] } { 2 \\Gamma ( t - \\frac { 1 } { 2 } ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "7637.png", "formula": "\\begin{align*} \\chi ^ { F g } & = \\prod _ { h \\in F } ( \\chi _ 1 ^ { - 1 } ) ^ { h g } \\circ ( \\chi _ 2 ^ { - 1 } ) ^ { h g } \\circ \\chi _ 1 ^ { h g } \\circ \\chi _ 2 ^ { h g } \\\\ & = \\prod _ { h \\in F } ( \\chi _ 1 ^ { - 1 } ) ^ { h g } \\circ \\prod _ { h \\in F } ( \\chi _ 2 ^ { - 1 } ) ^ { h g } \\circ \\prod _ { h \\in F } \\chi _ 1 ^ { h g } \\circ \\prod _ { h \\in F } \\chi _ 2 ^ { h g } \\\\ & = [ \\chi _ 1 ^ { F g } , \\chi _ 2 ^ { F g } ] \\end{align*}"}
{"id": "5796.png", "formula": "\\begin{align*} f ( x , y ) = g ^ { h _ s ( x , y ) } , \\end{align*}"}
{"id": "1383.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + \\bigtriangleup u - | u | ^ { p - 1 } u + i \\gamma u = f \\\\ u ( x , 0 ) = u _ 0 \\in H ^ 1 ( \\mathbb { T } ) , \\end{cases} \\end{align*}"}
{"id": "4821.png", "formula": "\\begin{align*} \\delta ( p , ( \\pi _ j ) _ j ) S ^ { d } ( p ) \\lesssim _ { n } \\frac h \\lambda + \\prod _ { j = 1 } ^ d s _ { k _ j , \\lambda } ( p , \\pi _ j ) \\end{align*}"}
{"id": "7748.png", "formula": "\\begin{align*} V _ { m } ( \\alpha ) = \\Big \\{ v _ { m } ( x , { \\bf y } ; \\alpha ) = \\displaystyle \\sum _ { k = 1 } ^ { m } \\sum _ { j = 1 } ^ { N _ h } { \\tilde v } _ { k , j } ^ \\alpha \\vartheta _ j ( x ) \\varphi _ k ( \\psi _ x ( { \\bf y } ) ) , \\ ( x , { \\bf y } ; \\alpha ) \\in \\Omega _ { 1 D } \\times \\Sigma _ x \\times { \\mathcal P } \\Big \\} , \\end{align*}"}
{"id": "1479.png", "formula": "\\begin{align*} \\psi _ s = \\sum _ { j = 1 } ^ { w - 1 } \\sum _ { k = 1 } ^ { r _ j } { { p } } _ { j , k } \\varphi _ { \\zeta _ j , k } + \\sum _ { k = 1 } ^ { s _ w } { { p } } _ { w , k } \\varphi _ { \\zeta _ w , k } \\enspace . \\end{align*}"}
{"id": "3063.png", "formula": "\\begin{align*} \\mathrm { E } [ \\mathbf { Y } | \\boldsymbol { a } < \\mathbf { Y } \\leq \\boldsymbol { b } ] & = \\mathrm { E } \\left [ \\left ( \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } \\mathbf { Z } + \\boldsymbol { \\mu } \\right ) | \\boldsymbol { \\xi _ { a } } < \\mathbf { Z } \\leq \\boldsymbol { \\xi _ { b } } \\right ] \\\\ & = \\boldsymbol { \\mu } + \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } \\mathrm { E } \\left [ \\mathbf { Z } | \\boldsymbol { \\xi _ { a } } < \\mathbf { Z } \\leq \\boldsymbol { \\xi _ { b } } \\right ] , \\end{align*}"}
{"id": "392.png", "formula": "\\begin{align*} 0 \\geq C _ 1 C _ 3 - \\frac { C _ 1 C _ 3 } { 2 } - \\frac { C _ 2 } { n } | S _ { o , c } | = c - \\frac { C _ 2 } { n } | S _ { o , c } | , \\end{align*}"}
{"id": "2815.png", "formula": "\\begin{align*} f _ * = \\min _ { x \\in \\mathbb { R } ^ d } f ( x ) = \\min _ { i \\in \\mathcal { I } } \\big \\{ f _ i - \\tfrac { 1 } { 2 L } \\| g _ i \\| ^ 2 \\big \\} \\end{align*}"}
{"id": "7383.png", "formula": "\\begin{align*} \\Delta \\bar { \\phi } - \\bar { \\phi } + p U _ 0 ^ { p - 1 } \\bar { \\phi } = 0 \\quad \\textrm { i n } \\ \\ \\mathbb { R } ^ N . \\end{align*}"}
{"id": "6828.png", "formula": "\\begin{align*} g _ { } ( t ) = g _ 0 + g _ 1 N ( t ) , \\ a ( t ) = a _ 0 + a _ 1 N ( t ) , g _ 0 , g _ 1 , a _ 0 , a _ 1 > 0 . \\end{align*}"}
{"id": "5668.png", "formula": "\\begin{align*} n _ k \\coloneqq | \\overline { J _ k } | = | J _ k | + 1 . \\end{align*}"}
{"id": "4185.png", "formula": "\\begin{align*} \\Big | \\Big ( \\frac { d } { d \\lambda } \\Big ) ^ \\alpha F ^ { ( \\iota ) } ( \\lambda ) \\Big | & = \\Big | \\Big ( \\frac { d } { d \\lambda } \\Big ) ^ \\alpha \\int _ { - 2 } ^ 2 2 ^ \\iota F ( \\tau ) \\check \\chi ( 2 ^ \\iota ( \\lambda - \\tau ) ) \\ , d \\tau \\Big | \\\\ & \\lesssim _ N 2 ^ { \\iota ( \\alpha + 1 ) } \\int _ { - 2 } ^ 2 \\frac { | F ( \\tau ) | } { ( 1 + 2 ^ \\iota | \\lambda - \\tau | ) ^ N } \\ , d \\tau . \\end{align*}"}
{"id": "1739.png", "formula": "\\begin{align*} \\log F ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) = & \\log F ( c z \\ , | \\ , c \\overline \\omega _ 1 , c \\omega _ 2 ) \\sim \\int _ C \\frac { e ^ { c z s } } { e ^ { c \\overline \\omega _ 1 s } - 1 } \\sum _ { k \\geq 0 } \\frac { B _ k \\cdot ( c \\omega _ 2 ) ^ { k - 1 } ( s ) ^ { k - 2 } } { k ! } \\ , d s \\\\ & = \\sum _ { k \\geq 0 } \\frac { B _ k \\cdot \\omega _ 2 ^ { k - 1 } } { k ! } \\cdot \\int _ C \\frac { e ^ { c z s } \\ , ( c s ) ^ { k - 2 } } { e ^ { c \\overline \\omega _ 1 s } - 1 } \\ d ( c s ) . \\end{align*}"}
{"id": "3251.png", "formula": "\\begin{align*} u _ { A , q } ( \\cdot , y ) = \\Phi ( \\cdot , y ) + u _ { A , q } ^ s ( \\cdot , y ) \\mathrm { i n } \\ ; \\R ^ 3 , \\end{align*}"}
{"id": "5271.png", "formula": "\\begin{align*} O _ i ( Q ) : = \\left \\langle \\prod _ { \\ell \\in I _ i } \\tau ^ { ( a _ \\ell , b _ \\ell ) } _ { d _ \\ell } \\sigma _ 1 ^ { k _ 1 ( i ) } \\sigma _ 2 ^ { k _ 2 ( i ) } \\sigma _ { 1 2 } \\right \\rangle ^ { \\mathbf { s } ^ { \\Gamma _ { 0 , k _ 1 ( i ) , k _ 2 ( i ) , 1 , I _ i } } , o } \\end{align*}"}
{"id": "4527.png", "formula": "\\begin{align*} \\tilde { \\nabla } u = \\tilde { g } ^ { u i } \\partial _ i = \\partial _ \\tau . \\end{align*}"}
{"id": "6278.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } q ^ { \\frac { k ( k - 3 ) } { 2 } } x ^ k A _ q ( q ^ k x ) = - \\ , _ 0 \\phi _ 1 ( - ; 0 ; q , - q ^ 2 x ) . \\end{align*}"}
{"id": "5084.png", "formula": "\\begin{align*} S ^ { n , 1 } _ \\tau : = n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { \\tau } ( t - s ) ^ \\alpha \\sigma ' ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) d s . \\end{align*}"}
{"id": "7968.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - 1 & 1 & 1 \\\\ 1 & 1 & 0 & 0 \\end{bmatrix} . \\end{align*}"}
{"id": "7096.png", "formula": "\\begin{align*} \\tau _ x ( f ) \\colon = \\int _ { ( k , v ) \\in \\mathcal { O } _ x } f ( k , v ) \\ ; d k d v . \\end{align*}"}
{"id": "7565.png", "formula": "\\begin{align*} \\gamma _ x ^ \\alpha : \\mathbb { R } \\to N : t \\mapsto \\varphi ( \\exp ( t v _ \\alpha ) , x ) , \\alpha = 1 , \\dots , r , \\end{align*}"}
{"id": "8052.png", "formula": "\\begin{align*} \\| X \\| ^ { 2 } : = g ( X , X ) , \\| X \\| : = \\sqrt { \\| X \\| ^ { 2 } } . \\end{align*}"}
{"id": "4648.png", "formula": "\\begin{align*} \\kappa _ \\beta = \\frac { 2 ^ \\alpha \\Gamma ( ( \\beta + \\alpha ) / 2 ) \\Gamma ( ( d - \\beta ) / 2 ) } { \\Gamma ( \\beta / 2 ) \\Gamma ( ( d - \\beta - \\alpha ) / 2 ) } , \\end{align*}"}
{"id": "2546.png", "formula": "\\begin{align*} r R \\Sigma ^ n Y = \\textbf { R } r \\Sigma ^ n Y \\cong \\Sigma ^ n \\textbf { R } r Y = \\Sigma ^ n r R Y , \\end{align*}"}
{"id": "6374.png", "formula": "\\begin{align*} \\bigl [ U ( \\sigma ) \\bigr ] \\ , ( \\widehat { \\tau } ) = \\widehat { \\sigma \\tau } , \\ \\ \\forall \\ , \\tau \\in S _ { \\infty } . \\end{align*}"}
{"id": "8893.png", "formula": "\\begin{align*} \\sideset { } { ' } \\sum _ { a } 2 f _ { i a } T _ { i a } ^ i + \\sideset { } { ' } \\sum _ { a < b } f _ { a b } \\left ( T _ { i b } ^ a + T _ { a b } ^ i + T _ { a i } ^ b \\right ) = - 2 \\ , \\partial _ i F , \\end{align*}"}
{"id": "5968.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\| u ( t ) \\| ^ 2 + | u ( t ) | _ 1 ^ 2 \\ , d t \\leq & C ^ 1 _ { T , \\beta } \\int _ 0 ^ T ( g _ { \\beta , T } ( t ) ) ^ { - 1 } \\| f ( t ) \\| ^ 2 \\ , d t \\\\ & + C ^ 2 _ { T , \\beta } \\| u ^ 0 \\| ^ 2 \\end{align*}"}
{"id": "584.png", "formula": "\\begin{align*} M _ 5 = f _ k ( i ) f _ k ( i + 1 ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( i ) \\tau _ 3 ( i + 1 ) } { \\sqrt { i } } \\Big ) . \\end{align*}"}
{"id": "3544.png", "formula": "\\begin{align*} \\chi _ { \\sigma _ i } ( h _ k ) = \\sum _ { l = 0 } ^ { k } \\binom { k } { l } \\cdot \\binom { n - k } { i - l } ( - 1 ) ^ l . \\end{align*}"}
{"id": "280.png", "formula": "\\begin{align*} \\Lambda ' _ i = \\Lambda _ i + \\pi ^ { \\nu } Z ^ { \\nu } _ i . \\end{align*}"}
{"id": "3842.png", "formula": "\\begin{align*} a _ { 2 } ^ j : = 2 ^ j ( \\kappa _ 2 + 1 / 2 ) _ { \\lfloor ( j + 1 ) / 2 \\rfloor } ( \\gamma _ 2 + 1 ) _ { \\lfloor j / 2 \\rfloor } . \\end{align*}"}
{"id": "7557.png", "formula": "\\begin{align*} \\| S ^ m ( f _ 1 , \\dots , f _ m ) \\| _ { L ^ { p , \\infty } ( \\R ) } \\leq C \\prod _ { i = 1 } ^ m \\| f _ i \\| _ { L ^ { p _ i } ( \\R ) } \\end{align*}"}
{"id": "8914.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\left ( \\limsup _ { n \\to \\infty } \\ \\lambda _ n ^ { - \\frac { 1 } { p - 2 } } \\max _ { d _ n ( x ) \\ge R \\lambda _ n ^ { - 1 / 2 } } u _ n ( x ) \\right ) = 0 . \\end{align*}"}
{"id": "4478.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\frac 1 2 \\\\ l \\end{array} \\right ) = \\frac { ( - 1 ) ^ { l - 1 } ( 2 l - 3 ) ! ! } { 2 ^ l l ! } \\in \\left [ - \\frac 1 8 , \\frac 1 2 \\right ] . \\end{align*}"}
{"id": "8874.png", "formula": "\\begin{align*} f _ \\lambda ( \\lambda _ i ) = \\prod _ { j ( \\neq i ) } ( \\lambda _ i - \\lambda _ j ) , \\ f _ { \\lambda \\lambda } ( \\lambda _ i ) = 2 \\sum _ { j ( \\neq i ) } \\prod _ { k ( \\neq i , j ) } ( \\lambda _ i - \\lambda _ k ) . \\end{align*}"}
{"id": "1145.png", "formula": "\\begin{align*} ( - a q ; q ) _ { \\infty } & = 1 + \\sum _ { k = 1 } ^ { \\infty } \\frac { a ^ k q ^ { ( 3 k ^ 2 - k ) / 2 } ( - a q ; q ) _ { k - 1 } ( 1 + a q ^ { 2 k } ) } { ( q ; q ) _ k } . \\end{align*}"}
{"id": "2875.png", "formula": "\\begin{align*} F ( x _ 1 , x _ 2 , \\dotsc , x _ { n } ) = \\big ( f ( x _ 1 , x _ 2 , \\dotsc , x _ { k } ) , f ( x _ 2 , x _ 3 , \\dotsc , x _ { k + 1 } ) , \\dotsc , f ( x _ n , x _ 1 , \\dotsc , x _ { k - 1 } ) \\big ) , \\end{align*}"}
{"id": "7781.png", "formula": "\\begin{align*} \\sin \\theta ( \\mathbf { x } , { \\mathcal C } _ 1 ) = \\min _ { \\mathbf { y } \\in { \\mathcal C } _ 1 } \\lVert \\mathbf { x } - \\mathbf { y } \\rVert _ 2 , \\end{align*}"}
{"id": "6970.png", "formula": "\\begin{align*} a _ n ^ N = ( \\lambda _ n ^ 2 - \\mu _ n ^ 2 ) \\prod _ { \\substack { k = 1 \\\\ k \\ne n } } ^ N \\left ( \\frac { \\lambda _ n ^ 2 - \\mu _ k ^ 2 } { \\lambda _ n ^ 2 - \\lambda _ k ^ 2 } \\right ) n \\le N , \\end{align*}"}
{"id": "9010.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { ( k + z ) ^ n } = \\frac { ( - 1 ) ^ { n } } { 2 ^ { n } \\Gamma ( n ) } \\left ( \\psi _ { n - 1 } \\left ( \\frac { z } { 2 } \\right ) - \\psi _ { n - 1 } \\left ( \\frac { z + 1 } { 2 } \\right ) \\right ) . \\end{align*}"}
{"id": "5728.png", "formula": "\\begin{align*} \\pi _ { [ a , i ] } \\cdot \\pi _ { [ i + 1 , b ] } & = \\pi _ { [ a , i ] } \\pi _ { [ i , b - 1 ] } + \\pi _ { [ a , i ] } \\pi _ { [ i + 1 , b - 1 ] } y _ { b } \\end{align*}"}
{"id": "1865.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { u p s t e p s } \\ , \\ , \\ , ( n , m ) \\rightarrow ( n + 1 , m + 1 ) , \\\\ & \\mbox { d o w n s t e p s } \\ , \\ , \\ , ( n , m ) \\rightarrow ( n + 1 , m - p ) \\ , \\ , \\ , \\mbox { b y $ p $ u n i t s } . \\end{aligned} \\end{align*}"}
{"id": "0.png", "formula": "\\begin{align*} g _ i = ( x _ i - i t ) \\left ( \\prod _ { k < i , j \\leq i } ( x _ k - x _ j - t ) \\right ) \\left ( \\prod _ { j > i } ( x _ i - x _ j - t ) \\right ) . \\end{align*}"}
{"id": "311.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sum _ { 1 \\le i \\le k } \\frac { \\sqrt { v _ i } - \\sqrt { v _ { i - 1 } } } { 1 + v _ { i - 1 } } = \\lim _ { k \\to \\infty } \\sum _ { 1 \\le i \\le k } \\int _ { v _ { i - 1 } } ^ { v _ i } \\frac { \\dd } { \\dd v } \\Big [ \\sqrt { v } \\Big ] \\frac { \\dd { v } } { 1 + v _ { i - 1 } } = \\int _ 0 ^ 1 \\frac { \\dd { v } } { 2 \\sqrt { v } ( 1 + v ) } = \\frac { \\pi } { 4 } . \\end{align*}"}
{"id": "5274.png", "formula": "\\begin{align*} \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { k _ 1 ( 0 ) + r } a ^ { \\widehat { Q } ^ { + r } } + \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { k _ 2 ( 0 ) + s } b ^ { \\widehat { Q } ^ { + s } } = 1 + \\sum _ { j = 1 } ^ h \\left ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { Q } _ j } + \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { Q } _ j } \\right ) . \\end{align*}"}
{"id": "2071.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : b r o w n i a n g i r s a n o v } B ^ i _ t : = \\overline { B } ^ i _ t - \\int _ 0 ^ t \\mathsf { X } ^ i _ s \\dd s \\end{align*}"}
{"id": "5573.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Phi ( K ^ { \\sigma ^ 2 } _ t f ( x ) ) \\nu _ { \\sigma ^ 2 } ( x ) d x & = \\int _ 0 ^ \\infty \\frac { 1 } { \\sigma ^ 2 } \\Phi \\left ( K _ t d _ { \\sigma ^ 2 } f \\left ( \\frac { x } { \\sigma ^ 2 } \\right ) \\right ) \\nu \\left ( \\frac { x } { \\sigma ^ 2 } \\right ) d x \\\\ & = \\int _ 0 ^ \\infty \\Phi ( K _ t d _ { \\sigma ^ 2 } f ( x ) ) \\nu ( x ) d x . \\end{align*}"}
{"id": "4626.png", "formula": "\\begin{align*} \\alpha = \\alpha _ 0 \\beta _ 1 \\alpha _ 1 \\cdots \\beta _ { n } \\alpha _ n \\end{align*}"}
{"id": "6031.png", "formula": "\\begin{align*} x ^ 4 + a x ^ 3 = x ^ 4 - 4 x ^ 3 + ( b + 6 ) x ^ 2 - ( 2 b + 4 ) x + ( b + c + 1 ) . \\end{align*}"}
{"id": "60.png", "formula": "\\begin{align*} P _ 1 \\unlhd { P _ 1 P _ 2 } \\unlhd { P _ 1 P _ 2 P _ 3 } \\unlhd \\cdots \\unlhd { P _ 1 P _ 2 \\cdots P _ s } = G \\end{align*}"}
{"id": "5394.png", "formula": "\\begin{align*} \\langle \\nabla ^ s u , \\nabla ^ s \\phi \\rangle _ { L ^ 2 ( \\R ^ { 2 n } ) } = \\langle ( - \\Delta ) ^ { s / 2 } u , ( - \\Delta ) ^ { s / 2 } \\phi \\rangle _ { L ^ 2 ( \\R ^ n ) } \\end{align*}"}
{"id": "1798.png", "formula": "\\begin{align*} \\eta ( x , z ) = \\| x \\| _ 1 + \\lambda \\sum _ { i = 1 } ^ m \\max \\{ 0 , ( A x - b ) _ i \\} + b ^ T z + \\kappa \\sum _ { i = 1 } ^ n \\max \\{ 0 , | ( A ^ T z ) _ i | - 1 \\} \\end{align*}"}
{"id": "4717.png", "formula": "\\begin{align*} B _ { k , n } : = \\{ \\omega ^ { - 1 } \\ : | \\ : \\omega \\in B _ { k , n } ^ { * } \\} . \\end{align*}"}
{"id": "2975.png", "formula": "\\begin{align*} F _ { m , Q } ( \\gamma _ 2 ^ { - 1 } z , s ) = \\sum _ { \\gamma \\neq I , \\gamma _ 2 ^ { - 1 } } f _ m ( \\gamma z , s ) \\end{align*}"}
{"id": "3620.png", "formula": "\\begin{align*} v ( \\Box \\phi , w ) & = \\inf \\limits _ { w ' \\in W } \\{ w R w ' \\rightarrow _ \\mathsf { G } v ( \\phi , w ' ) \\} , & v ( \\lozenge \\phi , w ) & = \\sup \\limits _ { w ' \\in W } \\{ w R w ' \\wedge _ \\mathsf { G } v ( \\phi , w ' ) \\} . \\end{align*}"}
{"id": "7114.png", "formula": "\\begin{align*} C ( \\tau , t , x ) = \\begin{cases} \\frac { \\partial \\tau ( t , x ) } { \\partial x } & \\boldsymbol { R } ( \\tau ; t , x ) = \\tau \\\\ \\frac { \\partial \\boldsymbol { R } ( \\tau ; t , x ) } { \\partial x } & \\boldsymbol { R } ( \\tau ; t , x ) \\neq \\tau \\end{cases} \\end{align*}"}
{"id": "1388.png", "formula": "\\begin{align*} v = W _ t ^ \\gamma u _ 0 - T [ W _ t ^ \\gamma u _ 0 , v , \\cdots , v ] + G + z , \\end{align*}"}
{"id": "1144.png", "formula": "\\begin{align*} \\sum _ { n , k , \\ell , j \\ge 0 } A ( n , k , \\ell , j ) x ^ k y ^ { \\ell } z ^ j q ^ n & = \\sum _ { k , \\ell \\ge 0 } \\frac { ( - z q ; q ) _ { \\ell } } { ( q ; q ) _ k ( q ^ 2 ; q ^ 2 ) _ { \\ell } } x ^ k y ^ { \\ell } q ^ { \\binom { k + \\ell + 1 } { 2 } + \\binom { \\ell + 1 } { 2 } } . \\end{align*}"}
{"id": "1770.png", "formula": "\\begin{align*} Q _ d ( \\alpha , A , z + a f _ k ) = K ( \\alpha , A , z + a f _ k ) \\cap \\left ( d \\widetilde { z } + \\operatorname { s p a n } \\{ \\widetilde { x } ^ t _ s : \\ ; s \\in \\mathbb { N } , \\ : t \\in A \\cup \\{ k \\} \\} \\right ) . \\end{align*}"}
{"id": "2695.png", "formula": "\\begin{align*} D _ r : = Q _ r ( h _ \\mathrm { M F } - \\mu _ + ) Q _ r , D _ \\ell : = Q _ \\ell ( h _ \\mathrm { M F } - \\mu _ + ) Q _ \\ell \\end{align*}"}
{"id": "7137.png", "formula": "\\begin{align*} \\mathcal { I } _ { D ^ I } & = \\langle x _ 1 ^ { m _ 1 } , x _ 2 ^ { m _ 2 } , x _ 3 ( - x _ 1 + x _ 2 + x _ 3 ) ( x _ 2 + x _ 3 ) ^ { m _ 3 - 2 } \\rangle , \\\\ \\mathcal { I } _ { D ^ { I \\ ! I } } & = \\langle x _ 1 ^ { m _ 1 } , x _ 2 ( x _ 1 + x _ 2 ) ^ { m _ 2 - 1 } , x _ 3 ( x _ 1 + x _ 2 + x _ 3 ) ^ { m _ 3 - 1 } \\rangle . \\end{align*}"}
{"id": "1265.png", "formula": "\\begin{align*} \\sigma ^ { * } ( \\mathcal { B } _ { 2 , k } ) & = \\left \\{ 2 \\cos \\left ( \\frac { h } { k + 1 } \\pi \\right ) \\colon 1 \\le h \\le k \\right \\} . \\end{align*}"}
{"id": "3989.png", "formula": "\\begin{align*} \\widehat E ^ \\dag = \\widehat E - \\{ e _ 0 \\} = \\{ e _ 1 , \\cdots , e _ r , \\widehat e _ { r + 1 } , \\cdots , \\widehat e _ { r + s } \\} . \\end{align*}"}
{"id": "277.png", "formula": "\\begin{align*} \\Lambda _ i ^ t A _ i \\Lambda _ i = ( \\Lambda _ i ' ) ^ t A _ i \\Lambda ' _ i = B , \\end{align*}"}
{"id": "5808.png", "formula": "\\begin{align*} n _ 0 = x _ 0 ^ 2 + x _ 0 y _ 0 + y _ 0 ^ 2 + z ^ 2 . \\end{align*}"}
{"id": "3021.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda } < \\hat { \\Lambda } \\cdot F = \\frac { \\mu } { n } . \\end{align*}"}
{"id": "6310.png", "formula": "\\begin{align*} S _ q ( x ) = \\frac { a ( a - 1 ) } { x ^ 2 } + \\frac { a b ( 1 + q ) } { q x } + \\frac { q ^ { 1 - n } ( [ n ] _ q - a ) - q ^ { - n } b x } { 1 - q } + \\frac { b ^ 2 } { q } . \\end{align*}"}
{"id": "5807.png", "formula": "\\begin{align*} n = x ^ 2 + x y + y ^ 2 + 1 6 z ^ 2 , \\end{align*}"}
{"id": "6591.png", "formula": "\\begin{align*} c _ 3 = \\frac 1 4 d ^ 2 - \\left ( 1 - \\frac { c _ 1 } { 2 d } \\right ) \\left ( \\frac 1 2 d + c _ 2 d \\right ) ^ 2 - \\frac { c _ 1 } { 2 d } \\left ( \\frac 1 2 d + c _ 2 d - \\frac 1 2 \\right ) ^ 2 > 0 . \\end{align*}"}
{"id": "8776.png", "formula": "\\begin{gather*} \\frac { 1 } { p _ 1 ' } + \\frac { 1 } { p _ 2 ' } + \\frac { 1 } { p } = 1 , \\phi _ 1 ( g ) \\phi _ 2 ( g ^ { - 1 } g ' ) = \\phi _ 1 ( g ) ^ { p _ 1 / p _ 2 ' } \\phi _ 2 ( g ^ { - 1 } g ' ) ^ { p _ 2 / p _ 1 ' } ( \\phi _ 1 ( g ) ^ { p _ 1 } \\phi _ 2 ( g ^ { - 1 } g ' ) ^ { p _ 2 } ) ^ { 1 / p } \\end{gather*}"}
{"id": "7402.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { { \\C } _ 1 } \\left ( \\sum _ { i = 1 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \\cdot t _ { i } \\right ) ^ 2 d y \\\\ & \\ge \\int _ { B _ 1 ( 0 ) } \\left ( f ( | z | ) ( z \\cdot t _ { 1 } ) \\right ) ^ 2 d z + o _ n ( 1 ) \\ge c _ 0 > 0 \\ \\textrm { f o r s o m e c o n s t a n t } \\ c _ 0 > 0 , \\end{aligned} \\end{align*}"}
{"id": "803.png", "formula": "\\begin{align*} f ( z ) = \\int _ { 0 } ^ { z } \\frac { G ( t ) \\psi ( \\omega ( t ) ) } { t } d t . \\end{align*}"}
{"id": "4984.png", "formula": "\\begin{align*} E [ ( N ^ { n , 1 } _ { t } ) ^ 2 ] = \\sum ^ { \\lfloor n t \\rfloor } _ { k = 0 } \\int ^ { 1 \\wedge ( n t - k ) } _ { 0 } \\left [ ( n t - k ) ^ { \\alpha } - ( n t - k - y ) ^ { \\alpha } \\right ] ^ 2 \\ , d y , \\end{align*}"}
{"id": "1686.png", "formula": "\\begin{align*} T x - T y = ( c ( x ) E x , s ( x ) F x ) - ( c ( y ) E y , s ( y ) F y ) \\end{align*}"}
{"id": "1461.png", "formula": "\\begin{align*} { { \\tilde { \\psi } } } _ { \\alpha , i , s } = { \\rm { E v a l } } _ { t _ { i , s } \\rightarrow \\alpha } \\bigcirc _ { { { w } } = 0 } ^ { s } ( \\theta _ { t _ { i , s } } + \\gamma _ { r - s + { { w } } } ) ^ { - 1 } : K [ t _ { i , s } ] _ { \\substack { 1 \\leq i \\leq m \\\\ 0 \\leq s \\leq r - 1 } } \\longrightarrow K [ t _ { i ' , s ' } ] _ { ( i ' , s ' ) \\neq ( i , s ) } ; \\ \\ t ^ k _ { i , s } \\mapsto \\dfrac { \\alpha ^ k } { \\prod _ { { { w } } = 0 } ^ s ( k + \\gamma _ { r - s + { { w } } } ) } \\enspace . \\end{align*}"}
{"id": "961.png", "formula": "\\begin{align*} L = L ( B ) = \\left \\{ B x \\mid x \\in \\mathbb { Z } ^ { n } \\right \\} , \\end{align*}"}
{"id": "34.png", "formula": "\\begin{align*} \\rho ' _ a = r \\circ \\rho _ a \\circ r ^ { - 1 } \\end{align*}"}
{"id": "1734.png", "formula": "\\begin{align*} F ( z + \\omega _ 2 \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) F ( z \\ , | \\ , \\overline \\omega _ 1 , - \\omega _ 2 ) = \\exp \\Bigg ( \\int _ { C } \\frac { e ^ { ( z + \\omega _ 2 ) s } } { ( e ^ { \\overline \\omega _ 1 s } - 1 ) ( e ^ { \\omega _ 2 s } - 1 ) } \\frac { d s } { s } + \\int _ { c \\cdot C } \\frac { e ^ { z s } } { ( e ^ { \\overline \\omega _ 1 s } - 1 ) ( e ^ { - \\omega _ 2 s } - 1 ) } \\frac { d s } { s } \\Bigg ) . \\end{align*}"}
{"id": "1036.png", "formula": "\\begin{align*} H ( t ) : = ( F * G ) ( t ) = \\int _ { 0 } ^ { t } F ( t - s ) G ( s ) \\ , \\textnormal { d } s . \\end{align*}"}
{"id": "5549.png", "formula": "\\begin{align*} \\mathbb { E } [ \\ell , k ] = \\left . \\left [ k \\dbinom { \\ell } { k } + \\sum \\limits _ { i = 0 } ^ { ( \\ell - k - 1 ) / 2 } \\dbinom { \\ell } { k + 2 i + 1 } \\right ] \\middle / \\dbinom { \\ell } { k } \\right . \\end{align*}"}
{"id": "2868.png", "formula": "\\begin{align*} P ^ f _ { H , d } \\colon = R _ d ^ { T ^ * } ( T P ^ f _ H ) \\ , . \\end{align*}"}
{"id": "4133.png", "formula": "\\begin{align*} S ^ { i , 1 } ( \\psi _ 1 , \\psi _ 2 , \\psi _ 3 ) & = \\textstyle \\sum _ \\ell ( d ( x ^ \\ell \\psi _ \\ell ) - x ^ \\ell d \\psi _ \\ell ) = \\sum _ \\ell d x ^ \\ell \\wedge \\psi _ \\ell , \\\\ S ^ { i , 2 } \\omega & = ( d x ^ 1 \\wedge \\omega , d x ^ 2 \\wedge \\omega , d x ^ 3 \\wedge \\omega ) . \\end{align*}"}
{"id": "4621.png", "formula": "\\begin{align*} \\bar c _ i ( \\alpha , g ) = \\bar c _ i ( \\alpha , h ) . \\end{align*}"}
{"id": "6233.png", "formula": "\\begin{align*} & \\int \\frac { x } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\widetilde { h } _ n ( x ; q ) d _ q x = \\frac { 1 - q } { [ n - 1 ] _ q ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\frac { 1 } { q } \\widetilde { h } _ n ( x ; q ) - q ^ { - n } [ n ] _ q x \\widetilde { h } _ { n - 1 } ( x ; q ) \\right ) , \\end{align*}"}
{"id": "2164.png", "formula": "\\begin{align*} \\frac { \\Im F _ { \\nu } ( i y ) } { y } \\int _ { \\mathbb { R } } \\frac { ( 1 + s ^ { 2 } ) \\ , d \\sigma _ { \\mu } ( s ) } { | F _ { \\nu } ( i y ) - s | ^ { 2 } } = \\frac { - \\Im G _ { \\nu } ( i y ) } { y } \\int _ { \\mathbb { R } } \\frac { ( 1 + s ^ { 2 } ) \\ , d \\sigma _ { \\mu } ( s ) } { | 1 - s G _ { \\nu } ( i y ) | ^ { 2 } } < 1 \\end{align*}"}
{"id": "7159.png", "formula": "\\begin{align*} ( g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 ) \\{ g _ { 1 1 } y _ 1 + ( g _ { 1 3 } + g _ { 2 3 } ) y _ 3 \\} ^ { a - 1 } = g _ { 2 3 } ( g _ { 1 3 } + g _ { 2 3 } ) ^ { a - 1 } y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } . \\end{align*}"}
{"id": "4732.png", "formula": "\\begin{align*} & C _ 1 \\cdot C _ { 2 e } = C _ { 1 2 e } + C _ e , \\\\ & C _ { 2 e } \\cdot C _ { e 2 } = \\frac { z - z ^ { - 1 } } { q - q ^ { - 1 } } C _ { 2 e 2 } , \\\\ & C _ e \\cdot C _ { 1 2 e } = \\frac { q ^ { 2 } z - q ^ { - 2 } z ^ { - 1 } } { q - q ^ { - 1 } } C _ e . \\end{align*}"}
{"id": "722.png", "formula": "\\begin{align*} A ( Y , Z ) = \\sum _ { i = 1 } ^ m g _ X ( Y , e _ i ) A ( e _ i , Z ) = \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ m g _ X ( Y , e _ i ) g _ X ( Z , e _ j ) A ( e _ i , e _ j ) . \\end{align*}"}
{"id": "6318.png", "formula": "\\begin{align*} L _ n ^ { \\alpha } ( x ; q ) : = \\frac { 1 } { ( q ; q ) _ n } \\ , _ 2 \\phi _ 1 \\left ( \\begin{array} { c c c c } q ^ { - n } , - x \\\\ 0 \\end{array} \\mid q ; q ^ { n + \\alpha + 1 } \\right ) , \\alpha > - 1 , n \\in \\mathbb { N } \\end{align*}"}
{"id": "6357.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\int _ { \\Omega } f ( x , v _ n ) ( v _ n - v ) d x \\right | & \\leqslant c _ 1 \\int _ { \\Omega } \\left | v _ n \\right | ^ { q ( x ) - 1 } \\left | v _ n - v \\right | d x \\\\ & \\leqslant c _ 1 \\left | \\left | | v _ n | ^ { q ( x ) - 1 } \\right | \\right | _ { \\frac { q ( x ) } { q ( x ) - 1 } } \\left | \\left | v _ n - v \\right | \\right | _ { q ( x ) } \\longrightarrow 0 . \\end{aligned} \\end{align*}"}
{"id": "5072.png", "formula": "\\begin{align*} T ^ n _ \\tau & : = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s \\left ( \\sigma ' ( X _ s ) \\right ) ^ 2 \\sigma ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) \\\\ & \\times \\left ( \\int _ 0 ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) d s . \\end{align*}"}
{"id": "3781.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { R } } _ { k , n } ( \\xi ) : = \\int _ { \\R ^ 3 } e ^ { i y \\cdot \\xi } 2 ^ { 3 k + 2 n } ( 1 + 2 ^ k | y \\cdot \\tilde { \\zeta } | ) ^ { - N _ 0 ^ 3 } ( 1 + 2 ^ { k + n } ( | y \\cdot \\tilde { \\zeta } _ 1 | + | y \\cdot \\tilde { \\zeta } _ 2 | ) ) ^ { - N _ 0 ^ 3 } d y = \\widehat { \\mathfrak { R } } ( 2 ^ { - k } ( \\xi \\cdot \\tilde { \\zeta } ) , 2 ^ { - k - n } ( \\xi \\cdot \\tilde { \\zeta } _ 1 ) , 2 ^ { - k - n } ( \\xi \\cdot \\tilde { \\zeta } _ 2 ) ) , \\end{align*}"}
{"id": "4328.png", "formula": "\\begin{align*} \\tilde { \\Omega } _ { L _ i } \\geq 0 , \\int _ { \\mathcal { M } } \\tilde { \\Omega } _ { L _ i } = \\int _ { L _ i } \\Omega , i = 0 , 1 , \\ldots N . \\end{align*}"}
{"id": "2186.png", "formula": "\\begin{align*} { \\rm r o w } ( A ) = \\left ( \\ \\sum _ { j \\in \\Z } a _ { 1 j } , \\ \\dots , \\ \\sum _ { j \\in \\Z } a _ { m j } \\right ) , { \\rm c o l } ( A ) = \\left ( \\ \\sum _ { i \\in \\Z } a _ { i 1 } , \\ \\dots , \\ \\sum _ { i \\in \\Z } a _ { i n } \\right ) \\end{align*}"}
{"id": "5555.png", "formula": "\\begin{align*} \\frac { 2 n \\dbinom { 4 n } { 2 n } + \\sum \\limits _ { i = 0 } ^ { n } \\dbinom { 4 n } { 2 i + 1 } } { \\dbinom { 4 n } { 2 n } } = 2 n + \\frac { 2 ^ { 4 n - 2 } } { \\dbinom { 4 n } { 2 n } } = k + \\frac { 2 ^ { 2 k - 2 } } { \\dbinom { 2 k } { k } } . \\end{align*}"}
{"id": "3209.png", "formula": "\\begin{align*} q _ { t + 1 } ^ { } & = [ q _ { t } ^ { } - \\sum _ { i = 1 } ^ n f ^ { } _ { t , i } \\tau ] ^ + \\\\ & ~ ~ + \\sum _ { k = 1 } ^ K \\Big ( \\big [ q ^ { } _ { k , t } - \\sum _ { i = 1 } ^ n f _ { k , t , i } ^ { } \\tau \\big ] ^ + ~ { \\rm m o d } ~ W \\Big ) , \\end{align*}"}
{"id": "1766.png", "formula": "\\begin{align*} | f _ k ( y _ n ^ * ) - \\sum _ { j = 1 } ^ { m _ 1 } y _ n ^ * ( x ^ k _ j ) | < \\epsilon / 4 . \\end{align*}"}
{"id": "2152.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } ( x - s ) ^ { - 2 } \\ , d \\nu ( s ) = 3 ^ { - 1 } \\int _ { - 1 } ^ { 2 } s ^ { 2 } ( x - s ) ^ { - 2 } \\ , d \\lambda ( s ) = \\begin{cases} + \\infty & x \\neq 0 ; \\\\ 1 & x = 0 , \\end{cases} \\end{align*}"}
{"id": "6372.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } ( w _ 1 , \\ldots , w _ d , 0 , \\ldots , 0 , \\ldots ) , \\mbox { w i t h $ d \\geq 2 $ a n d w i t h } \\\\ \\mbox { $ w _ 1 \\geq w _ 2 \\geq \\cdots \\geq w _ d > 0 $ s u c h t h a t $ \\sum _ { i = 1 } ^ d w _ d = 1 $ . } \\end{array} \\right . \\end{align*}"}
{"id": "1626.png", "formula": "\\begin{align*} r _ k ( M ) = \\bigcup _ { n < k } M ( n ) . \\end{align*}"}
{"id": "729.png", "formula": "\\begin{align*} ( \\partial _ s + \\Delta ) w = I _ 1 + I _ 2 . \\end{align*}"}
{"id": "7482.png", "formula": "\\begin{align*} \\mathrm { F i x } ( Q ) = \\cap _ { i = 1 } ^ { m } C _ { i } \\not = \\emptyset . \\end{align*}"}
{"id": "821.png", "formula": "\\begin{align*} Q B _ { \\beta , \\infty } : = \\bigcup _ { p > 0 } Q B _ { \\beta , p } \\end{align*}"}
{"id": "6507.png", "formula": "\\begin{align*} u = 0 \\omega _ b . \\end{align*}"}
{"id": "7745.png", "formula": "\\begin{align*} { } ^ L H _ S : = ( H ( E ) \\rtimes { } ^ L ( Z _ H ) _ S ) / Z _ H ( E ) , \\end{align*}"}
{"id": "5358.png", "formula": "\\begin{align*} a ( \\xi ) = \\frac { \\abs { \\xi _ n } ^ s } { \\abs { \\xi } ^ s } = \\left ( \\frac { \\xi _ n ^ 2 } { \\xi _ 1 ^ 2 + \\cdots + \\xi _ n ^ 2 } \\right ) ^ { s / 2 } , \\xi \\neq 0 , \\end{align*}"}
{"id": "294.png", "formula": "\\begin{align*} f ( q ) = \\frac { 1 } { q \\log q } = \\frac { 1 } { q } \\frac { e ^ \\gamma } { \\mu _ q } \\prod _ { p < q } \\Big ( 1 - \\frac { 1 } { p } \\Big ) = \\frac { e ^ \\gamma } { \\mu _ q } { \\rm d } ( { \\rm L } _ q ) . \\end{align*}"}
{"id": "6708.png", "formula": "\\begin{align*} \\phi ^ { ( 0 ) } \\Big ( + , \\frac { k } { 2 } \\Big ) \\Big \\vert _ { D _ { - k } \\oplus D _ k } = 0 , \\ ; \\ ; \\ ; k \\in 2 \\mathbb { Z } + 1 . \\end{align*}"}
{"id": "7191.png", "formula": "\\begin{align*} u _ { p + \\frac { [ j _ 1 + j _ 2 ] } { T } } w _ 1 ( n ) w _ 2 = \\sum _ { i \\in \\Lambda } x _ i ( n _ i ) w _ 2 , \\end{align*}"}
{"id": "787.png", "formula": "\\begin{align*} \\frac { \\beta ( 1 + ( 1 - 2 \\alpha ) r ^ m ) } { ( 1 - r ^ m ) ^ { 2 ( 1 - \\alpha ) + 1 } } & + \\frac { ( 1 - \\beta ) r ^ m } { ( 1 - r ^ m ) ^ { 2 ( 1 - \\alpha ) } } + \\sum _ { n = 2 } ^ { \\infty } \\prod _ { k = 0 } ^ { n - 2 } \\frac { k + 2 ( 1 - \\alpha ) } { k + 1 } \\phi _ { n } ( r ) \\\\ & = \\frac { 1 } { 4 ^ { 1 - \\alpha } } - \\phi _ { 1 } ( r ) . \\end{align*}"}
{"id": "5121.png", "formula": "\\begin{align*} A ^ k = \\sum _ { i = 0 } ^ { n - 1 } \\binom { k } { i } N ^ i \\end{align*}"}
{"id": "7006.png", "formula": "\\begin{align*} L _ { { \\bf k } } = L _ 1 ^ { k _ 1 } \\otimes \\ldots \\otimes L _ m ^ { k _ m } \\end{align*}"}
{"id": "440.png", "formula": "\\begin{align*} \\psi _ { \\mu } ( z ) = \\int _ { \\mathbb { R } _ { + } } \\frac { z t } { 1 - z t } \\ , d \\mu ( t ) , \\quad \\eta _ { \\mu } ( z ) = \\frac { \\psi _ { \\mu } ( z ) } { 1 + \\psi _ { \\mu } ( z ) } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
{"id": "8666.png", "formula": "\\begin{align*} Q ( x ) \\geq \\begin{cases} ( 1 + \\epsilon ) \\ , \\frac { ( d - 2 ) ^ 2 } { 4 | x | ^ 2 } & \\ d \\neq 2 \\ , , \\\\ ( 1 + \\epsilon ) \\ , \\frac { 1 } { 4 | x | ^ 2 ( \\ln | x | ) ^ 2 } & \\ d = 2 \\ , , \\end{cases} \\qquad \\ | x | \\geq R \\ , , \\end{align*}"}
{"id": "7142.png", "formula": "\\begin{align*} H ^ * ( X _ { D } ) & = \\mathbb { Z } [ x _ 1 , x _ 2 , x _ 3 ] / \\langle x _ 1 ^ { m _ 1 } , x _ 2 ^ { m _ 2 } , x _ 3 ( - x _ 1 + x _ 2 + x _ 3 ) ( x _ 2 + x _ 3 ) ^ { m _ 3 - 2 } \\rangle , \\\\ H ^ * ( X _ { \\widetilde { D } } ) & = \\mathbb { Z } [ y _ 1 , y _ 2 , y _ 3 ] / \\langle y _ 1 ^ { \\widetilde { m } _ 1 } , y _ 2 ^ { \\widetilde { m } _ 2 } , y _ 3 ( - y _ 1 + y _ 2 + y _ 3 ) ( y _ 2 + y _ 3 ) ^ { \\widetilde { m } _ 3 - 2 } \\rangle . \\end{align*}"}
{"id": "1859.png", "formula": "\\begin{align*} A _ { 0 } ( z ) & = \\frac { 1 } { z - a _ { 0 } ^ { ( 0 ) } - \\sum _ { j = 1 } ^ { p } a _ { 0 } ^ { ( j ) } \\ , A _ { j - 1 } ^ { ( 1 ) } ( z ) } \\\\ A _ { j } ( z ) & = A _ { 0 } ( z ) \\ , A ^ { ( 1 ) } _ { j - 1 } ( z ) 1 \\leq j \\leq p . \\end{align*}"}
{"id": "6623.png", "formula": "\\begin{align*} I _ { g , k , \\ell } ( c ) = h ^ 4 \\int _ { \\mathbb { R } ^ 4 } w ( x ) H ( h ^ { - 1 } k , F ( x ) ) e _ { k \\ell } ( - h c \\cdot g x ) \\ , d x = I _ { k , \\ell } ( { } ^ t g c ) . \\end{align*}"}
{"id": "1598.png", "formula": "\\begin{align*} f ( x ^ 1 ) f '' ( x ^ 1 ) + 3 f '^ 2 ( x ^ 1 ) + 3 = 0 , \\end{align*}"}
{"id": "3511.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast } & = \\frac { \\ln ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) - \\ln ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) } { \\sqrt { 2 \\pi } \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "6381.png", "formula": "\\begin{align*} \\tau _ { \\pi } = \\gamma _ 3 \\gamma _ 4 \\gamma _ 1 \\gamma _ 2 \\gamma _ 4 \\gamma _ 3 \\gamma _ 2 \\gamma _ 1 = ( 1 , 4 ) ( 1 , 5 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 5 ) ( 1 , 4 ) ( 1 , 3 ) ( 1 , 2 ) = ( 1 , 5 , 3 ) \\in S _ 5 \\subseteq S _ { \\infty } . \\end{align*}"}
{"id": "8149.png", "formula": "\\begin{align*} & \\| P _ { \\delta } ( W _ { n ; \\textrm { i n } } ) e ^ { - i H _ 0 t _ n } \\psi \\| = \\| P _ { \\delta } ( W _ { n ; \\textrm { i n } } ) e ^ { - i H _ 0 t _ n } \\chi _ { S _ R } \\psi \\| \\\\ & \\leq \\| \\chi _ { S _ R } e ^ { i H _ 0 t _ n } P _ { \\delta } ( W _ { n ; \\textrm { i n } } ) \\| _ \\textrm { o p } \\| \\psi \\| \\xrightarrow { n \\rightarrow \\infty } 0 \\end{align*}"}
{"id": "3589.png", "formula": "\\begin{align*} v _ 1 ( \\triangle ( p \\rightarrow { \\sim } \\neg p ) ) & = \\begin{cases} 1 & $ p $ \\\\ 0 & $ p $ \\end{cases} \\end{align*}"}
{"id": "4915.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial X } { \\partial t } = \\psi u ^ { \\alpha } \\rho ^ \\delta \\sigma _ { k } ^ { \\frac { \\beta } { k } } ( \\l ) \\nu ( x , t ) , \\\\ & X ( \\cdot , 0 ) = X _ 0 , \\end{cases} \\end{align*}"}
{"id": "1658.png", "formula": "\\begin{align*} \\mathbb { \\hat { E } } [ \\varphi ( X ) ] = \\mathbb { \\hat { E } } [ \\varphi ( Y ) ] . \\end{align*}"}
{"id": "5652.png", "formula": "\\begin{align*} a _ { i j } ( x ) D _ { i j } ( D _ { r } u ) ( x ) = D _ { r } g ( x ) \\quad \\ | x | > r _ 0 , \\end{align*}"}
{"id": "7604.png", "formula": "\\begin{gather*} g _ 1 ( t ) = 5 \\sin 1 0 t , g _ 2 ( t ) = 5 \\cos 1 0 t , g _ 3 ( t ) + g _ 7 ( t ) = - 1 , \\\\ g _ 4 ( t ) = 1 , g _ 5 = 1 , g _ 6 ( t ) + g _ 7 ( t ) = 1 , g _ 8 ( t ) = g _ 9 ( t ) = 0 . \\end{gather*}"}
{"id": "8936.png", "formula": "\\begin{align*} A ( \\cdot , u , p ) = g _ { x x } ( \\cdot , Y ( \\cdot , u , p ) , Z ( \\cdot , u , p ) ) , \\ B ( \\cdotp , u , p ) = \\det E ( \\cdot , Y ( \\cdot , u , p ) , Z ( \\cdot , u , p ) ) \\psi ( \\cdot , u , p ) . \\end{align*}"}
{"id": "5345.png", "formula": "\\begin{align*} 0 = - F ( u _ f - f ) = - B ^ * ( \\phi , u _ f - f ) = B ^ * ( \\phi , f ) = L ^ * ( \\phi , f ) + q ^ * ( \\phi , f ) = L ( f , \\phi ) \\end{align*}"}
{"id": "969.png", "formula": "\\begin{align*} \\overline { f } ^ { \\otimes m } = \\overbrace { \\overline { f } \\otimes \\overline { f } \\otimes \\cdots \\otimes \\overline { f } } ^ { m } , \\ \\ m \\in \\mathbb { Z } , \\ \\ m \\geq 1 . \\end{align*}"}
{"id": "70.png", "formula": "\\begin{align*} \\mathcal E ^ s : = \\{ \\mathcal E _ \\varphi ^ s \\} . \\end{align*}"}
{"id": "21.png", "formula": "\\begin{align*} \\rho _ { \\eta ^ n } ( X ) = \\alpha & \\iff V _ n ( P _ { n m _ 0 } ( X ) ) = \\alpha \\\\ & \\iff P _ { n m _ 0 } ( X ) = V _ n ^ { - 1 } ( \\alpha ) \\\\ & \\iff P _ { n m _ 0 } ( X ) - V _ n ^ { - 1 } ( \\alpha ) = 0 . \\end{align*}"}
{"id": "7075.png", "formula": "\\begin{align*} u _ { i } ( t ) = \\beta _ { i } u _ { i } ( t _ { k } ) + \\alpha _ { i } ( t - t _ { k } ) , \\forall t \\in ( t _ { k } , t _ { k + 1 } ] , \\end{align*}"}
{"id": "4288.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { 2 } \\mathfrak { a } ^ { T } K \\mathfrak { a } , \\prod \\limits _ { l = 1 } ^ m f _ { l } ^ { T } \\mathfrak { a } \\right ] = \\sum \\limits _ { i = 1 } ^ m \\prod \\limits _ { l = 1 } ^ m f _ { l } ^ { T } ( J K ) ^ { \\delta _ { i l } } \\mathfrak { a } . \\end{align*}"}
{"id": "3398.png", "formula": "\\begin{align*} { M } _ i ( e _ i ) = \\frac { 1 } { \\abs { G L _ { e _ i } ( \\kappa _ i ) } } \\left ( \\frac { \\abs { W _ i ^ \\tau } \\abs { H ^ 1 ( G , V _ i ^ \\vee ) } } { \\abs { H ^ 1 ( G , V _ i ) } } \\right ) ^ { e _ i } . \\end{align*}"}
{"id": "5343.png", "formula": "\\begin{align*} \\norm { u } _ { H ^ { s , p } ( \\Omega ) } \\vcentcolon = \\inf \\{ \\ , \\norm { w } _ { H ^ { s , p } ( \\R ^ n ) } \\ , ; \\ , w \\in H ^ { s , p } ( \\R ^ n ) , w | _ \\Omega = v \\ , \\} . \\end{align*}"}
{"id": "3259.png", "formula": "\\begin{align*} u - T _ { A , q } u = v \\mbox { i n } H ^ 1 ( D ) . \\end{align*}"}
{"id": "8962.png", "formula": "\\begin{align*} \\| u \\| ^ p _ { W ^ { k , p } _ \\delta ( M ) } : = \\sum _ { j = 0 } ^ { k } \\left \\vert \\left \\vert \\rho ^ { - \\delta - \\frac { n } { p } + j } \\left \\vert \\hat \\nabla ^ j u \\right \\vert _ { \\hat g } \\right \\vert \\right \\vert ^ p _ { L ^ p ( M , \\hat g ) } \\end{align*}"}
{"id": "8316.png", "formula": "\\begin{align*} \\mathfrak { T } ( t ) e ^ { z t } = \\sum _ { n = 0 } ^ { \\infty } \\frac { w _ n ( z ) } { n ! } t ^ n , \\end{align*}"}
{"id": "2919.png", "formula": "\\begin{align*} g _ i ( x ) & = ( \\textup { r o w $ i $ o f $ B $ } ) \\cdot ( g \\circ S ^ { 1 - i } ) ( x ) + d ' \\\\ & = \\sum _ { j = 1 } ^ n b _ j ( g \\circ S ^ { 1 - j - ( 1 - ( 1 + ( i - 1 ) k ) ) } ) ( x ) + d ' = \\sum _ { j = 1 } ^ n b _ j ( g \\circ S ^ { 1 - j + ( 1 - i ) k } ) ( x ) + d ' , \\end{align*}"}
{"id": "5774.png", "formula": "\\begin{align*} E ( G / A ) = E ( G / H ) E ( H / A ) \\neq 0 \\end{align*}"}
{"id": "2872.png", "formula": "\\begin{align*} r = \\frac { \\dim M - \\dim \\left ( \\ker \\omega _ x \\right ) } { 2 } + \\dim ( T _ x { \\mathcal L } \\cap \\ker \\omega _ x ) , \\hbox { f o r a l l } \\ ; x \\in M \\ , . \\end{align*}"}
{"id": "2091.png", "formula": "\\begin{align*} N _ { X _ { H , V } } ( p m , n ) = \\sum _ { u \\in \\{ 0 , \\dots , p - 1 \\} ^ n } \\left ( \\Pi _ { j = 0 } ^ { p - 1 } N _ { u ^ j } \\right ) ^ m . \\end{align*}"}
{"id": "7384.png", "formula": "\\begin{align*} \\psi _ n ( y ) = \\bar \\psi \\left ( y - h _ n t _ 1 \\right ) + \\bar \\psi \\left ( T _ 5 \\left ( y - h _ n t _ 1 \\right ) \\right ) + \\bar \\psi \\left ( T _ 9 \\left ( y - h _ n t _ 1 \\right ) \\right ) . \\end{align*}"}
{"id": "5405.png", "formula": "\\begin{align*} \\langle \\Theta \\nabla ^ s u , \\nabla ^ s \\phi \\rangle _ { L ^ 2 ( \\R ^ { 2 n } ) } = & \\frac { 1 } { 2 } \\int _ { \\R ^ n } \\gamma ^ { 1 / 2 } ( x ) \\left [ \\phi ( x ) ( - \\Delta ) ^ s ( \\gamma ^ { 1 / 2 } u ) ( x ) + u ( x ) ( - \\Delta ) ^ s ( \\gamma ^ { 1 / 2 } \\phi ) ( x ) \\right . \\\\ & \\quad \\left . - ( - \\Delta ) ^ { s } ( \\gamma ^ { 1 / 2 } u \\phi ) ( x ) - u ( x ) \\phi ( x ) ( - \\Delta ) ^ s m ( x ) \\right ] \\ , d x . \\end{align*}"}
{"id": "7904.png", "formula": "\\begin{align*} z ^ { - \\mu } z ^ { \\rho } : = \\exp ( - \\mu \\log z ) \\exp ( \\rho \\log z ) \\end{align*}"}
{"id": "4536.png", "formula": "\\begin{align*} D _ { m , n } ( x , y ) : = \\sigma _ { m , 2 m ; n , 2 n } ( f ; x , y ) - f ( x , y ) . \\end{align*}"}
{"id": "658.png", "formula": "\\begin{align*} Q ( g ) : = D L _ { ( g , P ( g ) ) } ( 0 , \\cdot ) . \\end{align*}"}
{"id": "5008.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) ) ^ 2 \\Psi ^ { n , 4 } _ s d s \\\\ & = \\frac { \\alpha } { 2 ( \\alpha + 1 ) ^ 2 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "3190.png", "formula": "\\begin{align*} g \\left ( x \\right ) = \\sum _ { z \\in \\mathbb { Z } ^ { d } } \\gamma ^ { d } f \\left ( x + \\gamma z \\right ) \\ , x \\in \\mathbb { R } ^ { d } \\ . \\end{align*}"}
{"id": "4160.png", "formula": "\\begin{align*} h _ \\ell ( t ) : = ( - 1 ) ^ \\ell ( 2 ^ \\ell \\ell ! \\sqrt { \\pi } ) ^ { - 1 / 2 } e ^ { t ^ 2 / 2 } \\Big ( \\frac { d } { d t } \\Big ) ^ \\ell ( e ^ { - t ^ 2 } ) , u \\in \\R . \\end{align*}"}
{"id": "362.png", "formula": "\\begin{align*} J = J ( n ) : = \\left \\lfloor 4 \\frac { \\log \\log n } { \\log ( b ( n ) ) } - 1 \\right \\rfloor , \\end{align*}"}
{"id": "4082.png", "formula": "\\begin{align*} E ( X _ 2 ^ 2 ) = \\varepsilon ^ 2 \\big [ P ( \\boldsymbol { X } = \\mathbf { s } ( 2 ) ) + P ( \\boldsymbol { X } = \\mathbf { s } ( 5 ) ) \\big ] , \\end{align*}"}
{"id": "8046.png", "formula": "\\begin{align*} \\int _ C \\frac { 1 } { e ^ { 2 \\pi i ( s + \\bar { f } ) } - 1 } q ^ { s + \\bar { f } } \\frac { \\Gamma ( 1 + \\frac { d H } { 2 \\pi i } + s d + \\bar { f } d ) } { \\Gamma ( 1 - \\frac { d ^ \\prime H } { 2 \\pi i } - s d ^ \\prime + \\bar { f } d ^ \\prime ) \\prod _ { i = 1 } ^ N \\Gamma ( 1 + \\frac { w _ i H } { 2 \\pi i } + w _ i s + \\bar { f } w _ i ) } , \\end{align*}"}
{"id": "6259.png", "formula": "\\begin{align*} & \\int f ( t ) h ( t / q ) \\left ( \\frac { 1 - t ( 1 + q ) } { q ( 1 - q ) t ^ 2 } u ( t / q ) + \\frac { 1 } { ( 1 - q ) ^ 2 t ^ 2 } \\right ) y ( t ) d _ q t \\\\ & = f ( x / q ) h ( x / q ) \\left ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \\right ) . \\end{align*}"}
{"id": "2714.png", "formula": "\\begin{align*} \\langle v , K _ { 1 1 } u \\rangle = \\ ; & \\frac { 1 } { 2 } \\langle v \\otimes u _ 1 \\ , , \\ , w \\ , u _ 1 \\otimes u \\rangle \\\\ \\langle v , K _ { 2 2 } u \\rangle = \\ ; & \\frac { 1 } { 2 } \\langle v \\otimes u _ 2 \\ , , \\ , w \\ , u _ 2 \\otimes u \\rangle \\\\ [ 2 m m ] \\langle v , K _ { 1 2 } u \\rangle = \\ ; & \\langle v \\otimes u _ 1 \\ , , \\ , w \\ , u _ 2 \\otimes u \\rangle . \\end{align*}"}
{"id": "6459.png", "formula": "\\begin{align*} \\Bar \\Phi \\circ Q _ \\mathfrak { g } = \\Bar { Q } \\circ \\Bar \\Phi . \\end{align*}"}
{"id": "8144.png", "formula": "\\begin{align*} & \\varphi _ { n ; \\textrm { o u t } } = P _ { \\delta } ( W _ { n , m ; \\textrm { o u t } } ) \\varphi _ n & & \\varphi _ { n ; \\textrm { i n } } = P _ { \\delta } ( W _ { n , m ; \\textrm { i n } } ) \\varphi _ n \\\\ & \\varphi _ { n ; \\textrm { s u r } } = P _ { \\delta } ( W _ { n , m ; \\textrm { s u r } } ) \\varphi _ n \\end{align*}"}
{"id": "3117.png", "formula": "\\begin{align*} a ( u , J I { w } ) - a _ { \\mathrm { p w } } ( { w } , z _ { \\mathrm { n c } } ) & = b ( \\lambda u , J I { w } ) - a _ { \\mathrm { p w } } ( I { w } , z _ { \\mathrm { n c } } ) = \\lambda b ( u , J I { w } - I { w } - \\delta I { w } ) . \\end{align*}"}
{"id": "5624.png", "formula": "\\begin{align*} X - 1 = \\big ( X ^ { \\frac 1 { 2 ^ n } } - 1 \\big ) \\prod _ { i = 1 } ^ n \\big ( X ^ { \\frac { 1 } { 2 ^ i } } + 1 \\big ) \\end{align*}"}
{"id": "8153.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } \\| e ^ { - i t H _ 0 } \\varphi - e ^ { - i t H } \\psi \\| = 0 \\end{align*}"}
{"id": "3129.png", "formula": "\\begin{align*} \\Lambda _ { \\mathrm { m o n } } ^ 2 : = \\frac { 1 + 1 / \\alpha + ( 1 + \\alpha ) ( \\Lambda _ 1 ^ 2 + \\Lambda _ 2 ^ 2 ) ( \\Lambda _ 3 + \\epsilon _ 3 ) } { 1 - ( 1 + \\alpha ) ( \\Lambda _ 1 ^ 2 + \\Lambda _ 2 ^ 2 ) \\widehat \\Lambda _ 3 } . \\end{align*}"}
{"id": "4151.png", "formula": "\\begin{align*} X _ j ^ \\mu = \\partial _ { x _ j } + \\tfrac i 2 \\mu ( [ x , X _ j ] ) . \\end{align*}"}
{"id": "1900.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { U } _ { [ n , j ] } } w ( \\gamma ) = \\sum _ { k \\in \\mathbb { Z } } a _ { - j } ^ { ( j ) } \\ , B _ { [ k - 1 , j - 1 ] } ^ { ( 1 ) } W _ { [ n - k - 1 , 0 ] } , 1 \\leq j \\leq p . \\end{align*}"}
{"id": "469.png", "formula": "\\begin{align*} \\beta _ i ( z ) & : = \\beta _ { i - 1 } ( z ) + \\sum _ { m = 1 } ^ { s _ i } \\frac 1 m \\sum _ { a | m } \\mu \\left ( \\frac m a \\right ) \\left ( \\alpha _ { i - 1 } ( z ^ { m / a } ) \\right ) ^ a \\ ! \\ ! , \\\\ \\alpha _ i ( z ) & : = 1 + ( k z - 1 ) \\cdot { \\mathcal E } ( \\beta _ i ( z ) ) , 1 \\le i \\le q . \\end{align*}"}
{"id": "585.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) = & f _ k ( i ) f _ k ( i + 1 ) f _ k ( j ) f _ k ( j + 1 ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( i ) \\tau _ 3 ( i + 1 ) \\tau _ 3 ( j ) \\tau _ 3 ( j + 1 ) } { \\sqrt { i } } \\Big ) \\\\ & + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) \\tau _ 3 ( j + 1 ) } { \\sqrt { j - i - 1 } } \\Big ) , \\end{align*}"}
{"id": "167.png", "formula": "\\begin{align*} \\| C ^ 3 _ { ( m ) } \\| _ { L ^ 1 } = \\frac { 1 } { L ^ { 2 d } } \\sum _ { k , p \\in \\Lambda ^ * } \\frac { 1 } { m ^ 2 + \\theta ( p ) } \\cdot \\frac { 1 } { m ^ 2 + \\theta ( q ) } \\cdot \\frac { 1 } { m ^ 2 + \\theta ( p + q ) } . \\end{align*}"}
{"id": "2332.png", "formula": "\\begin{align*} f = a _ 0 + a _ 1 Q + \\ldots + a _ r Q ^ r = l ( Q ) \\end{align*}"}
{"id": "6388.png", "formula": "\\begin{align*} \\Phi ( S ) = \\left \\{ \\begin{array} { l l } S \\setminus \\{ p \\} , & \\mbox { i f $ p \\in S $ , } \\\\ S \\cup \\{ p \\} , & \\mbox { i f $ p \\not \\in S $ } \\end{array} \\right \\} , \\ \\ \\mbox { f o r } S \\subseteq \\{ 1 , \\ldots , k \\} . \\end{align*}"}
{"id": "4513.png", "formula": "\\begin{align*} \\nabla ^ \\Sigma _ { \\alpha \\beta } F = | \\nabla u | ^ { - 2 } A _ { \\alpha \\beta } . \\end{align*}"}
{"id": "1800.png", "formula": "\\begin{align*} \\phi _ \\mathrm { d c v } ( x , z ) = \\phi _ + ( x , z ) - h ( x ) , \\phi _ \\mathrm { p c v } ( x , z ) = \\phi _ - ( x , z ) - h ( x ) . \\end{align*}"}
{"id": "7548.png", "formula": "\\begin{align*} \\gamma _ { p , i } ^ k & = 2 ^ { \\gamma _ { p , i } ^ \\star } - 1 , i \\in \\mathcal K , & s ^ k = 2 ^ { s ^ \\star } - 1 \\end{align*}"}
{"id": "7870.png", "formula": "\\begin{align*} x \\tilde \\delta _ \\alpha ( a ) = \\alpha ^ { - 1 / 2 } \\delta ( x \\zeta _ \\alpha ( a ) ) - \\alpha ^ { - 1 / 2 } \\delta ( x ) \\zeta _ \\alpha ( a ) = : A _ 1 - A _ 2 . \\end{align*}"}
{"id": "3857.png", "formula": "\\begin{align*} X \\otimes ^ G Y ( \\underbrace { \\omega _ g , . . . } _ { g \\in G } ) = X ( \\underbrace { \\omega _ { \\psi ^ { P , G } [ p ] } , . . . } _ { p \\in P } ) \\otimes Y ( \\underbrace { \\omega _ { \\psi ^ { Q , G } [ q ] } , . . . } _ { q \\in Q } ) . \\end{align*}"}
{"id": "6739.png", "formula": "\\begin{align*} \\mu \\int _ \\Omega \\varepsilon ( w _ \\delta - z _ \\delta ) : \\varepsilon v + \\nu \\int _ \\Omega ( \\mathsf { m } _ \\delta ( \\varepsilon w _ \\delta ) - \\mathsf { m } _ \\delta ( \\varepsilon z _ \\delta ) ) : \\varepsilon v = \\langle u _ 1 - u _ 2 , v \\rangle , \\forall v \\in Y . \\end{align*}"}
{"id": "638.png", "formula": "\\begin{align*} \\frac { k _ 1 ^ 4 + k _ 2 ^ 4 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 4 } & = \\frac { 1 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 2 } - \\frac { 2 z ^ 2 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 3 } \\\\ & + \\frac { z ^ 4 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 4 } - \\frac { 2 k _ 1 ^ 2 k _ 2 ^ 2 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 4 } , \\\\ \\frac { ( k _ 1 ^ 2 + k _ 2 ^ 2 ) ^ 2 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 4 } & = \\frac { 1 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 2 } - \\frac { 2 z ^ 2 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 3 } + \\frac { z ^ 4 } { ( k _ 1 ^ 2 + k _ 2 ^ 2 + z ^ 2 ) ^ 4 } . \\end{align*}"}
{"id": "7485.png", "formula": "\\begin{align*} \\partial _ \\mu ^ 2 E ( \\lambda , \\mu ) & = - \\lim _ { T \\to \\infty } \\frac 1 T \\partial _ \\mu ^ 2 ( \\ln Z ^ { ( W ) } _ T ( \\lambda , \\mu ) + 1 ) \\\\ & = - \\lim _ { T \\to \\infty } \\frac 1 T \\frac { \\partial _ \\mu ^ 2 Z ^ { ( W ) } _ T ( \\lambda , \\mu ) } { Z ^ { ( W ) } _ T ( \\lambda , \\mu ) } . \\end{align*}"}
{"id": "2672.png", "formula": "\\begin{align*} & a _ \\ell ( c _ i ^ 2 - c _ i ) - a _ k ( b _ i ^ 2 - b _ i ) = 0 i > \\ell \\\\ & a _ \\ell c _ i c _ j - a _ k b _ i b _ j = 0 i > \\ell , j > i . \\end{align*}"}
{"id": "4371.png", "formula": "\\begin{align*} & \\int _ { D _ 0 } | F _ j - ( 1 - b _ { t _ j , 1 } ( k \\Psi ) ) f _ j F ^ { 2 k } | ^ 2 e ^ { - k \\varphi + v _ { t _ j , 1 } ( k \\Psi ) - k \\Psi } \\\\ \\le & ( 2 - e ^ { - t _ j - 1 } ) \\int _ { D _ 0 } \\mathbb { I } _ { \\{ - t _ j - 1 < k \\Psi < - t _ j \\} } | f _ j | ^ 2 e ^ { - k \\Psi } \\\\ \\le & ( 2 e ^ { t _ j + 1 } - 1 ) \\int _ { \\{ k \\Psi < - t _ j \\} \\cap D _ 0 } | f _ j | ^ 2 , \\end{align*}"}
{"id": "6580.png", "formula": "\\begin{align*} \\{ f , g \\} ( x ) \\ ; : = \\ ; \\langle [ f ' ( x ) , g ' ( x ) ] , x \\rangle \\mbox { f o r $ x \\in U $ , } \\end{align*}"}
{"id": "788.png", "formula": "\\begin{align*} \\frac { \\beta ( 1 + r ^ m ) } { ( 1 - r ^ m ) ^ { 3 } } + \\frac { ( 1 - \\beta ) r ^ m } { ( 1 - r ^ m ) ^ { 2 } } + \\sum _ { n = 1 } ^ { \\infty } n \\phi _ { n } ( r ) = \\frac { 1 } { 4 } . \\end{align*}"}
{"id": "7574.png", "formula": "\\begin{align*} \\varphi ( \\exp ( \\lambda _ \\alpha v _ \\alpha ) , x ) = \\varphi _ \\alpha ( \\lambda _ \\alpha , x ) \\forall \\ , \\alpha = 1 , \\dots , r , \\forall x \\in N , \\end{align*}"}
{"id": "2891.png", "formula": "\\begin{align*} f _ 0 = f _ { k + 1 } ( x _ 1 , \\cdots , x _ { i - 1 } , 0 , x _ { i + 1 } , \\cdots , x _ { k + 1 } ) \\end{align*}"}
{"id": "248.png", "formula": "\\begin{align*} v = ( v _ 1 , \\ldots , v _ n ) ^ t : = \\delta ^ { ( s ) } c : = ( \\delta ^ { ( s ) } c _ 1 , \\ldots , \\delta ^ { ( s ) } c _ n ) ^ t \\in R ^ n \\end{align*}"}
{"id": "917.png", "formula": "\\begin{align*} P ( u ) \\cdot w = \\begin{bmatrix} 0 \\\\ g \\end{bmatrix} , w ( x ) \\perp \\mathrm { K e r } P ( u ) ( x ) , \\end{align*}"}
{"id": "2804.png", "formula": "\\begin{align*} & c _ 1 : = \\max _ { h \\in ( 0 , 1 ] } \\ , \\ , \\ , \\ , 2 h - h ^ 2 \\frac { - \\kappa } { 1 - \\kappa } = 2 - \\frac { - \\kappa } { 1 - \\kappa } \\\\ & c _ 2 : = \\max _ { h \\in [ 1 , \\bar { h } ( \\kappa ) ] } 2 h - h ^ 3 \\frac { - \\kappa } { 2 - ( 1 + \\kappa ) h } \\geq 2 - \\frac { \\kappa } { 2 - ( 1 + \\kappa ) } = c _ 1 \\end{align*}"}
{"id": "5562.png", "formula": "\\begin{align*} \\int s W _ + ( \\omega , \\omega ) d \\mu = \\int \\Big [ 8 | W _ + | ^ 2 - 4 | W _ + ( \\omega ) | ^ 2 + 2 [ W _ + ( \\omega , \\omega ) ] ^ 2 \\Big ] d \\mu . \\end{align*}"}
{"id": "37.png", "formula": "\\begin{align*} \\delta _ n ( \\beta ) : = \\dfrac { f ' ( v _ n ) } { \\beta } \\mod \\mathcal { D } _ n , \\end{align*}"}
{"id": "6729.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { } \\int _ { \\Omega } \\varepsilon ( z + t _ n w ) : \\varepsilon v = \\int _ { \\Omega } \\varepsilon ( z + t w ) : \\varepsilon v . \\end{align*}"}
{"id": "6057.png", "formula": "\\begin{align*} \\Lambda = ( \\lambda _ 1 , \\lambda _ 2 ) \\subseteq K \\end{align*}"}
{"id": "1487.png", "formula": "\\begin{align*} e _ { \\lambda } ^ { x } ( t ) = \\sum _ { n = 0 } ^ { \\infty } ( x ) _ { n , \\lambda } \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 1 1 , 1 4 ] ) . \\end{align*}"}
{"id": "2731.png", "formula": "\\begin{align*} \\sum _ { u = i } ^ s q _ { \\alpha _ u } \\alpha _ u \\alpha ^ * _ u - \\sum _ { v = i } ^ t \\beta _ v \\beta _ v ^ * \\end{align*}"}
{"id": "1678.png", "formula": "\\begin{align*} ( 1 - \\alpha ( 1 + ( 2 - \\alpha ) A ) ) ^ 2 \\| y _ 1 + y _ 2 \\| & \\le \\| O _ 1 y _ 1 + O _ 2 y _ 2 \\| \\\\ & \\le ( 1 + \\alpha ( 1 + ( 2 + \\alpha ) A ) ) ^ 2 \\| y _ 1 + y _ 2 \\| , \\end{align*}"}
{"id": "8849.png", "formula": "\\begin{align*} \\mu _ { \\beta , N } ( d x ) = \\frac { 1 } { Z _ { \\beta , N } } e ^ { - \\frac { \\beta } { 4 } \\sum _ { i = 1 } ^ N x _ i ^ 2 } \\prod _ { 1 \\le i < j \\le N } | x _ i - x _ j | ^ { \\beta } \\prod _ { i = 1 } ^ N d x _ i , \\end{align*}"}
{"id": "6930.png", "formula": "\\begin{align*} M f ( t ) = t f ( t ) \\end{align*}"}
{"id": "4199.png", "formula": "\\begin{align*} \\Vert g _ { \\le \\iota } ^ { ( 1 ) } \\Vert _ p = \\bigg \\Vert \\sum _ { \\ell = - 1 } ^ \\iota \\sum _ { m = 1 } ^ { M _ { \\ell } } \\tilde g _ { m } ^ { ( \\ell ) } \\bigg \\Vert _ p \\lesssim 2 ^ { - \\varepsilon \\iota } \\norm { F ^ { ( \\iota ) } } _ { L ^ 2 _ s } \\norm { f } _ p , \\end{align*}"}
{"id": "6087.png", "formula": "\\begin{align*} f ( x ) & = x ^ 4 - 2 x ^ 3 + x \\\\ & = ( x - 1 ) ^ 4 + 2 ( x - 1 ) ^ 3 - ( x - 1 ) \\\\ & = x ( x - 1 ) ( x ^ 2 - x - 1 ) \\end{align*}"}
{"id": "2045.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : k a c c h a o s } \\lim _ { N \\to + \\infty } \\langle f ^ N , \\varphi _ k \\otimes 1 ^ { \\otimes N - k } \\rangle = \\langle f ^ { \\otimes k } , \\varphi _ k \\rangle . \\end{align*}"}
{"id": "8974.png", "formula": "\\begin{align*} \\left ( g _ u \\circ f _ u \\right ) ( b _ n ) = k _ n . \\end{align*}"}
{"id": "5762.png", "formula": "\\begin{align*} G = N T S , \\end{align*}"}
{"id": "7927.png", "formula": "\\begin{align*} [ f ] \\in \\mathbb K / \\mathbb L \\mapsto v _ f = \\sum _ { i = 1 } ^ m \\lceil - ( D _ i \\cdot f ) \\rceil b _ i \\in N , \\end{align*}"}
{"id": "4593.png", "formula": "\\begin{align*} \\mathcal { J } = \\mathcal { J } _ \\delta : = \\{ i \\in [ N ] : \\ , \\gamma _ i ( 0 ) \\in \\mathcal { I } _ \\delta \\} , \\mathcal { I } _ \\delta : = ( - 2 + \\delta , 2 - \\delta ) , \\end{align*}"}
{"id": "2793.png", "formula": "\\begin{align*} s _ i ( h _ i , \\kappa ) : = 2 h _ i - h _ i ^ 2 \\frac { - \\kappa } { 2 \\min \\left ( 1 , \\frac { 1 } { h _ i } \\right ) - ( 1 + \\kappa ) } \\end{align*}"}
{"id": "3970.png", "formula": "\\begin{align*} \\alpha _ { a } ( e ) & = \\theta ( a ) e \\theta ( a ) ^ { * } & & \\\\ \\theta ( \\sigma a \\sigma ^ { - 1 } ) & = \\overline { \\alpha } _ { \\sigma } ( \\theta ( a ) ) & & \\end{align*}"}
{"id": "5639.png", "formula": "\\begin{align*} \\left | \\iint _ { Q _ T } ( \\vec { b } \\cdot \\nabla m ) \\Delta p \\right | = \\left | \\iint _ { Q _ T } - \\nabla ( \\vec { b } \\cdot \\nabla m ) \\cdot \\nabla p \\right | \\leq C \\iint _ { Q _ T } | \\nabla p | \\leq C . \\end{align*}"}
{"id": "8470.png", "formula": "\\begin{align*} A : & = \\{ i \\in \\{ 1 , 2 , \\ldots , n \\} : y _ i \\leq \\widetilde { y } _ { i ^ * } \\} \\\\ B : & = \\{ i \\in \\{ 1 , 2 , \\ldots , n \\} : y _ i \\geq \\widetilde { y } _ { i ^ { * } + 1 } \\} . \\end{align*}"}
{"id": "4772.png", "formula": "\\begin{align*} \\left | \\mathcal T _ { \\kappa , 2 } \\right | = \\sum _ { i = 0 } ^ { \\lfloor \\kappa / 2 \\rfloor } ( \\kappa - 2 i + 1 ) . \\end{align*}"}
{"id": "8647.png", "formula": "\\begin{align*} \\frac { d \\beta } { d t } ( t ) = \\beta ^ 2 ( t ) + \\omega ^ 2 ( t ) , \\frac { d \\alpha } { d t } ( t ) + 2 \\beta ( t ) = 0 , \\end{align*}"}
{"id": "8296.png", "formula": "\\begin{align*} \\hat { P } \\phi _ { 0 } ( x ) = \\hat { T } \\phi _ { 0 } ( x ) = \\phi _ { 0 } ( x ) \\end{align*}"}
{"id": "8870.png", "formula": "\\begin{align*} \\begin{vmatrix} \\det \\left ( ( \\lambda I _ N - H ) _ { k | k } \\right ) & \\det \\left ( ( \\lambda I _ N - H ) _ { k | k + 1 } \\right ) \\\\ \\det \\left ( ( \\lambda I _ N - H ) _ { k + 1 | k } \\right ) & \\det \\left ( ( \\lambda I _ N - H ) _ { k + 1 | k + 1 } \\right ) \\end{vmatrix} = f ( \\lambda ) \\det \\left ( ( \\lambda I _ N - H ) _ { k k + 1 | k k + 1 } \\right ) , \\end{align*}"}
{"id": "5515.png", "formula": "\\begin{align*} \\abs { h ( z ) - z } \\leq 6 \\sum _ { n = 2 } ^ \\infty \\left ( \\frac { 3 } { 2 } \\abs { z } \\right ) ^ n < 6 \\cdot \\frac { 9 } { 4 } \\abs { z } ^ 2 \\sum _ { n = 0 } ^ \\infty \\left ( \\frac { 3 } { 4 } \\right ) ^ n < 1 0 0 \\abs { z } ^ 2 . \\end{align*}"}
{"id": "529.png", "formula": "\\begin{align*} F ( \\dot { x } , x , t ) = \\frac { h _ 1 ( t ) h _ 2 ( t ) } { [ h _ 2 ( t ) x ( t ) + h _ 4 ( t ) ] ^ 2 } \\left [ \\dot x + \\left ( \\frac { \\dot { h } _ 2 ( t ) } { h _ 2 ( t ) } - \\frac { \\dot { h } _ 1 ( t ) } { h _ 1 ( t ) } \\right ) x \\right ] \\end{align*}"}
{"id": "4308.png", "formula": "\\begin{align*} \\mathcal { S } ( L ) = \\int _ L f _ L ( e ^ { - i \\hat { \\theta } } \\Omega ) - \\int _ { L _ 0 } f _ { L _ 0 } ( e ^ { - i \\hat { \\theta } } \\Omega ) - \\int _ { \\mathcal { C } } \\lambda \\wedge e ^ { - i \\hat { \\theta } } \\Omega . \\end{align*}"}
{"id": "8121.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial r } h ( r , s ) & = \\frac { d ^ { - 1 } ( \\log ^ { ( N ) } ( r \\exp ^ { ( N ) } ( s ^ d ) ) ) ^ { 1 / d - 1 } } { r \\prod _ { p = 1 } ^ { N - 1 } \\log ^ { ( p ) } ( r \\exp ^ { ( N ) } ( s ^ d ) ) } , \\\\ \\frac { \\partial } { \\partial s } h ( r , s ) & = \\frac { s ^ { d - 1 } ( \\log ^ { ( N ) } ( r \\exp ^ { ( N ) } ( s ^ d ) ) ) ^ { 1 / d - 1 } \\prod _ { p = 1 } ^ { N - 1 } \\exp ^ { ( p ) } ( s ^ d ) } { \\prod _ { p = 1 } ^ { N - 1 } \\log ^ { ( p ) } ( r \\exp ^ { ( N ) } ( s ^ d ) ) } . \\end{align*}"}
{"id": "562.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ j ) = \\mathbb { P } ( P _ i , P _ j \\ { \\rm { a r e \\ b o t h } } \\ k - { \\rm { f r e e } } ) . \\end{align*}"}
{"id": "5907.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { V \\in \\mathbb { V } _ { n , k } } \\rho _ \\mathrm { L P } ( \\mu _ { V X ^ { ( n ) } } , \\tilde { \\mu } _ { V X ^ { ( n ) } } ) = 0 . \\end{align*}"}
{"id": "7085.png", "formula": "\\begin{align*} e ( t _ { k + 1 } ) = \\left \\{ \\begin{aligned} & A _ { 1 } e ( t _ { k } ) & & \\beta ^ { \\top } e ( t _ { k } ) \\geq 0 \\\\ & A _ { 2 } e ( t _ { k } ) & & \\beta ^ { \\top } e ( t _ { k } ) < 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "5625.png", "formula": "\\begin{align*} V = \\left [ - \\frac { m \\nabla p _ { \\infty } } { ( m - \\varrho ^ E ) } - \\vec { b } \\right ] \\cdot \\vec { \\nu } , \\end{align*}"}
{"id": "2608.png", "formula": "\\begin{align*} 2 \\nu = r _ + + r _ - \\ , \\ \\ 3 \\nu = a + 2 b \\ , \\end{align*}"}
{"id": "8500.png", "formula": "\\begin{align*} \\rho ( \\varepsilon ) = \\frac { 2 b d k \\log ( \\frac { 4 b d } { \\varepsilon } ) } { \\varepsilon ^ 2 } + \\frac { 2 k ^ d \\log ( \\frac { 4 k ^ d } { \\varepsilon } ) } { \\varepsilon ^ 2 } + \\frac { \\log ( \\frac { 3 } { \\delta } ) } { 2 \\varepsilon ^ 2 } = n . \\end{align*}"}
{"id": "2256.png", "formula": "\\begin{align*} e _ \\delta ( z ) = \\frac { \\delta } { \\sqrt \\pi } \\int _ 0 ^ z e ^ { - \\frac { t ^ 2 } { 4 } } { \\rm { d } } t , e _ \\sigma ( z ) = \\frac { \\sigma } { \\sqrt \\pi } \\int _ 0 ^ z e ^ { - \\frac { t ^ 2 } { 4 } } { \\rm { d } } t , \\end{align*}"}
{"id": "8785.png", "formula": "\\begin{align*} \\| \\phi _ 1 * \\phi _ 2 \\| _ { q ( p _ 1 , p _ 2 ) } \\geq \\| \\phi _ 1 ^ * * \\phi _ 2 ^ * \\| _ { q ( p _ 1 , p _ 2 ) } \\geq C ( p _ 1 , p _ 2 ) \\| \\phi _ 1 ^ * \\| _ { p _ 1 } \\| \\phi _ 2 ^ * \\| _ { p _ 2 } = C ( p _ 1 , p _ 2 ) \\| \\phi _ 1 \\| _ { p _ 1 } \\| \\phi _ 2 \\| _ { p _ 2 } \\end{align*}"}
{"id": "3582.png", "formula": "\\begin{align*} \\chi _ o ( Q _ d ) = \\begin{cases} 2 & d \\ , ; \\\\ 4 & d \\ , . \\end{cases} \\end{align*}"}
{"id": "5699.png", "formula": "\\begin{align*} ( x _ i - x _ { i + 1 } ) e _ k ( x _ 1 , \\ldots , x _ i ) = 0 \\end{align*}"}
{"id": "5148.png", "formula": "\\begin{align*} \\mathcal { N } ( n , 2 c + 1 , \\{ 1 , 2 c \\} ) & = \\mathcal { N } ( 2 c q + r + 2 c , 2 c + 1 , \\{ 1 , 2 c \\} ) \\\\ & = \\mathcal { N } ( 2 c q + r + 6 , 7 , \\{ 1 , 2 c \\} ) & \\ref { t w o } \\\\ & = \\mathcal { N } ( 6 q + r ' + 6 , 7 , \\{ 1 , 6 \\} ) & \\ref { t h r e e } \\\\ & = \\mathcal { N } ( \\phi _ 1 ( n ) , 7 , \\{ 1 , 6 \\} ) & \\end{align*}"}
{"id": "7864.png", "formula": "\\begin{align*} \\psi ( u _ t T ) & = \\varphi ( e _ M u _ t T e _ M ) = \\varphi ( e _ M ( u _ t J u _ t J ) T ( J u _ t J ) ^ * e _ M ) = \\varphi ( \\alpha _ t ^ 0 e _ M T ( J u _ t J ) ^ * e _ M ) \\\\ & = \\varphi ( e _ M T ( J u _ t J ) ^ * e _ M \\alpha _ t ^ 0 ) = \\varphi ( e _ M T ( J u _ t J ) ^ * u _ t J u _ t J e _ M ) = \\psi ( T u _ t ) . \\end{align*}"}
{"id": "4230.png", "formula": "\\begin{align*} \\# \\left \\{ i : \\frac { d _ { i + 1 , k } } { d _ { i , k } } > \\exp \\left [ { - \\frac { 1 } { 2 \\psi ^ { - 1 } ( k ) } } \\right ] \\right \\} > \\frac { \\log e ^ { - 1 / 2 } } { \\log \\exp ( - k / ( 2 \\psi ^ { - 1 } ( k ) ) } = \\frac { 1 } { k } \\psi ^ { - 1 } ( k ) . \\end{align*}"}
{"id": "190.png", "formula": "\\begin{align*} \\P ( x ^ 1 _ { 1 : n } , x ^ 2 _ { 1 : n } , y _ { 1 : n } ) = \\prod _ { t = 1 } ^ n Q ( y _ t | & x ^ 1 _ t , x ^ 2 _ t ) q ^ 1 _ t ( x ^ 1 _ t | x ^ 1 _ { 1 : t - 1 } , y _ { 1 : t - 1 } ) \\times \\\\ & q ^ 2 _ t ( x ^ 2 _ t | x ^ 2 _ { 1 : t - 1 } , y _ { 1 : t - 1 } ) , \\end{align*}"}
{"id": "4318.png", "formula": "\\begin{align*} d F = \\Omega ( \\cdot , v _ 1 , \\ldots , v _ { n - 1 } ) . \\end{align*}"}
{"id": "6441.png", "formula": "\\begin{align*} \\eta ( e _ 1 , e _ 2 ) : = \\dfrac { \\partial \\{ F , G \\} } { \\partial y } \\ , \\mu \\otimes _ \\mathcal { O } \\dfrac { \\partial } { \\partial x } - \\dfrac { \\partial \\{ F , G \\} } { \\partial x } \\ , \\mu \\otimes _ \\mathcal { O } \\dfrac { \\partial } { \\partial y } . \\end{align*}"}
{"id": "6446.png", "formula": "\\begin{align*} \\ker \\left ( \\Gamma ( \\varphi ^ * E _ { - 2 } ) \\stackrel { \\varphi ^ * \\dd ^ { ( 2 ) } } { \\longrightarrow } \\Gamma ( \\varphi ^ * E _ { - 1 } ) \\right ) & \\longrightarrow \\ker \\left ( d t _ 1 | _ { \\Gamma ( \\ker d s _ 1 ) } \\right ) = \\Gamma ( \\ker d s _ 1 ) \\cap \\Gamma ( \\ker d t _ 1 ) \\subset \\mathfrak { X } ( B _ 2 ) \\end{align*}"}
{"id": "6732.png", "formula": "\\begin{align*} \\mu \\int _ { \\Omega } \\varepsilon z : \\varepsilon v + \\nu \\int _ { \\Omega } \\mathsf { m } ' \\left ( \\varepsilon y ; \\varepsilon z \\right ) : \\varepsilon v = \\int _ { \\Omega } h \\cdot v , \\forall v \\in Y , \\end{align*}"}
{"id": "1040.png", "formula": "\\begin{align*} \\phi ( t ) = \\| T ( t , A ) \\| _ { \\infty } \\psi ( t ) = \\| \\tilde { B } T ( t , A ) \\| _ q , \\end{align*}"}
{"id": "7787.png", "formula": "\\begin{align*} ( v ( E ) , v ( F ) ) : = \\chi ( E ^ \\vee \\otimes F ) \\end{align*}"}
{"id": "4715.png", "formula": "\\begin{align*} H _ { l , r } ^ { - } = \\begin{cases} H _ l ^ { - 1 } H _ { l + 1 } ^ { - 1 } \\cdots H _ { r } ^ { - 1 } & \\hbox { i f } l \\leq r , \\\\ H _ l ^ { - 1 } H _ { l - 1 } ^ { - 1 } \\cdots H _ { r } ^ { - 1 } & \\hbox { i f } l > r , \\end{cases} \\end{align*}"}
{"id": "4652.png", "formula": "\\begin{align*} \\delta \\in [ 0 , ( d - \\alpha ) / 2 ] , \\kappa = \\kappa _ \\delta , h ( x ) = h _ \\delta ( x ) = | x | ^ { - \\delta } , q ( x ) = q _ \\delta ( x ) = \\kappa | x | ^ { - \\alpha } . \\end{align*}"}
{"id": "3152.png", "formula": "\\begin{align*} \\mathit { M } _ { \\Phi } \\doteq \\left \\{ \\omega \\in E _ { 1 } : f _ { \\Phi } \\left ( \\omega \\right ) = \\inf \\ , f _ { \\Phi } ( E _ { 1 } ) = - \\mathrm { P } _ { \\Phi } \\right \\} \\end{align*}"}
{"id": "6948.png", "formula": "\\begin{align*} \\rho = \\sum _ { k = 1 } ^ n a _ k \\delta _ { \\xi _ k } , n = \\deg \\theta . \\end{align*}"}
{"id": "7771.png", "formula": "\\begin{align*} \\mathcal { N } \\bigg ( \\begin{bmatrix} \\mathbf { m } _ { j , } ^ k \\\\ [ 2 m m ] \\mathbf { m } ^ k _ { j , } \\end{bmatrix} , \\begin{bmatrix} Z ^ k _ { j , } & Z ^ k _ { j , } \\\\ [ 2 m m ] Z ^ { k } _ { j , } & Z ^ k _ { j , } \\end{bmatrix} \\bigg ) , \\end{align*}"}
{"id": "567.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i < j \\leq n } \\mathbb { E } ( X _ i X _ j ) = \\frac { 1 } { 2 } \\Big ( \\sum _ { 1 \\leq i \\leq n } f _ k ( i ) \\Big ) ^ 2 + O _ { \\alpha , \\varepsilon } ( n ^ { 3 / 2 + \\varepsilon } ) . \\end{align*}"}
{"id": "5424.png", "formula": "\\begin{align*} \\int _ { \\Omega _ e } ( - \\Delta ) ^ s m _ 0 f g \\ , d x = 0 \\end{align*}"}
{"id": "3094.png", "formula": "\\begin{align*} ( b \\rightharpoonup f ) ( m ) = \\sum \\limits _ { ( b ) } b _ 0 f ( S ^ { - 1 } b _ { - 1 } \\rightharpoonup m ) . \\end{align*}"}
{"id": "4231.png", "formula": "\\begin{align*} A = \\left \\{ k \\geq k _ 0 : \\frac { 1 } { k } | B \\cap \\{ k _ 0 , \\ldots , k \\} | \\geq \\frac { 1 } { 2 } \\right \\} . \\end{align*}"}
{"id": "3758.png", "formula": "\\begin{align*} \\beta _ t = \\sup _ { s \\in [ 0 , t ] } \\beta _ t ( s ) \\leq 1 - 2 \\epsilon , \\alpha _ t = \\sup _ { s \\in [ 0 , t ] } \\alpha _ t ( s ) \\leq ( 2 / 3 + \\iota ) - \\epsilon . \\end{align*}"}
{"id": "8916.png", "formula": "\\begin{align*} \\phi _ n ( x ) = e ^ { - \\gamma \\ , \\lambda _ n ^ { \\frac { 1 } { 2 } } | x - P ^ i _ n | } + e ^ { - \\gamma \\ , \\lambda _ n ^ { \\frac { 1 } { 2 } } | x - P ^ j _ n | } , \\end{align*}"}
{"id": "3030.png", "formula": "\\begin{align*} \\Omega _ { \\bullet } ( z , w ) : = & \\sum _ { \\lambda \\in \\mathcal { P } } \\frac { N _ { \\lambda } ( z w , z ^ 2 , w ^ 2 ) ^ { g - 1 } \\tilde { N } _ { \\lambda } ( z w , z ^ 2 , w ^ 2 ) } { \\tilde { N } _ { \\lambda } ( z ^ 2 , w ^ 2 ) } \\prod ^ k _ { j = 1 } H _ { \\lambda } ( \\Z _ j ; z ^ 2 , w ^ 2 ) \\\\ \\Omega _ { \\ast } ( z , w ) : = & \\sum _ { \\lambda \\in \\mathcal { P } } \\frac { N _ { \\lambda } ( z w , z ^ 2 , w ^ 2 ) ^ { 2 g - 1 } } { N _ { \\lambda } ( z ^ 2 , w ^ 2 ) } \\prod ^ k _ { j = 1 } H _ { \\lambda } ( \\Z _ j ; z ^ 2 , w ^ 2 ) ^ 2 , \\end{align*}"}
{"id": "2755.png", "formula": "\\begin{align*} c _ { n , \\sigma p } ^ { - 1 } K _ { j } ( x ) = & - \\int _ { B _ { 4 } \\setminus B _ { 3 } } \\frac { \\phi ^ { p - 1 } ( y ) } { | x - y | ^ { n + \\sigma p } } d y - \\int _ { B _ { R } \\setminus B _ { 4 } } \\frac { d y } { | x - y | ^ { n + \\sigma p } } \\\\ & + \\int _ { B _ { R } ^ { c } } \\frac { | \\mathcal { A } _ { \\varphi } ( y ) | ^ { p - 2 } \\mathcal { A } _ { \\varphi } ( y ) } { | x - y | ^ { n + \\sigma p } } d y : = \\sum ^ { 3 } _ { i = 1 } J _ { i } , \\end{align*}"}
{"id": "5630.png", "formula": "\\begin{align*} \\iint _ { Q _ T } \\varphi _ t ( \\varrho _ 1 - \\varrho _ 2 ) + \\nabla \\cdot ( m \\nabla \\varphi ) ( p _ 1 - p _ 2 ) - \\nabla \\varphi \\cdot \\vec { b } ( \\varrho _ 1 - \\varrho _ 2 ) + \\varphi f ( \\varrho _ 1 - \\varrho _ 2 ) d x d t = 0 . \\end{align*}"}
{"id": "8350.png", "formula": "\\begin{align*} \\sum \\limits _ { i } ^ { N } \\alpha _ i = 0 , \\end{align*}"}
{"id": "3389.png", "formula": "\\begin{align*} \\nu ( z ) : = \\langle \\phi ( z ) , \\nabla g ( z ) \\rangle , \\end{align*}"}
{"id": "7738.png", "formula": "\\begin{align*} H ^ i ( F , A ( 1 ) ) \\cong \\begin{cases} H _ 0 ( F , A ) & i = 2 \\\\ 0 & i \\ge 3 \\end{cases} \\end{align*}"}
{"id": "7898.png", "formula": "\\begin{align*} \\nabla _ { \\vec s , k } = \\nabla _ { \\frac { \\partial } { \\partial t _ { \\vec s , k } } } = \\frac { \\partial } { \\partial t _ { \\vec s , k } } + \\frac { 1 } { z } \\tilde { T } _ { \\vec s , k } \\star _ t \\end{align*}"}
{"id": "3623.png", "formula": "\\begin{align*} v ( \\Delta \\tau , w ) & = \\begin{cases} 1 & v ( \\tau , w ) = 1 \\\\ 0 & , \\end{cases} & & v ( \\Delta ^ \\neg \\phi , w ) & = \\begin{cases} ( 1 , 0 ) & v ( \\phi , w ) = ( 1 , 0 ) \\\\ ( 0 , 1 ) & . \\end{cases} \\end{align*}"}
{"id": "359.png", "formula": "\\begin{align*} q _ A = k + o ( 1 ) . \\end{align*}"}
{"id": "1119.png", "formula": "\\begin{align*} w _ t ^ * = g ^ { - 1 } ( s _ t | \\rho , \\lambda ) , ~ ~ ~ ~ ~ t = 1 , \\ldots , n , \\end{align*}"}
{"id": "4033.png", "formula": "\\begin{align*} \\sup _ { \\delta < | t | \\leq \\frac { \\pi } { h } } \\left \\lvert \\frac { \\phi ( s + \\mathrm { i } t ) } { \\phi ( s ) } \\right \\rvert & = \\sup _ { \\delta < | t | \\leq \\frac { \\pi } { h } } \\left \\lvert \\prod _ { i = 1 } ^ n \\frac { \\phi _ i ( s + \\mathrm { i } t ) } { \\phi _ i ( s ) } \\right \\rvert \\\\ & \\leq c _ 1 ^ n = o ( n ^ { - 1 / 2 } ) . \\end{align*}"}
{"id": "5571.png", "formula": "\\begin{align*} K ^ { \\phi } _ t p _ k ( x ) = \\sum _ { l = 0 } ^ k \\dbinom { k } { l } \\frac { W _ \\phi ( k + 1 ) } { W _ { \\phi } ( l + 1 ) } e ^ { - t l } \\left ( 1 - e ^ { - t } \\right ) ^ { k - l } p _ l ( x ) . \\end{align*}"}
{"id": "8014.png", "formula": "\\begin{align*} f _ - ^ * ( D _ - ) = \\sum _ { i = 1 } ^ m a _ { i , - } \\tilde { D } _ i - \\left ( \\sum _ { i = 1 } ^ m a _ { i , - } D _ i \\cdot e \\right ) E . \\end{align*}"}
{"id": "7816.png", "formula": "\\begin{align*} \\gamma \\star \\delta = \\rho _ { g ^ { - 1 } \\mu _ { t ^ { - 1 } } } ( \\rho _ { \\mu _ t g } ( \\gamma ) \\cup \\rho _ { \\mu _ t g } ( \\delta ) ) . \\end{align*}"}
{"id": "4746.png", "formula": "\\begin{align*} d _ { n + 1 } & = ( 2 n ^ 2 - ( n - 1 ) ^ 2 + 2 ) d _ 1 - v ( \\Delta ) \\left ( \\frac { 2 ( n ^ 2 - 1 ) - ( ( n - 1 ) ^ 2 - 1 ) + 2 } { 1 2 } \\right ) \\\\ & = ( n + 1 ) ^ 2 d _ 1 - v ( \\Delta ) \\left ( \\frac { ( n + 1 ) ^ 2 - 1 } { 1 2 } \\right ) . \\end{align*}"}
{"id": "6803.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } ( a u '' ) '' v \\ , d x = [ ( a u '' ) ' v ] ^ { x = 1 } _ { x = 0 } - [ a u '' v ' ] ^ { x = 1 } _ { x = 0 } + \\int _ { 0 } ^ { 1 } a u '' v '' d x . \\end{align*}"}
{"id": "6835.png", "formula": "\\begin{align*} p ( t , v , g ) = \\int _ 0 ^ { V _ F } \\int _ { - \\infty } ^ { \\infty } e ^ t p _ { } ( t , \\tilde { v } , e ^ t \\tilde { g } ) \\left [ \\sum _ { k = - \\infty } ^ { + \\infty } \\frac { 1 } { V _ F } e ^ { - k ^ 2 ( \\frac { 2 \\pi } { V _ F } ) ^ 2 D ( t ) + i k \\frac { 2 \\pi } { V _ F } ( v - \\tilde { v } ) + i k \\frac { 2 \\pi } { V _ F } ( g - e ^ t \\tilde { g } ) } \\bar { G } _ { t , k } ( g - \\tilde { g } ) \\right ] d \\tilde { v } d \\tilde { g } . \\end{align*}"}
{"id": "4077.png", "formula": "\\begin{align*} P _ 0 ( D ' _ k ) = \\sum \\limits _ { j = 0 } ^ n P _ 0 ( D _ j \\cap D ' _ k ) . \\end{align*}"}
{"id": "1835.png", "formula": "\\begin{align*} \\phi _ { j } ( z ) = \\int \\frac { d \\mu _ { j } ( x ) } { z - x } , 0 \\leq j \\leq p - 1 , \\end{align*}"}
{"id": "7113.png", "formula": "\\begin{align*} H ( t , x ) \\triangleq \\begin{bmatrix} \\vdots \\\\ h _ p ( \\tau _ i , t , x ) \\\\ \\vdots \\end{bmatrix} , \\forall \\tau _ i \\in \\boldsymbol { M } ( t , x ) , \\end{align*}"}
{"id": "6445.png", "formula": "\\begin{align*} \\ker ( t _ 0 ^ * \\rho ) & \\longrightarrow \\ker \\left ( d t _ 0 | _ { \\Gamma ( \\ker d s _ 0 ) } \\right ) = \\Gamma ( \\ker d s _ 0 ) \\cap \\Gamma ( \\ker d t _ 0 ) \\subset \\mathfrak { X } ( B _ 1 ) \\end{align*}"}
{"id": "2038.png", "formula": "\\begin{align*} \\lambda ( v , v _ * ) & = \\Phi ( | v - v _ * | ) \\int _ 0 ^ \\pi \\Sigma ( \\theta ) \\dd \\theta , \\\\ \\Gamma ( v , v _ * , \\dd z ' , \\dd v _ * , \\dd v _ * ' ) & = \\psi ( v , v _ * , \\cdot ) _ { \\# } \\left ( \\frac { \\Sigma } { \\int _ 0 ^ \\pi \\Sigma ( \\theta ) \\dd \\theta } \\right ) . \\end{align*}"}
{"id": "8885.png", "formula": "\\begin{align*} a _ { k + 1 } = \\left ( a _ k ^ 2 - | \\det ( a _ k ^ 2 I _ n - A ^ H A ) | \\left ( \\frac { n - 1 } { ( n + 1 ) a _ k ^ 2 - \\| A \\| _ F ^ 2 } \\right ) ^ { n - 1 } \\right ) ^ { 1 / 2 } , k = 1 , 2 , \\cdots . \\end{align*}"}
{"id": "2035.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : t r u e m a x w e l l m o l e c u l e s } \\Phi ( | u | ) = 1 , \\int _ 0 ^ \\pi \\Sigma ( \\theta ) \\dd \\theta = + \\infty . \\end{align*}"}
{"id": "6184.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\log \\Omega = \\frac { \\partial } { \\partial t } \\widehat { \\omega } _ { t } = - \\frac { 1 } { T _ { 0 } } ( \\omega _ { 0 } + \\psi ^ { \\ast } \\omega _ { N } ) \\in c _ { 1 } ^ { B } ( M ) , \\int _ { M } \\Omega \\wedge \\eta _ { 0 } = 1 . \\end{array} \\end{align*}"}
{"id": "4378.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde F _ j | ^ 2 \\\\ \\le & 2 \\int _ { \\{ \\Psi < - t _ 1 \\} } | ( 1 - b _ { t _ { 0 } , B _ { j } } ( \\Psi ) ) f _ { t _ { 0 } } | ^ { 2 } + 2 \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde { F } _ { j } - ( 1 - b _ { t _ { 0 } , B _ { j } } ( \\Psi ) ) f _ { t _ { 0 } } | ^ { 2 } \\\\ \\le & 2 \\int _ { \\{ \\Psi < - t _ 0 \\} } | f _ { t _ 0 } | ^ 2 + 2 ( e ^ { t _ 0 - t _ 1 + B _ j } - 1 ) \\frac { G ( t _ { 0 } ) - G ( t _ { 0 } + B _ { j } ) } { B _ { j } } . \\end{align*}"}
{"id": "7897.png", "formula": "\\begin{align*} \\mathbf t ( z ) = \\sum \\limits _ { l = 0 } ^ \\infty t _ l z ^ l . \\end{align*}"}
{"id": "723.png", "formula": "\\begin{align*} | A ( Y , Z ) | & \\le \\sum _ { i = 1 } ^ m \\left | g _ X ( Y , e _ i ) \\right | \\sqrt { \\sum _ { j = 1 } ^ m g _ X ( Z , e _ j ) ^ 2 } \\sqrt { \\sum _ { j = 1 } ^ m A ( e _ i , e _ j ) ^ 2 } \\\\ & \\le \\sqrt { \\sum _ { j = 1 } ^ m g _ X ( Z , e _ j ) ^ 2 } \\sqrt { \\sum _ { i = 1 } ^ m g _ X ( Y , e _ i ) ^ 2 } \\sqrt { \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ m A ( e _ i , e _ j ) ^ 2 } = | A | _ { g _ X } | Z | _ { g _ X } | Y | _ { g _ X } . \\end{align*}"}
{"id": "2220.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\mathrm { R i c } _ { \\omega ( t ) } ^ { T } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } u ( t ) = - \\widehat { \\omega } _ { \\infty } . \\end{array} \\end{align*}"}
{"id": "8241.png", "formula": "\\begin{align*} z = x + i y . \\end{align*}"}
{"id": "1319.png", "formula": "\\begin{align*} \\dot x & = v , \\\\ \\dot v & = - \\nabla \\Phi ( x ) . \\end{align*}"}
{"id": "3880.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\| a _ n - a \\| _ \\infty = 0 , \\sup _ \\Omega a < \\lambda _ 1 , \\sup _ { n \\in \\mathbb N } \\left [ \\| f _ n \\| _ 1 + \\| g _ n \\| _ \\infty \\right ] < + \\infty . \\end{align*}"}
{"id": "3245.png", "formula": "\\begin{align*} \\int _ { | a | \\geq 2 R _ 0 } \\int _ { | a | - R _ 0 } ^ \\infty \\ 1 ( m _ f ( r , a ) > \\kappa ) \\ , \\frac { d r } { r ^ { p + 1 } } \\ , d a & \\leq \\int _ { R _ 0 + ( \\gamma / \\kappa ) ^ { 1 / d } > | a | \\geq 2 R _ 0 } \\int _ { | a | - R _ 0 } ^ \\infty \\frac { d r } { r ^ { p + 1 } } \\ , d a \\\\ & = p ^ { - 1 } \\int _ { R _ 0 + ( \\gamma / \\kappa ) ^ { 1 / d } > | a | \\geq 2 R _ 0 } \\frac { d a } { ( | a | - R _ 0 ) ^ p } \\ , . \\end{align*}"}
{"id": "5137.png", "formula": "\\begin{align*} \\left [ \\frac { \\partial } { \\partial z } , z \\frac { \\partial } { \\partial w } \\right ] = \\frac { \\partial } { \\partial w } . \\end{align*}"}
{"id": "220.png", "formula": "\\begin{align*} p ( y ^ { n } _ { [ 1 : T ] } , x ^ { n } , z ^ { n } ) = p ( z ^ { n } ) p ( x ^ { n } | z ^ { n } ) p ( y ^ { n } _ { [ 1 : T ] } | x ^ { n } ) . \\end{align*}"}
{"id": "3500.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast \\ast } = & \\frac { \\Gamma \\left ( ( m + 1 ) / 2 \\right ) m ^ { 3 / 2 } \\left [ \\xi _ { p } \\left ( 1 + \\frac { \\xi _ { p } ^ { 2 } } { m } \\right ) ^ { - ( m - 3 ) / 2 } - \\xi _ { q } \\left ( 1 + \\frac { \\xi _ { q } ^ { 2 } } { m } \\right ) ^ { - ( m - 3 ) / 2 } \\right ] } { \\Gamma ( m / 2 ) ( m - 1 ) ( m - 3 ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } \\\\ & + \\frac { m ^ { 2 } } { ( m - 2 ) ( m - 4 ) } \\frac { F _ { Y _ { ( 2 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , ~ m > 4 , \\end{align*}"}
{"id": "7996.png", "formula": "\\begin{align*} H _ { ( \\mathbb P ^ 3 , K 3 ) } ( y ) = e ^ { \\frac { H \\log y } { 2 \\pi i } } \\sum _ { d \\geq 0 } y ^ d \\frac { \\Gamma ( 1 + \\frac { 4 H } { 2 \\pi i } + 4 d ) } { \\Gamma ( 1 + \\frac { H } { 2 \\pi i } + d ) ^ 4 } [ \\textbf { 1 } ] _ { 4 d } , \\end{align*}"}
{"id": "5069.png", "formula": "\\begin{align*} E [ | L ^ { n , 2 } _ \\tau + L ^ { n , 3 } _ \\tau | ^ 2 ] & = n ^ { 4 \\alpha + 2 } \\int _ { [ 0 , \\tau ] ^ 2 } \\gamma _ { s _ 1 } \\gamma _ { s _ 2 } \\mathbf { 1 } _ { | s _ 1 - s _ 2 | \\le \\frac 1 n } ( s _ 1 - \\eta _ n ( s _ 1 ) ) ^ \\alpha ( s _ 2 - \\eta _ n ( s _ 2 ) ) ^ \\alpha \\\\ & \\times E [ ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s _ 1 ) } ) \\Lambda ^ n _ { s _ 1 } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s _ 2 ) } ) \\Lambda ^ n _ { s _ 2 } ] d s _ 1 d s _ 2 \\end{align*}"}
{"id": "6953.png", "formula": "\\begin{align*} \\rho = \\sum _ { k \\ge 1 } a _ k \\delta _ { \\lambda _ k ^ 2 } \\end{align*}"}
{"id": "8937.png", "formula": "\\begin{align*} T u ( \\Omega ) : = Y ( \\cdot , u , D u ) ( \\Omega ) = \\Omega ^ * , \\end{align*}"}
{"id": "6301.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } p _ n ( x | q ) = - q ^ { - n } [ n ] _ q [ n + 1 ] _ q \\ , _ 2 \\phi _ 1 ( q ^ { 1 - n } , q ^ { n + 2 } ; q ^ 2 ; q , x ) , \\end{align*}"}
{"id": "2203.png", "formula": "\\begin{align*} C _ { \\tiny \\rm F i n s l e r } ( T ) = C _ { \\psi , \\varphi } ( \\mathbf { T } ) : = \\inf _ { | \\mathbf { y } ^ * | _ { f ( x ) , \\psi } = 1 } | \\mathbf { T } ^ * { \\mathbf { y } ^ * } | _ { x , \\varphi } \\end{align*}"}
{"id": "5039.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } E \\left [ | Q ^ { n , 5 } _ \\tau | ^ 2 \\right ] = 0 . \\end{align*}"}
{"id": "2801.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\nabla f ( x _ N ) \\| ^ 2 { } \\leq { } 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) \\max \\left \\{ ( 1 - h ) ^ { 2 N } \\ , , \\ , \\frac { 1 } { 1 + 2 N h } \\right \\} \\end{aligned} \\end{align*}"}
{"id": "2623.png", "formula": "\\begin{align*} | \\widetilde c _ p | = \\frac { 1 } { | c _ p | } > 1 \\ , \\end{align*}"}
{"id": "350.png", "formula": "\\begin{align*} A = A _ 1 \\sqcup . . . \\sqcup A _ \\ell , \\end{align*}"}
{"id": "4907.png", "formula": "\\begin{align*} \\mathrm { q a d j } ( \\lambda E _ n - A ) = \\sum _ { j = 1 } ^ n \\prod _ { k = 1 ; k \\ne j } ^ n ( \\lambda - \\lambda _ k ( A ) ) v _ j v _ j ^ * . \\end{align*}"}
{"id": "5459.png", "formula": "\\begin{align*} a a _ 0 b a _ 0 ^ { - 1 } = a ' x _ 0 b ' a _ 0 ^ { - 1 } , ( a , a ' \\in A , b , b ' \\in B ) \\end{align*}"}
{"id": "5083.png", "formula": "\\begin{align*} S ^ n = n ^ { \\alpha + \\frac 1 2 } \\int _ { [ \\tau _ 1 , \\tau _ 2 ] } ( t - s ) ^ \\alpha \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) d s \\end{align*}"}
{"id": "3418.png", "formula": "\\begin{align*} ( g h ) _ { 1 , 1 } = g _ { 1 , 1 } h _ { 1 , 1 } + g _ { 1 , 2 } h _ { 2 , 1 } + g _ { 1 , 3 } h _ { 3 , 1 } . \\end{align*}"}
{"id": "6047.png", "formula": "\\begin{align*} g _ { r , s , j } ( x ) = r ^ { j + \\sum _ { i = 1 } ^ { m } \\mu _ { i , j } } g _ j \\left ( r x + s \\right ) \\end{align*}"}
{"id": "5580.png", "formula": "\\begin{align*} I I \\le q \\ , \\mathfrak { b } \\int _ \\beta ^ { \\infty } \\lambda ^ { q - 2 } d \\lambda = \\frac { q } { 1 - q } \\mathfrak { b } \\beta ^ { q - 1 } . \\end{align*}"}
{"id": "4142.png", "formula": "\\begin{align*} \\varphi _ k ^ { | \\mu | } ( x ) = | \\mu | ^ { d _ 1 / 2 } \\varphi _ k ( | \\mu | ^ { 1 / 2 } x ) \\end{align*}"}
{"id": "138.png", "formula": "\\begin{align*} T _ { \\lambda ^ 3 } : = 5 4 \\lambda ^ 3 \\ , C \\star Q \\star S , Q : = S \\big ( S ^ 2 \\star S ^ 2 \\big ) . \\end{align*}"}
{"id": "4155.png", "formula": "\\begin{align*} X _ j ^ \\mu ( g \\circ T _ { \\bar \\mu } ^ { - 1 } ) ( T _ { \\bar \\mu } z ) & = \\nabla g ( z ) v ^ \\mu _ j + \\tfrac i 2 \\omega _ \\mu ( T _ { \\bar \\mu } z , X _ j ) g ( z ) \\\\ & = \\nabla g ( z ) v ^ \\mu _ j + \\tfrac i 2 | \\mu | \\omega ( z , v ^ \\mu _ j ) g ( z ) . \\end{align*}"}
{"id": "5063.png", "formula": "\\begin{align*} L ^ { n , 3 } _ \\tau & = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) ( s - \\eta _ n ( s ) ) ^ \\alpha \\\\ & \\times \\left [ ( W _ s - W _ { \\eta _ n ( s ) } ) \\left ( \\int _ { \\eta _ n ( s ) } ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) - \\int _ { \\eta _ n ( s ) } ^ s \\psi _ { n , 1 } ( u , s ) d u \\right ] d s . \\end{align*}"}
{"id": "5869.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p - q } } \\bigg ( \\int _ { x _ k } ^ { \\infty } u \\bigg ) ^ { \\frac { p } { p - q } } V _ r ( 0 , x _ k ) ^ { \\frac { p q } { p - q } } \\approx \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p - q } } \\bigg ( \\int _ { x _ k } ^ { \\infty } u \\bigg ) ^ { \\frac { p } { p - q } } V _ r ( x _ { k - 1 } , x _ k ) ^ { \\frac { p q } { p - q } } \\end{align*}"}
{"id": "6437.png", "formula": "\\begin{align*} \\varrho ( [ x , y ] _ \\mathfrak { g } ) - [ \\varrho ( x ) , \\varrho ( y ) ] = \\rho ( \\eta ( x , y ) ) . \\end{align*}"}
{"id": "1070.png", "formula": "\\begin{align*} \\lim _ { | y | \\to \\infty } \\| R ( x + i y , A ) \\| _ { \\infty } = 0 , \\end{align*}"}
{"id": "2776.png", "formula": "\\begin{align*} T _ 2 ( h _ i { } \\leq { } 1 , h _ { i - 1 } { } \\geq { } 1 ) = h _ { i - 1 } ( 1 + \\kappa ) \\tfrac { 1 - h _ { i - 1 } } { 2 - ( 1 + \\kappa ) h _ { i - 1 } } { } \\leq { } 0 \\end{align*}"}
{"id": "459.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( z \\Sigma _ { \\mu } ( z ) ) = z , z \\in \\rho \\mathbb { D } . \\end{align*}"}
{"id": "7955.png", "formula": "\\begin{align*} I _ { X _ - } ( y , z ) = z e ^ { t _ - / z } \\sum _ { d \\in \\mathbb K _ { - } } \\tilde y ^ { d } \\left ( \\prod _ { i = 0 } ^ { m } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\textbf { 1 } _ { [ - d ] } , \\end{align*}"}
{"id": "2086.png", "formula": "\\begin{align*} \\widetilde { N _ H } ( n ) = C a r d ( \\{ v : | v | = n , u \\not \\sqsubset v w _ 1 v u \\in \\mathcal { L } _ H \\} ) . \\end{align*}"}
{"id": "3299.png", "formula": "\\begin{align*} u ( 0 ) = u _ 0 , \\ ; \\partial _ t u ( 0 ) = v _ 0 \\Omega ^ - . \\end{align*}"}
{"id": "6563.png", "formula": "\\begin{align*} \\frac { f ( x + t y ) - f ( x ) } { t } \\ ; = \\ ; g ^ \\vee ( t ) \\ ; \\to \\ ; g ^ \\vee ( 0 ) \\end{align*}"}
{"id": "2958.png", "formula": "\\begin{align*} ( ( \\exp , 0 ) \\cdot X _ 0 - x _ 0 ) ( ( f , a ) ) = \\exp ( a ) . \\end{align*}"}
{"id": "6985.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { \\lambda _ N ^ 2 } \\prod \\limits _ { k = 1 } ^ { N - 1 } \\Bigg ( \\frac { \\lambda _ N ^ 2 + \\mu _ { k } ^ { 2 } } { \\lambda _ N ^ 2 + \\lambda _ { k } ^ { 2 } } \\Bigg ) = \\infty . \\end{align*}"}
{"id": "7323.png", "formula": "\\begin{align*} { \\sf t } _ 6 & = \\{ \\dot { \\sf M } + f ( u ) - f _ 2 ( u ) \\} ( \\chi _ { 4 , { \\sf a } } - \\chi _ 4 ) \\lesssim { \\bf 1 } _ { 1 < | x | < 4 } . \\end{align*}"}
{"id": "7242.png", "formula": "\\begin{align*} \\begin{aligned} ( \\rho _ \\alpha - \\alpha ( n - 1 ) ) x _ { u _ 1 } & = ( 1 - \\alpha ) ( k - 2 ) x _ { u _ 1 } + ( 1 - \\alpha ) \\sum _ { u u _ 1 \\in E ( H ) } x _ u \\\\ & \\leq ( 1 - \\alpha ) ( k - 2 ) x _ { u _ 1 } + ( 1 - \\alpha ) ( n - k + 1 ) x _ v \\end{aligned} \\end{align*}"}
{"id": "92.png", "formula": "\\begin{align*} \\forall x , y \\in \\Lambda , S _ t ( x , y ) : = \\big < \\varphi _ x \\varphi _ y \\big > _ { A , g , \\nu + 1 / t } . \\end{align*}"}
{"id": "7910.png", "formula": "\\begin{align*} J ( t , z ) = L ( t , z ) ^ { - 1 } [ \\textbf { 1 } ] _ { \\vec 0 } . \\end{align*}"}
{"id": "1925.png", "formula": "\\begin{align*} A _ { j } ( z ) = S _ { j } ( z ) , A _ { j } ^ { ( q ) } ( z ) = S _ { j } ^ { ( q ) } ( z ) , \\mbox { p r o v i d e d \\eqref { e q : r e d a n k } h o l d s } . \\end{align*}"}
{"id": "5694.png", "formula": "\\begin{align*} m _ { d + 1 , 1 ^ k } ( x _ 1 , x _ 2 , \\ldots , x _ i ) = \\begin{cases} m _ { d , 1 ^ k } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d \\ge 2 ) , \\\\ \\\\ ( k + 1 ) m _ { 1 ^ { k + 1 } } ( x _ 1 , x _ 2 , \\ldots , x _ i ) x _ { i + 1 } & ( d = 1 ) , \\end{cases} \\end{align*}"}
{"id": "4253.png", "formula": "\\begin{align*} p _ i = \\frac { 1 } { 2 } \\frac { \\widetilde { \\Phi } ( n - 1 ) + \\widetilde { \\Phi } ( n + 1 ) - 2 \\widetilde { \\Phi } ( n ) } { \\widetilde { \\Phi } ( n - 1 ) - \\widetilde { \\Phi } ( n + 1 ) } , \\end{align*}"}
{"id": "5923.png", "formula": "\\begin{align*} { \\lambda _ k } = \\mathop { \\lim } \\limits _ { T \\to \\infty } \\frac 1 T \\ln { D _ k } \\ , \\ \\ \\lambda _ 1 \\ge \\dots \\ge \\lambda _ k \\end{align*}"}
{"id": "6706.png", "formula": "\\begin{align*} \\mathbf { c } _ n ( \\lambda ) = \\mathbf { c } _ { - n } ( \\lambda ) = \\frac { 1 } { \\sqrt { \\pi } } \\frac { \\Gamma ( \\lambda ) \\Gamma ( \\lambda + \\frac { 1 } { 2 } ) } { \\Gamma ( \\lambda + \\frac { 1 + n } { 2 } ) \\Gamma ( \\lambda + \\frac { 1 - n } { 2 } ) } , \\ ; \\ ; \\ ; \\lambda \\in \\mathfrak { a } _ \\mathbb { C } ^ * . \\end{align*}"}
{"id": "8250.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { r r r r } - i \\ , \\frac { 3 a ^ { 2 } } { 2 \\beta \\hslash ^ { 2 } } a _ { 2 } + i a _ { 4 } & = & \\lambda a _ { 1 } ; \\\\ i \\ , \\frac { 3 a ^ { 2 } } { 2 \\beta \\hslash ^ { 2 } } a _ { 1 } - i a _ { 3 } & = & \\lambda a _ { 2 } ; \\\\ i a _ { 2 } & = & \\lambda a _ { 3 } ; \\\\ - i a _ { 1 } & = & \\lambda a _ { 4 } . \\end{array} \\right . , \\end{align*}"}
{"id": "5411.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } f _ { \\epsilon } ( x ) = - \\frac { C _ { n , s } } { 2 } \\int _ { \\R ^ n } \\frac { \\delta f _ { \\epsilon } ( x , y ) } { | y | ^ { n + 2 s } } \\ , d y \\end{align*}"}
{"id": "7581.png", "formula": "\\begin{align*} \\frac { d Y } { d t } = A ( t ) Y Y ( 0 ) = I , \\end{align*}"}
{"id": "2130.png", "formula": "\\begin{align*} \\left | \\frac { \\Re G _ { \\nu } ( z ) - G _ { \\nu } ^ { * } ( 0 ) } { \\Im G _ { \\nu } ( z ) } \\right | & = \\frac { | z | } { \\Im z } \\left | \\frac { \\Re G _ { \\nu } ( z ) - G _ { \\nu } ^ { * } ( 0 ) } { z } \\right | \\frac { \\Im z } { - \\Im G _ { \\nu } ( z ) } \\\\ & \\leq \\sqrt { 1 + \\delta ^ 2 } \\left | \\frac { G _ { \\nu } ( z ) - G _ { \\nu } ^ { * } ( 0 ) } { z } \\right | \\frac { 2 } { c } \\\\ & \\rightarrow 2 \\sqrt { 1 + \\delta ^ 2 } . \\end{align*}"}
{"id": "6833.png", "formula": "\\begin{align*} D ( t ) = \\int _ { 0 } ^ { t } a ( s ) d s - \\frac { ( \\int _ { 0 } ^ { t } e ^ { s - t } a ( s ) d s ) ^ 2 } { \\int _ { 0 } ^ { t } e ^ { 2 ( s - t ) } a ( s ) d s } \\geq 0 . \\end{align*}"}
{"id": "3064.png", "formula": "\\begin{align*} \\mathrm { E } \\left [ \\mathbf { Z } | \\boldsymbol { \\xi _ { a } } < \\mathbf { Z } \\leq \\boldsymbol { \\xi _ { b } } \\right ] = \\frac { \\boldsymbol { \\delta } } { F _ { \\mathbf { Z } } ( \\boldsymbol { \\xi _ { a } } , \\boldsymbol { \\xi _ { b } } ) } . \\end{align*}"}
{"id": "7834.png", "formula": "\\begin{align*} \\| T _ \\phi ( x _ n ) \\| _ 2 ^ 2 = \\langle T _ \\phi ^ * T _ \\phi \\widehat { x _ n } , \\widehat { x _ n } \\rangle \\leq \\| T _ \\phi ^ * T _ \\phi ( \\widehat { x _ n } ) \\| _ 1 \\to 0 . \\end{align*}"}
{"id": "169.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\frac { 1 } { 1 + \\| p \\| ^ 2 } \\frac { \\| \\tilde k - p \\| _ 2 } { \\big [ 1 + \\| \\tilde k - p \\| _ 2 ^ 2 \\big ] ^ 2 } \\ , d p = \\int _ { \\R ^ 3 } \\frac { 1 } { 1 + \\| \\tilde k + p \\| ^ 2 } \\frac { \\| p \\| _ 2 } { \\big [ 1 + \\| p \\| _ 2 ^ 2 \\big ] ^ 2 } \\ , d p \\leq \\frac { c } { 1 + \\| \\tilde k \\| _ 2 ^ 2 } . \\end{align*}"}
{"id": "8771.png", "formula": "\\begin{align*} C ( P ) : = \\frac { 1 } { B \\circ q ( P ) } \\prod _ { k = 1 } ^ N B ( p _ k ) . \\end{align*}"}
{"id": "4304.png", "formula": "\\begin{align*} \\langle h , k \\rangle = \\int _ L h k \\Omega . \\end{align*}"}
{"id": "3810.png", "formula": "\\begin{align*} \\sum _ { i = 0 , 1 , 2 } \\big | { } _ 1 ^ { 1 } H y p ^ { \\mu , i } _ { k , j ; n } ( t _ 1 , t _ 2 ) \\big | \\lesssim \\sum _ { l \\in [ n + \\epsilon M _ t / 2 , 2 ] \\cap \\Z } 2 ^ { k - j - l + 2 \\epsilon M _ t } \\big ( 2 ^ { - a _ p } 2 ^ { - k - n + \\epsilon M _ t } 2 ^ { 3 j + 2 l } \\big ) ^ { 1 / 2 } \\big ( 2 ^ { - a _ p } 2 ^ { - k - n + \\epsilon M _ t } 2 ^ { - j } \\big ) ^ { 1 / 2 } \\mathcal { M } ( C ) ( t _ 2 - t _ 1 ) \\end{align*}"}
{"id": "2984.png", "formula": "\\begin{align*} H _ m ( t ) = i t \\int _ 0 ^ \\pi e ( - m t \\cos \\theta ) \\phi _ { 2 , m } ( t \\sin \\theta , s ) e ^ { - i \\theta } \\ , d \\theta . \\end{align*}"}
{"id": "3491.png", "formula": "\\begin{align*} c _ { ( 1 ) } ^ { \\ast } & = \\frac { ( m - 1 ) \\Gamma ( 1 / 2 ) } { ( 2 \\pi ) ^ { 1 / 2 } m } \\left [ \\int _ { 0 } ^ { \\infty } u ^ { 1 / 2 - 1 } \\left ( 1 + \\frac { 2 t } { m } \\right ) ^ { - ( m - 1 ) / 2 } \\mathrm { d } u \\right ] ^ { - 1 } \\\\ & = \\frac { ( m - 1 ) } { m ^ { 3 / 2 } B ( \\frac { 1 } { 2 } , ~ \\frac { m - 2 } { 2 } ) } , ~ i f ~ m > 2 \\end{align*}"}
{"id": "5748.png", "formula": "\\begin{align*} ( i - a + 1 ) \\Big ( \\pi _ { [ a - 1 , b ] } + \\pi _ { [ a , b ] } y _ { b + 1 } \\Big ) = ( i - a + 1 ) \\pi _ { [ a , b + 1 ] } \\end{align*}"}
{"id": "6255.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } q ^ { \\frac { n ( n - 3 ) } { 2 } } x ^ n \\left ( 1 - q + q c + q ^ { n + 2 } x \\right ) A _ q ( q ^ n x ) = ( 1 - q ) A _ q ( \\frac { x } { q } ) - ( c q + x ) A _ q ( q x ) , \\end{align*}"}
{"id": "5622.png", "formula": "\\begin{align*} a \\big ( X ^ { 6 ^ k } \\big ) b \\big ( X ^ { 6 ^ k } \\big ) = \\big ( X ^ { 2 \\cdot 3 ^ k } + X ^ { 3 ^ k } + 1 \\big ) ^ { 2 ^ k } \\end{align*}"}
{"id": "5691.png", "formula": "\\begin{align*} x _ 1 ^ { d + 1 } + x _ 2 ^ { d + 1 } + \\cdots + x _ i ^ { d + 1 } = ( x _ 1 ^ d + x _ 2 ^ d + \\cdots + x _ i ^ d ) x _ { i + 1 } ( d \\ge 1 ) . \\end{align*}"}
{"id": "8244.png", "formula": "\\begin{align*} a _ { \\pm } = \\frac { \\sqrt { 1 + \\sqrt { 1 + \\left ( \\frac { 1 6 \\beta m \\lambda } { 3 } \\right ) ^ { 2 } } \\pm \\frac { 2 \\sqrt { 1 + \\sqrt { 1 + \\left ( \\frac { 1 6 \\beta m \\lambda } { 3 } \\right ) ^ { 2 } } } } { \\sqrt { 2 } } } } { 4 } \\end{align*}"}
{"id": "3026.png", "formula": "\\begin{align*} N _ { \\lambda } ( z , w ) = \\prod _ { x \\in \\lambda } ( z ^ { a ( x ) + 1 } - w ^ { l ( x ) } ) ( z ^ { a ( x ) } - w ^ { l ( x ) + 1 } ) , \\end{align*}"}
{"id": "6488.png", "formula": "\\begin{align*} u ( c _ 2 ) = u _ x ( c _ 2 ) = y ( c _ 2 ) = y _ x ( c _ 2 ) = 0 . \\end{align*}"}
{"id": "5308.png", "formula": "\\begin{align*} \\Psi ( \\xi ) = i \\langle a , \\xi \\rangle + \\frac 1 2 \\langle \\xi , Q \\xi \\rangle + \\int _ { \\R ^ d } \\left ( 1 - e ^ { i \\langle x , \\xi \\rangle } + i \\langle x , \\xi \\rangle 1 _ { \\{ | x | < 1 \\} } \\right ) \\nu ( d x ) . \\end{align*}"}
{"id": "1417.png", "formula": "\\begin{align*} g ( \\psi _ G ) = \\begin{cases} 4 , \\ i f \\ a t \\ l e a s t \\ o n e \\ o f \\ t h e \\ ( P _ 1 ) \\ c o n d i t i o n s \\ h o l d s \\\\ 8 , \\ i f \\ n o n e \\ o f \\ t h e \\ ( P _ 1 ) \\ c o n d i t i o n s \\ h o l d s \\end{cases} . \\end{align*}"}
{"id": "4811.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( Q ( u ) - Q _ n ( u ) ) ^ 2 d u = \\sum _ { k = 1 } ^ n \\int _ { ( k - 1 ) / n } ^ { k / n } ( Q ( u ) - q _ { n , k } ) ^ 2 d u . \\end{align*}"}
{"id": "611.png", "formula": "\\begin{align*} \\partial _ x ^ { 2 M + 1 } f ( x , y , 1 , z / n ) ^ { - \\alpha } & = \\sum _ { i = 0 } ^ { 2 M } A _ i ( x , y ) f ( x , y , 1 , z / n ) ^ { - \\alpha - i - 1 } , \\\\ \\partial _ x ^ { 2 M + 1 } \\partial _ y ^ { 2 M + 1 } f ( x , y , 1 , z / n ) ^ { - \\alpha } & = \\sum _ { i = 0 } ^ { 4 M } B _ i ( x , y ) f ( x , y , 1 , z / n ) ^ { - \\alpha - i - 2 } , \\end{align*}"}
{"id": "3974.png", "formula": "\\begin{align*} \\Pr ( \\Delta ( C _ M ) \\le \\delta ) = \\Pr ( X \\ge 1 ) . \\end{align*}"}
{"id": "1536.png", "formula": "\\begin{align*} \\mu ( A ) = \\Theta ( n e ^ { - \\alpha R } \\kappa \\sqrt { \\ss } ) \\int _ { \\widehat { r } } ^ R e ^ { - \\beta r _ 0 } \\sinh ( \\alpha r _ 0 ) d r _ 0 \\end{align*}"}
{"id": "4216.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { 1 \\vee \\log \\left ( \\phi ^ { - 1 } ( n ) \\right ) } = \\infty . \\end{align*}"}
{"id": "2678.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } ( n - \\ell - 2 ) ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\\\ & = ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } ( n - \\ell - 2 ) \\\\ & = ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\binom { n - k - 2 } { 2 } \\\\ & = ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\binom { n - 2 } { 3 } . \\end{align*}"}
{"id": "2773.png", "formula": "\\begin{align*} \\begin{aligned} T _ 1 \\left ( h _ i { } \\leq { } 1 \\right ) = - 1 + h _ i { } \\leq { } 0 \\end{aligned} \\end{align*}"}
{"id": "3222.png", "formula": "\\begin{align*} \\psi ( \\Phi v , w ) = - \\psi ( v , \\Phi w ) . \\end{align*}"}
{"id": "6878.png", "formula": "\\begin{align*} N ( t ) & = \\int _ { \\mathbb { R } } g _ + p ( t , g ) d g = \\int _ { \\mathbb { R } } g _ + d g \\int _ { \\mathbb { R } } G _ t ( g - y ) p _ { t , 0 } ( y ) d y \\\\ & = \\int _ { \\mathbb { R } } p _ { t , 0 } ( y ) d y \\int _ { \\mathbb { R } } g _ + G _ t ( g - y ) d g \\\\ & = \\int _ { \\mathbb { R } } ( N ( B ( t ) + y , C ( t ) ) ) p _ { t , 0 } ( y ) d y . \\end{align*}"}
{"id": "1992.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ { 0 , \\alpha } ( B _ { d / 8 } ( x ) ) } & \\le \\frac { C ( N , \\mu , \\alpha ) } { d ^ { \\alpha } } \\left ( \\inf _ { B _ { d / 8 } ( x ) } u + d \\| F \\| _ { L ^ N ( B _ { d / 4 } ( x ) ) } \\right ) \\\\ & \\le \\frac { C ( N , \\mu , \\alpha ) } { d ^ { \\alpha } } \\left ( u ( x ) + d \\| F \\| _ { L ^ N ( B _ { d / 4 } ( x ) ) } \\right ) . \\end{align*}"}
{"id": "8226.png", "formula": "\\begin{align*} \\left ( - \\frac { \\hslash ^ { 2 } } { 2 m } \\partial _ { x } ^ { 2 } + \\frac { \\beta \\hslash ^ { 4 } } { 3 m } \\partial _ { x } ^ { 4 } \\right ) \\phi _ { \\pm } = \\pm i \\lambda \\phi _ { \\pm } . \\end{align*}"}
{"id": "8456.png", "formula": "\\begin{align*} | g _ i ( x , y ) - g _ i ( x ' , y ' ) | \\leq L \\| ( x - x ' , y - y ' ) \\| , i = 1 , 2 . \\end{align*}"}
{"id": "4765.png", "formula": "\\begin{align*} & a _ \\kappa = \\sum _ { i , j \\in \\mathbb N \\colon \\atop { 3 ( i + j ) \\leq \\kappa } } ( \\kappa - 3 ( i + j ) + 1 ) , b _ \\kappa = \\sum _ { i , j \\in \\mathbb N \\colon \\atop { 2 ( i + j ) \\leq \\kappa } } \\binom { \\kappa - 2 ( i + j ) + 3 } { 3 } - a _ \\kappa , \\\\ & c _ \\kappa = \\frac 1 6 \\left ( \\binom { \\kappa + 7 } { 7 } - 3 b _ \\kappa - a _ \\kappa \\right ) . \\end{align*}"}
{"id": "5374.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ^ * + \\sum _ { | \\alpha | \\leq m } ( - 1 ) ^ { | \\alpha | } D ^ \\alpha ( a _ \\alpha u ^ * ) & = F ^ * \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u ^ * & = f ^ * \\mbox { i n } \\ ; \\ ; \\Omega _ e . \\end{align*}"}
{"id": "7192.png", "formula": "\\begin{align*} & ( z _ 0 + z _ 2 ) ^ { k + \\frac { j _ 2 } { T } } Y _ { M ^ 3 } ( u , z _ 0 + z _ 2 ) I ^ { \\circ } ( w _ 1 , z _ 2 ) w _ 2 \\\\ = & ( z _ 2 + z _ 0 ) ^ { k + \\frac { j _ 2 } { T } } I ^ { \\circ } ( Y _ { M ^ 1 } ( u , z _ 0 ) w _ 1 , z _ 2 ) w _ 2 \\end{align*}"}
{"id": "1769.png", "formula": "\\begin{align*} ( z + a f _ k ) ( y ^ * ) = \\lim _ n ( z + a f _ k ) ( y _ n ^ * ) = 0 \\end{align*}"}
{"id": "7856.png", "formula": "\\begin{align*} E _ { \\ell ^ \\infty \\Gamma } ( T _ \\xi ^ * T _ \\eta ) ( t ) = E _ { \\ell ^ \\infty \\Gamma } ( T _ { \\eta \\otimes \\overline { \\xi } } ) ( t ) = \\langle T _ { \\eta \\otimes \\overline { \\xi } } \\delta _ t , \\delta _ t \\rangle = \\langle u _ t ^ * ( \\eta \\otimes \\overline { \\xi } ) u _ t , \\eta \\otimes \\overline { \\xi } \\rangle . \\end{align*}"}
{"id": "98.png", "formula": "\\begin{align*} \\big < F ; G \\big > ^ h _ { A , g , \\nu } = \\big < F G \\big > ^ h _ { A , g , \\nu } - \\big < F \\big > ^ h _ { A , g , \\nu } \\big < G \\big > ^ h _ { A , g , \\nu } , F , G : \\R ^ \\Lambda \\rightarrow \\R , h \\in \\R ^ \\Lambda . \\end{align*}"}
{"id": "6765.png", "formula": "\\begin{align*} \\binom { k } { d } d \\mid \\gcd ( n , k ) \\end{align*}"}
{"id": "6471.png", "formula": "\\begin{align*} u ( \\cdot , 0 ) = u _ 0 ( \\cdot ) , \\ u _ t ( \\cdot , 0 ) = u _ 1 ( \\cdot ) , \\ y ( \\cdot , 0 ) = y _ 0 ( \\cdot ) y _ t ( \\cdot , 0 ) = y _ 1 ( \\cdot ) , \\ x \\in ( 0 , L ) . \\end{align*}"}
{"id": "7578.png", "formula": "\\begin{align*} E _ h = | | x _ { k + 1 } - x ( t _ { k + 1 } ) | | , \\end{align*}"}
{"id": "3564.png", "formula": "\\begin{align*} w _ 2 ( \\pi ) = \\dbinom { \\sum \\limits _ { i = 1 } ^ n c _ i N _ i } { 2 } \\left ( \\sum _ { l = 1 } ^ n v _ l ^ 2 \\right ) + \\left ( \\sum _ { i = 1 } ^ n c _ i N _ i '' \\right ) \\left ( \\sum \\limits _ { 1 \\leq l < k \\leq n } v _ l v _ k \\right ) . \\end{align*}"}
{"id": "1377.png", "formula": "\\begin{align*} \\mathfrak { F } ( v ) & = \\mathcal { E } [ v , \\cdots , v ] + \\mathcal { H L } _ B [ T [ W _ t u _ 0 , v , \\cdots , v ] , v , \\cdots , v ] \\\\ & \\qquad - \\frac { p - 1 } { 2 } T [ W _ t u _ 0 , v , \\left ( \\mathcal { R } ^ 2 + \\mathcal { N R } \\right ) [ v , \\cdots , v ] , v , \\cdots , v ] \\\\ & \\qquad + \\frac { p - 1 } { 2 } T [ W _ t u _ 0 , \\left ( \\mathcal { R } ^ 2 + \\mathcal { N R } \\right ) [ v , \\cdots , v ] , v , \\cdots , v ] , \\end{align*}"}
{"id": "1043.png", "formula": "\\begin{align*} T ( t , A + B ) x - T ( t , A ) x = \\sum _ { n = 1 } ^ { \\infty } S _ { n } ( t ) x , \\end{align*}"}
{"id": "1019.png", "formula": "\\begin{align*} \\mathsf { F r } _ x ( a ) = \\mathsf { F r } _ { x | _ S } ( a ) \\end{align*}"}
{"id": "604.png", "formula": "\\begin{align*} & \\zeta ( \\Delta _ n , s ) = V _ \\alpha ( s ) \\Bigg ( a ( s ) \\ , n ^ { \\alpha - 2 s } + \\sum _ { m = 0 } ^ { M - 1 } b _ m ( s ) \\ , n ^ { - 2 m } \\Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) , \\\\ & \\zeta ( \\widetilde { \\Delta } _ n , s ) = V _ \\alpha ( s ) \\Bigg ( \\widetilde { a } ( s ) \\ , n ^ { \\alpha - 2 s } + \\sum _ { m = 0 } ^ { M - 1 } \\widetilde { b } _ m ( s ) \\ , n ^ { - 2 m } \\Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) . \\end{align*}"}
{"id": "6770.png", "formula": "\\begin{align*} e ^ { x \\sqrt { n } ( \\sqrt { \\kappa _ n } - 1 ) } = \\sum _ { \\ell = 0 } ^ { L - 1 } \\beta _ \\ell ( x ) \\cdot n ^ { - \\ell / 2 } + O ( n ^ { - L / 2 } ) . \\end{align*}"}
{"id": "8158.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\lim \\limits _ { t \\rightarrow \\infty } \\| P _ \\delta ( W _ { n , m _ 0 ; \\textrm { s u r } } ) e ^ { - i t H _ 0 } \\varphi \\| = 0 \\end{align*}"}
{"id": "4395.png", "formula": "\\begin{align*} & \\int _ { D _ j } | F _ { m , \\epsilon , j } - ( 1 - v ' _ { \\epsilon } ( \\Psi _ m ) ) { f F ^ { 1 + \\delta } } | ^ { 2 } e ^ { v _ { \\epsilon } ( \\Psi _ m ) - \\varphi _ { m } - \\Psi _ m } \\\\ & \\leq \\int _ { D _ j } ( v '' _ { \\epsilon } ( \\Psi _ m ) ) | f F ^ { 1 + \\delta } | ^ 2 e ^ { - \\phi - \\varphi _ { m } - \\Psi _ m } . \\end{align*}"}
{"id": "5422.png", "formula": "\\begin{align*} 0 = & \\lim _ { k \\to \\infty } \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) u ^ { ( 1 ) } _ k u ^ { ( 2 ) } _ k \\ , d x + \\lim _ { k \\to \\infty } \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) u ^ { ( 1 ) } _ k g \\ , d x \\\\ & + \\lim _ { k \\to \\infty } \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) f u ^ { ( 2 ) } _ k \\ , d x + \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) f g \\ , d x . \\end{align*}"}
{"id": "2034.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : h a r d s p h e r e B } \\Phi ( | u | ) = | u | , \\Sigma ( \\theta ) = 1 . \\end{align*}"}
{"id": "3422.png", "formula": "\\begin{align*} n _ 1 = n ( 1 , 1 ) , n _ 2 = n ( \\zeta , 1 ) , n _ 3 = n ( 0 , 2 ) , n _ 4 = n _ 1 ^ t , n _ 5 = n _ 3 ^ t , \\end{align*}"}
{"id": "7449.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\abs { w ^ n } ^ 2 d x = o ( 1 ) \\int _ 0 ^ { \\infty } \\int _ 0 ^ L \\sigma ( s ) \\abs { \\eta ^ n _ x } ^ 2 d s d x = o ( 1 ) . \\end{align*}"}
{"id": "3183.png", "formula": "\\begin{align*} \\inf h _ { \\gamma _ { - } } \\left ( E _ { 1 } \\right ) = \\inf \\Gamma ( h _ { \\gamma _ { - } } ) \\left ( E _ { 1 } \\right ) \\ , \\end{align*}"}
{"id": "7130.png", "formula": "\\begin{align*} s _ 1 ^ { * } ( y ) : = 1 s _ n ^ { * } ( y ) : = \\sum _ { d } d s _ d ( y ^ { n / d } ) \\end{align*}"}
{"id": "1080.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } \\mathrm { e } ^ { - 2 b | \\mathbf { x } - \\mathbf { y } | ^ 2 / t } \\ , \\textnormal { d } \\mathbf { y } = \\int _ { \\mathbb { R } ^ { d } } \\mathrm { e } ^ { - 2 b | \\mathbf { z } | ^ 2 / t } \\ , \\textnormal { d } \\mathbf { z } = \\left ( \\frac { t \\pi } { 2 b } \\right ) ^ { d / 2 } . \\end{align*}"}
{"id": "7090.png", "formula": "\\begin{align*} \\delta ( t _ { k + 1 } ) & \\leq \\delta ( t _ { 0 } ) + 0 . 5 \\beta _ { \\max } \\mathsf { T } _ { \\max } \\sum ^ { k } _ { l = 0 } L \\beta ^ { l } _ { \\max } \\| e ( t _ { 0 } ) \\| \\\\ & \\leq \\delta ( t _ { 0 } ) + 0 . 5 \\| e ( t _ { 0 } ) \\| \\mathsf { T } _ { \\max } L \\sum ^ { k } _ { l = 0 } \\beta ^ { l + 1 } _ { \\max } \\\\ & \\leq \\delta ( t _ { 0 } ) + \\frac { L \\beta _ { \\max } \\mathsf { T } _ { \\max } \\| e ( t _ { 0 } ) \\| } { 2 ( 1 - \\beta _ { \\max } ) } . \\end{align*}"}
{"id": "3590.png", "formula": "\\begin{align*} v _ 1 ( \\alpha ) & = f _ \\alpha ( v _ 1 ( l _ 1 ) , \\ldots , v _ 1 ( l _ n ) ) \\\\ v _ 2 ( \\alpha ) & = f _ \\alpha ( v _ 2 ( l _ 1 ) , \\ldots , v _ 2 ( l _ n ) ) \\end{align*}"}
{"id": "3653.png", "formula": "\\begin{align*} H _ f ( m ) = m ^ { 1 - \\frac { p } { N } } H _ f ( 1 ) , 0 < H _ f ( 1 ) < + \\infty . \\end{align*}"}
{"id": "6951.png", "formula": "\\begin{align*} \\left ( U ^ * H _ u \\big | _ { E ( s ) } U f \\right ) ( \\xi ) = s \\bar \\xi \\overline { f ( \\xi ) } . \\end{align*}"}
{"id": "6135.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\end{gather*}"}
{"id": "1142.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { ( - z q ; q ) _ n } { ( q ; q ) _ n } q ^ { \\binom { n + 1 } { 2 } } = ( - q ; q ) _ { \\infty } ( - z q ^ 2 ; q ^ 2 ) _ { \\infty } . \\end{align*}"}
{"id": "8799.png", "formula": "\\begin{align*} \\phi _ 1 * \\phi _ 2 ( g ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } 1 _ { L _ 1 ( t _ 1 ) } * 1 _ { L _ 2 ( t _ 2 ) } ( g ) d t _ 1 d t _ 2 \\end{align*}"}
{"id": "3754.png", "formula": "\\begin{align*} \\| B _ { k ; j , l } ^ { e s s ; m } ( t , x ) \\| _ { L ^ 2 } \\lesssim \\sup _ { g \\in L ^ 2 , \\| g \\| _ { L ^ 2 } = 1 } \\big | \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } ( t - s ) \\big | { ( \\hat { v } \\times \\xi ) } \\big | \\big | \\overline { \\hat { g } ( \\xi ) } \\big | \\big | \\hat { f } ( s , \\xi , v ) \\big | \\varphi _ { m , k , l } ^ { e s s } ( \\omega , \\xi ) \\varphi _ { m ; - 1 0 M _ t } ( t - s ) \\varphi _ k ( \\xi ) \\end{align*}"}
{"id": "8522.png", "formula": "\\begin{align*} P _ { k } ( x ) & = x ^ { \\mathrm { p a r } _ k } \\sum _ { i = 0 } ^ n c _ i ( x ^ 2 - 1 ) ^ i , \\end{align*}"}
{"id": "1636.png", "formula": "\\begin{align*} U ^ * = \\{ t ^ * _ i : i \\le \\mathbf { i } \\} \\cup \\{ u ^ * : u \\in \\max ( D ) ^ + \\setminus B \\} , \\end{align*}"}
{"id": "5718.png", "formula": "\\begin{align*} \\pi _ { [ a , b ] } = \\pi _ { [ a - 1 , b - 1 ] } + \\pi _ { [ a , b - 1 ] } y _ { b } ( 1 \\le a \\le b \\le n - 1 ) \\end{align*}"}
{"id": "836.png", "formula": "\\begin{align*} p ( \\boldsymbol { s } ^ { r } | \\boldsymbol { s } ) = \\prod _ { m } \\mathop { p ( s _ { m } ^ { r } | s _ { m } ) } = \\prod _ { m } ( 1 - s _ { m } ) \\delta ( s _ { m } ^ { r } ) + s _ { m } \\rho _ { r } ^ { s _ { m } ^ { r } } ( 1 - \\rho _ { r } ) ^ { 1 - s _ { m } ^ { r } } , \\end{align*}"}
{"id": "3873.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ p G - \\lambda G ^ { p - 1 } = \\delta _ { x _ 0 } & \\Omega \\\\ G \\geq 0 & \\Omega \\\\ G = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
{"id": "553.png", "formula": "\\begin{align*} \\sum \\limits _ { 1 \\leq n \\leq N } f _ k ( n ) f _ k ( n + 1 ) = N \\sum _ { \\substack { w _ 1 \\leq N , w _ 2 \\leq N + 1 \\\\ ( w _ 1 , w _ 2 ) = 1 } } \\frac { g _ k ( w _ 1 ) g _ k ( w _ 2 ) } { ( w _ 1 w _ 2 ) ^ 2 } + O _ { \\varepsilon } ( N ^ { \\varepsilon } ) . \\end{align*}"}
{"id": "4281.png", "formula": "\\begin{align*} T _ \\xi ^ f ( \\phi , \\varepsilon , \\tau ( \\phi ) ) = T _ \\xi ^ f ( \\gamma _ 0 , a , [ b _ 0 ] ) . \\end{align*}"}
{"id": "3889.png", "formula": "\\begin{align*} \\eta _ \\epsilon = \\eta \\Omega \\setminus B _ { \\epsilon } ( 0 ) , \\eta _ \\epsilon = 0 B _ \\frac { \\epsilon } { 2 } ( 0 ) , | \\eta _ \\epsilon | + \\epsilon | \\nabla \\eta _ \\epsilon | \\leq C B _ { \\epsilon } ( 0 ) \\setminus B _ \\frac { \\epsilon } { 2 } ( 0 ) \\end{align*}"}
{"id": "8596.png", "formula": "\\begin{align*} \\sum _ { \\varepsilon _ j = 0 , 1 } \\prod _ { j = 1 } ^ l ( - 1 ) ^ { \\varepsilon _ j } r _ { k _ j } \\left ( \\alpha \\oplus \\frac { \\varepsilon _ j } { 2 ^ { k _ j } } \\right ) = \\prod _ { j = 1 } ^ l \\left ( r _ { k _ j } \\left ( \\alpha \\right ) - r _ { k _ j } \\left ( \\alpha \\oplus 2 ^ { - k _ j } \\right ) \\right ) = \\pm 2 ^ l . \\end{align*}"}
{"id": "4503.png", "formula": "\\begin{align*} \\Delta | \\nabla u | = \\frac 1 { | \\nabla u | } ( - K | \\nabla u | + | k | ^ 2 | \\nabla u | ^ 2 - \\langle \\div k , \\nabla u \\rangle | \\nabla u | ) \\end{align*}"}
{"id": "8383.png", "formula": "\\begin{align*} F ^ * ( k , 2 , 4 ) & = 1 2 8 \\\\ F ^ * ( k , 2 , 5 ) & = 2 5 6 \\\\ F ^ * ( 1 2 , 2 , 6 ) & = 7 6 8 , \\\\ F ^ * ( 1 3 , 2 , 7 ) & = 1 \\ , 5 3 6 , \\\\ F ^ * ( 1 3 , 2 , 6 ) & = 1 \\ , 0 2 4 . \\end{align*}"}
{"id": "1657.png", "formula": "\\begin{align*} \\kappa _ j ( x _ 1 , \\cdots , x _ { j - 1 } , B ) = & \\frac { \\overline { V } ( X _ 1 ) - \\bar { f } ( \\frac { j - 1 } { n } , \\hat { W ^ { ( n ) } } ) ^ 2 } { \\overline { V } ( X _ 1 ) - \\underline { V } ( X _ 1 ) } \\underline { P } ( B ) \\\\ & + \\frac { \\bar { f } ( \\frac { j - 1 } { n } , \\hat { W ^ { ( n ) } } ) ^ 2 - \\underline { V } ( X _ 1 ) } { \\overline { V } ( X _ 1 ) - \\underline { V } ( X _ 1 ) } \\overline { P } ( B ) , \\end{align*}"}
{"id": "6231.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } h _ n ( x ; q ) = [ n ] _ q h _ { n - 1 } ( \\frac { x } { q } ; q ) , \\end{align*}"}
{"id": "2043.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : c o m p a t i b i l i t y m e a s u r e s } \\forall \\varphi _ j \\in C _ b ( E ^ j ) , \\langle f ^ k , \\varphi _ j \\otimes 1 ^ { \\otimes ( k - j ) } \\rangle = \\langle f ^ j , \\varphi _ j \\rangle . \\end{align*}"}
{"id": "7506.png", "formula": "\\begin{align*} X _ { T , \\Lambda } = \\bigg \\{ u \\in C ( [ 0 , T ] ; C _ { \\Lambda } ( \\mathbb { R } ^ { N } ) ) : \\| \\Lambda u ( t ) \\| _ { L ^ { \\infty } ( \\mathbb { R } ^ { N } ) } \\leq \\delta , t \\in ( 0 , T ) \\bigg \\} \\end{align*}"}
{"id": "649.png", "formula": "\\begin{align*} \\frac { \\Omega ( 1 - s ) } { \\Omega ( s ) } = \\frac { \\xi _ 2 ( - s ) } { \\xi _ 2 ( s - 1 ) } = \\frac { \\xi _ 2 ( s + 1 ) } { \\xi _ 2 ( s - 1 ) } = \\frac { s ( s - 1 ) } { \\pi ^ 2 } \\cdot \\frac { \\zeta ( \\Delta , s + 1 ) } { \\zeta ( \\Delta , s - 1 ) } , \\end{align*}"}
{"id": "7441.png", "formula": "\\begin{align*} v = - f _ 1 z = - f _ 3 . \\end{align*}"}
{"id": "7662.png", "formula": "\\begin{align*} c ( \\mathcal E ) = 1 + n H + \\binom { n } { 2 } H ^ 2 + \\binom { n } { 3 } H ^ 3 + \\binom { n } { 4 } H ^ 4 \\end{align*}"}
{"id": "1976.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / 2 } } | f _ k ( x ) \\Phi ( x ) | \\ , d x & \\le \\| f _ k \\| _ { L ^ { N } ( B _ { 1 / 2 } ) } \\| \\Phi \\| _ { L ^ { N ' } ( B _ { 1 / 2 } ) } \\\\ & \\le \\frac { C ( N ) } { k } \\| \\Phi \\| _ { H ^ { 1 } ( B _ { 1 / 2 } ) } \\to 0 . \\\\ \\end{align*}"}
{"id": "8253.png", "formula": "\\begin{align*} i \\frac { 3 a ^ { 2 } } { 2 \\beta \\hslash ^ { 2 } } \\lambda a _ { 1 } + a _ { 2 } = \\lambda ^ { 2 } a _ { 2 } \\ , \\Rightarrow \\ , a _ { 2 } = i \\frac { 3 a ^ { 2 } } { 2 \\beta \\hslash ^ { 2 } } . \\frac { \\lambda } { \\lambda ^ { 2 } - 1 } a _ { 1 } . \\end{align*}"}
{"id": "6896.png", "formula": "\\begin{align*} \\rho ( t , v ) : = \\frac { 1 } { V _ F } \\sum _ { k = - \\infty } ^ { + \\infty } c _ k \\exp \\left ( i k \\frac { 2 \\pi } { V _ F } ( v - \\int _ 0 ^ t g _ { } ( s ) d s ) \\right ) , \\end{align*}"}
{"id": "6882.png", "formula": "\\begin{align*} N _ k ( t ) : = \\frac { 1 } { V _ F } \\int _ { 0 } ^ { \\infty } g p _ k ( t , g ) d g , k \\in \\mathbb { Z } , \\end{align*}"}
{"id": "7717.png", "formula": "\\begin{align*} T = \\sum _ { d _ 1 , \\ldots , d _ k \\le x } g _ k ( d _ 1 , \\ldots , d _ k ) \\Big ( \\frac { x } { d _ 1 \\cdots d _ k } P _ { 2 k - 1 } \\Big ( \\log \\frac { x } { d _ 1 \\cdots d _ k } \\Big ) + O \\Big ( \\Big ( \\frac { x } { d _ 1 \\cdots d _ k } \\Big ) ^ { \\theta _ { 2 k } + \\varepsilon } \\Big ) , \\end{align*}"}
{"id": "4002.png", "formula": "\\begin{align*} \\textstyle \\sum _ { j = 1 } ^ t p _ { i , j } = p _ i , ~ ~ i = 1 , \\cdots , s ; \\sum _ { i = 1 } ^ s p _ { i , j } = p ' _ j , ~ ~ j = 1 , \\cdots , t . \\end{align*}"}
{"id": "423.png", "formula": "\\begin{align*} H ' _ { \\mu } ( s + i t ) = 1 - \\int _ { \\mathbb { R } } \\frac { 1 + x ^ { 2 } } { ( s + i t - x ) ^ { 2 } } d \\sigma ( x ) , \\end{align*}"}
{"id": "6821.png", "formula": "\\begin{align*} \\begin{aligned} y '' ( \\delta ) & = y '' ( 1 ) - \\int _ \\delta ^ 1 \\Biggl ( y ''' ( 1 ) - \\int _ x ^ 1 y '''' ( s ) d s \\Biggr ) d x \\\\ & = y '' ( 1 ) - y ''' ( 1 ) + \\delta y ''' ( 1 ) + \\int _ \\delta ^ 1 y '''' ( s ) ( s - \\delta ) d s . \\end{aligned} \\end{align*}"}
{"id": "7809.png", "formula": "\\begin{align*} \\Theta _ n ( \\phi ) = B _ { \\delta / 2 } \\circ \\widetilde { \\Theta } _ n ( \\phi ) \\circ B _ { - \\delta / 2 } , \\end{align*}"}
{"id": "373.png", "formula": "\\begin{align*} e ^ { - k s } - \\sum _ { j = 1 } ^ \\infty ( - 1 ) ^ { j + 1 } \\frac { e ^ { - ( k + j ) s } } { j ! } = e ^ { - k s } e ^ { - e ^ { - s } } , \\end{align*}"}
{"id": "2698.png", "formula": "\\begin{align*} \\mathcal { N } _ 1 = a ^ * _ 1 a _ 1 , \\qquad \\mathcal { N } _ 2 = a ^ * _ 2 a _ 2 , \\qquad \\mathcal { N } _ \\perp = \\sum _ { m \\ge 3 } a ^ * _ m a _ m = N - \\mathcal { N } _ 1 - \\mathcal { N } _ 2 . \\end{align*}"}
{"id": "6033.png", "formula": "\\begin{align*} \\Lambda = \\left ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ m \\right ) \\end{align*}"}
{"id": "8263.png", "formula": "\\begin{align*} k = \\frac { \\sqrt { \\sqrt { 1 + \\frac { 1 6 } { 3 } \\beta m E } - 1 } } { 2 \\hslash \\sqrt { \\beta / 3 } } , k ' = \\frac { \\sqrt { \\sqrt { 1 + \\frac { 1 6 } { 3 } \\beta m E } + 1 } } { 2 \\hslash \\sqrt { \\beta / 3 } } . \\end{align*}"}
{"id": "544.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { g _ k ( n ) } { n ^ 2 } = \\prod _ { p } \\Big ( 1 - \\frac { 1 } { p ^ { 2 k } } \\Big ) = \\frac { 1 } { \\zeta ( 2 k ) } . \\end{align*}"}
{"id": "3909.png", "formula": "\\begin{align*} | W _ c ( \\mu , \\nu ) - W _ c ( \\tilde \\mu , \\tilde \\nu ) | & = \\left ( W _ c ( \\mu , \\nu _ t ) ^ { \\frac { 1 } { 2 } } + W _ c ( \\tilde \\mu , \\tilde \\nu ) ^ { \\frac { 1 } { 2 } } \\right ) \\left | W _ c ( \\mu , \\nu ) ^ { \\frac { 1 } { 2 } } - W _ c ( \\tilde \\mu , \\tilde \\nu ) ^ { \\frac { 1 } { 2 } } \\right | . \\end{align*}"}
{"id": "7281.png", "formula": "\\begin{align*} u = - { \\sf U } _ \\infty ( x ) - \\theta = - { \\sf U } _ \\infty ( x ) - ( \\theta _ 0 + \\theta _ 1 + \\cdots + \\theta _ L ) ( L \\gg 1 ) \\end{align*}"}
{"id": "6998.png", "formula": "\\begin{align*} \\alpha = P _ 1 \\widehat { d x _ 1 } \\wedge \\dots \\wedge d x _ n + \\dots + P _ n d x _ 1 \\wedge \\dots \\wedge \\ \\widehat { d x _ n } . \\end{align*}"}
{"id": "6730.png", "formula": "\\begin{align*} ( \\mathsf { m } ' ( \\varepsilon y ; \\varepsilon z ) - \\mathsf { m } ' ( \\varepsilon y ; \\varepsilon w ) ) : \\varepsilon ( z - w ) \\geq 0 | \\varepsilon y | = g . \\end{align*}"}
{"id": "5870.png", "formula": "\\begin{align*} D _ 2 \\approx \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p - q } } V _ r ( x _ { k - 1 } , x _ k ) ^ { \\frac { p q } { p - q } } \\sum _ { i = k } ^ { M - 1 } \\int _ { x _ i } ^ { x _ { i + 1 } } u ( s ) \\bigg ( \\int _ { s } ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } d s . \\end{align*}"}
{"id": "7307.png", "formula": "\\begin{align*} h [ \\epsilon ] & = - h _ 1 [ \\epsilon ] + h _ 1 ' [ \\epsilon ] \\\\ & \\lesssim { \\sf R } _ { \\sf i n } ^ { - \\frac { 1 } { 8 } ( 2 n - 9 ) } ( T - t ) ^ { { \\sf d } _ 1 } \\lambda ^ { - \\frac { n + 2 } { 2 } } \\sigma ( 1 + | y | ^ 2 ) ^ { - 1 - \\frac { 1 } { 1 6 } ( 2 n - 9 ) } { \\bf 1 } _ { | y | < 2 { \\sf R } _ { \\sf i n } } . \\end{align*}"}
{"id": "1981.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } a _ k + \\liminf _ { k \\to \\infty } b _ k & \\stackrel { * } { \\le } \\liminf _ { k \\to \\infty } ( a _ k + b _ k ) \\\\ & \\le \\limsup _ { k \\to \\infty } a _ k + \\liminf _ { k \\to \\infty } b _ k \\\\ & \\le \\limsup _ { k \\to \\infty } ( a _ k + b _ k ) \\\\ & \\stackrel { * * } { \\le } \\limsup _ { k \\to \\infty } a _ k + \\limsup _ { k \\to \\infty } b _ k . \\end{align*}"}
{"id": "3819.png", "formula": "\\begin{align*} { } ^ 1 _ 1 L a s t E l l _ { k _ 1 , j _ 1 ; n _ 1 } ^ { i , i _ 1 , \\mu _ 1 } ( t _ 1 , t _ 2 ) = \\sum _ { a = 1 , 2 } \\big | \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( t _ a ) \\cdot ( \\xi + \\eta ) + i t _ a \\mu _ 1 | \\eta | - i t _ a \\hat { v } \\cdot \\xi } ( \\hat { V } ( t _ a ) \\cdot ( \\xi + \\eta ) + \\mu _ 1 | \\eta | - \\hat { v } \\cdot \\xi ) ^ { - 1 } \\end{align*}"}
{"id": "1866.png", "formula": "\\begin{align*} a _ { n } = a _ { n } ^ { ( p ) } , n \\in \\mathbb { Z } . \\end{align*}"}
{"id": "451.png", "formula": "\\begin{align*} \\Phi ( z ) = z \\Sigma _ { \\nu } ( z ) \\Phi _ { n } ( z ) = z \\Sigma _ { \\nu _ { n } } ( z ) , n \\in \\mathbb { N } . \\end{align*}"}
{"id": "2580.png", "formula": "\\begin{align*} \\xi _ { x , y } = x \\ , \\omega _ 1 + y \\ , \\omega _ 2 \\in \\overline C \\ , \\backslash \\{ 0 \\} \\ . \\end{align*}"}
{"id": "7831.png", "formula": "\\begin{align*} \\mathcal S _ X = \\left \\{ \\left ( \\begin{smallmatrix} a & x \\\\ y ^ * & b \\end{smallmatrix} \\right ) \\mid a \\in C ^ * ( M , \\tilde M ) , b \\in C ^ * ( N , \\tilde N ) , x , y \\in X \\right \\} . \\end{align*}"}
{"id": "2166.png", "formula": "\\begin{align*} I _ { 2 } = 1 + \\int _ { \\mathbb { R } } \\frac { 1 + s ^ { 2 } } { s ^ { 2 } } \\ , d \\rho _ { \\nu } ( s ) = \\lim _ { y \\rightarrow 0 ^ { + } } \\frac { \\Im F _ { \\nu } ( i y ) } { y } \\in ( 1 , + \\infty ] . \\end{align*}"}
{"id": "4669.png", "formula": "\\begin{align*} l _ { i j } ( \\omega ) = \\sum \\limits _ { k = 1 } ^ d \\frac { \\partial ^ 2 \\psi ^ k } { \\partial \\omega _ i \\partial \\omega _ j } ( \\omega ) n ^ k ( \\omega ) , i , j = 1 , . . . , m - 1 ; \\end{align*}"}
{"id": "4643.png", "formula": "\\begin{align*} c _ 0 ( \\Z _ + ) \\ni x : = ( y _ k ) _ { k \\in \\Z _ + } \\mapsto J ^ { - 1 } x : = ( y _ k + y _ 0 ) _ { k \\in \\N } \\in c ( \\N ) . \\end{align*}"}
{"id": "2582.png", "formula": "\\begin{align*} \\begin{aligned} T : = & \\left \\{ \\small { \\begin{pmatrix} e ^ { i \\theta _ 1 } & 0 & 0 \\\\ 0 & e ^ { i \\theta _ 2 } & 0 \\\\ 0 & 0 & e ^ { i \\theta _ 3 } \\end{pmatrix} } : \\theta _ 1 + \\theta _ 2 + \\theta _ 3 = 0 \\right \\} \\\\ \\simeq \\ & S \\big ( U ( 1 ) \\times U ( 1 ) \\times U ( 1 ) \\big ) \\simeq U ( 1 ) \\times U ( 1 ) \\ . \\end{aligned} \\end{align*}"}
{"id": "8965.png", "formula": "\\begin{align*} \\int _ M \\rho ^ { - \\delta - \\frac { n } { p } + j } | \\hat \\nabla ^ j u | d V _ { \\hat g } = \\sum _ { i = 1 } ^ p \\int _ { f _ i ^ { - 1 } ( E _ i ) } f _ i ^ \\ast \\rho ^ { - \\delta - \\frac { n } { p } + j } | \\partial _ j f _ i ^ \\ast u | \\sqrt { \\det \\hat g _ { i j } } d \\mathbf { x } \\end{align*}"}
{"id": "6103.png", "formula": "\\begin{align*} 0 < \\alpha : = \\inf _ { v \\in H ^ 1 _ 0 ( \\Omega ) \\setminus \\{ 0 \\} } \\sup _ { v \\in H ^ 1 _ 0 ( \\Omega ) \\setminus \\{ 0 \\} } \\frac { a ( v , w ) } { \\| \\nabla v \\| \\ , \\| \\nabla w \\| } . \\end{align*}"}
{"id": "6824.png", "formula": "\\begin{align*} p ( t , 0 , g ) = p ( t , V _ F , g ) = 0 , 0 < g \\leq g _ F , t > 0 . \\end{align*}"}
{"id": "1424.png", "formula": "\\begin{align*} \\alpha _ { j } = \\phi _ 0 ^ { ( 1 - d ) } \\sum _ { n = - N } ^ { N } n \\phi _ j ( n ) , j \\not = 0 , \\textrm { a n d } \\alpha _ { 0 } = 0 . \\end{align*}"}
{"id": "2075.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : b a c k k o l m o g o r o v a p p e n d i x } \\frac { \\dd } { \\dd t } T _ t \\varphi = T _ t L \\varphi , \\frac { \\dd } { \\dd t } T _ t \\varphi = L T _ t \\varphi , \\end{align*}"}
{"id": "8195.png", "formula": "\\begin{align*} V _ { U } = \\left \\lbrace v \\in \\mathcal { H } \\left ( \\begin{array} { c c } U & - I \\\\ 0 & 0 \\end{array} \\right ) v = 0 \\right \\rbrace . \\end{align*}"}
{"id": "4941.png", "formula": "\\begin{align*} 0 = D _ i Q = \\sum \\theta _ k \\theta _ { k i } , \\\\ 0 \\ge D ^ 2 _ { i j } Q = \\theta _ { k i j } \\theta _ k + \\theta _ { k i } \\theta _ { k j } . \\end{align*}"}
{"id": "2509.png", "formula": "\\begin{align*} g ( A _ { X _ j } U , A _ { X _ j } V ) = n ^ 2 g ( H ' , U ) g ( H ' , V ) . \\end{align*}"}
{"id": "8532.png", "formula": "\\begin{align*} T _ 1 ( - ; p ) & = p \\\\ T _ n ( b _ 2 , b _ 3 , \\dots , b _ n ; p ) & = T _ { n - 1 } ( b _ 3 , \\dots , b _ n ; p ) p + b _ 2 T _ { n - 2 } ( b _ 4 , \\dots , b _ n ; p ) \\end{align*}"}
{"id": "5050.png", "formula": "\\begin{align*} K ^ { n , 1 } _ \\tau & = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ { ( \\delta + \\frac 1 n ) \\wedge \\tau } \\gamma _ s \\left ( \\sigma ' ( X _ s ) \\right ) ^ 2 \\\\ & \\times \\left [ \\left ( \\int _ { 0 } ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) ^ 2 - \\left ( \\int _ { \\eta _ n ( s ) - \\delta } ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) ^ 2 \\right ] \\ , d s . \\end{align*}"}
{"id": "2594.png", "formula": "\\begin{align*} B _ { k l } = \\sum _ { j = 1 } ^ 3 x ^ k _ { j } { x ^ l _ j } ^ \\ast \\ , \\ \\ C = x _ 1 ^ 1 x _ 3 ^ 2 - x _ 3 ^ 1 x _ 1 ^ 2 \\ , \\ \\ C ^ \\ast = { x _ 3 ^ 1 } ^ \\ast { x _ 2 ^ 2 } ^ \\ast - { x _ 2 ^ 1 } ^ \\ast { x _ 3 ^ 2 } ^ \\ast \\ , \\end{align*}"}
{"id": "5366.png", "formula": "\\begin{align*} M _ C ( H ^ s \\to H ^ { - s } ) \\vcentcolon = \\{ \\ , a \\in M ( H ^ s \\to H ^ { - s } ) \\ , ; \\ , \\norm { a } _ { s , - s } < C \\ , \\} \\end{align*}"}
{"id": "7899.png", "formula": "\\begin{align*} \\nabla _ { z \\partial z } = z \\frac { \\partial } { \\partial z } - \\frac { 1 } { z } E \\star _ t + \\mu , \\end{align*}"}
{"id": "5519.png", "formula": "\\begin{align*} ( f ' \\circ A ) ( z ) = ( B ' \\circ g ) ( z ) g ' ( z ) \\frac { ( z + i \\lambda ) ^ 2 } { 2 \\lambda i } . \\end{align*}"}
{"id": "6481.png", "formula": "\\begin{align*} u = y = 0 \\ \\ \\ \\ ( c _ 1 , c _ 2 ) . \\end{align*}"}
{"id": "7127.png", "formula": "\\begin{align*} \\mathbf { u } _ { i _ 1 i _ 2 } + \\cdots + \\mathbf { u } _ { i _ { \\ell - 1 } i _ \\ell } = \\begin{cases} \\boldsymbol { 0 } & i _ 1 = 0 i _ \\ell = n + 1 , \\\\ \\mathbf { u } _ { i _ 1 i _ \\ell } & \\end{cases} \\end{align*}"}
{"id": "5434.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { p } \\vcentcolon = \\frac { 1 } { p _ 1 } + \\frac { 1 } { r _ 2 } = \\frac { 1 } { p _ 2 } + \\frac { 1 } { r _ 1 } < 1 , \\end{align*}"}
{"id": "2494.png", "formula": "\\begin{align*} ( \\nabla F _ \\ast ) ( X , Y ) = { \\nabla } _ { F _ \\ast X } ^ N F _ \\ast Y - F _ \\ast ( { \\nabla } _ X ^ M Y ) , ~ \\forall X , Y \\in \\Gamma ( T M ) . \\end{align*}"}
{"id": "686.png", "formula": "\\begin{align*} \\left \\| e ^ { f _ o } v _ s - ( 4 \\pi ) ^ { - \\frac { n } { 2 } } \\right \\| ^ 2 _ { L ^ 2 ( d \\nu _ * ) } = \\sum _ { j = 1 } ^ { \\infty } c _ j ^ 2 e ^ { - 2 \\lambda _ j s } & \\leq e ^ { - 2 \\lambda _ 1 s } \\sum _ { j = 1 } ^ { \\infty } c _ j ^ 2 \\leq \\overline { C } e ^ { - 2 \\lambda _ 1 s } \\quad s \\geq 0 . \\end{align*}"}
{"id": "7418.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } f ( S _ k \\alpha ) = \\sum _ { m = 0 } ^ { n - 1 } \\sum _ { i = 2 } ^ { r _ m - 1 } T _ { m , i } ^ * + \\sum _ { i = 2 } ^ { R } T _ { n , i } ^ * + O ( N ^ { 1 / 2 - \\varepsilon } ) \\end{align*}"}
{"id": "4459.png", "formula": "\\begin{align*} F ( [ \\sigma ' ] ) \\circ F ( [ \\sigma ] ) & = ( [ d _ 2 d _ 2 \\sigma ' ] , [ d _ 0 d _ 0 \\sigma ' ] ) \\circ ( [ d _ 2 d _ 2 \\sigma ] , [ d _ 0 d _ 0 \\sigma ] ) \\\\ & = ( [ d _ 2 d _ 2 \\sigma '' ] , [ d _ 0 d _ 0 \\sigma '' ] ) \\\\ & = F ( [ \\sigma '' ] ) . \\end{align*}"}
{"id": "3103.png", "formula": "\\begin{align*} V ( \\mathcal { T } ) : = \\textit { C R } ^ 1 _ 0 ( \\mathcal { T } ) : = \\{ v \\in P _ 1 ( \\mathcal { T } ) : \\ & v \\textup { m i d } ( F ) F \\in \\mathcal { F } ( \\Omega ) \\\\ & v ( \\textup { m i d } ( F ) ) = 0 F \\in \\mathcal { F } ( \\partial \\Omega ) \\} . \\end{align*}"}
{"id": "242.png", "formula": "\\begin{align*} \\Gamma _ i ^ { ( s ) t } : = ( x ^ { ( p ^ s ) } ) ^ { - 1 } ( \\delta _ i ^ { ( s ) } ) ^ G x \\in \\textup { M a t } _ N ( \\mathcal A ) . \\end{align*}"}
{"id": "4102.png", "formula": "\\begin{align*} B { u } _ i ^ \\prime + ( A + B Q ) u _ i & \\ , = - H _ i ( u ^ \\prime + Q u ) , \\ i = 1 , 2 , \\\\ ( B + H _ 1 ) \\widetilde { u } \\ , ^ \\prime + \\bigl ( A + ( B + H _ 1 ) Q \\bigr ) \\widetilde { u } & \\ , = - H _ 2 ( u ^ \\prime _ + + Q u _ + ) , \\\\ B \\overline { \\overline { u } } \\ , ^ \\prime + ( A + B Q ) \\overline { \\overline { u } } & \\ , = - H _ 1 ( { u } _ 2 ' + Q { u } _ 2 ) - H _ 2 ( { u } _ 1 ' + Q { u } _ 1 ) . \\end{align*}"}
{"id": "7151.png", "formula": "\\begin{align*} H ^ * ( X _ { D } ) & = \\mathbb { Z } [ x _ 1 , x _ 2 , x _ 3 ] / \\langle x _ 1 ^ { a } , x _ 2 ( x _ 1 + x _ 2 ) ^ { b - 1 } , x _ 3 ( x _ 1 + x _ 2 + x _ 3 ) ^ { c - 1 } \\rangle , \\\\ H ^ * ( X _ { \\widetilde { D } } ) & = \\mathbb { Z } [ y _ 1 , y _ 2 , y _ 3 ] / \\langle y _ 1 ^ { a } , y _ 2 ( y _ 1 + y _ 2 ) ^ { c - 1 } , y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { b - 1 } \\rangle . \\end{align*}"}
{"id": "5524.png", "formula": "\\begin{align*} \\frac { \\abs { f ' ( z ) } } { \\abs { f ' ( z _ 0 ) } } = \\abs { \\frac { g ( \\textbf { z } _ 0 ) } { g ( \\textbf { z } ) } } ^ 2 \\abs { \\frac { g ' ( \\textbf { z } ) } { g ' ( \\textbf { z } _ 0 ) } } \\abs { \\frac { \\textbf { z } + \\lambda i } { \\textbf { z } _ 0 + \\lambda i } } ^ 2 . \\end{align*}"}
{"id": "3162.png", "formula": "\\begin{align*} \\mathit { \\Omega } _ { \\mathfrak { m } } ^ { \\flat } \\doteq \\left \\{ \\omega \\in E _ { 1 } : f _ { \\mathfrak { m } } ^ { \\flat } \\left ( \\omega \\right ) = \\inf \\ , f _ { \\mathfrak { m } } ^ { \\flat } ( E _ { 1 } ) = - \\mathrm { P } _ { \\mathfrak { m } } ^ { \\flat } \\right \\} \\end{align*}"}
{"id": "3984.png", "formula": "\\begin{align*} \\lambda _ { a _ \\delta } = \\begin{cases} 1 , & { \\rm i f } ~ { 0 } \\in I _ { \\bar { b } } ^ t + b _ k a _ \\delta ; \\\\ 0 , & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "4575.png", "formula": "\\begin{align*} r \\psi ( F _ * 1 ) = \\psi ( F _ * ^ e r ^ { p ^ e } ) = ( \\psi \\cdot r ^ { p ^ e - 1 } ) ( F _ * ^ e r ) \\notin Q , \\end{align*}"}
{"id": "6460.png", "formula": "\\begin{align*} \\Phi _ 0 ( [ x , y ] _ { \\mathfrak { g } } ) - [ \\Phi _ 0 ( x ) , \\Phi _ 0 ( y ) ] = [ Q , \\Phi _ 1 ( x , y ) ] . \\end{align*}"}
{"id": "5514.png", "formula": "\\begin{align*} \\abs { a _ n } = \\abs { \\frac { 1 } { 2 \\pi i } \\int _ { \\gamma } \\frac { h ( z ) } { z ^ { n + 1 } } d z } \\leq \\frac { 1 } { 2 \\pi } \\left ( \\frac { 3 } { 2 } \\right ) ^ n \\int _ 0 ^ { 2 \\pi } \\abs { h \\left ( \\frac { 2 } { 3 } e ^ { i \\theta } \\right ) } d \\theta . \\end{align*}"}
{"id": "2433.png", "formula": "\\begin{align*} \\overline { M _ f } = M _ f + \\sum _ { g \\prec f } r _ { g f } ( q ) M _ g , \\end{align*}"}
{"id": "7948.png", "formula": "\\begin{align*} f _ + ^ * ( K _ { X _ + } + D _ + ) = f _ - ^ * ( K _ { X _ - } + D _ - ) \\end{align*}"}
{"id": "8393.png", "formula": "\\begin{align*} f _ { \\alpha } ( r ) = \\begin{cases} r , & r \\geq 0 \\\\ - \\alpha r & r < 0 . \\end{cases} \\end{align*}"}
{"id": "7367.png", "formula": "\\begin{align*} t _ { k _ i } : = T _ k ^ { - 1 } \\ , t _ i . \\end{align*}"}
{"id": "4295.png", "formula": "\\begin{align*} | \\int _ L ( e ^ { - i \\hat { \\theta } } \\Omega ) | \\leq | \\int _ L \\Omega | \\leq \\int _ L d v o l ( L ) = ( L ) . \\end{align*}"}
{"id": "7285.png", "formula": "\\begin{align*} 0 = ( \\Delta - f _ 2 ' ( { \\sf U } _ \\infty ) ) \\theta _ 0 + f ( { \\sf U } _ \\infty ) = ( \\Delta - q L _ 1 ^ { q - 1 } | x | ^ { - 2 } ) \\theta _ 0 + L _ 1 ^ p | x | ^ { \\frac { 2 p } { 1 - q } } . \\end{align*}"}
{"id": "1306.png", "formula": "\\begin{align*} \\| \\mu ( E ) - \\nu ( E ) \\| _ { T V } = \\frac { 1 } { 2 } \\| f - g \\| _ { L ^ 1 } . \\footnotemark \\end{align*}"}
{"id": "6598.png", "formula": "\\begin{align*} [ ( V \\setminus L ) \\times ( V \\setminus L ) ] \\cap [ ( V \\setminus R ) \\times ( V \\setminus R ) ] = C \\times C , \\end{align*}"}
{"id": "7292.png", "formula": "\\begin{align*} \\theta _ 2 ( x ) & = { \\sf c } _ 2 | x | ^ 2 { \\sf U } _ \\infty ^ { p - 1 } | x | ^ \\frac { 2 ( p - q ) } { 1 - q } \\theta _ 0 + { \\sf h } _ 2 ( x ) = { a } _ 2 | x | ^ \\frac { 4 ( p - q ) } { 1 - q } \\theta _ 0 + { \\sf h } _ 2 ( x ) ( { a } _ 2 \\not = 0 ) , \\\\ { \\sf h } _ 2 ( x ) & = | x | ^ \\frac { 4 ( p - q ) } { 1 - q } \\theta _ 0 \\sum _ { i = 1 } ^ { N ' } b _ i | x | ^ \\frac { 2 ( p - q ) i } { 1 - q } . \\end{align*}"}
{"id": "5124.png", "formula": "\\begin{align*} W _ 0 \\subseteq W _ 1 \\subseteq \\ldots \\subseteq W _ n = S H _ * ( X ; \\Z ) , \\end{align*}"}
{"id": "2768.png", "formula": "\\begin{align*} \\begin{aligned} Z ( x _ i ) = f _ i - \\tfrac { \\mu } { 2 } \\| x _ i \\| ^ 2 \\nabla Z ( x _ i ) = g _ i - \\mu x _ i . \\end{aligned} \\end{align*}"}
{"id": "223.png", "formula": "\\begin{align*} F ( x ) = ( ( ( a _ 0 x + a _ 1 ) ^ { 2 ^ n - 2 } + a _ 2 ) ^ { 2 ^ n - 2 } + a _ 3 ) ^ { 2 ^ n - 2 } + a _ 4 \\end{align*}"}
{"id": "3763.png", "formula": "\\begin{align*} T _ { k , j ; n , l , r } ^ { \\mu , m , i } ( \\mathfrak { m } , E ) ( t , x , \\zeta ) = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big ( ( t - s ) \\mathfrak { K } ^ { \\mu , E } _ { k ; n } ( y , \\omega , v , \\zeta ) + \\mathfrak { K } ^ { e r r ; \\mu , E } _ { k ; n } ( y , v , \\zeta ) \\big ) f ( s , x - y + ( t - s ) \\omega , v ) \\varphi _ { j , n } ^ { i ; r } ( v , \\zeta ) \\end{align*}"}
{"id": "7051.png", "formula": "\\begin{align*} g _ 1 ( M ) = & g _ 1 ( M - 1 ) + g _ 1 ( M - 2 ) + F _ { M - 3 } + F _ { M - 4 } + \\cdots + F _ 1 + 1 \\\\ = & g _ 1 ( M - 1 ) + g _ 1 ( M - 2 ) + F _ { M - 1 } . \\end{align*}"}
{"id": "2641.png", "formula": "\\begin{align*} C _ o = D _ { x y } \\cap D _ z \\ , \\ D _ { z } \\backslash C _ o \\ , \\ D _ { x y } \\backslash C _ o \\ , \\ X \\backslash ( D _ { z } \\cup D _ { x y } ) \\ . \\end{align*}"}
{"id": "6052.png", "formula": "\\begin{align*} \\tilde { \\Lambda } = \\left ( \\tilde { \\lambda } _ 1 , \\tilde { \\lambda } _ 2 , \\ldots , \\tilde { \\lambda } _ { m } \\right ) \\end{align*}"}
{"id": "2890.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ m \\left ( p ^ { k m } - p ^ { k ( i - 1 ) } \\right ) , \\end{align*}"}
{"id": "6492.png", "formula": "\\begin{align*} u ( c _ 1 ) = u _ x ( c _ 1 ) = y ( c _ 1 ) = y _ x ( c _ 1 ) = y ( c _ 2 ) = y _ x ( c _ 2 ) = 0 . \\end{align*}"}
{"id": "2779.png", "formula": "\\begin{align*} \\begin{aligned} \\tfrac { 2 L \\sigma _ N } { B } = 1 + \\tfrac { h _ { N - 1 } } { 1 - \\kappa } \\ , \\ , , \\tfrac { \\alpha _ { N - 1 } } { B } = 1 \\end{aligned} \\end{align*}"}
{"id": "4948.png", "formula": "\\begin{align*} | A _ { n _ k } f ( x ) - A _ { n _ { k - 1 } } f ( x ) | & = \\bigg | \\sum _ { j \\in J ( k ) } ( A _ { m _ j } f ( x ) - A _ { m _ { j - 1 } } f ( x ) ) \\bigg | \\\\ & \\leq \\sum _ { j \\in J ( k ) } | A _ { m _ j } f ( x ) - A _ { m _ { j - 1 } } f ( x ) | . \\end{align*}"}
{"id": "7889.png", "formula": "\\begin{align*} \\mathfrak H _ { \\vec s } : = H ^ * ( D _ { I _ { \\vec s } } ) . \\end{align*}"}
{"id": "5393.png", "formula": "\\begin{align*} \\| \\nabla ^ s u \\| _ { L ^ 2 ( \\R ^ { 2 n } ) } = \\sqrt { \\frac { C _ { n , s } } { 2 } } [ u ] _ { W ^ { s , 2 } ( \\R ^ n ) } = \\| ( - \\Delta ) ^ { s / 2 } u \\| _ { L ^ 2 ( \\R ^ n ) } \\leq \\| u \\| _ { H ^ s ( \\R ^ n ) } . \\end{align*}"}
{"id": "7892.png", "formula": "\\begin{align*} \\tilde { T } _ { \\vec s } ^ k = [ T _ { I _ { \\vec s } } ^ k ] _ { \\vec s } . \\end{align*}"}
{"id": "7812.png", "formula": "\\begin{align*} \\eta _ Y ^ { - 1 } \\psi _ k \\eta _ { k - 1 } ( H ^ { 2 , 0 } ( X _ { k - 1 } ) ) = \\eta _ Y ^ { - 1 } ( \\psi _ k ( \\ell _ { k - 1 } ) ) = \\eta _ Y ^ { - 1 } ( \\psi ( \\ell _ 0 ) ) = \\eta _ Y ^ { - 1 } ( P ( Y , \\eta _ Y ) ) = H ^ { 2 , 0 } ( Y ) , \\end{align*}"}
{"id": "5579.png", "formula": "\\begin{align*} I \\le q \\ , \\Vert \\mu \\Vert \\int _ 0 ^ \\beta \\lambda ^ { q - 1 } \\ , d \\lambda = \\beta ^ q \\ , \\Vert \\mu \\Vert . \\end{align*}"}
{"id": "589.png", "formula": "\\begin{align*} \\zeta ( \\widetilde { \\Delta } _ n , s ) \\sim A ( s ) n ^ { \\alpha - 2 s } + B ( s ) + \\sum _ { j = 1 } ^ \\infty C _ j ( s ) n ^ { - 2 j } , \\end{align*}"}
{"id": "1570.png", "formula": "\\begin{align*} z ^ i _ { \\epsilon } : = \\frac { \\partial \\varphi ^ i } { \\partial x ^ { \\epsilon } } = \\delta _ { \\epsilon 1 } [ \\delta _ { i 1 } f ' ( x ^ 1 ) \\cos x ^ 2 + \\delta _ { i 2 } f ' ( x ^ 1 ) \\sin x ^ 2 + \\delta _ { i 3 } ] \\\\ + \\delta _ { \\epsilon 2 } [ - \\delta _ { i 1 } f ( x ^ 1 ) \\sin x ^ 2 + \\delta _ { i 2 } f ( x ^ 1 ) \\cos x ^ 2 ] , \\end{align*}"}
{"id": "3083.png", "formula": "\\begin{align*} \\lim _ { z _ { k } \\rightarrow + \\infty } z _ { k } H ( \\boldsymbol { z } ) \\overline { G } _ { n } \\left ( \\frac { 1 } { 2 } \\boldsymbol { z } ^ { T } \\boldsymbol { z } \\right ) = 0 , ~ k \\in \\{ 1 , 2 , \\cdots , n \\} \\end{align*}"}
{"id": "1397.png", "formula": "\\begin{align*} A = \\bigcap _ \\tau U _ \\tau \\subset \\bigcap _ \\tau \\left ( B _ \\varepsilon + B ( \\delta _ \\tau , 0 ) \\right ) = B _ \\varepsilon , \\end{align*}"}
{"id": "8138.png", "formula": "\\begin{align*} \\psi = \\psi ^ \\parallel \\otimes \\psi ^ \\perp \\end{align*}"}
{"id": "3965.png", "formula": "\\begin{align*} \\Bigl \\| \\sum _ { \\eta \\in G _ { u } } j ( b ) ( \\eta \\gamma ) ^ { * } \\ , j ( b ) ( \\eta \\gamma ) \\Bigr \\| = \\Bigl \\| \\sum _ { \\eta \\in G _ { v } } j ( b ) ( \\eta ) ^ { * } \\ , j ( b ) ( \\eta ) \\Bigr \\| \\le \\| b \\| _ { r } ^ { 2 } . \\end{align*}"}
{"id": "2241.png", "formula": "\\begin{align*} \\begin{cases} \\mathbf { U } ^ 0 \\cdot \\nabla _ { X , Y } \\mathbf { U } ^ 0 - \\mathbf { B } ^ 0 \\cdot \\nabla _ { X , Y } \\mathbf { B } ^ 0 + \\nabla _ { X , Y } P ^ 0 = 0 , \\\\ \\mathbf { U } ^ 0 \\cdot \\nabla _ { X , Y } \\mathbf { B } ^ 0 - \\mathbf { B } ^ 0 \\cdot \\nabla _ { X , Y } \\mathbf { U } ^ 0 = 0 , \\\\ \\nabla _ { X , Y } \\cdot \\mathbf { U } ^ 0 = 0 , \\quad \\nabla _ { X , Y } \\cdot \\mathbf { B } ^ 0 = 0 . \\end{cases} \\end{align*}"}
{"id": "7665.png", "formula": "\\begin{align*} E ( Y ; u , v ) = \\sum _ { p , q \\geq 0 } h ^ { p , q } ( Y ) ( - u ) ^ p ( - v ) ^ q \\ , \\in \\ , \\mathbb Z [ u , v ] . \\end{align*}"}
{"id": "1427.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde I _ { 1 , 2 , 1 } & \\leq ( 2 N + 1 ) ^ 2 + C \\beta ^ { - 1 / 2 } a ^ { - 1 } N ^ 3 \\log N \\\\ & \\leq \\tilde C \\big ( ( \\beta ^ { - 1 / 2 } a ^ { - 1 } N \\log N ) \\vee 1 \\big ) N ^ { 2 } . \\end{aligned} \\end{align*}"}
{"id": "5638.png", "formula": "\\begin{align*} \\mathcal { I } _ 3 = \\underbrace { \\iint _ { Q _ T } ( \\nabla p \\cdot \\vec { b } ) ( \\nabla \\cdot ( m \\nabla p ) ) } _ { \\mathcal { I } _ { 3 , 1 } } + \\underbrace { \\iint _ { Q _ T } m F ( \\nabla p \\cdot \\nabla \\vec { b } ) } _ { \\mathcal { I } _ { 3 , 2 } } . \\end{align*}"}
{"id": "8642.png", "formula": "\\begin{align*} H _ { n H } = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n p _ i ^ 2 + { U _ { n H } } ( x _ 1 , \\ldots , x _ n ) , \\end{align*}"}
{"id": "8225.png", "formula": "\\begin{align*} \\hat { H } ^ { \\dagger } \\phi _ { \\pm } = \\pm i \\lambda \\phi _ { \\pm } \\end{align*}"}
{"id": "324.png", "formula": "\\begin{align*} D _ v ( n ) = \\Big \\{ p \\mid n \\ ; : \\ , \\prod _ { q ^ e \\| n , q < p } q ^ e \\ \\le \\ p ^ v \\Big \\} . \\end{align*}"}
{"id": "5883.png", "formula": "\\begin{align*} \\sigma _ j = \\frac { 1 } { [ \\log _ 2 \\log _ 2 ( j + 3 ) ] ^ \\alpha } , j = 1 , 2 , \\dots , \\end{align*}"}
{"id": "3465.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } s ^ { - 1 / 2 } \\overline { G } _ { ( k ) } ( s ) \\mathrm { d } s < \\infty , ~ k = 1 , 2 \\cdots , n . \\end{align*}"}
{"id": "1886.png", "formula": "\\begin{align*} z A _ { 0 } ( z ) - 1 & = a _ { 0 } ^ { ( 0 ) } A _ { 0 } ( z ) + \\sum _ { j = 1 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { j } ( z ) = a _ { 0 } ^ { ( 0 ) } A _ { 0 } ( z ) + \\sum _ { j = 1 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { 0 } ( z ) A _ { j - 1 } ^ { ( 1 ) } ( z ) \\\\ & = A _ { 0 } ( z ) ( a _ { 0 } ^ { ( 0 ) } + \\sum _ { j = 1 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { j - 1 } ^ { ( 1 ) } ( z ) ) \\end{align*}"}
{"id": "4369.png", "formula": "\\begin{align*} & \\int _ { D _ 0 } | \\tilde F - ( 1 - b ( k \\Psi ) ) f F ^ { 2 k } | ^ 2 e ^ { - k \\varphi + v ( k \\Psi ) - k \\Psi } \\\\ \\le & ( 2 - e ^ { - t _ 0 - 1 } ) \\int _ { D _ 0 } \\mathbb { I } _ { \\{ - t _ 0 - 1 < k \\Psi < - t _ 0 \\} } | f | ^ 2 e ^ { - k \\Psi } , \\end{align*}"}
{"id": "6744.png", "formula": "\\begin{align*} \\langle \\hat h , v \\rangle _ { Y ' , Y } : = \\mu \\int _ { \\Omega } \\varepsilon \\xi : \\varepsilon v + \\nu \\int _ { \\Omega } \\mathsf { m } ' \\left ( \\varepsilon \\overline { y } ; \\varepsilon \\xi \\right ) : \\varepsilon v , \\forall v \\in Y . \\end{align*}"}
{"id": "4107.png", "formula": "\\begin{align*} \\big \\langle V ' ( \\vec p ) \\widehat { \\vec p } \\big ( u ' + Q u \\big ) , w \\rangle _ X = \\int _ D \\big ( \\widehat { \\rho } \\ , \\partial _ t \\vec v \\cdot { \\vec w } + S _ 0 + S _ 1 + \\cdots + S _ L \\big ) \\d x \\end{align*}"}
{"id": "7576.png", "formula": "\\begin{align*} \\frac { d x } { d t } = f ( t , x ) , t \\in [ a , b ] , x ( t ) \\in N , f \\in \\mathfrak { X } _ t ( N ) . \\end{align*}"}
{"id": "1918.png", "formula": "\\begin{align*} \\mathcal { I } _ { [ m , j ] } : = \\{ ( i _ { 1 } , \\ldots , i _ { m } ) \\in \\mathbb { Z } _ { \\geq 0 } ^ { m } : 0 \\leq i _ { 1 } \\leq j , \\ , \\ , \\mbox { a n d } \\ , \\ , 0 \\leq i _ { k } \\leq i _ { k - 1 } + p \\ , \\ , \\mbox { f o r e a c h } \\ , \\ , k = 2 , \\ldots , m \\} . \\end{align*}"}
{"id": "1282.png", "formula": "\\begin{align*} H e _ 0 ( x ) & = 1 , \\\\ H e _ 1 ( x ) & = x , \\\\ H e _ j ( x ) & = x H e _ { j - 1 } ( x ) - ( j - 1 ) H e _ { j - 2 } ( x ) ( \\forall j \\ge 2 ) . \\end{align*}"}
{"id": "2929.png", "formula": "\\begin{align*} P ( \\Pi ) = \\left \\{ y \\in [ 1 , m ] ^ m : \\begin{array} { r l c } \\sum _ { i \\in E } y _ i & = \\binom { m + 1 } { 2 } & \\\\ \\sum _ { i \\in S } y _ i & \\leq \\binom { m + 1 } { 2 } - \\binom { m + 1 - | S | } { 2 } , & \\forall S \\subseteq E : S \\neq \\emptyset \\end{array} \\right \\} . \\end{align*}"}
{"id": "1885.png", "formula": "\\begin{align*} A _ { [ n , 0 ] } = \\sum _ { \\gamma \\in \\mathcal { D } _ { [ n , 0 ] } } w ( \\gamma ) = \\sum _ { j = 0 } ^ { p } \\sum _ { \\gamma \\in \\mathcal { L } _ { [ n , j ] } } w ( \\gamma ) = \\sum _ { j = 0 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { [ n - 1 , j ] } , n \\geq 1 . \\end{align*}"}
{"id": "3994.png", "formula": "\\begin{align*} \\textstyle ( C \\beta ) ^ \\bot = \\bigoplus _ { i = 0 } ^ { r } ( C _ i \\beta _ i ) ^ { \\bot _ { A _ i } } , \\end{align*}"}
{"id": "7186.png", "formula": "\\begin{align*} & m = j + 1 + \\delta ( j _ 3 , j _ 4 ) - \\delta ( [ j _ 1 + j _ 3 ] , [ j _ 2 + j _ 4 ] ) + \\frac { [ j _ 1 + j _ 3 ] } { T } - \\frac { j _ 1 + j _ 3 } { T } \\end{align*}"}
{"id": "8920.png", "formula": "\\begin{align*} [ D ^ 2 u - A ( \\cdot , u , D u ) ] = B ( \\cdot , u , D u ) , \\end{align*}"}
{"id": "5410.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\frac { 1 } { 1 + | x | ^ { n + 2 s } } \\ , d x & = \\omega _ n \\int _ 0 ^ { \\infty } \\frac { r ^ { n - 1 } } { 1 + r ^ { \\frac { n + 2 s } { 2 } } } \\ , d r < \\infty \\end{align*}"}
{"id": "8660.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\underline f > \\tau \\} } ( r ) g ( r ) \\ , d r & = \\int _ { a _ \\tau } ^ \\infty g ( r ) \\ , d r \\leq a _ \\tau ^ { - \\alpha } \\sup _ { s > 0 } s ^ { \\alpha } \\int _ s ^ \\infty g ( r ) \\ , d r \\\\ & = \\alpha \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\underline f > \\tau \\} } ( r ) \\ , \\frac { d r } { r ^ { 1 + \\alpha } } \\ \\overline \\nu _ \\alpha ( g ) \\ , . \\end{align*}"}
{"id": "6778.png", "formula": "\\begin{align*} & M : = [ k + 1 ] , \\ n _ 1 = \\ldots = n _ k : = 1 , \\ n _ { k + 1 } : = \\beta + 1 , \\\\ & a _ { i 1 } : = \\alpha _ i \\ \\forall i \\in [ k ] , \\ a _ { k + 1 , 1 } : = 3 , \\ a _ { k + 1 , 2 } = \\ldots = a _ { k + 1 , \\beta + 1 } : = 1 , \\\\ & b : = \\beta + 2 , \\ c : = a , \\\\ & x ^ * _ { 1 1 } = \\ldots = x ^ * _ { k 1 } : = \\frac { 2 \\beta - 3 } { 6 \\beta } , \\ x ^ * _ { k + 1 , 1 } : = 1 , \\ x ^ * _ { k + 1 , 2 } = \\ldots = x ^ * _ { k + 1 , \\beta + 1 } : = \\frac { 1 } { 3 } . \\end{align*}"}
{"id": "5245.png", "formula": "\\begin{align*} \\langle \\tau ^ { ( m _ 1 , m _ 2 ) } _ 0 \\sigma _ 1 ^ { m _ 1 } \\sigma _ 2 ^ { m _ 2 } \\sigma _ { 1 2 } \\rangle ^ { \\mathbf { s } , o } = 1 . \\end{align*}"}
{"id": "6632.png", "formula": "\\begin{align*} \\int _ { B _ h } \\phi _ w \\big ( b ^ { - 1 } ( x ^ { - 1 } , u ) \\big ) \\ , d m _ p ( b ) = \\hbox { N } _ h \\left ( N , w _ x , h I \\ , \\hbox { m o d } \\ , \\ell \\right ) . \\end{align*}"}
{"id": "591.png", "formula": "\\begin{align*} \\widetilde \\Delta _ n u ( j _ 1 , j _ 2 ) : = & \\frac { n ^ 2 } { 4 \\pi ^ 2 } \\bigg ( \\frac { 1 0 } { 3 } u ( j _ 1 , j _ 2 ) - \\frac { 2 } { 3 } \\Big ( u ( j _ 1 + 1 , j _ 2 ) + u ( j _ 1 , j _ 2 + 1 ) \\\\ & + u ( j _ 1 - 1 , j _ 2 ) + u ( j _ 1 , j _ 2 - 1 ) \\Big ) - \\frac { 1 } { 6 } \\Big ( u ( j _ 1 - 1 , j _ 2 - 1 ) \\\\ & + u ( j _ 1 - 1 , j _ 2 + 1 ) + u ( j _ 1 + 1 , j _ 2 - 1 ) + u ( j _ 1 + 1 , j _ 2 + 1 ) \\Big ) \\bigg ) . \\end{align*}"}
{"id": "2475.png", "formula": "\\begin{align*} \\bar { p } = a \\Pi _ { \\alpha \\in \\left ( \\Delta _ 1 ^ { + } \\cap { \\text B } ^ { \\perp } \\right ) \\backslash \\pm { \\text B } } \\alpha \\bar { q } = b \\Pi _ { \\alpha \\in \\left ( \\Delta _ 1 ^ { + } \\cap { \\text B } ^ { \\perp } \\right ) \\backslash \\pm { \\text B } } \\alpha \\end{align*}"}
{"id": "4191.png", "formula": "\\begin{align*} f = \\sum _ { m = 1 } ^ { M _ { \\ell } } f _ { m } ^ { ( \\ell ) } f _ { m } ^ { ( \\ell ) } : = f \\chi _ { B _ { m } ^ { ( \\ell ) } } . \\end{align*}"}
{"id": "3175.png", "formula": "\\begin{align*} B _ { \\gamma } \\left ( a \\right ) = \\int _ { \\Theta _ { d } } \\hat { f } \\left ( - \\gamma ^ { - 1 } \\theta \\right ) \\mathrm { e } ^ { - i \\theta \\cdot a } \\mathrm { d } \\left ( \\mathrm { F } _ { \\ast } \\mathrm { P } \\right ) \\left ( \\theta \\right ) + R _ { f , a , \\gamma } \\ , \\end{align*}"}
{"id": "2818.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\big \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\big \\} { } \\leq { } \\frac { 2 L \\big [ f ( x _ 0 ) - f ( x _ N ) \\big ] } { \\sum \\limits _ { i = 0 } ^ { N - 1 } p ( h _ i , \\kappa ) } \\end{align*}"}
{"id": "2830.png", "formula": "\\begin{align*} x _ i = x _ 0 + \\tfrac { 1 } { L } \\sum \\limits _ { k = 0 } ^ { i - 1 } h _ k g _ k . \\end{align*}"}
{"id": "6695.png", "formula": "\\begin{align*} U _ 0 = G _ 0 \\supset \\dots \\supset U _ n \\supset G _ n \\supset V _ n \\supset D _ n \\supset U _ { n + 1 } \\supset \\dots \\ \\ \\ \\ M _ n \\subset U _ n . \\end{align*}"}
{"id": "4934.png", "formula": "\\begin{align*} h ^ i _ j & = \\frac { 1 } { \\rho ^ 2 \\sqrt { \\rho ^ 2 + | D \\rho | ^ 2 } } ( e ^ { i k } - \\frac { \\rho ^ i \\rho ^ k } { \\rho ^ 2 + | D \\rho | ^ 2 } ) ( - \\rho D _ k D _ j \\rho + 2 \\rho _ k \\rho _ j + \\rho ^ 2 e _ { k j } ) \\\\ & = - \\rho ^ { - 2 } \\rho ^ i _ { j } + \\rho ^ { - 1 } \\delta ^ i _ { j } . \\end{align*}"}
{"id": "7657.png", "formula": "\\begin{align*} W _ { ( \\rho _ 1 , \\rho _ 2 ) } = \\langle M _ { u _ i } ^ t \\cdot a _ j \\ , | \\ , 1 \\leq i , j \\leq 2 \\rangle \\end{align*}"}
{"id": "3444.png", "formula": "\\begin{align*} R ( N , \\Sigma ) : = \\{ & \\alpha \\in M \\ , | \\ , \\langle \\alpha , n ( \\alpha ) \\rangle = 1 \\textrm { f o r s o m e } n ( \\alpha ) \\in \\Sigma [ 1 ] \\\\ & \\textrm { a n d } \\langle \\alpha , n _ { j } \\rangle \\leq 0 \\textrm { f o r } n _ j \\in \\Sigma [ 1 ] \\setminus \\{ n ( \\alpha ) \\} \\} \\end{align*}"}
{"id": "3632.png", "formula": "\\begin{align*} u _ n \\coloneqq \\sum _ { i = 1 } ^ { n \\wedge k } m _ i \\delta _ { x _ i } + \\sum _ { i = 1 } ^ { ( n 2 ^ n ) ^ N } u ^ d _ + ( Q ^ n _ i ) \\delta _ { x ^ n _ i } - \\sum _ { i = 1 } ^ { ( n 2 ^ n ) ^ N } u ^ d _ - ( Q ^ n _ i ) \\delta _ { y ^ n _ i } , \\end{align*}"}
{"id": "7563.png", "formula": "\\begin{align*} \\frac { d x _ i } { d t } = \\eta _ i ( t , x ) , i = 1 , \\dots , n . \\end{align*}"}
{"id": "4280.png", "formula": "\\begin{align*} A ( \\rho ) = \\tau ( \\phi ) ^ { - 1 } \\rho ( \\tau ( \\phi ) ) \\zeta _ \\phi ^ { p , \\infty } ( \\rho ) \\end{align*}"}
{"id": "7393.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ 4 U _ { h , i } ( y ) & \\le M { \\min \\{ | y - h t _ 1 | ^ { - \\left ( \\frac { N - 1 } { 2 } \\right ) } , 1 \\} } e ^ { ( - 1 + \\eta ) | y - h t _ 1 | } \\sum _ { i = 2 } ^ 4 e ^ { - \\frac { \\eta } { 2 } | h t _ i - h t _ 1 | } \\\\ & = 3 M { \\min \\{ | y - h t _ 1 | ^ { - \\left ( \\frac { N - 1 } { 2 } \\right ) } , 1 \\} } e ^ { ( - 1 + \\eta ) | y - h t _ 1 | } e ^ { - \\sqrt { 2 } \\eta h } . \\end{align*}"}
{"id": "2246.png", "formula": "\\begin{align*} & ( u _ p ^ i , h _ p ^ i ) ( x , 0 ) = ( - \\overline u _ e ^ i ( x ) , - \\overline h _ e ^ i ( x ) ) , \\\\ & ( u _ p ^ i , v _ p ^ i , h _ p ^ i , g _ p ^ i ) ( x , \\infty ) = ( 0 , 0 , 0 , 0 ) . \\end{align*}"}
{"id": "2271.png", "formula": "\\begin{align*} & \\int _ 0 ^ r \\left [ ( - \\phi '' - \\frac { z } { 2 } \\phi ' + \\frac { z } { 2 } e _ \\sigma \\psi ' ) \\cdot ( z ^ { 2 } \\phi ) + ( - \\psi '' - \\frac { z } { 2 } \\psi ' + \\frac { z } { 2 } e _ \\sigma \\phi ' ) \\cdot ( z ^ { 2 } \\psi ) \\right ] { \\rm { d } } z \\\\ = & \\int _ 0 ^ r \\left ( | z ( \\phi ' , \\psi ' ) | ^ 2 + \\frac { 3 } { 4 } | z ( \\phi , \\psi ) | ^ 2 + 2 z ( \\phi ' \\phi + \\psi ' \\psi ) - \\frac { 3 e _ \\sigma + z e _ \\sigma ' } { 2 } z ^ { 2 } \\phi \\psi \\right ) . \\end{align*}"}
{"id": "5106.png", "formula": "\\begin{align*} \\Lambda ^ { ( 1 ) } _ { n , \\delta } & = \\exp \\left ( i \\mu Y ^ { n , 1 } _ { t } + i \\lambda n ^ { \\alpha + \\frac 1 2 } \\int _ { t - \\delta } ^ { \\eta _ n ( t ) } \\psi _ { n , 1 } ( s , \\eta _ n ( t ) ) d W _ s \\right ) \\\\ & \\times \\left [ \\exp \\left ( i \\lambda n ^ { \\alpha + \\frac 1 2 } \\int ^ { t - \\delta } _ 0 \\psi _ { n , 1 } ( s , \\eta _ n ( t ) ) d W _ s \\right ) - 1 \\right ] \\end{align*}"}
{"id": "130.png", "formula": "\\begin{align*} C \\star \\big ( S ^ 3 - C ^ 3 \\big ) \\star S = C \\star E \\big ( 3 C ^ 2 + 3 E C + E ^ 2 \\big ) \\star ( E + C ) . \\end{align*}"}
{"id": "3821.png", "formula": "\\begin{align*} \\sum _ { i = 0 , 1 , 2 , 3 , 4 } \\big | { } _ 0 ^ { 1 } E r r ^ i _ { k , j ; n } ( t _ 1 , t _ 2 ) \\big | \\lesssim \\mathcal { M } ( C ) 2 ^ { ( \\alpha ^ \\star + 3 \\iota + 1 4 0 \\epsilon ) M _ t - ( k + 2 n ) / 2 } \\lesssim 2 ^ { M _ t / 2 } \\mathcal { M } ( C ) . \\end{align*}"}
{"id": "575.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ { i + 1 } ) = f _ k ( i ) f _ k ( i + 1 ) + O _ { \\alpha , \\varepsilon } ( i ^ { - 1 / 2 + \\varepsilon } ) \\end{align*}"}
{"id": "1343.png", "formula": "\\begin{align*} - \\Delta _ x S + \\alpha S = \\rho ( t , x ) : = \\int _ { \\mathcal { V } } f ( t , x , v ) \\d v . \\end{align*}"}
{"id": "8070.png", "formula": "\\begin{align*} \\phi ( G / H \\times \\{ 0 \\} ) = G \\cdot x . \\end{align*}"}
{"id": "4409.png", "formula": "\\begin{align*} M _ 1 ( f , s _ n ) = M _ 1 ( g , 1 ) = 1 \\ \\ \\ \\mbox { a n d } \\ \\ \\ M _ 1 ( P _ { m _ n + K } f , s _ n ) = M _ 1 ( P _ K g , 1 ) > N . \\end{align*}"}
{"id": "1365.png", "formula": "\\begin{align*} \\begin{cases} i v _ t + \\bigtriangleup v \\pm \\mathcal { R } ^ 2 [ v ] \\pm \\mathcal { N R } [ v ] = 0 \\\\ v ( x , 0 ) = v _ 0 = u _ 0 \\in H ^ s _ x ( \\mathbb { T } ) . \\end{cases} \\end{align*}"}
{"id": "1146.png", "formula": "\\begin{align*} \\Delta = \\Delta _ 1 \\xrightarrow { \\gamma _ 1 } \\Delta _ 2 \\xrightarrow { \\gamma _ 2 } \\cdots \\xrightarrow { \\gamma _ { k - 1 } } \\Delta _ k = \\emptyset \\end{align*}"}
{"id": "7708.png", "formula": "\\begin{align*} \\widetilde { F } ( n ) = \\sum _ { d \\delta = n } g ( d ) \\tau _ k ( \\delta ) , \\end{align*}"}
{"id": "3345.png", "formula": "\\begin{align*} \\begin{pmatrix} \\cos ( \\frac { N \\pi } { d } ) & - \\sin ( \\frac { N \\pi } { d } ) & 0 & 0 \\\\ \\sin ( \\frac { N \\pi } { d } ) & \\cos ( \\frac { N \\pi } { d } ) & 0 & 0 \\\\ 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "2944.png", "formula": "\\begin{align*} [ F r e e ~ C h a n n e l ] ~ ~ \\quad & \\Omega _ F ( \\psi _ 0 ) \\equiv s \\lim _ { t \\to \\infty } U _ 0 ( 0 , t ) J _ c \\psi ( t ) \\\\ & \\psi ( t ) = U ( t , 0 ) \\psi _ 0 , \\\\ & J _ c = U _ 0 ( t , 0 ) J ( | x | / t ^ { \\alpha } \\leq 1 ) U _ 0 ( 0 , t ) , \\\\ & 0 \\leq \\alpha \\leq a ( n , p ) < 1 , \\end{align*}"}
{"id": "993.png", "formula": "\\begin{align*} B _ { \\alpha , \\beta } ^ { * } = \\operatorname { d i a g } \\left \\{ b _ { 1 } , b _ { 2 } , \\cdots , b _ { n } \\right \\} , \\end{align*}"}
{"id": "7157.png", "formula": "\\begin{align*} ( g _ { 2 1 } y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 ) \\{ ( g _ { 1 1 } + g _ { 2 1 } ) y _ 1 + ( g _ { 1 2 } + g _ { 2 2 } ) y _ 2 + ( g _ { 1 3 } + g _ { 2 3 } ) y _ 3 \\} ^ { a - 1 } \\\\ = g _ { 2 3 } ( g _ { 1 3 } + g _ { 2 3 } ) ^ { a - 1 } y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } . \\end{align*}"}
{"id": "7883.png", "formula": "\\begin{align*} \\vec s ^ j = ( s _ 1 ^ j , \\ldots , s _ n ^ j ) \\in \\mathbb Z ^ n , j = 1 , 2 , \\ldots , m , \\end{align*}"}
{"id": "8932.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , u , D u ) = \\psi ( \\cdot , u , D u ) , \\end{align*}"}
{"id": "7935.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m ( D _ i \\cdot e ) b _ i = 0 \\end{align*}"}
{"id": "704.png", "formula": "\\begin{align*} \\mu = \\frac { f ( u ) } { \\sqrt { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | ^ 2 } } \\Bigl ( \\partial _ t + \\frac { 1 } { f ( u ) ^ 2 } \\widetilde { g } ^ { i j } u _ j \\partial _ { i } \\Bigr ) . \\end{align*}"}
{"id": "1972.png", "formula": "\\begin{align*} A ( x , v ) & : = A _ + ( x ) \\chi _ { \\{ v > 0 \\} } + A _ - ( x ) \\chi _ { \\{ v < 0 \\} } , \\\\ f ( x , v ) & : = f _ + ( x ) \\chi _ { \\{ v > 0 \\} } + f _ - ( x ) \\chi _ { \\{ v < 0 \\} } , \\\\ \\lambda ( v ) & : = \\lambda _ + \\chi _ { \\{ v > 0 \\} } + \\lambda _ - \\chi _ { \\{ v \\le 0 \\} } . \\end{align*}"}
{"id": "5267.png", "formula": "\\begin{align*} \\mathbf { s } = \\mathbf { s } ^ \\Gamma = s ^ \\Gamma \\oplus \\bigoplus _ { i \\in \\left [ l \\right ] , ~ j \\in \\left [ d _ i + \\delta _ { i = 1 } \\right ] } s _ { i j } ^ \\Gamma \\end{align*}"}
{"id": "6763.png", "formula": "\\begin{align*} \\gcd \\left ( \\binom { \\alpha \\beta } { d _ 1 } , \\binom { \\alpha \\beta } { d _ 2 } , \\dots , \\binom { \\alpha \\beta } { d _ r } \\right ) \\alpha \\end{align*}"}
{"id": "2989.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty y ^ { s - \\frac 1 2 } K _ { i r } ( 2 \\pi | n | y ) \\ , d y = \\tfrac 1 4 ( \\pi | n | ) ^ { - s - \\frac 1 2 } \\Gamma ( \\tfrac s 2 + \\tfrac { i r } { 2 } + \\tfrac 1 4 ) \\Gamma ( \\tfrac s 2 - \\tfrac { i r } { 2 } + \\tfrac 1 4 ) . \\end{align*}"}
{"id": "5657.png", "formula": "\\begin{align*} \\sup _ { \\mathbb { R } ^ n } \\left | \\underline { u } ( x ) - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \\right | \\leq C , \\end{align*}"}
{"id": "4768.png", "formula": "\\begin{align*} F ( \\mathbf { H } _ ) = l F ( \\mathbf { R } _ 1 ) + ( l - 1 ) ( F ( \\mathbf { R } _ 2 ) - 2 F ( \\mathbf { R } _ 1 ) ) , \\\\ F ( \\mathbf { B } _ ) = l F ( \\mathbf { Q } _ 1 ) + ( l - 1 ) ( F ( \\mathbf { Q } _ 2 ) - 2 F ( \\mathbf { Q } _ 1 ) ) . \\end{align*}"}
{"id": "7705.png", "formula": "\\begin{align*} C _ { f , k } = \\prod _ p \\left ( 1 - \\frac 1 { p } \\right ) ^ k \\sum _ { \\nu _ 1 , \\ldots , \\nu _ k = 0 } ^ { \\infty } \\frac { f ( p ^ { \\max ( \\nu _ 1 , \\ldots , \\nu _ k ) } ) } { p ^ { ( r + 1 ) ( \\nu _ 1 + \\cdots + \\nu _ k ) } } . \\end{align*}"}
{"id": "3260.png", "formula": "\\begin{align*} \\tilde { u } _ { n } - T _ { A _ n , q _ n } \\tilde { u } _ { n } = \\frac { v _ { n } } { \\Vert u _ { n } \\Vert _ { H ^ 1 ( D ) } } = : \\tilde { v } _ { n } \\textrm { i n } H ^ 1 ( D ) , \\end{align*}"}
{"id": "1509.png", "formula": "\\begin{align*} \\Big ( x \\frac { d } { d x } \\Big ) _ { n , \\lambda } = \\Big ( x \\frac { d } { d x } \\Big ) \\Big ( x \\frac { d } { d x } - \\lambda \\Big ) \\cdots \\Big ( x \\frac { d } { d x } - ( n - 1 ) \\lambda \\Big ) , ( n \\ge 1 ) . \\end{align*}"}
{"id": "5064.png", "formula": "\\begin{align*} L ^ { n , 4 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) ( s - \\eta _ n ( s ) ) ^ \\alpha \\int _ { \\eta _ n ( s ) } ^ s \\psi _ { n , 1 } ( u , s ) d u d s . \\end{align*}"}
{"id": "3495.png", "formula": "\\begin{align*} L _ { 1 } = \\frac { \\Gamma \\left ( ( m + 1 ) / 2 \\right ) \\sqrt { m } \\left [ \\xi _ { p } \\left ( 1 + \\frac { \\xi _ { p } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } - \\xi _ { q } \\left ( 1 + \\frac { \\xi _ { q } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } \\right ] } { \\Gamma ( m / 2 ) ( m - 1 ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "4549.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 4 \\right ) } \\Vert _ { p } = \\mathcal { O } ( 1 ) \\left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \\right ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "7874.png", "formula": "\\begin{align*} f ( x _ 0 , t ) = \\left ( t ^ { - 1 } G ( x _ 0 , t ) \\right ) ^ { 2 - \\sigma } \\leq C _ { 1 0 } t ^ { \\sigma - 2 } . \\end{align*}"}
{"id": "3574.png", "formula": "\\begin{align*} n _ { \\pi } & = \\frac { 1 } { 2 } ( \\dim \\pi - \\chi _ { \\pi } ( h _ 2 ) ) \\\\ & = \\frac { 1 } { 2 } [ n - 2 ] _ q ! \\left ( T _ { n - 2 } T _ { n - 1 } - ( 1 + q ) \\sum _ { i < j } \\chi _ i ( - 1 ) \\chi _ j ( - 1 ) \\right ) . \\end{align*}"}
{"id": "2414.png", "formula": "\\begin{align*} K _ R ( x ) = 1 , ~ | x | < R ; K _ R ( x ) = 0 , ~ | x | \\ge R + 1 , \\end{align*}"}
{"id": "6614.png", "formula": "\\begin{align*} e _ { k \\ell } ( c \\cdot z ) & = e _ { k _ 1 \\ell _ 1 k _ 2 \\ell _ 2 } \\big ( c \\cdot k _ 2 \\ell _ 2 \\bar k _ 2 \\bar \\ell _ 2 z _ 1 + c \\cdot k _ 1 \\ell _ 1 \\bar k _ 1 \\bar \\ell _ 1 z _ 2 \\big ) \\\\ & = e _ { k _ 1 \\ell _ 1 } \\big ( ( \\bar k _ 2 \\bar \\ell _ 2 c ) \\cdot z _ 1 \\big ) e _ { k _ 2 \\ell _ 2 } \\big ( ( \\bar k _ 1 \\bar \\ell _ 1 c ) \\cdot z _ 2 \\big ) . \\end{align*}"}
{"id": "8246.png", "formula": "\\begin{align*} k _ { \\pm } = \\frac { \\sqrt { a _ { + } + b _ { + } } \\pm i \\sqrt { a _ { + } - b _ { + } } } { \\hslash \\sqrt { 2 \\beta / 3 } } , k ' _ { \\pm } = \\frac { \\sqrt { a _ { - } + b _ { - } } \\mp i \\sqrt { a _ { - } - b _ { - } } } { \\hslash \\sqrt { 2 \\beta / 3 } } \\end{align*}"}
{"id": "6435.png", "formula": "\\begin{align*} \\Phi _ { k - 1 } ( \\xi _ { i _ 1 } , \\ldots , \\xi _ { i _ k } ) = \\emph { p r } \\circ [ \\cdots [ [ Q ' , \\iota _ { \\xi _ { i _ 1 } } ] , \\iota _ { \\xi _ { i _ 2 } } ] , \\ldots , \\iota _ { \\xi _ { i _ k } } ] \\subset \\mathfrak { X } ( E ) [ 1 ] , \\ ; k \\in \\mathbb { N } , \\end{align*}"}
{"id": "5891.png", "formula": "\\begin{align*} I ( \\nu ) = - \\frac { 1 } { 2 } \\log \\mathrm { d e t } ( I _ k - A A ^ T ) \\end{align*}"}
{"id": "6689.png", "formula": "\\begin{align*} \\partial _ t U - \\partial _ { r r } U - A _ r ( \\partial _ r U ) + A _ 0 U = \\theta \\end{align*}"}
{"id": "1823.png", "formula": "\\begin{align*} Q _ { n } ( z ) = \\int \\frac { P _ { n } ( z ) - P _ { n } ( x ) } { z - x } \\ , d \\mu ( x ) . \\end{align*}"}
{"id": "2624.png", "formula": "\\begin{align*} c _ n = b _ n / | b _ n | = b _ { n - } / | b _ { n - } | \\end{align*}"}
{"id": "8712.png", "formula": "\\begin{align*} g _ { x ^ { 1 } } = g _ { x ^ { 2 } } = 0 , g _ { \\dot { x } ^ { 1 } } = \\frac { \\dot { x } ^ { 1 } } { \\sqrt { ( \\dot { x } ^ { 1 } ) ^ 2 + ( \\dot { x } ^ { 2 } ) ^ 2 } } + b \\cos ( t + c ) , g _ { \\dot { x } ^ { 2 } } = \\frac { \\dot { x } ^ { 2 } } { \\sqrt { ( \\dot { x } ^ { 1 } ) ^ 2 + ( \\dot { x } ^ { 2 } ) ^ 2 } } + b \\sin ( t + c ) . \\end{align*}"}
{"id": "8327.png", "formula": "\\begin{align*} \\delta ( a ) = \\sum _ { n \\in \\Z } \\delta _ n ( a ) \\end{align*}"}
{"id": "6480.png", "formula": "\\begin{align*} u ( c _ 1 ) = u _ x ( c _ 1 ) = y ( c _ 2 ) = y _ x ( c _ 2 ) = 0 . \\end{align*}"}
{"id": "1471.png", "formula": "\\begin{align*} { { \\tilde { \\psi } } } _ { \\alpha , s } \\bigcirc _ { w ' = 0 } ^ s ( \\theta _ { X _ s } + \\gamma _ { r - s + w ' } ) \\bigcirc _ { w = 1 } ^ { \\ell } ( \\theta _ { X _ s } + \\xi _ w ) ( P ) = { \\rm { E v a l } } _ { { { X _ s \\rightarrow \\alpha } } } \\bigcirc _ { w = 1 } ^ { \\ell } ( \\theta _ { X _ s } + \\xi _ w ) ( P ) \\enspace . \\end{align*}"}
{"id": "8851.png", "formula": "\\begin{align*} \\begin{dcases} & d X _ k ( t ) = d B _ { k k + 1 } ( t ) + \\dfrac { \\alpha _ k - 1 } { 2 } \\frac { 1 } { X _ k ( t ) } d t \\\\ & X _ k ( 0 ) = x _ k \\end{dcases} , \\end{align*}"}
{"id": "7347.png", "formula": "\\begin{align*} [ h _ p , a ] = [ [ h _ { p - 1 } , x ] y , a ] & = ( [ h _ { p - 1 } , x ] , a , y ) + [ [ h _ { p - 1 } , x ] , a ] y - [ h _ { p - 1 } , x ] [ a , y ] & \\\\ & \\in A _ { p + q } + [ h _ { p - 1 } , x ] A _ { q + 1 } \\subseteq A _ { p + q } . & \\end{align*}"}
{"id": "6755.png", "formula": "\\begin{align*} f ^ { * } \\left ( K _ Y + L _ { 1 0 0 } \\right ) = f ^ { * } \\left ( E + \\sum \\limits _ { i = 1 } ^ { n } { F _ { i i } } \\right ) . \\end{align*}"}
{"id": "6487.png", "formula": "\\begin{align*} u ( c _ 1 ) = u _ x ( c _ 1 ) = y ( c _ 1 ) = y _ x ( c _ 1 ) = 0 . \\end{align*}"}
{"id": "3567.png", "formula": "\\begin{align*} w _ 2 ( \\pi ) = \\begin{cases} \\dfrac { m _ { \\pi } } { 2 } a _ 2 , & \\ , \\ , q \\equiv 1 \\pmod 4 \\\\ \\\\ \\dbinom { m _ { \\pi } } { 2 } \\left ( \\sum \\limits _ { i = 1 } ^ n v _ i ^ 2 \\right ) , & \\ , \\ , q \\equiv 3 \\pmod 4 . \\end{cases} \\end{align*}"}
{"id": "2150.png", "formula": "\\begin{align*} \\gamma _ { \\mu _ { t } } = t / 2 , \\quad \\sigma _ { \\mu _ { t } } = ( t / 2 ) \\ , \\delta _ { 1 } , \\end{align*}"}
{"id": "1355.png", "formula": "\\begin{align*} \\widehat { T ( u ^ 1 , \\cdots , u ^ p ) } ( k ) = \\sum _ { k = k _ 1 - k _ 2 + \\cdots + k _ p } m ( k _ 1 , \\cdots , k _ n ) \\prod _ { \\substack { \\ell = 1 \\\\ o d d } } ^ p \\widehat { u ^ \\ell } _ { k _ \\ell } \\prod _ { \\substack { \\ell = 2 \\\\ e v e n } } ^ p \\overline { \\widehat { u ^ \\ell } _ { k _ \\ell } } \\end{align*}"}
{"id": "7543.png", "formula": "\\begin{align*} \\mathcal M _ 0 = [ \\vec r ^ 0 , \\vec s ^ 0 ] = \\{ \\vec x : r ^ 0 _ i \\le x _ i \\le s ^ 0 _ i \\} \\end{align*}"}
{"id": "7586.png", "formula": "\\begin{align*} \\Omega ( t ) = \\sum _ { k = 0 } ^ { \\infty } H _ k ( t ) , \\end{align*}"}
{"id": "4432.png", "formula": "\\begin{align*} u _ { i _ 1 \\cdots i _ m } = u _ { i _ { \\pi ( 1 ) } \\cdots i _ { \\pi ( m ) } } \\end{align*}"}
{"id": "1202.png", "formula": "\\begin{align*} \\begin{cases} & , \\\\ & | g _ t ( x ) | \\le C ( 1 + | x | ^ { K } ) | x | , \\ \\forall x \\in \\R ^ d , \\\\ & | \\nabla g _ t ( x ) | \\le C ( 1 + | x | ^ { K } ) \\ \\ x \\in \\R ^ d . \\end{cases} \\end{align*}"}
{"id": "5974.png", "formula": "\\begin{align*} \\left | \\frac 1 { 2 \\pi i } \\int _ { \\Gamma _ - \\cup \\Gamma _ + } e ^ { s t } K ( s ) d s \\right | & \\leq \\frac { C _ K } { \\pi } \\int _ { r _ 0 } ^ \\infty r ^ { - \\mu } e ^ { - r t \\cos \\varphi } d r \\\\ & = \\frac { C _ K } { \\pi } t ^ { \\mu - 1 } ( \\cos \\varphi ) ^ { \\mu - 1 } \\int _ { r _ 0 \\cos \\varphi } ^ \\infty r ^ { - \\mu } e ^ { - r } d r \\\\ & = \\frac { C _ K } { \\pi } t ^ { \\mu - 1 } ( \\cos \\varphi ) ^ { \\mu - 1 } \\Gamma ( 1 - \\mu , r _ 0 \\cos \\varphi ) , \\end{align*}"}
{"id": "3969.png", "formula": "\\begin{align*} j ( a ^ { * } ) ( \\gamma ) = j ( a ) ( \\gamma ^ { - 1 } ) ^ { * } j ( a * b ) ( \\gamma ) = \\sum _ { \\eta \\in G ^ { r ( \\gamma ) } } j ( a ) ( \\eta ) \\ , \\alpha _ { \\eta } \\bigl ( j ( b ) ( \\eta ^ { - 1 } \\gamma ) \\bigr ) . \\end{align*}"}
{"id": "6390.png", "formula": "\\begin{align*} | \\pi | _ 1 + | \\pi | _ 2 = | \\pi | , \\mbox { t h e t o t a l n u m b e r o f b l o c k s o f $ \\pi $ , w h i l e } \\end{align*}"}
{"id": "7814.png", "formula": "\\begin{align*} \\rho ' _ \\tau = \\rho _ { f ' } R ' = \\rho _ { f \\widetilde { H } ( R R ' ) } R ' = \\rho _ f \\rho _ { \\widetilde { H } ( R R ' ) } R ' = \\rho _ f R R \\rho _ { \\widetilde { H } ( R R ' ) } R ' = \\rho _ \\tau ( R \\rho _ { \\widetilde { H } ( R R ' ) } R ' ) . \\end{align*}"}
{"id": "5388.png", "formula": "\\begin{align*} B _ q ( u , v ) - \\lambda \\langle u , v \\rangle = \\tilde { F } ( v ) \\ \\ \\ v \\in \\widetilde { H } ^ s ( \\Omega ) \\end{align*}"}
{"id": "7013.png", "formula": "\\begin{align*} \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } S ( h ) = n ! \\cdot \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } V o l ( h ) \\end{align*}"}
{"id": "2113.png", "formula": "\\begin{align*} ( ( A _ G ) x ) ( v ) : = ( L _ G ( x ) ) ( v ) + r ( v ) x ( v ) , \\end{align*}"}
{"id": "8714.png", "formula": "\\begin{align*} P _ { j } = g _ { x ^ { j } } - \\frac { d } { d t } g _ { \\dot { x } ^ { j } } , \\textnormal { f o r } ~ ~ j = 1 , 2 , \\end{align*}"}
{"id": "7479.png", "formula": "\\begin{align*} P _ { n _ { n _ { k + 1 } ^ { \\prime } } } = T _ { h \\left ( n _ { n _ { k + 1 } ^ { \\prime } } \\right ) } \\cdots T _ { h \\left ( n _ { n _ { k } ^ { \\prime } } + M + 1 \\right ) } T _ { l _ { M } } \\cdots T _ { l _ { 1 } } P _ { n _ { n _ { k } ^ { \\prime } } } , \\end{align*}"}
{"id": "8527.png", "formula": "\\begin{align*} \\sum _ { i = m + 1 } ^ { n - 1 } z ^ { i } 4 ^ { i - m - 1 } \\int _ { - 1 } ^ 1 \\bigg ( ( t ^ 2 - 1 ) \\phi _ m ( t ) + 4 a _ { m + 1 } \\psi _ m ( t ) + 1 6 \\ : a _ { m + 1 } ^ 2 \\phi _ { m - 1 } ( t ) \\bigg ) \\bigg ( \\sum _ { j \\geq i + 1 } \\ldots \\bigg ) , \\end{align*}"}
{"id": "4458.png", "formula": "\\begin{align*} F ( [ s _ 1 s _ 1 f ] ) & = ( [ d _ 2 d _ 2 s _ 1 s _ 1 f ] , [ d _ 0 d _ 0 s _ 1 s _ 1 f ] \\coloneqq ( [ i d _ x ] , [ i d _ y ] ) . \\end{align*}"}
{"id": "7194.png", "formula": "\\begin{align*} \\langle u _ n f , I ^ { \\circ } ( w _ 1 , z ) w _ 2 \\rangle = & \\langle u ( n ) f , I ^ { \\circ } ( w _ 1 , z ) w _ 2 \\rangle \\\\ = & \\langle f , \\theta ( u ( n ) ) I ^ { \\circ } ( w _ 1 , z ) w _ 2 \\rangle \\end{align*}"}
{"id": "3607.png", "formula": "\\begin{align*} & \\langle y , ( - u _ i , 1 , 0 , 0 ) \\rangle \\leq 0 , \\ i = 1 , \\ldots , m , \\\\ & \\langle y , - e _ i \\rangle \\leq 0 , \\ , i = 1 , \\ldots , s , \\end{align*}"}
{"id": "4352.png", "formula": "\\begin{align*} u _ s = v _ { r - 1 } \\left ( a _ s g ^ s \\right ) = v _ { r - 1 } ( a _ s ) + s v _ { r - 1 } ( g ) = v _ { r - 1 } ( a _ s ) + s V _ r . \\end{align*}"}
{"id": "6035.png", "formula": "\\begin{align*} \\kappa _ i = \\sigma ( \\lambda _ i ) \\end{align*}"}
{"id": "2903.png", "formula": "\\begin{align*} A = \\begin{pmatrix} a & b & c & d \\\\ e & f & g & h \\\\ c & d & a & b \\\\ g & h & e & f \\end{pmatrix} , \\end{align*}"}
{"id": "6714.png", "formula": "\\begin{align*} \\delta _ n \\otimes \\delta _ m = \\bigoplus _ { 0 \\leq j \\leq \\min ( n , m ) } \\delta _ { n + m - 2 j } , \\ ; \\ ; \\ ; \\ , n , m \\in \\mathbb { N } _ 0 . \\end{align*}"}
{"id": "401.png", "formula": "\\begin{align*} f ( p , v ) = \\big ( \\sigma ( p ) , v - \\sf K ( p ) \\big ) . \\end{align*}"}
{"id": "3785.png", "formula": "\\begin{align*} \\sum _ { i = 3 , 4 } \\big | T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , E ) ( t , x , \\zeta ) + \\hat { \\zeta } \\times T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , B ) ( t , x , \\zeta ) \\big | \\end{align*}"}
{"id": "1089.png", "formula": "\\begin{align*} \\tilde \\lambda _ { j } = \\frac { \\gamma ^ { - j } \\kappa _ { k , j } } { j ! \\big ( ( k - j ) ! \\big ) ^ 2 } , \\end{align*}"}
{"id": "2880.png", "formula": "\\begin{align*} f _ i ( x ) = f \\circ S ^ { 1 - i } ( x ) = f ( x _ i , \\dotsc , x _ { i + n - 1 } ) , i \\in \\Z , \\end{align*}"}
{"id": "8099.png", "formula": "\\begin{align*} \\partial _ { 1 } f = \\partial _ { 2 } f = 0 \\end{align*}"}
{"id": "5794.png", "formula": "\\begin{align*} [ a , b ] ^ g = ( [ a , b ] ^ { a _ 1 } ) ^ { b _ 1 } \\in [ A , B ] , \\end{align*}"}
{"id": "8178.png", "formula": "\\begin{align*} g ( y ) = \\int \\limits _ { | x | > | y | } | \\varphi _ 0 ( x ) | ^ 2 \\ , d x \\end{align*}"}
{"id": "1133.png", "formula": "\\begin{align*} \\dot { C } ( \\lambda w ) = \\left ( \\zeta - \\frac { \\rho } { \\lambda } \\right ) w . \\end{align*}"}
{"id": "4258.png", "formula": "\\begin{align*} \\xi _ h : = \\Theta _ { { 2 \\pi \\over k } ( h - 1 ) } \\xi _ 1 , \\ h = 1 , \\ldots , k , \\ \\hbox { w i t h } \\ \\xi _ 1 : = ( 1 , 0 , 0 ) \\in \\mathbb R ^ 2 \\times \\mathbb R ^ { n - 2 } , \\ \\end{align*}"}
{"id": "8672.png", "formula": "\\begin{align*} \\left ( \\sum _ { 1 \\le i , j \\le 3 } C _ { i j } x _ i x _ j \\right ) \\cdot L ( x ) = 0 \\end{align*}"}
{"id": "1084.png", "formula": "\\begin{align*} \\Gamma \\left ( 1 + \\frac { d } { 2 } \\right ) ^ { 2 / d } = \\begin{cases} \\frac { \\pi } { 4 } & d = 1 \\\\ 1 & d = 2 \\\\ \\frac { 3 ^ { 2 / 3 } \\pi ^ { 1 / 3 } } { 2 ^ { 4 / 3 } } & d = 3 . \\end{cases} \\end{align*}"}
{"id": "2157.png", "formula": "\\begin{align*} \\left | \\frac { H ( z _ { 1 } ) - H ( z _ { 2 } ) } { z _ { 1 } - z _ { 2 } } - b \\right | = \\left | \\int _ { \\mathbb { R } } \\frac { 1 + s ^ { 2 } } { ( z _ { 1 } - s ) ( z _ { 2 } - s ) } \\ , d \\rho ( s ) \\right | \\leq b . \\end{align*}"}
{"id": "8723.png", "formula": "\\begin{align*} \\Psi + \\mu U = 0 , \\end{align*}"}
{"id": "8254.png", "formula": "\\begin{align*} a _ { 2 } ^ { 2 } = - a _ { 1 } ^ { 2 } , \\end{align*}"}
{"id": "2520.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { 1 } { 2 \\lambda ^ 2 } \\{ h ( F _ \\ast ( \\nabla _ X Z ) , F _ \\ast Y ) + h ( F _ \\ast ( \\nabla _ Y Z ) , F _ \\ast X ) \\} \\\\ + R i c ( X , Y ) + \\frac { \\mu } { \\lambda ^ 2 } h ( F _ \\ast X , F _ \\ast Y ) = 0 . \\end{array} \\end{align*}"}
{"id": "7914.png", "formula": "\\begin{align*} D _ - = D _ { - , 1 } + \\cdots + D _ { - , n } , \\end{align*}"}
{"id": "5720.png", "formula": "\\begin{align*} ( y _ i - y _ { i + 1 } ) \\cdot \\pi _ { [ a , b ] } = 0 ( a \\le i \\le b ) \\end{align*}"}
{"id": "5534.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\abs { w _ i - \\lambda \\frac { z _ i - a } { 1 - \\bar { a } z _ i } } ^ 2 . \\end{align*}"}
{"id": "3325.png", "formula": "\\begin{align*} \\mu _ 1 \\star \\mu _ 2 ( g ) = \\sum _ { h \\in G } \\mu _ 1 ( g ^ { - 1 } h ) \\mu _ 2 ( h ) \\end{align*}"}
{"id": "2274.png", "formula": "\\begin{align*} J _ 4 & = \\int _ 0 ^ r - \\frac { z ^ 3 } { 2 } ( e _ \\sigma + \\bar \\psi ) \\cdot ( e ' _ \\sigma \\phi + e ' _ \\delta \\psi ) { \\rm { d } } z \\\\ & \\leq \\Vert z ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert z ^ { 2 } ( e ' _ \\delta , e ' _ \\sigma ) \\Vert _ { L ^ 2 } \\cdot \\frac { 1 } { 2 } ( \\Vert e _ \\sigma \\Vert _ { L ^ \\infty } + \\Vert \\bar \\psi \\Vert _ { L ^ \\infty } ) , \\end{align*}"}
{"id": "4362.png", "formula": "\\begin{align*} v _ r ( \\pi _ r ) = e _ r \\ , v _ { r - 1 } ( \\pi _ r ) = e _ r . \\end{align*}"}
{"id": "3123.png", "formula": "\\begin{align*} \\widehat { \\eta } ^ 2 ( T ) : = | T | ^ { 2 { m } / 3 } \\Vert \\widehat { \\lambda } _ h \\widehat { u } _ { \\mathrm { n c } } \\Vert ^ 2 _ { L ^ 2 ( T ) } + | T | ^ { 1 / 3 } \\sum _ { F \\in \\widehat { \\mathcal { F } } ( T ) } \\Vert [ { D } ^ { m } _ { \\mathrm { p w } } \\widehat { u } _ { \\mathrm { n c } } ] _ F \\times \\nu _ F \\Vert ^ 2 _ { L ^ 2 ( F ) } . \\end{align*}"}
{"id": "2852.png", "formula": "\\begin{align*} & c _ 1 : = \\max _ { h \\in ( 0 , 1 ] } \\ , \\ , \\ , \\ , 2 h - h ^ 2 \\frac { - \\kappa } { 1 - \\kappa } = 2 - \\frac { - \\kappa } { 1 - \\kappa } \\\\ & c _ 2 : = \\max _ { h \\in [ 1 , \\bar { h } ( \\kappa ) ] } 2 h - h ^ 3 \\frac { - \\kappa } { 2 - ( 1 + \\kappa ) h } \\geq 2 - \\frac { \\kappa } { 2 - ( 1 + \\kappa ) } = c _ 1 . \\end{align*}"}
{"id": "7916.png", "formula": "\\begin{align*} \\angle _ I : = \\left \\{ \\sum _ { i \\in I } a _ i D _ i | a _ i \\in \\mathbb R _ { > 0 } \\right \\} \\subset \\mathbb L ^ \\vee \\otimes \\mathbb R . \\end{align*}"}
{"id": "2308.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\left ( \\frac { 1 } { y } \\int _ y ^ \\infty f ( t ) { \\rm d } t \\right ) ^ p y ^ \\alpha { \\rm d } y \\leq \\left ( \\frac { p } { \\alpha + 1 - p } \\right ) ^ p \\int _ 0 ^ \\infty f ( y ) ^ p y ^ \\alpha { \\rm d } y . \\end{align*}"}
{"id": "1850.png", "formula": "\\begin{align*} \\begin{aligned} w ( ( n , m ) \\rightarrow ( n + 1 , m + 1 ) ) & = 1 , \\\\ w ( ( n , m ) \\rightarrow ( n + 1 , m - j ) ) & = a _ { m - j } ^ { ( j ) } , 0 \\leq j \\leq p . \\end{aligned} \\end{align*}"}
{"id": "4450.png", "formula": "\\begin{align*} ( - 2 ) ^ m m ! \\ , I _ 0 ( ( R f ) _ { i _ 1 j _ 1 \\cdots i _ m j _ m } ) = J _ { i _ 1 j _ 1 } \\cdots J _ { i _ m j _ m } ( J _ m f ) , \\end{align*}"}
{"id": "5916.png", "formula": "\\begin{align*} Q ( f _ 1 , \\ldots , f _ n ) = 0 . \\end{align*}"}
{"id": "5896.png", "formula": "\\begin{align*} \\sigma _ { p , \\beta } ^ 2 : = \\Big ( \\frac { p } { \\beta } \\Big ) ^ { 2 / p } \\frac { \\Gamma ( 3 / p ) } { \\Gamma ( 1 / p ) } , \\end{align*}"}
{"id": "1571.png", "formula": "\\begin{align*} C = \\sqrt { \\det A } = \\sqrt { \\sum \\limits _ { k \\neq l } ^ { } \\left ( z ^ k _ 1 \\right ) ^ 2 \\left ( z ^ l _ 2 \\right ) ^ 2 - \\sum \\limits _ { k \\neq l } ^ { } z ^ k _ 1 z ^ k _ 2 z ^ l _ 1 z ^ l _ 2 } . \\end{align*}"}
{"id": "5168.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { m - 1 } p _ i \\cdot \\Bigg [ \\sum _ { j = 0 } ^ { m - 1 } \\sum _ { k = 0 } ^ { m - 1 } \\prod _ { i _ 2 = j + 1 } ^ { m - 1 } \\rho _ { i _ 2 + k } + \\sum _ { j = 0 } ^ { m - 1 } \\sum _ { k = 0 } ^ { m - 1 } \\prod _ { i _ 2 = j } ^ { m - 1 } \\rho _ { i _ 2 + k } \\Bigg ] . \\end{align*}"}
{"id": "8624.png", "formula": "\\begin{align*} \\mathcal { T } _ { i j } ( x , p ) \\underset { } { = } \\phi _ j ( [ \\tilde { p } _ { i j } ] _ + ) , \\end{align*}"}
{"id": "1460.png", "formula": "\\begin{align*} & \\sum _ { k = s + 1 } ^ { r n } a _ { k , s } \\psi _ { { i , 0 } } \\circ { \\mathcal { T } } _ { \\bold { c } } \\circ B ( \\theta _ t ) \\bigcirc _ { { { w } } = r + 1 } ^ { r - s - 1 + k } ( \\theta _ t + \\gamma _ { { w } } ) \\circ B ( \\theta _ t ) ^ { - 1 } ( t ^ n H _ { \\ell } ( t ) ) \\\\ & = \\sum _ { k = s + 1 } ^ { r n } a _ { k , s } [ \\alpha _ i ] \\circ { \\rm { E v a l } } _ { \\alpha _ i } \\bigcirc _ { { { w } } = r + 1 } ^ { r - s - 1 + k } ( \\theta _ t + \\gamma _ { { { w } } } ) ( t ^ n H _ { \\ell } ( t ) ) = 0 \\enspace . \\end{align*}"}
{"id": "8644.png", "formula": "\\begin{align*} U ( t ) = \\exp \\left ( { \\rm i } \\alpha ( t ) \\widehat { H } _ 2 \\right ) \\exp \\left ( { \\rm i } \\beta ( t ) \\widehat { H } _ 3 \\right ) , \\end{align*}"}
{"id": "6940.png", "formula": "\\begin{align*} \\rho = \\sum _ { k \\ge 1 } w _ k \\delta _ { \\lambda _ k } , \\rho ^ { [ 1 ] } = \\sum _ { k \\ge 1 } w _ k ^ { [ 1 ] } \\delta _ { \\mu _ k } \\end{align*}"}
{"id": "8863.png", "formula": "\\begin{align*} f _ \\lambda ( \\lambda ) & = N \\lambda ^ { N - 1 } + \\sum _ { k = 1 } ^ { N - 1 } ( - 1 ) ^ k ( N - k ) \\lambda ^ { N - k - 1 } \\sum _ { 1 \\le j _ 1 < \\cdots < j _ k \\le N } \\underset { 1 \\le l , m \\le k } { { \\rm d e t } } ( H _ { j _ l j _ m } ) , \\\\ f _ { \\lambda \\lambda } ( \\lambda ) & = N ( N - 1 ) \\lambda ^ { N - 2 } + \\sum _ { k = 1 } ^ { N - 2 } ( - 1 ) ^ k ( N - k ) ( N - k - 1 ) \\lambda ^ { N - k - 2 } \\sum _ { 1 \\le j _ 1 < \\cdots < j _ k \\le N } \\underset { 1 \\le l , m \\le k } { { \\rm d e t } } ( H _ { j _ l j _ m } ) . \\end{align*}"}
{"id": "5905.png", "formula": "\\begin{align*} \\mu _ { V X ^ { ( n ) } } ( A ) : = \\mathbb { P } ( V X ^ { ( n ) } \\in A ) = \\mathbb { P } \\Big ( \\sum _ { j = 1 } ^ n n ^ { \\kappa / p } \\frac { Z _ j } { ( \\lVert Z ^ { ( n ) } \\rVert _ p ^ p + W _ n ) ^ { 1 / p } } V _ { \\bullet , j } \\in A \\Big ) \\end{align*}"}
{"id": "4466.png", "formula": "\\begin{align*} 1 = \\sum \\limits _ { k \\in \\mathbb N } \\psi _ k ( \\xi ) + \\psi _ { \\leq 0 } ( \\xi ) . \\end{align*}"}
{"id": "8816.png", "formula": "\\begin{align*} \\phi _ 1 ^ * * \\phi _ 2 ^ * ( x ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } 1 _ { ( - J _ 1 ( t _ 1 ) , J _ 1 ( t _ 1 ) ) } * 1 _ { ( - J _ 2 ( t _ 2 ) , J _ 2 ( t _ 2 ) ) } ( x ) d t _ 1 d t _ 2 \\end{align*}"}
{"id": "6779.png", "formula": "\\begin{align*} \\sum _ { i \\in M _ P } \\sum _ { j \\in N _ i } a _ { i j } x _ { i j } + \\sum _ { \\substack { i j \\in P , \\\\ i \\in M _ P - M _ 0 } } ( b - s ) x _ { i j } \\leq b + ( | M _ P - M _ 0 | - 1 ) ( b - s ) \\end{align*}"}
{"id": "4254.png", "formula": "\\begin{align*} \\mathbb { P } ( ) \\leq \\frac { \\mathbb { G } ( n ) } { \\mathbb { G } ( m ) } = \\frac { \\widetilde { \\Phi } ( n ) } { \\widetilde { \\Phi } ( m ) } \\leq \\frac { f ( n ) } { f ( m ) } \\end{align*}"}
{"id": "4089.png", "formula": "\\begin{align*} E ( \\mathbf { S } _ n ) = \\tau \\mathbf { v } _ n = \\varepsilon \\sum \\limits _ { i = 1 } ^ 3 \\big [ p _ n ( i ) - p _ n ( i + 3 ) \\big ] \\mathbf { e } _ i = \\varepsilon \\sum \\limits _ { i = 1 } ^ 3 ( a _ i n + b _ i ) \\mathbf { e } _ i . \\end{align*}"}
{"id": "8389.png", "formula": "\\begin{align*} A _ 2 & = \\{ ( P _ f ) _ { f \\in G } \\in \\wp ( G ) ^ G \\ ; | \\ ; \\mbox { t h e r e e x i s t s $ f \\in G $ s u c h t h a t $ i d \\in P _ f \\cup P _ f ^ { - 1 } $ } \\} \\\\ & = \\bigcup _ { f \\in G } W _ { ( f , i d ) } . \\end{align*}"}
{"id": "6638.png", "formula": "\\begin{align*} Y = \\left ( \\rho X + \\lambda \\right ) \\ ; , \\ , \\end{align*}"}
{"id": "4121.png", "formula": "\\begin{align*} d ^ i ( K ^ i ) ^ { m } - ( K ^ { i + 1 } ) ^ { m } d ^ i = m S ^ i ( K ^ i ) ^ { m - 1 } , m \\geq 1 . \\end{align*}"}
{"id": "4289.png", "formula": "\\begin{align*} \\operatorname { t r } X \\mathcal { L } ( \\rho ) & = - \\operatorname { t r } ( X \\frac { 1 } { 2 } \\sum \\limits _ { j } [ C _ { j } , [ C _ { j } , \\rho ] ] ) = - \\frac { 1 } { 2 } \\sum \\limits _ { j } \\operatorname { t r } ( X ( C _ { j } ^ { 2 } \\rho - 2 C _ { j } \\rho C _ { j } + \\rho C _ { j } ^ { 2 } ) \\\\ & = - \\operatorname { t r } ( \\rho \\frac { 1 } { 2 } \\sum \\limits _ { j } [ C _ { j } , [ C _ { j } , X ] ] ) = \\operatorname { t r } \\mathcal { L } ( X ) \\rho . \\end{align*}"}
{"id": "6531.png", "formula": "\\begin{align*} \\lim \\limits _ { Q \\to \\infty } \\frac 1 Q \\sum \\limits _ { k : k \\geq 0 , k < Q } \\mu ( k ) k \\leq \\lim \\limits _ { Q \\to \\infty } \\sum \\limits _ { k : k \\geq 0 } \\mu ( k ) \\left [ \\frac k Q \\wedge 1 \\right ] = 0 . \\end{align*}"}
{"id": "8206.png", "formula": "\\begin{align*} \\psi ( - a / 2 ) = \\psi ( a / 2 ) = \\partial _ { x } ^ { 2 } \\psi ( - a / 2 ) = \\partial _ { x } ^ { 2 } \\psi ( a / 2 ) = 0 , \\end{align*}"}
{"id": "8742.png", "formula": "\\begin{align*} \\psi ( t ) = \\exp \\left ( \\int _ s ^ t A _ 1 ( \\tau ) d \\tau \\right ) . \\end{align*}"}
{"id": "8717.png", "formula": "\\begin{align*} E ( x , \\dot { x } , u ) = \\lambda \\left ( \\sqrt { ( u ^ { 1 } ) ^ { 2 } + ( u ^ { 2 } ) ^ { 2 } } - \\frac { \\dot { x } ^ { 1 } u ^ { 1 } + \\dot { x } ^ { 2 } u ^ { 2 } } { \\sqrt { ( \\dot { x } ^ { 1 } ) ^ { 2 } + ( \\dot { x } ^ { 2 } ) ^ { 2 } } } \\right ) = \\frac { \\lambda } { \\| \\dot { x } \\| } \\left ( \\| u \\| \\| \\dot { x } \\| - ( \\dot { x } ^ { 1 } u ^ { 1 } + \\dot { x } ^ { 2 } u ^ { 2 } ) \\right ) . \\end{align*}"}
{"id": "3769.png", "formula": "\\begin{align*} \\widetilde { K } ^ { \\star ; m , i ; a } _ { k , j ; n , l , r } ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } E B ^ a ( t , s , x - y , \\omega , v ) \\cdot \\nabla _ v \\big [ ( t - s ) \\big ( \\mathcal { K } ^ { E , i } _ { k , j , n } ( \\mathfrak { m } ) ( y , v , \\omega , \\zeta ) + \\hat { \\zeta } \\times \\mathcal { K } ^ { B , i } _ { k , j , n } ( \\mathfrak { m } ) ( y , v , \\omega , \\zeta ) \\big ) \\end{align*}"}
{"id": "3163.png", "formula": "\\begin{align*} \\left \\{ d _ { + } \\in L _ { + } ^ { 2 } ( \\mathbb { S } ; \\mathbb { C } ) : \\max _ { c _ { + } \\in L _ { + } ^ { 2 } ( \\mathbb { S } ; \\mathbb { C } ) } \\mathfrak { f } _ { \\mathfrak { m } } \\left ( c _ { - } , c _ { + } \\right ) = \\mathfrak { f } _ { \\mathfrak { m } } \\left ( c _ { - } , d _ { + } \\right ) \\right \\} \\end{align*}"}
{"id": "134.png", "formula": "\\begin{align*} \\Big \\| C \\star ( { \\bf 1 } ^ \\epsilon _ 0 \\| C ^ 3 \\| _ { L ^ 1 } - { \\bf 1 } ^ \\epsilon _ 0 \\| C _ \\infty ^ 3 \\| _ { L ^ 1 } ) \\star S \\Big \\| _ { L ^ 1 \\cap L ^ \\infty } \\leq \\frac { \\gamma _ t \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } } { m ^ 2 _ t } + \\gamma _ t \\Big ( \\frac { 1 } { m ^ 4 _ t } + \\frac { c } { m ^ 2 _ t } { \\bf 1 } _ { d = 2 } + \\frac { c } { m _ t } { \\bf 1 } _ { d = 3 } \\Big ) . \\end{align*}"}
{"id": "1029.png", "formula": "\\begin{align*} \\binom x { \\ 1 _ S } = \\begin{pmatrix} x _ 1 , & x _ 2 , & x _ 3 , & \\dots \\\\ \\ 1 _ S ( 1 ) , & \\ 1 _ S ( 2 ) , & \\ 1 _ S ( 3 ) , & \\dots \\end{pmatrix} , \\end{align*}"}
{"id": "7810.png", "formula": "\\begin{align*} f ( X _ 1 , \\eta _ 1 , X _ 2 , \\eta _ 2 ) = ( X _ 1 , \\gamma _ 1 ^ { - 1 } \\eta _ 1 , X _ 2 , \\gamma _ 2 \\eta _ 2 ) , \\end{align*}"}
{"id": "1214.png", "formula": "\\begin{align*} S ( t + s ) \\rho _ 0 = S ( t ) S ( s ) \\rho _ 0 ; \\ t , s > 0 , \\end{align*}"}
{"id": "3044.png", "formula": "\\begin{align*} \\sum _ { x \\in \\lambda } h ( x ) = | \\lambda | + n ( \\lambda ) + n ( \\lambda ^ { \\ast } ) . \\end{align*}"}
{"id": "8508.png", "formula": "\\begin{align*} t r ( A ( t , \\delta ) ^ { 1 6 } ) = t ^ { 1 6 } - 1 6 t ^ { 1 4 } \\delta + 1 0 4 t ^ { 1 2 } \\delta ^ 2 - 3 5 2 t ^ { 1 0 } \\delta ^ 3 + 6 6 0 t ^ 8 \\delta ^ 4 - 6 7 2 t ^ 6 \\delta ^ 5 + 3 3 6 t ^ 4 \\delta ^ 6 - 6 4 t ^ 2 \\delta ^ 7 + 2 \\delta ^ 8 . \\end{align*}"}
{"id": "3424.png", "formula": "\\begin{align*} \\Gamma = \\langle \\gamma _ 1 , \\ldots , \\gamma _ r | \\delta _ 1 = \\cdots = \\delta _ s = 1 \\rangle . \\end{align*}"}
{"id": "4049.png", "formula": "\\begin{align*} \\hat { \\beta } ( y ) + \\hat { \\beta } ( z ) = \\sum _ { i = 1 } ^ { n } \\hat { \\beta } ( ( y _ { i } + z _ { i } ) e _ { i } ( n ) ) . \\end{align*}"}
{"id": "986.png", "formula": "\\begin{align*} \\tau ( \\alpha R ) = L \\left ( H ^ { * } ( \\overline { \\alpha } ) \\right ) . \\end{align*}"}
{"id": "86.png", "formula": "\\begin{align*} \\forall \\varphi \\in \\R ^ { \\Lambda _ { \\epsilon , L } } , ( \\Delta ^ \\epsilon \\varphi ) _ x = \\epsilon ^ { - 2 } \\sum _ { y \\sim x } \\big [ \\varphi _ y - \\varphi _ x \\big ] , \\end{align*}"}
{"id": "2125.png", "formula": "\\begin{align*} f ^ { \\prime } ( \\alpha ) = c . \\end{align*}"}
{"id": "5250.png", "formula": "\\begin{align*} \\begin{aligned} { Q } ^ { + r } & = Q \\cup \\{ ( \\emptyset , r , 0 ) \\} \\\\ { Q } ^ { + s } & = Q \\cup \\{ ( \\emptyset , 0 , s ) \\} , \\end{aligned} \\end{align*}"}
{"id": "1736.png", "formula": "\\begin{align*} g _ { k - 2 } ( z , \\omega _ 1 , \\widetilde \\omega _ 1 ) = \\int _ C \\frac { - e ^ { ( z + \\overline \\omega _ 1 ) s } \\ , s ^ { k - 2 } } { ( e ^ { \\omega _ 1 s } - 1 ) ( e ^ { \\widetilde \\omega _ 1 s } - 1 ) } \\ d s , \\end{align*}"}
{"id": "8828.png", "formula": "\\begin{align*} f _ t \\circ \\left ( \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha ( t _ 1 , t _ 2 ) d t _ 1 d t _ 2 \\right ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } f _ { T ( t _ 1 , t _ 2 ) } \\circ ( \\alpha ( t _ 1 , t _ 2 ) ) ( x ) d t _ 1 d t _ 2 \\end{align*}"}
{"id": "5243.png", "formula": "\\begin{align*} v ' ( x ^ r + y ^ s ) = C \\left ( r ( k _ 2 + 1 ) x ^ { k _ 1 + r } y ^ { k _ 2 } - s ( k _ 1 + 1 ) x ^ { k _ 1 } y ^ { k _ 2 + s } \\right ) . \\end{align*}"}
{"id": "7772.png", "formula": "\\begin{align*} q ^ k _ { j , } ( \\alpha ^ * ) = q ^ k _ { j , } ( \\alpha ^ * ) - [ Z ( \\boldsymbol { \\alpha } , \\alpha ^ * ) ] ^ T \\ , [ Z _ { j , } ^ k ] ^ { - 1 } \\ , { \\bf m } ^ k _ { j , } , \\end{align*}"}
{"id": "2106.png", "formula": "\\begin{align*} ( L y _ i ) ( v ) & = - \\frac { \\delta _ E ( e _ 0 ) } { \\delta _ V ( v ) } \\frac { 1 } { | e _ 0 | } y _ i ( v ) . \\end{align*}"}
{"id": "6326.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + \\dfrac { x ( 1 + q ) } { q ^ 2 ( x ^ 2 - 1 ) } D _ { q ^ { - 1 } } y ( x ) - \\frac { [ n ] _ q [ n + 1 ] _ q } { q ^ { 1 + n } ( x ^ 2 - 1 ) } y ( x ) = 0 . \\end{align*}"}
{"id": "2673.png", "formula": "\\begin{align*} b _ j = \\frac { c _ j ( b _ i - 1 ) } { ( c _ i - 1 ) } . \\end{align*}"}
{"id": "6990.png", "formula": "\\begin{align*} \\partial _ { i _ 1 } ^ { k _ 1 } \\cdots \\partial _ { i _ r } ^ { k _ r } \\left ( I _ Q \\right ) ( f ) = 0 \\end{align*}"}
{"id": "692.png", "formula": "\\begin{align*} s = \\log ( \\lambda | t | ) , t = - \\lambda ^ { - 1 } e ^ s = - e ^ { s - s _ 0 } , \\end{align*}"}
{"id": "3266.png", "formula": "\\begin{align*} v _ j : = \\mathcal { S } f _ j \\quad \\mbox { i n } D \\mbox { a n d } u _ j : = v _ j + u ^ s _ j \\mbox { i n } D . \\end{align*}"}
{"id": "3438.png", "formula": "\\begin{align*} H _ Z ^ k ( X , \\mathcal { F } ) = 0 k = 0 , . . . , r - 1 . \\end{align*}"}
{"id": "957.png", "formula": "\\begin{align*} \\phi ( x ) = x ^ { n } - \\phi _ { n - 1 } x ^ { n - 1 } - \\cdots - \\phi _ { 1 } x - \\phi _ { 0 } \\in \\mathbb { Z } [ x ] , \\end{align*}"}
{"id": "3628.png", "formula": "\\begin{align*} H _ f ^ - ( x , m ) = \\inf \\{ H _ f ( x , m ) , f ' _ - ( x , 0 ^ + , 0 ) m \\} ? \\end{align*}"}
{"id": "6558.png", "formula": "\\begin{align*} \\beta ^ \\vee ( x ) : = \\beta ( x , \\cdot ) \\colon E _ j \\times \\cdots \\times E _ k \\to F \\end{align*}"}
{"id": "6426.png", "formula": "\\begin{align*} \\begin{cases} & \\Bar { \\Phi } ^ { ( r ) } \\colon S _ \\mathbb { K } ^ { r + 1 } ( \\mathfrak { g } [ 1 ] ) \\xrightarrow { \\Phi _ { r } } \\mathfrak { X } _ { - r } ( E ) [ 1 ] \\\\ & \\Bar { \\Psi } ^ { ( r ) } \\colon S _ \\mathbb { K } ^ { r + 1 } ( \\mathfrak { g } [ 1 ] ) \\xrightarrow { \\Psi _ { r } } \\mathfrak { X } _ { - r } ( E ) [ 1 ] \\end{cases} , \\ ; \\end{align*}"}
{"id": "6548.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\prod _ { i = 0 } ^ m ( 1 - r _ { m - i } ^ i ) < \\infty , \\end{align*}"}
{"id": "2239.png", "formula": "\\begin{align*} ( U , V ) | _ { Y = 0 } = ( u _ b , 0 ) = ( 1 - \\delta , 0 ) , ~ { \\rm f o r ~ } 0 < \\delta \\ll 1 , \\end{align*}"}
{"id": "4519.png", "formula": "\\begin{align*} F _ { \\alpha \\beta } = \\nabla ^ \\Sigma _ { \\alpha \\beta } F = | \\nabla u | ^ { - 2 } ( \\nabla _ 3 k _ { \\alpha \\beta } - \\nabla _ \\alpha k _ { \\beta 3 } ) = O ( | x | ^ { - \\tau - 2 } ) . \\end{align*}"}
{"id": "569.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ { i + 1 } ) = \\mathbb { P } ( P _ i , P _ { i + 1 } \\ { \\rm { a r e \\ b o t h } } \\ k - { \\rm { f r e e } } ) , \\end{align*}"}
{"id": "3912.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } K ( x ) d x = 1 , \\left \\| K \\right \\| _ \\infty < \\infty , \\mbox { s u p p } ( K ) \\subset [ - 1 , 1 ] , \\int _ \\mathbb { R } K ( x ) x ^ l d x = 0 , \\end{align*}"}
{"id": "7558.png", "formula": "\\begin{align*} S ^ m _ t ( f _ 1 , \\dots , f _ m ) ( x ) = \\int _ { \\S ^ { m - 1 } } \\prod _ { i = 1 } ^ m f ( x - t y _ i ) d \\sigma ( y ) , \\end{align*}"}
{"id": "8778.png", "formula": "\\begin{align*} \\| \\phi _ 1 * \\phi _ 2 \\| _ p ^ p \\geq \\int _ { G } ^ { } \\int _ { G } ^ { } \\phi _ 1 ( g ) ^ { p _ 1 } \\phi _ 2 ( g ^ { - 1 } g ' ) ^ { p _ 2 } d g d g ' = \\| \\phi _ 1 \\| _ { p _ 1 } ^ { p _ 1 } \\| \\phi _ 2 \\| _ { p _ 2 } ^ { p _ 2 } = 1 \\end{align*}"}
{"id": "2674.png", "formula": "\\begin{align*} a _ \\ell c _ j ( c _ i - b _ i ) ( c _ j + c _ i - 1 ) = 0 . \\end{align*}"}
{"id": "3258.png", "formula": "\\begin{align*} T _ { A , q } w ( x ) : = \\int _ { D } \\Phi ( x , y ) Q _ { A , q } w ( y ) d y , x \\in D . \\end{align*}"}
{"id": "2190.png", "formula": "\\begin{align*} d ( c _ k ) + k \\le d ( c _ { k - 1 } ) + k - 1 \\le \\dots \\le d ( c ) = d _ 0 ( c _ k ) + k . \\end{align*}"}
{"id": "8308.png", "formula": "\\begin{align*} C ^ { v } : = \\{ f : \\lVert f ( x + h ) - f ( x ) \\rVert = O ( v ( h ) ) \\} . \\end{align*}"}
{"id": "6416.png", "formula": "\\begin{align*} [ Q , \\Bar { P } ] = P \\qquad \\Bar { P } ^ { ( - 1 ) } = \\widetilde { P } ^ { ( - 1 ) } - [ Q , \\vartheta ] ^ { ( - 1 ) } = \\widetilde { P } ^ { ( - 1 ) } - [ Q ^ { ( 0 ) } , \\vartheta ] = 0 . \\end{align*}"}
{"id": "5432.png", "formula": "\\begin{align*} s = \\theta s _ 1 + ( 1 - \\theta ) s _ 2 \\quad \\frac { 1 } { p } = \\frac { \\theta } { p _ 1 } + \\frac { 1 - \\theta } { p _ 2 } , \\end{align*}"}
{"id": "243.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ { k ( s ) } : = ( \\Gamma _ i ^ { ( s ) } ) _ { j k } = ( \\Gamma _ i ^ { ( s ) t } ) _ { k j } \\in \\mathcal A . \\end{align*}"}
{"id": "8170.png", "formula": "\\begin{align*} \\| \\chi _ { S _ r ^ c } V \\| = r ^ { - 2 } \\in L ^ 1 ( r ) \\end{align*}"}
{"id": "4837.png", "formula": "\\begin{align*} \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } g . \\varphi _ k = f ^ * \\Big ( \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } g . \\varphi _ k ' \\Big ) \\in f ^ * R _ \\mathrm { r a t } ( G ' ) \\ ; . \\end{align*}"}
{"id": "2646.png", "formula": "\\begin{align*} u _ { x y } = l _ 1 ( x _ 0 , x _ 1 ) \\ , \\begin{vmatrix} x _ 0 & x _ 1 \\\\ a _ 0 & a _ 1 \\end{vmatrix} \\ , \\ u _ { y z } = l _ 2 ( y _ 0 , y _ 1 ) \\ , \\begin{vmatrix} y _ 0 & y _ 1 \\\\ b _ 0 & b _ 1 \\end{vmatrix} \\ , \\end{align*}"}
{"id": "4474.png", "formula": "\\begin{align*} \\Psi ( y ) = c _ 1 ^ \\pm e ^ y + c _ 2 ^ \\pm e ^ { - y } + q _ \\pm , y \\gtrless 0 , \\end{align*}"}
{"id": "8696.png", "formula": "\\begin{align*} A _ { m a x } = \\frac { ( 1 + b ) ^ 3 } { 2 } \\int \\limits _ { t _ { 0 } } ^ { t _ { 1 } } ( x ^ { 1 } \\dot { x } ^ { 2 } - x ^ { 2 } \\dot { x } ^ { 1 } ) d t . \\end{align*}"}
{"id": "4816.png", "formula": "\\begin{align*} \\delta ( p , \\pi _ 1 , \\ldots , \\pi _ d ) S ( p ) ^ d \\lesssim _ { n } \\prod _ { j = 1 } ^ d s _ { k _ j } ( p , e ( \\pi _ j ) ) , \\end{align*}"}
{"id": "7274.png", "formula": "\\begin{align*} u ( x , t ) & = \\pm \\eta ( t ) ^ \\frac { 2 } { 1 - q } { \\sf U } ( \\xi ) ( 1 + o ) = \\pm \\eta ( t ) ^ \\frac { 2 } { 1 - q } { \\sf U } _ \\infty ( \\xi ) ( 1 + o ) \\\\ & = \\pm { \\sf U } _ \\infty ( x ) ( 1 + o ) | \\xi | \\to \\infty . \\end{align*}"}
{"id": "3538.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast \\ast } = & \\frac { \\Gamma ( t - 2 ) \\left [ \\xi _ { p } \\left ( 1 + \\xi _ { p } ^ { 2 } \\right ) ^ { - ( t - 2 ) } - \\xi _ { q } \\left ( 1 + \\xi _ { q } ^ { 2 } \\right ) ^ { - ( t - 2 ) } \\right ] } { 4 \\Gamma ( t - \\frac { 1 } { 2 } ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } \\\\ & + \\frac { 1 } { ( 2 t - 5 ) ( 2 t - 3 ) } \\frac { F _ { Y _ { ( 2 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , ~ t > \\frac { 5 } { 2 } , \\end{align*}"}
{"id": "1514.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } \\Big ( ( 1 ) _ { p , \\lambda } + ( 2 ) _ { p , \\lambda } + \\cdots + ( n ) _ { p , \\lambda } \\Big ) x ^ { n } = \\sum _ { n = 1 } ^ { \\infty } x ^ { n } \\sum _ { k = 1 } ^ { n } ( k ) _ { p , \\lambda } \\\\ & = \\sum _ { k = 1 } ^ { \\infty } ( k ) _ { p , \\lambda } \\sum _ { n = k } ^ { \\infty } x ^ { n } = \\frac { 1 } { 1 - x } \\sum _ { k = 1 } ^ { \\infty } ( k ) _ { p , \\lambda } x ^ { k } = \\frac { 1 } { ( 1 - x ) ^ { 2 } } F _ { p , \\lambda } \\Big ( \\frac { x } { 1 - x } \\Big ) . \\end{align*}"}
{"id": "5158.png", "formula": "\\begin{align*} \\alpha _ i = \\begin{cases} n _ 1 - 2 & \\\\ n _ i & \\\\ - n _ i / 2 & \\\\ ( 2 - n _ { 2 k } ) / 2 & \\end{cases} \\end{align*}"}
{"id": "2658.png", "formula": "\\begin{align*} g _ \\alpha ( p ) & = ( n - k - 1 ) ( n - 2 ) p ^ { 2 n - k - 5 } \\\\ & + \\left ( ( n - k - 1 ) ( k - 1 ) + 1 + ( n - k - 2 ) ( n - k - 3 ) \\right ) p ^ { n - 3 } ( p - 1 ) \\\\ & + ( n - k - 2 ) p ^ { n - 3 } ( p - 1 ) ^ 2 + ( n - k - 2 ) p ^ { n - 2 } ( p - 1 ) \\\\ & + ( n - k - 2 ) ( n - k - 3 ) p ^ { n - 3 } ( p - 1 ) ( p - 2 ) . \\end{align*}"}
{"id": "7299.png", "formula": "\\begin{align*} ( V \\tilde v _ 1 ( \\eta ^ { - 1 } \\lambda _ i y , t ) , \\psi _ 2 ) _ { L _ y ^ 2 ( B _ { 4 { \\sf R } _ { \\sf i n } } ) } & = \\int _ { | y | < 4 { \\sf R } _ { \\sf i n } } V ( y ) \\tilde v _ 1 ( \\eta ^ { - 1 } \\lambda _ i y , t ) \\psi _ 2 ( y ) d y \\\\ & = \\eta ^ n \\lambda _ i ^ { - n } \\int _ { | \\xi | < 4 { \\sf R } _ { \\sf i n } \\eta \\lambda _ i ^ { - 1 } } V ( \\eta \\lambda _ i ^ { - 1 } \\xi ) \\tilde v _ 1 ( \\xi , t ) \\psi _ 2 ( \\eta \\lambda _ i ^ { - 1 } \\xi ) d \\xi . \\end{align*}"}
{"id": "3326.png", "formula": "\\begin{align*} \\beta = \\prod _ { i = 1 } ^ k \\gamma _ { j _ i } \\sigma _ { j _ i } \\gamma _ { j _ i } ^ { - 1 } \\end{align*}"}
{"id": "398.png", "formula": "\\begin{align*} m ^ \\# = \\frac { f \\cdot m } { \\int f d m } . \\end{align*}"}
{"id": "384.png", "formula": "\\begin{align*} h _ { t + s } ( o , y ) - h _ { t + s } ( o , z ) & = p _ { t + s } ( o , y ) - p _ { t + s } ( o , z ) \\\\ & = \\sum _ { w \\in V } p _ { t / 2 } ( o , w ) ( p _ { t / 2 + s } ( w , y ) - p _ { t / 2 + s } ( w , z ) ) \\\\ & = \\sum _ { w \\in V } h _ { t / 2 } ( o , w ) ( h _ { t / 2 + s } ( w , y ) - h _ { t / 2 + s } ( w , z ) ) \\end{align*}"}
{"id": "2782.png", "formula": "\\begin{align*} \\begin{aligned} T _ 3 ( h _ { N - 1 } { } \\geq { } 1 ) = 0 \\end{aligned} \\end{align*}"}
{"id": "2380.png", "formula": "\\begin{align*} \\nu ( L _ p ( h _ \\rho ) ) = \\beta _ b + b \\gamma _ \\rho . \\end{align*}"}
{"id": "8928.png", "formula": "\\begin{align*} B ( \\cdot , u , p ) = \\det E ( \\cdot , Y ( \\cdot , u , p ) , Z ( \\cdot , u , p ) ) \\psi ( \\cdot , u , p ) . \\end{align*}"}
{"id": "1088.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n ^ { - j } \\alpha _ { k , j } = \\lim _ { n \\to \\infty } ( n r ^ d ) ^ { - j } r ^ { d ( 2 k - 1 ) } \\kappa _ { k , j } , \\end{align*}"}
{"id": "225.png", "formula": "\\begin{align*} ( b + c ) a + b + c + 1 & = a ^ { - 1 } \\cdot a + a ^ { - 1 } + 1 = a ^ { - 1 } \\\\ ( b + 1 ) a + b + c + 1 & = a ( a ^ { - 1 } + c + 1 ) + a ^ { - 1 } + 1 = a ^ { - 1 } \\left ( ( c + 1 ) a ^ 2 + 1 \\right ) \\end{align*}"}
{"id": "4244.png", "formula": "\\begin{align*} \\mathbb { P } _ x ( X _ n = o ) \\leq \\sqrt { \\frac { 2 c ( o ) } { c _ { \\min } } } \\left ( \\exp \\left [ - \\frac { r ^ 2 } { 2 n } \\right ] + \\exp \\left [ - \\frac { c _ 1 ( \\log r ) ^ \\gamma n } { r ^ 2 } \\right ] \\right ) , \\end{align*}"}
{"id": "3814.png", "formula": "\\begin{align*} \\big | E r r ^ { \\mu , a ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) \\big | \\lesssim \\sum _ { i = 1 , 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } f ( t _ i , X ( t _ i ) - y , v ) \\big | K ^ a _ { k , n } ( y , v , V ( t _ i ) ) \\big | d y d v \\end{align*}"}
{"id": "4011.png", "formula": "\\begin{align*} x ( t ) = \\Phi ( t , t _ 0 ) x _ 0 + \\int ^ { t } _ { t _ 0 } \\Phi ( t , s ) B ( s ) u ( s ) d s . \\end{align*}"}
{"id": "7458.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ 0 ^ L \\abs { y ^ n } ^ 2 d x = 0 . \\end{align*}"}
{"id": "8391.png", "formula": "\\begin{align*} D ^ C & = \\{ ( P _ f ) _ { f \\in G } \\in \\wp ( G ) ^ G \\ ; | \\ ; \\mbox { $ \\exists g , h \\in G $ w i t h $ g \\neq h $ s u c h t h a t $ g h ^ { - 1 } \\notin P _ h $ a n d $ h g ^ { - 1 } \\notin P _ g $ } \\} \\\\ & = \\bigcup _ { g , h \\in G , g \\neq h } ( W _ { ( h , g h ^ { - 1 } ) } ^ C \\cap W _ { ( g , h g ^ { - 1 } ) } ^ C ) , \\end{align*}"}
{"id": "5316.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { N _ t } { v ( t ) } = 1 \\end{align*}"}
{"id": "8752.png", "formula": "\\begin{align*} \\begin{array} { l l } A _ n ( t ) = \\Pi _ { k = 1 } ^ n \\left [ ( I - \\kappa e ^ { - \\alpha _ k ( t , s ) } ) ^ { - 1 } \\partial _ t \\alpha _ k ( t , s ) \\right ] , \\end{array} \\end{align*}"}
{"id": "5885.png", "formula": "\\begin{align*} d _ { n - 1 } ( { \\mathcal K } , \\lambda ^ n ) _ X \\geq C '' \\frac { 1 } { [ \\log _ 2 n ] ^ \\alpha } , n = 2 , \\ldots . \\end{align*}"}
{"id": "6071.png", "formula": "\\begin{align*} f ( x ) = x ^ { p } ( x - 1 ) ^ { q } . \\end{align*}"}
{"id": "8898.png", "formula": "\\begin{align*} \\rho _ k = \\dfrac { 1 } { \\varepsilon } \\left ( \\dfrac { 1 } { \\widetilde { v } - \\varepsilon } \\sum _ { l = 1 } ^ n v ^ l \\dfrac { f ^ h _ { j i } } { f _ { j i } } ( w _ l ) - \\dfrac { f ^ h _ { j i } } { f _ { j i } } ( w _ k ) \\right ) , \\end{align*}"}
{"id": "2949.png", "formula": "\\begin{align*} X _ c \\cdot ( f , a ) = f ( c ) + X _ { c + a } . \\end{align*}"}
{"id": "5901.png", "formula": "\\begin{align*} \\xi _ n : = \\Big ( \\Big ( \\frac { 1 } { 1 + \\alpha } \\Big ) ^ { 1 / p } - \\frac { n ^ { 1 / p } } { ( \\lVert Z ^ { ( n ) } \\rVert _ p ^ p + W _ n ) ^ { 1 / p } } \\Big ) ^ 2 \\longrightarrow 0 \\end{align*}"}
{"id": "6386.png", "formula": "\\begin{align*} \\mathrm { W O T } - \\lim _ { n \\to \\infty } T _ { h _ o } ^ { ( n ) } = \\mathrm { W O T } - \\lim _ { n \\to \\infty } U ( \\gamma _ n ) = A _ 0 = T _ { h _ o } . \\end{align*}"}
{"id": "2378.png", "formula": "\\begin{align*} L ( h _ \\rho ) - L _ p ( h _ \\rho ) = \\sum _ { j \\in J } \\partial _ { j } L ( h _ \\theta ) ( h _ \\rho - h _ \\theta ) ^ { j } . \\end{align*}"}
{"id": "7505.png", "formula": "\\begin{align*} \\mathbf { S } _ { d , 0 } ( s ) \\mathcal { K } _ { t } u & = \\mathbf { S } _ { d , 0 } ( s ) \\mathbf { S } _ { d , 0 } ( t ) | \\cdot | ^ { \\alpha } | u | ^ p + \\mathbf { S } _ { d , 0 } ( s ) \\mathbf { S } _ { d , 0 } ( t ) \\zeta ( \\cdot ) \\mathbf { w } \\\\ & = \\mathbf { S } _ { d , 0 } ( t + s ) | \\cdot | ^ { \\alpha } | u | ^ p + \\mathbf { S } _ { d , 0 } ( t + s ) \\zeta ( \\cdot ) \\mathbf { w } \\\\ & = \\mathbf { S } _ { d , - \\alpha } ( t + s ) | u | ^ p + \\mathbf { S } _ { d , 0 } ( t + s ) \\zeta ( \\cdot ) \\mathbf { w } \\\\ & = \\mathcal { K } _ { t + s } u . \\end{align*}"}
{"id": "1256.png", "formula": "\\begin{align*} P ( T , x ) & = \\dfrac { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { W _ { l ( T ) } ^ { n ( T , 0 ) } \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) - 1 } W _ j ^ { n ( T , l ( T ) - j ) } } = \\dfrac { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) - j ) } } \\\\ & = \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) - n ( T , l ( T ) - j ) } . \\end{align*}"}
{"id": "3192.png", "formula": "\\begin{align*} F \\left ( f \\right ) \\left ( k \\right ) \\equiv \\hat { f } \\left ( k \\right ) \\doteq \\int _ { \\mathbb { R } ^ { d } } f \\left ( x \\right ) \\mathrm { e } ^ { - i k \\cdot x } \\mathrm { d } x = \\frac { ( - 1 ) ^ { n } } { k _ { j _ { 1 } } ^ { 2 } \\cdots k _ { j _ { n } } ^ { 2 } } F \\left ( \\frac { \\partial ^ { 2 n } f } { \\partial x _ { j _ { 1 } } ^ { 2 } \\cdots \\partial x _ { j _ { n } } ^ { 2 } } \\right ) \\left ( k \\right ) \\ , \\end{align*}"}
{"id": "5942.png", "formula": "\\begin{align*} { \\left ( { { \\eta _ 0 } } \\right ) _ { k p } } = \\left ( { \\frac { { d + 1 } } { 2 } - k } \\right ) { \\delta _ { k p } } \\end{align*}"}
{"id": "7681.png", "formula": "\\begin{align*} r ( x , y , Z ) : = r _ { \\mathrm { V } } ( x ) + r _ { \\mathrm { O } } ( x , y , Z ) \\end{align*}"}
{"id": "6605.png", "formula": "\\begin{align*} \\sum _ { \\pi : \\lambda _ { \\pi _ { \\infty } } \\in t \\Omega } \\chi _ { \\pi _ p } ( \\phi _ \\beta ) = \\Lambda _ \\Omega ( t ) \\int _ { \\chi \\in \\mathcal { X } _ p ^ { t e m p } / W } \\chi ( \\phi _ { \\beta } ) \\ , d \\nu _ p ( \\chi ) + O _ \\Omega \\Big ( \\| \\beta \\| _ { L ^ 1 } t ^ { d - \\delta } \\Big ) , \\end{align*}"}
{"id": "5981.png", "formula": "\\begin{align*} Q ^ \\beta \\hat U ( t _ { n + 1 } ) + A U _ { n + 1 } = Q ^ \\beta \\hat U ( t _ { n + 1 } ) + A \\hat U ( t _ { n + 1 } ) = f ( t _ { n + 1 } ) , \\end{align*}"}
{"id": "7990.png", "formula": "\\begin{align*} W : = \\Sigma \\circ h : X ^ \\vee \\rightarrow \\mathbb C . \\end{align*}"}
{"id": "4355.png", "formula": "\\begin{align*} y ^ { \\nu _ { r - 1 } ( a _ s ) } R _ { r - 1 } ( a _ s ) & + y ^ { \\nu _ { r - 1 } ( a ' _ s ) } R _ { r - 1 } ( a ' _ s ) \\\\ = & \\begin{cases} 0 , & \\mbox { i f } v _ { r - 1 } ( a _ s + a ' _ s ) > v _ { r - 1 } ( a _ s ) , \\\\ y ^ { \\nu _ { r - 1 } ( a _ s + a ' _ s ) } R _ r ( a _ s + a ' _ s ) , & \\mbox { i f } v _ { r - 1 } ( a _ s + a ' _ s ) = v _ { r - 1 } ( a _ s ) . \\end{cases} \\end{align*}"}
{"id": "183.png", "formula": "\\begin{align*} C _ t = ( t A + ( \\alpha - t ) ) ^ { - 1 } . \\end{align*}"}
{"id": "5155.png", "formula": "\\begin{align*} D / \\mathcal { W } _ { i , j } = \\begin{cases} | W _ j | d _ G ( w _ i , w _ j ) & \\\\ | W _ i | - 1 & \\\\ 2 ( | W _ i | - 1 ) & \\\\ \\end{cases} \\end{align*}"}
{"id": "5147.png", "formula": "\\begin{align*} \\phi _ 2 ( 2 c q _ 1 + a _ 1 + 2 c q _ 2 + a _ 2 ) & = 6 q _ 1 + 6 q _ 2 + a ' _ 1 + a ' _ 2 \\\\ & \\leq \\phi _ 2 ( 6 q _ 1 + 6 q _ 2 + s _ 1 + s _ 2 ) \\\\ & \\leq 6 q _ 1 + 6 q _ 2 + b ' _ 1 + b ' _ 2 \\\\ & = \\phi _ 2 ( 6 q _ 1 + b _ 1 + 6 q _ 2 + b _ 2 ) \\end{align*}"}
{"id": "6240.png", "formula": "\\begin{align*} & \\int \\dfrac { x } { ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\frac { q ^ { - 1 } - q ^ { - \\nu } [ \\nu ] _ q ^ 2 } { x } - \\frac { c q ^ { - \\nu } [ \\nu ] ^ 2 } { x ^ 2 } + c + q x \\right ) J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) d _ q x = \\\\ & \\frac { x } { q ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) - ( c + x ) D _ { q ^ { - 1 } } J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) \\right ) , \\end{align*}"}
{"id": "4758.png", "formula": "\\begin{align*} ( \\nu , s ) \\cdot ( \\mu , w ) = ( s \\mu + F \\nu - s w F ( s ) ^ { - 1 } ( \\nu ) , s w F ( s ) ^ { - 1 } ) . \\end{align*}"}
{"id": "8258.png", "formula": "\\begin{align*} \\lambda _ { 1 } = + \\lambda _ { + } : c \\lambda _ { 2 } = + \\lambda _ { - } : d \\lambda _ { 3 } = - \\lambda _ { + } : a \\lambda _ { 4 } = - \\lambda _ { - } : b . \\end{align*}"}
{"id": "3884.png", "formula": "\\begin{align*} - \\Delta _ p v ^ 0 - a | v ^ 0 | ^ { p - 2 } v ^ 0 = 0 \\qquad \\Omega \\end{align*}"}
{"id": "8551.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\left ( h ^ { - 1 } ( \\mu ( t + h ) - \\mu ( t ) ) - \\dot \\mu ( t ) \\right ) = 0 \\end{align*}"}
{"id": "5709.png", "formula": "\\begin{align*} e _ k ( y _ 1 , y _ 2 , \\ldots , y _ n ) = \\sum _ { \\substack { 0 \\le p \\le i , \\ 0 \\le q \\le n - i \\\\ p + q = k } } e _ { p } ( y _ 1 , y _ 2 , \\ldots , y _ i ) e _ { q } ( y _ { i + 1 } , y _ { i + 2 } , \\ldots , y _ n ) \\end{align*}"}
{"id": "7739.png", "formula": "\\begin{align*} \\Gamma ( B G , \\mathbb Z ( n ) ) : = \\lim _ { [ k ] \\in \\Delta ^ { \\mathrm { o p } } } \\Gamma ( G ^ { \\times k } , \\mathbb Z ( n ) ) , \\end{align*}"}
{"id": "2671.png", "formula": "\\begin{align*} & a _ \\ell ( c _ i ^ 2 - c _ i ) = 0 i > \\ell \\\\ & a _ \\ell c _ i c _ j = 0 i > \\ell , j > i . \\end{align*}"}
{"id": "8569.png", "formula": "\\begin{align*} \\left \\| \\sum _ { k = 1 } ^ n a _ k \\phi _ k \\right \\| _ p \\le c \\left \\| \\sum _ { k = 1 } ^ n a _ k \\phi _ k \\right \\| _ 2 \\end{align*}"}
{"id": "5077.png", "formula": "\\begin{align*} G ^ n _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s \\left ( \\sigma ' ( X _ s ) \\right ) ^ 2 M ^ { n , 1 } _ s M ^ { n , 2 } _ s d s . \\end{align*}"}
{"id": "1587.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { \\partial ^ 2 C ^ 2 } { \\partial z ^ i _ { \\epsilon } \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } v ^ i = f ( x ^ 1 ) \\left [ - f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \\right ] , \\end{align*}"}
{"id": "7141.png", "formula": "\\begin{align*} ( p x _ 1 + q x _ 2 + r x _ 3 ) ^ { m _ 3 } = \\alpha x _ 1 ^ { m _ 1 } + \\beta x _ 2 ^ { m _ 2 } + r ^ { m _ 3 } x _ 3 ( - x _ 1 + x _ 2 + x _ 3 ) ( x _ 2 + x _ 3 ) ^ { m _ 3 - 2 } \\end{align*}"}
{"id": "282.png", "formula": "\\begin{align*} \\mathfrak g ( S ) \\times \\mathfrak g ( S ) \\rightarrow \\mathfrak g ( S ) , \\ \\ ( a , b ) \\mapsto a + _ { { \\pi } } b : = a + b + \\pi a b . \\end{align*}"}
{"id": "6496.png", "formula": "\\begin{align*} \\int _ 0 ^ { c _ 1 } \\left ( \\abs { z } ^ 2 + \\abs { y _ x } ^ 2 \\right ) d x = o ( 1 ) \\int _ { c _ 2 } ^ L \\left ( \\abs { v } ^ 2 + a \\abs { u _ x } ^ 2 \\right ) d x = o ( 1 ) . \\end{align*}"}
{"id": "7531.png", "formula": "\\begin{align*} \\bigg \\| \\max _ { 1 \\le k \\le n } | T _ k - V _ { k } | \\bigg \\| _ { L ^ { \\frac { p } { 2 } } } \\le C n ^ { \\frac { 1 } { 2 } } \\max _ { 1 \\le k \\le n } \\big \\| \\zeta _ { k } \\big \\| _ { L ^ p } ^ 2 = C n ^ { - \\frac { 1 } { 2 } } \\| m \\| _ { L ^ p } ^ { 2 } . \\end{align*}"}
{"id": "1404.png", "formula": "\\begin{align*} \\| \\sin \\Theta ( Q , \\widetilde Q ) \\| = \\| Q _ \\perp ^ * \\widetilde Q \\| = \\| \\widetilde Q _ \\perp ^ * Q \\| , \\end{align*}"}
{"id": "6790.png", "formula": "\\begin{align*} \\widehat { ( r _ j f ) } ( \\xi ) = i ( 1 - \\phi ( \\xi ) ) \\xi _ i / | \\xi | \\widehat { f } ( \\xi ) . \\end{align*}"}
{"id": "4228.png", "formula": "\\begin{align*} \\psi ^ { - 1 } ( x ) = 8 \\phi ^ { - 1 } \\Bigl ( a \\left ( 8 a ^ { - 1 } ( x ) \\right ) \\Bigr ) , \\end{align*}"}
{"id": "2088.png", "formula": "\\begin{align*} h ( X ) = \\frac { \\log _ 2 ( \\widetilde { N _ H } ( q \\alpha ) ) } { ( q + r ) \\alpha } + \\frac { h ( W ) } { ( q + r ) \\alpha } = h . \\end{align*}"}
{"id": "1685.png", "formula": "\\begin{align*} T x = ( c ( x ) E x , s ( x ) F x ) . \\end{align*}"}
{"id": "4278.png", "formula": "\\begin{align*} A ( \\rho ) = t _ \\rho \\zeta _ \\phi ^ { p , \\infty } ( \\rho ) \\end{align*}"}
{"id": "510.png", "formula": "\\begin{align*} \\sum _ { r \\leq s } \\Phi ( x _ i ^ r ) & = \\sum _ { k = 0 } ^ n \\dim ( M _ i ^ k \\cap \\ker A _ i ^ s ) \\end{align*}"}
{"id": "2465.png", "formula": "\\begin{align*} \\sum _ { \\mu \\not = 0 } \\dim \\left [ \\gg ^ { \\mu } , x \\right ] = \\sum _ { \\mu \\not = 0 } \\dim \\gg _ { 1 } ^ { \\mu } = | \\Delta _ { 1 } \\backslash { A } ^ { \\perp } | . \\end{align*}"}
{"id": "8173.png", "formula": "\\begin{align*} \\varphi _ 0 ( x ) = C e ^ { - c | x | } | x | \\geq 1 \\end{align*}"}
{"id": "8161.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\| \\chi _ { S _ { v t } } e ^ { - i t H } \\psi \\| = \\| \\psi \\| \\end{align*}"}
{"id": "7871.png", "formula": "\\begin{align*} V ^ * A _ 1 & = x \\tilde \\Delta _ \\alpha ( a ) & & + \\alpha ^ { - 1 / 2 } \\Delta ^ { 1 / 2 } ( x ) \\zeta _ \\alpha ( a ) & & - \\alpha ^ { 1 / 2 } \\Gamma ( x , \\zeta _ \\alpha ( a ) ) \\\\ & = : B _ 1 & & + B _ 2 & & - B _ 3 . \\end{align*}"}
{"id": "8000.png", "formula": "\\begin{align*} f _ { ( X , K 3 ) } ( t ^ 4 , 0 ) = f _ { ( \\mathbb P ^ 3 , K 3 ) } ( t ) . \\end{align*}"}
{"id": "770.png", "formula": "\\begin{align*} T = \\frac { \\sum _ { m = 1 } ^ { M } ( T _ { { m } } - \\alpha ) ^ 2 } { \\sum _ { k = 1 } ^ { M } ( T _ { k } - \\alpha ) } + \\alpha . \\end{align*}"}
{"id": "4062.png", "formula": "\\begin{align*} & \\psi ^ i \\psi ^ j = - q _ { i j } ^ { - 1 } \\psi ^ j \\psi ^ i , & & \\sum _ { i = 1 } ^ r q _ i ^ { - 1 } \\psi ^ i \\psi ^ { - i } = 0 . \\end{align*}"}
{"id": "1816.png", "formula": "\\begin{align*} s _ { n } : = \\int x ^ { n } d \\mu ( x ) = \\langle J ^ { n } e _ { 0 } , e _ { 0 } \\rangle , n \\geq 0 , \\end{align*}"}
{"id": "3446.png", "formula": "\\begin{align*} & x _ 1 \\cdot \\frac { \\partial f } { \\partial x _ 1 } , . . . , x _ n \\cdot \\frac { \\partial f } { \\partial x _ n } , \\\\ & w _ { - \\alpha } ( f ) : = \\sum \\limits _ { m \\in \\Delta \\cap M } h t _ { - \\alpha } ( m ) \\cdot a _ m \\cdot x ^ { m - \\alpha } , \\alpha \\in R ( N , \\Sigma _ { C ( \\Delta ) } ) . \\end{align*}"}
{"id": "4522.png", "formula": "\\begin{align*} g = ( | \\nabla u | ^ { - 2 } + a ^ 2 + b ^ 2 ) d u ^ 2 + 2 a d u d x _ 1 + 2 b d u d x _ 2 + d x _ 1 ^ 2 + d x _ 2 ^ 2 , \\end{align*}"}
{"id": "8272.png", "formula": "\\begin{align*} \\psi _ { B _ { 0 } } = B _ { 0 } \\left [ \\frac { x } { a / 2 } - \\frac { \\sinh ( k ' x ) } { \\sinh ( k ' a / 2 ) } \\right ] , \\end{align*}"}
{"id": "3917.png", "formula": "\\begin{align*} \\Delta _ n \\underset { \\sim } { < } ( \\frac { 1 } { n } ) ^ { \\frac { \\bar { \\beta } } { 2 \\bar { \\beta } + d } ( \\frac { 1 } { \\beta _ 1 } + \\frac { 1 } { \\beta _ 2 } ) } \\mbox { f o r } \\beta _ 2 = \\beta _ 3 . \\end{align*}"}
{"id": "5157.png", "formula": "\\begin{align*} \\begin{pmatrix} n _ 1 - 2 & 1 & 0 & 0 & \\ldots & 0 & 0 & 0 \\\\ - 1 & - n _ 2 / 2 & 1 & 0 & \\ldots & 0 & 0 & 0 \\\\ 0 & - 1 & n _ 3 & 1 & \\ldots & 0 & 0 & 0 \\\\ 0 & 0 & - 1 & - n _ 4 / 2 & \\ldots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ldots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & 0 & \\ldots & - n _ { 2 k - 2 } / 2 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & \\ldots & - 1 & n _ { 2 k - 1 } & 1 \\\\ 0 & 0 & 0 & 0 & \\ldots & 0 & - 1 & ( 2 - n _ { 2 k } ) / 2 \\\\ \\end{pmatrix} , \\end{align*}"}
{"id": "7334.png", "formula": "\\begin{align*} t _ { i j } = \\begin{cases} v _ k , ~ j = i + k ~ ~ k \\in \\mathcal { I } , \\\\ 0 , \\quad , \\end{cases} \\end{align*}"}
{"id": "8808.png", "formula": "\\begin{align*} \\phi _ 1 ( g ' ) = \\int _ { 0 } ^ { \\infty } 1 _ { L _ 1 ( t _ 1 ) } ( g ' ) d t _ 1 , \\phi _ 2 ( g '^ { - 1 } g ) = \\int _ { 0 } ^ { \\infty } 1 _ { L _ 2 ( t _ 2 ) } ( g '^ { - 1 } g ) d t _ 2 \\end{align*}"}
{"id": "4473.png", "formula": "\\begin{align*} \\Omega _ t ^ \\pm = \\{ ( x , y ) \\in \\R ^ 2 : y \\gtrless \\varphi ( t , x ) \\} \\end{align*}"}
{"id": "447.png", "formula": "\\begin{align*} \\Omega _ { \\mu } \\cap \\mathbb { H } = \\{ r e ^ { i \\theta } : r > 0 , h _ { \\mu } ( r ) < \\theta < \\pi \\} . \\end{align*}"}
{"id": "7297.png", "formula": "\\begin{align*} \\begin{cases} ( \\lambda \\dot \\lambda - \\sigma ) ( \\Lambda _ y { \\sf Q } , \\psi _ 2 ) _ { L _ y ^ 2 ( B _ { 4 { \\sf R } _ { \\sf i n } } ) } + \\lambda ^ \\frac { n - 2 } { 2 } ( V ( \\eta ^ \\frac { 2 } { 1 - q } \\tilde v _ 1 + \\tilde w _ 1 ) , \\psi _ 2 ) _ { L _ y ^ 2 ( B _ { 4 { \\sf R } _ { \\sf i n } } ) } = 0 t \\in ( 0 , T ) , \\\\ \\lambda ( t ) = 0 t = T . \\end{cases} \\end{align*}"}
{"id": "4683.png", "formula": "\\begin{align*} q \\equiv 3 \\pmod 8 , \\left ( \\frac { p } { q } \\right ) = - 1 . \\end{align*}"}
{"id": "1961.png", "formula": "\\begin{align*} \\mathbf { d } _ { 3 } & = ( 0 , \\ldots , 0 , - a _ { 0 } ^ { ( 2 ) } , - a _ { 1 } ^ { ( 1 ) } , z - a _ { 2 } ^ { ( 0 ) } ) , \\\\ \\mathbf { v } _ { 3 } & = ( A _ { 0 } ^ { ( 3 ) } , \\ldots , A _ { p - 4 } ^ { ( 3 ) } , - \\sum _ { j = 3 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { j - 3 } ^ { ( 3 ) } , - \\sum _ { j = 2 } ^ { p } a _ { 1 } ^ { ( j ) } A _ { j - 2 } ^ { ( 3 ) } , - \\sum _ { j = 1 } ^ { p } a _ { 2 } ^ { ( j ) } A _ { j - 1 } ^ { ( 3 ) } ) . \\end{align*}"}
{"id": "835.png", "formula": "\\begin{align*} p ( x _ { m } ^ { c } | s _ { m } ^ { c } ) = ( 1 - s _ { m } ^ { c } ) \\delta ( x _ { m } ^ { c } ) + s _ { m } ^ { c } \\mathcal { C N } \\left ( x _ { m } ^ { c } ; 0 , ( \\sigma _ { m } ^ { c } ) ^ { 2 } \\right ) , \\end{align*}"}
{"id": "5369.png", "formula": "\\begin{align*} M _ 0 ( H ^ s \\to H ^ { - s } ) \\subset M _ \\Omega ( H ^ s \\to H ^ { - s } ) = M _ 0 ( H ^ s \\to H ^ { - s } ) + M _ { \\delta ( \\Omega ) } ( H ^ s \\to H ^ { - s } ) . \\end{align*}"}
{"id": "2867.png", "formula": "\\begin{align*} { \\mathcal L } _ H = \\left \\{ ( q , p , P _ q , P _ p ) \\ ; \\left | \\exists \\ ; u \\in U \\ ; \\mbox { s . t . } \\begin{array} { l } P _ q = \\frac { \\partial H } { \\partial q } ( q , p , u ) , P _ p = \\frac { \\partial H } { \\partial p } ( q , p , u ) , \\\\ \\frac { \\partial H } { \\partial u } ( q , p , u ) = 0 \\ , \\end{array} \\right . \\right \\} \\ , . \\end{align*}"}
{"id": "2854.png", "formula": "\\begin{align*} \\begin{aligned} c ' ( h ) = - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\left [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\right ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 \\end{aligned} . \\end{align*}"}
{"id": "864.png", "formula": "\\begin{align*} \\textrm { r a n k } \\left ( \\mathbf { V } \\right ) & = \\sum _ { m } \\frac { \\log \\left ( 1 + \\frac { \\lambda _ { m } } { \\varepsilon } \\right ) } { \\log \\left ( 1 + \\frac { 1 } { \\varepsilon } \\right ) } + o \\left ( \\varepsilon \\right ) \\\\ & = \\frac { M \\log \\left ( \\frac { 1 } { \\varepsilon } \\right ) + \\log \\left ( \\left | \\mathbf { V } + \\varepsilon \\mathbf { I } \\right | \\right ) } { \\log \\left ( 1 + \\frac { 1 } { \\varepsilon } \\right ) } + o \\left ( \\varepsilon \\right ) \\end{align*}"}
{"id": "8953.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } i \\partial _ t u + \\Delta u & = \\pm | u | ^ 2 u , ( t , x ) \\in \\R \\times \\R , \\\\ u ( 0 ) & = u _ 0 \\in M ^ s _ { p , 2 } ( \\R ) . \\end{array} \\right . \\end{align*}"}
{"id": "3070.png", "formula": "\\begin{align*} f _ { \\mathbf { M ^ { \\ast \\ast } } _ { - \\xi _ { \\boldsymbol { s } k } } } ( \\boldsymbol { v } ) = c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast \\ast } \\exp \\left \\{ - \\frac { 1 } { 2 } \\boldsymbol { v } ^ { T } \\boldsymbol { v } - \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } \\right \\} = \\phi _ { n - 1 } ( \\boldsymbol { v } ) , ~ \\boldsymbol { v } \\in \\mathbb { R } ^ { n - 1 } , \\end{align*}"}
{"id": "3940.png", "formula": "\\begin{align*} | k ( t , s ) | & \\le \\prod _ { l = 1 } ^ d \\left \\| K _ { h _ l ^ * } \\right \\| _ { \\infty } \\left \\| P _ { \\tilde { w } _ 1 - w _ d } g ( X _ { w _ d } ) - \\pi ( g ) \\right \\| _ { L ^ 1 } \\\\ & \\le \\frac { c } { \\prod _ { l = 1 } ^ d h _ l ^ * } e ^ { - \\rho ( \\tilde { w } _ 1 - w _ d ) } \\left \\| g \\right \\| _ \\infty \\\\ & \\le \\frac { c } { ( \\prod _ { l = 1 } ^ d h _ l ^ * ) ^ 2 } e ^ { - \\rho ( \\tilde { w } _ 1 - w _ d ) } . \\end{align*}"}
{"id": "2663.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 2 } ( n - 1 ) p ^ { 2 n - k - 5 } & = ( n - 1 ) \\left ( p ^ { 2 n - 6 } + p ^ { 2 n - 7 } + \\ldots + p ^ { n - 3 } \\right ) \\\\ & = ( n - 1 ) p ^ { n - 3 } \\sum _ { i = 0 } ^ { n - 3 } p ^ i \\\\ & = ( n - 1 ) p ^ { n - 3 } \\frac { 1 - p ^ { n - 2 } } { 1 - p } . \\end{align*}"}
{"id": "6110.png", "formula": "\\begin{align*} \\abs { U _ 0 } & \\leq 2 \\mu + \\mu ( s + 1 ) \\\\ & = 2 \\mu + \\mu \\abs { z _ r \\ldots z _ { r + s } } _ { Y \\cup \\mathcal { O } } \\\\ & \\leq 2 \\mu + \\mu ( \\lambda \\abs { z _ r \\ldots z _ { r + s } } _ { X \\cup \\mathcal { P } } + \\lambda c ) \\\\ & \\leq 2 \\mu + \\mu ( \\lambda ( \\abs { U _ 0 } _ { X \\cup \\mathcal { P } } + 2 \\mu ) + \\lambda c ) \\end{align*}"}
{"id": "2549.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } l ( X ) = ( 0 \\to 0 \\to \\cdots \\to X ) & \\hbox { } \\\\ & \\hbox { } \\\\ e ( X _ 1 \\to X _ 2 \\to \\cdots \\to X _ n ) = X _ n & \\hbox { } \\\\ & \\hbox { } \\\\ r ( X ) = ( X \\xrightarrow { 1 _ { X } } X \\xrightarrow { 1 _ { X } } \\cdots \\xrightarrow { 1 _ { X } } X ) & \\hbox { } \\end{array} \\right . \\end{align*}"}
{"id": "8599.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty t _ { n , k } a _ k w _ { m _ k } ( x ) \\to 0 \\end{align*}"}
{"id": "483.png", "formula": "\\begin{align*} N \\leq \\left ( \\frac { c + \\frac { 1 } { 2 } c ' } { \\frac { 1 } { 2 } c ' } \\right ) ^ \\rho = ( 1 + 2 c / c ' ) ^ \\rho . \\end{align*}"}
{"id": "5451.png", "formula": "\\begin{align*} l ^ B _ A ( m ) \\cdot r ^ B _ A ( m ) = l ^ B _ A ( m ) \\cdot q ( r ^ B _ A ( m ) ) \\cdot q ( r ^ B _ A ( m ) ) ^ { - 1 } \\cdot r ^ B _ A ( m ) \\end{align*}"}
{"id": "6397.png", "formula": "\\begin{align*} \\sum _ { q ' = 0 } ^ q \\binom { q } { q ' } \\cdot 2 ^ { q ' } \\cdot ( - 1 ) ^ { q ' } , \\end{align*}"}
{"id": "3419.png", "formula": "\\begin{align*} g = \\begin{pmatrix} a & * & * \\\\ b & * & * \\\\ c & * & * \\end{pmatrix} , a \\equiv 1 \\bmod 3 , b \\equiv c \\equiv 0 \\bmod \\sqrt { - 3 } . \\end{align*}"}
{"id": "6119.png", "formula": "\\begin{align*} P ' ( z ) = n a _ n ( z - \\omega _ 1 ) ( z - \\omega _ 2 ) \\cdots ( z - \\omega _ { n - 1 } ) . \\end{align*}"}
{"id": "8264.png", "formula": "\\begin{align*} \\psi _ { A _ { k } } = A _ { k } \\left [ \\frac { \\cos ( k x ) } { \\cos ( k a / 2 ) } - \\frac { \\cosh ( k ' x ) } { \\cosh ( k ' a / 2 ) } \\right ] \\end{align*}"}
{"id": "5126.png", "formula": "\\begin{align*} \\mathbb { L } f _ ! B ( G , G , * ) _ { \\mathrm { t r i v } } \\simeq f _ ! B ( G , G , * ) _ { \\mathrm { t r i v } } = B ( H , G , * ) _ { \\mathrm { t r i v } } \\end{align*}"}
{"id": "44.png", "formula": "\\begin{align*} \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { f ' ( v _ { n + 1 } + w ) } { f ( v _ { n + 1 } + w ) } = \\eta \\delta _ n ( \\beta ) . \\end{align*}"}
{"id": "1326.png", "formula": "\\begin{align*} \\rho _ f ( x ) : = \\int _ \\R f \\d v P _ f ( x ) : = \\rho _ f ( x ) T _ f ( x ) : = \\int _ \\R v ^ 2 f \\d v , \\end{align*}"}
{"id": "1065.png", "formula": "\\begin{align*} B R ( z , A ) = \\int _ 0 ^ \\infty \\mathrm { e } ^ { z s } \\tilde { B } T ( s , A ) \\ , \\mathrm { d } s , \\end{align*}"}
{"id": "1804.png", "formula": "\\begin{align*} \\nabla \\phi _ \\mathrm { p d } ( x ^ { ( k _ i ) } , z ^ { ( k _ i ) } ) - \\nabla \\phi _ \\mathrm { p d } ( x ^ { ( k _ i + 1 ) } , z ^ { ( k _ i + 1 ) } ) + \\begin{bmatrix} - A ^ T z ^ { ( k _ i + 1 ) } \\\\ A x ^ { ( k _ i + 1 ) } \\end{bmatrix} \\in \\begin{bmatrix} \\partial f ( x ^ { ( k _ i + 1 ) } ) + \\nabla h ( x ^ { ( k _ i + 1 ) } ) \\\\ \\partial g ^ \\ast ( z ^ { ( k _ i + 1 ) } ) \\end{bmatrix} , \\end{align*}"}
{"id": "2184.png", "formula": "\\begin{align*} F _ { \\mu \\boxplus \\nu } ^ { \\prime } ( s _ { 0 } ) = F _ { \\nu } ^ { \\prime } ( \\alpha ) \\omega ^ { \\prime } ( s _ { 0 } ) \\end{align*}"}
{"id": "1158.png", "formula": "\\begin{align*} T [ p q ] = T [ p ] T [ q ] \\end{align*}"}
{"id": "4233.png", "formula": "\\begin{align*} \\frac { d _ { i + 1 , k } } { d _ { i , k } } \\leq \\Psi ( k ) : = \\exp \\left [ - \\frac { k } { 2 \\psi ^ { - 1 } ( k ) } \\right ] \\end{align*}"}
{"id": "3082.png", "formula": "\\begin{align*} \\mathrm { M T C E } _ { \\boldsymbol { q } } ( \\mathbf { X } ) & = \\mathrm { E } \\left [ \\mathbf { X } | \\mathbf { X } > V a R _ { \\boldsymbol { q } } ( \\mathbf { X } ) \\right ] \\\\ & = \\mathrm { E } [ \\mathbf { X } | X _ { 1 } > V a R _ { q _ { 1 } } ( X _ { 1 } ) , \\cdots , X _ { n } > V a R _ { q _ { n } } ( X _ { n } ) ] , \\end{align*}"}
{"id": "5570.png", "formula": "\\begin{align*} Q ^ \\phi _ t p _ k ( x ) = \\sum _ { l = 0 } ^ k \\dbinom { k } { l } \\frac { W _ \\phi ( k + 1 ) } { W _ { \\phi } ( l + 1 ) } x ^ l t ^ { k - l } . \\end{align*}"}
{"id": "3553.png", "formula": "\\begin{align*} j _ n ^ * ( t _ l ) = \\begin{cases} t _ l ' , & \\\\ 0 , & \\end{cases} \\end{align*}"}
{"id": "2921.png", "formula": "\\begin{align*} f ( A x + e ) = g ( x ) + d \\end{align*}"}
{"id": "1910.png", "formula": "\\begin{align*} W _ { [ n , 0 ] } = a _ { 0 } ^ { ( 0 ) } W _ { [ n - 1 , 0 ] } + \\sum _ { j = 1 } ^ { p } \\sum _ { s = 0 } ^ { j } \\sum _ { \\ell \\in \\mathbb { Z } } \\sum _ { k \\in \\mathbb { Z } } a _ { - s } ^ { ( j ) } \\ , A _ { [ \\ell - 1 , j - s - 1 ] } ^ { ( 1 ) } B _ { [ k - \\ell - 1 , s - 1 ] } ^ { ( 1 ) } W _ { [ n - k - 1 , 0 ] } , \\end{align*}"}
{"id": "4991.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { \\tau \\wedge t } ( t - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\Theta ^ n _ s d s = 0 , \\end{align*}"}
{"id": "4567.png", "formula": "\\begin{align*} \\mathbf r ^ 1 & = d \\mathbf e ^ 1 \\times \\mathbf v = ( 0 , - d v _ 3 , d v _ 2 ) , \\\\ \\mathbf s ^ 1 & = ( \\mathbf r ^ 1 \\times \\mathbf v ) / d = ( \\mathbf e ^ 1 \\times \\mathbf v ) \\times \\mathbf v = \\left ( - v _ 2 ^ 2 - v _ 3 ^ 2 , v _ 1 v _ 2 , v _ 1 v _ 3 \\right ) . \\end{align*}"}
{"id": "3474.png", "formula": "\\begin{align*} & \\mathrm { D T V } _ { ( p , q ) } ( X ) = - \\mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \\mu ^ { 2 } + 2 \\mu \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sigma ^ { 2 } \\left ( L _ { 1 } + \\frac { c _ { 1 } } { c _ { ( 1 ) } ^ { \\ast } } L _ { 2 } \\right ) , \\end{align*}"}
{"id": "3540.png", "formula": "\\begin{align*} \\begin{aligned} a ( [ r ] \\boxplus [ { r ' } ] ) ) & = \\{ a [ { x + x ' } ] : x \\in r G , \\ x ' \\in r ' G \\} \\\\ & = \\{ [ { a x + a x ' } ] : x \\in r G , \\ x ' \\in r ' G \\} = [ a [ r ] \\boxplus a [ { r ' } ] ) ; \\end{aligned} \\end{align*}"}
{"id": "8007.png", "formula": "\\begin{align*} I _ { ( Z _ - , D _ { Z , - } ) } ( y , z ) = z e ^ { t _ - / z } \\sum _ { d \\in \\mathbb K _ { - } } \\tilde y ^ { d } \\left ( \\prod _ { i \\in M _ 0 } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\\\ \\left ( \\prod _ { 0 < a \\leq v _ 2 \\cdot d } ( v _ 2 + a z ) \\right ) \\textbf { 1 } _ { [ - d ] } I _ { D _ { Z , - } , d } , \\end{align*}"}
{"id": "1613.png", "formula": "\\begin{align*} F ( x , y ) = \\frac { 2 ( y ^ 1 ) ^ 2 + ( x ^ 1 ) ^ 2 ( y ^ 2 ) ^ 2 } { b y ^ 1 } , \\end{align*}"}
{"id": "8035.png", "formula": "\\begin{align*} I _ { D } ( y ) = \\lim _ { y _ { \\mathrm { r } + 1 } \\rightarrow 0 } L \\circ \\bar I _ { \\tilde D } ( y ) , \\end{align*}"}
{"id": "7960.png", "formula": "\\begin{align*} I _ { D _ + , d } = \\frac { 1 } { \\prod _ { i \\in I _ + , D _ i \\cdot d > 0 } ( \\bar D _ i + ( D _ i \\cdot d ) z ) } [ \\textbf { 1 } ] _ { ( - D _ i \\cdot d ) _ { i \\in I _ + } } , \\end{align*}"}
{"id": "6139.png", "formula": "\\begin{gather*} A ' _ 1 \\cap B _ 2 \\cap H _ { 1 , 2 } = \\{ e _ 1 , \\dots , e _ { d / 2 } \\} , A ' _ 2 \\cap B _ 1 \\cap H _ { 1 , 4 } = \\{ f _ 1 , \\dots , f _ { d / 2 } \\} , \\\\ A ' _ 1 \\cap B _ 2 \\cap H _ { 3 , 4 } = \\{ g _ 1 , \\dots , g _ { d / 2 } \\} , A ' _ 2 \\cap B _ 1 \\cap H _ { 2 , 3 } = \\{ h _ 1 , \\dots , h _ { d / 2 } \\} . \\end{gather*}"}
{"id": "8506.png", "formula": "\\begin{align*} t r ( A ( t , \\delta ) ^ { 1 1 } ) = t ^ { 1 1 } - 1 1 [ t ^ { 9 } \\delta - 4 t ^ { 7 } \\delta ^ 2 + 7 t ^ { 5 } \\delta ^ 3 - 5 t ^ 3 \\delta ^ 4 + t \\delta ^ 5 ] \\in S ^ { 1 1 } , \\ ; \\forall t , \\delta \\in R . \\end{align*}"}
{"id": "7180.png", "formula": "\\begin{align*} & z _ 0 ^ { - 1 } \\delta ( \\frac { z _ 1 - z _ 2 } { z _ 0 } ) ( \\frac { z _ 1 - z _ 2 } { z _ 0 } ) ^ { \\frac { j _ 1 } { T } } Y _ { M ^ 3 } ( u , z _ 1 ) I ( w _ 1 , z _ 2 ) w _ 2 \\\\ - & e ^ { \\frac { j _ 1 } { T } \\pi i } z _ 0 ^ { - 1 } \\delta ( \\frac { - z _ 2 + z _ 1 } { z _ 0 } ) ( \\frac { z _ 2 - z _ 1 } { z _ 0 } ) ^ { \\frac { j _ 1 } { T } } I ( w _ 1 , z _ 2 ) Y _ { M ^ 2 } ( u , z _ 1 ) w _ 2 \\\\ = & z _ 1 ^ { - 1 } \\delta ( \\frac { z _ 2 + z _ 0 } { z _ 1 } ) ( \\frac { z _ 2 + z _ 0 } { z _ 1 } ) ^ { \\frac { j _ 2 } { T } } I ( Y _ { M ^ 1 } ( u , z _ 0 ) w _ 1 , z _ 2 ) w _ 2 \\end{align*}"}
{"id": "5616.png", "formula": "\\begin{align*} f ( X ) D [ X ] = ( A P _ 1 ^ { r _ 1 } \\cdots P _ n ^ { r _ n } ) _ t = ( A _ t P _ 1 ^ { r _ 1 } \\cdots P _ n ^ { r _ n } ) _ t = ( P _ 1 ^ { r _ 1 } \\cdots P _ n ^ { r _ n } ) _ t . \\end{align*}"}
{"id": "6540.png", "formula": "\\begin{align*} \\P \\{ m \\} = \\prod \\limits _ { i = 0 } ^ { m - 1 } \\left ( 1 - \\P \\{ \\ell ^ { ( A ) } _ i > m - i \\} \\right ) \\geq \\exp \\left ( - g _ m \\sum \\limits _ { i = 0 } ^ { m - 1 } \\P \\{ \\ell ^ { ( A ) } _ i > m - i \\} \\right ) , \\end{align*}"}
{"id": "6791.png", "formula": "\\begin{align*} \\varphi _ k ( D ) = 2 ^ { - 2 \\ell k } ( - \\Delta ) ^ \\ell \\widetilde \\varphi _ j ( D ) \\end{align*}"}
{"id": "1262.png", "formula": "\\begin{align*} P ( \\mathcal { B } _ { d , k } , x ) & = E _ k ( x , d - 1 ) \\prod _ { j = 1 } ^ { k - 1 } E _ j ( x , d - 1 ) ^ { ( d - 2 ) ( d - 1 ) ^ { k - 1 - j } } . \\end{align*}"}
{"id": "3598.png", "formula": "\\begin{align*} & { \\rm ( a ) } \\ \\rho ( I ( G ) ) = 1 ; \\quad { \\rm ( b ) } \\ \\rho _ { i c } ( I ( G ) ) = 1 ; { \\rm ( c ) } \\ G \\mbox { i s b i p a r t i t e } ; \\quad { \\rm ( d ) } \\ \\rho ( I ( G ) ^ \\vee ) = 1 ; \\end{align*}"}
{"id": "2338.png", "formula": "\\begin{align*} I ( f ) = \\left \\{ b , 1 \\leq b \\leq \\deg ( f ) \\mid \\epsilon ( f ) = \\frac { \\nu ( f ) - \\nu ( \\partial _ b f ) } { b } \\right \\} . \\end{align*}"}
{"id": "513.png", "formula": "\\begin{align*} B \\wedge \\langle S \\rangle & = B \\wedge \\left ( \\bigvee _ { s \\in S } \\langle s \\rangle \\right ) \\\\ & = \\bigvee _ { s \\in S } ( B \\wedge \\langle s \\rangle ) \\end{align*}"}
{"id": "1433.png", "formula": "\\begin{align*} \\hat { s } _ { g } \\left ( q , r , \\tau _ t \\right ) & = \\\\ & \\frac { q \\exp \\left ( - \\frac { \\left ( r - d _ { g } \\right ) ^ 2 } { 2 \\tau _ t ^ 2 } \\right ) } { q \\exp \\left ( - \\frac { \\left ( r - d _ { g } \\right ) ^ 2 } { 2 \\tau _ t ^ 2 } \\right ) + ( 1 - q ) \\exp \\left ( - \\frac { r ^ 2 } { 2 \\tau _ t ^ 2 } \\right ) } , \\\\ \\end{align*}"}
{"id": "3210.png", "formula": "\\begin{align*} E ^ { } _ t = \\sum _ { i = 1 } ^ n \\tau \\xi ^ { } ( f ^ { } _ { t , i } ) ^ 3 , \\end{align*}"}
{"id": "3847.png", "formula": "\\begin{align*} \\Pi _ \\xi ^ { \\mu , \\nu } = \\Pi ^ { \\mu , \\nu } \\circ \\Big ( u + \\xi ( v - u ) \\Big ) ^ { - 1 } . \\end{align*}"}
{"id": "5832.png", "formula": "\\begin{align*} \\mu _ n ( \\alpha ) = - \\frac { \\log \\left | \\alpha - p _ n / q _ n \\right | } { \\log q _ n } , \\end{align*}"}
{"id": "6140.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\Delta t \\| u _ 0 \\| _ 2 ^ 3 } { \\nu \\ , h ^ { 1 / 2 } } = o ( 1 ) , \\end{aligned} \\end{align*}"}
{"id": "7219.png", "formula": "\\begin{align*} C _ 0 : = \\frac { 3 ^ { - | a | } 2 ^ { - | b | } } { 8 \\sqrt { 2 } } > 0 \\end{align*}"}
{"id": "4302.png", "formula": "\\begin{align*} \\Theta _ \\omega ( \\alpha ) = \\sum _ i \\arctan \\lambda _ i = \\hat { \\theta } \\mod \\pi \\Z . \\end{align*}"}
{"id": "1879.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\widetilde { \\mathcal { D } } _ { [ n , j , k ] } } w ( \\gamma ) = A _ { [ k , i ] } A _ { [ n - k - 1 , j - i - 1 ] } ^ { ( i + 1 ) } . \\end{align*}"}
{"id": "6743.png", "formula": "\\begin{align*} J ( S _ \\delta ( u _ \\delta ) , u _ \\delta ) \\leq j _ \\delta ( u _ \\delta ) \\leq j _ \\delta ( \\overline { u } ) = J ( S _ \\delta ( \\overline { u } ) , \\overline { u } ) . \\end{align*}"}
{"id": "6780.png", "formula": "\\begin{align*} \\sum _ { i \\in M _ P } \\sum _ { j \\in N _ i } a _ { i j } \\tilde x _ { i j } + ( b - s ) \\sum _ { i j \\in \\bar P } \\tilde x _ { i j } & \\leq \\sum _ { i \\in M _ P } \\sum _ { j \\in N _ i } a _ { i j } \\tilde x _ { i j } + ( b - s ) | \\bar P | \\\\ & \\leq s + ( b - s ) | \\bar P | \\\\ & = b + ( | \\bar P | - 1 ) ( b - s ) \\\\ & = b + ( | M _ P - M _ 0 | - 1 ) ( b - s ) . \\end{align*}"}
{"id": "382.png", "formula": "\\begin{align*} V ( s ) \\ge \\begin{cases} c ( m ) s ^ { m + 1 } & s \\le R \\\\ c ( m ) R s ^ { m } & s > R \\\\ \\end{cases} , \\end{align*}"}
{"id": "4046.png", "formula": "\\begin{align*} \\hat { \\beta } = A ^ { 0 } Y + ( Y ^ { \\prime } H _ { 1 } ^ { 0 } Y , \\ldots , Y ^ { \\prime } H _ { k } ^ { 0 } Y ) ^ { \\prime } , \\end{align*}"}
{"id": "8833.png", "formula": "\\begin{align*} \\| f _ t \\circ ( \\phi _ 1 * \\phi _ 2 ) \\| = \\lim _ { n \\to \\infty } \\left \\| f _ t \\circ \\left ( 1 _ { A _ 1 ^ { ( n ) } } * 1 _ { A _ 2 ^ { ( n ) } } \\right ) \\right \\| \\leq \\lim _ { n \\to \\infty } \\left \\| f _ t \\circ \\left ( 1 _ { A _ 1 ^ { ( n ) } } ^ * * 1 _ { A _ 2 ^ { ( n ) } } ^ * \\right ) \\right \\| = \\| f _ t \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) \\| . \\end{align*}"}
{"id": "8556.png", "formula": "\\begin{align*} \\Phi ( x ) + \\Psi ( \\phi ( x ) ) = x \\phi ( x ) \\ , x \\geq 0 \\ , \\end{align*}"}
{"id": "9017.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\left ( - \\frac { 1 } { 2 } \\right ) ^ k P _ k \\left ( n - 1 , \\frac { n } { 2 } \\right ) \\left ( \\zeta \\left ( k - n + 2 , \\frac { n } { 4 } \\right ) - \\zeta \\left ( k - n + 2 , \\frac { n + 2 } { 4 } \\right ) \\right ) = 2 ^ { 1 - n } \\Gamma ^ 2 \\left ( \\frac { n } { 2 } \\right ) . \\end{align*}"}
{"id": "7905.png", "formula": "\\begin{align*} \\nabla _ { \\vec s , k } \\left ( L ( t , z ) z ^ { - \\mu } z ^ { \\rho } \\alpha \\right ) = 0 , \\nabla _ { z \\partial z } \\left ( L ( t , z ) z ^ { - \\mu } z ^ { \\rho } \\alpha \\right ) = 0 \\end{align*}"}
{"id": "5186.png", "formula": "\\begin{align*} \\mathcal { R } : = \\{ \\hbox { $ q $ a h a l f - n o d e o f $ \\widehat { C } $ w h i c h i s o f m u l t i p l i c i t y $ 0 $ b u t n o t a n a n c h o r } \\} , \\end{align*}"}
{"id": "4975.png", "formula": "\\begin{align*} \\psi _ { n , 1 } ( s , t ) : = ( t - \\eta _ n ( s ) ) ^ { \\alpha } - ( t - s ) ^ { \\alpha } , 0 \\le s < t \\le T , \\end{align*}"}
{"id": "5166.png", "formula": "\\begin{align*} | M | & = p _ 0 p _ 1 \\cdots p _ { m - 1 } - ( 1 - p _ 0 ) ( 1 - p _ 1 ) \\cdots ( 1 - p _ { m - 1 } ) \\\\ & = \\prod _ { i = 0 } ^ { m - 1 } p _ i - \\prod _ { i ' = 0 } ^ { m - 1 } ( 1 - p _ { i ' } ) , \\end{align*}"}
{"id": "4003.png", "formula": "\\begin{align*} H _ { \\gamma } ( Y | X ) = \\textstyle \\sum _ { i = 1 } ^ s \\sum _ { j = 1 } ^ t - p _ { i , j } \\log _ { \\gamma } \\frac { p _ { i , j } } { p _ i } ; \\\\ I _ { \\gamma } ( X ; Y ) = \\textstyle \\sum _ { i = 1 } ^ s \\sum _ { j = 1 } ^ t - p _ { i , j } \\log _ { \\gamma } \\frac { p _ i p ' _ j } { p _ { i , j } } . \\end{align*}"}
{"id": "6569.png", "formula": "\\begin{align*} \\psi _ i ( \\psi ^ { - 1 } _ j ( x , v ) ) = ( x , g _ { i j } ( x ) ( v ) ) \\end{align*}"}
{"id": "3834.png", "formula": "\\begin{align*} E r r U ^ { 2 ; 0 } _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\sum _ { a = 1 , 2 } ( - 1 ) ^ a \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i \\widetilde { \\Phi } ^ 2 _ { \\mu _ 1 , \\mu _ 2 } ( \\xi , \\eta , \\sigma ; t _ a , X ( t _ a ) , v ) } ( \\Phi ^ 2 _ { \\mu _ 1 , \\mu _ 2 } ( \\xi , \\eta , \\sigma ; V ( t _ a ) , v ) ) ^ { - 1 } \\end{align*}"}
{"id": "5141.png", "formula": "\\begin{align*} \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( m + 1 ) \\oplus \\mathcal { G } ( m + 1 ) = 0 = \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( m ) \\oplus \\mathcal { G } ( m ) \\end{align*}"}
{"id": "4894.png", "formula": "\\begin{align*} \\begin{array} { l l } \\dot { x } _ i = u _ i , & i = 1 , \\dots , r , \\\\ \\dot { y } _ { i j } = x _ i u _ j - x _ j u _ i , & i < j \\\\ \\end{array} \\end{align*}"}
{"id": "4039.png", "formula": "\\begin{align*} a = \\frac { \\frac { t } { \\sqrt { n - 1 } } } { \\sqrt { 1 + \\frac { t ^ 2 } { n - 1 } } } . \\end{align*}"}
{"id": "1039.png", "formula": "\\begin{align*} \\phi ( t ) = \\| F ( t ) \\| _ q \\psi ( t ) = \\| G ( t ) \\| _ r . \\end{align*}"}
{"id": "3690.png", "formula": "\\begin{align*} U _ \\Gamma = \\{ ( x ' , x _ n ) : \\ , x ' \\in \\Gamma \\ , \\ , \\ , \\ , x _ n \\in ( - c , c ) \\} \\end{align*}"}
{"id": "4104.png", "formula": "\\begin{align*} \\widetilde { C } ( m , p ) : = C ( m , p ) ^ { - 1 } = C \\left ( \\frac { 1 } { 4 m } , \\frac { p - m } { m ( 3 p - 4 m ) } \\right ) . \\end{align*}"}
{"id": "6587.png", "formula": "\\begin{align*} q ( y ) = \\max \\{ q _ j ( y _ j ) \\colon j \\in \\Phi \\} \\quad \\mbox { f o r a l l $ \\ , y = ( y _ j ) _ { j \\in J } \\in F $ , } \\end{align*}"}
{"id": "728.png", "formula": "\\begin{align*} ( \\partial _ s + \\Delta ) w & = v ( \\partial _ s + \\Delta ) \\varphi + \\varphi ( \\partial _ s + \\Delta ) v - 2 g ( \\nabla \\varphi , \\nabla v ) \\\\ & = v ( \\partial _ s + \\Delta ) \\varphi + \\varphi ( \\partial _ s + \\Delta ) v - 2 \\rho \\lambda e ^ { \\lambda u } \\varphi g ( \\nabla u , \\nabla v ) . \\end{align*}"}
{"id": "3041.png", "formula": "\\begin{align*} \\chi = | \\mathfrak { S } _ + \\times \\mathfrak { S } _ - \\times W _ 1 | ^ { - 1 } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\substack { w = ( w _ + , w _ - , w _ 1 ) \\\\ \\in \\mathfrak { S } _ + \\times \\mathfrak { S } _ - \\times W _ 1 } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\varphi _ + ( w _ + ) \\varphi _ - ( w _ - ) \\varphi ( w _ 1 ) R ^ { G } _ { T _ { w } } \\theta _ { w } . \\end{align*}"}
{"id": "8758.png", "formula": "\\begin{align*} [ n + m ] + [ n - m ] & = [ n ] [ 2 ] _ { q ^ m } , \\\\ [ n + m ] [ n - m ] & = [ n ] ^ 2 - [ m ] ^ 2 , \\\\ [ m ] [ m + n ] - [ \\ell ] [ \\ell + n ] & = [ m - \\ell ] [ m + \\ell + n ] , \\\\ [ 2 n ] & = [ 2 ] [ n ] _ { q ^ 2 } . \\end{align*}"}
{"id": "8625.png", "formula": "\\begin{align*} \\beta ^ * - \\sigma & = \\beta ^ * ( I ^ * + R ^ * ) - \\delta I ^ * \\\\ 0 & = \\gamma I ^ * - \\omega R ^ * + \\delta R ^ * I ^ * \\end{align*}"}
{"id": "5672.png", "formula": "\\begin{align*} \\xi \\colon H _ * ( \\textstyle { \\prod _ { k = 1 } ^ { m } } F l _ { n _ k } ; \\Z ) \\cong H _ * ( F l _ { J } ; \\Z ) . \\end{align*}"}
{"id": "3688.png", "formula": "\\begin{align*} P ( h ) u _ h = E ( h ) u _ h \\ , \\ , M , \\\\ \\end{align*}"}
{"id": "1294.png", "formula": "\\begin{align*} \\mathcal { G } ( r ( T ) , x ) = \\frac { \\mathcal { G } _ 1 ( r ( T ) , x ) } { \\mathcal { G } _ 2 ( r ( T ) , x ) } , \\end{align*}"}
{"id": "8626.png", "formula": "\\begin{align*} \\begin{cases} r ^ * _ i < \\tilde { c } _ i & \\\\ r ^ * _ i = \\tilde { c } _ i & \\end{cases} , 1 \\leq i \\leq n \\end{align*}"}
{"id": "4141.png", "formula": "\\begin{align*} F ^ { ( \\iota ) } ( L ) = \\sum _ { \\ell = - 1 } ^ \\infty F ^ { ( \\iota ) } ( L ) \\chi _ \\ell ( L / U ) , \\end{align*}"}
{"id": "5893.png", "formula": "\\begin{align*} \\sigma _ { p , \\alpha } ^ 2 : = \\Big ( \\frac { p } { 1 + \\alpha } \\Big ) ^ { 2 / p } \\frac { \\Gamma ( 3 / p ) } { \\Gamma ( 1 / p ) } , \\end{align*}"}
{"id": "547.png", "formula": "\\begin{align*} f _ k ( n ) = \\sum _ { \\substack { r d \\mid n \\\\ d \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } \\frac { \\mu ( r ) } { r d } . \\end{align*}"}
{"id": "2766.png", "formula": "\\begin{align*} \\Sigma _ i = ( X _ i , U _ i , F _ i ) i \\in \\{ 1 , 2 \\} . \\end{align*}"}
{"id": "6800.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { H ^ i _ a ( 0 , 1 ) } : = \\sum _ { j = 0 } ^ { i - 1 } \\| u ^ { ( j ) } \\| ^ 2 _ { L ^ 2 ( 0 , 1 ) } + \\| \\sqrt { a } u ^ { ( i ) } \\| ^ 2 _ { L ^ 2 ( 0 , 1 ) } \\forall \\ ; u \\in H ^ i _ a ( 0 , 1 ) , \\end{align*}"}
{"id": "8740.png", "formula": "\\begin{align*} \\partial _ t ^ k u ( t ) = A _ k ( t ) u ( t ) ~ \\Leftrightarrow ~ \\partial _ t ^ k U ( t , s ) u _ s = A _ k ( t ) U ( t , s ) u _ s \\end{align*}"}
{"id": "1523.png", "formula": "\\begin{align*} \\cosh d _ H : = \\cosh r \\cosh r ' - \\sinh r \\sinh r ' \\cos ( \\theta { - } \\theta ' ) . \\end{align*}"}
{"id": "8971.png", "formula": "\\begin{align*} \\| u \\| ^ q _ { L ^ q ( \\Omega ) } + b \\| \\gamma u \\| ^ r _ { L ^ r ( \\Sigma ) } = 1 < 2 , \\end{align*}"}
{"id": "7326.png", "formula": "\\begin{align*} { \\sf t } _ 8 & = n _ 3 ( \\chi _ 4 - \\chi _ { 3 , { \\sf e } } ) \\lesssim { \\sf U } _ \\infty ^ q { \\bf 1 } _ { \\tfrac { 1 } { 2 } { \\sf r } _ 3 < | x | < 2 } . \\end{align*}"}
{"id": "4237.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { 1 \\vee \\log \\left ( \\phi ^ { - 1 } ( n ) \\right ) } = \\infty . \\end{align*}"}
{"id": "8261.png", "formula": "\\begin{align*} \\hat { H } \\phi ( x ) = E \\phi ( x ) \\end{align*}"}
{"id": "244.png", "formula": "\\begin{align*} ( ( \\phi _ i ^ { ( s ) } ) ^ G ( x _ { k j } ) ) ( 1 ) = \\delta _ { k j } + \\pi \\Gamma ^ { k ( s ) } _ { i j } ( 1 ) , \\end{align*}"}
{"id": "1305.png", "formula": "\\begin{align*} \\mu ( E ) = \\int _ E f ( \\mathrm { d } z ) , \\end{align*}"}
{"id": "7399.png", "formula": "\\begin{align*} \\psi _ n ( y ) & = \\psi _ n ( T _ 5 y ) = \\psi _ n ( T _ 2 T _ 5 y ) = \\psi _ n ( T _ 3 T _ 5 y ) = \\psi _ n ( T _ 4 T _ 5 y ) \\\\ & = \\psi _ n ( T _ 9 y ) = \\psi _ n ( T _ 2 T _ 9 y ) = \\psi _ n ( T _ 3 T _ 9 y ) = \\psi _ n ( T _ 4 T _ 9 y ) . \\end{align*}"}
{"id": "8633.png", "formula": "\\begin{align*} \\frac { d g } { d t } = - e ^ { - 2 \\gamma t } X _ 2 ^ R ( g ) - \\omega ^ 2 e ^ { 2 \\gamma t } X _ 1 ^ R ( g ) , g \\in \\widetilde { S L } ( 2 , \\mathbb { R } ) , t \\in \\mathbb { R } , \\end{align*}"}
{"id": "2859.png", "formula": "\\begin{align*} \\kappa = \\frac { \\mu } { \\delta ^ { - 1 } \\ , \\| A ^ T A \\| _ 2 + \\mu } \\Leftrightarrow \\frac { 1 } { \\kappa } = \\frac { 1 } { \\mu ^ { - 1 } \\ , \\delta ^ { - 1 } \\ , \\| A ^ T A \\| _ 2 + 1 } . \\end{align*}"}
{"id": "5452.png", "formula": "\\begin{align*} m = ( l ^ B _ A ( m ) q ( r ^ B _ A ( m ) ) ( q ( r ^ B _ A ( m ) ) ^ { - 1 } r ^ B _ A ( m ) ) \\end{align*}"}
{"id": "6203.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } f ( x ) = p ( x ) f ( x ) \\end{align*}"}
{"id": "1954.png", "formula": "\\begin{align*} \\langle \\mathcal { W } ^ { n } \\hat { e } _ { r } , \\hat { e } _ { 0 } \\rangle = \\sum _ { \\gamma \\in \\mathcal { S } _ { 3 } ( n , r ) } w ( \\gamma ) \\end{align*}"}
{"id": "7943.png", "formula": "\\begin{align*} y ^ d \\mapsto \\prod _ { i = 1 } ^ { l _ - } \\tilde y _ i ^ { p _ i ^ - \\cdot d } \\prod _ { j \\in S _ - } \\tilde x _ j ^ { D _ j \\cdot d } . \\end{align*}"}
{"id": "5695.png", "formula": "\\begin{align*} \\underbrace { \\varpi _ { a } \\varpi _ { a + 1 } \\cdots \\varpi _ { b } } _ { k } = k ! e _ k ( x _ 1 , x _ 2 , \\ldots , x _ b ) , \\end{align*}"}
{"id": "7188.png", "formula": "\\begin{align*} & m = j + \\delta ( j _ 3 , j _ 4 ) + 1 - \\delta ( [ j _ 1 + j _ 3 ] , j _ 4 ) + \\frac { [ j _ 1 + j _ 3 ] } { T } - \\frac { j _ 1 + j _ 3 } { T } \\end{align*}"}
{"id": "958.png", "formula": "\\begin{align*} E = Q [ \\theta ] = \\left \\{ \\sum _ { i = 0 } ^ { n - 1 } a _ { i } \\theta ^ { i } \\mid a _ { i } \\in Q \\right \\} . \\end{align*}"}
{"id": "6737.png", "formula": "\\begin{align*} \\mu \\int _ { \\Omega } \\varepsilon w : \\varepsilon v + \\nu \\int _ { \\Omega } \\mathsf { m } _ \\delta ' ( \\varepsilon y _ \\delta ) \\varepsilon w : \\varepsilon v = \\int _ { \\Omega } h \\cdot v , \\forall v \\in Y , \\end{align*}"}
{"id": "5665.png", "formula": "\\begin{align*} J = J _ 1 \\sqcup J _ 2 \\sqcup \\cdots \\sqcup J _ { m } , \\end{align*}"}
{"id": "6082.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 - x = x ( x - 1 ) ( x ^ 2 + x + 1 ) . \\end{align*}"}
{"id": "473.png", "formula": "\\begin{align*} { \\mathcal C } _ X ( A , z ) : = \\sum _ { n = 1 } ^ \\infty \\frac { c _ n ( A ) } { n ! } z ^ n , z \\in \\mathbb C . \\end{align*}"}
{"id": "2496.png", "formula": "\\begin{align*} g ( R ( U , V ) W , X ) = g ( ( \\nabla _ U T ) _ V W , X ) - g ( ( \\nabla _ V T ) _ U W , X ) , \\end{align*}"}
{"id": "4664.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\tau ( t ) t ^ { \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| u ( t , \\cdot ) - \\Psi _ t \\| _ { q , h } = \\infty . \\end{align*}"}
{"id": "1130.png", "formula": "\\begin{align*} E _ { \\mbox { \\tiny M D } } ( \\theta ) = \\sup _ { \\lambda \\ge 0 } \\sup _ { P \\le P _ w } \\left \\{ \\lambda ( \\sqrt { P _ s P } - \\theta ) - \\lim _ { n \\to \\infty } \\frac { C \\left ( \\lambda \\sqrt { P n } \\right ) } { n } - \\frac { \\lambda ^ 2 \\sigma _ N ^ 2 P } { 2 } \\right \\} . \\end{align*}"}
{"id": "1302.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\alpha _ j \\ \\mu ( T _ j , \\lambda ) - \\sum _ { j = 1 } ^ { k } \\mu ( T _ j , \\lambda ) & = \\sum _ { j = 1 } ^ { k } ( \\alpha _ j - 1 ) \\mu ( T _ j , \\lambda ) . \\end{align*}"}
{"id": "1977.png", "formula": "\\begin{align*} \\int _ { B _ 1 } \\Big ( A ( x ) \\nabla u \\cdot \\nabla ( \\eta u ) \\Big ) \\ , d x = - \\int _ { B _ 1 } f ( x ) ( \\eta u ) \\ , d x . \\end{align*}"}
{"id": "8340.png", "formula": "\\begin{align*} f ( u ) = \\frac { 1 } { ( 2 \\pi ) ^ 3 } \\int \\limits _ { 0 } ^ { \\infty } \\ , d \\nu \\ , \\nu ^ 2 \\ , \\int \\limits _ { \\mathbb { S } ^ 2 } \\ , d ^ 2 p \\ , \\mathcal { F } f ( p ; \\nu ) \\ , ( p \\cdot u ) ^ { - i \\nu - 1 } \\end{align*}"}
{"id": "4097.png", "formula": "\\begin{align*} B u ' ( t ) + A u ( t ) + B Q u ( t ) = f ( t ) , t \\in \\ , ] 0 , T [ , u ( 0 ) = u _ 0 , \\end{align*}"}
{"id": "2728.png", "formula": "\\begin{align*} \\mathbf { g r a d } ' \\gamma _ { t } ' \\circ \\gamma _ t = \\gamma ' \\circ \\gamma _ n \\mathbf { c u r l } . \\end{align*}"}
{"id": "2149.png", "formula": "\\begin{align*} d \\nu ( s ) = ( 3 / 8 ) [ s ^ { 2 } I _ { [ - 1 , 1 ] } ( s ) + s ^ { - 2 } I _ { \\mathbb { R } \\setminus [ - 1 , 1 ] } ( s ) ] \\ , d \\lambda ( s ) , \\end{align*}"}
{"id": "8489.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\left ( g ^ n _ i \\right ) ^ 2 \\frac { 1 } { n } \\rightarrow \\int _ 0 ^ 1 g ( x ) ^ 2 d x . \\end{align*}"}
{"id": "3181.png", "formula": "\\begin{align*} \\mathrm { P } _ { ( \\Phi , \\mathfrak { a } _ { \\mathfrak { b } } ) } = - \\inf _ { \\rho \\in E _ { 1 } } \\left \\{ f _ { \\Phi } \\left ( \\rho \\right ) + \\Delta _ { \\mathfrak { a } _ { \\mathfrak { b } } } \\left ( \\rho \\right ) \\right \\} \\ . \\end{align*}"}
{"id": "2939.png", "formula": "\\begin{align*} e ^ { i \\Delta t } F _ c ( \\frac { | x | } { ( t + 1 ) ^ { \\alpha } } \\leq 1 ) e ^ { - i \\Delta t } = F _ c ( \\frac { | x - 2 P t | } { ( t + 1 ) ^ { \\alpha } } \\leq 1 ) , \\end{align*}"}
{"id": "4751.png", "formula": "\\begin{align*} \\lambda ( u , s ) = \\mu ( s ) \\cdot \\lim _ { n \\to + \\infty } \\left ( \\frac { u _ 0 ^ { ( n ) } } { s _ 0 ^ { ( n ) } } \\right ) ^ { 2 ^ n } = \\mu ( s ) \\cdot \\lim _ { n \\to + \\infty } \\left ( \\frac { u _ 0 ^ { ( n ) } } { \\mu ( s ) } \\right ) ^ { 2 ^ n } . \\end{align*}"}
{"id": "4379.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde F - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f _ { t _ 0 } | ^ 2 \\\\ \\leq & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde F - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f _ { t _ 0 } | ^ 2 e ^ { - \\Psi + v _ { t _ 0 , B } ( \\Psi ) } \\\\ \\leq & ( e ^ { - t _ 1 } - e ^ { - t _ 0 - B } ) \\int _ { \\{ \\Psi < - t _ 1 \\} } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f _ { t _ 0 } | ^ 2 e ^ { - \\Psi } , \\end{align*}"}
{"id": "4699.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ 1 ( n ) \\ , q ^ n = \\frac { 1 } { ( q ; q ) ^ 2 _ \\infty } \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n q ^ { n ( n + 1 ) / 2 } . \\end{align*}"}
{"id": "1225.png", "formula": "\\begin{align*} \\mathcal { F } ( v , x ) & = \\dfrac { \\mathcal { F } _ 1 ( v , x ) } { \\mathcal { F } _ 2 ( v , x ) } , \\end{align*}"}
{"id": "4619.png", "formula": "\\begin{align*} \\phi [ \\alpha ] g [ \\alpha ] ^ { - 1 } g ^ { - 1 } \\phi ^ { - 1 } = [ \\alpha ] h [ \\alpha ] ^ { - 1 } h ^ { - 1 } \\end{align*}"}
{"id": "7492.png", "formula": "\\begin{align*} \\sum _ { m \\ge 0 } m ^ n t ^ m = \\frac { t A _ n ( t ) } { ( 1 - t ) ^ { n + 1 } } , \\end{align*}"}
{"id": "4048.png", "formula": "\\begin{align*} \\hat { \\beta } ( - z ) = - \\hat { \\beta } ( z ) z \\in \\mathbb { R } ^ { n } , \\end{align*}"}
{"id": "8811.png", "formula": "\\begin{align*} \\int _ { G } ^ { } 1 _ { L _ 1 ( t _ 1 ) } ( g ' ) 1 _ { L _ 2 ( t _ 2 ) } ( g '^ { - 1 } g ) d g ' = 1 _ { L _ 1 ( t _ 1 ) } * 1 _ { L _ 2 ( t _ 2 ) } ( g ) , \\end{align*}"}
{"id": "8466.png", "formula": "\\begin{align*} w _ \\ell & : = \\left ( \\sum _ { q \\in \\left [ b \\right ] } p _ { n } \\left ( q , \\ell \\right ) \\right ) \\\\ f _ { \\ell , 1 } & : = \\left ( \\sum _ { j \\in \\left [ b \\right ] } \\frac { p _ { n } \\left ( j , \\ell \\right ) } { \\sum _ { q \\in \\left [ b \\right ] } p _ { n } \\left ( q , \\ell \\right ) } h _ { 1 , b , j } \\right ) \\\\ f _ { \\ell , 2 } & : = h _ { 1 , b , \\ell } \\\\ f _ { i , j } & : = h _ { 1 , b , 1 } , \\forall j > 2 , \\forall i . \\end{align*}"}
{"id": "1700.png", "formula": "\\begin{align*} & A _ 1 : = \\{ y | | y - 2 ^ { | j | + 2 N } e _ 1 | \\le 2 ^ { 2 N - 1 } \\} , \\\\ & A _ 2 : = \\{ y | | y - 2 ^ { | j | + 2 N } e _ 1 | \\ge 2 ^ { 2 N - 1 } , | y - 2 ^ { | k | + 2 N } e _ 1 | \\ge 2 ^ { 2 N - 2 } \\} , \\\\ & A _ 3 : = \\{ y | | y - 2 ^ { | j | + 2 N } e _ 1 | \\ge 2 ^ { 2 N - 1 } , | y - 2 ^ { | k | + 2 N } e _ 1 | \\le 2 ^ { 2 N - 2 } \\} , \\end{align*}"}
{"id": "6841.png", "formula": "\\begin{align*} \\| p ( t , v , g ) - \\frac { 1 } { V _ F } p _ 0 ( t , g ) \\| _ { L ^ 1 ( ( 0 , V _ F ) \\times \\mathbb { R } ) } & \\leq \\sum _ { k \\neq 0 , k \\in \\mathbb { Z } } e ^ { - k ^ 2 ( \\frac { 2 \\pi } { V _ F } ) ^ 2 D ( t ) } = 2 \\sum _ { k = 1 } ^ { + \\infty } e ^ { - k ( \\frac { 2 \\pi } { V _ F } ) ^ 2 D ( t ) } \\\\ & = 2 \\frac { e ^ { - ( \\frac { 2 \\pi } { V _ F } ) ^ 2 D ( t ) } } { 1 - e ^ { - ( \\frac { 2 \\pi } { V _ F } ) ^ 2 D ( t ) } } . \\end{align*}"}
{"id": "7286.png", "formula": "\\begin{align*} \\theta _ 0 ( x ) = { a } _ 0 { \\sf U } _ \\infty ^ { p + 1 - q } = { a } _ 0 L _ 1 ^ { p + 1 - q } | x | ^ { \\frac { 2 p } { 1 - q } + 2 } \\end{align*}"}
{"id": "1389.png", "formula": "\\begin{align*} ( i \\partial _ t + \\partial _ x ^ 2 + i \\gamma ) T [ W _ t ^ \\gamma & u _ 0 , v , \\cdots , v ] = - \\widehat { \\mathcal { H L } _ B } [ W _ t ^ \\gamma u _ 0 , v \\cdots , v ] \\\\ & \\qquad - \\frac { p - 1 } { 2 } \\widehat { T } [ W _ t ^ \\gamma u _ 0 , ( i \\partial _ t + \\bigtriangleup ) v , v , \\cdots , v ] \\\\ & \\qquad + \\frac { p - 1 } { 2 } \\widehat { T } [ W _ t ^ \\gamma u _ 0 , v , ( i \\partial _ t + \\bigtriangleup ) v , v , \\cdots , v ] , \\end{align*}"}
{"id": "8477.png", "formula": "\\begin{align*} U & = ( n - k + 1 ) y _ k + \\frac { L ( n - k ) ( n - k + 1 ) } { 2 } \\\\ & > \\frac { L n ( n - 1 ) } { 2 } \\ge U \\end{align*}"}
{"id": "519.png", "formula": "\\begin{align*} \\Phi _ { g n } ( x , t ) = { \\frac { h _ 1 ( t ) } { h _ 2 ( t ) } } \\ln \\vert h _ 2 ( t ) x + h _ 4 ( t ) \\vert \\ , \\end{align*}"}
{"id": "7826.png", "formula": "\\begin{align*} C ^ * _ \\lambda \\mathbb F _ 2 \\otimes C ^ * _ \\rho \\mathbb F _ 2 \\ni \\sum _ { i = 1 } ^ n a _ i \\otimes x _ i \\mapsto \\sum _ { i = 1 } ^ n a _ i x _ i \\in \\mathbb B ( \\ell ^ 2 \\mathbb F _ 2 ) , \\end{align*}"}
{"id": "1909.png", "formula": "\\begin{align*} A _ { [ t , - 1 ] } ^ { ( 1 ) } = B _ { [ t , - 1 ] } ^ { ( 1 ) } : = \\delta _ { t , - 1 } = \\begin{cases} 1 , & t = - 1 , \\\\ 0 , & t \\neq - 1 . \\end{cases} \\end{align*}"}
{"id": "4248.png", "formula": "\\begin{align*} D ( m ) = 1 + \\sum _ { j = 1 } ^ { \\infty } \\prod _ { i = 1 } ^ { j } \\bigg ( \\frac { 1 } { E _ { m + i } } - 1 \\bigg ) . \\end{align*}"}
{"id": "8991.png", "formula": "\\begin{align*} \\exp \\left ( \\sum _ { r = 1 } ^ { \\infty } x _ r \\frac { t ^ r } { r ! } \\right ) = \\sum _ { n = 0 } ^ { \\infty } B _ { n } ( x _ 1 , x _ 2 , \\ldots , x _ n ) \\frac { t ^ n } { n ! } , \\end{align*}"}
{"id": "1604.png", "formula": "\\begin{align*} E = b ^ 2 f ^ 2 ( x ^ 1 ) . \\end{align*}"}
{"id": "1308.png", "formula": "\\begin{align*} H _ V = \\int _ 1 ^ t \\frac { \\d s } { V ( s ) } . \\end{align*}"}
{"id": "2591.png", "formula": "\\begin{align*} \\nu _ 1 = p + 2 q - ( r _ + + r _ - ) \\ \\ \\ \\ \\mbox { a n d } \\ \\ \\ \\ \\nu _ 2 = r _ + + r _ - - \\nu _ 3 \\ . \\end{align*}"}
{"id": "1003.png", "formula": "\\begin{align*} \\pi _ { N } ^ { \\boldsymbol { K } } ( a ) : = \\begin{cases} \\frac { K _ { N + 1 } ( N + 1 , a ) } { 1 - K _ { N + 1 } ( N + 1 , N + 1 ) } , & K _ { N + 1 } ( N + 1 , N + 1 ) < 1 , \\\\ 1 , & a = 0 , \\\\ 0 , & a \\neq 0 . \\end{cases} \\end{align*}"}
{"id": "5934.png", "formula": "\\begin{align*} w ^ { ( n ) } = \\mathop \\int { W ^ { \\left ( n \\right ) } } \\left ( { { t _ 1 } - { t _ 2 } , \\ldots , { t _ 1 } - { t _ n } } \\right ) d { t _ 2 } . . . d { t _ n } \\end{align*}"}
{"id": "1185.png", "formula": "\\begin{align*} \\| j _ { p } ( v ) \\| _ { L ^ { q } ( \\rho ; \\R ^ d ) } ^ { q } = \\| v \\| _ { L ^ p ( \\rho ; \\R ^ d ) } ^ p = \\int _ { \\R ^ d } \\langle j _ p ( v ) , v \\rangle d \\rho . \\end{align*}"}
{"id": "1187.png", "formula": "\\begin{align*} | h ( \\varphi ) | & \\le \\| \\varphi \\| _ { \\dot { H } ^ { 1 , q } ( \\rho ) } \\| h \\| _ { \\dot { H } ^ { - 1 , p } ( \\rho ) } \\\\ & = \\| \\nabla \\varphi \\| _ { L ^ q ( \\rho ; \\R ^ d ) } \\| h \\| _ { \\dot { H } ^ { - 1 , p } ( \\rho ) } . \\end{align*}"}
{"id": "3333.png", "formula": "\\begin{align*} E _ { \\gamma _ 1 , \\gamma _ 2 } = I m ( \\gamma _ 1 ^ { - 1 } - I d ) \\cap I m ( \\gamma _ 2 - I d ) \\end{align*}"}
{"id": "2257.png", "formula": "\\begin{align*} \\begin{cases} e '' _ \\delta + \\frac { z } { 2 } e ' _ \\delta = 0 , \\\\ e _ \\delta ( 0 ) = 0 , e _ \\delta ( \\infty ) = \\delta , \\end{cases} \\end{align*}"}
{"id": "5288.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } d x _ 1 \\wedge \\cdots \\wedge \\widehat { d x _ i } \\wedge \\cdots \\wedge d x _ n ) = & { } ( - 1 ) ^ { i - 1 } a _ i x _ 1 ^ { a _ i } \\cdots x _ i ^ { a _ i - 1 } \\cdots x _ n ^ { a _ n } \\Omega \\\\ & { } + ( - 1 ) ^ { i - 1 } \\hbar ^ { - 1 } r _ i x _ 1 ^ { a _ 1 } \\cdots x _ i ^ { r _ i + a _ i - 1 } \\cdots x _ n ^ { a _ n } \\Omega \\end{aligned} \\end{align*}"}
{"id": "5340.png", "formula": "\\begin{align*} [ u ] _ { W ^ { s , p } ( \\Omega ) } \\vcentcolon = \\left ( \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { n + s p } } \\ , d x d y \\right ) ^ { 1 / p } \\end{align*}"}
{"id": "1836.png", "formula": "\\begin{align*} H = ( h _ { i , j } ) _ { i , j = 0 } ^ { \\infty } = \\begin{pmatrix} a _ { 0 } ^ { ( 0 ) } & 1 & & & \\\\ \\vdots & a _ { 1 } ^ { ( 0 ) } & 1 & \\\\ a _ { 0 } ^ { ( p ) } & \\vdots & a _ { 2 } ^ { ( 0 ) } & \\ddots \\\\ & a _ { 1 } ^ { ( p ) } & \\vdots & \\ddots \\\\ & & a _ { 2 } ^ { ( p ) } & \\\\ & & & \\ddots \\\\ & \\end{pmatrix} \\end{align*}"}
{"id": "3860.png", "formula": "\\begin{align*} \\Big \\langle f , T _ 2 \\circledast T _ 2 \\Big \\rangle = \\Big \\langle f , T _ 1 \\Big \\rangle \\otimes \\Big \\langle f , T _ 2 \\Big \\rangle \\end{align*}"}
{"id": "3517.png", "formula": "\\begin{align*} c _ { 1 } = \\frac { 1 } { 2 } , ~ c _ { ( 1 ) } ^ { \\ast } = \\frac { 1 } { 4 } , ~ c _ { ( 2 ) } ^ { \\ast } = \\frac { 1 } { 1 6 } . \\end{align*}"}
{"id": "4909.png", "formula": "\\begin{align*} v _ { i j } = \\left \\{ \\begin{array} { l l } \\displaystyle \\sqrt { \\frac { | ( \\lambda _ i ( A ) E _ n - A ) _ { m m } | ^ { r o w } } { \\prod _ { k = 1 ; k \\ne i } ^ n ( \\lambda _ i ( A ) - \\lambda _ k ( A ) ) } } , & j = m , \\\\ & \\\\ \\displaystyle \\frac { - v _ { i m } ^ { - 1 } \\overline { | ( \\lambda _ i ( A ) E _ n - A ) _ { j m } | ^ { r o w } } } { \\prod _ { k = 1 ; k \\ne i } ^ n ( \\lambda _ i ( A ) - \\lambda _ k ( A ) ) } , & j \\ne m . \\end{array} \\right . \\end{align*}"}
{"id": "4880.png", "formula": "\\begin{align*} \\begin{aligned} S _ { k } ( N , \\alpha ) & \\ll N ^ { 1 - \\varrho + \\varepsilon } + \\frac { q ^ { \\varepsilon } N L ^ { c } } { \\sqrt { q ( 1 + | \\lambda | N ^ { k } ) } } , \\\\ & \\varrho = \\left \\{ \\begin{array} { l l l } { 1 / 8 } , & i f \\ k = 2 , \\\\ 1 / 1 4 , & i f \\ k = 3 . \\end{array} \\right . \\end{aligned} \\end{align*}"}
{"id": "26.png", "formula": "\\begin{align*} \\Tilde { \\Phi } _ L ( \\beta ) \\Tilde { \\sigma } _ j ^ { - 1 } ( \\xi ) & = \\Tilde { \\sigma } _ i ^ { - 1 } h _ i ( \\xi ) \\\\ & = \\Tilde { \\sigma } _ i ^ { - 1 } ( ( \\alpha , h _ i ) _ { M , n } + \\xi ) \\\\ & = ( \\alpha , h _ i ) _ { M , n } + \\Tilde { \\sigma } _ i ^ { - 1 } ( \\xi ) \\end{align*}"}
{"id": "7169.png", "formula": "\\begin{align*} V [ g ] = \\mathcal { L } ( V , g ) / D \\mathcal { L } ( V , g ) . \\end{align*}"}
{"id": "3143.png", "formula": "\\begin{align*} E _ { 1 } = \\overline { \\mathrm { c o } } \\left ( \\mathcal { E } \\left ( E _ { 1 } \\right ) \\right ) = \\overline { \\mathcal { E } \\left ( E _ { 1 } \\right ) } \\ , \\end{align*}"}
{"id": "5032.png", "formula": "\\begin{align*} \\xi _ { j , n } = \\int _ { \\frac j n } ^ { \\frac { j + 1 } n \\wedge \\tau } \\gamma _ s ( s - \\frac j n ) ^ { 2 \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\frac j n } ) \\left [ ( W _ s - W _ { \\frac j n } ) ^ 2 - ( s - \\frac j n ) \\right ] d s . \\end{align*}"}
{"id": "5637.png", "formula": "\\begin{align*} \\iint _ { Q _ T } p \\Delta p ( \\nabla p \\cdot \\nabla m ) = - \\iint _ { Q _ T } | \\nabla p | ^ 2 [ \\nabla p \\cdot \\nabla m ] - \\underbrace { \\iint _ { Q _ T } p [ \\nabla p \\cdot \\nabla ( \\nabla p \\cdot \\nabla m ) ] } _ { \\mathcal { J } _ { 2 , 2 } } . \\end{align*}"}
{"id": "3713.png", "formula": "\\begin{align*} { \\downarrow _ 1 } ( v \\Uparrow u ) = \\{ u ' \\in { \\downarrow _ 1 } ( v ) \\mid q ( u ) \\prec q ( u ' ) \\} \\cup \\{ u \\} . \\end{align*}"}
{"id": "6453.png", "formula": "\\begin{align*} \\Phi \\circ Q = Q ' \\circ \\Phi . \\end{align*}"}
{"id": "3807.png", "formula": "\\begin{align*} \\Phi _ 5 ( \\xi , \\eta , \\sigma , \\kappa , \\chi , \\zeta ) : = \\hat { \\zeta } \\cdot ( \\xi + \\eta + \\sigma + \\kappa + \\chi ) + \\mu _ 4 | \\chi | + \\mu _ 3 | \\kappa | + \\mu _ 2 | \\sigma | + \\mu _ 1 | \\eta | + \\mu | \\xi | . \\end{align*}"}
{"id": "270.png", "formula": "\\begin{align*} D ^ { \\nu } _ i + D _ i ^ { \\nu t } = C ^ { \\nu } _ i . \\end{align*}"}
{"id": "3340.png", "formula": "\\begin{align*} B ( 3 , 5 , 7 ) = \\sigma _ 2 \\sigma _ 1 ^ { - 1 } \\sigma _ 2 \\sigma _ 1 ^ { - 1 } \\sigma _ 2 \\sigma _ 1 \\sigma _ 2 ^ { - 1 } \\sigma _ 1 \\sigma _ 2 ^ { - 1 } \\sigma _ 1 \\end{align*}"}
{"id": "6418.png", "formula": "\\begin{align*} [ Q , \\Bar { \\Phi } ^ { ( n + 1 ) } + \\zeta ] = \\Bar \\Phi ^ { ( n ) } \\circ Q _ \\mathfrak { g } ^ { ( 1 ) } - \\Bar { Q } ^ { ( 1 ) } \\circ \\Bar \\Phi ^ { ( n ) } . \\end{align*}"}
{"id": "2357.png", "formula": "\\begin{align*} \\nu ( l ( h _ \\sigma ) ) = \\nu ( f ) = \\nu _ \\sigma ( f ) = \\nu ( a _ { \\sigma 0 } ( f ) ) \\end{align*}"}
{"id": "5483.png", "formula": "\\begin{align*} \\mathrm { v o l } ( N _ d ( \\mathrm { C H } ( S ) ) \\cap B ( x , \\rho ) ) & = \\mathrm { v o l } ( N _ d ( \\mathrm { C H } ( \\gamma S ) ) \\cap B ( \\gamma x , \\rho ) ) \\\\ & \\lesssim e ^ { a \\rho \\beta } V \\left ( 1 + \\frac { 1 } { a } \\log A ( 1 + A C ' ) \\right ) \\lesssim e ^ { a \\rho \\beta } , \\end{align*}"}
{"id": "233.png", "formula": "\\begin{align*} \\delta _ { i j } ^ { ( 2 s ) } a = \\frac { 1 } { \\pi } ( ( a ^ { p ^ s } + \\pi \\delta ^ { ( s ) } _ { i } a ) ^ { p ^ s } - a ^ { p ^ { 2 s } } ) + ( \\pi ^ { p ^ s - 1 } + \\delta ^ { ( s ) } _ { i } \\pi ) ( ( \\delta ^ { ( s ) } _ { j } a ) ^ { p ^ s } + \\pi ( \\delta ^ { ( s ) } ) _ { i j } a ) . \\end{align*}"}
{"id": "644.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { H _ n ( 1 - s ) } { H _ n ( s ) } = 1 . \\end{align*}"}
{"id": "6769.png", "formula": "\\begin{align*} \\frac { 1 } { \\kappa _ n } \\left ( 1 - \\frac { 1 } { x \\sqrt { n \\cdot \\kappa _ n } } \\right ) = \\sum _ { j = 0 } ^ { J - 1 } \\alpha _ j ( x ) \\cdot n ^ { - j / 2 } + O ( n ^ { - J / 2 } ) . \\end{align*}"}
{"id": "1577.png", "formula": "\\begin{align*} \\mathcal { F } ( x , z ) = \\frac { 2 C ^ 3 } { E } . \\end{align*}"}
{"id": "5291.png", "formula": "\\begin{align*} \\Xi _ j ^ r : = \\frac { 1 } { e ^ { \\pi i \\tfrac { j + 1 } { r } } \\hbar ^ { ( j + 1 ) / r } \\Gamma ( \\tfrac { j + 1 } { r } ) ( \\zeta ^ { j + 1 } - 1 ) } \\sum _ { k = 0 } ^ { r - 2 } ( \\zeta ^ { - k ( j + 1 ) } - \\zeta ^ { j + 1 } ) ( \\Psi _ { k + 1 } - \\Psi _ k ) \\end{align*}"}
{"id": "3319.png", "formula": "\\begin{align*} e ^ { h , u } _ n : = u ^ h _ n - R _ h u ( t _ n ) , e ^ { h , \\varphi } : = \\varphi ^ h _ n - P ^ 2 _ h \\varphi ( t _ n ) , e ^ { h , \\psi } : = \\psi ^ h _ n - P ^ 3 _ h \\psi ( t _ n ) , \\end{align*}"}
{"id": "8040.png", "formula": "\\begin{align*} \\bar { I } _ { K _ S } = - H \\frac { \\Gamma ( 1 + \\frac H z ) ^ 4 \\Gamma ( 1 - \\frac H z ) } { \\Gamma ( 1 + \\frac { 3 H } { z } ) } \\sum _ { m \\geq 0 } ( - 1 ) ^ m \\frac { ( 2 \\pi i ) e ^ { \\frac { 2 \\pi i m } { 3 } } } { e ^ { - 2 \\pi i \\frac { H } { z } } - e ^ { \\frac { 2 \\pi i m } { 3 } } } \\frac { q ^ { - \\frac m 3 } } { \\Gamma ( m ) \\Gamma ( 1 - \\frac m 3 ) ^ 4 \\Gamma ( 1 + \\frac m 3 ) } . \\end{align*}"}
{"id": "8116.png", "formula": "\\begin{align*} c _ { N + 1 , 0 } = \\int _ 0 ^ 1 \\frac { ( \\log u ^ { - 1 } ) ^ j } { j ! \\ , u } H _ { N - j } ( u ^ { 1 / ( N - j ) } ) \\ , \\dd u , j = 0 , \\ldots , N - 1 . \\end{align*}"}
{"id": "9004.png", "formula": "\\begin{align*} \\zeta ( a , b ) - \\zeta \\left ( a , b + \\frac { 1 } { 2 } \\right ) = 2 ^ a \\Phi ( - 1 , a , 2 b ) , \\end{align*}"}
{"id": "6006.png", "formula": "\\begin{align*} f _ 2 ( x ) = x ^ 4 - 4 x ^ 3 + 7 x ^ 2 - 6 x + 3 . \\end{align*}"}
{"id": "3322.png", "formula": "\\begin{align*} u _ 0 ( x ) = \\exp ( - 2 | x | ^ 2 ) , v _ 0 \\equiv 0 . \\end{align*}"}
{"id": "4771.png", "formula": "\\begin{align*} \\mathcal T _ { \\kappa , 2 } = \\bigcup _ { i = 0 } ^ { \\lfloor \\kappa / 2 \\rfloor } \\left \\{ [ n _ 0 , n _ 1 , n _ 2 , n _ 3 ] \\in \\mathcal S _ { \\kappa , 2 } \\colon n _ 1 = n _ 2 = i \\right \\} \\triangleq \\bigcup _ { i = 0 } ^ { \\lfloor \\kappa / 2 \\rfloor } \\mathcal { Q } _ { \\kappa , i } . \\end{align*}"}
{"id": "8378.png", "formula": "\\begin{align*} F ^ * ( k , 2 , 3 ) = 8 \\left \\lceil \\frac { k } { 4 } \\right \\rceil . \\end{align*}"}
{"id": "2850.png", "formula": "\\begin{align*} \\bar { x } _ i : = x _ i - \\frac { - \\kappa } { 1 - \\kappa } \\frac { h _ i } { L } U _ * \\in \\big [ x _ { i + 1 } , x _ { i } \\big ] , i \\in \\big \\{ 0 , \\dots , N - 1 \\big \\} . \\end{align*}"}
{"id": "1349.png", "formula": "\\begin{align*} s _ c : = \\frac { 1 } { 2 } - \\frac { 2 } { p - 1 } = \\frac { p - 5 } { 2 ( p - 1 ) } , \\end{align*}"}
{"id": "6650.png", "formula": "\\begin{align*} R & \\leq \\frac { 1 } { n \\log n } \\left [ n \\log P _ { \\ , } - n \\log r _ 0 - n \\log \\sqrt { \\pi } + \\frac { 1 } { 2 } n \\log \\frac { n } { 2 } - \\frac { n } { 2 } \\log e + o ( n ) - 0 . 5 9 9 n \\right ] \\\\ & = \\frac { 1 } { n \\log n } \\left [ \\left ( \\frac { 1 } { 2 } + \\left ( 1 + b \\right ) \\right ) \\ , n \\log n - n \\left ( \\log \\frac { \\lambda \\sqrt { \\pi e } } { 2 \\rho } + 1 . 0 5 9 9 \\right ) + o ( n ) \\right ] \\ ; , \\ , \\end{align*}"}
{"id": "3145.png", "formula": "\\begin{align*} \\lim \\limits _ { L \\rightarrow \\infty } \\hat { \\rho } ( \\left \\vert A _ { L } \\right \\vert ^ { 2 } ) = | \\hat { \\rho } ( A ) | ^ { 2 } \\ , \\mathrm { } A \\in \\mathcal { U } . \\end{align*}"}
{"id": "8465.png", "formula": "\\begin{align*} h & = \\sum _ { A \\in \\left [ B \\right ] ^ { d } } w _ A h _ { d , B , A } . \\end{align*}"}
{"id": "6107.png", "formula": "\\begin{align*} \\beta _ h & = \\inf _ { ( \\tau _ h , v _ h ) \\in S ( V _ h ) } \\sup _ { ( \\sigma _ h , u _ h ) \\in S ( V _ h ) } b _ 1 ( ( \\tau _ h , - v _ h ) , ( \\sigma _ h , - u _ h ) ) . \\end{align*}"}
{"id": "5508.png", "formula": "\\begin{align*} d ( f , g ) : = \\sum _ { k = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { k } } \\frac { d _ { k } ( f , g ) } { 1 + d _ { k } ( f , g ) } , \\end{align*}"}
{"id": "5458.png", "formula": "\\begin{align*} m = l ^ B _ A ( m ) r ^ B _ A ( m ) & = l ^ B _ A ( m ) q ( r ^ B _ A ( m ) ) q ( r ^ B _ A ( m ) ) ^ { - 1 } r ^ B _ A ( m ) \\\\ & \\stackrel { ( L 2 ) } = q ( l ^ B _ A ( m ) r ^ B _ A ( m ) ) q ( r ^ B _ A ( m ) ) ^ { - 1 } r ^ B _ A ( m ) \\\\ & = q ( m ) q ( r ^ B _ A ( m ) ) ^ { - 1 } r ^ B _ A ( m ) \\end{align*}"}
{"id": "2957.png", "formula": "\\begin{align*} Y \\cdot ( f , a ) = a + Y . \\end{align*}"}
{"id": "2800.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\| \\nabla f ( x _ i ) \\| ^ 2 { } \\leq { } { \\frac { 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) } { 1 + 2 \\sum \\limits _ { i = 0 } ^ { N - 1 } h _ i } } \\end{align*}"}
{"id": "1799.png", "formula": "\\begin{align*} \\nabla \\phi _ \\mathrm { p d } ( x ^ { ( k ) } , z ^ { ( k ) } ) - \\nabla \\phi _ \\mathrm { p d } ( x ^ { ( k + 1 ) } , z ^ { ( k + 1 ) } ) \\in \\begin{bmatrix} A ^ T z ^ { ( k + 1 ) } + \\partial f ( x ^ { ( k + 1 ) } ) + \\nabla h ( x ^ { ( k + 1 ) } ) \\\\ - A x ^ { ( k + 1 ) } + \\partial g ^ \\ast ( z ^ { ( k + 1 ) } ) \\end{bmatrix} . \\end{align*}"}
{"id": "4618.png", "formula": "\\begin{align*} \\phi = [ \\gamma _ h r \\gamma _ g ^ { - 1 } ] \\in \\mathcal L _ x ( M ) \\end{align*}"}
{"id": "2792.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\| \\nabla f ( x _ i ) \\| ^ 2 { } \\leq { } \\frac { 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\frac { h _ i ( 2 - h _ i ) ( 2 - \\kappa h _ i ) } { 2 - ( 1 + \\kappa ) h _ i } } \\end{align*}"}
{"id": "4635.png", "formula": "\\begin{align*} D ( A ^ { 2 } ) & = \\left \\{ x : = \\left ( x _ k \\right ) _ { k \\in \\N } \\in D ( A ) \\ , \\middle | \\ , A x \\in D ( A ) \\right \\} \\\\ & = \\left \\{ x : = \\left ( x _ k \\right ) _ { k \\in \\N } \\in c _ 0 \\ , \\middle | \\ , \\left ( w ^ { k } w ^ { k + 1 } x _ { k + 2 } \\right ) _ { k \\in \\N } \\in c _ 0 \\right \\} \\end{align*}"}
{"id": "6968.png", "formula": "\\begin{align*} \\Phi ( z ) = 1 - \\sum _ { n \\ge 1 } \\frac { a _ n } { \\lambda _ n ^ 2 - z } , \\end{align*}"}
{"id": "7594.png", "formula": "\\begin{align*} \\varphi ( Y , x _ 0 ) = \\frac { \\alpha x _ 0 + \\beta } { \\gamma x _ 0 + \\delta } . \\end{align*}"}
{"id": "4437.png", "formula": "\\begin{align*} & J _ m ( x , r \\xi ) = \\frac { r ^ { m } } { | r | } ( J _ m ^ k f ) ( x , \\xi ) \\mbox { f o r } r \\neq 0 , \\\\ & J _ m ( x + s \\xi , \\xi ) = J _ m ( x , \\xi ) . \\end{align*}"}
{"id": "1936.png", "formula": "\\begin{align*} C _ { [ m , j ] } & = [ w ^ { m ( p + 1 ) + j + 1 } ] \\ , h ( w ) ^ { j + 1 } \\\\ & = \\frac { j + 1 } { m ( p + 1 ) + j + 1 } \\ , [ t ^ { m ( p + 1 ) } ] \\ , ( t ^ { p + 1 } + 1 ) ^ { m ( p + 1 ) + j + 1 } \\\\ & = \\frac { j + 1 } { m ( p + 1 ) + j + 1 } \\binom { m ( p + 1 ) + j + 1 } { m } \\\\ & = \\frac { j + 1 } { m p + j + 1 } \\binom { m ( p + 1 ) + j } { m } \\end{align*}"}
{"id": "5837.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ M a _ k \\sup _ { N \\leq i \\leq k } b _ i \\approx \\sum _ { k = N } ^ M a _ k b _ k \\end{align*}"}
{"id": "3360.png", "formula": "\\begin{align*} \\psi ( f ) = f \\circ \\lambda , \\end{align*}"}
{"id": "8594.png", "formula": "\\begin{align*} r _ k ( x \\oplus 2 ^ { - s } ) = \\left \\{ \\begin{array} { r l l } r _ k ( x ) & & k \\neq s , \\\\ - r _ k ( x ) & & k = s , \\end{array} \\right . \\end{align*}"}
{"id": "6898.png", "formula": "\\begin{align*} N ( t ) = \\rho ( t , V _ F ) \\int _ 0 ^ { + \\infty } g \\frac { 1 } { \\sqrt { 2 \\pi a ( t ) } } \\exp \\left ( - \\frac { ( g - g _ { } ( t ) ) ^ 2 } { 2 a ( t ) } \\right ) d g , \\end{align*}"}
{"id": "889.png", "formula": "\\begin{align*} B ( z ) = \\frac { a - z } { 1 - \\bar a z } \\end{align*}"}
{"id": "8682.png", "formula": "\\begin{align*} f ( b ) : = \\begin{cases} \\frac { \\int \\limits _ { 0 } ^ { \\pi } \\sin ^ { n - 2 } ( t ) d t } { \\int \\limits _ { 0 } ^ { \\pi } \\frac { \\sin ^ { n - 2 } ( t ) } { [ \\phi ( b \\cos ( t ) ) ] ^ n } d t } , & \\mbox { i f } d V = d V _ { B H } , \\\\ \\frac { \\int \\limits _ { 0 } ^ { \\pi } \\sin ^ { n - 2 } ( t ) T ( b \\cos ( t ) ) d t } { \\int \\limits _ { 0 } ^ { \\pi } \\sin ^ { n - 2 } ( t ) d t } , & \\mbox { i f } d V = d V _ { H T } , \\end{cases} \\end{align*}"}
{"id": "3539.png", "formula": "\\begin{align*} \\boldsymbol { \\mu } = 1 0 ^ { - 3 } \\left ( \\begin{array} { c c c c c c c c c c c } - 1 . 1 4 0 6 7 7 \\\\ 5 . 8 9 6 2 4 0 \\\\ 2 . 1 0 7 3 4 3 \\end{array} \\right ) , \\mathbf { \\Sigma } = 1 0 ^ { - 4 } \\left ( \\begin{array} { c c c c c c c c c c c } 1 9 . 0 8 8 9 3 5 & 1 2 . 5 0 3 1 1 6 & - 3 . 7 2 0 4 9 2 \\\\ 1 2 . 5 0 3 1 1 6 & 2 0 . 2 6 8 8 1 6 & - 3 . 1 6 2 6 0 1 \\\\ - 3 . 7 2 0 4 9 2 & - 3 . 1 6 2 6 0 1 & 8 . 8 5 1 9 1 3 \\end{array} \\right ) . \\end{align*}"}
{"id": "8339.png", "formula": "\\begin{align*} \\textrm { t h e r e e x i s t s N s u c h t h a t } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( 1 + k ) ^ { 2 n } } { | \\mu ( k ) | ^ { 2 } } w ( k ) < \\infty \\textrm { f o r } n < N \\textrm { a n d i n f i n i t e f o r } n \\geq N . \\end{align*}"}
{"id": "3669.png", "formula": "\\begin{align*} \\big | \\mathfrak { H } ^ { \\tilde { j } , j ; 2 } _ { k , \\tilde { k } } ( t _ 1 , t _ 2 ) \\big | \\lesssim \\sum _ { a = 1 , 2 } \\int _ 0 ^ { t _ a } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { 0 } ^ { 2 \\pi } \\int _ 0 ^ { \\pi } 2 ^ { k + \\tilde { j } - 2 j + \\gamma M _ t + 9 0 \\epsilon M _ t } ( ( t _ a - \\tau ) 2 ^ k + 1 ) ( 1 + 2 ^ { k - 3 0 \\epsilon M _ t } | y | ) ^ { - N _ 0 ^ 3 } \\| E ( \\tau , \\cdot ) \\| _ { L ^ \\infty _ x } \\end{align*}"}
{"id": "6246.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } q ^ n \\left ( q - q ^ 3 - q ^ n x \\right ) ( - q ^ { - 3 } x ; q ) _ n A i _ q ( q ^ n x ) \\\\ & = \\frac { x } { 1 + q } \\ , _ 1 \\phi _ 1 ( 0 ; - q ^ 2 ; q , - x ) - \\left ( q ( 1 + q ) + \\frac { x } { q } \\right ) A i _ q ( \\frac { x } { q } ) . \\end{align*}"}
{"id": "7682.png", "formula": "\\begin{align*} x _ { k + 1 } : = x _ k , \\overline { y } _ { k + 1 } : = y _ k - \\frac { 1 } { \\sigma _ k } g ( x _ { k + 1 } ) , \\overline { Z } _ { k + 1 } : = \\left [ Z _ k - \\frac { 1 } { \\sigma _ k } X ( x _ { k + 1 } ) \\right ] _ { + } \\end{align*}"}
{"id": "677.png", "formula": "\\begin{align*} | D \\widehat \\psi _ s | _ { g _ o } ^ 2 = ( g _ { 0 } ) _ { a b } | _ { \\widehat \\psi _ s } \\frac { \\partial \\widehat \\psi _ s ^ a } { \\partial x ^ i } \\frac { \\partial \\widehat \\psi _ s ^ b } { \\partial x ^ j } g _ o ^ { i j } = { \\rm t r } _ { g _ o } ( \\widehat \\psi _ s ^ * g _ o ) . \\end{align*}"}
{"id": "43.png", "formula": "\\begin{align*} \\dfrac { ( \\Tilde { \\mathcal { N } } ( f ) \\circ \\rho _ { \\eta } ( X ) ) ' } { \\Tilde { \\mathcal { N } } ( f ) \\circ \\rho _ { \\eta } ( X ) } = \\eta \\dfrac { ( \\Tilde { \\mathcal { N } } ( f ) ) ' } { \\Tilde { \\mathcal { N } } ( f ) } \\circ \\rho _ { \\eta } ( X ) . \\end{align*}"}
{"id": "3308.png", "formula": "\\begin{align*} \\partial _ \\nu ^ + S ( \\partial _ t ) \\varphi = - \\frac 1 2 \\varphi + K ^ t ( \\partial _ t ) \\varphi . \\end{align*}"}
{"id": "8420.png", "formula": "\\begin{gather*} W _ D : = \\sum _ { i , \\chi } n _ { i , \\chi } C _ { \\varphi _ i } ( D ) \\chi \\end{gather*}"}
{"id": "6264.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { x } f ( t ) h ( t / q ) \\left ( - \\frac { q ( 1 + q ) + t } { q ^ 2 ( 1 - q ) t } u ( t / q ) + \\frac { 1 } { q ( 1 - q ) ^ 2 t } \\right ) y ( t ) \\ , d _ q t \\\\ & = \\frac { f ( x ) } { q ( 1 - q ) } \\sum _ { n = 0 } ^ { \\infty } q ^ { \\frac { n ( n - 3 ) } { 2 } } x ^ { n - 1 } \\left ( 1 - q + q ^ { n + 2 } x \\right ) y ( q ^ n x ) = G ( x ) . \\end{align*}"}
{"id": "8286.png", "formula": "\\begin{align*} \\frac { k ' a } { 2 } \\left [ \\tanh ( k ' a / 2 ) - \\coth ( k ' a / 2 ) \\right ] = 1 , \\end{align*}"}
{"id": "1569.png", "formula": "\\begin{align*} A = \\left ( A _ { \\epsilon \\delta } \\right ) = \\left ( \\sum \\limits _ { i = 1 } ^ { 3 } z ^ i _ { \\epsilon } z ^ i _ { \\delta } \\right ) . \\end{align*}"}
{"id": "6869.png", "formula": "\\begin{align*} H ( \\lambda ) : & = \\Phi ( \\lambda ) ( \\phi ( \\lambda ) + \\lambda ( 1 - \\Phi ( \\lambda ) ) ) - \\frac { 1 } { 2 } { \\phi ( \\lambda ) } . \\end{align*}"}
{"id": "5132.png", "formula": "\\begin{align*} \\omega = \\sum _ i [ \\alpha _ i ^ \\# , \\alpha _ i ] \\end{align*}"}
{"id": "1357.png", "formula": "\\begin{align*} \\overline { \\widehat { w } } _ k : = \\langle k \\rangle ^ { - s } \\langle \\tau - k ^ 2 \\rangle ^ { b ' } \\overline { \\widehat { v } } _ k , \\qquad \\widehat { z ^ \\ell } _ { k _ \\ell } : = \\langle k _ \\ell \\rangle ^ { s } \\langle \\tau _ \\ell + ( - 1 ) ^ { \\ell + 1 } k _ \\ell ^ 2 \\rangle ^ { b } \\widehat { u ^ \\ell } _ { k _ \\ell } , \\end{align*}"}
{"id": "1235.png", "formula": "\\begin{align*} \\det ( x I - B _ 1 ( T ) ) & = \\prod _ { j = 1 } ^ { l ( T ) } \\prod _ { v \\in V ( T , j ) } \\mathcal { F } ( v , x ) \\\\ & = \\prod _ { v \\in V ( T ) } \\mathcal { F } ( v , x ) . \\end{align*}"}
{"id": "8979.png", "formula": "\\begin{align*} F _ { q , r } ( u ) = E ( u + 1 ) - \\frac { n - 2 } { 2 q ( n - 1 ) } \\int _ M R ' | u + 1 | ^ q d V _ g - \\frac { n - 2 } { r } \\int _ { \\partial M } H ' | \\gamma ( u + 1 ) | ^ r d \\sigma _ g \\end{align*}"}
{"id": "5441.png", "formula": "\\begin{align*} \\frac { 1 } { p _ 0 } - \\frac { s _ 0 } { n } = \\frac { n + 2 s } { 2 n } , \\frac { 1 } { p _ 1 } - \\frac { s _ 1 } { n } = \\frac { 2 s } { n } , \\frac { 1 } { p _ 2 } - \\frac { s _ 2 } { n } = \\frac { n - 2 s } { 2 n } \\end{align*}"}
{"id": "3369.png", "formula": "\\begin{align*} a \\mapsto \\begin{pmatrix} a & 0 \\\\ 0 & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "679.png", "formula": "\\begin{align*} \\partial _ s \\left ( \\widehat \\psi _ { s * } X \\right ) = \\widehat \\psi _ { s * } [ X _ s , X ] = \\widehat \\psi _ { s * } [ X _ s - X , X ] \\end{align*}"}
{"id": "3228.png", "formula": "\\begin{align*} \\tilde { N } _ i ( E _ * ) = H ^ 0 ( \\textnormal { P E n d } _ { \\mathrm { S p } } ( E _ { * _ i } ) \\otimes K ( D ) ) . \\end{align*}"}
{"id": "4084.png", "formula": "\\begin{align*} \\{ \\mathbf { S } _ n = \\mathbf { s } ( j ) \\} = C ( n , j ) . \\end{align*}"}
{"id": "7629.png", "formula": "\\begin{align*} { \\rm K B S M } \\left ( L ( p , q ) \\right ) = \\frac { { \\rm K B S M } ( { \\rm S T } ) } { < a - b b m ( a ) > } , { \\rm w h e r e } \\ a \\ { \\rm b a s i s \\ e l e m e n t \\ o f \\ K B S M ( S T ) } . \\end{align*}"}
{"id": "8788.png", "formula": "\\begin{align*} H ( p , G ) = \\tilde { Y } _ O ( ( \\underbrace { p , p , \\cdots , p } _ { p ' / 2 } ) , G ) ^ { 2 / p ' } \\end{align*}"}
{"id": "688.png", "formula": "\\begin{align*} e ^ { f _ o ( \\cdot ) } v ( \\cdot , s _ 0 - s _ 0 ^ { \\theta / 4 } ) = e ^ { f _ o ( \\cdot ) } w ( \\cdot , s _ 0 - s _ 0 ^ { \\theta / 4 } ) . \\end{align*}"}
{"id": "8338.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\sum _ { j = 0 } ^ { \\infty } \\frac { 1 } { ( \\textrm { m a x } \\ , ( j , k ) + 1 ) ^ 2 } \\left | \\frac { \\mu ( j ) } { \\mu ( k ) } \\right | ^ 2 \\frac { w ( k ) } { w ' ( j ) } < \\infty , \\end{align*}"}
{"id": "2155.png", "formula": "\\begin{align*} \\mathcal { I } _ x ( f ( x ) ) = \\int _ \\mathbb { R } \\frac { 1 + s ^ 2 } { ( x - s ) ^ 2 + f ( x ) ^ 2 } \\ ; d \\rho ( s ) \\leq b , \\ ; \\ ; \\ ; \\ ; \\ ; x \\in \\mathbb { R } . \\end{align*}"}
{"id": "5448.png", "formula": "\\begin{align*} \\textsf { 1 } _ A \\stackrel { ( L 1 ) } = l ( \\textsf { 1 } _ M ) & = l ( m ^ { - 1 } m ) \\stackrel { ( L 3 ) } = l ( m ^ { - 1 } l ( m ) ) \\\\ & = l ( m ^ { - 1 } \\textsf { 1 } _ M ) = l ( m ^ { - 1 } ) , \\end{align*}"}
{"id": "5603.png", "formula": "\\begin{align*} x ^ 3 + y ^ 3 + n ^ 2 z ^ 3 = n x y z \\end{align*}"}
{"id": "6490.png", "formula": "\\begin{align*} \\int _ { c _ 1 } ^ { c _ 2 } \\abs { u } ^ 2 d x = \\int _ { c _ 1 } ^ { c _ 2 } \\abs { y } ^ 2 d x . \\end{align*}"}
{"id": "839.png", "formula": "\\begin{align*} \\lambda \\triangleq p \\left ( s _ { m } = 1 \\right ) = \\frac { \\rho _ { 0 , 1 } } { \\rho _ { 0 , 1 } + \\rho _ { 1 , 0 } } . \\end{align*}"}
{"id": "1511.png", "formula": "\\begin{align*} Y _ { \\lambda } ( x , p ) & = e ^ { x } \\phi _ { p , \\lambda } ( x ) + e ^ { x } \\int _ { 0 } ^ { x } \\phi _ { p , \\lambda } ( t ) d t \\\\ & = e ^ { x } \\phi _ { p , \\lambda } ( x ) + e ^ { x } \\sum _ { k = 0 } ^ { p } S _ { 2 , \\lambda } ( p , k ) \\int _ { 0 } ^ { x } t ^ { k } d t \\\\ & = e ^ { x } \\phi _ { p , \\lambda } ( x ) + e ^ { x } \\sum _ { k = 0 } ^ { p } S _ { 2 , \\lambda } ( p , k ) \\frac { x ^ { k + 1 } } { k + 1 } \\\\ & = e ^ { x } \\sum _ { k = 0 } ^ { p } \\bigg ( x { ^ k } + \\frac { x ^ { k + 1 } } { k + 1 } \\bigg ) S _ { 2 , \\lambda } ( p , k ) . \\end{align*}"}
{"id": "409.png", "formula": "\\begin{align*} G _ { \\mu } ( z ) = \\int _ { \\mathbb { R } } \\frac { d \\mu ( t ) } { z - t } , F _ { \\mu } ( z ) = \\frac { 1 } { G _ { \\mu } ( z ) } , z \\in \\mathbb { H } . \\end{align*}"}
{"id": "4484.png", "formula": "\\begin{align*} \\left | \\frac { x } t + p ' ( \\xi ) \\right | = | p ' ( \\xi ) - p ' ( \\xi _ 0 ) | \\gtrsim | p '' ( 2 ^ k ) | 2 ^ l \\gtrsim 2 ^ { k + l } ( 1 + 2 ^ { 2 k } ) ^ { - 5 / 2 } . \\end{align*}"}
{"id": "2870.png", "formula": "\\begin{align*} P ^ f _ H = \\{ ( x , 0 , p _ x , p _ y ) \\in { \\mathbb R } ^ 4 \\} \\equiv { \\mathbb R } ^ 3 \\end{align*}"}
{"id": "2232.png", "formula": "\\begin{align*} \\begin{array} { l l } \\left . \\partial _ s ^ 2 { \\rm L o g } \\left ( e ^ { \\int a _ 1 ( t , s ) d s } e ^ { \\int a _ 2 ( t , s ) d s } \\right ) \\right | _ { s = 0 } - \\left . \\partial _ s ^ 2 { \\rm L o g } \\left ( e ^ { \\int a _ 2 ( t , s ) d s } e ^ { - \\int a _ 1 ( t , s ) d s } \\right ) \\right | _ { s = 0 } = 0 \\end{array} \\end{align*}"}
{"id": "1301.png", "formula": "\\begin{align*} P ( T _ 0 , x ) & = \\dfrac { \\mathcal { G } _ 1 ( r ( T ) , x ) \\displaystyle \\prod _ { j = 1 } ^ { k } P ( T _ j , x ) ^ { \\alpha _ j } } { \\displaystyle \\prod _ { j = 1 } ^ { k } \\mathcal { G } _ 1 ( r ( T _ j ) , x ) } . \\end{align*}"}
{"id": "31.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { L , n } = T _ { L | K } ( \\lambda _ { \\rho } ( \\alpha ) \\psi _ { L , n } ( \\beta ) ) \\cdot _ { \\rho } ( v _ n ) . \\end{align*}"}
{"id": "421.png", "formula": "\\begin{align*} \\varphi _ { \\mu } ( z ) = c + \\int _ { \\mathbb { R } } \\frac { 1 + z x } { z - x } \\ , d \\sigma ( x ) , \\end{align*}"}
{"id": "4363.png", "formula": "\\begin{align*} q _ s = a _ s + a _ { s + 1 } g _ r + \\cdots + a _ m g _ r ^ { m - s } . \\end{align*}"}
{"id": "5997.png", "formula": "\\begin{align*} ( P _ h u , v ) _ { L ^ 2 ( \\Omega ) } = ( u , v ) _ { L ^ 2 ( \\Omega ) } v \\in V _ h . \\end{align*}"}
{"id": "7627.png", "formula": "\\begin{align*} P _ { \\vect { s ' } } ( f ( \\Delta _ \\Omega ^ { \\vect { s '' } } \\circ \\rho ) ) * I ^ { - \\vect { s '' } } _ \\Omega = c P _ { \\vect { s ' } - \\vect { s '' } } f \\end{align*}"}
{"id": "7821.png", "formula": "\\begin{align*} d \\Theta _ n \\circ A d _ \\phi = A d _ { \\Theta _ n ( \\phi ) } \\circ d \\Theta _ n \\end{align*}"}
{"id": "732.png", "formula": "\\begin{align*} \\frac { \\sum _ { i = 1 } ^ n a _ i } { \\sum _ { i = 1 } ^ n b _ i } \\le \\max _ { j = 1 , \\ldots , n } \\frac { a _ j } { b _ j } . \\end{align*}"}
{"id": "3221.png", "formula": "\\begin{align*} \\mathbb { P } ( H ^ 0 ( \\mathcal { M } , \\mathcal { L } ) ^ \\vee ) = \\mathbb { P } ( H ^ 0 ( \\mathcal { U } , \\mathcal { L } ) ^ \\vee ) \\cong \\mathbb { P } ( H ^ 0 ( \\mathcal { U } ' , \\mathcal { L } ' ) ^ \\vee ) = \\mathbb { P } ( H ^ 0 ( \\mathcal { M } ' , \\mathcal { L } ' ) ^ \\vee ) \\end{align*}"}
{"id": "7146.png", "formula": "\\begin{align*} ( g _ { 2 1 } y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 ) \\{ ( g _ { 1 1 } + g _ { 2 1 } ) y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 \\} ^ { b - 1 } \\\\ = \\alpha y _ 1 ^ a + \\beta y _ 2 ( y _ 1 + y _ 2 ) ^ { a - 1 } + g _ { 2 3 } ^ b y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { b - 1 } \\end{align*}"}
{"id": "3636.png", "formula": "\\begin{align*} \\norm { u \\wedge u ( \\cdot + \\delta e _ 2 ) } _ { L ^ 1 ( \\R ^ N ) } > m \\quad \\delta \\in ( 0 , \\delta _ 0 ) u \\in \\mathcal { M } ^ f _ { m + \\frac { \\eta } { 2 } } . \\end{align*}"}
{"id": "6639.png", "formula": "\\begin{align*} W ( y | x ) = \\frac { e ^ { - ( \\rho x + \\lambda ) } ( \\rho x + \\lambda ) ^ y } { y ! } \\ , . \\ , \\end{align*}"}
{"id": "5609.png", "formula": "\\begin{align*} 1 = \\left ( \\frac { - P _ 1 ^ 2 P _ 2 x z ^ 3 } { P _ 1 P _ 2 P _ 3 x z - y ^ 2 } \\right ) = ( - 1 ) ^ { ( P _ 1 P _ 2 P _ 3 x z - 1 ) / 2 } \\left ( \\frac { P _ 2 x z } { P _ 1 P _ 2 P _ 3 x z - y ^ 2 } \\right ) . \\end{align*}"}
{"id": "8344.png", "formula": "\\begin{align*} \\sum \\limits _ { i } \\alpha _ i = 0 \\end{align*}"}
{"id": "6373.png", "formula": "\\begin{align*} \\gamma _ 1 = ( 1 , 2 ) , \\ , \\gamma _ 2 = ( 1 , 3 ) , \\ldots , \\gamma _ n = ( 1 , n + 1 ) , \\ldots , \\end{align*}"}
{"id": "5371.png", "formula": "\\begin{align*} \\mathcal { P } _ { m , s } ( \\Omega ) \\vcentcolon = \\{ \\ , P = \\sum _ { | \\alpha | \\leq m } a _ \\alpha D ^ \\alpha \\ , ; \\ , \\norm { P } _ { m , s } < \\delta ( \\Omega ) , a _ { \\alpha } \\in M ( H ^ { s - | \\alpha | } \\rightarrow H ^ { - s } ) \\ , \\} \\end{align*}"}
{"id": "2825.png", "formula": "\\begin{align*} \\Delta _ { { \\mathcal { I } } } : = \\Big \\{ { \\alpha } \\in \\mathbb { R } ^ n : \\sum \\limits _ { i \\in \\mathcal { I } } \\alpha _ i = 1 , \\alpha _ i { } \\geq { } 0 , \\forall i \\in 0 , \\dots , n - 1 \\Big \\} . \\end{align*}"}
{"id": "6561.png", "formula": "\\begin{align*} \\frac { f ^ \\vee ( x + t y ) - f ^ \\vee ( x ) } { t } \\ ; = \\ ; g ^ \\vee ( t ) \\ ; \\to \\ ; g ^ \\vee ( 0 ) \\end{align*}"}
{"id": "725.png", "formula": "\\begin{align*} \\| \\nabla ^ 2 \\mathcal { H } \\| ^ 2 & = g ^ { i k } g ^ { j l } \\nabla ^ 2 \\mathcal { H } ( \\partial _ i , \\partial _ j ) \\nabla ^ 2 \\mathcal { H } ( \\partial _ k , \\partial _ l ) , \\\\ \\nabla ^ 2 \\mathcal { H } ( \\partial _ i , \\partial _ j ) & = \\partial _ i \\partial _ j \\mathcal { H } - \\nabla _ { \\nabla _ { \\partial _ i } \\partial _ j } \\mathcal { H } = \\mathcal { H } _ { i j } - \\Gamma _ { i j } ^ k \\mathcal { H } _ k . \\end{align*}"}
{"id": "527.png", "formula": "\\begin{align*} \\frac { 2 \\ddot { x } } { C _ 3 \\dot { x } ^ 2 } = F ( \\dot { x } , x , t ) \\ , \\end{align*}"}
{"id": "7849.png", "formula": "\\begin{align*} \\langle \\mu _ i ( x \\cdot \\phi _ 0 ( T ) - \\phi _ 0 ( T ) \\cdot x ) \\hat { a } , \\hat { b } \\rangle = \\mu _ i ( [ x , \\varphi _ 0 ] ) ( b ^ * T a ) \\to \\langle \\mu ( x \\cdot \\phi _ 0 ( T ) - \\phi _ 0 ( T ) \\cdot x ) \\hat { a } , \\hat { b } \\rangle . \\end{align*}"}
{"id": "8381.png", "formula": "\\begin{align*} F ( k , 2 , 4 ) & \\leq F ( 2 ^ m - 1 , 2 , 4 ) \\\\ & \\leq 2 ^ { 2 m - 1 } \\\\ & < 2 ^ { 2 m - 1 } + 2 ^ { m - 1 / 2 } + 2 \\\\ & \\leq k ^ 2 + k + 2 = 2 M ( k , 2 , 4 ) . \\end{align*}"}
{"id": "3738.png", "formula": "\\begin{align*} u = u \\cdot \\omega \\omega + \\sum _ { i = 1 , 2 , 3 } ( u \\cdot ( \\omega \\times \\mathbf { e } _ i ) ) ( \\omega \\times \\mathbf { e } _ i ) \\Longrightarrow B _ 3 = B \\cdot \\omega ( \\omega \\cdot \\mathbf { e } _ 3 ) - \\sum _ { i = 1 , 2 , 3 } ( B \\cdot ( \\omega \\times \\mathbf { e } _ i ) ) ( \\omega \\times \\mathbf { e } _ 3 ) \\cdot \\mathbf { e } _ i . \\end{align*}"}
{"id": "231.png", "formula": "\\begin{align*} \\Lambda _ i ^ { ( s ) } : = ( x ^ { ( p ^ s ) } ) ^ { - 1 } ( \\phi _ i ^ { ( s ) } ) ^ G ( x ) \\in \\textup { G L } _ N ( \\mathcal A ) \\end{align*}"}
{"id": "4461.png", "formula": "\\begin{align*} { } E ( - t ) = \\sum _ { n \\geq 0 } ( - 1 ) ^ n e _ n t ^ n = \\ , \\ , ( - \\sum _ { n \\geq 1 } \\frac { p _ n t ^ n } { n } ) = \\sum _ { \\lambda } ( - 1 ) ^ { ( 2 | \\lambda | - l ( \\lambda ) ) } \\frac { p _ { \\lambda } } { z _ { \\lambda } } t ^ { | \\lambda | } . \\end{align*}"}
{"id": "5822.png", "formula": "\\begin{align*} \\chi ( p _ n ) = 1 , \\chi ( p _ { n + 1 } ) = 1 , \\quad \\chi ( p _ { n + 2 } ) = - 1 . \\end{align*}"}
{"id": "7442.png", "formula": "\\begin{align*} \\eta ( \\cdot , s ) = s w ( x ) + \\int _ 0 ^ s f ^ 6 ( x , \\tau ) d \\tau . \\end{align*}"}
{"id": "3946.png", "formula": "\\begin{align*} { } | I _ 2 | & \\le \\frac { c } { T _ n } \\frac { 1 } { \\prod _ { l \\ge 3 } h _ l ^ * } \\int _ { h _ 1 ^ * h _ 2 ^ * \\sum _ { j = 1 } ^ d | \\log h _ j ^ * | } ^ { ( \\prod _ { j \\ge 3 } h _ j ^ * ) ^ { \\frac { 2 } { d - 2 } } } \\frac { 1 } { t \u2019 } d t \u2019 \\le \\frac { c } { T _ n } \\frac { \\sum _ { j = 1 } ^ 2 | \\log h _ j ^ * | } { \\prod _ { l \\ge 3 } h _ l ^ * } . \\end{align*}"}
{"id": "618.png", "formula": "\\begin{align*} f ( x , y , n , z ) ^ { - \\alpha } = ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { - \\alpha } \\sum _ { j = 0 } ^ \\infty { \\alpha + j - 1 \\choose j } \\Big ( \\sum _ { k = 0 } ^ \\infty d _ k \\frac { ( x ^ { 2 k + 4 } + y ^ { 2 k + 4 } ) } { ( x ^ 2 + y ^ 2 + z ^ 2 ) } n ^ { - 2 k - 2 } \\Big ) ^ { j } . \\end{align*}"}
{"id": "698.png", "formula": "\\begin{align*} \\Big \\| | t | ^ { - 1 } \\Phi ^ * _ { \\log | t | } g ^ \\lambda _ t - g _ o \\Big \\| _ { C ^ { k - 2 , \\gamma } _ { \\Phi ^ * _ { \\log | t | } g _ o } } & = | t | ^ { - 1 } \\Big \\| | t | \\Phi ^ * _ { - \\log | t | } ( | t | ^ { - 1 } \\Phi ^ * _ { \\log | t | } g ^ \\lambda _ t - g _ o ) \\Big \\| _ { C _ { g _ o } ^ { k - 2 , \\gamma } } \\\\ & \\le C ( A ) ( - \\log \\lambda ) ^ { - \\theta } \\quad t \\in I _ A . \\end{align*}"}
{"id": "1330.png", "formula": "\\begin{align*} F ( t , x , v ) = \\begin{cases} x / v , & \\mbox { f o r } x / v \\leq t , \\\\ t , & \\mbox { f o r } x / v > 0 , \\ , v > 0 \\\\ t , & \\mbox { f o r } v = 0 \\\\ t , & \\mbox { f o r } ( 1 - x ) / | v | > t , \\ , v < 0 \\\\ ( 1 - x ) / | v | & \\mbox { f o r } - ( 1 - x ) / v \\leq t . \\\\ \\end{cases} \\end{align*}"}
{"id": "8643.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n ( { \\widehat { x } } _ j { \\widehat { p } } _ j + { \\widehat { p } } _ j { \\widehat { x } } _ j ) = 2 \\sum _ { j = 1 } ^ n { \\widehat { x } } _ j { \\widehat { p } } _ j - { \\rm i } n \\widehat { I } , \\end{align*}"}
{"id": "2828.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { W } ( x _ i ) = f _ i \\nabla \\hat { W } ( x _ i ) = g _ i . \\end{aligned} \\end{align*}"}
{"id": "2343.png", "formula": "\\begin{align*} f = q Q ' + r \\mbox { w i t h } \\deg ( r ) < \\deg ( Q ' ) = \\deg ( Q _ 0 ) . \\end{align*}"}
{"id": "1761.png", "formula": "\\begin{align*} \\Omega ( \\alpha , A , z + a f _ k ) = \\{ z + a f _ k \\} \\cup \\bigcup _ { n = 1 } ^ \\infty \\Omega ( \\alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \\end{align*}"}
{"id": "7630.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\sigma _ { i , i + 1 } & : = & 1 + u \\ ( \\sigma _ i + \\sigma _ { i + 1 } ) + u ^ 2 \\ ( \\sigma _ i \\sigma _ { i + 1 } + \\sigma _ { i + 1 } \\sigma _ i ) + u ^ 3 \\ \\sigma _ i \\sigma _ { i + 1 } \\sigma _ i . \\\\ \\end{array} \\end{align*}"}
{"id": "7471.png", "formula": "\\begin{align*} T _ { q ^ { \\prime } } \\cdots T _ { p } : = \\begin{cases} I d & \\mathrm { i f } \\ , q ^ { \\prime } < p \\\\ T _ { q ^ { \\prime } } \\circ \\left ( T _ { q ^ { \\prime } - 1 } \\cdots T _ { p } \\right ) & \\mathrm { i f } \\ , q ^ { \\prime } \\ge p \\end{cases} . \\end{align*}"}
{"id": "1777.png", "formula": "\\begin{align*} g ( w ^ * ) = g ( w _ j ^ * ) = 0 . \\end{align*}"}
{"id": "3944.png", "formula": "\\begin{align*} \\tilde { I } _ 1 & \\le \\frac { c } { T _ n } \\frac { \\sum _ { j = 1 } ^ d | \\log h _ j ^ * | } { \\prod _ { l = 3 } ^ d h _ l ^ * } . \\end{align*}"}
{"id": "2959.png", "formula": "\\begin{align*} \\varphi ( z ) = 2 \\sqrt y \\sum _ { n \\neq 0 } a ( n ) K _ { i r } ( 2 \\pi | n | y ) e ( n x ) . \\end{align*}"}
{"id": "1547.png", "formula": "\\begin{align*} \\pi _ { m } = \\begin{cases} p ' _ { m } \\pi _ { m } + p ' _ { m + 1 } \\pi _ { m + 1 } , & \\\\ ( 1 - p ' _ { m - 1 } ) \\pi _ { m - 1 } + p ' _ { m + 1 } \\pi _ { m + 1 } , & \\\\ ( 1 - p ' _ { m - 1 } ) \\pi _ { m - 1 } + ( 1 - p ' _ { m } ) \\pi _ { m } , & \\end{cases} \\end{align*}"}
{"id": "5330.png", "formula": "\\begin{align*} E ^ 1 _ j = \\{ x _ { i _ j } \\in B \\} \\end{align*}"}
{"id": "7474.png", "formula": "\\begin{align*} \\cap _ { i = 1 } ^ { m } C _ { i } = \\cap _ { n = 0 } ^ { \\infty } \\mathrm { F i x } ( S _ { n } ) = \\cap _ { n = 0 } ^ { j _ { f } } \\mathrm { F i x } ( S _ { n } ) . \\end{align*}"}
{"id": "8122.png", "formula": "\\begin{align*} f ( x ) - f ( y ) & = \\biggl ( \\frac { x } { \\lvert x \\rvert } - \\frac { y } { \\lvert y \\rvert } \\biggr ) h \\biggl ( \\frac { \\lvert x \\rvert } { \\lVert x \\rVert } , \\lVert x \\rVert \\biggr ) + \\frac { y } { \\lvert y \\rvert } \\Biggl ( h \\biggl ( \\frac { \\lvert x \\rvert } { \\lVert x \\rVert } , \\lVert x \\rVert \\biggr ) - h \\biggl ( \\frac { \\lvert y \\rvert } { \\lVert y \\rVert } , \\lVert y \\rVert \\biggr ) \\Biggr ) , \\end{align*}"}
{"id": "8741.png", "formula": "\\begin{align*} \\partial _ t u ( t ) = A _ 1 ( t ) u ( t ) \\end{align*}"}
{"id": "4713.png", "formula": "\\begin{align*} B _ { k } ^ { * } = \\{ t _ { n - 1 } t _ { n - 2 } \\cdots t _ { 2 k } t _ { 2 k - 2 } t _ { 2 k - 4 } \\cdots t _ 2 \\ : | \\ : \\forall ~ j , ~ t _ j = 1 \\mbox { o r } t _ j = s _ { i _ j , j } \\mbox { f o r s o m e } 1 \\leq i _ j \\leq j \\} , \\end{align*}"}
{"id": "4749.png", "formula": "\\begin{align*} s _ b ^ { ( n + 1 ) } = \\frac { 1 } { 2 ^ g } \\sum _ { b _ 1 + b _ 2 = b } t _ { b _ 1 } ^ { ( n ) } t _ { b _ 2 } ^ { ( n ) } . \\end{align*}"}
{"id": "7510.png", "formula": "\\begin{align*} P ( E _ { x \\rightarrow y } | S _ { i } ) = { \\left ( \\frac { i } { m _ x } \\right ) } ^ { k _ x } , \\end{align*}"}
{"id": "2869.png", "formula": "\\begin{align*} H ( x , y , p _ x , p _ y , u _ 1 , u _ 2 ) = p _ x ( f ( x ) + u _ 1 ) + p _ y y - \\frac { 1 } { 2 } x ^ 2 - \\frac { 1 } { 2 } y ^ 2 - x u _ 1 - y u _ 2 - \\frac { 1 } { 2 } u _ 1 ^ 2 \\ , . \\end{align*}"}
{"id": "6395.png", "formula": "\\begin{align*} \\sum _ { \\substack { S \\subseteq \\{ 1 , \\ldots , k \\} \\ \\mathrm { w i t h } \\\\ \\rho \\wedge \\pi _ { { } _ S } = \\pi } } \\ ( - 1 ) ^ { | S | } \\ = \\ ( - 1 ) ^ { | \\pi | _ 1 / 2 } . \\end{align*}"}
{"id": "7580.png", "formula": "\\begin{align*} E _ \\mathcal { N } = \\max _ { k = 1 , \\ldots , \\mathcal { N } } | | x ( t _ k ) - x _ k | | . \\end{align*}"}
{"id": "3844.png", "formula": "\\begin{align*} b _ { k , n } ^ j = ( - 1 ) ^ { \\lfloor ( j + 1 ) / 2 \\rfloor } 2 ^ j ( \\kappa _ k + 1 / 2 ) _ { \\lfloor ( j + 1 ) / 2 \\rfloor } ( \\gamma _ { k } + n + k / 2 ) _ { \\lfloor j / 2 \\rfloor } . \\end{align*}"}
{"id": "4563.png", "formula": "\\begin{align*} R _ j : = \\{ v \\in F _ j : v F _ j F _ j \\setminus v \\subseteq F _ i 1 \\leq i < j \\} \\ , . \\end{align*}"}
{"id": "8653.png", "formula": "\\begin{align*} \\underline B : = \\sup _ { s > 0 } \\left ( \\int _ 0 ^ s V ( t ) ^ { - \\frac { 1 } { p - 1 } } d t \\right ) ^ { - 1 } \\int _ 0 ^ s W ( t ) \\left ( \\int _ 0 ^ t V ( t ' ) ^ { - \\frac { 1 } { p - 1 } } d t ' \\right ) ^ p d t \\ , . \\end{align*}"}
{"id": "6111.png", "formula": "\\begin{align*} W : = \\{ w \\in X ^ * \\mid \\abs { w } \\leq n , \\ , w \\in H \\} , \\end{align*}"}
{"id": "1718.png", "formula": "\\begin{align*} R _ { r _ n , \\beta ^ \\vee } ( t , q ^ { \\frac { 1 } { 2 } } ) : = B _ n ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) , \\ R _ { r _ n , \\delta ^ \\vee } ( t , q ^ { \\frac { 1 } { 2 } } ) : = D _ n ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) \\ . \\end{align*}"}
{"id": "7039.png", "formula": "\\begin{align*} A C ( r ) = \\frac { A } r \\int _ 0 ^ r e ^ { s A } \\ , d s = \\frac { e ^ { r A } - I } { r } \\to 0 = A P \\end{align*}"}
{"id": "3133.png", "formula": "\\begin{align*} | \\mathcal { T } _ \\ell | - | \\mathcal { T } _ 0 | \\le \\Lambda _ { \\mathrm { B D d V } } C _ c M ^ { 1 / s } \\sum _ { k = 0 } ^ { \\ell - 1 } \\eta _ k ^ { - 1 / s } . \\end{align*}"}
{"id": "6693.png", "formula": "\\begin{align*} \\overline { u } ( t , r ) = C _ 1 e ^ { \\gamma _ 0 t } [ h ( r ) - h ( \\delta _ 1 ) ] \\ , \\ 0 < r < \\delta _ 2 ( t ) , \\ \\ \\overline { v } ( t , r ) = C _ 2 e ^ { \\gamma _ 0 t } [ h ( r ) - h ( \\delta _ 2 ) ] , \\ 0 < r < \\delta _ 2 ( t ) , \\end{align*}"}
{"id": "2442.png", "formula": "\\begin{align*} L _ f = M _ f + \\sum _ { g \\prec f } \\ell _ { g f } ( q ) M _ g , \\ell _ { g f } ( q ) \\in q ^ { - 1 } \\Z [ q ^ { - 1 } ] . \\end{align*}"}
{"id": "5746.png", "formula": "\\begin{align*} \\pi _ i \\pi _ { [ a , b - 1 ] } y _ { b } = \\Big ( ( b - i ) \\pi _ { [ a - 1 , b - 1 ] } + ( i - a + 1 ) \\pi _ { [ a , b ] } \\Big ) y _ { b } \\end{align*}"}
{"id": "1242.png", "formula": "\\begin{align*} \\mathcal { F } ( v , x ) & = x - \\beta ( v ) - \\sum _ { w \\in c ( v ) } \\dfrac { 1 } { \\mathcal { F } ( w , x ) } \\\\ & = x - \\beta ( v _ 1 ) - \\dfrac { c _ j } { \\mathcal { F } ( v _ 2 , x ) } . \\end{align*}"}
{"id": "667.png", "formula": "\\begin{align*} \\left | \\left . \\int u _ { 1 , t } d \\nu ^ 2 _ { t } \\ , \\right | ^ { t = \\bar s } _ { t = s } - \\left . \\int u _ { 1 , t } v _ { 2 , t } \\ , d g _ { 1 , t } \\ , \\right | ^ { t = \\bar s } _ { t = s } \\right | & \\le C \\varepsilon \\end{align*}"}
{"id": "3429.png", "formula": "\\begin{align*} g \\equiv \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & \\sqrt { - 3 } x \\\\ 0 & \\sqrt { - 3 } y & 1 \\end{pmatrix} \\mod 3 . \\end{align*}"}
{"id": "512.png", "formula": "\\begin{align*} M ^ { \\leq 1 } & = \\mathrm { s p a n } ( u _ 1 ^ 1 , v _ 1 ^ 1 , w _ 1 ^ 1 ) , & M ^ { \\leq 2 } & = M , \\\\ M ^ 1 & \\cong M ^ { \\leq 1 } , & M ^ 2 & = M ^ { \\leq 2 } / M ^ { \\leq 1 } \\cong \\mathrm { s p a n } ( \\overline { v _ 1 ^ 2 + v _ 2 ^ 1 } ) . \\end{align*}"}
{"id": "1602.png", "formula": "\\begin{align*} \\mathcal { F } ( x , z ) = \\frac { 2 C ^ 3 } { E } . \\end{align*}"}
{"id": "2400.png", "formula": "\\begin{align*} & \\langle x , A _ n F _ n ( x ) \\rangle = \\langle x , A _ n ( x _ 1 ^ 3 , \\ldots , x _ { n - 1 } ^ 3 ) ^ \\top \\rangle - \\langle x , A _ n x \\rangle \\le - \\langle x , A _ n x \\rangle , \\end{align*}"}
{"id": "2902.png", "formula": "\\begin{align*} a _ 2 x _ 1 + ( a _ 1 + a _ 3 ) x _ 2 + \\cdots + ( a _ { k - 3 } + a _ { k - 2 } ) x _ { k - 2 } + a _ { k - 2 } x _ { k - 1 } = b + f ( a _ 1 , \\ldots , a _ k ) , \\end{align*}"}
{"id": "3285.png", "formula": "\\begin{align*} \\varphi _ 1 ( x , \\omega _ 1 ^ * ) = N _ { \\omega _ 1 ^ * } ^ { - 1 } ( \\omega _ 1 ^ * \\cdot A _ 1 ) , \\varphi _ 2 ( x , \\omega _ 1 ^ * ) = N _ { \\omega _ 2 ^ * } ^ { - 1 } ( - \\omega _ 2 ^ * \\cdot A _ 2 ) , \\end{align*}"}
{"id": "7796.png", "formula": "\\begin{align*} B _ { - \\lambda _ Y / r } \\left ( r \\alpha _ Y + \\lambda _ Y + \\frac { ( \\lambda _ Y , \\lambda _ Y ) } { 2 r } \\beta _ Y \\right ) = r \\alpha _ Y . \\end{align*}"}
{"id": "7001.png", "formula": "\\begin{align*} A = H _ { i _ 1 } \\cap \\ldots \\cap H _ { i _ n } \\end{align*}"}
{"id": "5058.png", "formula": "\\begin{align*} N ^ { n , 5 } _ \\tau : = \\frac 1 n \\sum _ { j = 0 } ^ { \\lfloor n \\tau \\rfloor } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\frac j n } ) \\int _ { 0 } ^ { 1 \\wedge ( n \\tau - j ) } \\gamma _ { \\frac { x + j } n } \\sum _ { k = j + 1 } ^ { \\infty } [ ( x + k ) ^ \\alpha - k ^ { \\alpha } ] ^ 2 d x = 0 . \\end{align*}"}
{"id": "905.png", "formula": "\\begin{align*} \\Psi ^ * _ t g _ { \\mathrm { c a n } } = g + \\frac { t } { 3 } ( \\frac { 1 } { 2 } \\mathrm { S c a l } _ g \\cdot g - \\mathrm { R i c } _ g ) + O ( t ^ 2 ) , \\end{align*}"}
{"id": "8042.png", "formula": "\\begin{align*} W = x _ 1 ^ { d / w _ 1 } + \\cdots + x _ n ^ { d / w _ N } , \\end{align*}"}
{"id": "164.png", "formula": "\\begin{align*} \\Lambda ^ * : = \\Big \\{ k \\in \\frac { 2 \\pi } { L } \\Z ^ d : \\forall i \\in \\{ 1 , \\dots , d \\} , \\ - \\frac { \\pi } { \\epsilon } < k _ i \\leq \\frac { \\pi } { \\epsilon } \\Big \\} , \\end{align*}"}
{"id": "2010.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m a r t i n g a l e p a r t i c l e s } M ^ { \\varphi _ N } _ t : = \\varphi _ N { \\left ( \\mathbf { X } ^ N _ t \\right ) } - \\varphi _ N { \\left ( \\mathbf { X } ^ N _ 0 \\right ) } - \\int _ 0 ^ t \\mathcal { L } _ N \\varphi _ N { \\left ( \\mathbf { X } ^ N _ s \\right ) } \\dd s , \\end{align*}"}
{"id": "6170.png", "formula": "\\begin{align*} \\psi ^ { \\ast } r _ { \\mathbf { S } _ { p } ^ { 1 } } ^ { 2 k } = | s | _ { h } ^ { 2 } . \\end{align*}"}
{"id": "3114.png", "formula": "\\begin{align*} \\alpha _ { j } = ( z _ { \\mathrm { n c } } , \\phi _ { j } ) _ { 1 + \\delta } = \\frac { 1 } { \\mu _ { j } } a _ { \\mathrm { p w } } ( z _ { \\mathrm { n c } } , \\phi _ { { j } } ) = \\frac { \\lambda } { \\mu _ { j } } ( u , \\phi _ { { j } } ) _ { 1 + \\delta } . \\end{align*}"}
{"id": "8478.png", "formula": "\\begin{align*} U & = ( n - k + 1 ) y _ k + \\frac { L ( n - k ) ( n - k + 1 ) } { 2 } \\\\ & = m y _ k + \\frac { L ( m - 1 ) m } { 2 } \\\\ & = m \\left ( y _ k + \\frac { L ( m - 1 ) } { 2 } \\right ) . \\end{align*}"}
{"id": "3404.png", "formula": "\\begin{align*} X \\begin{pmatrix} * & * & * \\\\ * & * & * \\\\ a & b & c \\end{pmatrix} = \\begin{cases} - a & \\\\ c & \\end{cases} \\end{align*}"}
{"id": "7306.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ 6 | g _ i | { \\bf 1 } _ { | \\xi | < 1 } + \\sum _ { i = 0 } ^ { 1 0 } | g _ i ' | { \\bf 1 } _ { | \\xi | < 1 } & \\leq { \\sf c } _ 0 ( T - t ) ^ { { \\sf c } _ 1 } \\lambda ^ { - \\frac { n + 2 } { 2 } } \\sigma ( 1 + | y | ^ 2 ) ^ { - ( 1 + { \\sf c } _ 2 ) } { \\bf 1 } _ { | \\xi | < 1 } . \\end{align*}"}
{"id": "3176.png", "formula": "\\begin{align*} \\Delta _ { A } \\left ( \\rho \\right ) = \\left \\Vert P _ { \\rho } \\pi _ { \\rho } \\left ( A \\right ) \\Omega _ { \\rho } \\right \\Vert _ { \\mathcal { H } _ { \\rho } } ^ { 2 } \\ , \\rho \\in E _ { 1 } \\ , \\end{align*}"}
{"id": "8096.png", "formula": "\\begin{align*} Z = f ( p \\partial _ { 1 } + q \\partial _ { 2 } ) . \\end{align*}"}
{"id": "5932.png", "formula": "\\begin{align*} \\langle { x ^ \\eta } \\left ( T \\right ) \\rangle = \\left \\langle { e ^ { \\eta \\mathop \\int \\limits _ 0 ^ T \\xi \\left ( t \\right ) d t } } \\right \\rangle = e ^ { W \\left [ { \\eta { { \\rm { I } } _ { \\left [ { 0 , 1 } \\right ] } } \\left ( t / T \\right ) } \\right ] } \\end{align*}"}
{"id": "6336.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } \\sin _ q z = \\cos _ q ( \\frac { z } { q } ) , D _ { q ^ { - 1 } } \\cos _ q z = - \\sin _ q ( \\frac { z } { q } ) . \\end{align*}"}
{"id": "1515.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( n ) _ { p , \\lambda } & \\bigg ( \\frac { 1 } { 1 - x } - \\sum _ { l = 0 } ^ { n } x ^ { l } \\bigg ) = \\sum _ { n = 1 } ^ { \\infty } ( n ) _ { p , \\lambda } \\frac { x ^ { n + 1 } } { 1 - x } \\\\ & = \\frac { x } { 1 - x } \\sum _ { n = 1 } ^ { \\infty } ( n ) _ { p , \\lambda } x ^ { n } = \\frac { x } { ( 1 - x ) ^ 2 } F _ { p , \\lambda } \\Big ( \\frac { x } { 1 - x } \\Big ) . \\end{align*}"}
{"id": "3313.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\tilde u + L \\tilde u = f \\end{align*}"}
{"id": "1239.png", "formula": "\\begin{align*} Q ( T , x ) & = \\prod _ { j = 1 } ^ { 8 } \\mathcal { H } ( u _ j , x ) \\\\ & = ( x - 1 ) ^ 5 \\ \\dfrac { x ^ 2 - 4 x + 1 } { x - 1 } \\ \\dfrac { x ^ 2 - 5 x + 1 } { x - 1 } \\ \\frac { x ^ 5 - 1 1 x ^ 4 + 3 8 x ^ 3 - 4 2 x ^ 2 + 8 x } { ( x ^ 2 - 4 x + 1 ) ( x ^ 2 - 5 x + 1 ) } \\\\ & = x ( x - 1 ) ^ 3 ( x ^ 4 - 1 1 x ^ 3 + 3 8 x ^ 2 - 4 2 x + 8 ) \\\\ & = x ( x - 1 ) ^ 3 ( x - 4 ) ( x ^ 3 - 7 x ^ 2 + 1 0 x - 2 ) . \\end{align*}"}
{"id": "1781.png", "formula": "\\begin{align*} y _ n ^ * = P _ n ^ * ( y ^ * ) - \\frac { 1 } { a } ( z + a f _ k ) ( P _ n ^ * ( y ^ * ) ) \\widetilde { x } ^ k _ n . \\end{align*}"}
{"id": "375.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } c _ i ^ 2 & = \\mathbb { P } _ { \\pi } [ T _ A > 0 ] = \\pi ( B ) c _ 1 ^ 2 / \\lambda _ 1 \\le \\mathbb { E } _ { \\pi } [ T _ A ] = \\sum _ { i = 1 } ^ { m } c _ i ^ 2 / \\lambda _ i \\le \\pi ( B ) / \\lambda _ 1 , \\\\ c _ 1 ^ 2 & = 1 - \\frac { \\| \\alpha / \\pi \\| _ 2 ^ 2 - 1 } { \\| \\alpha / \\pi \\| _ 2 ^ 2 } p : = \\sum _ { j = 2 } ^ m c _ j ^ 2 = \\pi ( B ) - c _ 1 ^ 2 = \\frac { \\| \\alpha / \\pi \\| _ 2 ^ 2 - 1 } { \\| \\alpha / \\pi \\| _ 2 ^ 2 } - \\pi ( A ) . \\end{align*}"}
{"id": "7498.png", "formula": "\\begin{align*} N ( t , t ) = t ^ { n + 1 } N ( 1 / t , 1 / t ) . \\end{align*}"}
{"id": "1572.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial z ^ i _ { \\epsilon } } = \\frac { 1 } { C } \\sum \\limits _ { i \\neq l } ^ { } \\left ( z ^ i _ 1 z ^ l _ 2 - z ^ i _ 2 z ^ l _ 1 \\right ) \\left ( \\delta _ { \\epsilon 1 } z ^ l _ 2 - \\delta _ { \\epsilon 2 } z ^ l _ 1 \\right ) . \\end{align*}"}
{"id": "1051.png", "formula": "\\begin{align*} \\theta ( t ) = \\int _ 0 ^ t \\phi ( t - s ) \\theta _ 0 ( s ) \\textnormal { d } s \\leq M _ \\alpha \\int _ 0 ^ t \\theta _ 0 ( s ) \\textnormal { d } s < \\infty , \\end{align*}"}
{"id": "2044.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : e m p i r i c a l m e a s u r e i n f i n i t e } \\mu _ { \\overline { \\mathcal { X } } { } ^ N } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\delta _ { \\overline { X } { } ^ i } . \\end{align*}"}
{"id": "1897.png", "formula": "\\begin{align*} \\mathcal { U } _ { [ n , j , k ] } : = \\{ \\gamma \\in \\mathcal { U } _ { [ n , j ] } : \\lambda ( \\gamma ) = k \\} , j \\leq k \\leq n - 1 . \\end{align*}"}
{"id": "2553.png", "formula": "\\begin{align*} \\dim Q ( p , q ) = \\dfrac { ( p + 1 ) ( q + 1 ) ( p + q + 2 ) } { 2 } \\ , \\end{align*}"}
{"id": "4914.png", "formula": "\\begin{align*} \\tau = \\begin{cases} t & \\alpha + \\delta + \\beta = 1 , \\\\ \\frac { \\log ( ( 1 - \\alpha - \\delta - \\beta ) \\eta t + 1 ) } { ( 1 - \\alpha - \\delta - \\beta ) \\eta } & \\alpha + \\delta + \\beta < 1 , \\\\ \\frac { \\log ( \\frac { T ^ * - t } { T ^ * } ) } { ( 1 - \\alpha - \\delta - \\beta ) \\eta } & \\alpha + \\delta + \\beta > 1 . \\end{cases} \\end{align*}"}
{"id": "6104.png", "formula": "\\begin{align*} 0 < \\beta : = \\inf _ { x \\in H \\setminus \\{ 0 \\} } \\sup _ { y \\in H \\setminus \\{ 0 \\} } \\frac { b ( x , y ) } { \\| x \\| _ { H } \\| y \\| _ H } = \\inf _ { y \\in H \\setminus \\{ 0 \\} } \\sup _ { x \\in H \\setminus \\{ 0 \\} } \\frac { b ( x , y ) } { \\| x \\| _ { H } \\| y \\| _ H } . \\end{align*}"}
{"id": "2930.png", "formula": "\\begin{align*} & & h _ i ^ 0 & = 0 , & \\forall i \\in [ m ] \\\\ & & h _ i ^ { j } & \\leq h _ i ^ { j + 1 } , & \\forall j \\in [ m - 1 ] , i \\in [ m ] \\\\ & & \\sum _ { i \\in [ m ] } h _ i ^ { j } & = j , & \\forall j \\in [ m ] \\\\ & & \\sum _ { k \\in [ j ] } h _ i ^ { k } & \\geq j - y _ i + 1 , & \\forall j \\in [ m ] , \\forall i \\in [ m ] \\end{align*}"}
{"id": "2751.png", "formula": "\\begin{align*} \\beta : = c _ { n , \\sigma p } \\left ( \\int _ { B _ { 4 } \\setminus B _ { 3 } } \\frac { \\phi ^ { p - 1 } ( y ) } { | y | ^ { n + \\sigma p } } d y + \\int _ { B _ { 6 } \\setminus B _ { 4 } } \\frac { d y } { | y | ^ { n + \\sigma p } } + \\int _ { B _ { 6 } ^ { c } } \\frac { ( 1 - \\psi ( y ) ) ^ { p - 1 } } { | y | ^ { n + \\sigma p } } d y \\right ) . \\end{align*}"}
{"id": "2022.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : p d m p f l o w } \\forall n \\in \\N , \\quad \\forall t \\in [ T ^ i _ { n } , T ^ i _ { n + 1 } ) , \\dd X ^ i _ t = a ( X ^ i _ t ) \\dd t , \\end{align*}"}
{"id": "5833.png", "formula": "\\begin{align*} r = \\frac { p _ 2 } { p _ 1 } , \\ , \\ , q = \\frac { q _ 2 } { p _ 1 } , \\ , \\ , p = \\frac { q _ 1 } { p _ 1 } \\end{align*}"}
{"id": "1656.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\mathbb { \\hat { E } } [ ( | X _ 1 | ^ 2 - \\lambda ) ^ + ] = 0 \\end{align*}"}
{"id": "6722.png", "formula": "\\begin{align*} \\prescript { } { n } { \\mathcal { A } } _ { m } \\ni \\varphi ( \\lambda ) = \\begin{cases} h ( \\lambda ) q _ { n , m } ( \\lambda ) , & m < n , \\\\ q _ { n , m } ( \\lambda ) h ( \\lambda ) , & m > n , \\\\ \\end{cases} \\ ; \\ ; \\ ; \\ ; \\lambda \\in \\mathfrak { a } ^ * _ \\mathbb { C } . \\end{align*}"}
{"id": "8458.png", "formula": "\\begin{align*} \\mathcal { H } ^ s _ { \\infty } ( E _ n ^ { ( 1 ) } ) \\ll \\begin{cases} \\left ( \\frac { \\delta } { 3 ^ n } \\right ) ^ s & \\tilde { \\delta } \\leq \\frac { 1 } { 3 ^ n } , \\\\ 3 ^ n \\cdot \\delta \\cdot \\left ( \\frac { \\delta } { 3 ^ n } \\right ) ^ s & \\tilde { \\delta } > \\frac { 1 } { 3 ^ n } . \\end{cases} \\end{align*}"}
{"id": "1136.png", "formula": "\\begin{align*} e ^ { - a x ^ 2 } = ( 4 \\pi a ) ^ { - 1 / 2 } \\int _ { - \\infty } ^ { \\infty } e ^ { - j q x } \\exp \\left \\{ - \\frac { q ^ 2 } { 4 a } \\right \\} \\mbox { d } q , ~ ~ ~ ~ ~ a > 0 , \\end{align*}"}
{"id": "4189.png", "formula": "\\begin{align*} M _ \\ell \\lesssim ( R / R _ \\ell ) ^ { d _ 1 } = 2 ^ { d _ 1 ( \\iota - \\ell ) } . \\end{align*}"}
{"id": "1610.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 E } { \\partial z ^ i _ { \\epsilon } \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } v ^ i = 2 b ^ 2 f ( x ^ 1 ) \\left [ 1 + 2 f '^ 2 ( x ^ 1 ) \\right ] . \\end{align*}"}
{"id": "4170.png", "formula": "\\begin{align*} F ( L , \\mathbf U ) f = f * \\mathcal K _ { F ( L , \\mathbf U ) } f \\in \\S ( G ) , \\end{align*}"}
{"id": "1834.png", "formula": "\\begin{align*} A _ { n } & = \\langle J ^ n e _ { 0 } , e _ { 0 } \\rangle , \\\\ A _ { n } ^ { ( 1 ) } & = \\langle J _ { 1 } ^ n e _ { 0 } , e _ { 0 } \\rangle , \\end{align*}"}
{"id": "7489.png", "formula": "\\begin{align*} G ( 1 ) = G ( 1 ) - G ( 0 ) = \\ln ( 2 ) - \\ln ( 1 ) = \\ln ( 2 ) , \\end{align*}"}
{"id": "8673.png", "formula": "\\begin{align*} I K = \\left \\{ \\sigma \\in \\Lambda ^ 2 X \\mid A ( \\sigma ) \\le 1 \\right \\} . \\end{align*}"}
{"id": "2625.png", "formula": "\\begin{align*} W _ A \\star W _ R = W _ { A R } \\ , \\end{align*}"}
{"id": "8431.png", "formula": "\\begin{align*} \\left \\| g ( j ) \\right \\| ^ 2 = \\sum \\limits _ { i \\in \\mathbb Z } { \\left | { g _ i ( j ) } \\right | ^ 2 } < \\infty \\left \\| h ( j ) \\right \\| ^ 2 = \\sum \\limits _ { i \\in \\mathbb Z } { \\sum \\limits _ { k \\in \\mathbb N } { \\left | { h _ { i , k } ( j ) } \\right | ^ 2 } } < \\infty . \\end{align*}"}
{"id": "3138.png", "formula": "\\begin{align*} \\mathrm { g } _ { \\theta } ( a _ { x , \\mathrm { s } } ) = \\mathrm { e } ^ { - i \\theta } a _ { x , \\mathrm { s } } \\ , x \\in \\mathbb { Z } ^ { d } , \\ \\mathrm { s } \\in \\mathrm { S } , \\end{align*}"}
{"id": "3631.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } u \\wedge u ^ t + \\int _ { \\R ^ N } u \\vee u ^ t = 2 \\int _ { \\R ^ N } u = 2 m . \\end{align*}"}
{"id": "6907.png", "formula": "\\begin{align*} \\mathcal { F } ^ { - 1 } _ { \\xi } [ \\hat { p } ( 0 , e ^ { - t } \\xi + \\mu ) ] & = \\mathcal { F } ^ { - 1 } _ { \\xi } [ \\hat { p } ( 0 , e ^ { - t } \\xi ) ] e ^ { - i \\mu e ^ t g } \\\\ & = e ^ t p ( 0 , e ^ t g ) e ^ { - i \\mu e ^ t g } . \\end{align*}"}
{"id": "6433.png", "formula": "\\begin{align*} \\Phi _ 0 ( x ) & : = [ Q , \\iota _ { \\nu _ 0 ( x ) } ] \\in \\mathfrak { X } _ { 0 } ( E ) [ 1 ] , \\ ; \\ ; x \\in \\mathfrak g [ 1 ] . \\\\ \\Phi _ 1 ( x , y ) & : = [ Q , \\iota _ { \\nu _ 1 ( x , y ) } ] ^ { ( - 1 ) } - \\sum _ { k \\geq 0 } [ [ Q , \\iota _ { \\nu _ 0 ( x ) } ] , \\iota _ { \\nu _ 0 ( y ) } ] ^ { ( k ) } \\in \\mathfrak { X } _ { - 1 } ( E ) [ 1 ] , \\ ; \\ ; x , y \\in \\mathfrak g [ 1 ] . \\end{align*}"}
{"id": "869.png", "formula": "\\begin{align*} v _ 0 * ^ { \\delta _ 1 } v _ 1 * ^ { \\delta _ 2 } \\cdots * ^ { \\delta _ m } v _ m = w _ 0 * ^ { \\delta _ 1 } w _ 1 * ^ { \\delta _ 2 } \\cdots * ^ { \\delta _ m } w _ m , \\end{align*}"}
{"id": "6647.png", "formula": "\\begin{align*} \\epsilon ' _ n = \\frac { P _ { \\ , } } { n ^ { 1 + b } } \\ , , \\ , \\end{align*}"}
{"id": "5719.png", "formula": "\\begin{align*} \\pi _ { [ a , a ] } = \\pi _ { [ a - 1 , a - 1 ] } + y _ { a } ( 1 \\le a \\le n - 1 ) \\end{align*}"}
{"id": "3978.png", "formula": "\\begin{align*} { \\cal G } _ t = \\big \\{ \\mbox { \\rm p r i m e $ p $ } \\ , \\big | \\ , q < p \\le t , ~ \\mu _ q ( p ) \\ge ( \\log _ q t ) ^ 2 \\big \\} ; \\\\ \\overline { \\cal G } _ t = \\big \\{ \\mbox { \\rm p r i m e $ p $ } \\ , \\big | \\ , q < p \\le t , ~ \\mu _ q ( p ) < ( \\log _ q t ) ^ 2 \\big \\} . \\end{align*}"}
{"id": "6667.png", "formula": "\\begin{align*} \\dfrac { \\partial i } { \\partial r } = \\dfrac { 1 } { 1 + \\xi _ 1 ' ( i ) ( h ( t ) - h _ 0 ) } : = \\sqrt { A _ 1 ( h ( t ) , i ) } , \\\\ - \\dfrac { 1 } { h ' ( t ) } \\dfrac { \\partial i } { \\partial t } = \\dfrac { \\xi _ 1 ( i ) } { 1 + \\xi _ 1 ' ( i ) ( h ( t ) - h _ 0 ) } : = C _ 1 ( h ( t ) , i ) , \\\\ \\dfrac { \\partial ^ 2 i } { \\partial r ^ 2 } = - \\dfrac { \\xi _ 1 '' ( i ) ( h ( t ) - h _ 0 ) } { [ 1 + \\xi _ 1 ' ( i ) ( h ( t ) - h _ 0 ) ] ^ 3 } : = B _ 1 ( h ( t ) , i ) , \\end{align*}"}
{"id": "4615.png", "formula": "\\begin{align*} g [ \\alpha ] ^ { - 1 } g ^ { - 1 } = [ ( a e ^ { - 1 } ) ( f ^ { - 1 } b ^ { - 1 } e ^ { - 1 } ) ( e a ^ { - 1 } ) ] \\end{align*}"}
{"id": "5745.png", "formula": "\\begin{align*} \\pi _ i \\pi _ { [ a - 1 , b - 1 ] } = ( b - i ) \\pi _ { [ a - 2 , b - 1 ] } + ( i - a + 2 ) \\pi _ { [ a - 1 , b ] } \\end{align*}"}
{"id": "3306.png", "formula": "\\begin{align*} V ( s ) \\colon & H ^ { - 1 / 2 } ( \\Gamma ) \\rightarrow H ^ { 1 / 2 } ( \\Gamma ) \\\\ K ( s ) \\colon & H ^ { 1 / 2 } ( \\Gamma ) \\rightarrow H ^ { 1 / 2 } ( \\Gamma ) \\\\ K ^ t ( s ) \\colon & H ^ { - 1 / 2 } ( \\Gamma ) \\rightarrow H ^ { - 1 / 2 } ( \\Gamma ) \\\\ W ( s ) \\colon & H ^ { 1 / 2 } ( \\Gamma ) \\rightarrow H ^ { - 1 / 2 } ( \\Gamma ) \\end{align*}"}
{"id": "2829.png", "formula": "\\begin{align*} P : = \\left [ \\begin{array} { l l l l l } g _ 0 & g _ 1 & \\dots & g _ N & x _ 0 \\end{array} \\right ] \\end{align*}"}
{"id": "4763.png", "formula": "\\begin{align*} \\mathcal K _ { \\kappa , 2 } = \\left \\{ \\left [ n _ 0 , n _ 1 , n _ 2 , n _ 3 \\right ] \\colon n _ i \\geq 0 , \\ ; \\sum _ { i = 0 } ^ { 3 } n _ i = \\kappa , \\ ; n _ 1 \\leq n _ 2 \\right \\} , \\end{align*}"}
{"id": "6750.png", "formula": "\\begin{align*} 2 K _ X & \\equiv f ^ * \\left ( 2 K _ Y + \\sum \\limits _ { \\sigma \\ne 0 } { D _ { \\sigma } } \\right ) ; \\\\ f _ { * } \\mathcal { O } _ X & = \\mathcal { O } _ Y \\oplus \\bigoplus \\limits _ { \\chi \\ne \\chi _ { 0 0 0 } } L _ { \\chi } ^ { - 1 } . \\end{align*}"}
{"id": "2476.png", "formula": "\\begin{align*} \\partial \\varphi \\left ( x \\right ) = x \\varphi \\left ( x \\right ) \\end{align*}"}
{"id": "481.png", "formula": "\\begin{align*} N \\leq \\left ( \\frac { 1 + \\frac { 1 } { 2 } \\sin \\theta } { \\frac { 1 } { 2 } \\sin \\theta } \\right ) ^ \\rho = ( 1 + 2 / \\sin \\theta ) ^ \\rho \\leq ( 1 + \\sqrt { 8 c } ) ^ \\rho . \\end{align*}"}
{"id": "4445.png", "formula": "\\begin{align*} \\Delta ^ m ( ^ s f ) = 2 ^ m \\delta _ e ^ m R f , \\end{align*}"}
{"id": "2988.png", "formula": "\\begin{align*} 1 2 \\pi ^ { 1 / 2 } | d | ^ { \\frac 3 2 } | b _ { \\psi } ( d ) | ^ 2 = \\frac { 1 } { \\langle \\varphi , \\varphi \\rangle } \\sum _ { Q \\in \\Gamma \\backslash \\mathcal Q _ { d ^ 2 } } \\chi _ d ( Q ) \\begin{dcases} \\int _ { C _ Q } \\varphi ( z ) y ^ { - 1 } \\ , | d z | & d > 0 , \\\\ \\int _ { C _ Q } i \\partial _ z \\varphi ( z ) \\ , d z & d < 0 . \\end{dcases} \\end{align*}"}
{"id": "7283.png", "formula": "\\begin{align*} ( \\Delta - f _ 2 ' ( { \\sf U } _ \\infty ) ) \\theta _ k & + \\sum _ { i = 1 } ^ N \\tfrac { f ^ { ( i ) } ( { \\sf U } _ \\infty ) } { i ! } ( ( \\theta _ 0 + \\cdots + \\theta _ { k - 1 } ) ^ i - ( \\theta _ 0 + \\cdots + \\theta _ { k - 2 } ) ^ i ) \\\\ & - \\sum _ { i = 2 } ^ N \\tfrac { f _ 2 ^ { ( i ) } ( { \\sf U } _ \\infty ) } { i ! } ( ( \\theta _ 0 + \\cdots + \\theta _ { k - 1 } ) ^ i - ( \\theta _ 0 + \\cdots + \\theta _ { k - 2 } ) ^ i ) = 0 . \\end{align*}"}
{"id": "7862.png", "formula": "\\begin{align*} [ \\Phi ( T ) , J x _ 1 J ] = \\lim _ i [ A _ i , J x _ 1 J ] \\otimes T = 0 , \\end{align*}"}
{"id": "2993.png", "formula": "\\begin{align*} \\sum _ { Q \\in \\Gamma \\backslash \\mathcal Q _ D } \\chi _ d ( Q ) \\int _ { C _ Q } i \\partial _ z \\varphi ( z ) y ^ { s } \\ , d z = \\pi ^ { - s - \\frac 1 2 } \\sqrt { | d | } \\ , \\Gamma ( \\tfrac s 2 + \\tfrac { i r } 2 + \\tfrac 3 4 ) \\Gamma ( \\tfrac s 2 - \\tfrac { i r } 2 + \\tfrac 3 4 ) L ( s + \\tfrac 1 2 , \\varphi \\times \\chi _ d ) . \\end{align*}"}
{"id": "2036.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m a x w e l l m o l e c u l e s c u t o f f } \\Phi ( | u | ) = 1 , \\int _ 0 ^ \\pi \\Sigma ( \\theta ) \\dd \\theta < + \\infty . \\end{align*}"}
{"id": "4577.png", "formula": "\\begin{align*} F _ * ^ e ( c ) \\notin ( J F _ * ^ e S ) : ( F _ * ^ e ( I ^ { [ p ^ e ] } : I ) ) = F _ * ^ e ( J ^ { [ p ^ e ] } ) : F _ * ^ e ( I ^ { [ p ^ e ] } : I ) . \\end{align*}"}
{"id": "5967.png", "formula": "\\begin{align*} g _ { \\beta , T } ( t ) = \\frac 1 { \\Gamma ( 1 - \\beta ) } \\left ( ( T - t ) ^ { - \\beta } + t ^ { - \\beta } \\right ) . \\end{align*}"}
{"id": "7271.png", "formula": "\\begin{align*} u ( x , t ) & = \\pm \\eta ( t ) ^ \\frac { 2 } { 1 - q } { \\sf U } ( \\xi ) + o ( \\eta ( t ) ^ \\frac { 2 } { 1 - q } ) | x | \\sim \\eta ( t ) , \\\\ u ( x , t ) & = \\pm { \\sf U } _ \\infty ( x ) + \\Theta _ J ( x , t ) + o ( \\Theta _ J ( x , t ) ) | x | \\sim \\sqrt { T - t } ( J \\in \\N ) . \\end{align*}"}
{"id": "154.png", "formula": "\\begin{align*} \\forall t > 0 , \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } \\leq c ( \\mu ) \\lambda , \\lim _ { \\mu \\rightarrow \\infty } c ( \\mu ) = 0 . \\end{align*}"}
{"id": "7468.png", "formula": "\\begin{align*} f \\left ( \\left \\{ 1 , \\dots , j _ { f } \\right \\} \\right ) = \\left \\{ 1 , \\dots , m \\right \\} . \\end{align*}"}
{"id": "2047.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : O m e g a N } \\Omega _ N \\left ( f ^ N , f \\right ) : = W _ p { \\left ( f ^ { N } , f ^ { \\otimes N } \\right ) } \\underset { N \\to + \\infty } { \\longrightarrow } 0 . \\end{align*}"}
{"id": "3693.png", "formula": "\\begin{align*} \\| h \\partial _ \\nu u _ { h } \\| _ { L ^ 2 ( \\Gamma ) } = o ( 1 ) . \\end{align*}"}
{"id": "329.png", "formula": "\\begin{align*} f ( A ) = \\sum _ p f ( A _ p ) & = \\sum _ { \\substack { p < x \\\\ p \\notin A } } f ( A _ p ) + \\sum _ { p \\in A , p \\ge x } f ( p ) \\\\ & \\le e ^ \\gamma \\sum _ { \\substack { p < x \\\\ p \\notin A } } { \\rm d } ( { \\rm L } _ p ) + \\sum _ { p \\in A , p \\ge x } f ( p ) \\\\ & \\le e ^ \\gamma \\sum _ { p } { \\rm d } ( { \\rm L } _ p ) + \\big ( e ^ \\gamma + o _ x ( 1 ) \\big ) \\sum _ { p \\ge x } { \\rm d } ( { \\rm L } _ p ) \\ \\le \\ e ^ \\gamma + o _ x ( 1 ) \\end{align*}"}
{"id": "8515.png", "formula": "\\begin{align*} \\epsilon _ p = \\frac { 2 p + 1 } { 2 } \\log \\left ( \\frac { p + 1 } { p } \\right ) - 1 = \\frac { 1 } { 2 y } \\log \\left ( \\frac { 1 + y } { 1 - y } \\right ) - 1 , \\end{align*}"}
{"id": "1324.png", "formula": "\\begin{align*} S _ \\tau f _ 0 ( z ) : = f _ \\tau ( z ) \\geq \\eta ( z ) \\int _ { \\{ \\tilde \\tau ( z ' ) \\leq R \\} } f _ 0 ( z ' ) \\d z ' . \\end{align*}"}
{"id": "6132.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\end{gather*}"}
{"id": "3530.png", "formula": "\\begin{align*} c _ { ( 1 ) } ^ { \\ast } = \\frac { 2 ( t - 1 ) } { B ( \\frac { 1 } { 2 } , t - \\frac { 3 } { 2 } ) } , ~ t > \\frac { 3 } { 2 } \\end{align*}"}
{"id": "3670.png", "formula": "\\begin{align*} \\mathfrak { M } ( \\xi , v , V ( s ) ) = \\frac { \\mathbf { e } _ 3 \\langle V ( s ) \\rangle } { | \\xi | ( | \\xi | + \\hat { V } ( s ) \\cdot \\xi ) } - \\frac { V _ 3 ( s ) } { | \\xi | ( | \\xi | + \\hat { V } ( s ) \\cdot \\xi ) ^ 2 } \\big ( \\xi - \\hat { V } ( s ) \\hat { V } ( s ) \\cdot \\xi \\big ) . \\end{align*}"}
{"id": "5530.png", "formula": "\\begin{align*} \\bar { \\psi } _ \\theta ( z ) = \\bar { f } _ \\theta ( \\gamma ) ( \\gamma ^ { - 1 } z ) , \\end{align*}"}
{"id": "6519.png", "formula": "\\begin{align*} \\mu ( ( b , \\infty ) ) \\geq ( \\ln b ) ^ { - a } . \\end{align*}"}
{"id": "1518.png", "formula": "\\begin{align*} D _ { r } F _ { n , \\lambda } ( x ) & = \\frac { 1 } { r ! } \\bigg ( \\frac { d } { d x } \\bigg ) ^ { r } \\bigg [ \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) k ! x ^ { k + r } \\bigg ] \\\\ & = \\frac { 1 } { r ! } \\bigg [ \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) k ! ( k + r ) _ { r } x ^ { k } \\bigg ] \\\\ & = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) \\binom { k + r } { k } k ! x ^ { k } . \\end{align*}"}
{"id": "1126.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { i = 0 } ^ { k - 1 } \\frac { \\lambda ^ 2 A ^ 2 [ ( 2 i + 1 ) / k - 1 ] ^ 2 } { [ ( \\sigma _ N ^ 2 + \\sigma _ Z ^ 2 ) \\lambda ^ 2 + \\rho ] ^ 2 } = P _ w . \\end{align*}"}
{"id": "358.png", "formula": "\\begin{align*} V ( r ) \\leq n \\frac { 3 r } { D } \\leq n \\frac { 3 D ^ { 1 / 2 } } { D } = 3 n \\frac { 1 } { D ^ { 1 / 2 } } \\leq \\frac { 3 n } { n ^ { 1 / 1 0 } } = 3 n ^ { 9 / 1 0 } . \\end{align*}"}
{"id": "1320.png", "formula": "\\begin{align*} X _ t ( x _ 0 , v _ 0 ) = x _ 0 + v _ 0 t + \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { s } \\nabla \\Phi ( X _ r ( x _ 0 , v _ 0 ) ) \\d r \\d s , \\end{align*}"}
{"id": "271.png", "formula": "\\begin{align*} ( F ^ { 1 } _ i ) _ { j k } = \\frac { 1 } { 2 } ( L _ { k i j } ( 1 ) + L _ { i j k } ( 1 ) - L _ { j k i } ( 1 ) ) . \\end{align*}"}
{"id": "625.png", "formula": "\\begin{align*} & F _ { 0 , 0 } ( x , y ) = \\widetilde { F } _ { 0 , 0 } ( x , y ) = 1 , \\\\ & F _ { 1 , 0 } ( x , y ) = 0 , F _ { 1 , 1 } ( x , y ) = \\frac { 2 } { 3 } \\pi ^ 2 \\Bigl ( x ^ 4 + y ^ 4 \\Bigr ) , \\\\ & \\widetilde { F } _ { 1 , 0 } ( x , y ) = 0 , \\widetilde { F } _ { 1 , 1 } ( x , y ) = \\frac { 2 } { 3 } \\pi ^ 2 \\Bigl ( x ^ 2 + y ^ 2 \\Bigr ) ^ 2 . \\end{align*}"}
{"id": "3593.png", "formula": "\\begin{align*} \\mu \\left ( \\bigcup \\limits _ { i = 1 } ^ { n } R _ i \\right ) = \\sum \\limits _ { J \\subseteq \\{ 1 , \\ldots , n \\} , J \\ne \\varnothing } ( - 1 ) ^ { | J | + 1 } \\mu \\left ( \\bigcap \\limits _ { j \\in J } R _ i \\right ) . \\end{align*}"}
{"id": "2502.png", "formula": "\\begin{align*} \\Delta f = d i v ( \\nabla f ) . \\end{align*}"}
{"id": "117.png", "formula": "\\begin{align*} ( { \\bf 1 } ^ \\epsilon _ 0 f ) ( x ) : = \\epsilon ^ { - d } { \\bf 1 } _ { x = 0 } f ( 0 ) . \\end{align*}"}
{"id": "5817.png", "formula": "\\begin{align*} p _ n = 8 n _ 1 + 1 , p _ { n + 1 } = 8 n _ 2 + 5 , p _ { n + 2 } = 8 n _ 3 + 7 \\end{align*}"}
{"id": "3365.png", "formula": "\\begin{align*} \\varphi _ n ( f ) ( t ) = ( u ( t ) \\otimes I _ { 2 ^ n } ) \\begin{pmatrix} f ( e ^ { \\pi i t } ) & 0 \\\\ 0 & f ( e ^ { \\pi i ( 1 + t ) } ) \\end{pmatrix} ( u ( t ) \\otimes I _ { 2 ^ n } ) ^ { \\ast } \\end{align*}"}
{"id": "5589.png", "formula": "\\begin{align*} b _ { i , j } = b _ { j , i } \\geq 0 \\ \\hbox { f o r a n y } \\ i , j = 1 , \\cdots , n ; b _ { i , i } = - \\sum _ { j = 1 , j \\neq i } ^ { } b _ { i , j } . \\end{align*}"}
{"id": "8247.png", "formula": "\\begin{align*} \\phi _ { \\pm } = A _ { \\pm } e ^ { k _ { \\pm } x } + B _ { \\pm } e ^ { - k _ { \\pm } x } + C _ { \\pm } e ^ { k ' _ { \\pm } x } + D _ { \\pm } e ^ { - k ' _ { \\pm } x } . \\end{align*}"}
{"id": "6768.png", "formula": "\\begin{align*} p ( n ) = \\frac { 1 } { 4 \\sqrt { 3 } n \\cdot \\kappa _ n } \\left ( 1 - \\frac { 1 } { x \\sqrt { n \\cdot \\kappa _ n } } \\right ) e ^ { x \\sqrt { n \\cdot \\kappa _ n } } \\left ( 1 + O ( e ^ { - x \\sqrt { n } / 2 } ) \\right ) \\end{align*}"}
{"id": "2389.png", "formula": "\\begin{align*} \\big ( E ( L ) \\big ) ( \\nu ) = \\frac { \\overline { \\eta } ( ( 1 , 0 ) L ^ { - 1 } ) } { | \\det ( L ) | } \\ , g ( \\nu ) , L \\in K _ 0 \\nu \\in \\R ^ * . \\end{align*}"}
{"id": "4882.png", "formula": "\\begin{align*} J = \\sum \\limits _ { 1 \\leq m _ { 1 } , \\ldots m _ { 4 } \\leq L } \\prod \\limits _ { i = 1 } ^ { 2 } r _ { i } ( 2 ^ { m _ { 1 } } + 2 ^ { m _ { 2 } } - 2 ^ { m _ { 3 } } - 2 ^ { m _ { 4 } } ) , \\end{align*}"}
{"id": "5971.png", "formula": "\\begin{align*} \\frac { ( 1 - \\epsilon _ 1 - \\epsilon _ 2 ) ( T / 2 ) ^ { - \\beta } } { \\Gamma ( 1 - \\beta ) } \\int _ 0 ^ T \\| u ( t ) \\| ^ 2 \\ , d t + \\int _ 0 ^ T | u ( t ) | _ 1 \\ , d t \\leq & \\frac 1 2 \\epsilon _ 1 ^ { - 1 } \\int _ 0 ^ T ( g _ { \\beta , T } ( t ) ) ^ { - 1 } \\| f ( t ) \\| ^ 2 \\ , d t \\\\ & + \\frac 1 2 \\epsilon _ 2 ^ { - 1 } \\frac { T ^ { 1 - \\beta } } { \\Gamma ( 2 - \\beta ) } \\| u ^ 0 \\| ^ 2 . \\end{align*}"}
{"id": "854.png", "formula": "\\begin{align*} \\nu _ { x _ { m } ^ { r } \\rightarrow f _ { m } ^ { r } } ( x _ { m } ^ { r } ) = \\mathcal { C N } ( x _ { m } ^ { r } ; x _ { B , m } ^ { r , p r i } , v _ { B , m } ^ { r , p r i } ) . \\end{align*}"}
{"id": "1791.png", "formula": "\\begin{align*} z _ { n _ k } ^ * \\overset { w ^ * } { \\underset { k \\rightarrow \\infty } { \\longrightarrow } } y ^ * - \\sum _ { m = 1 } ^ \\infty c _ m \\widetilde { u } _ m = : z ^ * . \\end{align*}"}
{"id": "7546.png", "formula": "\\begin{align*} | e _ k | = \\frac { \\Re \\{ e _ k \\} } { \\cos ( \\alpha _ k ^ * ) } = \\frac { \\Im \\{ e _ k \\} } { \\sin ( \\alpha _ k ^ * ) } \\end{align*}"}
{"id": "2508.png", "formula": "\\begin{align*} g ( A _ { X _ j } U , A _ { X _ j } V ) = \\sum \\limits _ { j = 1 } ^ { n } g ( A _ { X _ j } X _ j , U ) g ( A _ { X _ j } X _ j , V ) . \\end{align*}"}
{"id": "8561.png", "formula": "\\begin{align*} C ( \\lambda , 0 ) = \\int F ( \\lambda f ) \\ p ( x ) \\ \\mu ( d x ) \\leq 2 \\ , - 1 \\leq \\lambda \\leq 1 \\ . \\end{align*}"}
{"id": "8571.png", "formula": "\\begin{align*} \\{ e ^ { \\pm i n _ k x } : \\ , k = 1 , 2 , \\ldots \\} \\end{align*}"}
{"id": "5532.png", "formula": "\\begin{align*} \\int _ \\Omega \\abs { F ' } ^ p d x d y = \\int _ \\mathbb { D } \\abs { f ' } ^ { 2 - p } d x d y . \\end{align*}"}
{"id": "7071.png", "formula": "\\begin{align*} \\mathcal { S } _ 7 = \\{ W _ 2 = W _ 3 = W _ 4 = W _ 5 = W _ 6 = W _ 7 = 0 \\} , \\end{align*}"}
{"id": "5750.png", "formula": "\\begin{align*} & y _ i = \\pi _ i - \\pi _ { i - 1 } ( 1 \\le i \\le n - 1 ) , \\\\ & y _ n = - \\pi _ { n - 1 } \\end{align*}"}
{"id": "6065.png", "formula": "\\begin{align*} f ^ { ( j ) } ( \\lambda ) & = 0 \\quad \\qquad \\qquad \\\\ f ^ { ( p ) } ( \\lambda ) & \\neq 0 \\\\ f ^ { ( j ) } ( \\lambda ) & = ( j ) _ p g ^ { ( j - p ) } ( \\lambda ) . \\end{align*}"}
{"id": "6304.png", "formula": "\\begin{align*} S _ q ( x ) : = \\frac { 1 } { q } D _ { q ^ { - 1 } } u ( x ) + \\frac { 1 } { q } u ( x ) u ( x / q ) + A ( x ) u ( x / q ) + r ( x ) . \\end{align*}"}
{"id": "3876.png", "formula": "\\begin{align*} g \\in L ^ \\infty ( \\Omega ) \\cap W ^ { 1 , p } ( \\Omega ) \\hbox { $ p - $ h a r m o n i c i n } \\Omega , \\ g \\hbox { n o n - c o n s t a n t u n l e s s } g = 0 . \\end{align*}"}
{"id": "4857.png", "formula": "\\begin{align*} N = p _ { 1 } ^ { 3 } + p _ { 2 } ^ { 3 } + \\cdots + p _ { 8 } ^ { 3 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k _ { 2 } } } . \\end{align*}"}
{"id": "5537.png", "formula": "\\begin{align*} \\hat { B } ( z ) = \\lambda ^ B \\frac { z - a ^ B } { 1 - \\overline { a ^ B } z } . \\end{align*}"}
{"id": "3822.png", "formula": "\\begin{align*} { } _ { 1 } ^ \\kappa E l l H ^ { \\mu , 4 ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) : = { } _ { 1 ; 0 } ^ { \\kappa } E l l H ^ { \\mu , 4 ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) + { } _ { 1 ; 1 } ^ { \\kappa } E l l H ^ { \\mu , 4 ; l } _ { k , j , n } ( t _ 1 , t _ 2 ) + { } _ { 1 ; 2 } ^ { \\kappa } E l l H ^ { \\mu , 4 ; l } _ { k , j , n } ( t _ 1 , t _ 2 ) , \\end{align*}"}
{"id": "5362.png", "formula": "\\begin{align*} s _ { \\Tilde { \\theta } } = ( 1 - \\Tilde { \\theta } ) s _ 0 + \\Tilde { \\theta } s _ 1 = \\frac { r - s } { r - z } z + \\frac { s - z } { r - z } r = s . \\end{align*}"}
{"id": "5199.png", "formula": "\\begin{align*} k _ 1 ( \\Gamma ) = r ( J ) + ( p - 1 ) r + 1 k _ 2 ( \\Gamma ) = s ( J ) + ( \\ell _ 1 + \\ell _ 2 - | J | - p + 1 + \\sum _ { j \\in J } d _ j ) s + 1 , \\end{align*}"}
{"id": "4542.png", "formula": "\\begin{align*} \\Vert J _ { 2 2 } ^ { ( 1 ) } \\Vert _ { p } = \\mathcal { O } \\left ( ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) \\right ) . \\end{align*}"}
{"id": "1062.png", "formula": "\\begin{align*} - \\omega _ 0 ( A _ j ) = \\inf \\Re \\left ( \\operatorname { S p e c } ( A _ j ) \\right ) . \\end{align*}"}
{"id": "1707.png", "formula": "\\begin{align*} \\begin{aligned} & \\widehat { u ^ { 1 } _ 0 } = \\frac { \\ln N } { 2 \\sqrt { N } } \\sum _ { N \\le k \\le ( 1 + \\delta ) N } 2 ^ { \\frac { k } { 2 } } ( i \\phi ( \\xi + 2 ^ { k } e _ 1 ) - i \\phi ( \\xi - 2 ^ { k } e _ 1 ) ) , \\\\ & \\widehat { u ^ { 2 } _ 0 } = \\frac { \\ln N } { 2 \\sqrt { N } } \\sum _ { N \\le k \\le ( 1 + \\delta ) N } 2 ^ { \\frac { k } { 2 } } ( \\phi ( \\xi + 2 ^ { k } e _ 1 ) + \\phi ( \\xi - 2 ^ { k } e _ 1 ) ) . \\end{aligned} \\end{align*}"}
{"id": "517.png", "formula": "\\begin{align*} L _ { n } ( \\dot { x } , x ) = \\frac { a _ 1 \\dot x } { a _ 2 x + a _ 4 } , \\end{align*}"}
{"id": "541.png", "formula": "\\begin{align*} S _ 1 = \\frac { 1 } { d _ 1 \\cdots d _ l } \\sum \\limits _ { r _ j \\mid ( u _ j / d _ j ) , 1 \\leq j \\leq l } \\frac { \\mu ( r _ 1 ) \\cdots \\mu ( r _ l ) } { r _ 1 \\cdots r _ l } + O _ { \\alpha , l } \\Big ( \\frac { 1 } { \\sqrt { n } } \\prod \\limits _ { 1 \\leq j \\leq l } \\tau _ 2 ( u _ j / d _ j ) \\Big ) . \\end{align*}"}
{"id": "4187.png", "formula": "\\begin{align*} \\chi _ { B _ { 3 R } ^ { d _ { \\mathrm { C C } } } ( 0 ) } ( F ^ { ( \\iota ) } \\psi ) ( \\sqrt L ) f & = \\chi _ { B _ { 3 R } ^ { d _ { \\mathrm { C C } } } ( 0 ) } \\bigg ( \\sum _ { \\ell = - 1 } ^ \\iota + \\sum _ { \\ell = \\iota + 1 } ^ \\infty \\bigg ) F _ \\ell ^ { ( \\iota ) } ( L , U ) f \\\\ & = : g _ { \\le \\iota } + g _ { > \\iota } . \\end{align*}"}
{"id": "230.png", "formula": "\\begin{align*} u & = \\frac 1 2 \\ln \\left \\{ \\left [ ( x ^ 1 ) ^ 2 + ( x ^ 3 ) ^ 2 \\right ] \\left [ ( x ^ 2 ) ^ 2 + ( x ^ 4 ) ^ 2 \\right ] \\right \\} , \\\\ [ 4 p t ] v & = \\arctan \\frac { x ^ 1 x ^ 4 + x ^ 2 x ^ 3 } { x ^ 3 x ^ 4 - x ^ 1 x ^ 2 } + x ^ 0 , \\\\ [ 4 p t ] w & = x ^ 0 . \\end{align*}"}
{"id": "6939.png", "formula": "\\begin{align*} \\Sigma ^ * \\Sigma - \\Sigma ^ * \\Sigma \\Sigma ^ * \\Sigma = \\Sigma ^ * \\big ( q q ^ * \\big ) \\Sigma = \\Sigma ^ * q ( \\Sigma ^ * q ) ^ * = 0 , \\end{align*}"}
{"id": "2882.png", "formula": "\\begin{align*} F ( x ) & = \\left ( f ( x ) , f \\circ S ^ { - 1 } ( x ) , f \\circ S ^ { - 2 } ( x ) , \\dotsc , f \\circ S ^ { - n + 1 } ( x ) \\right ) \\\\ & = \\left ( f ( x _ 1 , x _ 2 \\ldots , x _ n ) , f ( x _ 2 , \\ldots , x _ { n } , x _ 1 ) , \\ldots , f ( x _ n , x _ 1 , \\ldots , x _ { n - 1 } ) \\right ) . \\end{align*}"}
{"id": "308.png", "formula": "\\begin{align*} f ( A ) & = \\sum _ { 0 \\le i \\le k } f ( A _ { ( i ) } ) \\ \\le \\ \\frac { e ^ \\gamma } { m _ q } \\sum _ { 0 \\le i \\le k } \\frac { { \\rm d } ( { \\rm L } _ { A _ { ( i ) } } ) } { 1 + v _ i } . \\end{align*}"}
{"id": "5498.png", "formula": "\\begin{align*} ( \\omega - \\omega _ { \\theta } ) ^ T A _ { \\theta } ( \\omega - \\omega _ { \\theta } ) = - ( 1 - \\gamma ) | | \\Phi ( \\omega - \\omega _ { \\theta } ) | | _ { \\rm D } ^ 2 - \\gamma | | \\Phi ( \\omega - \\omega _ { \\theta } ) | | _ { \\rm D i r } ^ 2 , \\end{align*}"}
{"id": "492.png", "formula": "\\begin{align*} f _ i ( b _ 1 \\otimes b _ 2 ) & = \\begin{cases} b _ 1 \\otimes f _ i ( b _ 2 ) & \\varepsilon _ i ( b _ 1 ) < \\varphi _ i ( b _ 2 ) \\\\ f _ i ( b _ 1 ) \\otimes b _ 2 & \\varepsilon _ i ( b _ 1 ) \\geq \\varphi _ i ( b _ 2 ) \\end{cases} \\end{align*}"}
{"id": "6649.png", "formula": "\\begin{align*} \\rho \\left | u _ { i _ 1 , t } - u _ { i _ 2 , t } \\right | & = \\left | v _ { i _ 1 , t } - v _ { i _ 2 , t } \\right | \\\\ & \\stackrel { ( a ) } { > } \\epsilon ' _ n v _ { i _ 1 , t } \\\\ & \\stackrel { ( b ) } { > } \\lambda \\epsilon ' _ n \\ ; , \\ , \\end{align*}"}
{"id": "8593.png", "formula": "\\begin{align*} \\sum _ { \\varepsilon _ j = 0 , 1 } ( - 1 ) ^ { \\varepsilon _ 1 + \\ldots + \\varepsilon _ l } w _ n \\left ( \\alpha \\oplus \\frac { \\varepsilon _ 1 } { 2 ^ { k _ 1 } } \\oplus \\ldots \\oplus \\frac { \\varepsilon _ l } { 2 ^ { k _ l } } \\right ) = \\left \\{ \\begin{array} { l l l } 0 & & n \\neq m , \\\\ \\pm 2 ^ l & & n = m . \\end{array} \\right . \\end{align*}"}
{"id": "1203.png", "formula": "\\begin{align*} 0 \\le \\zeta _ j \\le 1 , \\ \\zeta _ j ( x ) = 1 \\ , \\ \\sup _ { j , x } | \\nabla \\zeta _ j ( x ) | < \\infty . \\end{align*}"}
{"id": "4604.png", "formula": "\\begin{align*} \\{ \\lambda \\in P _ + \\mid \\mathcal { C } _ { X ^ + } ^ \\lambda \\neq 0 \\} = \\{ \\lambda \\in P _ + \\mid \\mathcal { C } _ { X ^ - } ^ \\lambda \\neq 0 \\} = R , \\end{align*}"}
{"id": "6712.png", "formula": "\\begin{align*} \\varphi ( \\lambda _ 1 , \\lambda _ 2 ) : = h ( \\lambda _ 1 , \\lambda _ 2 ) \\cdot q _ { l , n } ( \\lambda _ 1 , \\lambda _ 2 ) . \\end{align*}"}
{"id": "170.png", "formula": "\\begin{align*} \\hat f ( k ) & \\leq c \\| k \\| _ 2 \\int _ 0 ^ 1 d \\alpha \\int _ { \\R ^ 3 } \\frac { 1 } { 1 + \\| q \\| _ 2 ^ 2 } \\cdot \\frac { 1 } { 1 + \\| \\alpha k - q \\| _ 2 ^ 2 } \\ , d q \\\\ & = : \\| k \\| _ 2 \\int _ 0 ^ 1 d \\alpha \\ , I ( \\alpha k ) . \\end{align*}"}
{"id": "3762.png", "formula": "\\begin{align*} T _ { k , j ; n , l , r } ^ { \\mu , m , i } ( \\mathfrak { m } , B ) ( t , x , \\zeta ) = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big ( ( t - s ) \\mathfrak { K } ^ { \\mu , B } _ { k ; n } ( y , \\omega , v , \\zeta ) + \\mathfrak { K } ^ { e r r ; \\mu , B } _ { k ; n } ( y , v , \\zeta ) \\big ) f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "2474.png", "formula": "\\begin{align*} W _ A = \\left \\{ w \\in W \\mid w \\left ( { A } \\right ) \\subset { A } \\cup - { A } \\right \\} . \\end{align*}"}
{"id": "4591.png", "formula": "\\begin{align*} c _ { i j } ( t ) : = \\frac { 1 } { N ( \\lambda _ i ( t ) - \\lambda _ j ( t ) ) ^ 2 } . \\end{align*}"}
{"id": "7109.png", "formula": "\\begin{align*} \\beta _ \\mu ^ t = t ^ 2 \\cdot \\beta _ \\mu . \\end{align*}"}
{"id": "8925.png", "formula": "\\begin{align*} A ( \\cdot , u , p ) = g _ { x x } ( \\cdot , Y ( \\cdot , u , p ) , Z ( \\cdot , u , p ) ) . \\end{align*}"}
{"id": "2253.png", "formula": "\\begin{align*} \\begin{cases} ( 1 - \\delta + \\phi _ * ) ^ 2 \\phi '' _ * + \\frac { z } { 2 } ( 1 - \\delta + \\phi _ * ) \\phi ' _ * - \\frac { z } { 2 } \\psi _ * \\psi ' _ * \\\\ = ( 1 - \\delta + \\phi _ * ) ( | \\psi _ * ' | ^ 2 - | \\phi _ * ' | ^ 2 ) , \\\\ ( 1 - \\delta + \\phi _ * ) ^ 2 \\psi _ * '' + \\frac { z } { 2 } ( 1 - \\delta + \\phi _ * ) \\psi _ * ' - \\frac { z } { 2 } \\psi _ * \\phi ' _ * = 0 , \\\\ \\phi _ * ( 0 ) = 0 , \\phi _ * ( \\infty ) = \\delta , \\\\ \\psi _ * ( 0 ) = 0 , \\psi _ * ( \\infty ) = \\sigma . \\end{cases} \\end{align*}"}
{"id": "7450.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\abs { u ^ n _ x } ^ 2 d x = o ( 1 ) . \\end{align*}"}
{"id": "3054.png", "formula": "\\begin{align*} f _ { \\boldsymbol { X } ^ { \\ast } } ( \\boldsymbol { x } ) = \\frac { c _ { n } ^ { \\ast } } { \\sqrt { | \\boldsymbol { \\Sigma } | } } \\overline { G } _ { n } \\left \\{ \\frac { 1 } { 2 } ( \\boldsymbol { x } - \\boldsymbol { \\mu } ) ^ { T } \\mathbf { \\Sigma } ^ { - 1 } ( \\boldsymbol { x } - \\boldsymbol { \\mu } ) \\right \\} , ~ \\boldsymbol { x } \\in \\mathbb { R } ^ { n } \\end{align*}"}
{"id": "8126.png", "formula": "\\begin{align*} H = H _ 0 + V \\end{align*}"}
{"id": "8428.png", "formula": "\\begin{align*} u _ i \\left ( s \\right ) = \\xi _ i r ( s ) = j \\in S , \\end{align*}"}
{"id": "5947.png", "formula": "\\begin{align*} { \\rm { \\Delta } } _ { i j . . . k p } ^ { \\left ( n \\right ) } = w _ { i j . . . k p } ^ { \\left ( n \\right ) } \\delta \\left ( { { t _ 1 } - { t _ 2 } } \\right ) . . . \\delta \\left ( { { t _ 1 } - { t _ n } } \\right ) \\end{align*}"}
{"id": "8072.png", "formula": "\\begin{align*} \\phi ( N \\times { \\mathbb R } ^ { \\dim M - 1 } ) \\subset U ^ { + } : = \\{ x \\in M \\mid \\operatorname { s g n } ( Z ) > 0 \\} . \\end{align*}"}
{"id": "659.png", "formula": "\\begin{align*} Q ( g ) & = Q ( g _ o ) + Q ( g ) - Q ( g _ o ) \\\\ & = Q ( g _ o ) \\circ \\left [ I + Q ( g _ o ) ^ { - 1 } \\circ \\left ( Q ( g ) - Q ( g _ o ) \\right ) \\right ] , \\end{align*}"}
{"id": "3566.png", "formula": "\\begin{align*} w ( \\pi ) = \\begin{cases} ( 1 + \\delta ( s _ 1 + s _ 2 ) ) ( ( 1 + t _ 1 ) ( 1 + t _ 2 ) ) ^ { c _ 1 ' / 2 } ( 1 + t _ 1 + t _ 2 ) ^ { c _ 2 ' / 2 } , & q \\equiv 1 \\pmod 4 , \\\\ ( 1 + v _ 1 ) ^ { c _ 1 ' } ( 1 + v _ 2 ) ^ { c _ 1 ' } ( 1 + v _ 1 + v _ 2 ) ^ { c _ 2 ' } , & q \\equiv 3 \\pmod 4 . \\end{cases} \\end{align*}"}
{"id": "3410.png", "formula": "\\begin{align*} t _ { - 1 } = \\begin{pmatrix} - 1 \\\\ & 1 \\\\ & & - 1 \\end{pmatrix} . \\end{align*}"}
{"id": "6819.png", "formula": "\\begin{align*} \\begin{cases} u _ t ( t , x ) + a u _ { x x x x } ( t , x ) = h ( t , x ) , & ( t , x ) \\in ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u _ { x x } ( t , 0 ) = u _ { x x } ( t , 1 ) = 0 , & t \\in ( 0 , T ) , \\\\ u _ { x x x } ( t , 0 ) = u _ { x x x } ( t , 1 ) = 0 , & t \\in ( 0 , T ) , \\\\ u ( 0 , x ) = u _ 0 ( x ) , & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
{"id": "2811.png", "formula": "\\begin{align*} \\begin{aligned} c ' ( h ) = - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\left [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\right ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 \\end{aligned} \\end{align*}"}
{"id": "4688.png", "formula": "\\begin{align*} & ( - 1 ) ^ { k - 1 } \\left ( C _ m ( n ) - \\sum _ { j = - k } ^ k ( - 1 ) ^ j u _ m \\big ( n - j ( 3 j - 1 ) / 2 \\big ) \\right ) \\geqslant 0 . \\end{align*}"}
{"id": "4341.png", "formula": "\\begin{align*} \\partial \\overline { \\mathcal { M } } ( p _ 1 , \\ldots p _ k , q ) = & \\bigcup \\overline { \\mathcal { M } ( p _ l , r ) / \\R } \\times \\overline { \\mathcal { M } } ( p _ 1 , \\ldots p _ { l - 1 } , r , p _ { l + 1 } , \\ldots p _ k , q ) \\\\ & \\cup \\bigcup \\overline { \\mathcal { M } } ( p _ 1 , \\ldots , p _ j , r , p _ { l + j + 1 } , \\ldots p _ k , q ) \\times \\overline { \\mathcal { M } } ( p _ { j + 1 } , \\ldots p _ { l + j } , r ) . \\end{align*}"}
{"id": "689.png", "formula": "\\begin{align*} \\int _ { M } v _ s \\ , d g _ o = \\int _ { M } v _ { s _ 0 - s _ 0 ^ { \\theta / 4 } } \\ , d g _ o = \\int _ { M } w _ { s _ 0 - s _ 0 ^ { \\theta / 4 } } \\ , d g _ o = 1 \\pm O ( s _ 0 ^ { - \\theta / 2 } ) . \\end{align*}"}
{"id": "3379.png", "formula": "\\begin{align*} \\rho \\ , L = \\sqrt { \\frac { 2 A } { A + G } } L = \\sqrt { \\frac { 2 L ^ 2 A } { A + G } } \\leq \\sqrt { L A } \\leq A \\end{align*}"}
{"id": "2422.png", "formula": "\\begin{align*} \\| Q _ n \\pi ( a ) - Q _ n \\pi ( a ) Q _ n \\| & = \\| V _ n ^ * P _ n V _ n \\pi ( a ) - V _ n ^ * P _ n V _ n \\pi ( a ) V _ n ^ * P _ n V _ n \\| \\\\ & = \\| V _ n ^ * P _ n \\sigma ( a ) V _ n - V _ n ^ * P _ n \\sigma ( a ) P _ n V _ n \\| \\\\ & \\leq \\| P _ n \\sigma ( a ) - P _ n \\sigma ( a ) P _ n \\| \\approx 0 , \\end{align*}"}
{"id": "2747.png", "formula": "\\begin{align*} ( a + b ) ^ { p - 2 } = & a ^ { p - 2 } + \\sum ^ { p - 2 } _ { j = 1 } C _ { p - 2 } ^ { j } a ^ { p - 2 - j } b ^ { j } \\leq ( 1 + \\varepsilon ) a ^ { p - 2 } + C ( p ) b ^ { p - 2 } \\sum ^ { p - 2 } _ { j = 1 } \\varepsilon ^ { - \\frac { p - 2 - k } { k } } \\\\ \\leq & ( 1 + \\varepsilon ) a ^ { p - 2 } + \\frac { C ( p ) } { \\varepsilon ^ { p - 3 } } b ^ { p - 2 } , a , b \\geq 0 . \\end{align*}"}
{"id": "950.png", "formula": "\\begin{align*} \\Phi _ R ( x ) : = \\eta \\left ( \\frac { x _ 1 } { R } \\right ) \\zeta \\left ( \\frac { x ' } { R ^ { \\alpha } } \\right ) , x ' = ( x _ 2 , x _ 3 ) , R > 1 , \\alpha > 0 , \\end{align*}"}
{"id": "8436.png", "formula": "\\begin{align*} & d ( \\theta _ n u ( t ) ) = - \\theta _ n \\nu A u ( t ) d t - \\theta _ n \\lambda ( r ( t ) ) u ( t ) d t + \\theta _ n f ( t , u ( t ) ) d t \\\\ & \\quad + \\theta _ n g ( t ) d t + \\sum _ { k = 1 } ^ \\infty ( \\theta _ n h _ k ( t ) + \\theta _ n \\sigma _ k ( t , u ( t ) ) ) d W _ k ( t ) , t > s . \\end{align*}"}
{"id": "6323.png", "formula": "\\begin{align*} & \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + \\dfrac { 1 - q x ^ 2 ( 1 - q ) } { x } D _ { q } y ( x ) + \\dfrac { q x ^ 2 - q ^ { 1 - v } [ v ] ^ 2 _ q } { x ^ 2 } y ( x ) = 0 . \\end{align*}"}
{"id": "7803.png", "formula": "\\begin{align*} \\psi = \\eta _ Y \\widetilde { H } _ 0 ( \\phi ) \\eta _ X ^ { - 1 } . \\end{align*}"}
{"id": "4103.png", "formula": "\\begin{align*} B ( \\widetilde { u } - { u } _ 2 ) ^ \\prime + ( A + B Q ) ( \\widetilde { u } - { u } _ 2 ) = - H _ 1 ( \\widetilde { u } \\ , ^ \\prime + Q \\widetilde { u } ) - H _ 2 \\big ( ( u _ + - u ) ' + Q ( u _ + - u ) \\big ) \\end{align*}"}
{"id": "1318.png", "formula": "\\begin{align*} \\varphi ( x , v ) = H ( x , v ) + \\frac { x \\cdot v } { 4 } + \\frac { | x | ^ 2 } { 8 } , \\end{align*}"}
{"id": "1121.png", "formula": "\\begin{align*} g ( w | \\rho , \\lambda ) = \\sigma _ Z ^ 2 \\lambda w + \\left ( \\frac { \\rho } { \\lambda } + \\sigma _ N ^ 2 \\lambda \\right ) \\cdot w = \\left [ ( \\sigma _ N ^ 2 + \\sigma _ Z ^ 2 ) \\lambda + \\frac { \\rho } { \\lambda } \\right ] \\cdot w , \\end{align*}"}
{"id": "8307.png", "formula": "\\begin{align*} b _ n = \\frac { N ! } { \\pi n } \\sum _ { k = 0 } ^ { N - 1 } \\frac { ( - 1 ) ^ k F _ k ( n ) } { ( N - k ) ! } \\sin \\left ( \\frac { \\pi } { 2 } k \\right ) . \\end{align*}"}
{"id": "6118.png", "formula": "\\begin{align*} P ( z ) = a _ n ( z - \\lambda _ 1 ) ( z - \\lambda _ 2 ) \\cdots ( z - \\lambda _ n ) . \\end{align*}"}
{"id": "3746.png", "formula": "\\begin{align*} I _ { j ; l ; 0 } ^ { m ; p , q } ( t , x ) : = \\int _ { 0 } ^ { t } \\int _ { \\R ^ 3 } \\int _ { 0 } ^ { 2 \\pi } \\int _ 0 ^ \\pi 2 ^ { m - j - l + \\max \\{ l , p \\} } \\big [ \\sum _ { \\begin{subarray} { c } \\tilde { j } \\in [ ( 1 + 2 \\epsilon ) M _ t , \\infty ) \\cap \\Z _ + , \\tilde { l } \\in [ - \\tilde { j } , 2 ] \\cap \\Z , \\tilde { m } \\in [ - 1 0 M _ t , \\epsilon M _ t ] \\cap \\Z \\end{subarray} } | B _ { \\tilde { j } , \\tilde { l } } ^ { \\tilde { m } } ( s , x + ( t - s ) \\omega ) \\end{align*}"}
{"id": "213.png", "formula": "\\begin{align*} \\norm { p _ { X } p _ { Y | X } - q _ { X } p _ { Y | X } } _ { T V } = \\norm { p _ { X } - q _ { X } } _ { T V } . \\end{align*}"}
{"id": "8440.png", "formula": "\\begin{align*} D _ { \\beta } ( \\varphi , x _ 0 ) = \\left \\{ x \\in [ 0 , 1 ] : | T _ { \\beta } ^ n x - x _ 0 | < \\varphi ( n ) n \\in \\N \\right \\} \\end{align*}"}
{"id": "7508.png", "formula": "\\begin{align*} E _ { x \\leftrightarrow y } = E _ { x \\rightarrow y } \\land E _ { y \\rightarrow x } , \\end{align*}"}
{"id": "388.png", "formula": "\\begin{align*} q _ A = \\frac { 2 } { 1 + \\frac { G ( x , y ) } { G ( o , o ) } } . \\end{align*}"}
{"id": "572.png", "formula": "\\begin{align*} M _ 2 = \\sum _ { \\substack { 0 \\leq l \\leq i \\\\ \\gcd ( l , i ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } \\\\ \\gcd ( l + 1 , i + 1 ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } C _ { \\alpha } ( i , l ) . \\end{align*}"}
{"id": "5425.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) u ^ { ( 1 ) } _ f u ^ { ( 2 ) } _ g \\ , d x = \\int _ { \\Omega _ e } ( q _ 1 - q _ 2 ) f g \\ , d x = 0 , \\end{align*}"}
{"id": "229.png", "formula": "\\begin{align*} 0 & = c ( c \\gamma + \\gamma + 1 ) ^ 2 + c \\gamma ( c \\gamma + c + 1 ) ( c \\gamma + \\gamma + 1 ) + \\gamma ( c \\gamma + c + 1 ) ^ 2 \\\\ & = \\gamma ^ 3 c ^ 3 + \\gamma ^ 2 c ^ 2 + ( \\gamma + 1 ) c + \\gamma \\end{align*}"}
{"id": "2470.png", "formula": "\\begin{align*} \\left ( u , \\left [ x , z \\right ] \\right ) = \\left ( \\left [ u , x \\right ] , z \\right ) . \\end{align*}"}
{"id": "6077.png", "formula": "\\begin{align*} \\Lambda _ 2 = ( 0 , 2 , 3 , - 6 ) \\end{align*}"}
{"id": "595.png", "formula": "\\begin{align*} \\sigma ( \\widetilde \\Delta _ n ) = \\Bigg \\{ & \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi k _ 1 } { n } \\Big ) + \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi k _ 2 } { n } \\Big ) \\\\ & \\left . - \\frac { 2 n ^ 2 } { 3 \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi k _ 1 } { n } \\Big ) \\sin ^ 2 \\Big ( \\frac { \\pi k _ 2 } { n } \\Big ) \\right | \\forall _ { j = 1 , 2 } : k _ j = 0 \\ , , \\cdots , n - 1 \\Bigg \\} . \\end{align*}"}
{"id": "7542.png", "formula": "\\begin{align*} \\gamma _ { c , k } & = \\frac { | { \\vec { h } } _ { k } ^ { H } \\vec { p } _ { c } | ^ 2 } { \\sum \\limits _ { j \\in \\mathcal { K } } | \\vec { h } _ { k } ^ { H } \\vec { p } _ { j } | ^ 2 + 1 } , & \\gamma _ { p , k } & = \\frac { | { \\vec { h } } _ { k } ^ { H } \\vec { p } _ { k } | ^ 2 } { \\sum \\limits _ { j \\in \\mathcal { K } \\setminus k } | \\vec { h } _ { k } ^ { H } \\vec { p } _ { j } | ^ 2 + 1 } . \\end{align*}"}
{"id": "5911.png", "formula": "\\begin{align*} f ( z ) & = ( f _ 1 ( z ) , \\ldots , f _ p ( z ) , f _ { p + 1 } , \\ldots , f _ n ( z ) ) , \\\\ T _ 1 ( r ) & : = \\sum _ { j = 1 } ^ p T \\left ( r , e ^ { f _ j } \\right ) , \\\\ T _ 2 ( r ) & : = \\frac 1 { 4 \\pi } \\int _ { | z | = r } \\sum _ { j + 1 } ^ n | f _ j ( z ) | ^ 2 \\ , d \\theta - \\frac 1 { 4 \\pi } \\int _ { | z | = 1 } \\sum _ { j + 1 } ^ n | f _ j ( z ) | ^ 2 \\ , d \\theta , \\\\ T _ { \\exp _ A f } ( r ) & = T _ 1 ( r ) + T _ 2 ( r ) . \\end{align*}"}
{"id": "1550.png", "formula": "\\begin{align*} & \\P _ { \\pi } ( X _ { 1 } = 1 , X _ { 1 + d } = 1 , T _ Y > d ) \\\\ & \\le \\P _ { \\pi } ( X _ { 1 } = 1 , X _ { 1 + d } = 1 , T _ Y > d , \\mathcal { E } ) + \\P _ { \\pi } ( X _ 1 = 1 , \\overline { \\mathcal { E } } ) \\\\ & \\le \\P _ { \\pi } ( X _ { 1 } = 1 , X _ { 1 + d } = 1 , T _ Y > d , \\mathcal { E } ) + e ^ { - c _ 1 d } \\P _ { \\pi } ( X _ 1 = 1 ) . \\end{align*}"}
{"id": "7182.png", "formula": "\\begin{align*} & ( z _ 0 + z _ 2 ) ^ { k + \\frac { j _ 2 } { T } } Y _ { M ^ 3 } ( u , z _ 0 + z _ 2 ) I ( w _ 1 , z _ 2 ) w _ 2 \\\\ = & ( z _ 2 + z _ 0 ) ^ { k + \\frac { j _ 2 } { T } } I ( Y _ { M ^ 1 } ( u , z _ 0 ) w _ 1 , z _ 2 ) w _ 2 , \\end{align*}"}
{"id": "4719.png", "formula": "\\begin{align*} e ( H _ { 2 } ^ { - 1 } H _ { 3 } ^ { - 1 } H _ { 1 } H _ { 2 } ) e H _ { 2 } ^ { - 1 } H _ { 3 } ^ { - 1 } H _ { 1 } H _ { 2 } = e ( H _ { 2 } ^ { - 1 } H _ { 3 } ^ { - 1 } H _ { 1 } H _ { 2 } ) e H _ { 2 } ^ { - 1 } H _ { 1 } ^ { - 1 } H _ { 3 } H _ { 2 } . \\end{align*}"}
{"id": "1483.png", "formula": "\\begin{align*} \\left | { { \\lambda _ 0 } } + \\sum _ { i = 1 } ^ m \\sum _ { s = 0 } ^ { r - 1 } { { \\lambda _ { i , s } } } F _ { s } ( x , \\alpha _ i / \\beta ) \\right | _ { v _ 0 } > C ( \\boldsymbol { \\eta } , \\boldsymbol { \\zeta } , \\boldsymbol { \\alpha } , \\beta , \\varepsilon ) { \\mathrm { H } } _ { v _ 0 } ( { { \\boldsymbol { \\lambda } } } ) { \\mathrm { H } } ( { { \\boldsymbol { \\lambda } } } ) ^ { - \\mu ( \\boldsymbol { \\eta } , \\boldsymbol { \\zeta } , \\boldsymbol { \\alpha } , \\beta , \\varepsilon ) } \\enspace , \\end{align*}"}
{"id": "6089.png", "formula": "\\begin{align*} \\delta \\lVert \\alpha \\rVert _ { 1 } \\leq & \\rho _ { M } ( T ( \\alpha ) ) = \\sup _ { \\theta \\in M } \\lvert \\theta ( T ( \\alpha ) ) \\rvert \\\\ = & \\sup _ { \\theta \\in M } \\lvert ( T ^ { * } ( \\theta ) ) ( \\alpha ) \\rvert < \\lvert \\varphi ( \\alpha ) \\rvert \\\\ \\leq & \\lVert \\varphi \\rVert _ { 1 } \\lVert \\alpha \\rVert _ { 1 } \\leq \\delta \\lVert \\alpha \\rVert _ { 1 } , \\end{align*}"}
{"id": "795.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ { \\infty } | b _ k | r ^ k \\leq \\sum _ { n = N } ^ { \\infty } | a _ n | r ^ n = \\sum _ { n = N } ^ { \\infty } \\frac { | \\tilde { b } _ n | } { n } r ^ n , \\end{align*}"}
{"id": "5215.png", "formula": "\\begin{align*} \\int _ { \\Xi } x ^ { n r + k } e ^ { x ^ r / \\hbar } d x = { } & ( - 1 ) ^ n \\hbar ^ n \\left ( \\prod _ { i = 1 } ^ n ( i - 1 + \\frac { k + 1 } { r } ) \\right ) \\int _ { \\Xi } x ^ { k } e ^ { x ^ r / \\hbar } d x \\\\ = { } & ( - 1 ) ^ n \\hbar ^ n \\frac { \\Gamma \\left ( n + \\frac { k + 1 } { r } \\right ) } { \\Gamma \\left ( \\frac { k + 1 } { r } \\right ) } \\int _ { \\Xi } x ^ { k } e ^ { x ^ r / \\hbar } d x . \\end{align*}"}
{"id": "1725.png", "formula": "\\begin{align*} \\begin{aligned} x _ 1 = \\exp ( 2 \\pi i z / \\overline \\omega _ 1 ) , & x _ 2 = \\exp ( 2 \\pi i z / \\omega _ 2 ) , \\\\ q _ 1 = \\exp ( 2 \\pi i { \\omega _ 2 } / { \\overline \\omega _ 1 } ) , & q _ 2 = \\exp ( 2 \\pi i { \\omega _ 1 } / { \\omega _ 2 } ) , \\widetilde { q } _ 2 = \\exp ( 2 \\pi i { \\widetilde \\omega _ 1 } / { \\omega _ 2 } ) . \\end{aligned} \\end{align*}"}
{"id": "8655.png", "formula": "\\begin{align*} \\underline B ' : = \\sup _ { s > 0 } \\left ( \\int _ s ^ \\infty V ( t ) ^ { - \\frac { 1 } { p - 1 } } d t \\right ) ^ { - 1 } \\int _ s ^ \\infty W ( t ) \\left ( \\int _ t ^ \\infty V ( t ' ) ^ { - \\frac { 1 } { p - 1 } } d t ' \\right ) ^ p d t \\ , . \\end{align*}"}
{"id": "8055.png", "formula": "\\begin{align*} V = - \\operatorname { g r a d } p \\end{align*}"}
{"id": "3409.png", "formula": "\\begin{align*} \\Psi ( t _ z , n ) = \\log ( z ) - 2 \\pi i \\cdot n . \\end{align*}"}
{"id": "2471.png", "formula": "\\begin{align*} \\rho = \\frac { 1 } { 2 } \\sum _ { \\alpha \\in \\Delta _ { 0 } ^ { + } } \\alpha - \\frac { 1 } { 2 } \\sum _ { \\alpha \\in \\Delta _ { 1 } ^ { + } } \\alpha . \\end{align*}"}
{"id": "1469.png", "formula": "\\begin{align*} \\bigcirc _ { w = 0 } ^ i ( \\theta _ t + \\gamma _ { r - i + w } ) ^ { - 1 } \\circ ( \\theta _ t + \\xi _ { i , 1 } ) \\circ \\cdots \\circ ( \\theta _ t + \\xi _ { i , \\ell _ i } ) = \\bigcirc _ { w ' = 0 } ^ j ( \\theta _ t + \\gamma _ { r - j + w ' } ) ^ { - 1 } \\circ ( \\theta _ t + \\xi ' _ { j , 1 } ) \\circ \\cdots \\circ ( \\theta _ t + \\xi ' _ { j , \\ell ' _ j } ) \\enspace , \\end{align*}"}
{"id": "2133.png", "formula": "\\begin{align*} \\partial \\Omega \\cap \\mathbb { R } = A \\cup B \\cup C . \\end{align*}"}
{"id": "3485.png", "formula": "\\begin{align*} L _ { 1 } = \\frac { \\xi _ { p } \\phi ( \\xi _ { p } ) - \\xi _ { q } \\phi ( \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , ~ L _ { 1 } ^ { \\ast } = \\frac { \\xi _ { p } ^ { 2 } \\phi ( \\xi _ { p } ) - \\xi _ { q } ^ { 2 } \\phi ( \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "8929.png", "formula": "\\begin{align*} T u ( \\Omega ) : = Y ( \\cdot , u , D u ) ( \\Omega ) = \\Omega ^ * , \\end{align*}"}
{"id": "8613.png", "formula": "\\begin{align*} Q _ { n e w } = \\alpha Q _ { o l d } \\end{align*}"}
{"id": "6720.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ m h _ l ( \\lambda ^ 2 + k ^ 2 ) ( k \\lambda ) ^ l = 0 , \\ ; \\forall \\vert k \\vert \\leq m , \\lambda \\in \\mathfrak { a } ^ * _ \\mathbb { C } , h _ l = 0 , l = 0 , \\dots , m . \\end{align*}"}
{"id": "2202.png", "formula": "\\begin{align*} C ^ * ( \\mathbf { T } ) = C ( \\mathbf { T } ) = \\| \\mathbf { T } ^ { - 1 } \\| ^ { - 1 } \\end{align*}"}
{"id": "4595.png", "formula": "\\begin{align*} \\mathcal { E } : = N ^ { n \\xi } \\left ( \\frac { N ^ \\epsilon \\ell } { K } + \\frac { N T _ 1 } { \\ell } + \\frac { N \\eta } { \\ell } + \\frac { N ^ \\epsilon } { \\sqrt { N \\eta } } + \\frac { 1 } { \\sqrt { K } } \\right ) , \\end{align*}"}
{"id": "8442.png", "formula": "\\begin{align*} \\mathcal { L } ( D _ { \\beta } ( \\varphi , x _ 0 ) ) = \\begin{cases} 0 & \\sum _ { n = 1 } ^ { \\infty } \\varphi ( n ) < \\infty , \\\\ 1 & \\sum _ { n = 1 } ^ { \\infty } \\varphi ( n ) = \\infty . \\end{cases} \\end{align*}"}
{"id": "7701.png", "formula": "\\begin{align*} ( F * G ) ( n _ 1 , \\ldots , n _ k ) = \\sum _ { d _ 1 \\mid n _ 1 , \\ldots , d _ k \\mid n _ k } F ( d _ 1 , \\ldots , d _ k ) G ( n _ 1 / d _ 1 , \\ldots , n _ k / d _ k ) , \\end{align*}"}
{"id": "4858.png", "formula": "\\begin{align*} \\begin{cases} N _ { 1 } = p _ { 1 } + p _ { 2 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k } } \\\\ N _ { 2 } = p _ { 3 } + p _ { 4 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k } } \\end{cases} \\end{align*}"}
{"id": "7881.png", "formula": "\\begin{align*} \\vec s = ( s _ 1 , \\ldots , s _ n ) \\in \\mathbb Z ^ n . \\end{align*}"}
{"id": "410.png", "formula": "\\begin{align*} \\lim _ { z \\to \\infty } \\frac { F _ { \\mu } ( z ) } { z } = 1 , \\end{align*}"}
{"id": "8566.png", "formula": "\\begin{align*} w _ m ( x ) = r _ { k _ 1 } ( x ) \\ldots r _ { k _ l } ( x ) \\end{align*}"}
{"id": "8837.png", "formula": "\\begin{align*} \\int _ { G } ^ { } f \\circ ( \\phi _ 1 * \\phi _ 2 ) ( g ) d g = f _ + ' ( 0 ) \\| \\phi _ 1 * \\phi _ 2 \\| + \\int _ { 0 } ^ { \\infty } \\| f _ t \\circ ( \\phi _ 1 * \\phi _ 2 ) \\| d \\nu ( t ) \\end{align*}"}
{"id": "8896.png", "formula": "\\begin{align*} T _ { i j } ^ k = \\delta _ i ^ k \\rho _ j - \\delta _ j ^ k \\rho _ i . \\end{align*}"}
{"id": "6275.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } L _ n ^ { \\alpha } ( x ; q ) = \\dfrac { - q ^ { \\alpha + 1 } } { ( 1 - q ) } L _ { n - 1 } ^ { \\alpha + 1 } ( x ; q ) , \\end{align*}"}
{"id": "7094.png", "formula": "\\begin{align*} \\varphi _ t ( k , v ) = k \\exp _ G ( t v ) , \\ ( k _ 1 , v _ 1 , t ) \\cdot _ t ( k _ 2 , v _ 2 , t ) = \\varphi ^ { - 1 } _ t \\big ( \\varphi _ t ( k _ 1 , v _ 1 ) \\cdot \\varphi _ t ( k _ 2 , v _ 2 ) \\big ) . \\end{align*}"}
{"id": "1562.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\varphi } ( v ) = \\frac { 1 } { \\mathcal { F } } \\left \\lbrace \\frac { \\partial ^ 2 \\mathcal { F } } { \\partial z ^ i _ { \\epsilon } \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\mathcal { \\varphi } ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } \\right \\rbrace v ^ i . \\end{align*}"}
{"id": "2498.png", "formula": "\\begin{align*} g ( g r a d ~ f , X ) = X ( f ) , ~ f o r ~ X \\in \\Gamma ( T M ) . \\end{align*}"}
{"id": "7335.png", "formula": "\\begin{align*} ( [ x , y ] , z , w ) = 0 \\mbox { i f } \\mathsf { c h a r } ( F ) \\neq 2 , 3 . \\end{align*}"}
{"id": "1432.png", "formula": "\\begin{align*} \\hat { s } _ { g } \\left ( q , r , \\tau _ t \\right ) & = \\\\ & \\frac { q \\exp \\left ( - \\frac { \\left ( r - d _ { g } \\right ) ^ 2 } { 2 \\tau _ t ^ 2 } \\right ) } { q \\exp \\left ( - \\frac { \\left ( r - d _ { g } \\right ) ^ 2 } { 2 \\tau _ t ^ 2 } \\right ) + ( 1 - q ) \\exp \\left ( - \\frac { r ^ 2 } { 2 \\tau _ t ^ 2 } \\right ) } , \\\\ \\end{align*}"}
{"id": "6740.png", "formula": "\\begin{align*} j _ \\delta ' ( u _ \\delta ) h & = J _ y ( y _ \\delta , u _ \\delta ) S _ \\delta ' ( u _ \\delta ; h ) + J _ u ( y _ \\delta , u _ \\delta ) h + ( u - \\overline { u } , h ) \\\\ & = ( y _ \\delta - z _ d , S _ \\delta ' ( u _ \\delta ; h ) ) + \\alpha ( u , h ) + ( u - \\overline { u } , h ) \\end{align*}"}
{"id": "2558.png", "formula": "\\begin{align*} r _ + = q \\ , \\ r _ - = \\nu _ 3 = 0 \\implies ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) = ( p + q , q , 0 ) \\ , \\ J = q / 2 \\ . \\end{align*}"}
{"id": "4550.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 3 \\right ) } \\Vert _ { p } = \\mathcal { O } \\left ( \\frac { 1 } { m } \\right ) ( v ( | z _ { 1 } | ) + ( v ( | z _ { 2 } | ) ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "7022.png", "formula": "\\begin{align*} \\dot { \\hat { \\tau } } ^ * = - \\frac { \\alpha _ x ( 2 \\nu _ x \\tau ^ * + \\xi _ x ) + \\alpha _ y ( 2 \\nu _ y \\tau ^ * + \\xi _ y ) + \\nu _ x ^ 2 + \\nu _ y ^ 2 } { \\nu _ x ^ 2 + \\nu _ y ^ 2 + \\varepsilon } \\end{align*}"}
{"id": "2805.png", "formula": "\\begin{align*} \\begin{aligned} h _ * = \\arg \\max _ { 1 { } \\leq { } h { } \\leq { } \\bar { h } ( \\kappa ) } 2 h + \\frac { \\kappa h ^ 3 } { 2 - h \\left ( 1 + \\kappa \\right ) } \\end{aligned} \\end{align*}"}
{"id": "3408.png", "formula": "\\begin{align*} T = \\left \\{ t _ z : z \\in \\C ^ \\times \\right \\} , t _ z = \\begin{pmatrix} \\frac { 1 } { \\bar z } \\\\ & \\frac { \\bar z } { z } \\\\ & & z \\end{pmatrix} . \\end{align*}"}
{"id": "8005.png", "formula": "\\begin{align*} D _ { Z , - } = ( D _ { 1 , - } + D _ - ) \\cap Z _ - = D _ { 1 , - } \\cap Z _ - + \\sum _ { i \\in \\{ M _ + \\cup M _ - \\} \\setminus S _ - } \\bar D _ i \\cap Z _ - . \\end{align*}"}
{"id": "4425.png", "formula": "\\begin{align*} ( \\beta + 3 ) d ' - \\beta d + 1 = 2 i \\leq 2 d . \\end{align*}"}
{"id": "4655.png", "formula": "\\begin{align*} \\rho _ { s t } ( t ^ { 1 / \\alpha } x , t ^ { 1 / \\alpha } y ) = t ^ { \\frac { 2 \\delta - d } { \\alpha } } \\rho _ s ( x , y ) , s > 0 . \\end{align*}"}
{"id": "1930.png", "formula": "\\begin{align*} h ( w ) : = S _ { 0 } \\left ( \\frac { 1 } { w } \\right ) = \\sum _ { m = 0 } ^ { \\infty } C _ { [ m , 0 ] } w ^ { m ( p + 1 ) + 1 } . \\end{align*}"}
{"id": "3391.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\sup _ { y : \\| y \\| _ d \\leq b ( x ) } \\bigl | \\mu ( x , y ) - \\gamma \\sigma ^ 2 \\bigr | = 0 . \\end{align*}"}
{"id": "4372.png", "formula": "\\begin{align*} & \\int _ { D _ 0 } | F _ j | ^ 2 e ^ { - k \\varphi } \\\\ \\le & 2 \\int _ { D _ 0 } | ( 1 - b _ { t _ j , 1 } ( k \\Psi ) ) f _ j F ^ { 2 k } | ^ 2 e ^ { - k \\varphi } + 2 \\int _ { D _ 0 } | F _ j - ( 1 - b _ { t _ j , 1 } ( k \\Psi ) ) f _ j F ^ { 2 k } | ^ 2 e ^ { - k \\varphi } \\\\ \\le & ( 2 e ^ { t _ j + 1 } + 1 ) \\int _ { \\{ k \\Psi < - t _ j \\} \\cap D _ 0 } | f _ j | ^ 2 . \\end{align*}"}
{"id": "7092.png", "formula": "\\begin{align*} \\bigl ( ( y _ 6 - y _ 5 - 2 z _ 6 , y _ 7 - y _ 5 - 2 z _ 7 , y _ 8 - y _ 5 - 2 z _ 8 ) \\cdot ( d ) \\cdot D _ 1 \\cdots D _ 4 \\bigr ) ^ r \\subseteq J \\ k = 5 , \\end{align*}"}
{"id": "5100.png", "formula": "\\begin{align*} M ^ { n , 3 } _ s = \\int _ { \\eta _ n ( s ) - \\delta } ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u . \\end{align*}"}
{"id": "7560.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { \\sqrt { m } } - y _ m \\right | & = \\frac { 1 } { | \\frac 1 { \\sqrt { m } } + y _ m | } \\left | \\frac { 1 } { m } - y _ m ^ 2 \\right | \\leq \\sqrt { m } \\left | \\frac { 1 } { m } - \\left ( 1 - \\sum _ { j \\le { m - 1 } } y _ j ^ 2 \\right ) \\right | \\cr & \\leq 3 \\sum _ { j \\le m - 1 } \\left | y _ j - \\frac { 1 } { \\sqrt { m } } \\right | < 3 ( m - 1 ) \\frac { 1 } { 3 0 0 m \\cdot x \\sqrt { m } } . \\cr & < \\frac { 1 } { 1 0 0 x \\sqrt { m } } . \\end{align*}"}
{"id": "7513.png", "formula": "\\begin{align*} \\sum _ { i } \\mbox { P r } ( C _ { i } ) \\sum _ { j = 0 } ^ { k } \\binom { m - i } { j } \\left ( \\frac { \\binom { k } { j } ( ( j ! ) ( i ^ { ( k - j ) } ) ) } { m ^ { k } } \\right ) ^ { 2 } , \\end{align*}"}
{"id": "1997.png", "formula": "\\begin{align*} L _ n ^ { ( - n - s - 1 ) } ( x ) = \\displaystyle \\sum _ { j = 0 } ^ { n } ( - 1 ) ^ n \\frac { ( n + s - j ) ! } { ( n - j ) ! s ! } \\frac { x ^ j } { j ! } . \\end{align*}"}
{"id": "1129.png", "formula": "\\begin{align*} E _ { \\mbox { \\tiny M D } } ( \\theta ) = \\sup _ { \\lambda \\ge 0 } \\sup _ { P \\le P _ w } \\left \\{ \\lambda ( \\sqrt { P _ s P } - \\theta ) - C \\left ( \\lambda \\sqrt { P } \\right ) - \\frac { \\lambda ^ 2 \\sigma _ N ^ 2 P } { 2 } \\right \\} . \\end{align*}"}
{"id": "4134.png", "formula": "\\begin{align*} D ^ { 1 } \\left ( \\begin{array} { c } u \\\\ 0 \\\\ v \\end{array} \\right ) = \\left ( \\begin{array} { c } 0 \\\\ 0 \\\\ ( d T ) ^ { 2 } d u + d { v } \\end{array} \\right ) , \\end{align*}"}
{"id": "1499.png", "formula": "\\begin{align*} \\frac { d } { d x } e ^ { x ( e _ { \\lambda } ( t ) - 1 ) } & = ( e _ { \\lambda } ( t ) - 1 ) e ^ { x ( e _ { \\lambda } ( t ) - 1 ) } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( 1 ) _ { n - k , \\lambda } \\phi _ { k , \\lambda } ( x ) - \\phi _ { n , \\lambda } ( x ) \\bigg ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "695.png", "formula": "\\begin{align*} d \\mu _ t : = u ( \\cdot , t ) d g _ t : = ( 4 \\pi | t | ) ^ { - \\frac { n } { 2 } } e ^ { - f _ t } d g _ t , t \\in I \\end{align*}"}
{"id": "8780.png", "formula": "\\begin{align*} Y _ R ( P , G ) \\geq C ( P ) > 1 = \\tilde { Y } _ R ( P , G ) \\end{align*}"}
{"id": "2457.png", "formula": "\\begin{align*} x ' = \\Sigma x _ { \\beta _ i } ' = \\Sigma c _ { i } x _ { \\beta _ i } \\end{align*}"}
{"id": "3254.png", "formula": "\\begin{align*} u _ { A , q } ^ s ( x , d ) = \\frac { e ^ { i k \\vert x \\vert } } { \\vert x \\vert } \\Big ( u _ { A , q } ^ \\infty ( \\hat { x } , d ) + O \\Big ( \\frac { 1 } { \\vert x \\vert } \\Big ) \\Big ) , \\end{align*}"}
{"id": "7064.png", "formula": "\\begin{align*} \\Delta ( r _ 0 ) = \\left ( \\Pi ^ - \\right ) ^ { - 1 } ( r _ 0 ) - \\Pi ^ + ( r _ 0 ) = \\sum _ { k = 2 } ^ { \\infty } W _ k r _ 0 ^ k , \\end{align*}"}
{"id": "5139.png", "formula": "\\begin{align*} \\mathcal { G } ( n + 1 ) = \\mathcal { G } ( n ) \\oplus 1 \\ ; \\ n \\neq 0 \\bmod 2 c \\ ; \\ \\ ; \\ n \\neq 4 c + 1 \\bmod 1 2 c \\end{align*}"}
{"id": "3480.png", "formula": "\\begin{align*} g _ { 1 } ( u ) = \\overline { G } _ { ( 1 ) } ( u ) = \\overline { G } _ { ( 2 ) } ( u ) = \\exp \\{ - u \\} , \\end{align*}"}
{"id": "5310.png", "formula": "\\begin{align*} F _ { \\phi _ 1 , \\dots , \\phi _ n } ( \\mu ) = \\prod _ { i = 1 } ^ n \\langle \\phi _ i , \\mu \\rangle . \\end{align*}"}
{"id": "7063.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } r ^ \\pm } { \\mathrm { d } \\theta } \\ ! = \\ ! \\frac { \\sum _ { k = 2 } ^ { n } R ^ \\pm _ k ( \\theta ) r ^ k } { 1 + \\sum _ { k = 2 } ^ { n } \\Theta ^ \\pm _ k ( \\theta ) r ^ { k - 1 } } = \\sum _ { k = 2 } ^ { \\infty } S ^ \\pm _ k ( \\theta ) r ^ k , \\end{align*}"}
{"id": "1099.png", "formula": "\\begin{align*} G ( V , V ) G ( \\mbox { g r a d } f , P _ U U ) = G ( U , U ) G ( \\mbox { g r a d } f , P _ V V ) , \\end{align*}"}
{"id": "8310.png", "formula": "\\begin{align*} \\sum _ { j = m } ^ { \\infty } c _ j = O \\left ( v \\left ( \\frac { 1 } { m } \\right ) \\right ) , \\end{align*}"}
{"id": "7647.png", "formula": "\\begin{align*} p _ g ( S _ { n , 4 , 1 } ) = n \\binom { n } { 4 } - ( n + 1 ) \\binom { n - 1 } { 4 } . \\end{align*}"}
{"id": "2584.png", "formula": "\\begin{align*} \\left \\{ \\small { \\begin{pmatrix} 1 & 0 \\\\ 0 & U \\end{pmatrix} } : \\ U \\in S U ( 2 ) \\right \\} \\subset S U ( 3 ) \\end{align*}"}
{"id": "702.png", "formula": "\\begin{align*} g ^ { i j } = \\frac { 1 } { f ( u ) ^ 2 } \\widetilde { g } ^ { i j } + \\frac { 1 } { f ( u ) ^ 2 } \\frac { \\widetilde { g } ^ { j l } u _ l \\widetilde { g } ^ { i m } u _ m } { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | ^ 2 } . \\end{align*}"}
{"id": "6353.png", "formula": "\\begin{align*} c _ 2 | t | ^ { q ( x ) } \\leqslant F ( x , t ) : = \\int _ { 0 } ^ { t } f ( x , \\tau ) d \\tau , \\end{align*}"}
{"id": "8949.png", "formula": "\\begin{align*} \\begin{array} { l l } \\ ! \\ ! & \\ ! \\ ! A _ { i j } ( x _ 0 , u ( x _ 0 ) , D u ( x _ 0 ) ) - A _ { i j } ( x _ 0 , u _ 0 ( x _ 0 ) , D u _ 0 ( x _ 0 ) ) \\\\ = \\ ! \\ ! & \\ ! \\ ! D _ u A _ { i j } ( x _ 0 , \\hat u ( x _ 0 ) , D u ( x _ 0 ) ) ( u - u _ 0 ) ( x _ 0 ) + D _ { p _ k } A _ { i j } ( x _ 0 , u _ 0 ( x _ 0 ) , \\hat p ( x _ 0 ) ) D _ k ( u - u _ 0 ) ( x _ 0 ) , \\end{array} \\end{align*}"}
{"id": "404.png", "formula": "\\begin{align*} \\frac { d \\big ( \\phi ^ c _ { - t } \\big ) _ * \\nu ^ { \\ss } _ p } { d \\nu ^ { \\ss } _ { \\phi ^ c _ t p } } = e ^ { - \\delta t } . \\end{align*}"}
{"id": "341.png", "formula": "\\begin{align*} r ( g ) = d - 1 , \\end{align*}"}
{"id": "129.png", "formula": "\\begin{align*} \\psi : = C ^ 3 - { \\bf 1 } ^ \\epsilon _ 0 \\| C ^ 3 \\| _ { L ^ 1 } . \\end{align*}"}
{"id": "932.png", "formula": "\\begin{align*} E _ c ( \\Psi _ { t , \\eta _ i } ^ { q ( t ) } ) ( 0 , - \\frac { 1 } { 2 } h ) = E ( \\Psi _ { t , \\eta _ i } ^ { q ( t ) } ) ( 0 , - \\frac { 1 } { 2 } h + k \\cdot g ) + Q ( v _ k , v _ k ) = v _ k , \\end{align*}"}
{"id": "6313.png", "formula": "\\begin{align*} T _ q ( x ) = \\frac { a ( a - 1 ) } { q x ^ 2 } + \\frac { a b ( 1 + q ) } { q x } + \\frac { [ n ] _ q - q ^ n ( q ^ { - 1 } a + b x ) } { 1 - q } + b ^ 2 . \\end{align*}"}
{"id": "5531.png", "formula": "\\begin{align*} \\bar { \\Omega } ( \\theta ) = \\bar { \\psi } _ \\theta ( \\mathbb { H } \\setminus \\Sigma _ B ) \\cup \\bigcup _ { \\gamma \\in \\bar { \\Gamma } ( \\theta ) } \\gamma G _ \\theta . \\end{align*}"}
{"id": "5499.png", "formula": "\\begin{align*} \\widetilde { \\nabla } ( \\theta , \\omega _ { \\theta } ) = 0 . \\end{align*}"}
{"id": "2326.png", "formula": "\\begin{align*} D \\stackrel { \\eqref { l o w e r } } { > } C F - D E \\geq ( E + 1 ) F - F E = F \\geq D , \\end{align*}"}
{"id": "2295.png", "formula": "\\begin{align*} \\int _ 1 ^ x I _ 2 { \\rm d } x \\leq \\delta _ 0 ( \\Vert ( \\varphi , \\psi ) x ^ { - \\frac { 3 } { 4 } - \\sigma _ i } \\Vert _ { L _ x ^ 2 L _ y ^ 2 } ^ 2 + \\Vert ( u , h ) x ^ { - \\frac { 3 } { 4 } - \\sigma _ i } \\Vert _ { L _ x ^ 2 L _ y ^ 2 } ^ 2 ) + C . \\end{align*}"}
{"id": "2762.png", "formula": "\\begin{align*} \\nabla K _ { j } ( 0 ) = 0 , \\end{align*}"}
{"id": "3570.png", "formula": "\\begin{align*} w _ 2 ( \\pi ) = \\begin{cases} \\dfrac { m _ { \\pi } } { 2 } a _ 2 , & q \\equiv 1 \\mod 4 , \\\\ \\\\ \\dbinom { m _ \\pi } { 2 } ( \\sum v _ i ^ 2 ) , & q \\equiv 3 \\mod 4 , \\end{cases} \\end{align*}"}
{"id": "299.png", "formula": "\\begin{align*} C _ a ^ v \\ : = \\ \\big \\{ c \\in \\N \\ ; : \\ ; p \\mid c \\implies p \\in [ P ( a ^ * ) , P ( a ^ * ) ^ { 1 / \\sqrt { v } } ) \\big \\} . \\end{align*}"}
{"id": "4788.png", "formula": "\\begin{align*} \\{ B G \\} \\L ^ d = \\{ [ V _ K / G ] \\} - \\sum _ { 0 < n _ f < m } ( - 1 ) ^ { n _ f } \\{ B N _ G ( f ) \\} \\L ^ { d _ f } - ( - 1 ) ^ m \\sum _ { n _ f = m } \\{ [ V _ { H _ f } / N _ G ( f ) ] \\} , \\end{align*}"}
{"id": "1467.png", "formula": "\\begin{align*} g ( \\underline { X } , \\underline { Y } ) = g ( \\alpha , \\beta , \\underline { X } , \\underline { Y } ) = \\prod _ { { { 0 } } \\leq i , j \\leq { { r - 1 } } } ( X _ i - Y _ j ) \\prod _ { { { 0 } } \\leq i < j \\leq { { r - 1 } } } [ ( X _ j - X _ i ) ( Y _ j - Y _ i ) ] \\enspace . \\end{align*}"}
{"id": "5379.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ^ * + \\sum _ { | \\alpha | \\leq m } ( - 1 ) ^ { | \\alpha | } D ^ \\alpha ( a _ \\alpha u ^ * ) & = 0 \\mbox { i n } \\ ; \\ ; \\Omega , u ^ * - f \\in \\widetilde H ^ s ( \\Omega ) \\end{align*}"}
{"id": "1374.png", "formula": "\\begin{align*} \\left | \\underset { \\Gamma _ { 2 p - 1 } } { \\int \\sum } \\overline { \\widehat { \\eta } } _ k \\widehat { ( W _ t u _ 0 ) } _ { k _ 1 } \\prod _ { \\substack { j = 3 \\\\ o d d } } ^ { 2 p - 1 } \\widehat { v } _ { k _ j } \\prod _ { \\substack { l = 2 \\\\ e v e n } } ^ { 2 p - 1 } \\overline { \\widehat { v } } _ { k _ l } \\ , d \\Gamma \\right | \\lesssim \\| \\eta \\| _ { X ^ { - s - \\varepsilon , 1 / 2 - } } \\| u _ 0 \\| _ { H ^ s _ x } \\| v \\| ^ { 2 p - 2 } _ { X ^ { s , 1 / 2 + } _ T } . \\end{align*}"}
{"id": "7413.png", "formula": "\\begin{align*} \\sum _ { h = 1 } ^ H \\frac { 1 } { h \\| h \\alpha \\| } \\ll \\left \\{ \\begin{array} { l l } \\log ^ 2 H & \\mathrm { i f } \\ , \\ , \\ , \\gamma = 1 , \\\\ H ^ { \\gamma - 1 } & \\mathrm { i f } \\ , \\ , \\ , \\gamma > 1 \\end{array} \\right . \\end{align*}"}
{"id": "8959.png", "formula": "\\begin{align*} \\| u \\| _ { X ^ p _ \\Delta ( I ; E ) } = \\| e ^ { - i t \\Delta } u \\| _ { X ^ p ( I ; E ) } \\end{align*}"}
{"id": "633.png", "formula": "\\begin{align*} | R _ N ^ 1 ( s , n ) | & \\leqslant C \\log ( n ) \\int _ 0 ^ n z ^ { 2 \\alpha - 2 s - 1 } \\leq C , \\ \\textup { f o r } \\ N = M + 1 , \\\\ | R _ { N ' } ^ 2 ( s , n ) | & \\leqslant C \\log ( n ) \\int _ 0 ^ n z ^ { 2 \\alpha - 2 s - 1 } \\leq C , \\ \\textup { f o r } \\ N ' = 2 ( M + 1 ) . \\end{align*}"}
{"id": "6559.png", "formula": "\\begin{align*} f ^ \\wedge \\colon U \\times E _ 1 \\times \\cdots \\times E _ k \\to F \\ , , f ^ \\wedge ( x , v ) : = f ( x ) ( v ) \\quad \\mbox { f o r $ x \\in U $ , $ v \\in E _ 1 \\times \\cdots \\times E _ k $ } \\end{align*}"}
{"id": "6265.png", "formula": "\\begin{align*} h ( q ^ n x ) = \\frac { ( q x ; q ) _ n } { x ^ n } h ( x ) ( n \\in \\mathbb { N } _ 0 ) . \\end{align*}"}
{"id": "296.png", "formula": "\\begin{align*} f ( A _ n ) = \\sum _ { a \\in A _ n } \\frac { 1 } { a \\log a } \\ & \\le \\ \\frac { e ^ { \\gamma } } { m _ q } \\ , \\frac { { \\rm d } ( { \\rm L } _ { A _ n } ) } { 1 + v } . \\end{align*}"}
{"id": "5696.png", "formula": "\\begin{align*} \\prod _ { i \\in J } \\varpi _ i = \\prod _ { k = 1 } ^ m \\Big ( \\prod _ { i \\in J _ k } \\varpi _ i \\Big ) = \\prod _ { k = 1 } ^ m \\Big ( | J _ k | ! e _ { | J _ k | } ( x _ 1 , x _ 2 , \\ldots , x _ { \\max J _ k } ) \\Big ) \\end{align*}"}
{"id": "475.png", "formula": "\\begin{align*} \\beta _ 1 ( z ) & : = \\displaystyle \\sum _ { m = 1 } ^ { s _ 1 } \\frac { z ^ m } m , \\\\ \\beta _ i ( z ) & : = \\displaystyle \\beta _ { i - 1 } ( z ) + \\sum _ { m = 1 } ^ { s _ i } \\frac { ( 1 + ( z - 1 ) \\exp ( \\beta _ { i - 1 } ( z ) ) ) ^ m } m , 2 \\le i \\le q . \\end{align*}"}
{"id": "2966.png", "formula": "\\begin{align*} J ( \\gamma , z ) = \\frac { \\theta ^ * ( \\gamma z ) } { \\theta ^ * ( z ) } , \\theta ^ * ( z ) = y ^ { 1 / 4 } \\sum _ { n \\in \\Z } e ( n ^ 2 z ) , \\end{align*}"}
{"id": "7755.png", "formula": "\\begin{align*} { \\bf U } ^ k ( \\alpha _ i ) = \\displaystyle \\sum _ { j = 1 } ^ { N _ h } T _ j ^ k ( \\alpha _ i ) { \\boldsymbol \\xi } _ j k = 1 , \\ldots , m , \\ i = 1 , \\ldots , p . \\end{align*}"}
{"id": "4140.png", "formula": "\\begin{align*} U = ( - ( U _ 1 ^ 2 + \\dots + U _ { d _ 2 } ^ 2 ) ) ^ { 1 / 2 } \\end{align*}"}
{"id": "2267.png", "formula": "\\begin{align*} I _ 4 & = \\int _ 0 ^ r - \\frac { z } { 2 } ( e _ \\sigma + \\bar \\psi ) \\cdot ( e ' _ \\sigma \\phi + e ' _ \\delta \\psi ) { \\rm { d } } z \\\\ & \\leq \\frac { 1 } { 2 } \\Vert ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert z ( e ' _ \\delta , e ' _ \\sigma ) \\Vert _ { L ^ 2 } \\cdot \\left ( \\Vert e _ \\sigma \\Vert _ { L ^ \\infty } + \\Vert \\bar \\psi \\Vert _ { L ^ \\infty } \\right ) , \\end{align*}"}
{"id": "5643.png", "formula": "\\begin{align*} \\sigma _ k ( \\lambda ( D ^ 2 u ) ) = 1 \\end{align*}"}
{"id": "478.png", "formula": "\\begin{align*} S \\not \\subseteq \\bigcup _ { i = 1 } ^ n B ( x _ i , \\sin \\theta ) . \\end{align*}"}
{"id": "6059.png", "formula": "\\begin{align*} \\binom { j } { k _ 1 , k _ 2 , \\ldots , k _ { m } } = \\frac { j ! } { k _ 1 ! k _ 2 ! \\cdots k _ { m } ! } . \\end{align*}"}
{"id": "2606.png", "formula": "\\begin{align*} h ( \\vec x , \\vec p ) = \\sum _ { i = 1 } ^ 3 \\left ( \\dfrac { p _ i ^ 2 } { 2 m } + \\dfrac { 1 } { 2 } m \\omega x _ i ^ 2 \\right ) \\ , \\end{align*}"}
{"id": "2758.png", "formula": "\\begin{align*} - \\gamma \\leq J _ { 2 } \\leq - \\int _ { B _ { R - 2 } \\setminus B _ { 6 } } \\frac { d y } { | y | ^ { n + \\sigma p } } = - ( 6 ^ { - \\sigma p } - ( R - 2 ) ^ { - \\sigma p } ) \\gamma . \\end{align*}"}
{"id": "8093.png", "formula": "\\begin{align*} Z = f ( p \\partial _ { 1 } + q \\partial _ { 2 } ) . \\end{align*}"}
{"id": "221.png", "formula": "\\begin{align*} F ( x ) = ( \\cdots ( ( a _ 0 x + a _ 1 ) ^ { 2 ^ n - 2 } + a _ 2 ) ^ { 2 ^ n - 2 } \\cdots + a _ m ) ^ { 2 ^ n - 2 } + a _ { m + 1 } , \\end{align*}"}
{"id": "5208.png", "formula": "\\begin{align*} { H } ^ \\Lambda = F _ \\Lambda ^ * \\big ( \\boxplus _ { i = 1 } ^ v H ^ { \\Xi _ i } \\big ) , \\end{align*}"}
{"id": "2506.png", "formula": "\\begin{align*} g ( ( \\nabla _ X A ) _ W X , W ) = g ( \\nabla _ X A _ W X , W ) - g ( A _ { \\mathcal { H } \\nabla _ X W } X , W ) - g ( A _ W A _ X X , W ) . \\end{align*}"}
{"id": "7129.png", "formula": "\\begin{align*} n ( 1 + y ) s _ n ( y ) = y s _ n ^ { * } ( y ) - s _ { n - 1 } ^ { * } ( y ) + ( 1 + y ) \\sum _ { i = 1 } ^ { n - 1 } s _ { i } ^ { * } ( y ) s _ { n - i } ( y ) , \\end{align*}"}
{"id": "867.png", "formula": "\\begin{align*} s x y & = \\pi ( s ^ \\prime x ^ \\prime y ^ \\prime ) & ( \\ ; \\pi \\ ; ) \\\\ & = \\pi ( x ^ \\prime y ^ \\prime t ^ \\prime ) & ( \\ ; s ^ \\prime x ^ \\prime y ^ \\prime = x ^ \\prime y ^ \\prime t ^ \\prime ) \\\\ & = x y t , & \\end{align*}"}
{"id": "1833.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { B } _ { n } } w ( \\gamma ) = \\sum _ { \\gamma \\in \\mathcal { B } _ { n } } w ( \\gamma _ { 1 } ) w ( \\gamma _ { 2 } ) w ( \\gamma _ { 3 } ) w ( \\gamma _ { 4 } ) = a _ { 0 } \\sum _ { \\gamma \\in \\mathcal { B } _ { n } } w ( \\gamma _ 2 ) w ( \\gamma _ 4 ) = a _ { 0 } \\sum _ { k = 0 } ^ { n - 2 } A _ { k } ^ { ( 1 ) } A _ { n - k - 2 } , \\end{align*}"}
{"id": "345.png", "formula": "\\begin{align*} d ^ { ( \\infty ) } ( t ) = n p _ t ( o , o ) - 1 . \\end{align*}"}
{"id": "7472.png", "formula": "\\begin{align*} h \\left ( \\left \\{ n , \\dots , n + M - 1 \\right \\} \\right ) = h \\left ( \\mathbb { N } \\right ) . \\end{align*}"}
{"id": "582.png", "formula": "\\begin{align*} M _ 4 = f _ k ( j ) f _ k ( j + 1 ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) \\tau _ 3 ( j + 1 ) } { \\sqrt { j - i - 1 } } \\Big ) . \\end{align*}"}
{"id": "5607.png", "formula": "\\begin{align*} 1 = \\left ( \\frac { - 2 P _ 1 ^ 2 P _ 2 x z ^ 3 } { 2 P _ 1 P _ 2 P _ 3 x z - y ^ 2 } \\right ) = \\left ( \\frac { - 2 P _ 2 x z } { 2 P _ 1 P _ 2 P _ 3 x z - y ^ 2 } \\right ) . \\end{align*}"}
{"id": "3087.png", "formula": "\\begin{align*} \\boldsymbol { \\mu } = \\left ( \\begin{array} { c c c c c c c c c c c } 3 \\\\ 4 \\end{array} \\right ) , \\mathbf { \\Sigma } = \\left ( \\begin{array} { c c c c c c c c c c c } 0 . 9 & 0 . 5 \\\\ 0 . 5 & 0 . 7 \\end{array} \\right ) , ~ \\boldsymbol { \\gamma } = \\left ( \\begin{array} { c c c c c c c c c c c } 1 \\\\ 2 \\end{array} \\right ) . \\end{align*}"}
{"id": "7651.png", "formula": "\\begin{align*} Z = \\left \\{ P \\in G _ { n , s , m } \\ , \\colon \\ , \\omega \\big | _ { P } \\equiv 0 \\right \\} \\end{align*}"}
{"id": "6718.png", "formula": "\\begin{align*} \\varphi \\in \\prescript { } { m } { } _ { m } h _ l \\in ( \\mathbb { C } ) , \\ ; \\ ; \\ ; l = 0 , \\dots , m . \\end{align*}"}
{"id": "3381.png", "formula": "\\begin{align*} \\beta : = \\limsup _ { x \\to \\infty } \\frac { x b ' ( x ) } { b ( x ) } . \\end{align*}"}
{"id": "4927.png", "formula": "\\begin{align*} R _ { i j k l } = \\bar { R } _ { \\alpha \\beta \\gamma \\delta } X _ { ; i } ^ \\alpha X _ { ; j } ^ \\beta X _ { ; k } ^ \\gamma X _ { ; l } ^ \\delta + h _ { i l } h _ { j k } - h _ { i k } h _ { j l } , \\end{align*}"}
{"id": "1178.png", "formula": "\\begin{align*} \\sqrt { n } ( \\hat { \\mu } _ n - \\mu ) * \\gamma _ { \\sigma } = \\tau _ { a } \\big ( \\sqrt { n } ( \\hat { \\mu } _ n ^ { - a } - \\mu ^ { - a } ) * \\gamma _ { \\sigma } \\big ) \\stackrel { d } { \\to } \\tau _ a G ^ { \\circ } _ { \\mu ^ { - a } } = : G _ { \\mu } \\ \\dot { H } ^ { - 1 , p } ( \\mu * \\gamma _ { \\sigma } ) . \\end{align*}"}
{"id": "3937.png", "formula": "\\begin{align*} V a r ( \\hat { \\pi } ^ a _ { h ^ * , T _ n } ( x ) ) & = \\frac { 2 } { T _ n ^ 2 } \\int _ 0 ^ { T _ n } \\int _ 0 ^ { t } k ( t , s ) 1 _ { s < t } d s d t . \\end{align*}"}
{"id": "4497.png", "formula": "\\begin{align*} \\zeta _ 1 = \\sqrt { \\frac { 1 } { \\mathfrak { A } t } } x _ 1 , \\zeta _ 2 = \\sqrt { \\frac { 1 } { \\mathfrak { A } t } } x _ 2 , \\end{align*}"}
{"id": "2309.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\left ( \\frac { 1 } { y } \\int _ 0 ^ y f ( t ) { \\rm d } t \\right ) ^ p y ^ \\alpha { \\rm d } y \\leq \\left ( \\frac { p } { p - \\alpha - 1 } \\right ) ^ p \\int _ 0 ^ \\infty f ( y ) ^ p y ^ \\alpha { \\rm d } y . \\end{align*}"}
{"id": "6958.png", "formula": "\\begin{align*} F _ 1 & = \\frac { F } { 1 - F } \\intertext { s o } \\frac { F } { F _ 1 } & = 1 - F . \\end{align*}"}
{"id": "5336.png", "formula": "\\begin{align*} \\tau _ k = \\begin{cases} t - \\delta _ { n _ k + 1 } & n _ k < \\infty , \\\\ t & n _ k = \\infty . \\end{cases} \\end{align*}"}
{"id": "8994.png", "formula": "\\begin{align*} \\lambda _ n = \\frac { 1 } { n } \\delta _ { n - 1 } + \\frac { n - 1 } { n } \\lambda _ { n - 1 } , n \\geq 2 . \\end{align*}"}
{"id": "3019.png", "formula": "\\begin{align*} P _ F ( t ) \\cdot \\hat { L } _ { x y } = \\frac { 1 } { n } \\left ( \\frac { 3 } { 4 ( 4 n + 1 ) } - t \\right ) , \\ \\ P _ F ( t ) \\cdot \\hat { R } _ 0 = P _ F ( t ) \\cdot \\hat { R } _ 1 = \\frac { 3 } { 4 ( 4 n + 1 ) } - t . \\end{align*}"}
{"id": "6027.png", "formula": "\\begin{align*} a = - \\frac { 3 } { 2 } , b = \\frac { 3 } { 2 } , c = - \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "3897.png", "formula": "\\begin{align*} \\Lambda = \\| \\mathcal G \\| ^ \\frac { 1 } { p - 1 } _ { q _ 0 , B _ 2 ( 0 ) } < + \\infty \\end{align*}"}
{"id": "845.png", "formula": "\\begin{align*} \\hat { p } \\left ( s _ { m } ^ { r } | \\boldsymbol { y } \\right ) = \\frac { \\pi _ { s ^ { r } , m } ^ { i n } \\pi _ { s ^ { r } , m } ^ { o u t } } { \\pi _ { s ^ { r } , m } ^ { i n } \\pi _ { s ^ { r } , m } ^ { o u t } + ( 1 - \\pi _ { s ^ { r } , m } ^ { i n } ) ( 1 - \\pi _ { s ^ { r } , m } ^ { o u t } ) } , \\forall m , \\end{align*}"}
{"id": "4830.png", "formula": "\\begin{align*} R _ \\mathrm { r a t } ( G ) = \\big \\{ \\rho \\in R ( G ) \\ ; : \\ ; \\chi _ \\rho ( g ) \\in \\Q g \\in G \\big \\} \\ ; . \\end{align*}"}
{"id": "8587.png", "formula": "\\begin{align*} m = n _ l ( m ) - n _ { l - 1 } ( m ) - \\ldots - n _ { 1 } ( m ) \\end{align*}"}
{"id": "2685.png", "formula": "\\begin{align*} T : = \\exp \\Big \\{ - \\frac { 2 } { 1 + s / 2 } \\big ( L / 2 \\big ) ^ { 1 + s / 2 } \\Big \\} , \\end{align*}"}
{"id": "1964.png", "formula": "\\begin{align*} \\mathbf { d } _ { k } : = ( 0 , \\ldots , 0 , - a _ { 0 } ^ { ( k - 1 ) } , - a _ { 1 } ^ { ( k - 2 ) } , \\ldots , - a _ { k - 2 } ^ { ( 1 ) } , z - a _ { k - 1 } ^ { ( 0 ) } ) \\end{align*}"}
{"id": "1772.png", "formula": "\\begin{align*} y ^ * = b \\widetilde { z } + \\sum _ { j = 1 } ^ m \\left ( c _ j \\widetilde { x } ^ k _ j + v _ j ^ * \\right ) , \\end{align*}"}
{"id": "5195.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i \\in I } 2 d _ i + \\frac { 2 \\sum _ { i \\in I ( \\Gamma ) } a _ i + ( k _ 1 ( \\Gamma ) + k _ { 1 2 } ( \\Gamma ) - 1 ) ( r - 2 ) } { r } & + \\frac { 2 \\sum _ { i \\in I ( \\Gamma ) } b _ i + ( k _ 2 ( \\Gamma ) + k _ { 1 2 } ( \\Gamma ) - 1 ) ( s - 2 ) } { s } \\\\ & = 2 | I | + k _ 1 ( \\Gamma ) + k _ 2 ( \\Gamma ) + k _ { 1 2 } ( \\Gamma ) - 3 . \\end{aligned} \\end{align*}"}
{"id": "7303.png", "formula": "\\begin{align*} \\alpha _ 1 & \\lesssim \\int _ 0 ^ \\infty e ^ { \\mu _ 1 s _ 1 } \\{ ( \\lambda \\dot \\lambda - \\sigma ) + \\lambda ^ \\frac { n - 2 } { 2 } \\eta ^ \\frac { 2 } { 1 - q } ( T - t ) ^ { { \\sf d } _ 1 } + ( T - t ) ^ { { \\sf d } _ 1 } \\sigma \\} d s _ 1 \\\\ & \\lesssim ( T - t ) ^ { { \\sf d } _ 1 } \\sigma | _ { t = 0 } \\int _ 0 ^ \\infty e ^ { \\mu _ 1 s _ 1 } d s _ 1 \\\\ & \\lesssim ( T - t ) ^ { { \\sf d } _ 1 } \\sigma | _ { t = 0 } . \\end{align*}"}
{"id": "2916.png", "formula": "\\begin{align*} c _ n = 2 ^ { \\frac { n } { 2 } } c _ { \\frac { n } { 2 } } = 2 ^ { \\frac { n } { 2 } + \\frac { n } { 4 } } c _ { \\frac { n } { 4 } } = \\ldots = 2 ^ { \\frac { n } { 2 } + \\frac { n } { 4 } + \\cdots + \\frac { n } { 2 ^ t } } c _ { \\frac { n } { 2 ^ t } } . \\end{align*}"}
{"id": "4847.png", "formula": "\\begin{align*} ( X , { \\delta _ X } \\dot { D f } \\wedge \\delta _ X D g ) & = ( X , { \\delta _ X } { D ^ 2 f } ( \\dot X , \\cdot ) \\wedge \\delta _ X D g ) = - ( X , \\delta _ X D g \\wedge { \\delta _ X } { D ^ 2 f } ( \\dot X , \\cdot ) ) \\\\ & = - ( d _ X \\rho _ X \\delta _ X d D g , { D ^ 2 f } ( \\dot X , \\cdot ) ) = - ( \\nu _ { X } D g , { D ^ 2 f } ( \\dot X , \\cdot ) ) \\\\ & = - ( \\dot X , { D ^ 2 f } ( \\nu _ { X } D g , \\cdot ) ) \\end{align*}"}
{"id": "5815.png", "formula": "\\begin{align*} \\beta ( s ) = \\frac { 3 } { 4 } \\zeta ( s ) - 2 \\sum _ { n \\geq 1 } \\frac { 1 } { ( 4 n + 3 ) ^ s } , \\end{align*}"}
{"id": "5230.png", "formula": "\\begin{align*} v = g x ^ a y ^ b ( ( b + 1 ) x \\partial _ x - ( a + 1 ) y \\partial _ y ) \\in \\mathfrak { g } _ { A , ( a , b ) } , \\end{align*}"}
{"id": "4171.png", "formula": "\\begin{align*} d _ { \\mathrm { C C } } ( a g , a h ) = d _ { \\mathrm { C C } } ( g , h ) \\quad a , g , h \\in G . \\end{align*}"}
{"id": "415.png", "formula": "\\begin{align*} F _ { \\mu } ( z ) + \\varphi _ { \\mu } ( F _ { \\mu } ( z ) ) = z \\end{align*}"}
{"id": "5654.png", "formula": "\\begin{align*} | \\overline { w } ( x ) | = \\begin{cases} O ( | x | ^ { 2 - \\beta } + | x | ^ { - 2 } + | x | ^ { 2 - n } ) , & \\ \\min \\{ \\beta , 4 \\} \\neq n , \\\\ O ( | x | ^ { 2 - n } \\ln | x | ) , & \\ \\min \\{ \\beta , 4 \\} = n , \\end{cases} \\end{align*}"}
{"id": "7676.png", "formula": "\\begin{align*} F ( x ; \\sigma , y , Z ) : = f ( x ) + \\frac { 1 } { 2 \\sigma } \\norm { \\sigma y - g ( x ) } ^ 2 + \\frac { 1 } { 2 \\sigma } \\norm { \\left [ \\sigma Z - X ( x ) \\right ] _ { + } } _ { \\mathrm { F } } ^ 2 , \\end{align*}"}
{"id": "3947.png", "formula": "\\begin{align*} { } I _ 3 & \\le \\frac { c } { T _ n ^ 2 } \\int _ 0 ^ { T _ n } \\int _ { \\frac { 1 } { 2 } ( \\prod _ { l \\ge 3 } h _ l ^ * ) ^ { \\frac { 2 } { d - 2 } } } ^ D ( \\frac { c } { s '^ { \\frac { d } { 2 } } } + c ) d s ' d t = \\frac { c } { T _ n } ( ( \\prod _ { l \\ge 3 } h _ l ^ * ) ^ { - 1 } + D ) , \\end{align*}"}
{"id": "470.png", "formula": "\\begin{align*} \\ln ^ { ( 0 ) } x : = x , & \\ln ^ { ( s + 1 ) } x : = \\ln ( \\ln ^ { ( s ) } x ) , \\\\ \\exp ^ { ( 0 ) } x : = x , & \\exp ^ { ( s + 1 ) } x : = \\exp ( \\exp ^ { ( s ) } x ) , \\qquad s = 0 , 1 , 2 , \\dots \\end{align*}"}
{"id": "7017.png", "formula": "\\begin{align*} S _ i ( t ) = \\{ S _ { s , i } ( t ) \\cap S _ { 0 , i } ( t ) \\} , \\end{align*}"}
{"id": "1010.png", "formula": "\\begin{align*} x = [ 0 ; a _ 1 , a _ 2 , \\dots ] = \\smash { \\frac 1 { a _ 1 + \\frac 1 { a _ 2 + \\frac 1 { a _ 3 + \\cdots } } } } \\phantom { \\frac 9 { 9 _ y } } \\end{align*}"}
{"id": "701.png", "formula": "\\begin{align*} g _ { i j } = - u _ i u _ j + f ( u ) ^ 2 \\widetilde { g } _ { i j } . \\end{align*}"}
{"id": "913.png", "formula": "\\begin{align*} ( n I _ n - J _ n ) \\cdot \\begin{bmatrix} h _ { 1 , 1 1 } \\\\ h _ { 1 , 2 2 } \\\\ \\cdots \\\\ h _ { 1 , n n } \\end{bmatrix} = ( n I _ n - J _ n ) \\cdot \\begin{bmatrix} a _ { 1 1 } \\\\ a _ { 2 2 } \\\\ \\cdots \\\\ a _ { n n } \\end{bmatrix} . \\end{align*}"}
{"id": "7227.png", "formula": "\\begin{align*} u ^ 0 ( x , t ) = \\frac { 1 } { 2 } \\{ f ( x + t ) + f ( x - t ) \\} \\ge \\frac { 1 } { 2 } f ( x - t ) \\mbox { f o r } \\ ( x , t ) \\in \\R \\times [ 0 , \\infty ) . \\end{align*}"}
{"id": "7571.png", "formula": "\\begin{align*} \\frac { d g } { d t } = \\sum _ { \\alpha = 1 } ^ r b _ \\alpha ( t ) X _ \\alpha ^ R ( g ) , g \\in G , t \\in \\mathbb { R } , \\end{align*}"}
{"id": "7752.png", "formula": "\\begin{align*} u _ { m } ( x , { \\bf y } ; \\alpha ) = \\displaystyle \\sum _ { k = 1 } ^ { m } \\sum _ { j = 1 } ^ { N _ h } { \\tilde u } _ { k , j } ^ \\alpha \\vartheta _ j ( x ) \\varphi _ k ( \\psi _ x ( { \\bf y } ) ) \\end{align*}"}
{"id": "8085.png", "formula": "\\begin{align*} Z = f ( r , \\theta ) \\partial _ { \\theta } . \\end{align*}"}
{"id": "7757.png", "formula": "\\begin{align*} S _ j = R _ j D _ j P _ j ^ T , \\end{align*}"}
{"id": "6204.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + p ( x ) D _ q y ( x ) + r ( x ) y ( x ) = 0 , \\end{align*}"}
{"id": "342.png", "formula": "\\begin{align*} s ( g ) = d - 2 , \\end{align*}"}
{"id": "2021.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m a i n j u m p g e n e r a t o r } L _ \\mu \\varphi ( x ) = a \\cdot \\nabla \\varphi ( x ) + \\lambda ( x , \\mu ) \\int _ { E } \\{ \\varphi ( y ) - \\varphi ( x ) \\} P _ \\mu ( x , \\dd y ) , \\end{align*}"}
{"id": "1891.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { P } _ { [ n , j , k ] } } w ( \\gamma ) = W _ { [ k , i ] } A _ { [ n - k - 1 , j - i - 1 ] } ^ { ( i + 1 ) } . \\end{align*}"}
{"id": "4123.png", "formula": "\\begin{align*} T ^ { i - 1 , j + 1 } \\circ T ^ { i , j } = 0 , \\end{align*}"}
{"id": "8125.png", "formula": "\\begin{align*} F ( t ) & = P f ( \\cdot , t ) , \\\\ F _ n ( t ) & = P _ n f ( \\cdot , t ) . \\end{align*}"}
{"id": "6541.png", "formula": "\\begin{align*} a _ i ^ { - 1 } = \\frac { \\left ( \\AA ( i ) \\right ) ^ i } { i ! } \\geq \\exp \\left \\{ - \\AA ( m - i , m ) \\right \\} \\frac { \\left ( \\AA ( m - i , m ) \\right ) ^ { i } } { i ! } . \\end{align*}"}
{"id": "8223.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c } \\phi ( + a / 2 ) = e ^ { i \\varphi } . \\phi ( - a / 2 ) \\\\ \\partial _ { x } \\phi ( + a / 2 ) = e ^ { i \\varphi } . \\partial _ { x } \\phi ( - a / 2 ) \\\\ \\partial _ { x } ^ { 2 } \\phi ( + a / 2 ) = e ^ { i \\varphi } . \\partial _ { x } ^ { 2 } \\phi ( - a / 2 ) \\\\ \\partial _ { x } ^ { 3 } \\phi ( + a / 2 ) = e ^ { i \\varphi } . \\partial _ { x } ^ { 3 } \\phi ( - a / 2 ) \\end{array} \\right . . \\end{align*}"}
{"id": "4438.png", "formula": "\\begin{align*} ( J _ m ^ k f ) ( x , r \\xi ) & = \\frac { r ^ { m - k } } { | r | } ( J _ m ^ k f ) ( x , \\xi ) \\mbox { f o r } r \\neq 0 \\\\ ( J _ m ^ k f ) ( x + s \\xi , \\xi ) & = \\sum \\limits _ { l = 0 } ^ k \\binom { k } { l } ( - s ) ^ { k - l } ( J _ m ^ l f ) ( x , \\xi ) \\mbox { f o r } s \\in \\mathbb { R } . \\end{align*}"}
{"id": "7328.png", "formula": "\\begin{align*} { \\sf t } _ 9 & = \\{ f ( { \\sf U } _ \\infty + \\Theta _ { \\sf a } + \\theta ) - f _ 2 ( { \\sf U } _ \\infty + \\Theta _ { \\sf a } + \\theta ) + f _ 2 ( { \\sf U } _ \\infty ) + { \\sf U } _ \\infty ^ { q - 1 } ( \\Theta _ { \\sf a } + \\theta ) + P _ 1 \\theta \\} \\\\ & \\times ( 1 - \\chi _ { 2 , { \\sf a } } ) \\chi _ { 3 , { \\sf e } } . \\end{align*}"}
{"id": "8798.png", "formula": "\\begin{align*} \\phi ( g ) = \\int _ { 0 } ^ { \\infty } 1 _ { \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) } ( g ) d t \\end{align*}"}
{"id": "7762.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\displaystyle \\ \\ \\sum _ { j = 1 } ^ L \\lambda _ j ^ 2 \\ \\ } { \\displaystyle \\sum _ { j = 1 } ^ { N _ h } \\lambda _ j ^ 2 } \\ge \\epsilon _ 1 , \\frac { \\ \\ \\displaystyle \\sum _ { j = 1 } ^ { \\mu _ j } d _ { j , k } ^ 2 \\ \\ } { \\displaystyle \\sum _ { j = 1 } ^ { m } d _ { j , k } ^ 2 } \\ge \\epsilon _ 2 , \\end{align*}"}
{"id": "4936.png", "formula": "\\begin{align*} \\theta = \\log W + p ( u ) + N \\rho , \\end{align*}"}
{"id": "2281.png", "formula": "\\begin{align*} h _ e ^ 1 = \\sigma u _ e ^ 1 . \\end{align*}"}
{"id": "7453.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\abs { u _ x } ^ 2 d x = o ( 1 ) . \\end{align*}"}
{"id": "599.png", "formula": "\\begin{align*} V _ \\alpha ( s ) = 2 \\ , \\frac { \\sin ( \\pi s ) } { \\pi } \\frac { \\Gamma ( 1 - s ) \\Gamma ( \\alpha ) } { \\Gamma ( \\alpha - s ) } . \\end{align*}"}
{"id": "5939.png", "formula": "\\begin{align*} \\left \\langle A _ { { i _ 1 } { j _ 1 } } \\left ( { { t _ 1 } } \\right ) . . . A _ { { i _ n } { j _ n } } \\left ( { { t _ n } } \\right ) \\right \\rangle _ c = W _ { { i _ 1 } { j _ 1 } . . . { i _ n } { j _ n } } ^ { \\left ( n \\right ) } \\left ( { { t _ 1 } - { t _ 2 } , \\ldots , { t _ 1 } - { t _ n } } \\right ) \\end{align*}"}
{"id": "2554.png", "formula": "\\begin{align*} \\omega > \\tau \\ \\ \\mbox { i f } \\ \\ \\omega - \\tau = c _ 1 \\alpha _ 1 + c _ 2 \\alpha _ 2 \\ \\ \\mbox { f o r n o n n e g a t i v e i n t e g e r s } \\ ( c _ 1 , c _ 2 ) \\ne ( 0 , 0 ) . \\end{align*}"}
{"id": "1771.png", "formula": "\\begin{align*} y _ n ^ * = y ^ * - \\frac { 1 } { a } ( z + a f _ k ) ( y ^ * ) \\widetilde { x } ^ k _ n , \\end{align*}"}
{"id": "7408.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { k = 1 } ^ N f ( Z _ k ) \\to \\int _ 0 ^ 1 f ( x ) \\ , \\mathrm { d } x \\textrm { a . s . } \\end{align*}"}
{"id": "2385.png", "formula": "\\begin{align*} \\int _ { G _ 1 } f \\ , d \\mu _ { G _ 1 } = \\int _ { - \\infty } ^ \\infty \\int _ { - \\infty } ^ \\infty f [ x , a ] \\ , \\frac { d x \\ , d a } { a ^ 2 } , \\quad f \\in C _ c ( G _ 1 ) . \\end{align*}"}
{"id": "3099.png", "formula": "\\begin{align*} a ( u , v ) = \\lambda \\ , b ( u , v ) v \\in V \\end{align*}"}
{"id": "7358.png", "formula": "\\begin{align*} \\Phi ( E _ { \\lambda , c } [ f ] ) \\Delta = \\Delta E _ { \\lambda , c - \\hbar } [ f ' ] , \\ \\Phi ( F _ { \\lambda , c } [ f ] ) \\Delta = \\Delta E _ { \\lambda , c - \\hbar } [ f ' ] \\end{align*}"}
{"id": "281.png", "formula": "\\begin{align*} J ^ r ( R _ { \\pi } [ y ] ) : = R _ { \\pi } [ ( \\delta ^ { ( s ) } ) _ \\mu y _ j \\ | \\ \\mu \\in { \\mathbb M } _ n ^ r , j \\in \\{ 1 , \\ldots , N \\} ] ^ { \\widehat { \\ } } \\end{align*}"}
{"id": "6500.png", "formula": "\\begin{align*} \\int _ { ( 0 , b _ 1 + \\varepsilon ) \\cup ( b _ 2 - \\varepsilon , c _ 1 ) } \\abs { v } ^ 2 d x + \\int _ { ( 0 , b _ 1 ) \\cup ( b _ 2 , c _ 1 ) } | u _ x | ^ 2 d x = o ( 1 ) . \\end{align*}"}
{"id": "3915.png", "formula": "\\begin{align*} a _ j = \\frac { \\bar { \\beta } _ 3 } { \\beta _ j ( 2 \\bar { \\beta } _ 3 + d - 2 ) } \\mbox { f o r a n y } j \\in \\{ 1 , . . . , d \\} . \\end{align*}"}
{"id": "3722.png", "formula": "\\begin{align*} \\phi _ { \\lambda C _ 1 , \\lambda C _ 2 } = \\lambda \\circ \\phi _ { C _ 1 , C _ 2 } \\circ \\lambda ^ { - 1 } \\end{align*}"}
{"id": "35.png", "formula": "\\begin{align*} ( x , x ) _ { \\rho , L , n } = 0 . \\end{align*}"}
{"id": "7493.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } A _ n ( t ) \\frac { z ^ n } { n ! } = \\frac { t - 1 } { t - e ^ { ( t - 1 ) z } } . \\end{align*}"}
{"id": "1213.png", "formula": "\\begin{align*} T _ n : = \\left \\{ t \\in \\R ^ { d _ 0 } : \\abs { t } \\leq C ^ { - 1 } ( 4 \\big \\| \\mathbb { G } _ n ^ { ( \\sigma ) } \\big \\| _ { \\dot H ^ { - 1 , p } ( \\mu * \\gamma _ { \\sigma } ) } + 2 \\lambda _ n ) \\right \\} . \\end{align*}"}
{"id": "6452.png", "formula": "\\begin{align*} \\ell _ 2 ( e _ 1 , f e _ 2 ) = \\rho ( e _ 1 ) [ f ] e _ 2 + f \\ell _ 2 ( e _ 1 , e _ 2 ) \\end{align*}"}
{"id": "1226.png", "formula": "\\begin{align*} \\deg \\mathcal { F } _ 1 ( v , x ) & = \\deg \\mathcal { F } _ 2 ( v , x ) + 1 . \\end{align*}"}
{"id": "6168.png", "formula": "\\begin{align*} \\Psi ^ { \\ast } ( d i s t ^ { 2 } ( \\Psi ( x ) , \\mathbf { S } ^ { 3 } ) ) = d i s t ^ { 2 } ( x , \\mathbf { S } _ { p } ^ { 1 } ) . \\end{align*}"}
{"id": "8621.png", "formula": "\\begin{align*} \\sum _ \\xi m _ \\xi [ \\mathfrak { M } ^ { e v } _ { \\xi } ] = \\sum _ \\tau \\frac { m _ { \\tau } } { | A u t ( \\tau ) | } i _ * [ \\mathfrak { M } ^ { e v } ( \\mathcal { X } , \\tau ) _ { s , r e d } ] . \\end{align*}"}
{"id": "7875.png", "formula": "\\begin{align*} R + 1 > r _ { i + 1 } & = r _ 1 - \\Lambda _ 0 \\sqrt { a _ 0 } \\cdot \\sum _ { k = 1 } ^ i \\sqrt { t _ k } \\\\ & \\geq R + 3 - \\Lambda _ 0 \\sqrt { a _ 0 t _ i } \\cdot \\sum _ { k = 0 } ^ { \\infty } ( 1 + \\mu ) ^ { - k } \\\\ & = R + 3 - \\sqrt { t _ i } \\cdot \\frac { \\Lambda _ 0 \\sqrt { a _ 0 } ( 1 + \\mu ) } { \\mu } . \\end{align*}"}
{"id": "1415.png", "formula": "\\begin{align*} \\delta _ 0 = \\max _ { t _ 0 \\in \\mathbb { C } } \\left \\{ \\max \\left \\{ \\min _ { \\lambda \\in S ( \\Lambda _ 1 ) } | \\lambda - t _ 0 | - \\max _ { \\mu \\in S ( \\Lambda _ 2 ) } | \\mu - t _ 0 | , \\min _ { \\lambda \\in S ( \\Lambda _ 2 ) } | \\lambda - t _ 0 | - \\max _ { \\mu \\in S ( \\Lambda _ 1 ) } | \\mu - t _ 0 | \\right \\} \\right \\} , \\end{align*}"}
{"id": "4335.png", "formula": "\\begin{align*} \\left ( e ^ { - i \\hat { \\theta } } \\int _ { \\mathcal { E } _ k } \\Omega \\right ) \\leq 0 , \\forall k = 1 , 2 , \\ldots N - 1 . \\end{align*}"}
{"id": "5235.png", "formula": "\\begin{align*} f ^ { b - a } + ( a - b ) q f ' f ^ { b - a - 1 } = 1 . \\end{align*}"}
{"id": "8677.png", "formula": "\\begin{align*} \\Psi \\circ L _ t = \\Psi \\circ L _ 0 \\quad . \\end{align*}"}
{"id": "7596.png", "formula": "\\begin{align*} \\frac { d x } { d t } = 2 t - \\frac { x } { t } + \\frac { x ^ 2 } { t ^ 3 } \\ , , t \\geq 1 . \\end{align*}"}
{"id": "3967.png", "formula": "\\begin{align*} f ( z \\cdot e ) = z f ( e ) . \\end{align*}"}
{"id": "2018.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n l i n e a r i n t r o } \\partial _ t f _ t + \\nabla \\cdot ( f _ t K \\star f _ t ) = \\sigma \\Delta f _ t , \\end{align*}"}
{"id": "5888.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 ^ { k \\times \\infty } : = \\{ A = ( A _ { i j } ) _ { i , j = 1 } ^ { k , \\infty } \\colon ( A _ { i j } ) _ { j \\in \\mathbb { N } } \\in \\ell _ 2 , i = 1 , \\ldots , k \\} \\end{align*}"}
{"id": "989.png", "formula": "\\begin{align*} B ^ { * } = \\operatorname { d i a g } \\left \\{ b _ { 1 1 } , b _ { 2 2 } , \\cdots , b _ { n n } \\right \\} . \\end{align*}"}
{"id": "6811.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ 1 ( a u '' ) ' v ' d x & = ( a u '' v ' ) ( x _ 0 - \\delta ) - ( a u '' v ' ) ( 0 ) - \\int _ 0 ^ { x _ 0 - \\delta } a u '' v '' d x \\\\ & + \\int _ { x _ 0 - \\delta } ^ { x _ 0 + \\delta } ( a u '' ) ' v ' d x + ( a u '' v ' ) ( 1 ) - ( a u '' v ' ) ( x _ 0 + \\delta ) \\\\ & - \\int _ { x _ 0 + \\delta } ^ 1 a u '' v '' d x . \\end{aligned} \\end{align*}"}
{"id": "124.png", "formula": "\\begin{align*} \\| T _ { \\lambda ^ 0 } \\| _ { L ^ 1 \\cap L ^ \\infty } \\leq | \\mu - m ^ 2 | \\Big [ \\frac { \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } } { m ^ 2 _ t } + \\frac { 1 } { m ^ 4 _ t } + \\frac { c } { m ^ 2 _ t } { \\bf 1 } _ { d = 2 } + \\frac { c } { m _ t } { \\bf 1 } _ { d = 3 } \\Big ] . \\end{align*}"}
{"id": "1465.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { r - 1 } \\Psi _ { \\alpha } \\circ \\theta _ { X _ s } ( X ^ { i _ 0 } _ 0 \\cdots X ^ { i _ { r - 1 } } _ { r - 1 } ) = \\sum _ { s = 0 } ^ { r - 1 } \\frac { i _ s \\alpha ^ { \\vert \\underline i \\vert } } { \\prod _ { s ' = 0 } ^ { r - 1 } ( i _ { s ' } + \\gamma _ { r - s ' } ) \\cdots ( i _ { s ' } + \\gamma _ r ) } = \\vert \\underline { i } \\vert \\frac { \\alpha ^ { \\vert \\underline { i } \\vert } } { \\prod _ { s ' = 0 } ^ { r - 1 } ( i _ { s ' } + \\gamma _ { r - s ' } ) \\cdots ( i _ { s ' } + \\gamma _ r ) } \\enspace . \\end{align*}"}
{"id": "6067.png", "formula": "\\begin{align*} f ^ { ( p ) } ( x ) & = \\sum _ { k = 0 } ^ p \\binom { p } { k } ( p ) _ k ( x - \\lambda ) ^ { p - k } g ^ { ( p - k ) } ( x ) \\\\ & = ( x - \\lambda ) \\sum _ { k = 0 } ^ { p - 1 } \\binom { p } { k } ( p ) _ k ( x - \\lambda ) ^ { p - 1 - k } g ( x ) + p ! g ( x ) \\end{align*}"}
{"id": "3796.png", "formula": "\\begin{align*} E r r ^ 0 _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\sum _ { i = 0 , 1 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( t _ i ) \\cdot ( \\xi + \\eta + \\sigma ) + i \\mu _ 2 t _ i | \\sigma | + i \\mu t _ i | \\xi | + i \\mu _ 1 t _ i | \\eta | } \\clubsuit K _ { k _ 2 , j _ 2 ; n _ 2 } ^ { \\mu _ 2 , i _ 2 } ( t _ i , \\sigma , V ( t _ i ) ) \\cdot { } ^ 2 \\clubsuit K ( t _ i , \\xi , \\eta , X ( t _ i ) , V ( t _ i ) ) \\end{align*}"}
{"id": "7691.png", "formula": "\\begin{align*} & \\ ; \\ ; \\norm { \\mathrm { s v e c } ( \\left [ - X ( x _ k ) \\right ] _ { + } - \\left [ \\sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) \\right ] _ { + } ) } \\\\ = & \\ ; \\ ; \\norm { \\left [ - X ( x _ k ) \\right ] _ { + } - \\left [ \\sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) \\right ] _ { + } } _ { \\mathrm { F } } \\\\ \\leq & \\ ; \\ ; \\norm { \\sigma _ { k - 1 } Z _ { k - 1 } } _ { \\mathrm { F } } = \\sigma _ { k - 1 } \\norm { Z _ { k - 1 } } _ { \\mathrm { F } } , \\end{align*}"}
{"id": "111.png", "formula": "\\begin{gather*} S _ t ( x ) : = \\big < \\varphi _ 0 \\varphi _ x \\big > ^ { \\epsilon , L } _ { \\lambda , \\mu + 1 / t , m ^ 2 } , x \\in \\Lambda _ { \\epsilon , L } , \\\\ C _ t ( x ) : = \\Big ( - \\Delta ^ \\epsilon + m ^ 2 + \\frac { 1 } { t } \\Big ) ^ { - 1 } ( 0 , x ) , x \\in \\Lambda _ { \\epsilon , L } , \\end{gather*}"}
{"id": "108.png", "formula": "\\begin{align*} \\nu _ c ( g ) = \\inf \\{ \\nu \\in \\R : \\sup _ \\Lambda \\chi ^ \\Lambda ( g , \\nu ) < + \\infty \\} , \\end{align*}"}
{"id": "2630.png", "formula": "\\begin{align*} \\max _ { S \\in \\mathcal { B } ( \\bigcap _ { i = 1 } ^ m \\mathcal { F } _ i ) } | S | \\geqslant \\max _ { A \\subseteq E } \\biggl ( 2 \\cdot \\sum _ { i = 1 } ^ m r _ i ( E ) - ( m - 1 ) | A | - \\sum _ { i = 1 } ^ m r _ i ( E - A \\cap T _ i ^ c ) \\biggr ) \\end{align*}"}
{"id": "3975.png", "formula": "\\begin{align*} \\Pr \\big ( \\Delta ( C _ M ) > \\delta \\big ) & = \\Pr \\big ( \\Delta ( C _ M ) > \\delta \\mid { \\rm r a n k } ( M ) = k \\big ) \\Pr \\big ( { \\rm r a n k } ( M ) = k \\big ) \\\\ & ~ ~ + \\Pr \\big ( \\Delta ( C _ M ) > \\delta \\mid { \\rm r a n k } ( M ) < k \\big ) \\Pr \\big ( { \\rm r a n k } ( M ) < k \\big ) . \\end{align*}"}
{"id": "3727.png", "formula": "\\begin{align*} \\psi _ { k } ( x ) : = \\tilde { \\psi } ( | x | / 2 ^ k ) - \\tilde { \\psi } ( | x | / 2 ^ { k - 1 } ) , \\psi _ { \\leq k } ( x ) : = \\sum _ { l \\leq k } \\psi _ { l } ( x ) , \\psi _ { \\geq k } ( x ) : = 1 - \\psi _ { \\leq k - 1 } ( x ) , \\psi _ { [ k _ 1 , k _ 2 ] } ( x ) = \\sum _ { k \\in [ k _ 1 , k _ 2 ] \\cap \\Z } \\psi _ k ( x ) . \\end{align*}"}
{"id": "6021.png", "formula": "\\begin{align*} f ^ { ( j ) } ( x ) = g _ j ( x ) \\prod _ { i = 1 } ^ { m } ( x - \\lambda _ i ) ^ { \\mu _ { i , j } } \\end{align*}"}
{"id": "4650.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } t ^ { \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| u ( t , \\cdot ) - A \\Psi _ t \\| _ { q , h } = 0 . \\end{align*}"}
{"id": "5900.png", "formula": "\\begin{align*} \\mathbb { E } \\Big \\lVert \\sum _ { j = 1 } ^ n Z _ j V _ { \\bullet , j } \\Big \\lVert _ 2 ^ 2 = \\mathbb { E } \\Big [ \\sum _ { i , j = 1 } ^ n Z _ i Z _ j \\langle V _ { \\bullet , i } , V _ { \\bullet , j } \\rangle _ 2 \\Big ] = \\mathbb { E } [ Z _ 1 ^ 2 ] \\sum _ { j = 1 } ^ n \\langle V _ { \\bullet , i } , V _ { \\bullet , j } \\rangle _ 2 = k \\mathbb { E } [ Z _ 1 ^ 2 ] . \\end{align*}"}
{"id": "2737.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } ( - \\Delta ) ^ { \\sigma } _ { p } u _ { i } ( x ) = ( - \\Delta ) ^ { \\sigma } _ { p } u ( x ) - \\theta , \\end{align*}"}
{"id": "5275.png", "formula": "\\begin{align*} \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { k _ 1 ( 0 ) + r } a ^ { \\widehat { Q } ^ { + r } } + \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { k _ 2 ( 0 ) + s } b ^ { \\widehat { Q } ^ { + s } } = 1 + \\sum _ { j = 2 } ^ h \\left ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { Q } _ j } + \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { Q } _ j } \\right ) . \\end{align*}"}
{"id": "4856.png", "formula": "\\begin{align*} N = p _ { 1 } ^ { 2 } + p _ { 2 } ^ { 2 } + p _ { 3 } ^ { 2 } + p _ { 4 } ^ { 2 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k _ { 1 } } } . \\end{align*}"}
{"id": "2196.png", "formula": "\\begin{align*} b _ i \\le b ' _ { i + \\rho _ 1 - \\rho _ 2 + ( \\rho _ 2 - \\rho _ 1 + \\eta ) } = b ' _ { i + \\eta } ( i \\in \\Z ) . \\end{align*}"}
{"id": "5804.png", "formula": "\\begin{align*} a ( v ) = \\sum _ { i = 1 } ^ { v - 1 } ( 1 ) = v - 1 , \\end{align*}"}
{"id": "4232.png", "formula": "\\begin{align*} \\phi ( n ) \\geq \\phi \\left ( \\frac { 1 } { 8 } \\psi ^ { - 1 } ( a ( \\lfloor k / 4 \\rfloor ) ) \\right ) = a \\left ( 8 \\lfloor k / 4 \\rfloor \\right ) \\geq a ( 2 k ) , \\end{align*}"}
{"id": "286.png", "formula": "\\begin{align*} f ( A ) = \\sum _ { a \\in A } \\frac { 1 } { a \\log a } = \\sum _ { a \\in A } \\int _ 1 ^ \\infty a ^ { - t } \\dd { t } = \\int _ 1 ^ \\infty f _ t ( A ) \\dd { t } , \\end{align*}"}
{"id": "6207.png", "formula": "\\begin{align*} \\int f ( x ) \\left ( \\frac { d ^ 2 h } { d x ^ 2 } + p ( x ) \\frac { d h } { d x } + r ( x ) h ( x ) \\right ) y ( x ) d x = f ( x ) \\left ( \\frac { d h } { d x } y ( x ) - h ( x ) \\frac { d y } { d x } \\right ) , \\end{align*}"}
{"id": "757.png", "formula": "\\begin{align*} f ( t , x , w ) = \\sin ( \\lVert x \\rVert + w ) , g ( x ) = \\lVert x \\rVert ^ 2 , \\end{align*}"}
{"id": "9005.png", "formula": "\\begin{align*} \\zeta ( - n , x ) - \\zeta \\left ( - n , x + \\frac { 1 } { 2 } \\right ) = \\frac { 1 } { n + 1 } B _ { n + 1 } \\left ( x + \\frac { 1 } { 2 } \\right ) - \\frac { 1 } { n + 1 } B _ { n + 1 } ( x ) , \\end{align*}"}
{"id": "2367.png", "formula": "\\begin{align*} \\nu _ \\sigma ( l ( h _ \\sigma ) ) = \\nu _ \\sigma ( f ) . \\end{align*}"}
{"id": "1022.png", "formula": "\\begin{align*} B _ n = \\begin{pmatrix} 1 , & * ^ n , & 1 \\\\ 1 , & 0 ^ n , & 1 \\end{pmatrix} \\end{align*}"}
{"id": "1703.png", "formula": "\\begin{align*} \\begin{aligned} & \\big ( \\sum _ { - \\delta N \\le j \\le 0 } 2 ^ { - \\frac { 1 } { 2 } j q } I ^ q _ j \\big ) ^ { \\frac { 1 } { q } } \\ge \\frac { C } { ( \\ln N ) ^ 3 N ^ { \\frac { 1 } { 2 } } } ( \\sum _ { - \\delta N \\le j \\le 0 } 1 ) ^ { \\frac { 1 } { q } } = \\frac { C N ^ { \\frac { 1 } { q } - \\frac { 1 } { 2 } } } { ( \\ln N ) ^ 3 } . \\end{aligned} \\end{align*}"}
{"id": "6994.png", "formula": "\\begin{align*} F ( \\Delta ) = \\int _ \\Delta Q \\omega \\end{align*}"}
{"id": "2719.png", "formula": "\\begin{align*} b _ { r , \\alpha } : = \\Theta \\ , a ( u _ { r , \\alpha } ) b _ { r , \\alpha } ^ * : = \\Theta ^ { - 1 } a ^ * ( u _ { r , \\alpha } ) c _ { \\ell , \\alpha } : = \\Theta ^ { - 1 } a ( u _ { \\ell , \\alpha } ) c ^ * _ { \\ell , \\alpha } : = \\Theta a ^ * ( u _ { \\ell , \\alpha } ) . \\end{align*}"}
{"id": "5044.png", "formula": "\\begin{align*} \\gamma ( \\frac { \\lfloor n \\tau \\rfloor } n ) \\left ( \\tau - \\frac { \\lfloor n \\tau \\rfloor } n \\right ) ^ { 2 \\alpha + 2 } = \\mathbf { 1 } _ { \\{ \\frac { \\lfloor n \\tau \\rfloor } n \\le t _ 1 \\wedge t _ 2 \\} } \\left ( t _ 1 - \\frac { \\lfloor n \\tau \\rfloor } n \\right ) ^ { \\alpha } \\left ( t _ 2 - \\frac { \\lfloor n \\tau \\rfloor } n \\right ) ^ { \\alpha } \\left ( \\tau - \\frac { \\lfloor n \\tau \\rfloor } n \\right ) ^ { 2 \\alpha + 2 } . \\end{align*}"}
{"id": "99.png", "formula": "\\begin{align*} \\forall x , y \\in \\Lambda , 0 \\leq \\big < \\varphi _ x ; \\varphi _ y \\big > ^ { h } _ { A , g , \\nu } \\leq \\big < \\varphi _ x ; \\varphi _ y \\big > ^ { 0 } _ { A , g , \\nu } = \\big < \\varphi _ x \\varphi _ y \\big > ^ { 0 } _ { A , g , \\nu } . \\end{align*}"}
{"id": "7969.png", "formula": "\\begin{align*} D _ 1 = H , D _ 2 = H - P , D _ 3 = D _ 4 = P . \\end{align*}"}
{"id": "1376.png", "formula": "\\begin{align*} \\left | \\underset { \\Gamma _ { 2 p - 1 } } { \\int \\sum } \\overline { \\widehat { \\eta } } _ k \\prod _ { \\substack { j = 1 \\\\ o d d } } ^ { 2 p - 1 } \\widehat { v } _ { k _ j } \\prod _ { \\substack { j = 2 \\\\ e v e n } } ^ { 2 p - 1 } \\overline { \\widehat { v } } _ { k _ j } \\ , d \\Gamma \\right | \\lesssim \\| \\eta \\| _ { X ^ { - s , 1 / 2 - } } \\| v \\| _ { X ^ { s , 1 / 2 - } _ T } \\| v \\| ^ { 2 p - 2 } _ { X ^ { s , 1 / 2 + } _ T } . \\end{align*}"}
{"id": "374.png", "formula": "\\begin{align*} \\P _ y ( T _ A = T _ x ) & = \\P _ y ( T _ A = T _ x ; T _ A \\le t ^ * ) + \\P _ y ( T _ A = T _ x ; T _ A > t ^ * ) \\\\ & = o ( 1 ) + \\P _ \\pi ( T _ A = T _ x ) + o ( 1 ) \\\\ & = 1 / k + o ( 1 ) , \\end{align*}"}
{"id": "8176.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\| e ^ { - i t H } \\phi - J U _ 0 ( t ) f \\| = 0 \\end{align*}"}
{"id": "8259.png", "formula": "\\begin{align*} u ^ { \\dagger } . A . v = \\left ( \\begin{array} { c } u _ { + } \\\\ u _ { - } \\end{array} \\right ) ^ { \\dagger } . ( D P ) ^ { \\dagger } . \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) . ( D P ) . \\left ( \\begin{array} { c } v _ { + } \\\\ v _ { - } \\end{array} \\right ) = 0 , \\end{align*}"}
{"id": "3320.png", "formula": "\\begin{align*} P _ h ^ 2 \\colon H ^ { - 1 / 2 } ( \\Gamma ) \\to X _ h ^ { - \\frac 1 2 } P _ h ^ 3 \\colon H ^ { 1 / 2 } ( \\Gamma ) \\to X _ h ^ { \\frac 1 2 } \\end{align*}"}
{"id": "4568.png", "formula": "\\begin{align*} \\gcd \\left ( d ( v _ 2 ^ 2 + v _ 3 ^ 2 ) , d ( v _ 1 ^ 2 + v _ 3 ^ 2 ) , d ( v _ 1 ^ 2 + v _ 2 ^ 2 ) \\right ) & = d \\gcd \\left ( v _ 2 ^ 2 + v _ 3 ^ 2 , v _ 1 ^ 2 + v _ 3 ^ 2 , v _ 1 ^ 2 + v _ 2 ^ 2 \\right ) \\\\ { } & = d \\gcd \\left ( v _ 2 ^ 2 + v _ 3 ^ 2 , v _ 1 ^ 2 + v _ 3 ^ 2 , v _ 1 ^ 2 + v _ 2 ^ 2 , v _ 2 ^ 2 - v _ 3 ^ 2 , v _ 1 ^ 2 - v _ 3 ^ 2 , v _ 1 ^ 2 - v _ 2 ^ 2 \\right ) \\\\ { } & = d \\gcd \\left ( 2 v _ 1 ^ 2 , 2 v _ 2 ^ 2 , 2 v _ 3 ^ 2 , v _ 2 ^ 2 + v _ 3 ^ 2 , v _ 1 ^ 2 + v _ 3 ^ 2 , v _ 1 ^ 2 + v _ 2 ^ 2 \\right ) . \\end{align*}"}
{"id": "4114.png", "formula": "\\begin{align*} S ^ { i , j - 1 } K ^ { i , j } = K ^ { i + 1 , j - 1 } S ^ { i , j } , \\end{align*}"}
{"id": "6424.png", "formula": "\\begin{align*} \\left [ Q , \\Phi _ 1 ( [ [ x , y ] _ \\mathfrak { g } , z ] _ \\mathfrak { g } ) \\right ] + \\circlearrowleft ( x , y , z ) & = - \\left [ \\Phi _ 0 \\left ( [ x , y ] _ \\mathfrak { g } \\right ) , \\Phi _ 0 ( z ) \\right ] + \\circlearrowleft ( x , y , z ) \\\\ & = [ [ \\Phi _ 0 ( z ) , Q ] , \\Phi _ 1 ( x , y ) ] - [ [ \\Phi _ 1 ( x , y ) , \\Phi _ 0 ( z ) ] , Q ] + \\circlearrowleft ( x , y , z ) \\\\ & = [ Q , [ \\Phi _ 0 ( x ) , \\Phi _ 1 ( y , z ) ] ] + \\circlearrowleft ( x , y , z ) . \\end{align*}"}
{"id": "4052.png", "formula": "\\begin{align*} T ^ { ( a b ) } = \\sum _ { i , j , l , m } T ^ { i j } _ { l m } ( 1 \\otimes \\cdots \\otimes E _ i ^ l \\otimes \\cdots \\otimes E _ j ^ m \\otimes \\cdots \\otimes 1 ) , \\end{align*}"}
{"id": "4357.png", "formula": "\\begin{align*} v _ { r - 1 } ( b _ { 0 , 0 } ) = v _ { r - 1 } ( a _ { s _ 0 } ) + v _ { r - 1 } ( a ' _ { \\ell _ 0 } ) , \\end{align*}"}
{"id": "1869.png", "formula": "\\begin{align*} R _ { [ n , j ] } = S _ { [ n , j ] } = T _ { [ n , j ] } = 0 \\mbox { i f } \\ , \\ , n \\not \\equiv j \\ , \\ , \\mbox { m o d } \\ , \\ , ( p + 1 ) . \\end{align*}"}
{"id": "4042.png", "formula": "\\begin{align*} E e e ^ { \\prime } = \\sigma ^ { 2 } \\Sigma , \\end{align*}"}
{"id": "8345.png", "formula": "\\begin{align*} p ^ \\mu \\tilde { f } _ \\mu = 0 , \\end{align*}"}
{"id": "7669.png", "formula": "\\begin{align*} E ( S _ { 5 , 5 , 1 } ; u , v ) = 1 + 2 u v + 2 u ^ 2 v ^ 2 + u ^ 3 v ^ 3 - ( 5 u ^ 3 + 1 5 1 u ^ 2 v + 1 5 1 u v ^ 2 + 5 v ^ 3 ) , \\end{align*}"}
{"id": "8207.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { r r r r } 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 0 & - 1 \\\\ - 1 & 0 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 \\end{array} \\right ) \\end{align*}"}
{"id": "7454.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\abs { y _ x } ^ 2 d x = o ( 1 ) . \\end{align*}"}
{"id": "1288.png", "formula": "\\begin{align*} \\sigma ^ { * } ( \\mathcal { A } _ k ) & = \\bigcup _ { j = 2 } ^ { k } \\{ x \\in \\mathbb { R } \\colon H e _ j ( x ) = 0 \\} . \\end{align*}"}
{"id": "1625.png", "formula": "\\begin{align*} \\ell _ A = \\max \\{ | t | : t \\in A \\} \\mathrm { \\ \\ a n d \\ \\ } \\max ( A ) = \\{ s \\in A : | s | = \\ell _ A \\} . \\end{align*}"}
{"id": "1775.png", "formula": "\\begin{align*} w ^ * = b \\widetilde { z } + \\sum _ { j = 1 } ^ m w _ j ^ * , \\end{align*}"}
{"id": "5169.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ m \\sum _ { k = 0 } ^ { m - 1 } \\prod _ { i = 0 } ^ { r - 1 } \\rho _ { i + k } . \\end{align*}"}
{"id": "1063.png", "formula": "\\begin{align*} \\lim _ { | y | \\to \\infty } \\| R ( x + i y , A _ j ) \\| _ { \\infty } = 0 \\qquad \\forall x < - \\omega _ 0 ( A _ j ) . \\end{align*}"}
{"id": "472.png", "formula": "\\begin{align*} \\beta _ 1 ( z ) & = \\psi _ k ( 1 ) z + \\psi _ k ( 2 ) z ^ 2 + \\dots + \\psi _ k ( s _ 1 ) z ^ { s _ 1 } ; \\\\ { \\mathcal E } ( \\beta _ 1 ( z ) ) & = \\frac { 1 } { ( 1 - z ) ^ { \\psi _ k ( 1 ) } \\cdots ( 1 - z ^ { s _ 1 } ) ^ { \\psi _ k ( s _ 1 ) } } \\approx \\frac { \\mu } { ( 1 - t ) ^ M } , t \\to 1 { - } 0 ; \\\\ M & = \\psi _ k ( 1 ) + \\dots + \\psi _ k ( s _ 1 ) = \\dim _ K F ( { \\bf { N } } _ { s _ 1 } , k ) , \\mu = \\prod _ { q = 2 } ^ { s _ 1 } q ^ { - \\psi _ k ( q ) } . \\end{align*}"}
{"id": "5418.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s m - \\frac { ( - \\Delta ) ^ s m _ 1 } { \\gamma _ 1 ^ { 1 / 2 } } m & = 0 \\quad \\Omega , \\\\ m & = m _ 0 \\quad \\Omega _ e . \\end{align*}"}
{"id": "8700.png", "formula": "\\begin{align*} f = \\frac { ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } } { 2 } ( x ^ { 1 } \\dot { x } ^ { 2 } - x ^ { 2 } \\dot { x } ^ { 1 } ) , g = \\sqrt { ( \\dot { x } ^ { 1 } ) ^ 2 + ( \\dot { x } ^ { 2 } ) ^ 2 } + b ( \\cos \\theta ( x ^ 1 , x ^ 2 ) \\dot { x } ^ 1 + \\sin \\theta ( x ^ 1 , x ^ 2 ) \\dot { x } ^ 2 ) . \\end{align*}"}
{"id": "3545.png", "formula": "\\begin{align*} \\chi _ { \\pi } ( h _ i ) = \\sum _ { j = 0 } ^ n c _ j \\cdot \\chi _ { \\sigma _ j } ( h _ i ) \\end{align*}"}
{"id": "7362.png", "formula": "\\begin{align*} \\hat { V } ( y ) = V _ 0 + \\frac { a } { | y | ^ m } + O \\left ( \\frac { 1 } { | y | ^ { m + \\theta } } \\right ) \\end{align*}"}
{"id": "1000.png", "formula": "\\begin{align*} \\frac { \\P ( S _ { N + 1 } \\in \\cdot \\cap \\{ 0 , 1 , \\ldots , N \\} \\ | \\ S _ N ) } { \\P ( S _ { N + 1 } \\in \\{ 0 , 1 , \\ldots , N \\} \\ | \\ S _ N ) } = \\delta _ { S _ { N } } ( \\cdot ) \\end{align*}"}
{"id": "4900.png", "formula": "\\begin{align*} \\lvert v _ { i , j } \\rvert ^ 2 \\prod _ { k = 1 ; k \\ne i } ^ { n - 1 } ( { \\lambda _ i } ( A ) - { \\lambda _ k } ( A ) ) = \\prod _ { k = 1 } ^ { n - 1 } ( { \\lambda _ i } ( A ) - { \\lambda _ k } ( M _ j ) ) , \\end{align*}"}
{"id": "7030.png", "formula": "\\begin{align*} T _ { \\alpha } f \\geq T f - ( c - \\epsilon ) u \\geq c u - ( c - \\epsilon ) u = \\epsilon u \\end{align*}"}
{"id": "1750.png", "formula": "\\begin{align*} \\frac { D _ { n + 1 } ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) } { D _ n ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) } = & \\frac { D _ 0 \\big ( v + ( n + 1 ) w - ( n + 1 ) t \\tau / 2 , w , t \\big ) \\cdot B _ 0 \\big ( v + ( n + 1 ) w - n t \\tau / 2 , w + t \\tau / 2 , t \\big ) } { D _ 0 \\big ( v + n w - n t \\tau / 2 , w , t \\big ) } \\cdot \\\\ & \\prod _ { k = 0 } ^ { n - 1 } \\frac { B _ 0 \\big ( v + ( n + 1 ) w + ( 2 - n + 2 k ) t \\tau / 2 , w + t \\tau / 2 , t \\big ) } { B _ 0 \\big ( v + n w + ( 1 - n + 2 k ) t \\tau / 2 , w + t \\tau / 2 , t \\big ) } . \\end{align*}"}
{"id": "8088.png", "formula": "\\begin{align*} \\nabla _ { Z } Z = - \\operatorname { g r a d } p \\end{align*}"}
{"id": "1047.png", "formula": "\\begin{align*} \\sum _ { n = r + 1 } ^ \\infty \\| S _ n ( t ) \\| _ 1 < \\infty . \\end{align*}"}
{"id": "1932.png", "formula": "\\begin{align*} [ w ^ { n } ] \\ , h ( w ) = \\frac { 1 } { n } \\ , [ t ^ { n - 1 } ] \\ , \\phi ( t ) ^ { n } , n \\geq 1 . \\end{align*}"}
{"id": "52.png", "formula": "\\begin{align*} \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { ( f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ) ' ( v _ { n + 1 } + w ) } { f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ( v _ { n + 1 } + w ) } = \\eta \\delta _ n ( \\beta ) . \\end{align*}"}
{"id": "344.png", "formula": "\\begin{align*} Q ( A , B ) = \\sum _ { a \\in A , b \\in B } \\pi ( a ) P ( a , b ) . \\end{align*}"}
{"id": "6051.png", "formula": "\\begin{align*} \\Lambda = \\left ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ m \\right ) \\end{align*}"}
{"id": "1354.png", "formula": "\\begin{align*} \\Phi ( k , k _ 1 , \\cdots , k _ p ) & = k ^ 2 - k _ 1 ^ 2 + k _ 2 ^ 2 - \\cdots - k _ p ^ 2 , \\\\ k & = k _ 1 - k _ 2 + \\cdots + k _ p . \\end{align*}"}
{"id": "546.png", "formula": "\\begin{align*} | g _ k ( n ) | \\leq \\tau _ 2 ( n ) = O _ { \\varepsilon } ( n ^ { \\varepsilon } ) . \\end{align*}"}
{"id": "8028.png", "formula": "\\begin{align*} V _ { + , \\bar { \\tilde D } _ { m + 2 } , r } : = p ^ * _ + ( \\mathcal O _ { Y _ + } ( - 1 ) ) , V _ { - , r } : = p ^ * _ - ( \\mathcal O _ { Y _ - } ( - 1 ) ) . \\end{align*}"}
{"id": "7698.png", "formula": "\\begin{align*} G _ { f , k } ( n ) = \\sum _ { d ^ k \\delta = n } ( \\mu * f ) ( d ) \\tau _ k ( \\delta ) , \\end{align*}"}
{"id": "8326.png", "formula": "\\begin{align*} \\| \\delta _ n ( a ) \\| _ { M , N } = \\left \\| \\int _ 0 ^ 1 e ^ { 2 \\pi i n \\theta } \\rho _ \\theta ^ { - 1 } \\delta \\rho _ \\theta ( a ) \\ , d \\theta \\right \\| _ { M , N } \\le \\int _ 0 ^ 1 \\| \\rho _ \\theta ^ { - 1 } \\delta \\rho _ \\theta ( a ) \\| _ { M , N } \\ , d \\theta \\le C \\| a \\| _ { M ' , N ' } \\ , . \\end{align*}"}
{"id": "4659.png", "formula": "\\begin{align*} \\| \\Psi _ { s t } \\| _ { q , h } = t ^ { - \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| \\Psi _ s \\| _ { q , h } \\end{align*}"}
{"id": "3037.png", "formula": "\\begin{align*} R ^ H _ { T _ w } \\theta ( u s ) = \\frac { 1 } { | C _ H ( s ) ^ { \\circ F } | } \\sum _ { \\{ h \\in H ^ { F } \\mid h s h ^ { - 1 } \\in T _ { w } \\} } Q ^ { C _ H ( s ) ^ { \\circ } } _ { C _ { h ^ { - 1 } T _ { w } h } ( s ) } ( u ) \\theta ( h s h ^ { - 1 } ) . \\end{align*}"}
{"id": "6834.png", "formula": "\\begin{align*} p ( t , v , g ) = \\frac { 1 } { V _ F } \\sum _ { k = - \\infty } ^ { + \\infty } \\exp \\left ( - k ^ 2 ( \\frac { 2 \\pi } { V _ F } ) ^ 2 D ( t ) + i \\frac { 2 \\pi } { V _ F } k v + i \\frac { 2 \\pi } { V _ F } k g \\right ) ( p _ { t , k } * \\bar { G } _ { t , k } ) ( g ) . \\end{align*}"}
{"id": "2522.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { 1 } { 2 \\lambda ^ 2 } \\{ h ( \\nabla _ { \\tilde { X } } ^ N \\tilde { Z } , \\tilde { Y } ) + h ( \\nabla _ { \\tilde { Y } } ^ N \\tilde { Z } , \\tilde { X } ) \\} + R i c ( X , Y ) + \\frac { \\mu } { \\lambda ^ 2 } h ( \\tilde { X } , \\tilde { Y } ) = 0 . \\end{array} \\end{align*}"}
{"id": "2536.png", "formula": "\\begin{align*} R i c ( X _ 1 , Y _ 1 ) = a _ 1 a _ 3 R i c ( e _ 1 , e _ 1 ) + ( a _ 1 a _ 4 + a _ 2 a _ 3 ) R i c ( e _ 1 , e _ 2 ) + a _ 2 a _ 4 R i c ( e _ 2 , e _ 2 ) . \\end{align*}"}
{"id": "806.png", "formula": "\\begin{align*} | f ( z ^ m ) | + \\sum _ { n = N } ^ { \\infty } | a _ n | | z | ^ n \\leq d ( 0 , \\partial { \\Omega } ) \\end{align*}"}
{"id": "8909.png", "formula": "\\begin{align*} \\begin{cases} - U '' + \\lambda U = \\rho U ^ { p - 1 } & , \\\\ U > 0 & , \\\\ \\sum _ { i = 1 } ^ m U _ i ' ( 0 ) = 0 , \\end{cases} \\end{align*}"}
{"id": "8753.png", "formula": "\\begin{align*} { \\mathcal U } _ n ( t , s ) = \\exp \\left ( t A _ n ^ { 1 / n } \\right ) = \\exp \\left ( t ~ \\left \\{ \\Pi _ { k = 1 } ^ n \\left [ ( I - \\kappa e ^ { - a _ k ( t , s ) } ) ^ { - 1 } \\partial _ t a _ k ( t , s ) \\right ] \\right \\} ^ { 1 / n } \\right ) , \\end{align*}"}
{"id": "5051.png", "formula": "\\begin{align*} K ^ { n , 2 } _ \\tau & = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ { ( \\delta + \\frac 1 n ) \\wedge \\tau } \\gamma _ s \\left ( \\sigma ' ( X _ s ) \\right ) ^ 2 \\\\ & \\times \\left [ \\left ( \\int _ { \\eta _ n ( s ) - \\delta } ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) ^ 2 - \\left ( \\int _ { \\eta _ n ( s ) - \\delta } ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( s ) - \\delta } ) \\ , d W _ u \\right ) ^ 2 \\right ] \\ , d s . \\end{align*}"}
{"id": "5785.png", "formula": "\\begin{align*} E ( ( A R ) / A ) = E ( R / ( R \\cap A ) ) \\neq 0 . \\end{align*}"}
{"id": "1212.png", "formula": "\\begin{align*} \\| E \\| _ { L ^ { p } ( \\rho ; \\R ^ d ) } = \\sup \\left \\{ \\int _ { \\R ^ { d } } \\langle \\nabla \\varphi , E \\rangle d \\rho : \\varphi \\in C _ { 0 } ^ { \\infty } , \\| \\nabla \\varphi \\| _ { L ^ { q } ( \\rho ) } \\le 1 \\right \\} , \\end{align*}"}
{"id": "5318.png", "formula": "\\begin{align*} Z _ t : = \\lim _ { n \\to \\infty } n ^ { - 1 } Z ^ { ( n ) } _ t . \\end{align*}"}
{"id": "2033.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : c o l l i s i o n k e r n e l B } B ( u , \\sigma ) = \\Phi ( | u | ) \\Sigma ( \\theta ) , \\end{align*}"}
{"id": "7042.png", "formula": "\\begin{align*} g ( M ) & = g ( M - 1 ) + g ( M - 2 ) + F _ { M - 1 } , \\\\ h ( M ) & = h ( M - 1 ) + h ( M - 2 ) + F _ { M - 2 } , \\\\ g _ 1 ( M ) & = g _ 1 ( M - 1 ) + g _ 1 ( M - 2 ) + F _ { M - 1 } , \\\\ h _ 1 ( M ) & = h _ 1 ( M - 1 ) + h _ 1 ( M - 2 ) + F _ { M - 1 } . \\end{align*}"}
{"id": "8182.png", "formula": "\\begin{align*} & \\hat { \\eta } _ { x , p ; \\delta } ( \\xi ) = e ^ { - i x \\xi } \\hat { \\eta } _ \\delta ( \\xi - p ) \\end{align*}"}
{"id": "4331.png", "formula": "\\begin{align*} \\vec { H } = J \\nabla \\theta _ t , \\end{align*}"}
{"id": "7997.png", "formula": "\\begin{align*} H _ { ( Q _ 4 , K 3 ) } ( y ) = e ^ { \\frac { H \\log y } { 2 \\pi i } } \\sum _ { d \\geq 0 } y ^ d \\frac { \\Gamma ( 1 + \\frac { 4 H } { 2 \\pi i } + 4 d ) } { \\Gamma ( 1 + \\frac { H } { 2 \\pi i } + d ) ^ 4 } [ \\textbf { 1 } ] _ { d } . \\end{align*}"}
{"id": "5167.png", "formula": "\\begin{align*} | M ^ { ( m ) } | & = \\prod _ { j = 0 } ^ { m - 2 } p _ j + \\prod _ { j = 0 } ^ { m - 3 } p _ j ( 1 - p _ { m - 1 } ) + \\prod _ { j = 0 } ^ { m - 4 } p _ j ( 1 - p _ { m - 2 } ) ( 1 - p _ { m - 1 } ) \\\\ & + \\cdots + p _ 0 \\prod _ { j = 2 } ^ { m - 1 } ( 1 - p _ { j } ) + \\prod _ { j = 1 } ^ { m - 1 } ( 1 - p _ j ) \\\\ & = \\sum _ { j = 0 } ^ { m - 1 } \\prod _ { i _ 1 = 0 } ^ { j - 1 } p _ { i _ 1 } \\prod _ { i _ 2 = j + 1 } ^ { m - 1 } ( 1 - p _ { i _ 2 } ) , \\end{align*}"}
{"id": "3689.png", "formula": "\\begin{align*} \\| h \\partial _ \\nu u _ { h } \\| _ { L ^ 2 ( \\Gamma ) } = O ( 1 ) . \\end{align*}"}
{"id": "8906.png", "formula": "\\begin{align*} \\begin{cases} - u _ n '' + \\lambda _ n u _ n = \\rho _ n u _ n ^ { p - 1 } & , \\\\ u _ n > 0 & , \\\\ \\sum _ { \\rm { e } \\succ \\rm { v } } u _ { e , n } ' ( \\rm { v } ) = 0 & \\forall \\rm { v } \\in \\mathcal { V } , \\end{cases} \\end{align*}"}
{"id": "4793.png", "formula": "\\begin{align*} \\Phi ( \\tilde { a } , \\tilde { m } ) = \\int _ { 0 } ^ { t } \\mathcal { G } ( t - s ) \\left ( \\begin{array} { c c c } 0 \\\\ { g ( a _ { L } + \\tilde { a } , m _ { L } + \\tilde { m } ) } \\\\ \\end{array} \\right ) d s . \\end{align*}"}
{"id": "8730.png", "formula": "\\begin{align*} c _ 1 \\cos t _ 1 + c _ 2 \\sin t _ 1 + \\frac { \\mu a ^ 2 } { \\lambda } = 0 , \\end{align*}"}
{"id": "3335.png", "formula": "\\begin{align*} s i g n ( \\sigma _ i \\beta \\sigma _ j \\beta ^ { - 1 } ) = M e y e r ( \\sigma _ i , \\beta \\sigma _ j \\beta ^ { - 1 } ) \\end{align*}"}
{"id": "815.png", "formula": "\\begin{align*} K _ 2 = 1 + T _ { p - 1 } ( u ; x _ 0 , \\bar R _ 0 ) ^ { p - 1 } + T _ { q - 1 } ( u ; x _ 0 , \\bar R _ 0 ) ^ { q - 1 } + \\| u \\| ^ { \\frac { ( \\ell _ 1 + j _ \\infty ) ( \\ell _ 1 - 1 ) } { \\ell _ 1 - 2 } } _ { L ^ \\infty ( B _ { \\bar R _ 0 } ( x _ 0 ) ) } + \\| u \\| ^ { q - 1 } _ { L ^ \\infty ( B _ { R _ 0 } ) } + \\| f \\| _ { L ^ \\infty ( B _ { \\bar R _ 0 } ( x _ 0 ) ) } \\end{align*}"}
{"id": "5027.png", "formula": "\\begin{align*} Q ^ { n , 1 } _ \\tau = : Q ^ { n , 3 } _ \\tau + Q ^ { n , 4 } _ \\tau , \\end{align*}"}
{"id": "1715.png", "formula": "\\begin{align*} \\Phi _ { \\ell _ 2 } ( t ) = \\Phi _ { \\ell _ 1 } ( t ) \\circ \\mathbb { S } _ q ( \\Delta ) = \\Phi _ { \\ell _ 1 } ( t ) \\circ \\mathbb { S } _ q ( \\tilde { \\ell } _ 1 ) \\circ \\cdots \\circ \\mathbb { S } _ q ( \\tilde { \\ell } _ n ) , \\end{align*}"}
{"id": "7249.png", "formula": "\\begin{align*} b _ { \\xi } ( p , p _ 0 ) = b _ { \\xi } ( p , p _ 0 ' ) + b _ { \\xi } ( p _ 0 ' , p _ 0 ) \\end{align*}"}
{"id": "3952.png", "formula": "\\begin{align*} \\widehat { \\xi ^ \\mathbf { G } } ( s _ 1 , \\dots , s _ { d - 1 } ) = \\int _ { \\R ^ { d ( d - 1 ) } } \\phi ( \\widehat { y } ^ 1 ( s _ 1 , \\dots , s _ { d - 1 } ) ) \\prod _ { j = 1 } ^ { d - 1 } g _ { \\tilde { a } ( \\widehat { y } ^ j ( s _ 1 , \\dots , s _ { d - 1 } ) ) } ( z ^ { j + 1 } ) d z ^ 2 \\dots d z ^ d . \\end{align*}"}
{"id": "3011.png", "formula": "\\begin{align*} S _ { S _ n , \\frac { 1 } { 2 } W } ( L _ { x y } ) = \\frac { 3 n + 1 } { 2 ( 2 n + 1 ) } . \\end{align*}"}
{"id": "2941.png", "formula": "\\begin{align*} D _ { - \\Delta } F _ c ( \\frac { | x - 2 P t | } { ( t + 1 ) ^ { \\alpha } } \\leq 1 ) = - \\alpha \\frac { | x - 2 P t | } { ( t + 1 ) ^ { 1 + \\alpha } } F _ c ' \\geq 0 . \\end{align*}"}
{"id": "3787.png", "formula": "\\begin{align*} { } _ { } ^ z T _ { k , j ; n , l } ^ { T ; \\mu ; m , 2 } ( B ) ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big [ i \\mu \\mathbf { P } _ 3 \\big ( \\omega ^ { m ; B } _ { j , l } ( t - s , v , \\omega ) \\big ) K _ { k ; n } ^ { } ( 1 ) ( y , \\zeta ) + \\mathbf { P } _ 3 \\big ( c ^ { q ; m , B } _ { j , l } ( t - s , v , \\omega ) \\big ) K _ { k ; n } ^ { q } ( 1 ) ( y , \\zeta ) \\end{align*}"}
{"id": "6593.png", "formula": "\\begin{align*} ( 1 - \\tau _ i ) \\phi ( f ) & = \\phi ( f - f \\circ \\tau _ i ) \\\\ & = \\langle M _ f v , v \\rangle - \\langle M _ { f \\circ \\tau _ i } v , v \\rangle \\\\ & = \\langle M _ f v , v \\rangle - \\langle M _ f \\tau _ i v , \\tau _ i v \\rangle \\\\ & = \\langle M _ f v , v - \\tau _ i v \\rangle + \\langle M _ f ( v - \\tau _ i v ) , \\tau _ i v \\rangle . \\end{align*}"}
{"id": "6926.png", "formula": "\\begin{align*} f - q | _ g = & \\ - L ^ 2 ( L ( x ) y ) z + L ( L ^ 2 ( x ) y z ) \\\\ \\equiv & \\ - L ( L ^ 2 ( x ) y ) z + L ^ 3 ( x ) y z \\\\ \\equiv & \\ - L ^ 3 ( x ) y z + L ^ 3 ( x ) y z \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "4148.png", "formula": "\\begin{align*} X _ j f ( x , u ) & = \\frac { d } { d t } f ( ( x , u ) ( t X _ j , 0 ) ) \\big | _ { t = 0 } \\\\ & = \\partial _ { x _ j } f ( x , u ) + \\frac 1 2 \\sum _ { k = 1 } ^ { d _ 2 } \\langle U _ k , [ x , X _ j ] \\rangle \\partial _ { u _ k } f ( x , u ) , \\\\ U _ k f ( x , u ) & = \\partial _ { u _ k } f ( x , u ) . \\end{align*}"}
{"id": "6564.png", "formula": "\\begin{align*} h \\colon ( U \\times E ^ k ) \\times W \\to F \\ , , h ( ( x , y ) , v ) : = d ( f ^ \\wedge ) ( ( x , v ) , ( y , 0 ) ) \\ , . \\end{align*}"}
{"id": "5114.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\exp \\left ( - \\frac { \\lambda ^ 2 } 2 n ^ { 2 \\alpha + 1 } \\int _ { t - \\delta } ^ { \\eta _ n ( t ) } \\psi _ { n , 1 } ^ 2 ( s , \\eta _ n ( t ) ) d s \\right ) = \\exp \\left ( - \\frac { \\lambda ^ 2 } 2 \\kappa _ 1 ^ 2 \\right ) \\end{align*}"}
{"id": "5525.png", "formula": "\\begin{align*} \\frac { \\abs { f ' ( z ) } } { \\abs { f ' ( z _ 0 ) } } = \\left . \\abs { \\frac { \\textbf { z } _ 0 - \\alpha } { \\textbf { z } _ 0 + \\lambda i } } ^ 2 \\middle / \\abs { \\frac { \\textbf { z } - \\alpha } { \\textbf { z } + \\lambda i } } ^ 2 \\right . . \\end{align*}"}
{"id": "8973.png", "formula": "\\begin{align*} \\| k _ n u \\| ^ q _ { L ^ q ( \\Omega ) } + b \\| \\gamma ( k _ n u ) \\| ^ r _ { L ^ r ( \\Sigma ) } = k _ n ^ q \\| u \\| ^ q _ { L ^ q ( \\Omega ) } + k _ n ^ r b \\| u \\| ^ r _ { L ^ r ( \\Sigma ) } = 1 . \\end{align*}"}
{"id": "155.png", "formula": "\\begin{align*} \\| E \\| _ { L ^ 1 } \\leq \\tilde G _ { d , t , \\mu } ( \\lambda ) + \\sum _ { i = 1 } ^ 3 \\lambda ^ i P ^ { ( i ) } _ { d , t , \\mu , \\mu , c ( \\mu ) \\lambda } \\big ( \\| E \\| _ { L ^ 1 } \\big ) . \\end{align*}"}
{"id": "8452.png", "formula": "\\begin{align*} F \\supset \\bigcap _ { N = 1 } ^ { \\infty } \\bigcup _ { n = N } ^ { \\infty } \\bigcup _ { w \\in \\Sigma _ { \\beta _ 1 } ^ n , v \\in \\Sigma _ { \\beta _ 2 } ^ n } \\tilde { J } _ { n , \\beta _ 1 } ( w ) \\times \\tilde { J } _ { n , \\beta _ 2 } ( v ) : = \\widetilde { F } . \\end{align*}"}
{"id": "8357.png", "formula": "\\begin{align*} ( \\tilde { f } ^ { | u \\rangle } , \\tilde { f } ^ { | v \\rangle } ) _ { { } _ { \\mathfrak { J } } } = - 4 \\pi \\lambda \\textrm { c o t h } \\lambda , \\end{align*}"}
{"id": "5095.png", "formula": "\\begin{align*} E [ | P ^ { n , 1 } _ \\tau | ^ 2 ] = n ^ { 2 \\alpha + 1 } \\int _ { [ 0 , \\tau ] ^ 2 } ( t - s _ 1 ) ^ { \\alpha } ( t - s _ 2 ) ^ { \\alpha } E \\left [ ( \\sigma ' \\sigma ) ( X _ { \\eta _ n ( s _ 1 ) } ) ( \\sigma ' \\sigma ) ( X _ { \\eta _ n ( s _ 2 ) } ) \\Xi ^ { n , 2 } _ { s _ 1 } \\Xi ^ { n , 2 } _ { s _ 2 } \\right ] d s _ 1 d s _ 2 \\end{align*}"}
{"id": "7428.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\left ( u ( x , 0 ) , u _ t ( x , 0 ) , y ( x , 0 ) , y _ t ( x , 0 ) \\right ) = ( u _ 0 ( x ) , u _ 1 ( x ) , y _ 0 ( x ) , y _ 1 ( x ) ) , & x \\in ( 0 , L ) , \\\\ \\left ( w ( x , 0 ) , w ( x , - s ) \\right ) = ( w _ 0 ( x ) , \\phi _ 0 ( x , s ) ) , & x \\in ( 0 , L ) , \\ s > 0 , \\end{array} \\right . \\end{align*}"}
{"id": "3273.png", "formula": "\\begin{align*} \\Vert u _ 1 u _ 2 \\Vert _ { L ^ 1 ( B ) } \\leq C , \\Vert u _ j \\Vert _ { H ^ 2 ( B ) } \\leq C s ^ 2 e ^ { \\Lambda s } , \\ , j = 1 , 2 . \\end{align*}"}
{"id": "1820.png", "formula": "\\begin{align*} \\int \\frac { 1 } { z - x } d \\mu ( x ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { s _ { n } } { z ^ { n + 1 } } \\end{align*}"}
{"id": "6557.png", "formula": "\\begin{align*} d ^ k f ( x , y _ 1 , \\ldots , y _ k ) : = ( D _ { y _ k } \\cdots D _ { y _ 1 } ) ( f ) ( x ) \\end{align*}"}
{"id": "1553.png", "formula": "\\begin{align*} \\prod _ { b \\in \\lambda } \\dfrac { 1 - q ^ { c o a r m _ \\lambda ( b ) } t ^ { n - c o l e g _ \\lambda ( b ) } } { 1 - q ^ { a r m _ \\lambda ( b ) } t ^ { l e g _ \\lambda ( b ) + 1 } } = \\prod _ { 1 \\leq i < j \\leq n } \\prod _ { r = 0 } ^ { \\lambda _ i - \\lambda _ j - 1 } \\dfrac { 1 - q ^ r t ^ { j - i + 1 } } { 1 - q ^ r t ^ { j - i } } . \\end{align*}"}
{"id": "2209.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial } { \\partial t } \\omega ( t ) = - \\mathrm { R i c } _ { \\omega ( t ) } ^ { T } - \\omega ( t ) , \\omega ( 0 ) = \\omega _ { 0 } . \\end{array} \\end{align*}"}
{"id": "700.png", "formula": "\\begin{align*} F ( p , s ) = ( u ( p , s ) , p ) . \\end{align*}"}
{"id": "3960.png", "formula": "\\begin{align*} \\lim _ { n } \\| f ( \\gamma ) ^ { * } f ( \\gamma ) e _ { n } \\| _ { A _ { u } } = \\| f ( \\gamma ) ^ { * } f ( \\gamma ) \\| _ { A _ { u } } = \\| f ( \\gamma ) \\| ^ { 2 } _ { B _ { \\gamma } } . \\end{align*}"}
{"id": "126.png", "formula": "\\begin{align*} \\big \\| T _ \\lambda \\| _ { L ^ \\infty } & \\leq 3 \\lambda \\big ( \\| E \\| _ { L ^ \\infty } + \\eta _ t \\big ) \\Big ( \\| C \\| _ { L ^ 1 } \\| E \\| _ { L ^ \\infty } + \\| C \\| _ { L ^ 2 } ^ 2 \\Big ) \\\\ & \\leq 3 \\lambda \\big ( \\eta _ t + \\| E \\| _ { L ^ \\infty } \\big ) \\Big ( \\frac { \\| E \\| _ { L ^ \\infty } } { m ^ 2 _ t } + \\frac { c } { m ^ 2 _ t } { \\bf 1 } _ { d = 2 } + \\frac { c } { m _ t } { \\bf 1 } _ { d = 3 } \\Big ) , \\end{align*}"}
{"id": "6679.png", "formula": "\\begin{align*} \\begin{aligned} \\| z _ 1 - z _ 2 \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } & \\leq C _ 2 ( C _ { 0 2 } \\| g _ 1 - g _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + T _ 0 ^ { \\frac { \\beta _ 0 + 1 } { 2 } } \\| w _ 1 - w _ 2 \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } \\\\ & + \\| z _ { 0 1 } - z _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } + \\| w _ { 0 1 } - w _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } ) . \\\\ \\end{aligned} \\end{align*}"}
{"id": "479.png", "formula": "\\begin{align*} B ( x _ i , \\frac { 1 } { 2 } \\sin \\theta ) \\cap B ( x _ j , \\frac { 1 } { 2 } \\sin \\theta ) = \\emptyset . \\end{align*}"}
{"id": "218.png", "formula": "\\begin{align*} L _ { B } ( x ^ { n } , b _ { \\mathcal { T } } ) = \\sum _ { \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } } Z _ { \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } } ( x ^ n , b _ { \\mathcal { T } } ) , \\end{align*}"}
{"id": "3774.png", "formula": "\\begin{align*} H _ { k , j ; n , l , r } ^ { \\mu , m , i ; n o n , a } ( t , x , \\zeta ) = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\varphi _ { m ; - 1 0 M _ t } ( t - s ) E B ^ a ( t , s , x - y + ( t - s ) \\omega , \\omega , v ) \\cdot \\nabla _ v \\big ( ( t - s ) \\mathfrak { H } ^ { \\mu , E , i } _ { k , j ; n , l , r } ( y , \\omega , v , \\zeta ) \\end{align*}"}
{"id": "4704.png", "formula": "\\begin{align*} & ( - 1 ) ^ { k } \\left ( \\bigg ( \\sum _ { n = 1 } ^ \\infty u _ m ( n ) \\ , q ^ n \\bigg ) \\bigg ( \\sum _ { n = - k } ^ { k } ( - 1 ) ^ { n } \\ , q ^ { n ( 3 n - 1 ) / 2 } \\bigg ) - \\sum _ { n = 0 } ^ \\infty C _ m ( n ) \\ , q ^ n \\right ) \\\\ & = \\left ( \\sum _ { n = 0 } ^ \\infty C _ m ( n ) \\ , q ^ n \\right ) \\left ( \\sum _ { n = 0 } ^ \\infty \\widetilde { P } _ k ( n ) \\ , q ^ n \\right ) , \\end{align*}"}
{"id": "4959.png", "formula": "\\begin{align*} \\kappa _ 1 = \\sqrt { \\sum _ { k = 1 } ^ \\infty \\int _ 0 ^ 1 [ k ^ \\alpha - ( k - x ) ^ \\alpha ] ^ 2 d x } \\end{align*}"}
{"id": "1873.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { D } _ { [ n , j , k ] } } w ( \\gamma ) = A _ { [ k , 0 ] } \\ , A _ { [ n - k - 1 , j - 1 ] } ^ { ( 1 ) } . \\end{align*}"}
{"id": "2403.png", "formula": "\\begin{align*} J _ 1 ( t ) : & = \\langle E ( t ) , F _ n ( U ( t ) ) - F _ n ( \\mathbb U ( t ) ) \\rangle , \\\\ J _ 2 ( t ) : & = n \\| ( - A _ n ) ^ { - \\frac { 1 } { 2 } } \\{ \\Sigma _ n ( \\mathbb U ( t ) ) - \\Sigma _ n ( U ( t ) ) \\} \\| ^ 2 _ { \\mathrm F } , \\\\ J _ 3 ( t ) : & = n \\| E ( t ) ^ \\top ( - A _ n ) ^ { - 1 } \\{ \\Sigma _ n ( \\mathbb U ( t ) ) - \\Sigma _ n ( U ( t ) ) \\} \\| ^ 2 . \\end{align*}"}
{"id": "8876.png", "formula": "\\begin{align*} \\sum _ { 1 \\le j _ 1 < \\cdots < j _ k \\le N } \\ \\prod _ { \\ell = 1 } ^ k \\lambda _ { j _ \\ell } = \\sum _ { 1 \\le j _ 1 < \\cdots < j _ k \\le N } \\underset { 1 \\le \\ell , m \\le k } { { \\rm d e t } } ( A _ { j _ \\ell j _ m } ) , \\end{align*}"}
{"id": "4245.png", "formula": "\\begin{align*} \\mathbf { P } _ x ( \\tau _ \\dagger = n \\mid X ) = K ( X _ { n - 1 } ) \\prod _ { i = 0 } ^ { n - 2 } ( 1 - K ( X _ i ) ) . \\end{align*}"}
{"id": "7342.png", "formula": "\\begin{align*} & f ( w , x , y , z ) + f ( z , w , x , y ) & \\\\ = & ( w x , y , z ) - x ( w , y , z ) - ( x , y , z ) w + ( z w , x , y ) - w ( z , x , y ) - ( w , x , y ) z & \\\\ = & ( w x , y , z ) - x ( w , y , z ) - ( x , y , z ) w + ( z w , x , y ) - ( w x , y , z ) + ( w , x y , z ) - ( w , x , y z ) & \\\\ = & - x ( w , y , z ) - ( x , y , z ) w + ( z w , x , y ) + ( w , x y , z ) - ( w , x , y z ) & \\\\ = & - x ( w , y , z ) - ( x , y , z ) w + ( x y , z , w ) - ( x , y z , w ) + ( x , y , z w ) = 0 . & \\end{align*}"}
{"id": "4200.png", "formula": "\\begin{align*} \\tilde g _ { m } ^ { ( \\ell ) } : = \\chi _ { \\tilde B _ m ^ { ( \\ell ) } \\cap B _ { 3 R } ^ { d _ { \\mathrm { C C } } } ( 0 ) } F _ \\ell ^ { ( \\iota ) } ( L , U ) f _ { m } ^ { ( \\ell ) } \\end{align*}"}
{"id": "4453.png", "formula": "\\begin{align*} m ! \\ , & N _ 0 ( ( R f ^ { i _ 1 \\dots i _ k } ) _ { p _ 1 q _ 1 \\dots p _ { m - k } q _ { m - k } } ) \\\\ & = \\sigma ( i _ 1 \\dots i _ k ) \\ , \\sum \\limits _ { r = 0 } ^ k ( - 1 ) ^ r \\binom { k } { r } \\frac { \\partial ^ { r } } { \\partial x _ { i _ 1 } \\dots \\partial x _ { i _ r } } ( R ^ k ( G _ { m - r } ) ) _ { p _ 1 q _ 1 \\dots p _ { m - k } q _ { m - k } i _ { r + 1 } \\dots i _ k } , \\end{align*}"}
{"id": "8877.png", "formula": "\\begin{align*} \\underset { 1 \\le \\ell , m \\le k } { { \\rm d e t } } ( A _ { j _ \\ell j _ m } ) = \\det \\begin{pmatrix} a _ { j _ 1 j _ 1 } & a _ { j _ 1 j _ 2 } & \\cdots & a _ { j _ 1 j _ k } \\\\ a _ { j _ 2 j _ 1 } & a _ { j _ 2 j _ 2 } & \\cdots & a _ { j _ 2 j _ k } \\\\ \\vdots & & \\ddots & \\vdots \\\\ a _ { j _ k j _ 1 } & a _ { j _ k j _ 2 } & \\cdots & a _ { j _ k j _ k } \\end{pmatrix} . \\end{align*}"}
{"id": "5661.png", "formula": "\\begin{align*} \\sup _ { \\mathbb { R } ^ n } \\left | u ^ 0 _ \\infty ( x ) - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \\right | \\leq C _ 1 . \\end{align*}"}
{"id": "501.png", "formula": "\\begin{align*} \\Psi = e _ i ^ { \\varepsilon _ i ( \\Phi ) } ( \\Phi ) . \\end{align*}"}
{"id": "1612.png", "formula": "\\begin{align*} f ( x ^ 1 ) = \\frac { x ^ 1 } { \\sqrt { 2 } } + c , \\end{align*}"}
{"id": "5443.png", "formula": "\\begin{align*} \\textsf { F A C } ( M ) = \\bigsqcup _ { A \\subsetneqq M } \\textsf { F A C } ( A / M ) . \\end{align*}"}
{"id": "8068.png", "formula": "\\begin{align*} g ( \\operatorname { g r a d } ( \\| Z \\| ^ { 2 } ) + 2 \\nabla _ { Z } Z , 2 \\nabla _ { Z } Z ) = - F f ^ { 2 } \\| X \\| ^ { 2 } \\| \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) \\| ^ { 2 } \\end{align*}"}
{"id": "1421.png", "formula": "\\begin{align*} \\begin{aligned} A ^ { ( 1 ) } _ { N , d , D } & = \\left \\{ R _ { N , d , D } > K _ { 0 } \\Big ( \\frac { \\beta + \\gamma } { \\beta } \\Big ) ^ { 1 / 2 } N ^ { 4 / 3 } ( \\log N ) ^ { 4 / 3 } \\right \\} , \\\\ A ^ { ( 2 ) } _ { N , d , D } & = \\left \\{ R _ { N , d , D } < \\varepsilon _ 0 \\frac { \\gamma } { \\beta + \\gamma } N ^ { 4 / 3 } ( \\log N ) ^ { - 2 / 3 } \\right \\} . \\end{aligned} \\end{align*}"}
{"id": "7095.png", "formula": "\\begin{align*} \\tau _ x ( f ) : = \\int _ { G / G _ x } f ( g \\theta g ^ { - 1 } ) d g . \\end{align*}"}
{"id": "4285.png", "formula": "\\begin{align*} \\langle \\otimes _ { l = 1 } ^ { m } \\mathfrak { a } \\rangle _ { t } = \\exp \\biggl ( - \\frac { t } { 2 } \\sum \\limits _ { j = 1 } ^ N \\sum \\limits _ { i , p = 1 } ^ m \\otimes _ { l = 1 } ^ m ( J K _ { j } ) ^ { \\delta _ { i l } + \\delta _ { p l } } \\biggr ) \\langle \\otimes _ { l = 1 } ^ { m } \\mathfrak { a } \\rangle _ { 0 } , \\end{align*}"}
{"id": "3861.png", "formula": "\\begin{align*} f _ 1 \\ast f _ 2 = \\circledast \\circ f _ 1 \\tilde { \\otimes } f _ 2 \\circ \\Delta \\end{align*}"}
{"id": "7590.png", "formula": "\\begin{align*} \\frac { d x } { d t } = b _ 0 ( t ) + b _ 1 ( t ) x + b _ 2 ( t ) x ^ 2 \\ , , \\end{align*}"}
{"id": "2210.png", "formula": "\\begin{align*} h _ { j } ( p ) = 0 , h _ { j \\overline { l } } ( p ) = \\delta _ { j } ^ { l } , d h _ { j \\overline { l } } ( p ) = 0 . \\end{align*}"}
{"id": "1395.png", "formula": "\\begin{align*} \\| L _ { t } ^ { - 1 } [ S _ t u _ 0 ] W ^ \\gamma _ t u _ 0 \\| _ { H ^ 1 _ x ( \\mathbb { T } ) } = \\| W ^ \\gamma _ t u _ 0 \\| _ { H ^ 1 _ x ( \\mathbb { T } ) } \\lesssim e ^ { - \\gamma t } \\to 0 . \\end{align*}"}
{"id": "5619.png", "formula": "\\begin{align*} B = ( B _ 1 P _ 1 ^ { r _ 1 } ) _ t = ( B _ 2 P _ 1 ^ { r _ 1 } P _ 2 ^ { r _ 2 } ) _ t = \\dots = ( B _ { n } P _ { 1 } ^ { r _ { 1 } } P _ { 2 } ^ { r _ { 2 } } \\cdots P _ { n } ^ { r _ { n } } ) _ { t } . \\end{align*}"}
{"id": "2861.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { L } ( x ) = - \\tfrac { 1 } { N } \\sum _ { i = 1 } ^ N y _ i \\log \\big ( \\sigma ( A _ i x ) \\big ) + ( 1 - y _ i ) \\log \\big ( 1 - \\sigma ( A _ i x ) \\big ) . \\end{aligned} \\end{align*}"}
{"id": "3079.png", "formula": "\\begin{align*} c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast \\ast } = & \\frac { \\Gamma \\left ( \\frac { n - 1 } { 2 } \\right ) } { ( 2 \\pi ) ^ { ( n - 1 ) / 2 } } \\bigg \\{ \\int _ { 0 } ^ { \\infty } t ^ { \\frac { n - 3 } { 2 } } \\left [ ( 2 + 3 \\sqrt { 2 } ) \\left ( t + \\frac { \\xi _ { \\boldsymbol { s } , k } ^ { 2 } } { 2 } \\right ) + 3 \\right ] \\exp \\left ( - \\sqrt { 2 t + \\xi _ { \\boldsymbol { s } , k } ^ { 2 } } \\right ) \\mathrm { d } t \\bigg \\} ^ { - 1 } \\end{align*}"}
{"id": "15.png", "formula": "\\begin{align*} \\delta _ n ( \\beta ) : = \\dfrac { f ' ( v _ n ) } { \\beta } \\mod \\mathcal { D } _ n . \\end{align*}"}
{"id": "7526.png", "formula": "\\begin{align*} n = m _ s = m _ { s - 1 } \\frac { b _ { s - 1 } ^ * } { c _ { s } ^ * } \\geq h _ { s - 1 } \\left ( 1 - \\frac { s - 1 } { n + 2 ( s - 1 ) - 2 } \\right ) . \\end{align*}"}
{"id": "2415.png", "formula": "\\begin{align*} \\partial _ t u _ R + \\Delta ^ 2 u _ R = \\Delta f _ R ( u _ R ) + \\sigma ( u _ R ) \\dot { W } , R \\ge 1 \\end{align*}"}
{"id": "8886.png", "formula": "\\begin{align*} \\sigma _ 1 \\leqslant a _ 2 = \\left ( a _ 1 ^ 2 - \\left | \\operatorname { d e t } \\left ( a _ 1 ^ 2 - A ^ { H } A \\right ) \\right | \\left ( \\frac { n - 1 } { ( n + 1 ) a _ 1 ^ 2 - \\| A \\| _ { F } ^ { 2 } } \\right ) ^ { n - 1 } \\right ) ^ { 1 / 2 } \\leqslant a _ 1 \\end{align*}"}
{"id": "488.png", "formula": "\\begin{align*} ( X : B ) = \\bigcap _ { b \\in B } ( X + b ) = \\bigcap _ { i = 0 } ^ { r } ( X + b _ i ) \\end{align*}"}
{"id": "6731.png", "formula": "\\begin{align*} \\langle T ( z ) , v \\rangle _ { Y ' , Y } = \\langle l , v \\rangle _ { Y ' , Y } , \\forall v \\in Y , \\end{align*}"}
{"id": "3534.png", "formula": "\\begin{align*} L _ { 1 } = \\frac { \\Gamma ( t - 1 ) \\left [ \\xi _ { p } \\left ( 1 + \\xi _ { p } ^ { 2 } \\right ) ^ { - ( t - 1 ) } - \\xi _ { q } \\left ( 1 + \\xi _ { q } ^ { 2 } \\right ) ^ { - ( t - 1 ) } \\right ] } { 2 \\Gamma ( t - \\frac { 1 } { 2 } ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , ~ L _ { 2 } & = \\frac { F _ { Y _ { ( 1 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "4602.png", "formula": "\\begin{align*} \\langle \\cdot | \\cdot \\rangle _ { \\lambda } ^ i \\colon \\left ( { \\sf C } _ \\lambda ^ i \\otimes L _ \\lambda \\right ) \\otimes \\left ( { \\sf C } _ { \\lambda ^ \\dagger } ^ i \\otimes L _ { \\lambda ^ \\dagger } \\right ) \\rightarrow \\C , ( i = 1 , 2 ) \\end{align*}"}
{"id": "7497.png", "formula": "\\begin{align*} N _ n ( t , t ) = \\sum _ { r = 1 } ^ n \\frac { 1 } { n } \\binom { n } { r } \\binom { n } { r - 1 } t ^ { r } . \\end{align*}"}
{"id": "4108.png", "formula": "\\begin{align*} \\Phi '' ( \\vec p ) [ \\widehat { \\vec p } _ 1 , \\widehat { \\vec p } _ 2 ] = F '' ( V ( \\vec p ) ) [ V ' ( \\vec p ) \\widehat { \\vec p } _ 1 , V ' ( \\vec p ) \\widehat { \\vec p } _ 2 ] + F ' ( V ( \\vec p ) ) V '' ( \\vec p ) [ \\widehat { \\vec p } _ 1 , \\widehat { \\vec p } _ 2 ] \\end{align*}"}
{"id": "3580.png", "formula": "\\begin{align*} w _ 4 ( \\pi ) = \\begin{cases} \\sum \\limits _ { i = 1 } ^ 4 t _ i ^ 2 , & , \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "3546.png", "formula": "\\begin{align*} m _ { i j } = \\chi _ { \\sigma _ j } ( h _ i ) . \\end{align*}"}
{"id": "2355.png", "formula": "\\begin{align*} a _ j h _ \\rho ^ j = a _ { \\rho 0 j } + a _ { \\rho 1 j } Q _ \\rho + \\ldots + a _ { \\rho l j } Q _ \\rho ^ l \\end{align*}"}
{"id": "7847.png", "formula": "\\begin{align*} \\| L _ { \\xi p _ k } - L _ { \\xi _ n p _ k } \\| = \\sup _ { a \\in N , \\| a \\| _ 2 \\leq 1 } \\| ( \\xi - \\xi _ n ) p _ k a \\| = \\sup _ { a \\in N , \\| a \\| _ 2 \\leq 1 } \\varphi _ n ( p _ k a ^ * a p _ k ) \\to 0 . \\end{align*}"}
{"id": "4512.png", "formula": "\\begin{align*} \\bar R _ { \\alpha \\beta \\alpha 3 } & = R _ { \\alpha \\beta \\alpha 3 } + k _ { \\alpha 3 } k _ { \\beta \\alpha } - k _ { \\alpha \\alpha } k _ { \\beta 3 } \\\\ & = \\nabla _ \\alpha k _ { \\alpha \\beta } - \\nabla _ \\beta k _ { \\alpha \\alpha } = 0 , \\end{align*}"}
{"id": "6294.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } u ( x ) + p ( x ) u ( x / q ) + r ( x ) = 0 , \\end{align*}"}
{"id": "4186.png", "formula": "\\begin{align*} F _ \\ell ^ { ( \\iota ) } ( \\lambda , \\rho ) = ( F ^ { ( \\iota ) } \\psi ) ( \\sqrt \\lambda ) \\chi _ \\ell ( \\lambda / \\rho ) \\quad \\rho \\neq 0 \\end{align*}"}
{"id": "6218.png", "formula": "\\begin{align*} \\frac { D _ { q ^ { - 1 } } D _ q h ( x ) } { h ( x / q ) } & = D _ { q ^ { - 1 } } u ( x ) + \\frac { D _ { q ^ { - 1 } } h ( x ) } { h ( x / q ) } u ( x ) \\\\ & = D _ { q ^ { - 1 } } u ( x ) + u ( x ) u ( \\frac { x } { q } ) . \\end{align*}"}
{"id": "6213.png", "formula": "\\begin{align*} \\int _ 0 ^ a f ( t ) d _ q t : = ( 1 - q ) a \\sum _ { n = 0 } ^ \\infty q ^ n f ( a q ^ n ) , \\ , \\ , a \\in \\mathbb R , \\end{align*}"}
{"id": "6049.png", "formula": "\\begin{align*} \\left ( f _ { r , s } \\right ) _ { r ' , s ' } ( x ) = f ( x ) \\end{align*}"}
{"id": "7291.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ N \\sum _ { l = 1 } ^ i c _ { i , l } ( { \\sf U } _ \\infty ^ { - 1 } \\theta _ 0 ) ^ { i - 2 } ( \\theta _ 0 ^ { - 1 } \\theta _ 1 ) ^ { l - 1 } & = \\sum _ { i = 0 } ^ { N ' } c _ i | x | ^ \\frac { 2 ( p - q ) i } { 1 - q } , \\\\ \\sum _ { i = 3 } ^ N \\sum _ { l = 1 } ^ i d _ { i , l } ( { \\sf U } _ \\infty ^ { - 1 } \\theta _ 0 ) ^ { i - 3 } ( \\theta _ 0 ^ { - 1 } \\theta _ 1 ) ^ { l - 1 } & = \\sum _ { i = 0 } ^ { N ' } d _ i | x | ^ \\frac { 2 ( p - q ) i } { 1 - q } . \\end{align*}"}
{"id": "5784.png", "formula": "\\begin{align*} E ( G / H ^ 0 ) = E ( G / H ) E ( H / H ^ 0 ) = \\pm | H / H ^ 0 | = \\pm 1 . \\end{align*}"}
{"id": "3704.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\lim _ { \\alpha \\to 0 } \\frac { i } h \\int _ { \\Omega _ \\Gamma } [ - h ^ 2 \\Delta - 1 , \\chi _ \\alpha ( x _ n ) h D _ n ] u _ h \\overline { u _ h } d x = \\int _ { S ^ * _ \\Gamma M \\backslash S ^ * \\Gamma } | \\xi _ n | d \\mu ^ \\perp , \\end{align*}"}
{"id": "1257.png", "formula": "\\begin{align*} \\Phi & = \\{ l ( T ) \\} \\cup \\{ j \\in \\mathbb { N } \\colon 1 \\le j \\le l ( T ) - 1 , c _ { l ( T ) - j } > 1 \\} . \\end{align*}"}
{"id": "8967.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\left \\vert \\int _ \\Omega R u ^ 2 d V _ g \\right \\vert & = & \\left \\vert \\int _ \\Omega R ( ( \\xi _ { \\tilde K } + \\xi _ E ) u ) ^ 2 d V _ g \\right \\vert \\\\ & \\leq & \\left \\vert \\int _ \\Omega R \\xi _ { \\tilde K } ^ 2 u ^ 2 d V _ g \\right \\vert + \\left \\vert 2 \\int _ \\Omega R \\xi _ E \\xi _ { \\tilde K } u ^ 2 d V _ g \\right \\vert + \\left \\vert \\int _ \\Omega R \\xi _ E ^ 2 u ^ 2 d V _ g \\right \\vert . \\end{array} \\end{align*}"}
{"id": "669.png", "formula": "\\begin{align*} \\left | \\Box ^ * _ { g _ { 1 , t } } v _ { 2 , t } \\right | & = \\left | \\Box ^ * _ { g _ { 1 , t } } v _ { 2 , t } - \\Box ^ * _ { g _ { 2 , t } } v _ { 2 , t } \\right | \\\\ & \\le \\left | \\Delta _ { g _ { 1 , t } } v _ { 2 , t } - \\Delta _ { g _ { 2 , t } } v _ { 2 , t } \\right | + | R _ { g _ { 1 , t } } - R _ { g _ { 2 , t } } | v _ { 2 , t } \\\\ & \\le C \\varepsilon \\left ( | \\nabla ^ 2 v _ { 2 , t } | _ { g _ { 2 , t } } + | \\nabla v _ { 2 , t } | _ { g _ { 2 , t } } + v _ { 2 , t } \\right ) . \\end{align*}"}
{"id": "7830.png", "formula": "\\begin{align*} \\varphi ( x ) = \\langle \\pi ( x ) \\eta _ 0 , \\xi _ 0 \\rangle = \\langle P _ { \\mathcal L _ 1 } \\pi ( x ) \\eta _ 0 , \\xi _ 0 \\rangle = \\langle \\tilde \\pi ( x ) \\eta _ 0 , \\xi \\rangle . \\end{align*}"}
{"id": "2198.png", "formula": "\\begin{align*} & ( \\underbrace { + } _ { a _ { u j } } , \\ \\underbrace { - \\dots - } _ { a _ { u + 1 \\ , j + 1 } } , \\ \\cdots \\ , \\ \\underbrace { + \\dots + } _ { a _ { w - 1 \\ , j } } , \\ \\underbrace { - \\dots - } _ { a _ { w \\ , j + 1 } } ) . \\end{align*}"}
{"id": "2299.png", "formula": "\\begin{align*} \\begin{cases} ( u , v , h , g ) | _ { y = 0 } = ( u , v , h , g ) | _ { y \\rightarrow \\infty } = ( 0 , 0 , 0 , 0 ) , \\\\ ( u , v , h , g ) | _ { x = 1 } = ( u , v , h , g ) | _ { x \\rightarrow \\infty } = ( 0 , 0 , 0 , 0 ) . \\end{cases} \\end{align*}"}
{"id": "518.png", "formula": "\\begin{align*} \\Phi _ { n } ( x ) = \\frac { a _ 1 } { a _ 2 } \\ln \\vert a _ 2 x + a _ 4 \\vert \\ . \\end{align*}"}
{"id": "3899.png", "formula": "\\begin{align*} \\sup _ { B _ 1 ( 0 ) } u \\leq \\lim _ { j \\to + \\infty } \\Phi _ \\rho ( \\mu _ { j + 1 } , h ^ j _ 1 ) \\leq C _ 1 \\Phi _ \\rho ( p _ 1 , 2 ) , C _ 1 = C ^ \\frac { \\kappa } { p _ 1 ( \\kappa - 1 ) } ( 2 ^ { \\beta } \\kappa ^ { \\nu } ) ^ { \\frac { 1 } { p _ 1 } \\sum _ j \\frac { j } { \\kappa ^ j } } \\end{align*}"}
{"id": "7023.png", "formula": "\\begin{align*} \\hat \\tau \\leq - 2 \\frac { \\xi _ x \\nu _ x + \\xi _ y \\nu _ y } { \\nu _ x ^ 2 + \\nu _ y ^ 2 } = 2 \\tau ^ * . \\end{align*}"}
{"id": "4562.png", "formula": "\\begin{align*} \\mathrm { E h r } _ P ( x ) = \\sum _ { t \\geq 0 } | t P \\cap \\Z ^ { n } | x ^ t = \\frac { \\sum _ { i = 0 } ^ d h _ i ^ * x ^ i } { ( 1 - x ) ^ { d + 1 } } \\ , . \\end{align*}"}
{"id": "1896.png", "formula": "\\begin{align*} \\mathcal { U } _ { [ n , j ] } = \\bigcup _ { k = j } ^ { n - 1 } \\mathcal { U } _ { [ n , j , k ] } \\end{align*}"}
{"id": "5602.png", "formula": "\\begin{align*} x y z = a b ^ 2 , x + y + z = a b c . \\end{align*}"}
{"id": "7877.png", "formula": "\\begin{align*} D _ { Z , + } = ( D _ { 2 , + } + D _ + ) \\cap Z _ + \\end{align*}"}
{"id": "5151.png", "formula": "\\begin{align*} [ D \\vec { x } ] _ { v _ 1 } - [ D \\vec { x } ] _ { v _ 2 } = & \\sum _ { j = 1 } ^ n d _ G ( v _ 1 , v _ j ) \\vec { x } _ { v _ j } - \\sum _ { j = 1 } ^ n d _ G ( v _ 2 , v _ j ) \\vec { x } _ { v _ j } \\\\ = & d _ G ( v _ 1 , v _ 1 ) \\vec { x } _ { v _ 1 } + d _ G ( v _ 1 , v _ 2 ) \\vec { x } _ { v _ 2 } - d _ G ( v _ 1 , v _ 2 ) \\vec { x } _ { v _ 1 } - d _ G ( v _ 2 , v _ 2 ) \\vec { x } _ { v _ 2 } \\\\ = & d _ G ( v _ 1 , v _ 2 ) ( \\vec { x } _ { v _ 2 } - \\vec { x } _ { v _ 1 } ) . \\end{align*}"}
{"id": "6429.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\dd \\Theta _ t } { \\dd t } & = \\Bar { Q } \\circ H _ t + H _ t \\circ Q _ \\mathfrak g \\\\ \\Xi _ 0 & = \\Bar \\Xi _ 1 \\end{cases} \\end{align*}"}
{"id": "5339.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { s , p } ( \\Omega ) } \\vcentcolon = \\left ( \\| u \\| _ { L ^ p ( \\Omega ) } ^ p + [ u ] _ { W ^ { s , p } ( \\Omega ) } ^ p \\right ) ^ { 1 / p } , \\end{align*}"}
{"id": "2642.png", "formula": "\\begin{align*} g _ 1 = \\begin{vmatrix} x _ 0 & x _ 1 \\\\ a _ 0 & a _ 1 \\end{vmatrix} \\ , \\ g _ 2 = \\begin{vmatrix} y _ 0 & y _ 1 \\\\ b _ 0 & b _ 1 \\end{vmatrix} \\ , \\ g _ 3 = \\begin{vmatrix} z _ 0 & z _ 1 \\\\ c _ 0 & c _ 1 \\end{vmatrix} \\ . \\end{align*}"}
{"id": "63.png", "formula": "\\begin{align*} \\| A \\cdot u \\| _ r ^ 2 & = \\sum _ { k \\in \\Z } ( 1 + | k | ) ^ { 2 r } \\Big | \\sum _ { h \\in \\Z } A _ { k - h } u _ h \\Big | ^ 2 \\\\ \\ & \\leq \\sum _ { k \\in \\Z } ( 1 + | k | ) ^ { 2 r } \\Big ( \\sum _ { h \\in \\Z } \\frac { 1 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } \\Big ) \\Big ( \\sum _ { h \\in \\Z } \\| A _ { k - h } \\| ^ 2 ( 1 + | k - h | ) ^ { 2 s } | u _ h | ^ 2 ( 1 + | h | ) ^ { 2 r } \\Big ) . \\end{align*}"}
{"id": "8658.png", "formula": "\\begin{align*} \\sup \\left \\{ \\int _ 0 ^ \\infty f g \\ , d r : \\ f \\geq 0 \\ , , \\ \\underline \\mu _ \\alpha ( f ) \\leq 1 \\right \\} = \\alpha \\ , \\overline \\nu _ \\alpha ( g ) \\end{align*}"}
{"id": "8897.png", "formula": "\\begin{align*} f _ { i j } ( v ) \\ , \\partial _ k F ( v ) + f _ { j k } ( v ) \\ , \\partial _ i F ( v ) + f _ { k i } ( v ) \\ , \\partial _ j F ( v ) = 0 \\end{align*}"}
{"id": "712.png", "formula": "\\begin{align*} & R ^ N ( \\partial _ t , \\partial _ i , \\partial _ j , \\partial _ k ) = R ^ N _ { 0 i j k } = 0 , \\\\ & R ^ N ( \\partial _ t , \\partial _ i , \\partial _ j , \\partial _ t ) = R ^ N _ { 0 i j 0 } = - f ( t ) f '' ( t ) \\widetilde { g } _ { i j } . \\end{align*}"}
{"id": "1434.png", "formula": "\\begin{align*} x _ i x _ j = q ^ { h _ { i j } } x _ j x _ i , i , j = 1 , 2 , \\cdots , n , \\end{align*}"}
{"id": "8824.png", "formula": "\\begin{align*} f _ t \\left ( \\int _ { \\Omega } ^ { } \\alpha ( \\omega ) d \\omega \\right ) = \\int _ { \\Omega } ^ { } \\alpha ( \\omega ) d \\omega - t = \\int _ { \\Omega } ^ { } ( \\alpha ( \\omega ) - T ( \\omega ) ) d \\omega \\leq \\int _ { \\Omega } ^ { } f _ { T ( \\omega ) } \\circ \\alpha ( \\omega ) d \\omega . \\end{align*}"}
{"id": "4559.png", "formula": "\\begin{align*} \\Lambda ( G ) = \\mathrm { s p a n } _ K \\{ \\varepsilon _ 1 , \\varepsilon _ 2 , \\varepsilon _ 3 , \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 , \\alpha _ 1 \\alpha _ 2 \\} \\ , , \\end{align*}"}
{"id": "3806.png", "formula": "\\begin{align*} E r r ^ 0 _ { i , i _ 1 , i _ 2 , i _ 3 , i _ 4 } ( t _ 1 , t _ 2 ) = \\sum _ { b = 0 , 1 } \\int _ { ( \\R ^ 3 ) ^ 5 } e ^ { i X ( t _ b ) \\cdot ( \\xi + \\eta + \\sigma + \\kappa + \\chi ) + i t _ b ( \\mu _ 4 | \\chi | + \\mu _ 3 | \\kappa | + \\mu _ 2 | \\sigma | + \\mu | \\xi | + \\mu _ 1 | \\eta | ) } \\big ( \\Phi _ 5 ( \\xi , \\eta , \\sigma , \\kappa , \\chi , V ( t _ b ) ) \\big ) ^ { - 1 } \\end{align*}"}
{"id": "5423.png", "formula": "\\begin{align*} \\int _ { \\Omega _ e } ( q _ 1 - q _ 2 ) f g \\ , d x = 0 \\end{align*}"}
{"id": "8301.png", "formula": "\\begin{align*} \\hat { P } \\phi _ { n } ^ { \\pm } ( x ) = \\hat { T } \\phi _ { n } ^ { \\pm } ( x ) = \\phi _ { n \\pm 1 } ^ { \\mp } ( x ) . \\end{align*}"}
{"id": "5360.png", "formula": "\\begin{align*} s _ { \\Tilde { \\theta } } = ( 1 - \\Tilde { \\theta } ) s _ 0 + \\Tilde { \\theta } s _ 1 = \\frac { t - s } { t - r } r + \\frac { s - r } { t - r } t = s . \\end{align*}"}
{"id": "6657.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } u _ t ( t , r ) = \\Delta u ( t , r ) - \\lambda ( t , x ) | \\nabla u ( t , r ) | ^ { \\alpha } + a ( t , x ) v ^ { p } ( t , r ) , & t > 0 , \\ 0 < r < h ( t ) , \\\\ v _ t ( t , r ) = \\Delta v ( t , r ) - \\lambda ( t , x ) | \\nabla v ( t , r ) | ^ { \\alpha } + a ( t , x ) u ^ { p } ( t , r ) , & t > 0 , \\ 0 < r < g ( t ) , \\end{array} \\right . \\end{align*}"}
{"id": "185.png", "formula": "\\begin{align*} \\frac { 1 } { \\gamma } \\leq \\| \\dot C _ 0 \\| \\int _ { 0 } ^ { \\beta } e ^ { 2 \\int _ 0 ^ t \\chi _ s \\ , d s } \\ , d t = \\frac { 1 } { \\alpha ^ 2 } \\int _ { 0 } ^ { \\beta } e ^ { 2 \\int _ 0 ^ t \\chi _ s \\ , d s } \\ , d t . \\end{align*}"}
{"id": "903.png", "formula": "\\begin{align*} \\left ( r \\tilde { A } \\right ) '' - \\dfrac { 2 } { r } f ^ { 2 } \\tilde { A } = - \\dfrac { 1 } { 4 } g ^ { 2 } \\tilde { \\rho } ^ { 2 } r \\left ( B - \\tilde { A } \\right ) \\end{align*}"}
{"id": "6613.png", "formula": "\\begin{align*} \\hbox { N } _ h ( N , w ; \\xi ) = c _ h h ^ { - 2 } \\sum _ { c \\in \\mathbb { Z } ^ 4 } \\sum _ { k = 1 } ^ \\infty ( k \\ell ) ^ { - 4 } S _ { k } ( c ; \\xi ) I _ { k , \\ell } ( c ) , \\end{align*}"}
{"id": "7050.png", "formula": "\\begin{align*} \\sum _ { M \\geq 0 } F _ M \\ , z ^ M & = \\frac { z ( 1 - z - z ^ 2 ) } { ( 1 - z - z ^ 2 ) ^ 2 } , \\\\ \\sum _ { M \\geq 0 } M F _ M \\ , z ^ M & = \\frac { z + z ^ 3 } { ( 1 - z - z ^ 2 ) ^ 2 } , \\\\ \\sum _ { M \\geq 0 } \\sum _ { j = 0 } ^ M F _ j F _ { M - j } \\ , z ^ M & = \\frac { z ^ 2 } { ( 1 - z - z ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "8727.png", "formula": "\\begin{align*} \\int \\limits _ { a } ^ { c } U \\omega d t = 0 , \\end{align*}"}
{"id": "2926.png", "formula": "\\begin{align*} \\operatorname { i n t } ( F , G ) = \\operatorname { i n t } ( F ) \\cdot ( A , B ) = ( A , B ) \\cdot \\operatorname { i n t } ( G ) = \\operatorname { i n t } ( F ) \\cdot ( A , B ) \\cdot \\operatorname { i n t } ( G ) . \\end{align*}"}
{"id": "5655.png", "formula": "\\begin{align*} \\sup _ { D _ s } \\left | u _ s ( x ) - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \\right | < C _ 1 , \\end{align*}"}
{"id": "790.png", "formula": "\\begin{align*} e ^ { D r ^ m } ( \\beta + ( 1 - \\beta ( 1 - D ) ) r ^ m ) + \\sum _ { n = 2 } ^ { \\infty } \\prod _ { k = 0 } ^ { n - 2 } \\frac { D } { k + 1 } \\phi _ { n } ( r ) = e ^ { - D } . \\end{align*}"}
{"id": "929.png", "formula": "\\begin{align*} E _ c ( \\Psi _ t ) ( 0 , - \\frac { 1 } { 2 } f ) = E ( \\Psi _ t ) ( 0 , - \\frac { 1 } { 2 } f + k \\cdot g ) = v - Q ( v , v ) , \\end{align*}"}
{"id": "8130.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\int _ 0 ^ T \\| \\chi _ K \\psi _ t \\| \\ , d t = 0 \\end{align*}"}
{"id": "4443.png", "formula": "\\begin{align*} \\L I _ m f , g \\R = \\L f , I _ m ^ * g \\R \\end{align*}"}
{"id": "5115.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\limsup _ { n \\rightarrow \\infty } \\left | E \\left [ \\exp \\left ( i \\mu n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { t - \\delta } ( t - s ) ^ \\alpha \\xi ^ n _ s d W _ s \\right ) \\right ] - E \\left [ \\exp \\left ( i \\mu Y ^ { n , 1 } _ t \\right ) \\right ] \\right | = 0 . \\end{align*}"}
{"id": "3166.png", "formula": "\\begin{align*} e _ { ( \\cdot ) } \\left ( \\omega \\right ) \\doteq \\omega \\left ( \\mathfrak { e } _ { ( \\cdot ) } \\right ) = d _ { + } \\mathfrak { a } _ { + } \\end{align*}"}
{"id": "3761.png", "formula": "\\begin{align*} \\widetilde { T } _ { k , j ; n } ^ { b i l ; \\mu , i } ( \\mathfrak { m } , U ) ( t , x , \\zeta ) = \\int _ { 0 } ^ { t } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big ( E ( s , x - y + ( t - s ) \\omega ) + \\hat { v } \\times B ( s , x - y + ( t - s ) \\omega ) \\big ) \\cdot \\nabla _ v \\big ( ( t - s ) \\mathcal { K } ^ { U , i } _ { k , j , n } ( \\mathfrak { m } ) ( y , v , \\omega , \\zeta ) \\end{align*}"}
{"id": "2388.png", "formula": "\\begin{align*} ( \\chi _ { ( 1 , 0 ) } \\otimes \\rho ) \\left [ \\begin{pmatrix} x _ 1 \\\\ x _ 2 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ u & v \\end{pmatrix} \\right ] f ( t ) = | v | ^ { - 1 / 2 } e ^ { 2 \\pi i x _ 1 } f \\left ( \\frac { t - u } { v } \\right ) , t \\in \\R . \\end{align*}"}
{"id": "777.png", "formula": "\\begin{align*} \\beta | f ' ( z ^ m ) | + ( 1 - \\beta ) | f ( z ^ m ) | + \\sum _ { n = 1 } ^ { \\infty } | a _ n | \\phi _ { n } ( r ) \\leq d ( 0 , \\partial { \\Omega } ) \\end{align*}"}
{"id": "2857.png", "formula": "\\begin{align*} l _ 0 ( x ) = | x | _ 0 = \\left \\{ \\begin{array} { l l } 1 & x \\neq 0 \\\\ 0 & x = 0 \\end{array} \\right . . \\end{align*}"}
{"id": "2716.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\bigg | \\Big \\langle \\Phi , \\Big ( { U } _ N ( & H _ N - H _ { 2 \\mathrm { - m o d e } } ) { U } _ N ^ * - \\mathbb { H } - \\mu _ + \\mathcal { N } _ \\perp \\Big ) \\Phi \\Big \\rangle \\\\ & - \\frac { \\lambda } { \\sqrt { 2 ( N - 1 ) } } \\sum _ { m \\ge 3 } \\Big \\langle \\Phi , \\Big ( \\big ( w _ { + 1 - m } \\ , \\Theta + w _ { + 2 - m } \\ , \\Theta ^ { - 1 } \\big ) a _ m \\mathfrak { D } + \\mathrm { h . c . } \\Big ) \\Phi \\big \\rangle \\bigg | = 0 . \\end{align*}"}
{"id": "4725.png", "formula": "\\begin{align*} e _ { ( k + 1 ) } = \\jmath ( e _ { ( k + 1 ) } ) = \\Big ( \\frac { q - q ^ { - 1 } } { z - z ^ { - 1 } } \\Big ) ^ { k - 1 } e _ { ( k ) } H _ { 2 k } ^ { - 1 } H _ { 2 k - 1 } ^ { - 1 } H _ { 2 k + 1 } H _ { 2 k } e _ { ( k ) } . \\end{align*}"}
{"id": "2519.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\{ g ( \\nabla _ X \\xi , Y ) + g ( \\nabla _ Y \\xi , X ) \\} + R i c ( X , Y ) + \\mu g ( X , Y ) = 0 . \\end{align*}"}
{"id": "3859.png", "formula": "\\begin{align*} c \\Big ( T _ 1 , T _ 2 , T _ 3 \\Big ) : = \\Big \\langle \\Delta [ T _ 1 ] , T _ 2 \\times ^ { H ^ { T _ 1 } } T _ 3 \\Big \\rangle . \\end{align*}"}
{"id": "4696.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty C _ 0 ( n ) \\ , q ^ n = \\frac { - 1 } { ( q ; q ) _ \\infty } \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ { n } q ^ { n ( n + 1 ) / 2 } . \\end{align*}"}
{"id": "5791.png", "formula": "\\begin{align*} [ A , B ] = \\bigcup _ { n \\geq 1 } [ A , B ] _ n \\end{align*}"}
{"id": "8951.png", "formula": "\\begin{align*} D _ i ( u - u _ 0 ) ( x _ 0 ) = \\kappa ( u - u _ 0 ) ( x _ 0 ) D _ i \\phi ( x _ 0 ) , { \\rm f o r } \\ i = 1 , \\cdots , n . \\end{align*}"}
{"id": "4600.png", "formula": "\\begin{align*} \\frac { 1 } { a ( k + h ^ \\vee ) } + \\frac { 1 } { b ( \\ell + h ^ \\vee ) } = c n \\end{align*}"}
{"id": "5827.png", "formula": "\\begin{align*} \\frac { y _ { n } } { x _ { n } } = \\left ( \\frac { p _ n ^ 2 } { p _ n ^ 2 - 1 } \\right ) \\prod _ { k \\leq n - 1 } \\frac { p _ k ^ 2 } { p _ k ^ 2 - 1 } < \\frac { y _ { n + 1 } } { x _ { n + 1 } } = \\left ( \\frac { p _ { n + 1 } ^ 2 } { p _ { n + 1 } ^ 2 - 1 } \\right ) \\left ( \\frac { p _ n ^ 2 } { p _ n ^ 2 - 1 } \\right ) \\prod _ { k \\leq n - 1 } \\frac { p _ k ^ 2 } { p _ k ^ 2 - 1 } . \\end{align*}"}
{"id": "7160.png", "formula": "\\begin{align*} ( g _ { 2 1 } y _ 1 + g _ { 2 3 } y _ 3 ) \\{ g _ { 1 2 } y _ 2 + ( g _ { 1 3 } + g _ { 2 3 } ) y _ 3 \\} ^ { a - 1 } = g _ { 2 3 } ( g _ { 1 3 } + g _ { 2 3 } ) ^ { a - 1 } y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } . \\end{align*}"}
{"id": "1837.png", "formula": "\\begin{align*} \\begin{cases} h _ { j - 1 , j } = 1 , & j \\geq 1 , \\\\ h _ { j + k , j } = a _ { j } ^ { ( k ) } , & 0 \\leq k \\leq p , j \\geq 0 , \\\\ h _ { i , j } = 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
{"id": "1474.png", "formula": "\\begin{align*} \\left . { D _ { u , m } } \\right | _ { \\alpha _ m = 0 } & = \\displaystyle c _ { u , m } \\prod _ { i = 1 } ^ { m - 1 } ( - \\alpha _ i ) ^ { ( 2 n + 1 ) r ^ 2 } \\prod _ { 1 \\le i < j \\le m - 1 } ( \\alpha _ { j } - \\alpha _ { i } ) ^ { ( 2 n + 1 ) r ^ 2 } \\enspace \\\\ & = \\Psi _ { \\boldsymbol { 1 } } \\left ( Q _ { u , m - 1 } ( \\boldsymbol { t } ) { { \\mathfrak { A } } } ( \\boldsymbol { t } ) \\left . { { \\mathfrak { B } } } ( \\boldsymbol { t } ) \\right | _ { \\alpha _ m = 0 } \\right ) \\enspace . \\end{align*}"}
{"id": "2622.png", "formula": "\\begin{align*} \\widetilde c _ n = 1 / c _ n \\ . \\end{align*}"}
{"id": "7532.png", "formula": "\\begin{align*} \\mu ( | T _ { n } - 1 | > 1 ) & \\le \\mathbf { E } | T _ { n } - 1 | ^ { \\frac { p } { 2 } } \\le 2 ^ { \\frac { p } { 2 } - 1 } \\left \\{ \\mathbf { E } | T _ { n } - V _ { n } | ^ { \\frac { p } { 2 } } + \\mathbf { E } | V _ { n } - 1 | ^ { \\frac { p } { 2 } } \\right \\} \\\\ & \\le C n ^ { - \\frac { p } { 4 } } ( \\| m \\| _ { L ^ p } ^ { p } + \\| v \\| _ { \\eta } ^ { p } ) . \\\\ \\end{align*}"}
{"id": "56.png", "formula": "\\begin{align*} ( \\alpha , \\pi _ n ) _ { L , n } = ( \\alpha _ m , v _ m ) _ { E _ { \\rho } ^ m , m } & = - ( v _ m , 1 + b _ m ) _ { E _ { \\rho } ^ m , m } \\\\ & = - ( \\Phi _ { E _ { \\rho } ^ m } ( 1 + b _ m ) ( v _ { 2 m } ) - v _ { 2 m } ) \\\\ & = - ( \\Phi _ K ( \\operatorname { N } _ m ( 1 + b _ m ) ) ( v _ { 2 m } ) - v _ { 2 m } ) . \\end{align*}"}
{"id": "1912.png", "formula": "\\begin{align*} w ( ( n , m ) \\rightarrow ( n + 1 , m + 1 ) ) & = 1 , \\\\ w ( ( n , m ) \\rightarrow ( n + 1 , m - p ) ) & = a _ { m - p } . \\end{align*}"}
{"id": "1167.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\Phi ( t _ n h _ { n , 1 } , t _ n h _ { n , 2 } ) } { t _ n } = \\| h _ 1 - h _ 2 \\| _ { \\dot { H } ^ { - 1 , p } ( \\mu * \\gamma _ { \\sigma } ) } . \\end{align*}"}
{"id": "7893.png", "formula": "\\begin{align*} \\Phi _ 0 ( t ) = \\sum _ { m \\geq 3 } \\sum _ { \\beta } \\frac { 1 } { m ! } \\langle t , \\cdots , t \\rangle _ { 0 , m , \\beta } ^ { X _ { D , \\infty } } q ^ { \\beta } . \\end{align*}"}
{"id": "1250.png", "formula": "\\begin{align*} Q ( T , x ) & = \\prod _ { j = 1 } ^ { l ( T ) } Y _ j ^ { n ( T , l ( T ) + 1 - j ) - n ( T , l ( T ) - j ) } . \\end{align*}"}
{"id": "4861.png", "formula": "\\begin{align*} \\begin{cases} N _ { 1 } = p _ { 1 } + p _ { 2 } ^ { 3 } + p _ { 3 } ^ { 3 } + p _ { 4 } ^ { 3 } + p _ { 5 } ^ { 3 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k } } \\\\ N _ { 2 } = p _ { 6 } + p _ { 7 } ^ { 3 } + p _ { 8 } ^ { 3 } + p _ { 9 } ^ { 3 } + p _ { 1 0 } ^ { 3 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k } } . \\end{cases} \\end{align*}"}
{"id": "6307.png", "formula": "\\begin{align*} \\int F ( x ) k ( q x ) T _ q ( x ) y ( x ) d _ q x = F ( x ) k ( x ) \\left ( y ( x ) u ( x ) - D _ { q ^ { - 1 } } y ( x ) \\right ) . \\end{align*}"}
{"id": "3174.png", "formula": "\\begin{align*} B _ { \\gamma } \\left ( a \\right ) = \\int _ { \\Theta _ { d } } \\sum _ { z \\in \\mathfrak { L } } \\gamma ^ { d } f \\left ( \\gamma a + \\gamma z \\right ) \\mathrm { e } ^ { i \\theta \\cdot z } \\mathrm { d } \\left ( \\mathrm { F } _ { \\ast } \\mathrm { P } \\right ) \\left ( \\theta \\right ) \\ . \\end{align*}"}
{"id": "131.png", "formula": "\\begin{align*} \\big \\| C \\star \\big ( S ^ 3 - C ^ 3 \\big ) \\star S \\| _ { L ^ \\infty } & \\leq c \\| E \\| _ { L ^ \\infty } \\Big ( \\frac { 1 } { m ^ 4 _ t } { \\bf 1 } _ { d = 2 } + \\frac { 1 } { m ^ 2 _ t } { \\bf 1 } _ { d = 3 } \\Big ) + c \\| E \\| _ { L ^ \\infty } ^ 2 \\Big ( \\frac { 1 } { m ^ 4 _ t } { \\bf 1 } _ { d = 2 } + \\frac { 1 } { m ^ 3 _ t } { \\bf 1 } _ { d = 3 } \\Big ) \\\\ & + \\| E \\| _ { L ^ \\infty } ^ 3 \\Big ( \\frac { 4 } { m ^ 4 _ t } + \\frac { \\| E \\| _ { L ^ 1 } } { m ^ 2 _ t } \\Big ) . \\end{align*}"}
{"id": "4329.png", "formula": "\\begin{align*} 2 A \\sum _ 1 ^ N | F ( p _ j ) | \\leq 2 A N \\frac { ( F ( p ) ) } { \\sin \\epsilon } = \\frac { 2 A N } { \\sin \\epsilon } ( \\tilde { \\Omega } _ { L _ 0 } ) . \\end{align*}"}
{"id": "2483.png", "formula": "\\begin{align*} K _ r ( \\tau ( x , y ) ) = \\left ( \\sum _ { i = 1 } ^ { n - 2 } \\frac { 3 } { \\varepsilon } ( \\varphi _ i ^ 1 ( x , y ) - \\Phi _ i ( x , y ) ) ^ 2 \\right ) ^ r < \\left ( \\frac 1 3 \\right ) ^ r . \\end{align*}"}
{"id": "2504.png", "formula": "\\begin{align*} 2 A _ Y X = - \\left \\{ ( \\nu [ X , Y ] - \\lambda ^ 2 g ( X , Y ) \\left ( \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) \\right \\} - 2 \\lambda ^ 2 g ( X , Y ) \\left ( \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
{"id": "5457.png", "formula": "\\begin{align*} r ^ C _ A ( b ) \\stackrel { ( \\ref { s p l . c o m . 1 } ) } = & l ^ B _ A ( r ^ C _ A ( b ) ) r ^ B _ A ( r ^ C _ A ( b ) ) \\\\ = & \\ ; l ^ B _ A ( r ^ C _ A ( b ) ) b \\stackrel { ( \\ref { s p l . c o m } ) } = l ^ B _ A ( r ^ C _ A ( b ) ) l ^ C _ A ( b ) r ^ C _ A ( b ) . \\end{align*}"}
{"id": "1621.png", "formula": "\\begin{align*} P _ { o u t } ( i ) = \\textrm { P r } \\bigg [ \\gamma _ i < \\gamma _ { t h } ( i ) \\bigg ] . \\end{align*}"}
{"id": "8382.png", "formula": "\\begin{align*} | \\mathcal { G } ( 4 ) _ { < 2 ^ \\mu } | & \\geq \\sum _ { } ^ { \\mu } 2 ^ m \\left ( 1 - \\frac { 1 } { \\sqrt { 2 } } \\right ) - 1 \\\\ & = \\sum _ { \\ell = 0 } ^ { \\mu / 2 - 2 } 2 ^ { 2 \\ell + 4 } \\left ( 1 - \\frac { 1 } { \\sqrt { 2 } } \\right ) - 1 \\\\ & = \\frac { \\mu } { 2 } - 1 + 2 ^ 4 \\left ( 1 - \\frac { 1 } { \\sqrt { 2 } } \\right ) \\frac { 4 ^ { \\mu / 2 - 1 } - 1 } { 3 } . \\end{align*}"}
{"id": "3095.png", "formula": "\\begin{align*} \\rho ( f ) ( x ) = \\sum _ { ( x ) , ( f ( x _ 0 ) ) } ( S x _ { - 1 } ) f ( x _ 0 ) _ { - 1 } \\otimes f ( x _ 0 ) _ 0 \\end{align*}"}
{"id": "8319.png", "formula": "\\begin{align*} G ( t , x ) e ^ { z t } = \\sum _ { n = 0 } ^ { \\infty } \\bar { P } _ n ( x , z ) \\frac { t ^ n } { n ! } . \\end{align*}"}
{"id": "670.png", "formula": "\\begin{align*} \\nabla v _ { 2 , t } = - v _ { 2 , t } \\nabla f _ t , \\nabla ^ 2 v _ { 2 , t } = v _ { 2 , t } \\left ( \\nabla f _ t \\otimes \\nabla f _ t - \\nabla ^ 2 f _ t \\right ) . \\end{align*}"}
{"id": "2245.png", "formula": "\\begin{align*} & ( v _ e ^ i , g _ e ^ i ) ( x , 0 ) = - ( v _ p ^ { i - 1 } , g _ p ^ { i - 1 } ) ( x , 0 ) , \\\\ & ( v _ e ^ i , g _ e ^ i ) \\rightarrow ( 0 , 0 ) { \\rm ~ a s ~ } Y \\rightarrow \\infty . \\end{align*}"}
{"id": "8841.png", "formula": "\\begin{align*} f _ { ( n ) + } ' ( 0 ) = n f \\left ( \\frac { 1 } { n } \\right ) > - \\infty . \\end{align*}"}
{"id": "968.png", "formula": "\\begin{align*} \\overline { f } \\otimes \\overline { g } = H ^ { * } ( \\overline { f } ) \\overline { g } \\end{align*}"}
{"id": "819.png", "formula": "\\begin{align*} S _ { \\psi } f ( x ) : = \\frac { 1 } { \\Psi ( x ) } \\int _ 0 ^ { x } f ( t ) \\psi ( t ) d t \\end{align*}"}
{"id": "991.png", "formula": "\\begin{align*} F ( B ^ { * } ) = \\left \\{ \\sum _ { i = 1 } ^ { n } x _ { i } \\overline { b } _ { i } ^ { * } \\mid 0 \\leq x _ { i } < 1 \\ \\ x _ { i } \\in \\mathbb { R } \\right \\} . \\end{align*}"}
{"id": "5224.png", "formula": "\\begin{align*} \\Theta _ k : = \\bigoplus _ { i = 1 } ^ n R _ k \\partial _ { x _ i } . \\end{align*}"}
{"id": "285.png", "formula": "\\begin{align*} f ( A ) : = \\sum _ { a \\in A } \\frac { 1 } { a \\log a } \\ < \\ \\infty , \\end{align*}"}
{"id": "4126.png", "formula": "\\begin{align*} G ^ { i } = - \\sum _ { k = 0 } ^ { \\infty } ( T ^ { i } d ^ { i - 1 } ) ^ { k } T ^ { i } , \\end{align*}"}
{"id": "5574.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ { \\frac { \\alpha } { 2 } } u = \\sigma u ^ q + \\mu , \\ , & u > 0 \\ , \\Omega , \\\\ u = 0 & , \\end{cases} \\end{align*}"}
{"id": "8793.png", "formula": "\\begin{align*} \\phi _ 1 ^ * = \\frac { 1 _ { ( - 1 , 1 ) } } { 2 } , \\phi _ 2 ^ * = y _ 1 1 _ { ( - 5 , 2 \\lambda - 3 ) \\cup ( 3 - 2 \\lambda , 5 ) } + y _ 2 1 _ { ( 2 \\lambda - 3 , 3 - 2 \\lambda ) } \\end{align*}"}
{"id": "11.png", "formula": "\\begin{align*} a \\cdot _ { \\rho } x = \\rho _ a ( x ) ~ ~ \\forall x \\in \\Bar { \\Omega } . \\end{align*}"}
{"id": "843.png", "formula": "\\begin{align*} \\boldsymbol { V } _ { A } ^ { p o s t } = \\left ( \\frac { \\mathbf { F } ^ { H } \\mathbf { F } } { \\sigma _ { n } ^ { 2 } } + ( \\boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \\right ) ^ { - 1 } . \\end{align*}"}
{"id": "3357.png", "formula": "\\begin{align*} F _ m ( \\iota ) = \\begin{cases} & : 1 \\leq m \\leq 2 n a - 1 , \\\\ 0 & : . \\end{cases} \\end{align*}"}
{"id": "4661.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } t ^ { \\frac { d } { \\alpha } \\left ( 1 - \\frac { 1 } { q } \\right ) - \\frac { \\delta } { \\alpha } } \\| u ( t , \\cdot ) - A \\Psi _ t ( \\cdot ) \\| _ { q , H _ t } = 0 , \\end{align*}"}
{"id": "5876.png", "formula": "\\begin{align*} d _ n ^ \\gamma ( { \\mathcal K } ) _ X : = \\inf _ { k \\leq n } \\inf _ { \\| \\cdot \\| _ { Y _ k } } d ^ \\gamma ( { \\mathcal K } , Y _ k ) _ X , \\end{align*}"}
{"id": "5217.png", "formula": "\\begin{align*} \\sum _ { j \\in J } a _ j = r ( J ) + \\ell _ { 1 } ( J ) r , \\sum _ { j \\in J } b _ j = s ( J ) + \\ell _ { 2 } ( J ) s . \\end{align*}"}
{"id": "3991.png", "formula": "\\begin{align*} b _ i \\overline b _ i = - e _ i , ~ ~ i = 0 , 1 , \\cdots , r ; ~ ~ ~ ~ ~ ~ b _ { r + j } \\overline b _ { r + j } = - \\widehat e _ { r + j } , ~ ~ j = 1 , \\cdots , s . \\end{align*}"}
{"id": "2937.png", "formula": "\\begin{align*} & e ^ { i H _ a t } \\{ i [ H _ a , J _ a ] + \\frac { \\partial J _ a } { \\partial t } \\} U ( t , 0 ) \\psi ( 0 ) - e ^ { i H _ a t } i J _ a ( H ( t ) - H _ a ) U ( t , 0 ) \\psi ( 0 ) = \\\\ & e ^ { i H _ a t } D _ { H _ a } ( J _ a ) U ( t , 0 ) \\psi ( 0 ) + ( - i ) e ^ { i H _ a t } J _ a N ( x , t , | \\psi | ) U ( t , 0 ) \\psi ( 0 ) . \\\\ & H ( t ) = - \\Delta + N ( x , t , | \\psi | ) , \\\\ & H _ a \\equiv - \\Delta , . \\end{align*}"}
{"id": "769.png", "formula": "\\begin{align*} | g _ m | ^ 2 \\sim \\frac { L \\beta _ { P I } \\beta _ { I S } } { 2 } \\chi _ { 2 } ^ 2 \\left ( \\frac { 2 | h _ { P S } | ^ 2 } { L \\beta _ { P I } \\beta _ { I S } } \\right ) , m = 1 , 2 , \\ldots , M . \\end{align*}"}
{"id": "1117.png", "formula": "\\begin{align*} C ( v ) = \\ln \\left ( \\frac { \\sinh ( z _ 0 v ) } { z _ 0 v } \\right ) . \\end{align*}"}
{"id": "7702.png", "formula": "\\begin{align*} \\sum _ { n _ 1 , \\ldots , n _ k = 1 } ^ { \\infty } \\frac { ( \\mu * F ) ( n _ 1 , \\ldots , n _ k ) } { n _ 1 \\cdots n _ k } \\end{align*}"}
{"id": "6160.png", "formula": "\\begin{align*} \\begin{aligned} & \\| u ^ { n } _ { h } \\| ^ { 2 } _ { 2 } \\leq K _ { 4 } , \\\\ & \\sum _ { m = 1 } ^ { N } \\| u ^ { m } _ { h } - u ^ { m - 1 } _ { h } \\| ^ { 2 } _ { 2 } \\leq 3 2 K _ { 4 } , \\\\ & \\Delta t \\sum _ { m = 1 } ^ { N } \\| \\nabla u ^ { m } _ { h } \\| ^ { 2 } _ { 2 } \\leq 4 K _ { 4 } . \\end{aligned} \\end{align*}"}
{"id": "5831.png", "formula": "\\begin{align*} \\beta ( 2 ) = 0 . 9 1 5 9 6 5 5 9 4 1 7 7 2 1 9 0 1 5 0 5 4 6 0 3 5 1 4 9 3 2 3 8 4 1 1 0 7 7 4 1 4 9 3 7 4 2 8 1 6 7 \\ldots , \\end{align*}"}
{"id": "91.png", "formula": "\\begin{align*} \\forall \\varphi \\in \\R ^ \\Lambda , V _ 0 ( \\varphi ) = \\sum _ { x \\in \\Lambda } ( \\frac 1 4 g \\varphi ^ 4 _ x + \\frac 1 2 \\nu \\varphi ^ 2 _ x ) . \\end{align*}"}
{"id": "7843.png", "formula": "\\begin{align*} \\langle \\xi _ 1 \\otimes \\eta _ 1 , \\xi _ 2 \\otimes \\eta _ 2 \\rangle = \\langle ( L _ { \\xi _ 2 } ^ * L _ { \\xi _ 1 } ) \\eta _ 1 , \\eta _ 2 \\rangle \\end{align*}"}
{"id": "6554.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\mu ( k ) ( 1 - { E _ 2 ( i , m ) } ) ^ k & \\leq \\sum _ { k = 0 } ^ \\infty \\mu ( k ) \\left ( 1 - \\left ( \\frac { 1 } { 2 A ( m + 1 ) } \\right ) ^ { i + 2 } \\right ) ^ k \\\\ & \\leq \\sum _ { k = 0 } ^ \\infty \\mu ( k ) \\left ( 1 - \\left ( \\frac { 1 } { 2 A ( m + 2 ) } \\right ) ^ { i + 2 } \\right ) ^ k . \\end{align*}"}
{"id": "6331.png", "formula": "\\begin{align*} \\sin _ q z & : = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( q ; q ^ 2 ) _ \\infty } ( z ) ^ { 1 / 2 } J ^ { ( 1 ) } _ { 1 / 2 } ( 2 z ; q ^ 2 ) = \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n \\frac { z ^ { 2 n + 1 } } { [ 2 n + 1 ] _ q ! } , \\end{align*}"}
{"id": "4938.png", "formula": "\\begin{align*} \\theta = \\log h _ 1 ^ 1 + p ( u ) + N \\rho . \\end{align*}"}
{"id": "491.png", "formula": "\\begin{align*} B _ w ( \\lambda ) : = \\bigcup _ { m _ k \\in \\mathbb { N } } f _ { i _ 1 } ^ { m _ 1 } f _ { i _ 2 } ^ { m _ 2 } \\cdots f _ { i _ \\ell } ^ { m _ \\ell } b _ \\lambda \\subset B ( \\lambda ) \\end{align*}"}
{"id": "1380.png", "formula": "\\begin{align*} \\widehat { W _ t ^ \\gamma u _ 0 } = e ^ { - t ( i k ^ 2 + \\gamma ) } \\widehat { u _ 0 } . \\end{align*}"}
{"id": "7632.png", "formula": "\\begin{align*} \\begin{array} { l l l l } ( i ) & \\Sigma _ { n } & = & \\{ t _ { i _ { 1 } } ^ { k _ { 1 } } \\ldots t _ { i _ { r } } ^ { k _ { r } } \\cdot \\sigma \\} , \\ { \\rm w h e r e } \\ 0 \\le i _ { 1 } < \\ldots < i _ { r } \\le n - 1 , \\\\ ( i i ) & \\Sigma ^ { \\prime } _ { n } & = & \\{ { t ^ { \\prime } _ { i _ 1 } } ^ { k _ { 1 } } \\ldots { t ^ { \\prime } _ { i _ r } } ^ { k _ { r } } \\cdot \\sigma \\} , \\ { \\rm w h e r e } \\ 0 \\le i _ { 1 } < \\ldots < i _ { r } \\le n - 1 , \\\\ \\end{array} \\end{align*}"}
{"id": "5836.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ M a _ k ^ \\alpha \\bigg ( \\sum _ { i = N } ^ k b _ i \\bigg ) ^ \\alpha \\approx \\sum _ { k = N } ^ M a _ k ^ \\alpha b _ k ^ \\alpha , \\end{align*}"}
{"id": "5474.png", "formula": "\\begin{align*} \\mathrm { d i s t } _ { M \\setminus N _ s ( K ) } ( x , y ) & \\gtrsim \\mathrm { d i s t } _ { \\mathbb { D } \\setminus B ( 0 , s ) } ( A , B ) \\gtrsim e ^ { a s } \\mathrm { d i s t } ( 0 , C ) = e ^ { a s } \\mathrm { d i s t } ( r ( x ) , r ( y ) ) . \\end{align*}"}
{"id": "7511.png", "formula": "\\begin{align*} P ' ( f p ) = \\sum _ { x \\in N _ { s o r t e d } } \\sum _ { y \\in S _ x } f _ { f p } ( m _ { x } , \\gamma _ { x } , k _ { x } ) f _ { f p } ( m _ { y } , \\gamma _ { y } , k _ { y } ) . \\end{align*}"}
{"id": "8222.png", "formula": "\\begin{align*} U = e ^ { i \\varphi } \\ , \\left ( \\begin{array} { r r r r } 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\end{array} \\right ) . \\end{align*}"}
{"id": "1527.png", "formula": "\\begin{align*} \\Delta _ { r a d } = \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + \\frac { \\alpha } { 2 } \\frac { 1 } { \\tanh ( \\alpha r ) } \\frac { \\partial } { \\partial r } \\end{align*}"}
{"id": "3371.png", "formula": "\\begin{align*} \\min \\dim ( C ) = \\min \\{ C \\} . \\end{align*}"}
{"id": "4897.png", "formula": "\\begin{align*} H _ u ( \\lambda ) = u _ 1 h _ 1 ( \\lambda ) + \\dots + u _ r h _ r ( \\lambda ) , \\lambda \\in T ^ * G . \\end{align*}"}
{"id": "3286.png", "formula": "\\begin{align*} \\mathcal { J } ( \\xi , s ) = 2 s \\int _ { \\R ^ 3 } \\overline { \\omega } \\cdot A ( x ) e ^ { i x \\cdot \\xi } d x . \\end{align*}"}
{"id": "6776.png", "formula": "\\begin{align*} \\sum _ { i \\in M _ C } \\sum _ { j = 1 } ^ { r _ i - 1 } a _ { i r _ i } x ^ * _ { i j } + \\sum _ { i \\in M _ C } \\sum _ { j = r _ i } ^ { n _ i } \\max \\left \\{ a _ { i j } , b - \\sum _ { k \\in M _ C - i } a _ { k r _ k } \\right \\} x ^ * _ { i j } > b ? \\end{align*}"}
{"id": "4094.png", "formula": "\\begin{align*} \\mathrm { P f } \\left [ M _ { k , l } \\right ] _ { k , l = 1 } ^ { 2 N } \\doteq \\frac { 1 } { 2 ^ { N } N ! } \\sum _ { \\pi \\in \\mathcal { S } _ { 2 N } } \\left ( - 1 \\right ) ^ { \\pi } \\prod \\limits _ { j = 1 } ^ { N } M _ { \\pi \\left ( 2 j - 1 \\right ) , \\pi \\left ( 2 j \\right ) } \\end{align*}"}
{"id": "4587.png", "formula": "\\begin{align*} - \\frac { 1 } { m } = m + z , \\Im m \\Im z > 0 . \\end{align*}"}
{"id": "3575.png", "formula": "\\begin{align*} \\dim \\pi _ 1 - \\chi _ { \\pi _ 1 } ( h _ 1 ) & = ( q + 1 ) ( q ^ 2 + q + 1 ) - ( q + 1 ) ( 1 + \\chi _ { \\pi _ 1 } ( - 1 ) + \\chi _ { \\pi _ 1 } ^ { - 1 } ( - 1 ) ) \\\\ & = ( q + 1 ) ( q ^ 2 + q + 1 ) - ( q + 1 ) ( 1 + 2 ( - 1 ) ^ j ) \\\\ & = ( q + 1 ) ( q ^ 2 + q - 2 ( - 1 ) ^ j ) \\end{align*}"}
{"id": "5561.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } | \\nabla \\omega | ^ 2 = W _ + ( \\omega , \\omega ) - \\frac { s } { 3 } ~ , \\end{align*}"}
{"id": "2827.png", "formula": "\\begin{align*} \\hat { W } ( y ) : = Z ( y ) + \\frac { \\mu } { 2 } \\| y \\| ^ 2 , \\end{align*}"}
{"id": "4871.png", "formula": "\\begin{align*} \\begin{aligned} & C _ { 1 } ( \\chi , a ) = \\sum \\limits _ { h = 1 } ^ { q } \\overline { \\chi } ( h ) e \\left ( \\frac { a h } { q } \\right ) , C _ { 1 } ( q , a ) = C _ { 1 } ( \\chi ^ { 0 } , a ) , \\\\ & C _ { 3 } ( \\chi , a ) = \\sum \\limits _ { h = 1 } ^ { q } \\overline { \\chi } ( h ) e \\left ( \\frac { a h ^ { 3 } } { q } \\right ) , C _ { 3 } ( q , a ) = C _ { 3 } ( \\chi ^ { 0 } , a ) , \\end{aligned} \\end{align*}"}
{"id": "2848.png", "formula": "\\begin{align*} U _ * = \\sqrt { \\frac { 2 L \\ \\Delta } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } h _ i \\big ( 2 - h _ i \\frac { - \\kappa } { 1 - \\kappa } \\big ) } } . \\end{align*}"}
{"id": "614.png", "formula": "\\begin{align*} g ( x , y , n , z ) & : = \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi x } { n } \\Big ) + \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi y } { n } \\Big ) \\\\ & - \\frac { 2 n ^ 2 } { 3 \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi x } { n } \\Big ) \\sin ^ 2 \\Big ( \\frac { \\pi y } { n } \\Big ) + z ^ 2 . \\end{align*}"}
{"id": "3001.png", "formula": "\\begin{align*} \\delta ( Q ( f ) ) \\leq \\max _ t \\delta _ { - t G } ( Q ^ { ( t ) } ) = \\delta _ G ( Q ) \\ . \\end{align*}"}
{"id": "3031.png", "formula": "\\begin{align*} & \\# \\left \\{ ( A _ i , B _ i ) _ i ( X _ j ) _ j \\in G ^ { 2 g } \\times \\prod _ { j = 1 } ^ { k } C _ j \\Big \\arrowvert A _ 1 \\sigma ( B _ 1 ) A _ 1 ^ { - 1 } B _ 1 ^ { - 1 } \\prod _ { i = 2 } ^ { g } [ A _ i , B _ i ] \\prod _ { j = 1 } ^ { k } X _ j = 1 \\right \\} \\\\ = & ( n ' \\ast n ^ { g - 1 } \\ast 1 _ { C _ 1 } \\ast \\cdots \\ast 1 _ { C _ { k } } ) ( 1 ) . \\end{align*}"}
{"id": "3615.png", "formula": "\\begin{align*} & \\langle e _ { i _ 1 } + \\cdots + e _ { i _ k } , e _ { j _ 1 } + \\cdots + e _ { j _ d } \\rangle = | \\{ i _ 1 , \\ldots , i _ k \\} \\cap \\{ j _ 1 , \\ldots , j _ d \\} | = \\\\ & k + d - | \\{ i _ 1 , \\ldots , i _ k \\} \\cup \\{ j _ 1 , \\ldots , j _ d \\} | \\geq k + d - s . \\end{align*}"}
{"id": "3184.png", "formula": "\\begin{align*} \\Gamma ( h _ { \\gamma _ { - } } ) \\left ( \\rho \\right ) = f _ { \\Phi + \\Phi ^ { - \\mathfrak { b } _ { - } , \\gamma _ { - } } } \\left ( \\rho \\right ) + \\int _ { \\mathbb { S } } \\left \\vert \\rho \\left ( \\mathfrak { e } _ { \\Psi } \\right ) \\right \\vert ^ { 2 } \\mathfrak { a } _ { \\mathfrak { b } _ { + } } \\left ( \\mathrm { d } \\Psi \\right ) \\ , \\rho \\in E _ { 1 } \\ . \\end{align*}"}
{"id": "4815.png", "formula": "\\begin{align*} \\alpha _ { n + 1 } = \\left ( \\begin{matrix} x _ n \\\\ y _ n \\end{matrix} \\right ) - \\begin{bmatrix} 2 x _ n y _ n & x _ n ^ 2 \\\\ 1 & 0 \\end{bmatrix} ^ { - 1 } \\begin{bmatrix} x _ n ^ 2 y _ n + 5 \\\\ x _ n + 1 \\end{bmatrix} \\end{align*}"}
{"id": "5067.png", "formula": "\\begin{align*} \\Lambda ^ n _ s & = ( W _ s - W _ { \\eta _ n ( s ) } ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) \\\\ & + ( W _ s - W _ { \\eta _ n ( s ) } ) \\left ( \\int _ { \\eta _ n ( s ) } ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) - \\int _ { \\eta _ n ( s ) } ^ s \\psi _ { n , 1 } ( u , s ) d u . \\end{align*}"}
{"id": "5294.png", "formula": "\\begin{align*} \\frac { ( r - 2 ) k + 2 \\sum _ i a _ i } { r } = 2 l + k - 2 \\end{align*}"}
{"id": "5658.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow s _ 0 ^ - } \\overline { u } ' ( \\tau ) = \\lim _ { \\tau \\rightarrow s _ 0 ^ + } \\overline { u } ' ( \\tau ) = 0 . \\end{align*}"}
{"id": "4431.png", "formula": "\\begin{align*} ( u \\otimes v ) ( x _ 1 , \\cdots , x _ m , x _ { m + 1 } , \\cdots , x _ { m + k } ) = u ( x _ 1 , \\cdots , x _ m ) v ( x _ { m + 1 } , \\cdots , x _ { m + k } ) . \\end{align*}"}
{"id": "6347.png", "formula": "\\begin{align*} ( \\widehat { \\varPhi } ^ * _ { x , s } ) ^ { - 1 } ( t ) = \\int _ { 0 } ^ { t } \\dfrac { \\widehat { \\varPhi } _ { x } ^ { - 1 } ( \\tau ) } { \\tau ^ { \\frac { N + s } { N } } } d \\tau . \\end{align*}"}
{"id": "6227.png", "formula": "\\begin{align*} \\int x ^ { n - 2 } ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty h _ n ( x ; q ) d _ q x = \\frac { x ^ n ( x ^ 2 ; q ^ 2 ) _ \\infty } { [ n - 1 ] _ q } \\left ( \\frac { h _ n ( \\frac { x } { q } ; q ) } { x } - \\frac { 1 } { q } h _ { n - 1 } ( \\frac { x } { q } ; q ) \\right ) . \\end{align*}"}
{"id": "3316.png", "formula": "\\begin{align*} W _ 0 ^ m ( \\R ) \\coloneqq \\{ & g \\in C ^ { m - 1 } ( \\R ) \\ ; : \\ ; g \\equiv 0 ( - \\infty , 0 ) , \\\\ & g , g ^ { ( m ) } \\in L ^ 1 _ { } ( \\R ) \\} . \\end{align*}"}
{"id": "7912.png", "formula": "\\begin{align*} f _ + ^ * ( K _ { X _ + } + D _ + ) = f _ - ^ * ( K _ { X _ - } + D _ - ) . \\end{align*}"}
{"id": "5263.png", "formula": "\\begin{align*} P _ 1 = \\{ ( J \\setminus A _ i , k _ 1 ( 1 ) , k _ 2 ( 1 ) ) \\} \\end{align*}"}
{"id": "215.png", "formula": "\\begin{align*} \\pi _ { \\bar { Y } _ { \\mathcal { T } } , \\bar { X } } ( a _ { \\mathcal { T } } , b ) = \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } ( a _ { \\mathcal { T } } | b ) \\pi _ { \\bar { X } } ( b ) , \\end{align*}"}
{"id": "824.png", "formula": "\\begin{align*} 1 \\le [ w ] _ { Q B _ { \\beta , \\psi , \\infty } } : = \\inf \\left \\{ [ w ] _ { Q B _ { \\beta , \\psi , p } } : w \\in Q B _ { \\beta , \\psi , p } , ~ p > 0 \\right \\} . \\end{align*}"}
{"id": "4932.png", "formula": "\\begin{align*} F ( [ a _ { j } ^ i ] ) = f ( \\mu _ 1 , \\cdots , \\mu _ n ) , \\end{align*}"}
{"id": "2748.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\lim _ { i \\rightarrow \\infty } \\int _ { B _ { R } ^ { c } ( 0 ) } \\frac { u _ { i } ^ { p - 1 } ( x ) } { | x - y | ^ { n + \\sigma p } } d y = & u ^ { p - 1 } ( x ) \\lim _ { R \\rightarrow \\infty } \\int _ { B _ { R } ^ { c } ( 0 ) } \\frac { d y } { | x - y | ^ { n + \\sigma p } } \\\\ \\leq & u ^ { p - 1 } ( x ) \\lim _ { R \\rightarrow \\infty } \\int _ { B _ { R - | x | } ^ { c } ( x ) } \\frac { d y } { | x - y | ^ { n + \\sigma p } } = 0 . \\end{align*}"}
{"id": "4115.png", "formula": "\\begin{align*} S ^ { i , j } = d ^ { i , j - 1 } K ^ { i , j } - K ^ { i + 1 , j } d ^ { i , j } . \\end{align*}"}
{"id": "1006.png", "formula": "\\begin{align*} \\{ X _ M ( 0 ) = a _ 0 , \\ldots X _ M ( \\ell ) = a _ { \\ell } \\} \\end{align*}"}
{"id": "4361.png", "formula": "\\begin{align*} \\xi _ r ^ { s _ j } \\ , \\sigma _ { r - 1 } ^ { - \\nu _ { r - 1 } ( a _ { s _ j } ) } \\tau _ { r - 1 } ( a _ { s _ j } ) = \\tau _ r ( f ) \\ , \\sigma _ r ^ j , \\end{align*}"}
{"id": "4086.png", "formula": "\\begin{align*} E ( \\mathbf { R } _ n ) = \\varepsilon n \\left ( \\frac { n - 1 } { 2 } \\boldsymbol { a } + \\boldsymbol { b } \\right ) . \\end{align*}"}
{"id": "7382.png", "formula": "\\begin{align*} \\begin{aligned} 0 = \\int _ { \\mathbb { R } ^ N } f ( | y | ) \\left ( y \\cdot t _ { 1 } \\right ) \\bar { \\phi } ( y ) d y . \\end{aligned} \\end{align*}"}
{"id": "8736.png", "formula": "\\begin{align*} v = p ^ 2 + q ^ 2 . \\end{align*}"}
{"id": "8628.png", "formula": "\\begin{align*} ( \\dot { \\psi } _ \\phi f ) = \\frac { d } { d t } \\bigg | _ { t = 0 } f ( \\phi + t \\psi ) , \\forall f \\in C ^ { \\infty } ( U ) . \\end{align*}"}
{"id": "2516.png", "formula": "\\begin{align*} - g ( T _ U V , X ) + R i c ^ \\nu ( U , V ) + \\mu g ( U , V ) = 0 . \\end{align*}"}
{"id": "6567.png", "formula": "\\begin{align*} d ^ r ( \\beta \\circ f ) ( x , y _ 1 , \\ldots , y _ r ) = \\sum _ { ( I _ 1 , \\ldots , I _ r ) } \\beta ( d ^ { | I _ 1 | } f _ 1 ( x , y _ { I _ 1 } ) , \\ldots , d ^ { | I _ k | } f _ k ( x , y _ { I _ k } ) ) \\end{align*}"}
{"id": "1184.png", "formula": "\\begin{align*} j _ { p } ( v ) : = \\begin{cases} | v | ^ { p - 2 } v & \\ v \\neq 0 \\\\ 0 & \\end{cases} . \\end{align*}"}
{"id": "4392.png", "formula": "\\begin{align*} & \\eta \\sqrt { - 1 } \\partial \\bar \\partial ( \\varphi _ m + \\Psi _ m ) + v ' _ { \\epsilon } ( \\Psi _ m ) \\sqrt { - 1 } \\partial \\bar \\partial \\Psi _ m \\\\ \\ge & \\frac { 1 } { \\delta } \\sqrt { - 1 } \\partial \\bar \\partial ( \\varphi _ m + \\Psi _ m ) + v ' _ { \\epsilon } ( \\Psi _ m ) \\sqrt { - 1 } \\partial \\bar \\partial \\Psi _ m \\\\ \\ge & 0 \\end{align*}"}
{"id": "6036.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { k = 0 } ^ n c _ k x ^ k \\in K [ x ] \\end{align*}"}
{"id": "6883.png", "formula": "\\begin{align*} N _ 0 ( t ) & = \\frac { 1 } { V _ F } \\int _ { \\mathbb { R } } g _ + p _ 0 ( t , g ) d g = \\frac { 1 } { V _ F } \\int _ { \\mathbb { R } } g _ + \\int _ { \\mathbb { R } } p _ { t , 0 } ( y ) \\bar { G } _ { t , 0 } ( g - y ) d y \\\\ & = \\int _ { \\mathbb { R } } p _ { t , 0 } ( y ) d y \\int _ { \\mathbb { R } } \\frac { 1 } { V _ F } g _ + \\bar { G } _ { t , 0 } ( g - y ) d g \\\\ & = \\int _ { \\mathbb { R } } p _ { t , 0 } ( y ) N ( B ( t ) + y , C ( t ) ) d y . \\end{align*}"}
{"id": "6315.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } q ^ { \\frac { k ( k - 3 ) } { 2 } } x ^ k \\left ( 1 + q ^ { 2 + k } [ n - 1 ] _ q x \\right ) S _ n ( q ^ k x ; q ) = S _ n ( \\frac { x } { q } ; q ) + \\frac { x } { 1 - q } S _ { n - 1 } ( q x ; q ) . \\end{align*}"}
{"id": "7440.png", "formula": "\\begin{align*} - { \\mathcal { A } } _ m U = F . \\end{align*}"}
{"id": "106.png", "formula": "\\begin{align*} \\mu ^ \\Lambda _ { g , \\nu } ( d \\varphi ) = e ^ { - \\frac 1 2 ( \\varphi , - \\Delta \\varphi ) - \\sum _ { x \\in \\Lambda } ( \\frac 1 4 g \\varphi ^ 4 + \\nu \\varphi ^ 2 ) } , \\end{align*}"}
{"id": "3427.png", "formula": "\\begin{align*} F \\begin{pmatrix} g _ { 1 , 1 } & g _ { 1 , 2 } & g _ { 1 , 3 } \\\\ g _ { 2 , 1 } & g _ { 2 , 2 } & g _ { 2 , 3 } \\\\ g _ { 3 , 1 } & g _ { 3 , 2 } & g _ { 3 , 3 } \\end{pmatrix} = \\begin{pmatrix} \\frac { g _ { 1 , 2 } } { \\sqrt { - 3 } } & \\frac { g _ { 1 , 3 } } { \\sqrt { - 3 } } & \\frac { g _ { 2 , 1 } } { \\sqrt { - 3 } } & \\frac { g _ { 3 , 1 } } { \\sqrt { - 3 } } \\end{pmatrix} . \\end{align*}"}
{"id": "884.png", "formula": "\\begin{align*} V = V _ 0 \\supseteq V _ 1 \\supseteq V _ 2 \\supseteq \\cdots \\end{align*}"}
{"id": "7845.png", "formula": "\\begin{align*} \\langle ( x T _ \\xi ^ * y ) \\widehat { a b } , \\hat { c } \\rangle = \\langle \\xi y a b , x ^ * c \\xi \\rangle = \\langle ( x L _ \\xi y ) \\hat { a } , c \\xi b ^ * \\rangle . \\end{align*}"}
{"id": "7644.png", "formula": "\\begin{align*} g ( S _ n ) = n \\binom { n } { 3 } - ( n + 1 ) \\binom { n - 1 } { 3 } , \\deg ( S ) = \\binom { n + 1 } { 2 } . \\end{align*}"}
{"id": "5510.png", "formula": "\\begin{align*} \\mathcal { F } = \\left \\{ f : \\Omega \\to \\mathrm { D o m e } ( \\Omega ) : f ( x ) = x x \\in \\partial \\Omega f \\right \\} . \\end{align*}"}
{"id": "6946.png", "formula": "\\begin{align*} Q ^ * Q R Q ^ * Q & = R ^ { - 1 / 2 } R _ 1 ^ 2 R ^ { - 1 / 2 } = R ^ { - 1 / 2 } ( R ^ 2 - p p ^ * ) R ^ { - 1 / 2 } \\\\ & = R - R ^ { - 1 / 2 } p ( R ^ { - 1 / 2 } p ) ^ * \\end{align*}"}
{"id": "5917.png", "formula": "\\begin{align*} F ( z ) = \\sum _ { \\nu > 0 } c _ \\nu z ^ \\nu + \\sum _ { \\nu \\leq 0 } c _ \\nu z ^ \\nu = F _ { \\mathrm { m } } ( z ) + F _ { \\mathrm { b } } ( z ) . \\end{align*}"}
{"id": "2767.png", "formula": "\\begin{align*} \\Delta _ { { \\mathcal { I } } } : = \\Big \\{ { \\alpha } \\in \\mathbb { R } ^ n : \\sum \\limits _ { i \\in \\mathcal { I } } \\alpha _ i = 1 , \\alpha _ i { } \\geq { } 0 , \\forall i \\in 0 , \\dots , n - 1 \\Big \\} \\end{align*}"}
{"id": "5252.png", "formula": "\\begin{align*} \\begin{aligned} A ^ 1 _ { Q , j , R _ 0 } & = \\{ \\Gamma _ { \\widehat Q _ j , j , 1 , k _ 1 ( 0 ) , k _ 2 ( 0 ) , R _ 0 , S _ 0 } : 0 \\leq S _ 0 \\leq k _ 2 ( j ) \\} ; \\\\ A ^ 2 _ { Q , j , S _ 0 } & = \\{ \\Gamma _ { \\widehat Q _ j , j , 2 , k _ 1 ( 0 ) , k _ 2 ( 0 ) , R _ 0 , S _ 0 } : 0 \\leq R _ 0 \\leq k _ 1 ( j ) \\} . \\end{aligned} \\end{align*}"}
{"id": "6193.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } ( \\omega _ { N } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\phi _ { T _ { 0 } , \\varepsilon } ) ^ { 2 } \\wedge \\eta _ { N } = C _ { \\varepsilon } \\Omega _ { \\varepsilon } \\wedge \\eta _ { N } ; \\sup ( \\phi _ { T _ { 0 } , \\varepsilon } - \\phi _ { T _ { 0 } } ) = \\sup ( \\phi _ { T _ { 0 } } - \\phi _ { T _ { 0 } , \\varepsilon } ) \\end{array} \\end{align*}"}
{"id": "3431.png", "formula": "\\begin{align*} g * z = \\frac { 1 } { A + B z } \\cdot ( C + D z ) . \\end{align*}"}
{"id": "2356.png", "formula": "\\begin{align*} \\nu _ \\rho ( f ) = \\nu ( a _ 0 ) = \\nu ( l ( h _ \\rho ) ) \\end{align*}"}
{"id": "7564.png", "formula": "\\begin{align*} g = \\prod _ { \\alpha = 1 } ^ { r } \\exp ( \\lambda _ \\alpha v _ \\alpha ) , \\end{align*}"}
{"id": "3411.png", "formula": "\\begin{align*} G = \\langle g _ 1 , \\ldots , g _ n | r _ 1 , \\ldots , r _ m \\rangle , \\end{align*}"}
{"id": "8910.png", "formula": "\\begin{align*} - \\tilde u '' + \\tilde \\lambda \\tilde u = \\tilde u ^ { p - 1 } , \\tilde u \\ge 0 \\end{align*}"}
{"id": "1450.png", "formula": "\\begin{align*} & \\vec { p } _ { \\ell } ( z ) = { } ^ t \\Biggl ( P _ { \\ell } ( z ) , { P _ { \\ell , 1 , r - 1 } ( z ) , \\ldots , P _ { \\ell , 1 , 0 } ( z ) } , \\ldots , { P _ { \\ell , m , r - 1 } ( z ) , \\ldots , P _ { \\ell , m , 0 } ( z ) } \\Biggr ) \\enspace , \\end{align*}"}
{"id": "5674.png", "formula": "\\begin{align*} c _ k ( E _ n ^ * ) = e _ k ( \\tau _ 1 , \\tau _ 2 , \\ldots , \\tau _ n ) ( 1 \\le k \\le n ) , \\end{align*}"}
{"id": "5257.png", "formula": "\\begin{align*} W ^ { \\mathbf { s } _ 1 } = g ( W ^ { \\mathbf { s } _ 0 } ) . \\end{align*}"}
{"id": "2607.png", "formula": "\\begin{align*} J _ \\gamma = \\gamma - 1 + k \\ . \\end{align*}"}
{"id": "6073.png", "formula": "\\begin{align*} N _ 1 = N _ { f _ 1 } ( \\Lambda _ 1 ) \\end{align*}"}
{"id": "291.png", "formula": "\\begin{align*} \\delta ( S ) & = \\lim _ { x \\to \\infty } \\frac { 1 } { \\log x } \\sum _ { n \\in S , n \\le x } \\frac { 1 } { n } , \\qquad & \\Delta ( S ) = \\lim _ { x \\to \\infty } \\frac { 1 } { \\log \\log x } \\sum _ { n \\in S , 1 < n \\le x } \\frac { 1 } { n \\log n } , \\end{align*}"}
{"id": "4029.png", "formula": "\\begin{align*} b _ 0 & = a _ 0 \\\\ b _ 1 & = \\frac { 5 } { 2 } a _ 0 ^ 2 + a _ 1 , \\end{align*}"}
{"id": "5910.png", "formula": "\\begin{align*} S _ { \\exp _ A f } ( r ) = O ( \\log ^ + T _ { \\exp _ A f } ( r ) ) + O ( \\log r ) + O ( 1 ) ~ ~ | | , \\end{align*}"}
{"id": "2791.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\| \\nabla f ( x _ i ) \\| ^ 2 { } \\leq { } \\frac { 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left ( 2 h _ i - h _ i ^ 2 \\frac { - \\kappa } { 1 - \\kappa } \\right ) } \\end{align*}"}
{"id": "8781.png", "formula": "\\begin{align*} \\frac { 1 } { q ( q ( p _ 1 , p _ 2 ) , p _ 3 ) } = \\frac { 1 } { q ( p _ 1 , p _ 2 ) } + \\frac { 1 } { p _ 3 } - 1 = \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } + \\frac { 1 } { p _ 3 } - 2 = \\frac { 1 } { q ( p _ 1 , p _ 2 , p _ 3 ) } , \\end{align*}"}
{"id": "1160.png", "formula": "\\begin{align*} \\delta ( p _ { \\vec { k } } ) & = \\frac { 1 } { k ! } \\sum _ { \\sigma } \\delta \\left [ z _ { 1 \\sigma ( 1 ) } \\right ] \\cdots \\delta \\left [ z _ { 1 \\sigma ( k ) } \\right ] , \\end{align*}"}
{"id": "8320.png", "formula": "\\begin{align*} G ( t , x ) \\frac { \\left ( e ^ { t } - 1 \\right ) } { t - 2 \\pi i k } = - \\sum _ { n = 0 } ^ { \\infty } s _ n t ^ n , \\end{align*}"}
{"id": "2333.png", "formula": "\\begin{align*} \\nu _ Q ( f ) : = \\min _ { 0 \\leq i \\leq r } \\{ \\nu ( a _ i Q ^ i ) \\} . \\end{align*}"}
{"id": "8981.png", "formula": "\\begin{align*} F _ { q , r } ( u _ k ) \\leq F _ { q , r } ( 0 ) : = B , \\end{align*}"}
{"id": "4012.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } ( { t } ) & = f ( x ( t ) , u ( t ) ) , \\end{aligned} \\end{align*}"}
{"id": "5830.png", "formula": "\\begin{align*} \\chi ( p _ n ) = - 1 , \\chi ( p _ { n + 1 } ) = - 1 , \\chi ( p _ { n + 2 } ) = - 1 , \\end{align*}"}
{"id": "8927.png", "formula": "\\begin{align*} \\det D Y ( \\cdot , u , D u ) = \\psi ( \\cdot , u , D u ) , \\end{align*}"}
{"id": "6899.png", "formula": "\\begin{align*} \\rho ( 0 , v ) = \\rho _ { } ( v ) , v \\in [ 0 , V _ F ] . \\end{align*}"}
{"id": "8944.png", "formula": "\\begin{align*} t \\int _ { \\Omega } \\{ e ^ { [ \\tau ( 1 - t ) + \\epsilon ] ( u - u _ 0 ) } - 1 \\} f + ( 1 - t ) \\int _ { \\Omega ^ * } \\{ e ^ { [ \\tau ( 1 - t ) + \\epsilon ] ( u - u _ 0 ) \\circ ( T u _ 0 ) ^ { - 1 } } - 1 \\} f ^ * = 0 , \\end{align*}"}
{"id": "3055.png", "formula": "\\begin{align*} f _ { \\boldsymbol { X } ^ { \\ast \\ast } } ( \\boldsymbol { x } ) = \\frac { c _ { n } ^ { \\ast \\ast } } { \\sqrt { | \\boldsymbol { \\Sigma } | } } \\overline { \\mathcal { G } } _ { n } \\left \\{ \\frac { 1 } { 2 } ( \\boldsymbol { x } - \\boldsymbol { \\mu } ) ^ { T } \\mathbf { \\Sigma } ^ { - 1 } ( \\boldsymbol { x } - \\boldsymbol { \\mu } ) \\right \\} , ~ \\boldsymbol { x } \\in \\mathbb { R } ^ { n } . \\end{align*}"}
{"id": "7037.png", "formula": "\\begin{align*} C ( r ) f : = \\frac 1 r \\int _ 0 ^ r e ^ { s A } f \\ , d s f \\in E \\end{align*}"}
{"id": "4400.png", "formula": "\\begin{align*} \\Vert f \\Vert _ p = \\left ( \\frac { 1 } { 2 \\pi } \\int _ 0 ^ R \\int _ 0 ^ { 2 \\pi } | f ( r e ^ { i \\varphi } ) | ^ p d \\varphi d \\mu ( r ) \\right ) ^ { 1 / p } \\end{align*}"}
{"id": "4090.png", "formula": "\\begin{align*} E ( \\mathbf { R } _ n ) _ 1 = \\varepsilon n \\left [ \\frac { n - 1 } { 2 } a _ 1 + b _ 1 \\right ] . \\end{align*}"}
{"id": "203.png", "formula": "\\begin{align*} H _ 1 : \\ ( X ^ { n } , Z ^ { n } _ { 1 } , Z ^ { n } _ { 2 } ) \\sim \\prod _ { i = 1 } ^ { n } q _ { X , Z _ 1 , Z _ 2 } . \\end{align*}"}
{"id": "684.png", "formula": "\\begin{align*} e ^ { f _ o } v _ s = \\sum _ { j = 0 } ^ { \\infty } c _ j e ^ { - \\lambda _ j s } \\varphi _ j . \\end{align*}"}
{"id": "207.png", "formula": "\\begin{align*} \\beta _ { n , i } ( f ^ { n } , g ^ { n } _ { i } ) \\coloneqq \\Pr \\left ( \\hat { H } _ { i } = 0 | H = 1 \\right ) i \\in \\{ 1 , 2 \\} , \\end{align*}"}
{"id": "5839.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ M a _ k ^ \\alpha \\bigg ( \\sum _ { i = k } ^ M b _ i \\bigg ) ^ \\alpha \\approx \\sum _ { k = N } ^ M a _ k ^ \\alpha b _ k ^ \\alpha \\end{align*}"}
{"id": "4967.png", "formula": "\\begin{align*} C ^ n _ t = n ^ { \\alpha + \\frac 1 2 } \\int ^ t _ 0 ( t - s ) ^ { \\alpha } [ \\sigma ( X ^ n _ { \\eta _ n ( s ) } ) - \\sigma ( X ^ n _ s ) ] \\ , d W _ s \\end{align*}"}
{"id": "5462.png", "formula": "\\begin{align*} k = k _ 2 a _ 1 a _ 2 = \\textsf { 1 } _ M \\end{align*}"}
{"id": "2994.png", "formula": "\\begin{align*} ( 1 - \\epsilon ) h _ { X , \\mathcal { O } ( D ) } ( \\phi ) < \\sum _ { j = 1 } ^ q N ^ { ( 1 ) } ( D _ j , \\phi ) + 2 r \\max \\{ 0 , g ( C ) - 1 \\} . \\end{align*}"}
{"id": "3112.png", "formula": "\\begin{align*} \\big ( 1 + b ( u , v ) \\big ) \\Vert u - v \\Vert ^ 2 _ { L ^ 2 ( \\Omega ) } = 2 \\min _ { t \\in \\mathbb { R } } { \\Vert u - t v \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 } . \\end{align*}"}
{"id": "8284.png", "formula": "\\begin{align*} \\psi _ { A } = A \\left [ \\frac { \\cosh ( k ' x ) } { \\cosh ( k ' a / 2 ) } - \\frac { x } { a / 2 } \\frac { \\sinh ( k ' x ) } { \\sinh ( k ' a / 2 ) } \\right ] \\end{align*}"}
{"id": "2090.png", "formula": "\\begin{align*} \\sigma ^ { ( a n , b m ) } ( y ) = y . \\end{align*}"}
{"id": "5975.png", "formula": "\\begin{align*} | k ( t ) | \\leq \\frac { C _ K } { \\pi } t ^ { \\mu - 1 } ( \\cos \\varphi ) ^ { \\mu - 1 } \\Gamma ( 1 - \\mu , 0 ) = \\frac { C _ K } { \\pi } t ^ { \\mu - 1 } ( \\cos \\varphi ) ^ { \\mu - 1 } \\Gamma ( 1 - \\mu ) . \\end{align*}"}
{"id": "5500.png", "formula": "\\begin{align*} g ( \\theta , \\omega ) = \\hat { g } ( \\theta , \\Phi \\omega ) , \\end{align*}"}
{"id": "3601.png", "formula": "\\begin{align*} { \\rm R C } ( I ) = \\left ( \\bigcap _ { i = 1 } ^ { s + 1 } H ^ + _ { e _ i } \\right ) \\bigcap \\left ( \\bigcap _ { i = 1 } ^ m H ^ + _ { ( \\gamma _ i , - d _ i ) } \\right ) \\bigcap \\left ( \\bigcap _ { i = m } ^ p H ^ + _ { ( \\gamma _ i , - d _ i ) } \\right ) , \\end{align*}"}
{"id": "8472.png", "formula": "\\begin{align*} y ^ * _ i = ( i - 1 ) L + \\left [ U / n - ( n - 1 ) L / 2 \\right ] . \\end{align*}"}
{"id": "245.png", "formula": "\\begin{align*} \\Lambda _ i ^ { ( s ) } : = ( x ^ { ( p ^ s ) } ) ^ { - 1 } ( \\phi _ i ^ { ( s ) } ) ^ G ( x ) = 1 + \\pi \\Gamma _ i ^ { ( s ) t } \\in \\textup { G L } _ N ( \\mathcal A ) . \\end{align*}"}
{"id": "5265.png", "formula": "\\begin{align*} g ( W ^ \\nu ) = W ^ { g ( \\nu ) } . \\end{align*}"}
{"id": "4435.png", "formula": "\\begin{align*} ( W f ) _ { i _ 1 \\dots i _ m j _ 1 \\dots j _ m } & = 2 ^ m \\sigma ( i _ 1 \\dots i _ m ) \\sigma ( j _ 1 \\dots j _ m ) ( R f ) _ { i _ 1 j _ 1 \\dots i _ m j _ m } \\\\ ( R f ) _ { i _ 1 j _ 1 \\dots i _ m j _ m } & = \\frac { 1 } { ( m + 1 ) } \\alpha ( i _ 1 j _ 1 ) \\dots \\alpha ( i _ m j _ m ) ( W f ) _ { i _ 1 \\dots i _ m j _ 1 \\dots j _ m } . \\end{align*}"}
{"id": "5461.png", "formula": "\\begin{align*} k _ 1 a ^ { - 1 } = a _ 2 k _ 2 . \\end{align*}"}
{"id": "5026.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } E \\left [ | Q ^ { n , 1 } _ \\tau | ^ 2 \\right ] = 0 . \\end{align*}"}
{"id": "6398.png", "formula": "\\begin{align*} \\sigma _ { \\pi } : = \\prod _ { \\substack { 1 \\leq i \\leq \\ell \\\\ \\mathrm { w i t h } \\ a _ i \\neq b _ i } } \\ ( a _ i , b _ i ) \\in S _ k \\subseteq S _ { \\infty } . \\end{align*}"}
{"id": "5550.png", "formula": "\\begin{align*} \\mathbb { E } [ \\ell , k ] = \\left . \\left [ k \\dbinom { \\ell } { k } + \\sum \\limits _ { i = 0 } ^ { ( k ' - 1 ) / 2 } \\dbinom { \\ell } { k ' - 2 i - 1 } \\right ] \\middle / \\dbinom { \\ell } { k } \\right . \\end{align*}"}
{"id": "1751.png", "formula": "\\begin{align*} \\frac { D _ { n + 1 } ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) } { D _ n ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) } = \\prod _ { k = 0 } ^ { n - 1 } ( 1 - ( q ^ { \\frac { 1 } { 2 } } ) ^ { 1 - n + 2 k } x y ^ { n } ) ^ { - 1 } . \\end{align*}"}
{"id": "7707.png", "formula": "\\begin{align*} \\sum _ { n _ 1 \\cdots n _ k \\le x } \\tau ( [ n _ 1 , \\ldots , n _ k ] ) = x \\ , Q _ { \\tau , 2 k - 1 } ( \\log x ) + O \\big ( x ^ { \\theta _ { 2 k } + \\varepsilon } \\big ) , \\end{align*}"}
{"id": "6454.png", "formula": "\\begin{align*} \\Phi = \\sum _ { k \\in \\mathbb { Z } } \\Phi ^ { ( k ) } \\end{align*}"}
{"id": "1817.png", "formula": "\\begin{align*} s _ { 1 } & = b _ { 0 } \\\\ s _ { 2 } & = b _ { 0 } ^ 2 + a _ { 0 } \\\\ s _ { 3 } & = b _ { 0 } ^ 3 + 2 a _ { 0 } b _ { 0 } + a _ { 0 } b _ { 1 } \\\\ s _ { 4 } & = b _ { 0 } ^ 4 + 3 a _ { 0 } b _ { 0 } ^ 2 + 2 a _ { 0 } b _ { 0 } b _ { 1 } + a _ { 0 } b _ { 1 } ^ { 2 } + a _ { 0 } ^ 2 + a _ { 0 } a _ { 1 } . \\end{align*}"}
{"id": "7775.png", "formula": "\\begin{align*} \\tilde { \\bf { u } } _ m ( \\alpha ) = A ^ { - 1 } _ m ( \\alpha ) { \\bf f } _ m ( \\alpha ) + A ^ { - 1 } _ m ( \\alpha ) \\boldsymbol { \\eta } _ m = { \\bf u } _ m ( \\alpha ) + A ^ { - 1 } _ m ( \\alpha ) \\boldsymbol { \\eta } _ m . \\end{align*}"}
{"id": "2485.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( L _ \\xi g ) + R i c + \\mu g = 0 , \\end{align*}"}
{"id": "4492.png", "formula": "\\begin{align*} \\hat v ( t , \\xi ) = e ^ { i \\Theta ( t , \\xi ) } \\hat { h } ( t , \\xi ) . \\end{align*}"}
{"id": "1848.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { E } _ { u } & : = \\{ ( n , m ) \\rightarrow ( n + 1 , m + 1 ) : ( n , m ) \\in \\mathcal { V } \\} \\\\ \\mathcal { E } _ { \\ell } & : = \\{ ( n , m ) \\rightarrow ( n + 1 , m ) : ( n , m ) \\in \\mathcal { V } \\} \\\\ \\mathcal { E } _ { d } & : = \\{ ( n , m ) \\rightarrow ( n + 1 , m - j ) : ( n , m ) \\in \\mathcal { V } , \\ , \\ , \\ , 1 \\leq j \\leq p \\} . \\end{aligned} \\end{align*}"}
{"id": "2204.png", "formula": "\\begin{align*} C ^ * _ { \\tiny \\rm F i n s l e r } ( T ) = C ^ * _ { \\psi , \\varphi } ( \\mathbf { T } ) : = \\inf _ { \\| \\mathbf { u } \\| _ { x , \\varphi } = 1 } \\| \\mathbf { T } { \\mathbf { u } } \\| _ { f ( x ) , \\psi } . \\end{align*}"}
{"id": "87.png", "formula": "\\begin{align*} a ^ \\epsilon ( \\lambda , m ^ 2 ) : = - 3 \\lambda \\big ( - \\Delta ^ \\epsilon + m ^ 2 ) ^ { - 1 } ( 0 , 0 ) + 6 \\lambda ^ 2 \\big \\| \\big ( - \\Delta ^ \\epsilon + m ^ 2 \\big ) ^ { - 1 } ( 0 , \\cdot ) \\big \\| ^ 3 _ { L ^ 3 } . \\end{align*}"}
{"id": "1030.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\# \\{ i \\in \\N : s _ i \\le N x _ { s _ i } = b \\} } { \\# \\{ i \\in \\N : s _ i \\le N \\} } \\end{align*}"}
{"id": "4354.png", "formula": "\\begin{align*} \\tilde { c } _ s ( f ) + \\tilde { c } _ s ( h ) = \\tilde { c } _ s ( f + h ) , \\forall \\ , s \\ge 0 . \\end{align*}"}
{"id": "6102.png", "formula": "\\begin{align*} R ( e _ n , e _ { n + 2 } , e _ n , e _ { n + 2 } ) = a ^ 2 c \\end{align*}"}
{"id": "7430.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ m ( t ) = c ( m - 1 ) \\int _ 0 ^ L \\abs { \\omega _ x } ^ 2 d x + c m \\int _ 0 ^ { + \\infty } \\int _ 0 ^ L \\sigma ' ( s ) \\abs { \\eta _ x } ^ 2 d x d s . \\end{align*}"}
{"id": "2575.png", "formula": "\\begin{align*} \\mathbf S _ { 1 2 3 } ^ 0 : = \\mathbf S _ { 1 2 3 } | _ { \\mathcal H ^ 0 } \\ . \\end{align*}"}
{"id": "4807.png", "formula": "\\begin{align*} \\max _ { 1 \\le i \\le n } \\bigg | \\sum _ { k = 1 } ^ i \\widetilde \\eta _ { n , k } \\bigg | = O _ P \\big ( n ^ { - 1 / 2 } ( \\ln \\ln n ) ^ { 1 / 2 } \\big ) \\quad \\mbox { a s \\ } n \\to \\infty . \\end{align*}"}
{"id": "4994.png", "formula": "\\begin{align*} \\widehat { \\Theta } ^ n _ s & = \\sigma ( X _ s ) \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u + X ^ n _ { \\eta _ n ( s ) } - X ^ n _ s \\\\ & = \\sigma ( X _ s ) \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u + \\int ^ s _ { \\eta _ n ( s ) } \\left ( s - \\eta _ n ( u ) \\right ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\\\ & + \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\ , \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u , \\end{align*}"}
{"id": "1543.png", "formula": "\\begin{align*} \\widetilde { \\Delta } _ r ( f ) : = \\frac { 1 } { 2 } f '' + \\frac { \\alpha } { 2 } f ' , \\end{align*}"}
{"id": "4005.png", "formula": "\\begin{align*} \\kappa _ { \\bar \\epsilon _ j } ^ { - 1 } h _ L ( a _ { \\epsilon _ j } ) = \\kappa _ { \\bar \\epsilon _ j } ^ { - 1 } h _ R ( p _ { \\epsilon _ j } F ^ { - 2 } a ) = \\kappa _ { \\bar \\epsilon _ j } ^ { - 1 } h _ R ( p _ { \\epsilon _ j } E ^ R _ { \\epsilon _ j } ( F ^ { - 2 } ) a ) = \\kappa _ { \\bar \\alpha } ^ { - 1 } h _ R ( p _ \\alpha E ^ R _ { \\epsilon _ j } ( F ^ { - 2 } ) a ) . \\end{align*}"}
{"id": "5284.png", "formula": "\\begin{align*} \\Gamma _ { P , j } : = \\Gamma _ { 0 , k _ 1 ( j ) , k _ 2 ( j ) , 1 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I _ j } } ; O _ j ( P ) : = \\left \\langle \\prod _ { i \\in { I _ j } } \\tau ^ { ( a _ i , b _ i ) } _ { d _ i } \\sigma _ 1 ^ { k _ 1 ( j ) } \\sigma _ 2 ^ { k _ 2 ( j ) } \\sigma _ { 1 2 } \\right \\rangle ^ { \\mathbf { s } ^ { \\Gamma _ { P , j } } , o } \\end{align*}"}
{"id": "5333.png", "formula": "\\begin{align*} A ^ * \\phi ( x ) = \\nabla \\cdot ( Q \\nabla \\phi ) ( x ) - b \\cdot \\nabla \\phi ( x ) + \\int _ { \\R ^ d } [ \\phi ( x - y ) - \\phi ( x ) + \\nabla \\phi ( x ) y 1 _ { ( 0 , 1 ] } ( | y | ) ] \\nu ( d y ) . \\end{align*}"}
{"id": "7289.png", "formula": "\\begin{align*} { \\sf U } _ \\infty ^ q \\theta _ 0 = { a } _ 0 { \\sf U } _ \\infty ^ { p + 1 } \\Leftrightarrow { \\sf U } _ \\infty ^ { - 1 } \\theta _ 0 = { a } _ 0 { \\sf U } _ \\infty ^ { p - q } . \\end{align*}"}
{"id": "6399.png", "formula": "\\begin{align*} a _ 0 = b _ 0 : = k + 1 , \\end{align*}"}
{"id": "1895.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { U } _ { [ n , 0 ] } } w ( \\gamma ) = a _ { 0 } ^ { ( 0 ) } W _ { [ n - 1 , 0 ] } . \\end{align*}"}
{"id": "4496.png", "formula": "\\begin{align*} \\Phi ( \\xi , \\eta _ 1 , \\eta _ 2 ) & = \\frac { 3 \\xi } { ( 1 + \\xi ^ 2 ) ^ { 5 / 2 } } \\zeta _ 1 \\zeta _ 2 + O \\bigg ( \\zeta _ 1 ^ 3 + \\zeta _ 2 ^ 3 \\bigg ) = \\frac { 3 \\xi } { ( 1 + \\xi ^ 2 ) ^ { 5 / 2 } } \\zeta _ 1 \\zeta _ 2 + O \\left ( [ \\varrho _ 1 ( t ) ] ^ 3 \\right ) . \\end{align*}"}
{"id": "8688.png", "formula": "\\begin{align*} h = f + \\lambda g . \\end{align*}"}
{"id": "5282.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ h \\Big ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { P } ^ { C o n t } _ j } + & \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { P } ^ { C o n t } _ j } \\Big ) \\\\ & - \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { r + k _ 1 ( 0 ) } a ^ { \\widehat { P } ^ { C o n t , + r } } - \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { s + k _ 2 ( 0 ) } b ^ { \\widehat { P } ^ { C o n t , + s } } = - 1 . \\end{align*}"}
{"id": "7512.png", "formula": "\\begin{align*} p _ { s u c c } = \\sum _ { i = 1 } ^ { m i n ( m , ( 2 | E | - \\gamma _ { R S U } ) k ) } \\mbox { P r } ( C _ { i } ) \\sum _ { j = 0 } ^ { k } \\binom { m - i } { j } ( p _ { m a t c h , j } ) ^ { 2 } , \\end{align*}"}
{"id": "4045.png", "formula": "\\begin{align*} E _ { F } \\hat { \\beta } = \\beta ( F ) F \\in \\mathbf { F } _ { 2 } ( I _ { n } ) . \\end{align*}"}
{"id": "1552.png", "formula": "\\begin{align*} c ( b ) & = c o a r m _ \\lambda ( b ) - c o l e g _ \\lambda ( b ) \\\\ h ( b ) & = a r m _ \\lambda ( b ) + l e g _ \\lambda ( b ) + 1 \\end{align*}"}
{"id": "8924.png", "formula": "\\begin{align*} g ( x , Y , Z ) = u , g _ x ( x , Y , Z ) = p , \\end{align*}"}
{"id": "4731.png", "formula": "\\begin{align*} ( 1 ) ~ ~ \\overline { C ^ * _ { d } } & = C ^ * _ { d } , \\\\ ( 2 ) ~ ~ C ^ * _ { d } & = H _ { d } + \\sum \\limits _ { \\substack { d ' \\in I _ { k , n } \\\\ \\ell ( d ' ) < \\ell ( d ) } } p ^ * _ { d ' , d } H _ { d ' } , ~ \\mbox { w h e r e } p ^ * _ { d ' , d } \\in q \\mathbb { Z } [ q ] . \\end{align*}"}
{"id": "6128.png", "formula": "\\begin{align*} | F \\cap H _ j | { } & { } = | F \\cap ( H _ i \\cup H _ j ) | + | F \\cap ( H _ i \\cap H _ j ) | - | F \\cap H _ i | \\\\ { } & { } \\leq | F | + | H _ i \\cap H _ j | - r _ i \\\\ { } & { } \\leq r + ( r _ i + r _ j - r ) - r _ i \\\\ { } & { } = r _ j , \\end{align*}"}
{"id": "6066.png", "formula": "\\begin{align*} f ^ { ( j ) } ( x ) & = \\sum _ { k = 0 } ^ j \\binom { j } { k } ( p ) _ k ( x - \\lambda ) ^ { p - k } g ^ { ( j - k ) } ( x ) \\\\ & = ( x - \\lambda ) \\sum _ { k = 0 } ^ j \\binom { j } { k } ( p ) _ k ( x - \\lambda ) ^ { p - 1 - k } g ^ { ( j - k ) } ( x ) \\end{align*}"}
{"id": "1180.png", "formula": "\\begin{align*} \\varphi _ { x } ( y ) = \\frac { g _ { x } ^ { p - 1 } ( y ) } { \\| g _ { x } \\| _ { L ^ p ( \\gamma _ { \\sigma } ) } ^ { p - 1 } } g _ { x } ( y ) = \\frac { \\phi _ { \\sigma } ( x - y ) } { \\phi _ { \\sigma } ( y ) } = e ^ { - | x | ^ 2 / ( 2 \\sigma ^ 2 ) + \\langle x , y \\rangle / \\sigma ^ 2 } , y \\in \\R ^ d . \\end{align*}"}
{"id": "870.png", "formula": "\\begin{align*} \\left \\langle S ~ | ~ ( s t ) _ { m _ { s t } } = ( t s ) _ { m _ { t s } } \\ ; \\ ; s , t \\in S \\ ; \\ ; s \\neq t \\right \\rangle , \\end{align*}"}
{"id": "1228.png", "formula": "\\begin{align*} \\deg \\mathcal { F } _ 1 ( w , x ) = \\deg \\mathcal { F } _ 2 ( w , x ) + 1 . \\end{align*}"}
{"id": "5759.png", "formula": "\\begin{align*} \\pi _ { [ n - 1 ] } \\cdot \\pi _ { i _ n } = 0 \\end{align*}"}
{"id": "499.png", "formula": "\\begin{align*} b = \\begin{psmallmatrix} & 0 & \\\\ 0 & & 0 \\\\ & 1 & \\end{psmallmatrix} \\otimes \\begin{psmallmatrix} & 0 & \\\\ 0 & & 1 \\\\ & 1 & \\end{psmallmatrix} \\otimes \\begin{psmallmatrix} & 0 & \\\\ 1 & & 1 \\\\ & 1 & \\end{psmallmatrix} \\otimes \\begin{psmallmatrix} & 1 & \\\\ 1 & & 1 \\\\ & 1 & \\end{psmallmatrix} \\end{align*}"}
{"id": "2160.png", "formula": "\\begin{align*} \\left | H ( w ) \\right | = \\lim _ { k \\rightarrow \\infty } | H \\left ( \\omega ( z _ { n ( k ) } ) \\right ) | = \\lim _ { k \\rightarrow \\infty } | z _ { n ( k ) } | = + \\infty . \\end{align*}"}
{"id": "3443.png", "formula": "\\begin{align*} & H ^ k ( \\tilde { \\mathbb { P } } , \\mathcal { O } ( \\tilde { Y } + K _ { \\tilde { \\mathbb { P } } } ) ) \\cong H ^ k ( \\mathbb { P } , \\mathcal { O } ( Y + K _ { \\mathbb { P } } ) ) = 0 k > 0 \\\\ & H ^ k ( \\tilde { \\mathbb { P } } , \\mathcal { O } ( \\tilde { Y } + K _ { \\tilde { \\mathbb { P } } } - D _ j ) ) = 0 k > 0 \\end{align*}"}
{"id": "5335.png", "formula": "\\begin{align*} \\lim _ { b \\to 0 ^ + } N _ \\delta ( b ) = N _ \\delta \\end{align*}"}
{"id": "465.png", "formula": "\\begin{align*} \\eta _ { \\nu _ { n } } ( \\Phi _ { n } ( z ) ) = z \\end{align*}"}
{"id": "8583.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 e ^ { 2 \\pi i m ( x + u ) } & \\prod _ { k = 1 } ^ s \\cos ( 2 \\pi n _ { k _ j } u ) d u \\\\ & = 2 ^ { - s } \\int _ 0 ^ 1 e ^ { 2 \\pi i m ( x + u ) } ( \\cos 2 \\pi m u + \\cos ( - 2 \\pi m u ) ) d u \\\\ & = 2 ^ { 1 - s } \\int _ 0 ^ 1 e ^ { 2 \\pi i m ( x + u ) } \\cos ( 2 \\pi m u ) d u \\\\ & = 2 ^ { 1 - s } e ^ { 2 \\pi i m x } \\int _ 0 ^ 1 \\cos ^ 2 ( 2 \\pi m u ) d u \\\\ & + i \\cdot 2 ^ { 1 - s } e ^ { 2 \\pi i m x } \\int _ 0 ^ 1 \\sin ( 2 \\pi m u ) \\cos ( 2 \\pi m u ) d u \\\\ & = 2 ^ { - s } e ^ { 2 \\pi i m x } , \\end{align*}"}
{"id": "8056.png", "formula": "\\begin{align*} X _ { 1 } = h X _ { 2 } \\end{align*}"}
{"id": "3361.png", "formula": "\\begin{align*} F _ m ( \\psi ) ( x , y ) = ( F _ m ( ) ( x ) , F _ m ( \\widehat { \\psi } ) ( y ) ) . \\end{align*}"}
{"id": "7909.png", "formula": "\\begin{align*} \\mathcal L _ { X , D } = \\bigcup _ { t } z L ( t , - z ) ^ { - 1 } \\mathcal H _ + . \\end{align*}"}
{"id": "1276.png", "formula": "\\begin{align*} \\mathcal { E } ( \\mathcal { B } _ { d , k } ) & = \\sum _ { j = 0 } ^ { k - 1 } ( d - 1 ) ^ { k - 1 - j } \\Psi ( E _ { j + 1 } ( x , d - 1 ) ) - \\sum _ { j = 1 } ^ { k - 1 } ( d - 1 ) ^ { k - 1 - j } \\Psi ( E _ j ( x , d - 1 ) ) . \\end{align*}"}
{"id": "1452.png", "formula": "\\begin{align*} \\Delta ( z ) = ( - 1 ) ^ { r m } \\cdot \\left ( \\sum _ { \\ell = 0 } ^ { r m } P _ { \\ell } ( z ) \\Delta _ { 1 , \\ell + 1 } ( z ) \\right ) = ( - 1 ) ^ { r m } \\times \\ P _ { r m } ( z ) \\Delta _ { 1 , r m + 1 } ( z ) \\in K \\enspace . \\end{align*}"}
{"id": "7445.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle w = \\frac { 1 } { \\tilde { m } } \\left [ \\Lambda ^ m ( x ) - m \\int _ 0 ^ \\infty \\sigma ( s ) \\int _ 0 ^ s f ^ 6 ( \\tau ) d \\tau d s \\right ] , \\end{array} \\end{align*}"}
{"id": "2001.png", "formula": "\\begin{align*} G _ 1 ( x ) = n ! G ( x ) \\end{align*}"}
{"id": "6895.png", "formula": "\\begin{align*} \\begin{aligned} p ( t , v , g ) : = \\left ( \\frac { 1 } { V _ F } \\sum _ { k = - \\infty } ^ { + \\infty } c _ k \\exp \\left ( i k \\frac { 2 \\pi } { V _ F } ( v - \\int _ 0 ^ t g _ { } ( s ) d s ) \\right ) \\right ) \\frac { 1 } { \\sqrt { 2 \\pi a ( t ) } } \\exp \\left ( - \\frac { ( g - g _ { } ( t ) ) ^ 2 } { 2 a ( t ) } \\right ) , \\end{aligned} \\end{align*}"}
{"id": "7252.png", "formula": "\\begin{align*} \\liminf _ n - \\Delta _ 2 \\varphi ( \\gamma _ { 2 n } ( \\omega ) ) = 0 \\end{align*}"}
{"id": "6603.png", "formula": "\\begin{align*} I ( h ) : = \\int _ { D _ h } q _ v ^ { \\sum _ i m _ i t _ i } \\ , d t . \\end{align*}"}
{"id": "3125.png", "formula": "\\begin{align*} t _ 2 : = & \\big ( \\lambda _ h u _ { \\mathrm { n c } } , ( I - \\widehat { I } ) ( u - \\widehat { u } _ { \\mathrm { n c } } ) \\big ) _ { 1 + \\delta } \\le 2 \\Vert h _ { \\mathcal { T } } ^ { m } \\lambda _ h u _ { \\mathrm { n c } } \\Vert _ { L ^ 2 ( \\mathcal { T } \\setminus \\widehat { \\mathcal { T } } ) } \\Vert h _ { \\mathcal { T } } ^ { - { m } } ( I - \\widehat { I } ) ( u - \\widehat { u } _ { \\mathrm { n c } } ) \\Vert _ { L ^ 2 ( \\Omega ) } . \\end{align*}"}
{"id": "7765.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { n } \\beta _ { j , \\ell } ^ k \\ , [ \\alpha ^ * ] ^ \\ell k = 1 , \\ldots , m , \\ j = 1 , \\ldots , L \\end{align*}"}
{"id": "6775.png", "formula": "\\begin{align*} & \\sum _ { j \\in N _ { i ' } } \\max \\left \\{ a _ { i ' j } , b - \\sum _ { k \\in M _ C - i ' } a _ { k t _ k } \\right \\} x _ { i ' j } \\\\ & + \\sum _ { i \\in M _ C - i ' } \\left ( \\sum _ { j = 1 } ^ { t _ i } a _ { i t _ i } \\max \\left \\{ 1 , \\frac { a _ { i j } } { b - \\sum _ { k \\in M _ C - { i , i ' } } a _ { k t _ k } - a _ { i ' n _ { i ' } } } \\right \\} x _ { i j } + \\sum _ { j = t _ i + 1 } ^ { n _ i } a _ { i j } x _ { i j } \\right ) \\leq b \\end{align*}"}
{"id": "1649.png", "formula": "\\begin{align*} A \\ , = \\ , \\begin{pmatrix} \\lambda _ 1 & 1 & 0 & 0 & 0 & 0 & \\dots & 0 & 0 \\\\ 0 & 0 & \\lambda _ 2 & 1 & 0 & 0 & \\dots & 0 & 0 \\\\ 0 & 0 & 0 & 0 & \\lambda _ 3 & 1 & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & 0 & 0 & 0 & \\dots & \\lambda _ g & 1 \\\\ \\end{pmatrix} . \\end{align*}"}
{"id": "3225.png", "formula": "\\begin{align*} h : \\mathcal { N } _ { \\mathrm { S p } } ( 2 m , \\alpha , L ) & \\longrightarrow \\mathcal { A } \\coloneqq \\bigoplus _ { i = 1 } ^ { m } H ^ 0 ( X , K ^ { 2 i } ( D ^ { 2 i - 1 } ) ) \\\\ ( E _ * , \\varphi , \\Phi ) & \\longmapsto ( s _ 2 , s _ 4 , \\dots , s _ { 2 m } ) . \\end{align*}"}
{"id": "5364.png", "formula": "\\begin{align*} \\norm { ( - \\Delta ) ^ { s / 2 } u } _ { L ^ 2 ( \\R ^ n ) } & \\geq t \\norm { ( - \\Delta ) ^ { s / 2 } u } _ { L ^ 2 ( \\R ^ n ) } + \\frac { 1 - t } { C ( s , \\Omega ) } \\norm { u } _ { L ^ 2 ( \\R ^ n ) } \\\\ & = \\frac { 1 } { C ( s , \\Omega ) + 1 } \\left ( \\norm { ( - \\Delta ) ^ { s / 2 } u } _ { L ^ 2 ( \\R ^ n ) } + \\norm { u } _ { L ^ 2 ( \\R ^ n ) } \\right ) \\\\ & \\geq \\sqrt { \\delta ( \\Omega ) } \\norm { u } _ { H ^ s ( \\R ^ n ) } , \\end{align*}"}
{"id": "2479.png", "formula": "\\begin{align*} \\sup _ { ( x , y ) \\in \\bar B } \\sum _ { i = 1 } ^ { n - 2 } ( \\varphi _ i ^ 1 ( x , y ) - \\Phi _ i ( x , y ) ) ^ 2 < \\frac { \\varepsilon } { 9 } . \\end{align*}"}
{"id": "1778.png", "formula": "\\begin{align*} g ( w ^ * ) = ( z + a f _ k ) ( w ^ * ) = b + \\sum _ { j = 1 } ^ m a c _ j = 0 \\end{align*}"}
{"id": "5614.png", "formula": "\\begin{align*} ( D : _ K A ) = \\{ x \\in K \\mid x A \\subseteq D \\} . \\end{align*}"}
{"id": "3137.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } a _ { x , \\mathrm { s } } a _ { y , \\mathrm { t } } + a _ { y , \\mathrm { t } } a _ { x , \\mathrm { s } } & = & 0 \\\\ a _ { x , \\mathrm { s } } ^ { \\ast } a _ { y , \\mathrm { t } } + a _ { y , \\mathrm { t } } a _ { x , \\mathrm { s } } ^ { \\ast } & = & \\delta _ { x , y } \\delta _ { \\mathrm { s } , \\mathrm { t } } \\mathfrak { 1 } \\end{array} \\right . , x , y \\in \\Lambda , \\mathrm { s } , \\mathrm { t } \\in \\mathrm { S } , \\end{align*}"}
{"id": "7427.png", "formula": "\\begin{align*} \\bold { q } ( t ) = - \\beta \\theta _ x ( x , t ) - \\int _ 0 ^ { \\infty } g ( s ) \\theta _ x ( x , t - s ) d s , \\end{align*}"}
{"id": "3241.png", "formula": "\\begin{align*} p ^ { - 1 } R ^ { - p } & = p ^ { - 1 } \\left ( \\frac { \\sqrt { \\frac { \\kappa } { C A } + \\left ( \\frac { c _ d ' | \\nabla f ( a ) | } { 2 C A } \\right ) ^ 2 } + \\frac { c _ d ' | \\nabla f ( a ) | } { 2 C A } } { \\frac \\kappa { C A } } \\right ) ^ p \\\\ & = \\frac { c _ { d , p } } { \\kappa ^ p } \\left ( \\sqrt { \\tfrac { C A } { c _ d '^ 2 } \\kappa + \\left ( \\tfrac 1 2 | \\nabla f ( a ) | \\right ) ^ 2 } + \\tfrac 1 2 | \\nabla f ( a ) | \\right ) ^ p . \\end{align*}"}
{"id": "7459.png", "formula": "\\begin{align*} | \\lambda ^ n | \\to \\infty \\| U ^ n \\| _ { \\mathcal { H } } = \\| ( u ^ n , v ^ n , y ^ n , z ^ n , w ^ n , \\eta ^ n ) \\| _ { \\mathcal { H } } = 1 , \\end{align*}"}
{"id": "7573.png", "formula": "\\begin{align*} \\frac { d x } { d t } = \\sum _ { \\alpha = 1 } ^ { r } b _ \\alpha ( t ) X _ \\alpha , \\end{align*}"}
{"id": "895.png", "formula": "\\begin{align*} \\norm { h } ^ 2 = u ( a ) . \\end{align*}"}
{"id": "4960.png", "formula": "\\begin{align*} Y ^ { \\infty , 1 } _ t = \\kappa _ 2 \\int _ 0 ^ t ( t - s ) ^ \\alpha ( \\sigma ' \\sigma ) ( X _ s ) d B _ s + \\int _ 0 ^ t ( t - s ) ^ \\alpha \\sigma ' ( X _ s ) Y ^ { \\infty , 1 } _ s d W _ s , \\end{align*}"}
{"id": "7634.png", "formula": "\\begin{align*} z = - \\frac { 1 } { u ( 1 + u ^ 2 ) } . \\end{align*}"}
{"id": "2557.png", "formula": "\\begin{align*} \\begin{cases} T _ \\pm : ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) \\mapsto ( \\nu _ 1 \\pm 1 , \\nu _ 2 \\mp 1 , \\nu _ 3 ) \\ , \\\\ U _ \\pm : ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) \\mapsto ( \\nu _ 1 , \\nu _ 2 \\pm 1 , \\nu _ 3 \\mp 1 ) \\ , \\\\ V _ \\pm : ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) \\mapsto ( \\nu _ 1 \\pm 1 , \\nu _ 2 , \\nu _ 3 \\mp 1 ) \\ , \\end{cases} \\end{align*}"}
{"id": "4972.png", "formula": "\\begin{align*} \\widetilde { A } ^ n _ t = n ^ { \\alpha + \\frac 1 2 } \\sigma ( X _ t ) \\int ^ t _ 0 [ ( t - \\eta _ n ( s ) ) ^ { \\alpha } - ( t - s ) ^ { \\alpha } ] d W _ s , \\end{align*}"}
{"id": "8564.png", "formula": "\\begin{align*} D _ 4 F _ p \\big ( u ( t ) , \\zeta ( t ) , v ( t ) , c ( t ) \\big ) \\big [ \\dot c ( t ) \\big ] = \\nabla F _ p \\big ( u ( t ) , \\zeta ( t ) , v ( t ) , c ( t ) \\big ) \\cdot \\dot c ( t ) \\ . \\end{align*}"}
{"id": "6190.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } ( \\frac { \\partial } { \\partial t } - \\Delta _ { B } ) \\log | X | _ { \\omega } ^ { 2 } = \\frac { 1 } { | X | _ { \\omega } ^ { 2 } } \\left ( - g ^ { T i \\overline { j } } g _ { k \\overline { l } } ^ { T } ( \\partial _ { i } ^ { k } X ) ( \\overline { \\partial _ { j } ^ { l } X } ) + \\frac { \\left \\vert \\bigtriangledown ^ { T } | X | _ { \\omega } ^ { 2 } \\right \\vert _ { \\omega } ^ { 2 } } { | X | _ { \\omega } ^ { 2 } } \\right ) \\leq 0 . \\end{array} \\end{align*}"}
{"id": "3492.png", "formula": "\\begin{align*} c _ { ( 2 ) } ^ { \\ast } & = \\frac { ( m - 1 ) ( m - 3 ) \\Gamma ( 1 / 2 ) } { ( 2 \\pi ) ^ { 1 / 2 } m ^ { 2 } } \\left [ \\int _ { 0 } ^ { \\infty } u ^ { 1 / 2 - 1 } \\left ( 1 + \\frac { 2 t } { m } \\right ) ^ { - ( m - 3 ) / 2 } \\mathrm { d } u \\right ] ^ { - 1 } \\\\ & = \\frac { ( m - 1 ) ( m - 3 ) } { m ^ { 5 / 2 } B ( \\frac { 1 } { 2 } , ~ \\frac { m - 4 } { 2 } ) } , ~ i f ~ m > 4 , \\end{align*}"}
{"id": "5453.png", "formula": "\\begin{align*} a = a \\textsf { 1 } _ A & = a q ( k ) \\stackrel { ( L 2 ) } = q ( a k ) \\\\ & = q ( a ' k ' ) \\stackrel { ( L 2 ) } = a ' q ( k ' ) = a ' \\textsf { 1 } _ A = a ' . \\end{align*}"}
{"id": "8348.png", "formula": "\\begin{align*} \\widetilde { f } = { w _ 1 } ^ + \\tilde { f } _ + + { w _ 1 } ^ - \\tilde { f } _ - + w _ { r ^ { - 2 } } \\tilde { f } _ { 0 + } + w _ { r ^ { 2 } } \\tilde { f } _ { 0 - } , \\end{align*}"}
{"id": "3029.png", "formula": "\\begin{align*} \\tilde { N } _ { \\lambda } ( u , z , w ) = \\ ! \\ ! \\ ! \\prod _ { \\substack { x \\in \\lambda \\\\ h ( x ) \\equiv 0 \\ ! \\ ! \\ ! \\mod 2 } } \\ ! \\ ! \\ ! ( z ^ { a ( x ) + 1 } - u w ^ { l ( x ) } ) ( z ^ { a ( x ) } - u ^ { - 1 } w ^ { l ( x ) + 1 } ) . \\end{align*}"}
{"id": "848.png", "formula": "\\begin{align*} \\boldsymbol { V } _ { B \\rightarrow A } ^ { e x t } & = \\left ( ( \\boldsymbol { V } _ { B } ^ { p o s t } ) ^ { - 1 } - ( \\boldsymbol { V } _ { B } ^ { p r i } ) ^ { - 1 } \\right ) ^ { - 1 } , \\\\ \\boldsymbol { x } _ { B \\rightarrow A } ^ { e x t } & = \\boldsymbol { V } _ { B \\rightarrow A } ^ { e x t } \\left ( ( \\boldsymbol { V } _ { B } ^ { p o s t } ) ^ { - 1 } \\boldsymbol { x } _ { B } ^ { p o s t } - ( \\boldsymbol { V } _ { B } ^ { p r i } ) ^ { - 1 } \\boldsymbol { x } _ { B } ^ { p r i } \\right ) , \\end{align*}"}
{"id": "1733.png", "formula": "\\begin{align*} F ( z + \\omega _ 2 \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) = \\exp \\Bigg ( \\int _ { C } \\frac { e ^ { ( z + \\omega _ 2 ) s } } { ( e ^ { \\overline \\omega _ 1 s } 1 - 1 ) ( e ^ { \\omega _ 2 s } - 1 ) } \\frac { d s } { s } \\Bigg ) , 0 < ( z ) < ( \\overline \\omega _ 1 ) . \\end{align*}"}
{"id": "2361.png", "formula": "\\begin{align*} f - l ( h _ \\sigma ) = \\sum _ { i = 1 } ^ { \\deg _ X ( f ) } \\partial _ i l ( h _ \\sigma ) Q _ \\sigma ^ i \\end{align*}"}
{"id": "4822.png", "formula": "\\begin{align*} \\sum _ { p \\in \\pi _ j \\cap J } s _ { k _ j , \\lambda } ( p , \\pi _ j ) = \\frac { 1 } { { \\lambda + k _ j \\choose k _ j } } \\sum _ { p \\in \\pi _ j \\cap J } \\tilde S _ { k _ j } ( p , \\pi _ j ( p ) , \\alpha , \\lambda ) = 1 . \\end{align*}"}
{"id": "4350.png", "formula": "\\begin{align*} \\dfrac { V _ r } { e _ 1 \\cdots e _ { r - 1 } } = \\sum _ { 1 \\le j < r } \\dfrac { m _ r } { m _ j } \\ , \\dfrac { h _ j } { e _ 1 \\cdots e _ j } . \\end{align*}"}
{"id": "1669.png", "formula": "\\begin{align*} F = \\tilde { B } _ { k + 1 } \\setminus \\bigcup _ { i = 1 } ^ k \\tilde { B _ i } . \\end{align*}"}
{"id": "2899.png", "formula": "\\begin{align*} h \\left ( ( x _ 1 , x _ 2 , x _ 3 ) + ( a _ 1 , a _ 2 , a _ 3 ) \\right ) + h ( x _ 1 , x _ 2 , x _ 3 ) = b , \\end{align*}"}
{"id": "4747.png", "formula": "\\begin{align*} v ( \\psi _ { n } ( P ) ) & = \\widehat { \\lambda } ( n P ) + \\frac { n ^ 2 - 1 } { 1 2 } v ( \\Delta ) - n ^ 2 \\widehat { \\lambda } ( P ) \\\\ & = \\frac { v ( \\Delta ) } { 1 2 } - \\frac { c _ v ( n P ) } 2 + \\frac { ( n P . O ) _ v } 2 + \\frac { n ^ 2 - 1 } { 1 2 } v ( \\Delta ) - n ^ 2 \\frac { v ( \\Delta ) } { 1 2 } + n ^ 2 \\frac { c _ v ( P ) } 2 - \\frac { n ^ 2 ( P . O ) _ v } 2 \\\\ & = \\frac { ( n P . O ) _ v } 2 - \\frac { n ^ 2 ( P . O ) _ v } 2 + \\frac { 1 } { 2 } n ^ 2 c _ v ( P ) - \\frac { 1 } { 2 } c _ v ( n P ) . \\end{align*}"}
{"id": "7205.png", "formula": "\\begin{align*} \\psi _ 1 ( s ) = s ^ { - p a } \\log ( 2 + s ) \\ \\mbox { a n d } \\ \\psi _ 2 ( s ) = s ^ { - b } \\log ^ { p - 1 } ( 2 + s ) . \\end{align*}"}
{"id": "3608.png", "formula": "\\begin{align*} \\langle ( y _ 1 , \\ldots , y _ s ) , \\gamma _ j \\rangle = d _ j - y _ { s + 3 } \\end{align*}"}
{"id": "5320.png", "formula": "\\begin{align*} Z _ t = \\lim _ { n \\to \\infty } n ^ { - 1 } \\sum _ { i = 1 } ^ n \\delta _ { X _ i ( t ) } \\end{align*}"}
{"id": "3818.png", "formula": "\\begin{align*} \\forall i , i _ 1 \\in \\{ 0 , 1 , 2 , 3 , 4 \\} , \\mu _ 1 \\in \\{ + , - \\} , | H y p E l l _ { k _ 1 , j _ 1 ; n _ 1 } ^ { i , i _ 1 , \\mu _ 1 } ( t _ 1 , t _ 2 ) | \\leq \\sum _ { a = 1 , 2 , 3 } | { } ^ 1 _ a L a s t E l l _ { k _ 1 , j _ 1 ; n _ 1 } ^ { i , i _ 1 , \\mu _ 1 } ( t _ 1 , t _ 2 ) | , \\end{align*}"}
{"id": "693.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } ( \\log \\lambda ) ^ { \\theta / 4 } = \\tfrac { 1 } { 2 } s _ 0 ^ { \\theta / 4 } \\gg \\log A . \\end{align*}"}
{"id": "816.png", "formula": "\\begin{align*} K _ 2 ( u , m , R _ 0 ) : = 1 + T _ { p - 1 } ( u ; R _ 0 ) ^ { p - 1 } + T _ { q - 1 } ( u ; R _ 0 ) ^ { q - 1 } + \\| u \\| ^ { \\frac { m ( p - 1 ) } { p - 2 } } _ { L ^ \\infty ( B _ { R _ 0 } ) } + \\| u \\| ^ { q - 1 } _ { L ^ \\infty ( B _ { R _ 0 } ) } + \\| f \\| _ { L ^ \\infty ( B _ { R _ 0 } ) } . \\end{align*}"}
{"id": "1488.png", "formula": "\\begin{align*} e ^ { x ( e _ { \\lambda } ( t ) - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 1 4 , 1 5 ] ) . \\end{align*}"}
{"id": "4791.png", "formula": "\\begin{align*} \\rho _ { \\lambda } = \\rho ( \\lambda ^ { 2 } t , \\lambda x ) , u _ { \\lambda } = \\lambda u ( \\lambda ^ { 2 } t , \\lambda x ) , \\end{align*}"}
{"id": "8468.png", "formula": "\\begin{align*} \\frac { d } { d \\left \\| f _ j \\right \\| _ 2 ^ 2 } \\left ( \\prod _ { i = 1 } ^ d \\left \\| f _ i \\right \\| _ 2 ^ 2 - \\prod _ { i = 1 } ^ d \\left ( \\left \\| f _ i \\right \\| _ 2 ^ 2 - \\frac { L ^ 2 } { 1 2 b ^ 2 } \\right ) \\right ) & = \\prod _ { i \\neq j } \\left \\| f _ i \\right \\| _ 2 ^ 2 - \\prod _ { i \\neq j } \\left ( \\left \\| f _ i \\right \\| _ 2 ^ 2 - \\frac { L ^ 2 } { 1 2 b ^ 2 } \\right ) \\\\ & \\ge 0 . \\end{align*}"}
{"id": "9.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 + i & 1 & 0 \\\\ 1 + i & 4 i & 0 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "4058.png", "formula": "\\begin{align*} & \\sigma _ a s _ { ( k ) } = s _ { ( k ) } \\sigma _ a = s _ { ( k ) } , & & \\epsilon _ a s _ { ( k ) } = s _ { ( k ) } \\epsilon _ a = 0 , & & a = 1 , \\ldots , k - 1 . \\end{align*}"}
{"id": "8011.png", "formula": "\\begin{align*} H _ { ( Z _ - , D _ { Z , - } ) } ( y ) = e ^ { \\frac { t _ - } { 2 \\pi i } } \\sum _ { d \\in \\mathbb K _ { - } } y ^ d \\left ( \\frac { \\prod _ { i = 1 } ^ 2 \\Gamma ( 1 + \\frac { v _ i } { z } + v _ i \\cdot d ) } { \\prod _ { i \\in M _ 0 } \\Gamma ( 1 + \\frac { \\bar D _ i } { z } + D _ i \\cdot d ) } \\right ) \\textbf { 1 } _ { [ d ] } [ \\textbf { 1 } ] _ { ( D _ i \\cdot d ) _ { i \\in I _ - } , v _ 1 \\cdot d } . \\end{align*}"}
{"id": "2146.png", "formula": "\\begin{align*} \\nu = \\frac { 1 8 } { \\pi ^ { 4 } } \\sum \\nolimits _ { m \\in \\mathbb { Z } } \\frac { 1 } { m ^ 2 } \\left \\{ \\sum \\nolimits _ { n \\in \\mathbb { N } } \\frac { 1 } { 2 ^ { n } n ^ { 2 } } \\sum \\nolimits _ { j = 1 } ^ { 2 ^ { n } } \\delta _ { m + j 2 ^ { - n } } \\right \\} \\end{align*}"}
{"id": "3596.png", "formula": "\\begin{align*} & \\rho ( I ) : = \\sup \\left \\{ { n } / { r } \\ \\left . \\right | \\ , I ^ { ( n ) } \\not \\subset I ^ r \\right \\} , \\\\ & \\ \\ \\ \\ \\ \\widehat { \\rho } ( I ) : = \\sup \\left \\{ { n } / { r } \\ \\left . \\right | \\ , I ^ { ( n t ) } \\not \\subset I ^ { r t } t \\gg 0 \\right \\} , \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\rho _ { i c } ( I ) : = \\sup \\left \\{ { n } / { r } \\ \\left . \\right | \\ , I ^ { ( n ) } \\not \\subset \\overline { I ^ { r } } \\right \\} , \\ . \\end{align*}"}
{"id": "8174.png", "formula": "\\begin{align*} \\tilde { \\Omega } = \\lim \\limits _ { t \\rightarrow \\infty } e ^ { i t H } J U _ 0 ( t ) \\end{align*}"}
{"id": "8287.png", "formula": "\\begin{align*} \\frac { k ' a } { 2 } \\left [ \\tanh ( k ' a / 2 ) - \\coth ( k ' a / 2 ) \\right ] = - 1 \\end{align*}"}
{"id": "8706.png", "formula": "\\begin{align*} ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } r + \\lambda \\left ( \\frac { r } { \\sqrt { r ^ 2 + \\dot { r } ^ 2 } } + b ( \\left ( - \\dot { r } \\sin ( \\theta - t ) + r \\cos ( \\theta - t ) ) \\theta _ { r } + \\sin ( \\theta - t ) \\right ) \\right ) \\\\ - \\lambda \\frac { d } { d t } \\left ( \\frac { \\dot { r } } { \\sqrt { r ^ 2 + \\dot { r } ^ 2 } } + b \\cos ( \\theta - t ) \\right ) = 0 . \\end{align*}"}
{"id": "7221.png", "formula": "\\begin{align*} S _ p : = \\sum _ { j = 1 } ^ \\infty \\frac { j } { p ^ j } < \\infty . \\end{align*}"}
{"id": "4257.png", "formula": "\\begin{align*} \\Theta _ \\theta : = \\left ( \\begin{matrix} \\cos \\theta & \\sin \\theta & 0 \\\\ \\\\ - \\sin \\theta & \\cos \\theta & 0 \\\\ \\\\ 0 & 0 & I _ { n - 2 \\times n - 2 } \\end{matrix} \\right ) . \\end{align*}"}
{"id": "651.png", "formula": "\\begin{align*} q ( a , t ) = \\frac { \\pi ^ 2 } { \\abs { ( a + i b ) ( a + i b - 1 ) } } \\end{align*}"}
{"id": "6101.png", "formula": "\\begin{align*} \\theta _ { n \\ , \\ , n + 1 } = 0 \\theta _ { n \\ , \\ , n + 2 } = b \\ , \\omega _ { n + 1 \\ , \\ , n + 2 } \\end{align*}"}
{"id": "3359.png", "formula": "\\begin{align*} F _ m ( \\varphi ) = F _ m ( \\varphi _ 1 ) + F _ m ( \\varphi _ 2 ) . \\end{align*}"}
{"id": "6137.png", "formula": "\\begin{align*} | A ' _ 1 \\cap B _ 2 \\cap H _ 1 | { } & { } \\geq | A ' _ 2 \\cap B _ 1 \\cap H _ 1 | = | A ' _ 2 \\cap B _ 1 \\cap H _ 4 | \\\\ { } & { } \\geq | A ' _ 1 \\cap B _ 2 \\cap H _ 4 | = | A ' _ 1 \\cap B _ 2 \\cap H _ 3 | \\\\ { } & { } \\geq | A ' _ 2 \\cap B _ 1 \\cap H _ 3 | = | A ' _ 2 \\cap B _ 1 \\cap H _ 2 | \\\\ { } & { } \\geq | A ' _ 1 \\cap B _ 2 \\cap H _ 2 | = | A ' _ 1 \\cap B _ 2 \\cap H _ 1 | . \\end{align*}"}
{"id": "7654.png", "formula": "\\begin{align*} \\begin{pmatrix} \\lambda _ { \\delta } \\\\ \\mu _ { \\delta } \\end{pmatrix} , \\begin{pmatrix} \\lambda _ { \\theta } \\\\ \\mu _ { \\theta } \\end{pmatrix} \\end{align*}"}
{"id": "1831.png", "formula": "\\begin{align*} A _ { n } = b _ { 0 } A _ { n - 1 } + a _ { 0 } \\sum _ { k = 0 } ^ { n - 2 } A _ { k } ^ { ( 1 ) } A _ { n - k - 2 } , n \\geq 1 . \\end{align*}"}
{"id": "7246.png", "formula": "\\begin{align*} & \\mathcal { M } _ { 0 , \\Gamma } ( \\hat \\mu ) = ( | \\hat { \\mu } ( v _ { j } ) | ^ { 2 } ) _ { j = 1 } ^ { n } , \\\\ & \\mathcal { M } _ { 1 , \\Gamma } ( \\hat \\mu ) = ( | \\hat { \\mu } ( v _ { j } ) - \\hat { \\mu } ( v _ { k } ) | ^ { 2 } ) _ { \\{ v _ j , v _ k \\} \\in E , j < k } , \\\\ \\quad & \\mathcal { M } _ { 2 , \\Gamma } ( \\hat \\mu ) = ( | \\hat { \\mu } ( v _ { j } ) - i \\hat { \\mu } ( v _ { k } ) | ^ { 2 } ) _ { \\{ v _ j , v _ k \\} \\in E , j < k } . \\end{align*}"}
{"id": "2477.png", "formula": "\\begin{align*} Y _ { x } : = \\left \\{ y \\in \\left ( \\gg _ { x } \\right ) _ { 1 } \\mid \\left [ y , y \\right ] = 0 \\right \\} . \\end{align*}"}
{"id": "5348.png", "formula": "\\begin{align*} ( q _ 1 - q _ 2 ) ( v , v ^ * ) = \\lim _ { k \\to \\infty } ( q _ 1 - q _ 2 ) ( u _ { f _ k } - f _ k , u _ { g _ k } ^ * - g _ k ) = \\lim _ { k \\to \\infty } ( q _ 1 - q _ 2 ) ( u _ { f _ k } , u _ { g _ k } ^ * ) = 0 \\end{align*}"}
{"id": "368.png", "formula": "\\begin{align*} \\sum _ { m = D ^ 2 } ^ { \\infty } [ \\tilde p ^ \\# _ m ( A , A ) - \\tilde \\pi ( A ) ] \\lesssim \\frac { \\log ( D / \\delta ) } { f ( n ) } . \\end{align*}"}
{"id": "1689.png", "formula": "\\begin{align*} \\int _ I D T ( p ( t ) ) u \\ d t = \\int _ { I _ 1 } D T ( p ( t ) ) u \\ d t + \\int _ { I _ 2 } D T ( p ( t ) ) u \\ d t , \\end{align*}"}
{"id": "2537.png", "formula": "\\begin{align*} R i c ( e _ 1 , e _ 1 ) = - e ^ { - 6 x _ 2 } - e ^ { - 2 x _ 2 } - e ^ { - 4 x _ 2 } + 1 , \\end{align*}"}
{"id": "200.png", "formula": "\\begin{align*} x ' ( \\{ v _ 1 , v _ 2 \\} ) = x ' ( \\{ v _ 1 , v _ 3 \\} ) = 1 , \\end{align*}"}
{"id": "2979.png", "formula": "\\begin{align*} 2 i \\partial _ z F _ { m , Q } ( z , s ) = \\sum _ { \\substack { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma \\\\ \\gamma \\neq \\gamma _ 1 , \\gamma _ 2 } } f _ { 2 , m } ( \\gamma z , s ) \\frac { d ( \\gamma z ) } { d z } , \\end{align*}"}
{"id": "2337.png", "formula": "\\begin{align*} \\epsilon ( f ) = \\max _ { 1 \\leq b \\leq \\deg ( f ) } \\left \\{ \\frac { \\nu ( f ) - \\nu ( \\partial _ b f ) } { b } \\right \\} , \\end{align*}"}
{"id": "6069.png", "formula": "\\begin{align*} f ^ { ( j ) } ( \\lambda ) = \\binom { j } { p } p ! g ^ { ( j - p ) } ( \\lambda ) = ( j ) _ p g ^ { ( j - p ) } ( \\lambda ) . \\end{align*}"}
{"id": "6887.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } | y p _ { t , k } ( y ) | d y & = e ^ { - t } \\int _ { \\mathbb { R } } | y p _ { 0 , k } ( y ) | d y \\\\ & \\leq e ^ { - t } \\int _ { \\mathbb { R } } \\int _ { 0 } ^ { V _ F } | y | p _ { } ( v , y ) d v d y . \\end{align*}"}
{"id": "6543.png", "formula": "\\begin{align*} \\P \\{ \\theta _ \\infty < \\infty \\} = 1 . \\end{align*}"}
{"id": "372.png", "formula": "\\begin{align*} \\sum _ { t = \\delta ^ 2 } ^ { D ^ 2 } \\tilde p ^ \\# _ t ( A , A ) \\lesssim \\sum _ { t = \\delta ^ 2 } ^ { D ^ 2 } \\frac { 1 } { t f ( n ) } \\lesssim \\frac { \\log ( D / \\delta ) } { f ( n ) } . \\end{align*}"}
{"id": "4702.png", "formula": "\\begin{align*} ( - 1 ) ^ { k - 1 } \\left ( \\Big ( \\sum _ { n = 0 } ^ \\infty u _ m ( n ) \\ , q ^ n \\Big ) \\Big ( \\sum _ { n = 1 - k } ^ k ( - 1 ) ^ n q ^ { n ( 3 n - 1 ) / 2 } \\Big ) - \\sum _ { n = 0 } ^ \\infty C _ m ( n ) \\ , q ^ { n } \\right ) \\\\ = \\left ( \\sum _ { n = 0 } ^ \\infty C _ m ( n ) \\ , q ^ { n } \\right ) \\left ( \\sum _ { n = 0 } ^ \\infty M _ k ( n ) \\ , q ^ n \\right ) . \\end{align*}"}
{"id": "2464.png", "formula": "\\begin{align*} \\oplus _ { \\mu \\not = 0 } \\gg _ { 1 } ^ { \\mu } = \\oplus _ { \\alpha \\in \\Delta _ { 1 } \\backslash { A } ^ { \\perp } } \\gg _ { \\alpha } \\end{align*}"}
{"id": "6944.png", "formula": "\\begin{align*} \\Sigma ^ * R ^ { 1 / 2 } = R ^ { 1 / 2 } A . \\end{align*}"}
{"id": "3702.png", "formula": "\\begin{align*} \\frac { 1 } { 2 T } \\mu ( \\Lambda _ { S ^ * _ \\Gamma M \\backslash S ^ * \\Gamma , \\ , T } ) = 0 . \\end{align*}"}
{"id": "2375.png", "formula": "\\begin{align*} J : = \\{ 1 , \\ldots , d \\} \\setminus I . \\end{align*}"}
{"id": "4197.png", "formula": "\\begin{align*} \\kappa _ \\gamma ( a , b ) \\lesssim _ N \\sum _ { \\ell = - 1 } ^ \\iota 2 ^ { - \\iota \\gamma N + \\iota Q / 2 - \\ell d _ 2 / 2 } \\norm { F ^ { ( \\iota ) } } _ 2 \\lesssim \\norm { F ^ { ( \\iota ) } } _ 2 . \\end{align*}"}
{"id": "4407.png", "formula": "\\begin{align*} t _ { n , k } = \\left \\{ \\begin{array} { l l } { \\displaystyle \\frac { k - [ m _ n ] } { [ m _ n ] - [ m _ { n - 1 } ] } } , & \\ \\mbox { i f } \\ m _ { n - 1 } < | k | \\leq m _ n , \\\\ & \\\\ { \\displaystyle \\frac { [ m _ { n + 1 } ] - k } { [ m _ { n + 1 } ] - [ m _ n ] } } , & \\ \\mbox { i f } \\ m _ { n } < | k | \\leq m _ { n + 1 } . \\end{array} \\right . \\end{align*}"}
{"id": "4483.png", "formula": "\\begin{align*} \\Big | \\frac { x } t + p ' ( \\xi ) \\Big | = | p ' ( \\xi ) - p ' ( \\xi _ 0 ) | \\gtrsim | p '' ( 2 ^ k ) | 2 ^ k \\gtrsim 2 ^ { 2 k } ( 1 + 2 ^ { 2 k } ) ^ { - 5 / 2 } . \\end{align*}"}
{"id": "8298.png", "formula": "\\begin{align*} \\hat { H } \\phi _ { 0 } ( x ) = E _ { 0 } \\phi _ { 0 } = 0 \\end{align*}"}
{"id": "3240.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\ 1 ( c _ d ' r | \\nabla f ( a ) | + C A r ^ 2 > \\kappa ) \\ , \\frac { d r } { r ^ { p + 1 } } = p ^ { - 1 } R ^ { - p } \\ , . \\end{align*}"}
{"id": "3588.png", "formula": "\\begin{align*} ( a _ 1 , a _ 2 ) \\wedge ( b _ 1 , b _ 2 ) & = ( a _ 1 \\wedge b _ 1 , a _ 2 \\vee b _ 2 ) \\\\ ( a _ 1 , a _ 2 ) \\vee ( b _ 1 , b _ 2 ) & = ( a _ 1 \\vee b _ 1 , a _ 2 \\wedge b _ 2 ) \\\\ \\neg ( a _ 1 , a _ 2 ) & = ( a _ 2 , a _ 1 ) \\\\ ( a _ 1 , a _ 2 ) \\leq ( b _ 1 , b _ 2 ) & \\mbox { i f f } a _ 1 \\leq _ { [ 0 , 1 ] } b _ 1 \\mbox { a n d } b _ 2 \\leq _ { [ 0 , 1 ] } a _ 2 \\\\ ( a _ 1 , a _ 2 ) \\leq _ i ( b _ 1 , b _ 2 ) & \\mbox { i f f } a _ 1 \\leq _ { [ 0 , 1 ] } b _ 1 \\mbox { a n d } a _ 2 \\leq _ { [ 0 , 1 ] } b _ 2 \\end{align*}"}
{"id": "2488.png", "formula": "\\begin{align*} T _ E E ' = \\mathcal { H } \\nabla _ { \\nu E } \\nu E ' + \\nu \\nabla _ { \\nu E } \\mathcal { H } E ' , \\end{align*}"}
{"id": "696.png", "formula": "\\begin{align*} s _ 1 : = \\inf \\{ s ' > 0 : Z \\ge \\zeta [ s ' , s ] \\} . \\end{align*}"}
{"id": "2426.png", "formula": "\\begin{align*} & K _ a v _ b = q ^ { \\delta _ { a b } } v _ b , E _ a v _ b = \\delta _ { a + 1 , b } v _ a , \\ \\ F _ a v _ b = \\delta _ { a , b } v _ { a + 1 } , \\\\ & K _ a w _ b = q ^ { - \\delta _ { a b } } w _ b , E _ a w _ b = \\delta _ { a , b } w _ { a + 1 } , F _ a w _ b = \\delta _ { a + 1 , b } w _ { a } . \\end{align*}"}
{"id": "1630.png", "formula": "\\begin{align*} \\max ( A ) = \\{ s \\in A : | s | = \\ell _ A \\} . \\end{align*}"}
{"id": "3729.png", "formula": "\\begin{align*} \\mathcal { S } ^ \\infty : = \\{ m ( \\xi ) : m : \\R ^ 3 \\longrightarrow \\R , \\| \\mathcal { F } ^ { - 1 } [ m ] ( x ) \\| _ { L ^ 1 _ x } < + \\infty \\} . \\end{align*}"}
{"id": "3068.png", "formula": "\\begin{align*} \\mathrm { ( I I ) } ~ & \\mathrm { M D T C o v } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { Y } ) = \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } \\left [ \\frac { \\boldsymbol { \\Omega } } { F _ { \\mathbf { Z } } ( \\boldsymbol { \\xi _ { a } } , \\boldsymbol { \\xi _ { b } } ) } - \\frac { \\boldsymbol { \\delta } \\boldsymbol { \\delta } ^ { T } } { F _ { \\mathbf { Z } } ^ { 2 } ( \\boldsymbol { \\xi _ { a } } , \\boldsymbol { \\xi _ { b } } ) } \\right ] \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
{"id": "1783.png", "formula": "\\begin{align*} K ^ { ( \\alpha - 1 ) } \\subseteq \\bigcap _ { n = 1 } ^ \\infty \\operatorname { K e r } ( u _ n + a _ n f _ { q ( n ) } ) . \\end{align*}"}
{"id": "6633.png", "formula": "\\begin{align*} \\| \\phi _ w \\| _ r ^ r = \\int _ { G _ \\infty \\times G _ S } \\phi _ w ^ r \\ , d ( m _ \\infty \\times m _ p ) = \\| w \\| ^ r _ { L ^ r ( G _ \\infty ) } < \\infty . \\end{align*}"}
{"id": "3988.png", "formula": "\\begin{align*} 1 = \\sum _ { e \\in \\widehat E } e ; ~ ~ ~ ~ ~ \\overline e = e , ~ \\forall ~ e \\in \\widehat E ; ~ ~ ~ ~ ~ e e ' = \\begin{cases} e , \\ ! & e = e ' ; \\\\ 0 , \\ ! & e \\ne e ' ; \\end{cases} ~ \\forall ~ e , e ' \\in \\widehat E ; \\end{align*}"}
{"id": "4970.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\in [ 0 , T ] } E [ | C ^ n _ t - \\widetilde { C } ^ n _ t | ^ 2 ] = 0 \\end{align*}"}
{"id": "4863.png", "formula": "\\begin{align*} \\alpha = a / q + \\theta , | \\theta | \\leq 1 / q Q \\end{align*}"}
{"id": "1298.png", "formula": "\\begin{align*} P ( T _ 0 , x ) & = \\mathcal { G } ( r ( T ) , x ) \\prod _ { j = 1 } ^ { k } P ( T _ j , x ) ^ { \\alpha _ j } . \\end{align*}"}
{"id": "8407.png", "formula": "\\begin{gather*} \\hat \\phi ( \\tau , z ) = \\hat h ( \\tau ) ^ { \\rm t } \\theta _ { m } ( \\tau , z ) \\end{gather*}"}
{"id": "5576.png", "formula": "\\begin{align*} \\widetilde { G } ( x , y ) = \\frac { G ( x , y ) } { m ( x ) \\ , m ( y ) } , x , y \\in \\Omega , \\end{align*}"}
{"id": "156.png", "formula": "\\begin{align*} \\chi ^ { \\epsilon , L } _ { \\infty } : = \\epsilon ^ d \\sum _ { x \\in \\Lambda _ { \\epsilon , L } } \\big < \\varphi _ 0 \\varphi _ x \\big > ^ { \\epsilon , L } _ { \\lambda , \\mu } \\leq \\bar \\chi . \\end{align*}"}
{"id": "7829.png", "formula": "\\begin{align*} s _ \\omega ^ \\rho ( x ) = \\inf \\{ \\omega ( a ^ * a ) ^ { 1 / 2 } \\| y \\| \\rho ( b ^ * b ) ^ { 1 / 2 } \\} , \\end{align*}"}
{"id": "6420.png", "formula": "\\begin{align*} [ Q , Y ( x ) + \\widehat { \\varrho ( x ) } ) ] = 0 . \\end{align*}"}
{"id": "7035.png", "formula": "\\begin{align*} \\mu _ 1 \\sin ( \\mu _ 1 \\pi ) = - \\beta _ 1 \\qquad \\mu _ 2 \\sin ( \\mu _ 2 \\pi ) = - \\beta _ 2 , \\end{align*}"}
{"id": "825.png", "formula": "\\begin{align*} w ( x ) = \\Psi ^ \\alpha ( x ) \\psi ( x ) v ( x ) \\end{align*}"}
{"id": "6766.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty p _ k ( n ) q ^ n = \\prod _ { j = 1 } ^ k \\frac 1 { 1 - q ^ j } . \\end{align*}"}
{"id": "7209.png", "formula": "\\begin{align*} \\chi _ 0 ( y , s ) : = \\left \\{ \\begin{array} { l l } 1 & \\mbox { i f } \\ | y | \\le s + R , \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
{"id": "4641.png", "formula": "\\begin{align*} A x = \\lambda x \\end{align*}"}
{"id": "449.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\min _ { 1 \\le i \\le k _ { n } } \\mu _ { n , i } ( ( 1 - \\varepsilon , 1 + \\varepsilon ) ) = 1 \\end{align*}"}
{"id": "6226.png", "formula": "\\begin{align*} & \\int \\frac { ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty } { ( q ^ { - ( n + 1 ) } x ^ 2 ; q ^ 2 ) _ \\infty } h _ n ( x ; q ) d _ q x = \\\\ & \\frac { ( x ^ 2 ; q ^ 2 ) _ \\infty } { [ n + 1 ] _ q ( q ^ { - ( n + 1 ) } x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( x h _ n ( \\frac { x } { q } ; q ) - q ^ { n } ( 1 - q ^ n ) h _ { n - 1 } ( \\frac { x } { q } ; q ) \\right ) , \\end{align*}"}
{"id": "7478.png", "formula": "\\begin{align*} \\cap _ { n = 0 } ^ { \\infty } \\mathrm { F i x } ( S _ { n } ) \\subset \\cap _ { n = 0 } ^ { j _ { f } } \\mathrm { F i x } ( S _ { n } ) \\subset \\cap _ { i = 1 } ^ { m } C _ { i } \\end{align*}"}
{"id": "8636.png", "formula": "\\begin{align*} \\frac { \\partial \\psi } { \\partial t } = - { \\rm i } \\widehat H _ 0 ( t ) \\psi , \\psi \\in \\mathcal { H } , \\end{align*}"}
{"id": "8152.png", "formula": "\\begin{align*} \\psi = \\Omega ^ - \\varphi \\end{align*}"}
{"id": "3647.png", "formula": "\\begin{align*} H ( m ) = \\theta \\frac { H ( m ) } { m } + ( m - \\theta ) \\frac { H ( m ) } { m } \\leq \\theta \\frac { H ( \\theta ) } { \\theta } + ( m - \\theta ) \\frac { H ( m - \\theta ) } { m - \\theta } . \\end{align*}"}
{"id": "3060.png", "formula": "\\begin{align*} c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast \\ast } = \\frac { \\Gamma \\left ( ( n - 1 ) / 2 \\right ) } { ( 2 \\pi ) ^ { ( n - 1 ) / 2 } } \\left [ \\int _ { 0 } ^ { \\infty } x ^ { ( n - 3 ) / 2 } \\overline { \\mathcal { G } } _ { n } \\left ( \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } + x \\right ) \\mathrm { d } x \\right ] ^ { - 1 } \\end{align*}"}
{"id": "1708.png", "formula": "\\begin{align*} I _ 2 \\le & \\frac { C ( \\ln N ) ^ 2 } { N } \\sum _ { k = N } ^ { N + \\log _ 4 ( \\ln 4 \\cdot \\ln N ) } \\iint \\phi ( \\xi ) \\phi ( \\xi - \\eta + 2 ^ { k } e _ 1 ) \\phi ( \\eta - 2 ^ { k } e _ 1 ) \\dd \\eta \\dd \\xi \\le \\frac { C ( \\ln N ) ^ 3 } { N } . \\end{align*}"}
{"id": "5557.png", "formula": "\\begin{align*} \\tau ( M ) = \\frac { 1 } { 1 2 \\pi ^ 2 } \\int _ M \\left ( | W _ + | ^ 2 - | W _ - | ^ 2 \\right ) d \\mu _ g \\end{align*}"}
{"id": "2421.png", "formula": "\\begin{align*} & - [ 2 m - 1 ] _ q P _ q ( 2 m - k - 2 , k - 1 ) + [ 2 m - 2 k ] _ q P _ q ( 2 m - k - 1 , k - 1 ) \\\\ & \\qquad \\qquad = ( - [ 2 m - 1 ] _ q [ 2 m - 2 k ] _ q + [ 2 m - k - 1 ] _ q [ 2 m - 2 k ] _ q ) P _ q ( 2 m - k - 2 , k - 2 ) \\\\ & \\qquad \\qquad = - q ^ { 2 m - k - 1 } [ k ] _ q [ 2 m - 2 k ] _ q P _ q ( 2 m - k - 2 , k - 2 ) . \\end{align*}"}
{"id": "7525.png", "formula": "\\begin{align*} P _ { s } ( u ) P _ { s - 1 } ( r ) - P _ { s } ( r ) P _ { s - 1 } ( u ) = 0 , \\end{align*}"}
{"id": "5189.png", "formula": "\\begin{align*} W ( x _ 1 , \\ldots , x _ a ) = \\sum _ { i = 1 } ^ a x _ i ^ { r _ i } , \\end{align*}"}
{"id": "5810.png", "formula": "\\begin{align*} n = 4 x _ 0 ^ 2 + 1 6 y ^ 2 + 4 8 z ^ 2 . \\end{align*}"}
{"id": "8604.png", "formula": "\\begin{align*} \\dot { x } ( t ) = & \\ ; ( J - R ) Q ( x ( t ) - C ^ { T } u ( t ) ) , \\\\ y ( t ) = & \\ ; C x ( t ) + D u ( t ) , \\end{align*}"}
{"id": "1705.png", "formula": "\\begin{align*} I I I _ j \\le \\frac { C 2 ^ { 2 j } 2 ^ { - 2 N } } { N ^ { \\frac { 1 } { 2 } } ( \\ln N ) ^ 4 } + C 2 ^ { \\frac { j } { 2 } } \\sum _ { k = 1 } ^ \\infty \\frac { k ( t _ 0 2 ^ { 2 N } ) ^ { k + 1 } } { ( k + 1 ) ! } \\le \\frac { C ( 2 ^ { 2 j } 2 ^ { - 2 N } + 2 ^ { { \\frac { 1 } { 2 } } j } ) } { N ^ { \\frac { 1 } { 2 } } ( \\ln N ) ^ 4 } . \\end{align*}"}
{"id": "8362.png", "formula": "\\begin{align*} \\Big \\langle \\sum \\limits _ { i = 1 } ^ { m } \\alpha _ i \\tilde { f } ^ { | u _ i \\rangle } \\Big | \\sum \\limits _ { i = 1 } ^ { m } \\beta _ j \\tilde { f } ^ { | v _ j \\rangle } \\Big \\rangle _ { { } _ { \\mathfrak { J } , t } } = \\sum \\limits _ { i , j = 1 } ^ { m } \\overline { \\alpha _ i } \\beta _ j \\langle u _ i | v _ j \\rangle _ t , \\end{align*}"}
{"id": "637.png", "formula": "\\begin{align*} & \\frac { F _ { 1 , 1 } ( x , y ) } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { 3 } } = \\frac { 2 } { 3 } \\pi ^ 2 \\frac { x ^ 4 + y ^ 4 } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { 3 } } , \\\\ & \\frac { \\widetilde { F } _ { 1 , 1 } ( x , y ) } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { 3 } } = \\frac { 2 } { 3 } \\pi ^ 2 \\frac { ( x ^ 2 + y ^ 2 ) ^ 2 } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { 3 } } , \\end{align*}"}
{"id": "144.png", "formula": "\\begin{align*} & t \\in ( 0 , \\infty ) \\mapsto \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } ; \\\\ & \\lim _ { t \\to 0 } \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } = 0 . \\end{align*}"}
{"id": "6862.png", "formula": "\\begin{align*} b ^ * = \\frac { 1 } { a _ 1 } ( a _ 1 g _ 0 + g _ 1 ( c ^ * - a _ 0 ) ) = g _ 0 + \\frac { g _ 1 } { a _ 1 } ( c ^ * - a _ 0 ) = \\beta ( c ^ * ) . \\end{align*}"}
{"id": "6707.png", "formula": "\\begin{align*} \\Gamma ( \\lambda + a ) = ( \\lambda + ( a - 1 ) ) \\Gamma ( \\lambda + ( a - 1 ) ) , \\ ; \\ ; \\ ; a \\in \\mathbb { Z } , \\lambda \\in \\mathfrak { a } _ \\mathbb { C } ^ * \\end{align*}"}
{"id": "3282.png", "formula": "\\begin{align*} u _ { A , q } ^ { \\infty } ( \\hat { x } , d ) = \\frac { 1 } { 4 \\pi } \\int _ D e ^ { - i k \\hat { x } \\cdot y } Q _ { A , q } u _ { A , q } ( y , d ) d y . \\end{align*}"}
{"id": "8009.png", "formula": "\\begin{align*} I _ { D _ { Z , - } , d } = \\frac { \\prod _ { 0 < a < v _ 1 \\cdot d } ( v _ 1 + a z ) } { \\prod _ { i \\in I _ - , D _ i \\cdot d > 0 } ( \\bar D _ i + ( D _ i \\cdot d ) z ) } [ \\textbf { 1 } ] _ { ( - D _ i \\cdot d ) _ { i \\in I _ - } , v _ 1 \\cdot d } . \\end{align*}"}
{"id": "6045.png", "formula": "\\begin{align*} f ^ { ( j ) } ( x ) = g _ { j } ( x ) \\prod _ { i = 1 } ^ { m } ( x - \\lambda _ i ) ^ { \\mu _ { i , j } } \\end{align*}"}
{"id": "966.png", "formula": "\\begin{align*} H ^ { * } ( \\overline { f } ) = \\left [ \\ \\overline { f } , H \\overline { f } , \\cdots , H ^ { n - 1 } \\overline { f } \\ \\right ] _ { n \\times n } \\in \\mathbb { R } ^ { n \\times n } \\end{align*}"}
{"id": "4908.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } | ( \\lambda _ i ( A ) E _ n - A ) _ { p p } | ^ { r o w } & = \\prod _ { k = 1 ; k \\ne i } ^ n ( \\lambda _ i ( A ) - \\lambda _ k ( A ) ) | v _ { i p } | ^ 2 , \\\\ - | ( \\lambda _ i ( A ) E _ n - A ) _ { q p } | ^ { r o w } & = \\prod _ { k = 1 ; k \\ne i } ^ n ( \\lambda _ i ( A ) - \\lambda _ k ( A ) ) v _ { i p } v _ { i q } , p \\ne q . \\end{array} \\right . \\end{align*}"}
{"id": "576.png", "formula": "\\begin{align*} \\mathbb { V } ( \\overline { T } _ n ) = \\frac { 1 } { n ^ 2 } \\Big ( \\sum _ { 1 \\leq i , j \\leq n } \\mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) - \\mathbb { E } ^ 2 ( \\sum _ { 1 \\leq i \\leq n } X _ i X _ { i + 1 } ) \\Big ) . \\end{align*}"}
{"id": "202.png", "formula": "\\begin{align*} H _ 0 : \\ ( X ^ { n } , Z ^ { n } _ { 1 } , Z ^ { n } _ { 2 } ) \\sim \\prod _ { i = 1 } ^ { n } p _ { X , Z _ 1 , Z _ 2 } , \\end{align*}"}
{"id": "5428.png", "formula": "\\begin{align*} 0 \\leq \\psi _ 0 \\leq 1 , \\psi _ 0 ( \\xi ) = 1 \\quad | \\xi | \\leq 1 \\psi _ 0 ( \\xi ) = 0 | \\xi | \\geq 2 \\end{align*}"}
{"id": "7648.png", "formula": "\\begin{align*} S = S _ { n , s , m } = D _ { n - 1 } ( \\varphi ) \\hookrightarrow \\mathbb P ^ s . \\end{align*}"}
{"id": "4525.png", "formula": "\\begin{align*} \\tilde { g } = 2 d \\tau d u + ( | \\nabla u | ^ 2 + a ^ 2 + b ^ 2 ) d u ^ 2 + 2 a d u d x _ 1 + 2 b d u d x _ 2 + d x _ 1 ^ 2 + d x _ 2 ^ 2 \\end{align*}"}
{"id": "3640.png", "formula": "\\begin{align*} \\limsup _ { \\ell \\to \\infty } \\sum _ { 0 \\le i < k _ { \\sigma ( \\ell ) } } \\mu _ { \\sigma ( \\ell ) } ( B ^ i _ { \\sigma ( \\ell ) } ) \\le \\sum _ { 0 \\le i < k } \\norm { \\mu ^ i } + \\limsup _ { \\ell \\to \\infty } ( \\ell \\wedge k ) 2 ^ { - \\ell } = \\sum _ { 0 \\le i < k } \\norm { \\mu ^ i } , \\end{align*}"}
{"id": "8996.png", "formula": "\\begin{align*} \\Delta _ n = \\frac { n - 2 } { n - 1 } \\Delta _ { n - 2 } - \\frac { 1 } { ( n - 2 ) ( n - 1 ) } \\chi _ { n - 2 } - \\frac { 1 } { ( n - 2 ) ( n - 1 ) } \\lambda _ 1 , n \\geq 3 . \\end{align*}"}
{"id": "1369.png", "formula": "\\begin{align*} \\sup _ { k , \\tau } \\sum _ { \\substack { | k _ 2 | \\gtrsim | k | \\\\ k _ 2 ^ * \\ll | k | \\\\ | \\Phi | \\gtrsim k ^ 2 } } \\frac { \\langle k \\rangle ^ { 2 s + 2 \\varepsilon - 2 + } } { \\prod _ { i = 1 } ^ p \\langle k _ i \\rangle ^ { 2 s } } \\lesssim \\langle k \\rangle ^ { 2 \\varepsilon - 2 + ( p - 1 ) \\max ( 1 - 2 s , 0 ) + } , \\end{align*}"}
{"id": "4517.png", "formula": "\\begin{align*} d ( F _ 1 d x _ 1 + F _ 2 d x _ 2 ) = & \\frac { \\partial F _ 1 } { \\partial x _ 2 } d x _ 2 \\wedge d x _ 1 + \\frac { \\partial F _ 2 } { \\partial x _ 1 } d x _ 1 \\wedge d x _ 2 \\\\ = & ( | \\nabla u | ^ { - 2 } A _ { 1 2 } - | \\nabla u | ^ { - 2 } A _ { 1 2 } ) d x _ 2 \\wedge d x _ 1 = 0 . \\end{align*}"}
{"id": "4317.png", "formula": "\\begin{align*} _ J ( L ) = \\int _ L e ^ { - i \\theta } \\Omega \\geq | \\int _ { L _ 1 } \\Omega | + | \\int _ { L _ 2 } \\Omega | . \\end{align*}"}
{"id": "7764.png", "formula": "\\begin{align*} Q _ j ^ k ( \\alpha _ i ) = \\sum _ { \\ell = 0 } ^ { n } \\beta _ { j , \\ell } ^ k \\ , \\alpha _ i ^ \\ell + \\tilde \\eta , k = 1 , \\ldots , m , \\ j = 1 , \\ldots , L , \\ i = 1 , \\ldots , p , \\end{align*}"}
{"id": "2634.png", "formula": "\\begin{align*} r ( K ) & = r | _ C ( K ) . \\end{align*}"}
{"id": "8104.png", "formula": "\\begin{align*} \\omega = p ^ { * } ( \\omega _ { 0 } ) \\wedge q ^ { * } ( \\mu ) . \\end{align*}"}
{"id": "7452.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\abs { w _ x } ^ 2 d x = o ( 1 ) \\int _ 0 ^ L \\abs { w } ^ 2 d x = o ( 1 ) . \\end{align*}"}
{"id": "3283.png", "formula": "\\begin{align*} u ^ s _ { A , q } ( x , y ) = - \\frac { k ^ 2 } { 4 \\pi } \\underset { \\underset { ( \\ell _ 2 , m _ 2 ) \\in \\Gamma } { ( \\ell _ 1 , m _ 1 ) \\in \\Gamma } } { \\sum } i ^ { \\ell _ 1 - \\ell _ 2 } \\mu _ { ( \\ell _ 1 , m _ 1 ; \\ell _ 2 , m _ 2 ) } h ^ { ( 1 ) } _ { \\ell _ 1 } ( k \\vert x \\vert ) h ^ { ( 1 ) } _ { \\ell _ 2 } ( k \\vert y \\vert ) Y ^ { m _ 1 } _ { \\ell _ 1 } \\left ( \\hat x \\right ) Y ^ { m _ 2 } _ { \\ell _ 2 } \\left ( \\hat y \\right ) , \\end{align*}"}
{"id": "1439.png", "formula": "\\begin{align*} & { \\mathrm { H } } ( \\boldsymbol { \\beta } ) = \\prod _ { v \\in { { \\mathfrak { M } } } _ K } \\max \\{ 1 , | \\beta _ 0 | _ v , \\ldots , | \\beta _ m | _ v \\} \\enspace , \\end{align*}"}
{"id": "3654.png", "formula": "\\begin{align*} \\Tilde { f } ( u , \\xi ) = \\bigl ( 1 + u ^ { p ( \\frac { 1 } { p ^ \\star } - 1 ) } \\bigr ) \\abs { \\xi } ^ p \\end{align*}"}
{"id": "807.png", "formula": "\\begin{align*} 1 + z l _ { 0 } '' ( z ) / l _ { 0 } ' ( z ) = \\psi ( z ) . \\end{align*}"}
{"id": "5420.png", "formula": "\\begin{align*} \\langle \\Lambda _ { \\gamma _ 1 } f , g \\rangle = \\langle \\Lambda _ { \\gamma _ 2 } f , g \\rangle \\Leftrightarrow \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) u ^ { ( 1 ) } _ f u ^ { ( 2 ) } _ g \\ , d x = 0 , \\end{align*}"}
{"id": "6581.png", "formula": "\\begin{align*} d ^ k ( f ' ) ( x , y _ 1 , \\ldots , y _ k ) \\ ; = \\ ; d ^ { k + 1 } f ( x , y _ 1 , \\ldots , y _ k , \\cdot ) \\colon E \\to F \\end{align*}"}
{"id": "4345.png", "formula": "\\begin{align*} m _ k ^ { b _ 0 , \\ldots b _ k } ( p _ k , \\ldots p _ 1 ) = \\sum m _ l ( b _ k , \\ldots b _ k , p _ k , b _ { k - 1 } , \\ldots , p _ { k - 1 } , \\ldots , p _ 1 , b _ 0 , \\ldots b _ 0 ) . \\end{align*}"}
{"id": "8764.png", "formula": "\\begin{align*} & q ^ { r - 4 c - 2 a } [ 2 \\ell + 1 - r ] \\frac { q ^ { 4 c + 2 - 2 r } K ^ { - 2 } - 1 } { q ^ { 2 - 2 r } K ^ { - 2 } - 1 } + q ^ { 1 + 2 \\ell - 4 c - a } [ a ] \\frac { q ^ { 4 c + 2 - 2 r } K ^ { - 2 } - 1 } { q ^ { 2 - 2 r } K ^ { - 2 } - 1 } \\\\ & + q ^ { - 2 c + r - 1 - 2 \\ell - a } [ r - 2 c - a ] + q ^ { - 2 c + 1 - 2 \\ell } [ 2 c ] \\frac { q ^ { - 2 a } K ^ { - 2 } - 1 } { q ^ { 2 - 2 r } K ^ { - 2 } - 1 } - q ^ { - 4 c + 1 } [ 2 \\ell ] \\frac { q ^ { 4 c } - 1 } { q ^ { 2 - 2 r } K ^ { - 2 } - 1 } \\\\ & = [ 2 \\ell + 1 ] . \\end{align*}"}
{"id": "5005.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) ) ^ 2 \\Psi ^ { n , 1 } _ s d s \\\\ & = \\kappa _ 3 \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "5980.png", "formula": "\\begin{align*} K _ 1 ( s ) = ( s ^ \\beta { + } A ) ^ { - 1 } . \\end{align*}"}
{"id": "7348.png", "formula": "\\begin{align*} & x \\ast [ y , [ z , u ] ] + y \\ast [ x , [ z , u ] ] & \\\\ = & x ( y [ z , u ] - [ z , u ] y ) + ( y [ z , u ] - [ z , u ] y ) x + y ( x [ z , u ] - [ z , u ] x ) + ( x [ z , u ] - [ z , u ] x ) y & \\\\ = & ( x y ) [ z , u ] - x ( [ z , u ] y ) + y ( [ z , u ] x ) - [ z , u ] ( y x ) & \\\\ & + ( y x ) [ z , u ] - y ( [ z , u ] x ) + x ( [ z , u ] y ) - [ z , u ] ( x y ) & \\\\ = & ( x y + y x ) [ z , u ] - [ z , u ] ( x y + y x ) & \\\\ = & [ x \\ast y , [ z , u ] ] . & \\end{align*}"}
{"id": "7726.png", "formula": "\\begin{align*} e ^ T B _ { S S } e \\geq e ^ T \\prod _ { t = 1 } ^ k A _ { S S } ( t ) e \\geq \\min _ { \\substack { \\forall t : X ( t ) \\geq \\gamma I , X ( t ) e \\leq e \\cr e ^ T \\sum _ { t = 1 } ^ k X ( t ) e \\geq k | S | - \\Lambda } } e ^ T \\prod _ { t = 1 } ^ k X ( t ) e \\geq ? ? \\end{align*}"}
{"id": "7598.png", "formula": "\\begin{align*} \\frac { d \\Gamma } { d t } = G _ 1 ( t ) + G _ 2 ( t ) \\Gamma + \\Gamma G _ 3 ( t ) + \\Gamma G _ 4 ( t ) \\Gamma , \\end{align*}"}
{"id": "4752.png", "formula": "\\begin{align*} \\rho = \\min \\left \\{ \\frac { m _ 0 } { 2 } , \\frac { m _ 1 } { 2 } \\sqrt { \\frac { m _ 0 } { 2 M _ 0 + m _ 0 } } , \\cdots , \\frac { m _ { \\infty } } { 2 } \\prod _ { j = 0 } ^ { n _ 0 - 1 } \\sqrt { \\frac { m _ j } { 2 M _ 0 + m _ j } } \\right \\} . \\end{align*}"}
{"id": "6306.png", "formula": "\\begin{align*} T _ q ( x ) : = D _ { q } u ( x ) + u ( x ) u ( q x ) + \\tilde { A } ( x ) u ( q x ) + r ( x ) . \\end{align*}"}
{"id": "4059.png", "formula": "\\begin{align*} & Q _ { \\vec q } = \\sum _ { i , j } \\varepsilon _ i \\varepsilon _ j q _ i q _ j ^ { - 1 } E _ i ^ j \\otimes E _ { - i } ^ { - j } , & & ( Q _ { \\vec q } ) ^ { i l } _ { j m } = \\varepsilon _ i \\varepsilon _ j q _ i q _ j ^ { - 1 } \\delta ^ { i , - l } \\delta _ { j , - m } , \\end{align*}"}
{"id": "4471.png", "formula": "\\begin{align*} b _ k = ( \\sum \\limits _ { i = 1 } ^ k \\frac 1 i ) ( k ! ) ^ { - 2 } 4 ^ { - k } , I _ 0 ( x ) = \\sum \\limits _ { k = 0 } ^ \\infty \\frac { 1 } { ( k ! ) ^ 2 4 ^ k } x ^ { 2 k } . \\end{align*}"}
{"id": "2242.png", "formula": "\\begin{align*} ( \\mathbf { U } ^ 0 \\cdot \\mathbf { n } ) | _ { Y = 0 } = ( \\mathbf { B } ^ 0 \\cdot \\mathbf { n } ) | _ { Y = 0 } = 0 . \\end{align*}"}
{"id": "5170.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { m - 2 } \\sum _ { k = 0 } ^ { m - 1 } \\prod _ { i _ 2 = j + 1 } ^ { m - 1 } \\rho _ { i _ 2 + k } + \\sum _ { k = 0 } ^ { m - 1 } 1 = \\sum _ { j = 0 } ^ { m - 2 } \\sum _ { k = 0 } ^ { m - 1 } \\prod _ { i _ 2 = j + 1 } ^ { m - 1 } \\rho _ { i _ 2 + k } + m . \\end{align*}"}
{"id": "3434.png", "formula": "\\begin{align*} & x _ 1 \\cdot \\frac { \\partial f } { \\partial x _ 1 } , . . . , x _ n \\cdot \\frac { \\partial f } { \\partial x _ n } , \\\\ & w _ { - \\alpha } ( f ) : = \\sum \\limits _ { m \\in \\Delta \\cap M } h t _ { - \\alpha } ( m ) \\cdot a _ m \\cdot x ^ { m - \\alpha } , \\alpha \\in R ( N , \\Sigma _ { C ( \\Delta ) } ) . \\end{align*}"}
{"id": "8840.png", "formula": "\\begin{align*} f _ { ( n ) } ( y ) : = \\left \\{ \\begin{aligned} & n f \\left ( \\frac { 1 } { n } \\right ) y & & \\ ; y \\leq \\frac { 1 } { n } \\\\ & f ( y ) & & \\ ; y \\geq \\frac { 1 } { n } \\end{aligned} \\right . . \\end{align*}"}
{"id": "5670.png", "formula": "\\begin{align*} F l _ { J } \\coloneqq X _ { w _ J } \\cong \\prod _ { k = 1 } ^ m F l _ { n _ k } , \\end{align*}"}
{"id": "8537.png", "formula": "\\begin{align*} S _ n & = \\big ( n + \\frac { 1 } { 2 } \\big ) - n + B ^ \\prime - r _ n \\\\ n ! & = C n ^ { n + 1 / 2 } e ^ { - n } e ^ { r _ n } \\end{align*}"}
{"id": "8437.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { \\varepsilon _ 2 \\to \\varepsilon _ 1 } \\mathop { \\sup } \\limits _ { ( \\xi , j ) \\in K \\times S } P \\left ( { \\| { u ^ { \\varepsilon _ 2 } \\left ( { t , s , \\xi , j } \\right ) - u ^ { \\varepsilon _ 1 } \\left ( { t , s , \\xi , j } \\right ) } \\| \\ge \\eta } \\right ) = 0 . \\end{align*}"}
{"id": "6384.png", "formula": "\\begin{align*} \\mathrm { S O T } - \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n U ( \\gamma _ i ) = A _ 0 . \\end{align*}"}
{"id": "4091.png", "formula": "\\begin{align*} E ( \\mathbf { R } _ n ) _ 2 = \\varepsilon n \\left [ \\frac { n - 1 } { 2 } a _ 2 + b _ 2 \\right ] , \\end{align*}"}
{"id": "6687.png", "formula": "\\begin{align*} & X _ t - \\Delta X + \\lambda | \\nabla X | ^ { \\alpha } = 2 M K _ 1 h ' ( t ) [ 1 - K _ 1 ( h ( t ) - r ) ] + 2 M K _ 1 ^ 2 - \\dfrac { N - 1 } { r ^ 2 } 2 M K _ 1 [ K _ 1 ( h ( t ) - r ) - 1 ] \\\\ & + \\lambda ( 2 M K _ 1 ) ^ { \\alpha } [ 1 - K _ 1 ( h ( t ) - r ) ] ^ { \\alpha } \\geq 2 M K _ 1 ^ 2 \\geq 2 M ( h _ 0 + 1 - K _ 1 ^ { - 1 } ) ^ { - \\gamma } K _ 1 ^ 2 ( r + 1 ) ^ { \\gamma } \\geq a v ^ p = u _ t - \\Delta u + \\lambda | \\nabla u | ^ { \\alpha } . \\end{align*}"}
{"id": "2484.png", "formula": "\\begin{align*} h ( F _ \\ast X , F _ \\ast Y ) = \\lambda ^ 2 ( p ) g ( X , Y ) ~ ~ X , Y \\in \\Gamma ( T _ p M ) . \\end{align*}"}
{"id": "8441.png", "formula": "\\begin{align*} R _ { \\beta } ( \\varphi ) = \\left \\{ x \\in [ 0 , 1 ] : | T _ { \\beta } ^ n x - x | < \\varphi ( n ) n \\in \\N \\right \\} , \\end{align*}"}
{"id": "3672.png", "formula": "\\begin{align*} H ^ { j ; 1 } _ { k , \\tilde { k } } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\mathcal { K } _ { \\tilde { k } , k } ( y , v , V ( s ) ) f ( s , X ( s ) - y , v ) \\varphi _ { j } ( v ) d y d v d s , \\end{align*}"}
{"id": "3157.png", "formula": "\\begin{align*} f _ { \\mathfrak { m } } ^ { \\sharp } \\doteq \\Delta _ { \\mathfrak { a } } + f _ { \\Phi } = \\Delta _ { \\mathfrak { a } } + e _ { \\Phi } - \\beta ^ { - 1 } s \\ . \\end{align*}"}
{"id": "7313.png", "formula": "\\begin{align*} \\alpha _ j & = - \\sum _ { i = 0 } ^ J { \\sf A } _ { i j } ^ { - 1 } T ^ { \\frac { \\gamma } { 2 } + j } \\int _ { - \\log T } ^ \\infty e ^ { ( \\frac { \\gamma } { 2 } + j ) \\tau _ 1 } ( T - t ) ( G , e _ i ) _ \\rho d \\tau _ 1 \\\\ & \\lesssim T ^ { \\frac { \\gamma } { 2 } + j } \\int _ 0 ^ T ( T - t ) ^ { - 1 + J - j + { \\sf c } _ 1 } d t \\lesssim T ^ { \\frac { \\gamma } { 2 } + J + { \\sf c } _ 1 } . \\end{align*}"}
{"id": "4467.png", "formula": "\\begin{align*} \\frac 1 { p _ 1 } + \\frac 1 { p _ 2 } + \\dotsb + \\frac 1 { p _ n } = \\frac 1 p , \\end{align*}"}
{"id": "1340.png", "formula": "\\begin{align*} \\phi ( x , v ) = \\left ( 1 - \\gamma m - \\beta \\gamma \\psi ( m ) m \\right ) e ^ { - \\gamma M ( x ) } \\end{align*}"}
{"id": "6257.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } q ^ { \\frac { n ( n - 3 ) } { 2 } } \\left ( 1 - q ^ 2 + q ^ { n + 2 } x \\right ) ( x ; q ) _ n A _ q ( q ^ n x ) = \\frac { x ( 1 + q ) - q } { x } A _ q ( \\frac { x } { q } ) - x A _ q ( q x ) . \\end{align*}"}
{"id": "274.png", "formula": "\\begin{align*} Z ^ { \\nu } _ i : = B ^ { - 1 } ( D ^ { \\nu t } _ i + F _ i ^ { \\nu t } ) \\end{align*}"}
{"id": "6624.png", "formula": "\\begin{align*} \\left | \\sum _ { | c | > h ^ \\delta } \\sum _ { k = 1 } ^ \\infty k ^ { - 4 } S _ { k } ( c ; \\xi ) I _ { k , \\ell } ( { } ^ t g c ) \\right | \\ll _ { w , \\ell , \\theta } \\| g \\| _ E ^ \\theta h ^ { 5 - ( \\theta - 4 ) \\delta } . \\end{align*}"}
{"id": "7733.png", "formula": "\\begin{align*} \\tilde B _ { i i } ( k _ 1 : k _ 0 ) & \\geq \\eta _ i 0 \\leq k _ 0 \\leq k _ 1 \\leq \\tilde k _ L , \\cr \\tilde B _ { j j } ( k _ 1 : k _ 0 ) & \\geq \\tilde \\eta _ j 0 \\leq k _ 0 \\leq k _ 1 \\leq \\sigma , \\cr \\sum _ { k = 0 } ^ { \\sigma - 1 } B _ { j i } ( k ) & \\geq \\tilde \\delta , \\end{align*}"}
{"id": "956.png", "formula": "\\begin{align*} | x | = \\left ( \\sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } , \\ \\ \\ x ^ { \\prime } = \\left ( x _ { 1 } , x _ { 2 } , \\cdots , x _ { n } \\right ) \\in \\mathbb { R } ^ { n } . \\end{align*}"}
{"id": "570.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ { i + 1 } ) = ( 1 - \\alpha ) M _ 1 + \\alpha M _ 2 , \\end{align*}"}
{"id": "8135.png", "formula": "\\begin{align*} & \\eta _ { x , p ; \\delta } ( y ) = e ^ { i p ( y - x ) } \\eta _ \\delta ( y - x ) \\end{align*}"}
{"id": "3328.png", "formula": "\\begin{align*} \\Delta _ { \\hat { \\beta } } ( t ) = \\frac { 1 - t } { 1 - t ^ n } d e t ( { \\mathcal B } _ { - t } ( \\beta ) - I ) \\end{align*}"}
{"id": "8850.png", "formula": "\\begin{align*} d \\lambda _ i ( t ) = \\sqrt { 2 } d B _ i ( t ) + { \\beta } \\sum _ { j ( \\neq i ) } \\frac { 1 } { \\lambda _ i ( t ) - \\lambda _ j ( t ) } d t , \\ \\ \\beta > 0 , \\ \\ i = 1 , \\dots , N . \\end{align*}"}
{"id": "3697.png", "formula": "\\begin{align*} \\{ | ( \\xi ' , \\xi _ n ) | _ x ^ 2 + V ( x ) - E ( h ) , \\chi _ \\alpha ( x _ n ) \\xi _ n \\} = 2 \\chi _ \\alpha ' ( x _ n ) \\xi _ n ^ 2 - \\chi _ \\alpha ( x _ n ) \\partial _ { x _ n } ( R + V ) , \\end{align*}"}
{"id": "4087.png", "formula": "\\begin{align*} p _ { n + 1 } ( j ) = p _ n ( j ) + \\gamma f ( j ) . \\end{align*}"}
{"id": "2291.png", "formula": "\\begin{align*} v _ { e x } ^ i - u _ { e Y } ^ i = 0 , u _ { e x } ^ i + v _ { e Y } ^ i = 0 . \\end{align*}"}
{"id": "4391.png", "formula": "\\begin{align*} 2 \\mathrm { R e } ( \\bar \\partial _ { \\Phi } ^ * \\alpha , \\alpha \\llcorner ( \\bar \\partial \\eta ) ^ { \\sharp } ) _ { D _ j , \\Phi } \\geq & - \\int _ { D _ j } g ^ { - 1 } | \\bar \\partial _ { \\Phi } ^ * \\alpha | ^ 2 e ^ { - \\Phi } \\\\ & + \\sum _ { j , k = 1 } ^ n \\int _ { D _ j } - g ( \\partial _ j \\eta ) ( \\bar \\partial _ k \\eta ) \\alpha _ { \\bar j } \\bar \\alpha _ { \\bar k } e ^ { - \\Phi } , \\end{align*}"}
{"id": "6637.png", "formula": "\\begin{align*} \\Phi ( H ) = \\left [ \\begin{array} { c c } H & 0 \\\\ 0 & H \\end{array} \\right ] \\quad { \\rm a n d \\ h e n c e } \\Phi ^ \\dagger \\left ( \\left [ \\begin{array} { c c } Y _ { 1 } & 0 \\\\ 0 & Y _ { 2 } \\end{array} \\right ] \\right ) = Y _ { 1 } + Y _ { 2 } \\ . \\end{align*}"}
{"id": "4310.png", "formula": "\\begin{align*} \\mathcal { S } _ { L _ 0 } ( L ) = \\mathcal { S } _ { L _ 0 ' } ( L ) + \\mathcal { S } _ { L _ 0 } ( L _ 0 ' ) . \\end{align*}"}
{"id": "3257.png", "formula": "\\begin{align*} - \\Delta u ^ s - k ^ 2 u ^ s = Q _ { A , q } ( u ^ s + v ) \\mathrm { i n } \\ ; \\R ^ 3 , \\end{align*}"}
{"id": "1296.png", "formula": "\\begin{align*} \\mathcal { G } ( r ( T ) , x ) & = \\dfrac { P ( T , x ) } { \\displaystyle \\prod _ { j = 1 } ^ { d } P ( T _ j , x ) } . \\end{align*}"}
{"id": "2964.png", "formula": "\\begin{align*} \\varphi ( z ) = 2 \\sqrt y \\sum _ { n \\neq 0 } a _ \\varphi ( n ) K _ { i r } ( 2 \\pi | n | y ) e ( n x ) \\end{align*}"}
{"id": "5735.png", "formula": "\\begin{align*} \\pi _ i \\coloneqq \\pi _ { \\{ i \\} } = y _ 1 + y _ 2 + \\cdots + y _ i ( 1 \\le i \\le n - 1 ) \\end{align*}"}
{"id": "593.png", "formula": "\\begin{align*} \\sigma ( \\Delta _ n ) = \\bigg \\{ \\left . \\frac { n ^ 2 } { \\pi ^ 2 } \\sum _ { j = 1 } ^ \\alpha \\sin ^ 2 \\Big ( \\frac { \\pi k _ j } { n } \\Big ) \\right | \\forall _ { j = 1 , \\ldots , \\alpha } : k _ j = 0 \\ , , \\cdots , n - 1 \\bigg \\} \\end{align*}"}
{"id": "5259.png", "formula": "\\begin{align*} g : = g _ { 2 ^ { | I | } - 1 } \\circ \\cdots \\circ g _ 1 . \\end{align*}"}
{"id": "8709.png", "formula": "\\begin{align*} \\gamma _ 0 = ( a \\cos t , a \\sin t ) \\end{align*}"}
{"id": "8021.png", "formula": "\\begin{align*} 0 \\longrightarrow \\tilde { \\mathbb { L } } : = ( \\tilde { \\beta } ) \\longrightarrow \\mathbb { Z } ^ { m + 2 } \\stackrel { \\tilde { \\beta } } { \\longrightarrow } N \\oplus \\mathbb Z \\longrightarrow 0 , \\end{align*}"}
{"id": "3219.png", "formula": "\\begin{align*} \\dim \\mathcal { M } _ { } ( 2 m , \\alpha , L ) = m ( 2 m + 1 ) ( g - 1 ) + m ^ 2 n , \\end{align*}"}
{"id": "6393.png", "formula": "\\begin{align*} k = b _ 1 > b _ 2 > \\cdots > b _ { \\ell } . \\end{align*}"}
{"id": "4010.png", "formula": "\\begin{align*} t _ { k } ^ { ( h ) } = h + 2 H k = t _ { k - 1 } ^ { ( h ) } + 2 H . \\end{align*}"}
{"id": "1985.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / 2 } } \\big ( | \\nabla U _ k | ^ 2 - | \\nabla V _ { k , r } | ^ 2 \\big ) \\ , d x & = \\int _ { B _ { 1 / 2 } } \\big ( \\nabla ( U _ k - V _ { k , r } ) \\cdot \\nabla ( U _ k + V _ { k , r } ) \\big ) \\ , d x . \\end{align*}"}
{"id": "6300.png", "formula": "\\begin{align*} & \\int \\frac { x ( q x ; q ) _ \\infty } { ( q r _ n x ; q ) _ \\infty } p _ n ( x | q ) d _ q x \\\\ & = \\frac { q ^ { n } x ( x ; q ) _ \\infty } { [ n ] _ q [ n + 1 ] _ q ( r _ n x ; q ) _ \\infty } \\left ( \\frac { 1 } { q } ( 1 - x ) \\ , _ 2 \\phi _ 1 ( q ^ { - n + 1 } , q ^ { n + 2 } ; q ^ 2 ; q , x ) - p _ n ( \\frac { x } { q } | q ) \\right ) . \\end{align*}"}
{"id": "3111.png", "formula": "\\begin{align*} 0 < 1 - \\lambda _ h ( j ) \\kappa _ { m } ^ 2 h _ { \\max } ^ { 2 { m } } \\le 1 - \\kappa _ { m } ^ 2 h _ { \\max } ^ { 2 m } \\Vert w _ { \\mathrm { p w } } \\Vert _ { L ^ 2 ( \\Omega ) } \\le \\Vert v _ { \\mathrm { p w } } - \\kappa _ { m } ^ 2 h _ { \\mathcal { T } } ^ { 2 m } w _ { \\mathrm { p w } } \\Vert _ { L ^ 2 ( \\Omega ) } = \\Vert v _ { \\mathrm { n c } } \\Vert _ { L ^ 2 ( \\Omega ) } . \\end{align*}"}
{"id": "4303.png", "formula": "\\begin{align*} Z _ { V } ( E ) = - \\int _ V e ^ { - \\sqrt { - 1 } \\omega _ { X ^ \\vee } } c h ( E ) \\in \\R _ { > 0 } e ^ { \\sqrt { - 1 } \\phi _ V ( E ) } , \\phi _ V ( E ) \\in ( 0 , \\pi ) , \\end{align*}"}
{"id": "557.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i ) = \\sum _ { \\substack { 0 \\leq l \\leq i \\\\ \\gcd ( l , i ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } C _ { \\alpha } ( i , l ) = f _ k ( i ) + O _ \\alpha \\Big ( \\frac { \\tau _ 3 ( i ) } { \\sqrt { i } } \\Big ) , \\end{align*}"}
{"id": "2296.png", "formula": "\\begin{align*} & ( \\tilde { u _ p ^ n } , \\tilde { v _ p ^ n } , \\tilde { h _ p ^ n } , \\tilde { g _ p ^ n } ) | _ { y \\rightarrow \\infty } = ( 0 , 0 , 0 , 0 ) , \\\\ & ( \\tilde { u _ p ^ n } , \\tilde { v _ p ^ n } , \\tilde { h _ p ^ n } , \\tilde { g _ p ^ n } ) | _ { y = 0 } = ( - \\overline u _ e ^ n ( x ) , 0 , - \\overline h _ e ^ n ( x ) , 0 ) . \\end{align*}"}
{"id": "2895.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ { k } ) = x _ 1 + x _ 2 + x _ { k - 1 } + x _ { k } + \\sum _ { i = 1 } ^ { k - 2 } x _ i x _ { i + 1 } . \\end{align*}"}
{"id": "284.png", "formula": "\\begin{align*} \\begin{array} { r c l l l } \\Lambda ^ { ( s ) t } _ { \\pi , i } A _ i ^ { ( s ) } \\Lambda _ { \\pi , i } ^ { ( s ) } & = & B ^ { ( s ) } , & \\ & \\ \\\\ \\ & \\ & \\ & \\ & \\ \\\\ ( \\Lambda ^ { ( s ) } _ { \\pi , i } - 1 ) _ { k j } - ( \\Lambda ^ { ( s ) } _ { \\pi , j } - 1 ) _ { k i } & = & \\pi \\cdot L ^ { k ( s ) } _ { \\pi , i j } ( \\Lambda ^ { ( s ) } _ { \\pi } ) & = & p \\cdot L ^ { k ( s ) } _ { p , i j } ( \\Lambda ^ { ( s ) } _ { \\pi } ) . \\end{array} \\end{align*}"}
{"id": "648.png", "formula": "\\begin{align*} \\Omega ( s ) = \\frac { 1 } { 3 } \\ , s ( s - 1 ) \\ , \\pi \\ , \\xi _ 2 ( s - 1 ) . \\end{align*}"}
{"id": "5337.png", "formula": "\\begin{align*} \\nabla ^ s u ( x , y ) = \\sqrt { \\frac { C _ { n , s } } { 2 } } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { n / 2 + s + 1 } } ( x - y ) , \\end{align*}"}
{"id": "4390.png", "formula": "\\begin{align*} v _ { \\epsilon } ( t ) : = \\int _ { 0 } ^ { t } \\left ( \\int _ { - \\infty } ^ { t _ { 1 } } \\left ( \\frac { 1 } { B - 4 \\epsilon } \\mathbb { I } _ { ( - t _ { 0 } - B + 2 \\epsilon , - t _ { 0 } - 2 \\epsilon ) } * \\rho _ { \\frac { 1 } { 4 } \\epsilon } \\right ) ( s ) d s \\right ) d t _ { 1 } , \\end{align*}"}
{"id": "3188.png", "formula": "\\begin{align*} \\sum _ { z \\in \\mathbb { Z } ^ { d } } \\left \\vert f \\left ( z \\right ) \\right \\vert = \\left \\vert f \\left ( 0 \\right ) \\right \\vert + \\sum _ { n = 1 } ^ { d } \\sum _ { \\mathcal { I } \\in 2 ^ { \\left \\{ 1 , \\dots , d \\right \\} } : \\left \\vert \\mathcal { I } \\right \\vert = n } \\ \\sum _ { z \\in \\mathbb { Z } ^ { n } \\backslash \\left \\{ 0 \\right \\} } \\left \\vert f \\circ \\xi _ { \\mathcal { I } } \\left ( z \\right ) \\right \\vert \\ . \\end{align*}"}
{"id": "5775.png", "formula": "\\begin{align*} E ( G / S ' ) = E ( G / H ) E ( H / S ) \\neq 0 . \\end{align*}"}
{"id": "8533.png", "formula": "\\begin{align*} T _ 0 & = 1 ; \\\\ T _ 1 ( - ; p ) & = p ; \\\\ T _ 2 ( a _ 2 ; p ) & = p ^ 2 + a _ 2 ; \\\\ T _ 3 ( a _ 2 , a _ 3 ; p ) & = p ^ 3 + p ( a _ 2 + a _ 3 ) ; \\\\ T _ 4 ( a _ 2 , a _ 3 , a _ 4 ; p ) & = p ^ 4 + ( a _ 2 + a _ 3 + a _ 4 ) p ^ 2 + a _ 2 a _ 4 ; \\\\ T _ 5 ( a _ 2 , a _ 3 , a _ 4 , a _ 4 ; p ) & = p ^ 5 + ( a _ 2 + a _ 3 + a _ 4 + a _ 5 ) p ^ 3 + ( a _ 2 a _ 4 + a _ 2 a _ 3 + a _ 3 a _ 5 ) p . \\end{align*}"}
{"id": "5649.png", "formula": "\\begin{align*} \\Big | \\bar { u } _ R ( y ) - \\frac { 1 } { 2 } y ^ T A y \\Big | = | w _ R ( y ) | \\leq \\frac { 1 6 c _ 1 } { R ^ 2 } \\Big | x + \\frac { R } { 4 } y \\Big | ^ { 2 - \\varepsilon } \\leq C ( A , \\varepsilon , c _ 1 ) R ^ { - \\varepsilon } , \\end{align*}"}
{"id": "8062.png", "formula": "\\begin{align*} \\operatorname { H e s s } _ { p } ( V , W ) : = g ( \\nabla _ { V } \\operatorname { g r a d } ( p ) , W ) \\end{align*}"}
{"id": "5985.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } j ^ { - \\beta - 1 } ( n - j ) ^ { \\beta - 1 } = O ( n ^ { \\beta - 1 } ) . \\end{align*}"}
{"id": "5687.png", "formula": "\\begin{align*} ( d , 1 ^ k ) = ( d , \\underbrace { 1 , 1 , \\ldots , 1 } _ { k } , 0 , 0 , \\cdots ) . \\end{align*}"}
{"id": "2843.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { h \\big [ \\kappa h ^ 2 - 2 h ( 1 + \\kappa ) + 4 \\big ] } { 2 L \\big [ 2 - h ( 1 + \\kappa ) \\big ] } \\min \\{ \\| g _ 0 \\| ^ 2 \\ , , \\ , \\| g _ 1 \\| ^ 2 \\} { } \\leq { } f _ { 0 } - f _ { 1 } , \\end{aligned} \\end{align*}"}
{"id": "7172.png", "formula": "\\begin{align*} V [ g ] = V [ g ] _ - \\oplus V [ g ] _ 0 \\oplus V [ g ] _ + , \\end{align*}"}
{"id": "2079.png", "formula": "\\begin{align*} \\hat { S } _ n ( f ) = \\varphi \\left ( X _ 1 , \\ , f ( X _ 1 ) , \\ , \\hdots , \\ , X _ n , \\ , f ( X _ n ) \\right ) , \\end{align*}"}
{"id": "7210.png", "formula": "\\begin{align*} I ( x , t ) : = \\int _ { 0 } ^ { t } \\int _ { x - t + s } ^ { x + t - s } \\frac { \\chi _ 0 ( y , s ) } { ( 1 + | s + | y | | ) ^ { 1 + a } ( 1 + | s - | y | | ) ^ { 1 + b } } d y d s . \\end{align*}"}
{"id": "5442.png", "formula": "\\begin{align*} s _ 1 + s _ 2 - s _ 0 = s , \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } - \\frac { 1 } { p _ 0 } = \\frac { s } { n } . \\end{align*}"}
{"id": "340.png", "formula": "\\begin{align*} c _ 0 h + c _ 1 g _ 1 + \\ldots + c _ n g _ n = 0 , \\end{align*}"}
{"id": "5277.png", "formula": "\\begin{align*} \\Gamma _ { P ^ { C o n t } } : = \\bigsqcup _ { j = 0 } ^ h \\Gamma _ { 0 , k _ 1 ( j ) , k _ 2 ( j ) , 1 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I _ j } } . \\end{align*}"}
{"id": "4839.png", "formula": "\\begin{align*} \\tau ^ \\mathrm { a n / t o p } ( M / S , \\rho \\oplus \\rho ' ) = \\tau ^ \\mathrm { a n / t o p } ( M / S , \\rho ) + \\tau ^ \\mathrm { a n / t o p } ( M / S , \\rho ' ) \\end{align*}"}
{"id": "8445.png", "formula": "\\begin{align*} [ 0 , 1 ] = \\bigcup _ { w \\in \\Sigma _ { \\beta } ^ n } I _ { n , \\beta } ( w ) \\end{align*}"}
{"id": "4693.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ m ( n ) \\ , q ^ n = \\sum _ { k = 0 } ^ \\infty \\frac { q ^ { m k } } { ( q ; q ) ^ 2 _ k } . \\end{align*}"}
{"id": "6509.png", "formula": "\\begin{align*} u = \\frac { \\partial u } { \\partial \\nu } = 0 \\Gamma _ 1 . \\end{align*}"}
{"id": "3387.png", "formula": "\\begin{align*} g ( x , y ) : = x + \\gamma \\frac { \\| y \\| _ d ^ 2 } { \\tilde b ( x ) } . \\end{align*}"}
{"id": "2904.png", "formula": "\\begin{align*} C _ k ( a _ 1 , \\ldots , a _ n ) = \\begin{pmatrix} a _ 1 & a _ 2 & \\cdots & a _ n \\\\ a _ { n - k + 1 } & a _ { n - k + 2 } & \\cdots & a _ { n - k } \\\\ a _ { n - 2 k + 1 } & a _ { n - 2 k + 2 } & \\cdots & a _ { n - 2 k } \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ a _ { k + 1 } & a _ { k + 2 } & \\cdots & a _ k \\end{pmatrix} . \\end{align*}"}
{"id": "2667.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 2 } p ^ { n - 3 } ( p - 2 ) ( p - 3 ) \\binom { n - k - 2 } { 2 } & = p ^ { n - 3 } ( p - 2 ) ( p - 3 ) \\sum _ { i = 0 } ^ { n - 3 } \\binom { i } { 2 } \\\\ & = p ^ { n - 3 } ( p - 2 ) ( p - 3 ) \\binom { n - 2 } { 3 } \\end{align*}"}
{"id": "560.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i \\leq n } \\mathbb { E } ( { X _ i } ^ 2 ) = \\sum _ { 1 \\leq i \\leq n } \\mathbb { E } ( { X _ i } ) = O ( n ) , \\end{align*}"}
{"id": "287.png", "formula": "\\begin{align*} t \\ge \\tau : = 1 . 1 4 0 3 \\cdots , \\end{align*}"}
{"id": "1832.png", "formula": "\\begin{align*} A _ { n } = \\sum _ { \\gamma \\in \\mathcal { M } _ { n } } w ( \\gamma ) = \\sum _ { \\gamma \\in \\mathcal { A } _ { n } } w ( \\gamma ) + \\sum _ { \\gamma \\in \\mathcal { B } _ { n } } w ( \\gamma ) . \\end{align*}"}
{"id": "8169.png", "formula": "\\begin{align*} \\sup \\limits _ { y } | V ( x , y ) | = \\begin{cases} | x | ^ { - 2 } & | x | > 1 \\\\ 1 & | x | < 1 \\end{cases} \\end{align*}"}
{"id": "2362.png", "formula": "\\begin{align*} \\nu _ \\sigma \\left ( f - l ( h _ \\sigma ) \\right ) = \\beta _ b + b \\gamma _ \\sigma \\end{align*}"}
{"id": "2231.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { \\partial \\hat { \\rho } ( t , 0 ) } { \\partial t } = \\frac { i } { \\hbar } [ \\hat { \\rho } ( t , 0 ) , \\hat { H } ( t ) ] = \\left . \\frac { i } { \\hbar } \\partial _ s ^ 2 { \\rm L o g } \\left ( e ^ { \\int \\hat { \\rho } ( t , s ) d s } e ^ { \\int \\hat { H } ( t ) d s } \\right ) \\right | _ { s = 0 } , \\end{array} \\end{align*}"}
{"id": "433.png", "formula": "\\begin{align*} Q = \\{ s + i t : s \\in [ A , B ] , | t - f _ { \\nu } ( s _ { 0 } ) | \\le \\varepsilon \\} \\end{align*}"}
{"id": "3104.png", "formula": "\\begin{align*} \\delta : = \\frac { 1 } { 1 - \\lambda _ h \\kappa _ { m } ^ 2 h _ { \\mathcal { T } } ^ { 2 { m } } } - 1 = \\frac { \\lambda _ h \\kappa _ { m } ^ 2 h _ { \\mathcal { T } } ^ { 2 { m } } } { 1 - \\lambda _ h \\kappa _ { m } ^ 2 h _ { \\mathcal { T } } ^ { 2 { m } } } = \\lambda _ h \\kappa _ { m } ^ 2 h _ { \\mathcal { T } } ^ { 2 { m } } ( 1 + \\delta ) \\in P _ 0 ( \\mathcal { T } ) . \\end{align*}"}
{"id": "1794.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m i n i m i z e } & f ( x ) + g ( y ) + h ( x ) \\\\ \\mbox { s u b j e c t t o } & A x = y , \\end{array} \\end{align*}"}
{"id": "1435.png", "formula": "\\begin{align*} W H W ^ T = d i a g ( m _ 1 S , \\dots , m _ N S , 0 , \\dots , 0 ) . \\end{align*}"}
{"id": "5522.png", "formula": "\\begin{align*} \\frac { \\abs { f ' ( z ) } } { \\abs { f ' ( z _ 0 ) } } = \\frac { \\abs { g ' ( \\textbf { z } ) } } { \\abs { g ' ( \\textbf { z } _ 0 ) } } \\frac { \\abs { \\textbf { z } + \\lambda i } ^ 2 } { \\abs { \\textbf { z } _ 0 + \\lambda i } ^ 2 } \\geq \\frac { \\abs { g ' ( \\textbf { z } ) } } { \\abs { g ' ( \\textbf { z } _ 0 ) } } \\approx 1 , \\end{align*}"}
{"id": "5110.png", "formula": "\\begin{align*} \\Lambda ^ { ( 2 ) } _ n = \\Lambda ^ { ( 3 ) } _ { n , \\delta } + \\Lambda ^ { ( 4 ) } _ { n , \\delta } , \\end{align*}"}
{"id": "9003.png", "formula": "\\begin{align*} P _ { n - 2 r , n } \\left ( \\frac { n } { 2 } \\right ) = P _ { 2 r - 1 } \\left ( n - 1 , \\frac { n } { 2 } \\right ) = 0 . \\end{align*}"}
{"id": "3557.png", "formula": "\\begin{align*} J m = \\sum \\limits _ { j = 1 } ^ n \\binom { n - 1 } { j - 1 } c _ j . \\end{align*}"}
{"id": "8836.png", "formula": "\\begin{align*} f \\circ ( \\phi _ 1 * \\phi _ 2 ) ( g ) = f _ + ' ( 0 ) \\phi _ 1 * \\phi _ 2 ( g ) + \\int _ { 0 } ^ { \\infty } f _ t \\circ ( \\phi _ 1 * \\phi _ 2 ) ( g ) d \\nu ( t ) \\end{align*}"}
{"id": "1644.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x } ( 4 p _ t - 6 p p _ x - p _ { x x x } ) \\ , = \\ , 3 p _ { y y } , \\end{align*}"}
{"id": "2074.png", "formula": "\\begin{align*} \\label [ I e q ] { t r a n s i t i o n s e m i } T _ t \\varphi ( x ) : = \\int _ E \\varphi ( y ) P _ t ( x , \\dd y ) , \\end{align*}"}
{"id": "3559.png", "formula": "\\begin{align*} J m & = ( 1 / 2 ^ n ) J M a \\\\ & = ( 1 / 2 ^ n ) [ 2 ^ { n - 1 } , - 2 ^ { n - 1 } , 0 , \\ldots , 0 ] { } ^ t [ \\chi _ { \\pi } ( h _ 0 ) , \\chi _ { \\pi } ( h _ 1 ) , \\ldots , \\chi _ { \\pi } ( h _ n ) ] \\\\ & = ( 1 / 2 ) ( \\chi _ { \\pi } ( h _ 0 ) - \\chi _ { \\pi } ( h _ 1 ) ) . \\\\ \\end{align*}"}
{"id": "7226.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left \\{ \\log \\left ( \\frac { 1 + t - x } { 1 + l _ n R } \\right ) \\right \\} ^ { a _ n } \\quad \\mbox { i n } \\ D _ { n } \\end{align*}"}
{"id": "7722.png", "formula": "\\begin{align*} \\pi _ { S } ^ T ( k + 1 ) A _ { S \\bar { S } } ( k ) e + \\pi _ { \\bar { S } } ^ T ( k + 1 ) \\left ( e - A _ { \\bar { S } S } ( k ) e \\right ) = \\pi _ { \\bar { S } } ^ T ( k ) e \\end{align*}"}
{"id": "105.png", "formula": "\\begin{align*} \\dot \\kappa ^ { \\epsilon , L } _ t : = \\frac { 1 } { t } - \\frac { \\chi ^ { \\epsilon , L } _ t ( \\lambda , \\mu , m ^ 2 ) } { t ^ 2 } . \\end{align*}"}
{"id": "2099.png", "formula": "\\begin{align*} ( L ( x ) , x ) _ V & = \\sum \\limits _ { v \\in V } \\delta _ V ( v ) [ \\mathcal { Q } ( x ) ( v ) - n ( v ) x ( v ) ) ] x ( v ) \\\\ & = \\sum \\limits _ { v \\in V } \\delta _ V ( v ) [ ( a v g ^ * \\circ a v g ) ( x ) ( v ) - n ( v ) x ( v ) ) ] x ( v ) \\\\ & = \\sum \\limits _ { v \\in V } \\sum \\limits _ { e \\in E _ v } \\frac { \\delta _ E ( e ) } { | e | } [ ( a v g ( x ) ) ( e ) - x ( v ) ) ] x ( v ) . \\end{align*}"}
{"id": "1946.png", "formula": "\\begin{align*} \\langle \\mathcal { H } _ { q } ^ { n } \\ , e _ { r } , e _ { 0 } \\rangle = \\sum _ { \\gamma \\in \\mathcal { S } _ { 1 } ( n , r ) } w ( \\gamma ) \\end{align*}"}
{"id": "1323.png", "formula": "\\begin{align*} G _ V ( u ) = \\int _ 1 ^ u \\frac { \\d s } { V ( s ) } = \\int _ 1 ^ u \\frac { \\d s } { 1 + s ^ { \\xi } } \\sim 1 + u ^ { 1 - \\xi } , \\end{align*}"}
{"id": "2888.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d t _ i s _ i = n . \\end{align*}"}
{"id": "3922.png", "formula": "\\begin{align*} | k ( t _ j ) | \\le V a r ( \\mathbb { K } _ h ( x - X _ 0 ) ) ^ \\frac { 1 } { 2 } V a r ( \\mathbb { K } _ h ( x - X _ { t _ j } ) ) ^ \\frac { 1 } { 2 } \\le \\int _ { \\mathbb { R } ^ d } ( \\mathbb { K } _ h ( x - y ) ) ^ 2 \\pi ( y ) d y \\le \\frac { c } { \\prod _ { l = 1 } ^ d h _ l } . \\end{align*}"}
{"id": "3705.png", "formula": "\\begin{align*} \\phi _ h ( x _ 1 , x _ 2 ) = e ^ { \\frac { i } h x _ 1 } , h ^ { - 1 } \\in 2 \\pi \\mathbb { Z } . \\end{align*}"}
{"id": "1312.png", "formula": "\\begin{align*} f _ \\infty ( x , v ) = \\frac { \\rho } { ( 2 \\pi T ) ^ { d / 2 } } e ^ { - \\frac { | v - u | ^ 2 } { 2 T } } : = M _ { \\rho , T , u } ( x , v ) , \\end{align*}"}
{"id": "7025.png", "formula": "\\begin{align*} d _ i & = d _ 0 + U ( - \\Delta _ d , \\Delta _ d ) , \\\\ s _ i & = s _ 0 + U ( - \\Delta _ s , \\Delta _ s ) , \\end{align*}"}
{"id": "8639.png", "formula": "\\begin{align*} \\frac { d U } { d t } ( t ) U ^ \\dagger ( t ) = P ( { \\rm A d } _ { U ( t ) } { \\rm i } \\widehat H ( t ) ) . \\end{align*}"}
{"id": "3066.png", "formula": "\\begin{align*} & \\mathrm { M D T C o v } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { X } ) = \\mathrm { E } \\left [ ( \\mathbf { X } - \\mathrm { M D T E } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { X } ) ) ( \\mathbf { X } - \\mathrm { M D T E } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { X } ) ) ^ { T } | \\boldsymbol { a } < \\mathbf { X } \\leq \\boldsymbol { b } \\right ] . \\end{align*}"}
{"id": "7646.png", "formula": "\\begin{align*} e _ { \\mathrm { t o p } } ( S _ { s , n } ) = \\begin{cases} 4 n ^ 2 - 2 n ^ 3 + ( 3 n - 4 ) \\binom { n } { 2 } - \\binom { n } { 3 } & \\textrm { i f } s = 3 \\\\ n ^ 2 ( 1 0 - 1 0 n + 3 n ^ 2 ) + \\binom { n } { 2 } ( - 1 0 + 1 5 n - 6 n ^ 2 ) + \\binom { n } { 3 } ( 4 n - 5 ) - \\binom { n } { 4 } & \\textrm { i f } s = 4 \\end{cases} \\end{align*}"}
{"id": "2368.png", "formula": "\\begin{align*} \\nu _ \\sigma ( f ) = \\nu ( l ( h _ \\sigma ) ) = \\beta _ b + b \\gamma _ \\sigma . \\end{align*}"}
{"id": "7930.png", "formula": "\\begin{align*} M _ 0 = \\{ i \\in \\{ 1 , \\ldots , m \\} : D _ i \\cdot e = 0 \\} ; \\end{align*}"}
{"id": "3421.png", "formula": "\\begin{align*} n ( z , x _ 0 - 2 x ) \\cdot g = \\begin{pmatrix} a '' & * & * \\\\ b ' & * & * \\\\ c & * & * \\end{pmatrix} , a '' = a ' - \\sqrt { - 3 } x c . \\end{align*}"}
{"id": "975.png", "formula": "\\begin{align*} H ^ { * } ( \\overline { f } ) = f ( H ) = f _ { 0 } I _ { n } + f _ { 1 } H + \\cdots + f _ { n - 1 } H ^ { n - 1 } . \\end{align*}"}
{"id": "617.png", "formula": "\\begin{align*} \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\left ( \\frac { \\pi u } { n } \\right ) = \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ { k - 1 } } { ( 2 k ) ! } \\frac { 2 ^ { 2 k - 1 } \\pi ^ { 2 k - 2 } } { n ^ { 2 k - 2 } } u ^ { 2 k } . \\end{align*}"}
{"id": "3828.png", "formula": "\\begin{align*} E r r M ^ 0 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\sum _ { a = 1 , 2 } ( - 1 ) ^ a \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( t _ a ) \\cdot \\xi - i t \\hat { v } \\cdot \\eta + i t _ a \\mu _ 1 | \\xi - \\eta | } ( \\hat { V } ( t _ a ) \\cdot \\xi - \\hat { v } \\cdot \\eta + \\mu _ 1 | \\xi - \\eta | ) ^ { - 1 } \\hat { g } ( t _ a , \\eta , v ) \\end{align*}"}
{"id": "8756.png", "formula": "\\begin{align*} \\Delta ( F ^ { ( n ) } ) = \\sum _ { a = 0 } ^ n q ^ { a ( n - a ) } F ^ { ( a ) } \\otimes F ^ { ( n - a ) } K ^ { - a } . \\end{align*}"}
{"id": "250.png", "formula": "\\begin{align*} v ' _ { c } = 0 ; \\end{align*}"}
{"id": "4875.png", "formula": "\\begin{align*} \\int _ { \\mathfrak { M } _ { i } } f _ { i } ( \\alpha ) S _ { i } ^ { 2 } ( \\alpha ) T _ { i } ^ { 2 } ( \\alpha ) e ( - n \\alpha ) d \\alpha = \\frac { 1 } { 3 ^ { 4 } } \\mathfrak { S } ( n ) \\mathfrak { J } _ { i } ( n ) + O ( N _ { i } ^ { \\frac { 1 1 } { 9 } } L ^ { - 1 } ) \\end{align*}"}
{"id": "5984.png", "formula": "\\begin{align*} C _ { \\beta , \\varphi } = \\frac 1 { \\pi } ( \\cos \\varphi ) ^ { \\beta - 1 } \\Gamma ( 1 - \\beta ) . \\end{align*}"}
{"id": "4981.png", "formula": "\\begin{align*} \\widetilde { Y } ^ n _ t - Y ^ n _ t = R ^ n _ t + \\int _ 0 ^ t ( t - s ) ^ { \\alpha } \\sigma ' ( X _ s ) ( \\widetilde { Y } ^ n _ s - Y ^ n _ s ) \\ , d W _ s , \\end{align*}"}
{"id": "68.png", "formula": "\\begin{align*} F _ 1 ( k ) & = \\sum _ { h = 0 } ^ { k - 1 } \\frac { \\big | | k | ^ { 2 ( s - 1 ) } k - | h | ^ { 2 ( s - 1 ) } h \\big | ^ 2 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } , \\\\ F _ 2 ( k ) & = \\sum _ { h \\geq k } \\frac { \\big | | k | ^ { 2 ( s - 1 ) } k - | h | ^ { 2 ( s - 1 ) } h \\big | ^ 2 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } , \\\\ F _ 3 ( k ) & = \\sum _ { h < 0 } \\frac { \\big | | k | ^ { 2 ( s - 1 ) } k - | h | ^ { 2 ( s - 1 ) } h \\big | ^ 2 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } , \\end{align*}"}
{"id": "5082.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { \\tau } ( t - s ) ^ \\alpha \\sigma ' ( X _ s ) \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\ , \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) d s = 0 . \\end{align*}"}
{"id": "4993.png", "formula": "\\begin{align*} \\Theta ^ n _ s & = \\sigma ( X _ s ) \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u + X ^ n _ { \\eta _ n ( s ) } - X ^ n _ s \\\\ & = \\sigma ( X _ s ) \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u + \\int ^ s _ { \\eta _ n ( s ) } \\left ( s - \\eta _ n ( u ) \\right ) ^ { \\alpha } \\sigma ( X ^ n _ { \\eta _ n ( u ) } ) \\ , d W _ u \\\\ & + \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\ , \\sigma ( X ^ n _ { \\eta _ n ( u ) } ) \\ , d W _ u . \\end{align*}"}
{"id": "7115.png", "formula": "\\begin{align*} h ( \\tau , p ( \\tau ; t , x ) ) = h ( \\tau , p ( \\tau ; \\sigma , p ( \\sigma ; t , x ) ) ) . \\end{align*}"}
{"id": "1681.png", "formula": "\\begin{align*} ( 1 - \\alpha ( 1 + ( 2 - \\alpha ) A ) ) \\| w _ 1 + y _ 2 \\| & \\le \\| w _ 1 + O _ 2 y _ 2 \\| \\\\ & \\le ( 1 + \\alpha ( 1 + ( 2 + \\alpha ) A ) ) \\| w _ 1 + y _ 2 \\| . \\end{align*}"}
{"id": "2451.png", "formula": "\\begin{align*} & x _ { i + 1 } \\cdot x _ i = y _ { i } \\cdot y _ { i + 1 } = 0 , \\\\ & y _ { i + 1 } x _ { i + 1 } = - x _ i y _ i , \\\\ & y _ 0 x _ 0 = - ( x _ { - 1 } y _ { - 1 } ) ^ 2 , \\\\ & ( y _ 1 x _ 1 ) ^ 2 = - x _ 0 y _ 0 , \\end{align*}"}
{"id": "7253.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } - \\Delta _ k \\varphi ( \\gamma _ { n _ j } ( \\omega ) ) = 0 \\end{align*}"}
{"id": "4330.png", "formula": "\\begin{align*} \\partial _ t \\vec { x } = \\vec { H } . \\end{align*}"}
{"id": "3879.png", "formula": "\\begin{align*} - \\Delta _ p G _ n - \\lambda G _ n ^ { p - 1 } = f _ n \\Omega . \\end{align*}"}
{"id": "8541.png", "formula": "\\begin{align*} 3 ( a ) \\ \\mathrm { f o r } \\ k + 1 & = z ( 3 ( a ) \\ \\mathrm { f o r } \\ k ) + a _ { k + 1 } ( 3 ( b ) \\ \\mathrm { f o r } \\ k ) + a _ { k + 1 } ^ 2 ( 3 ( a ) \\ \\mathrm { f o r } \\ k - 1 ) \\\\ 3 ( b ) \\ \\mathrm { f o r } \\ k + 1 & = ( 2 z + 1 ) ( 3 ( a ) \\ \\mathrm { f o r } \\ k ) + a _ { k + 1 } ( 3 ( b ) \\ \\mathrm { f o r } \\ k ) + a _ { k + 1 } ( 3 ( c ) \\ \\mathrm { f o r } \\ k ) \\\\ 3 ( c ) \\ \\mathrm { f o r } \\ k + 1 & = - ( 3 ( a ) \\ \\mathrm { f o r } \\ k ) - a _ { k + 1 } ( 3 ( c ) \\ \\mathrm { f o r } \\ k ) \\end{align*}"}
{"id": "579.png", "formula": "\\begin{align*} M _ 3 = \\sum _ { \\substack { 0 \\leq m _ 2 \\leq j - i - 1 \\\\ \\gcd ( l + m _ 1 + m _ 2 , j ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } \\\\ \\gcd ( l + m _ 1 + m _ 2 , j + 1 ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } C _ { \\alpha } ( j - i - 1 , m _ 2 ) \\end{align*}"}
{"id": "838.png", "formula": "\\begin{align*} p ( \\boldsymbol { s } ) = p \\left ( s _ { 1 } \\right ) \\prod _ { m = 1 } ^ { M - 1 } \\mathop { p ( s _ { m + 1 } | s _ { m } ) } , \\end{align*}"}
{"id": "3380.png", "formula": "\\begin{align*} \\rho \\ , H = \\sqrt { \\frac { 2 H ^ 2 A } { A + G } } \\leq \\sqrt { \\frac { H ^ 2 A } { G } } = \\sqrt { H G } \\leq G , \\end{align*}"}
{"id": "4367.png", "formula": "\\begin{align*} \\sum \\nolimits _ { S , m } a _ { S , m } \\ , \\zeta _ { S , m } = 0 , \\end{align*}"}
{"id": "6626.png", "formula": "\\begin{align*} \\hbox { N } _ h ( N , w _ g ; \\xi ) = & c _ h h ^ { - 2 } \\ell ^ { - 4 } \\sum _ { k = 1 } ^ \\infty k ^ { - 4 } S _ { k } ( 0 ; \\xi ) I _ { k , \\ell } ( 0 ) \\\\ & + O _ { w , \\ell , \\theta , \\epsilon } \\big ( \\| g \\| _ E ^ \\theta h ^ { 3 - ( \\theta - 4 ) \\delta } + \\| g \\| _ E h ^ { 3 / 2 + 3 \\delta + \\epsilon } \\big ) . \\end{align*}"}
{"id": "5437.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { 1 } { p _ 1 } + \\frac { 1 } { r } \\end{align*}"}
{"id": "4164.png", "formula": "\\begin{align*} f \\times _ \\lambda g ( z ) = \\int _ { \\R ^ { 2 n } } f ( z - w ) g ( w ) e ^ { \\frac i 2 \\lambda \\omega ( z , w ) } \\ , d w , z \\in \\R ^ { 2 n } , \\end{align*}"}
{"id": "432.png", "formula": "\\begin{align*} \\nu _ { n } = \\mu _ { n , 1 } \\boxplus \\cdots \\boxplus \\mu _ { n , k _ { n } } , \\end{align*}"}
{"id": "8622.png", "formula": "\\begin{align*} k _ \\tau [ \\mathfrak { M } ^ { e v } ( \\mathcal { X } , \\tau ) ] = m _ \\tau k _ { * , r e d } [ \\mathfrak { M } ^ { e v } ( \\mathcal { X } , \\tau ) _ { s , r e d } ] . \\end{align*}"}
{"id": "6200.png", "formula": "\\begin{align*} & \\int f ( x ) \\Big ( \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q h ( x ) + p ( x ) D _ { q ^ { - 1 } } h ( x ) + r ( x ) h ( x ) \\Big ) y ( x ) d _ q x \\\\ & = f ( x / q ) \\Big ( y ( x / q ) D _ { q ^ { - 1 } } h ( x ) - h ( x / q ) D _ { q ^ { - 1 } } y ( x ) \\Big ) \\end{align*}"}
{"id": "7202.png", "formula": "\\begin{align*} \\langle f , I ^ { \\circ } ( w _ 1 , z _ 2 ) u _ { k - 1 + \\frac { j _ 2 } { T } } w _ 2 \\rangle = 0 . \\end{align*}"}
{"id": "7033.png", "formula": "\\begin{align*} u _ 1 ( x ) = \\cos \\left ( \\frac { \\pi } { 2 } x \\right ) u _ 2 ( x ) = \\sin \\left ( \\frac { \\pi } { 2 } x \\right ) \\end{align*}"}
{"id": "3618.png", "formula": "\\begin{align*} V ( \\mathcal { Q } ( I ) ) = \\{ ( 0 , \\ , 0 , \\ , 1 , \\ , 1 ) , \\ , ( 0 , \\ , 1 , \\ , 0 , \\ , 1 ) , \\ , ( 1 , \\ , 0 , \\ , 0 , \\ , 1 ) , \\ , ( 1 , \\ , 1 , \\ , 1 , \\ , 0 ) , \\ , ( { 1 } / { 3 } , \\ , { 1 } / { 3 } , \\ , { 1 } / { 3 } , \\ , { 2 } / { 3 } ) \\} , \\end{align*}"}
{"id": "5977.png", "formula": "\\begin{align*} C _ { \\beta , \\varphi } = \\frac 1 { \\pi } ( \\cos \\varphi ) ^ { \\beta - 1 } \\Gamma ( 1 - \\beta ) . \\end{align*}"}
{"id": "71.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } P _ 1 e = T Q _ 1 B f , \\\\ Q _ 0 B f = 0 . \\end{array} \\right . \\end{align*}"}
{"id": "4201.png", "formula": "\\begin{align*} \\norm { g _ { \\le \\iota } ^ { ( 1 ) } } _ p ^ p \\lesssim _ \\iota ( \\iota + 2 ) ^ { p - 1 } \\sum _ { \\ell = - 1 } ^ \\iota \\sum _ { m = 1 } ^ { M _ { \\ell } } \\Vert \\tilde g _ { m } ^ { ( \\ell ) } \\Vert _ p ^ p . \\end{align*}"}
{"id": "7962.png", "formula": "\\begin{align*} \\hat { \\Gamma } _ { ( X _ + , D _ + ) } : = \\bigoplus _ { \\vec d \\in ( \\mathbb Z ) ^ { | I _ + | } , f \\in \\mathbb K _ { + } / \\mathbb L } \\prod _ { i \\in M _ 0 } \\Gamma ( 1 + \\bar D _ i - \\langle D _ i \\cdot f \\rangle ) \\prod _ { i \\in I _ + , d _ i < 0 } \\frac { 1 } { \\bar D _ i - d _ i } \\textbf { 1 } _ { f } [ \\textbf 1 ] _ { ( d _ i ) _ { i \\in I _ + } } . \\end{align*}"}
{"id": "7014.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ r ) = ( x _ 1 v _ 1 + \\ldots + x _ r v _ r ) ^ n \\in A _ n \\simeq k . \\end{align*}"}
{"id": "4849.png", "formula": "\\begin{align*} & ( X , \\delta _ X f \\wedge \\delta _ X ( D ^ 2 g ( \\nu _ { X } D f , \\cdot ) - D ^ 2 f ( \\nu _ { X } D g , \\cdot ) ) ) \\\\ & = D ^ 2 g ( \\nu _ { X } D f , \\nu _ { X } D h ) - D ^ 2 f ( \\nu _ { X } D g , \\nu _ { X } D h ) + \\circlearrowright , \\end{align*}"}
{"id": "7994.png", "formula": "\\begin{align*} W : = \\Sigma \\circ h : X ^ \\vee \\rightarrow \\mathbb C . \\end{align*}"}
{"id": "4968.png", "formula": "\\begin{align*} Z ^ n _ t = n ^ { \\alpha + \\frac 1 2 } \\int ^ t _ 0 ( t - s ) ^ { \\alpha } [ \\sigma ( X ^ n _ { s } ) - \\sigma ( X _ { s } ) ] \\ , d W _ s . \\end{align*}"}
{"id": "4974.png", "formula": "\\begin{align*} \\widetilde { Z } ^ n _ t = n ^ { \\alpha + \\frac 1 2 } \\int ^ t _ 0 ( t - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\left ( X ^ n _ { s } - X _ { s } \\right ) \\ , d W _ s . \\end{align*}"}
{"id": "4723.png", "formula": "\\begin{align*} & e _ { ( l + 1 ) } \\\\ = & \\Big ( \\frac { q - q ^ { - 1 } } { z - z ^ { - 1 } } \\Big ) ^ { l - 1 } \\Big ( \\frac { q - q ^ { - 1 } } { z - z ^ { - 1 } } \\Big ) ^ { l - 2 } e _ { ( l - 1 ) } H _ { 2 l - 2 } H _ { 2 l - 1 } H _ { 2 l - 3 } ^ { - 1 } H _ { 2 l - 2 } ^ { - 1 } e _ { ( l - 1 ) } \\\\ & \\times H _ { 2 l } H _ { 2 l + 1 } H _ { 2 l - 1 } ^ { - 1 } H _ { 2 l } ^ { - 1 } e _ { ( l ) } \\\\ = & \\Big ( \\frac { q - q ^ { - 1 } } { z - z ^ { - 1 } } \\Big ) ^ { l - 1 } e _ { ( l ) } H _ { 2 l } H _ { 2 l + 1 } H _ { 2 l - 1 } ^ { - 1 } H _ { 2 l } ^ { - 1 } e _ { ( l ) } . \\end{align*}"}
{"id": "5087.png", "formula": "\\begin{align*} S ^ { n , M , 2 } _ \\tau = n ^ { \\alpha + \\frac 1 2 } \\sum _ { i = 0 } ^ { M - 1 } \\int _ { \\tau _ i } ^ { \\tau _ { i + 1 } } ( t - s ) ^ \\alpha [ \\sigma ' ( X _ s ) - \\sigma ' ( X _ { \\tau _ i } ) ] \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) d s \\end{align*}"}
{"id": "5357.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } \\norm { ( - \\Delta _ { x _ n } ) ^ { s / 2 } u ( x ' , x _ n ) } _ { L ^ p ( \\R ) } ^ p d x ' = \\norm { ( - \\Delta _ { x _ n } ) ^ { s / 2 } u } _ { L ^ p ( \\R ^ n ) } ^ p . \\end{align*}"}
{"id": "4347.png", "formula": "\\begin{align*} \\mathrm { d e p t h } \\ , ( R / \\big < J , y \\big > ) & = 1 + \\mathrm { d e p t h } \\ , ( K [ x _ { k + 1 } , \\ldots , x _ { n } ] / I ( D _ { F } ) ) \\\\ & = 1 + \\mathrm { d i m } \\ , ( K [ x _ { k + 1 } , \\ldots , x _ { n } ] / I ( D _ { F } ) ) \\\\ & = 1 + ( n - k ) - ( g - k ) \\\\ & = 1 + n - g . \\end{align*}"}
{"id": "6977.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { \\infty } \\left ( \\frac { z - \\mu _ k ^ 2 } { z - \\lambda _ k ^ 2 } \\right ) & = 1 - \\sum \\limits _ { k = 1 } ^ { \\infty } a _ { k } \\frac { ( \\lambda _ { k } ^ 2 - z ) + z } { \\lambda _ { k } ^ 2 ( \\lambda _ { k } ^ 2 - z ) } \\\\ & = 1 - \\sum \\limits _ { k \\geq 1 } \\frac { a _ k } { \\lambda _ k ^ 2 } - \\sum \\limits _ { k \\geq 1 } \\frac { a _ k z } { \\lambda _ k ^ 2 ( \\lambda _ k ^ 2 - z ) } , \\end{align*}"}
{"id": "8143.png", "formula": "\\begin{align*} W _ { n , m ; \\mathrm { f a r } } = W _ { n , m ; \\mathrm { o u t } } \\sqcup W _ { n , m ; \\mathrm { i n } } \\end{align*}"}
{"id": "7337.png", "formula": "\\begin{align*} ( [ w , x ] , y , z ) = [ w , ( x , y , z ) ] + [ x , ( w , y , z ) ] \\mbox { i f } \\mathsf { c h a r } ( F ) = 2 ; \\end{align*}"}
{"id": "2360.png", "formula": "\\begin{align*} \\beta _ j : = \\nu _ \\sigma \\left ( \\partial _ j l ( h _ \\sigma ) \\right ) = \\nu \\left ( \\partial _ j l ( h _ \\sigma ) \\right ) \\mbox { f o r e v e r y } j , 1 \\leq j \\leq \\deg _ X ( l ) \\mbox { a n d } \\rho < \\sigma . \\end{align*}"}
{"id": "8570.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty a _ n r _ n ( x ) \\end{align*}"}
{"id": "4794.png", "formula": "\\begin{align*} e ^ { \\sqrt { c _ { 0 } t } \\Lambda _ 1 } T _ { a } b = \\sum _ { j \\in \\mathbb { Z } } W _ { j } \\ , \\ , \\ , \\mbox { w i t h } \\ , \\ , \\ , W _ { j } \\triangleq \\mathcal { B } _ t ( \\dot S _ { j - 1 } A , \\dot \\Delta _ j B ) . \\end{align*}"}
{"id": "5927.png", "formula": "\\begin{align*} w ( \\eta ) = \\sup \\limits _ { a } \\left ( \\eta a - J ( a ) \\right ) \\end{align*}"}
{"id": "6159.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| u ^ { 1 } _ { h } \\| ^ { 2 } _ { 2 } + \\frac { \\nu \\Delta t } { 4 } \\| \\nabla u ^ { 1 } _ { h } \\| ^ { 2 } _ { 2 } \\leq \\frac { 1 } { 2 } \\| u _ { 0 } \\| ^ { 2 } _ { 2 } + \\frac { \\nu \\Delta t } { 4 } \\| \\nabla u ^ { 0 } _ { h } \\| ^ { 2 } _ { 2 } \\leq \\big ( \\frac { 1 } { 2 } + \\frac { \\nu \\Delta t } { 4 h ^ { 2 } } \\big ) \\| u _ { 0 } \\| ^ { 2 } _ { 2 } : = K _ { 3 } , \\end{align*}"}
{"id": "865.png", "formula": "\\begin{align*} d _ i \\Delta ^ { n - i } e _ i & = \\Delta ^ { n - i } \\phi ^ { n - i } ( d _ i ) e _ i & ( \\ ; d _ i \\Delta = \\Delta \\phi ( d _ i ) ) \\\\ & = \\Delta ^ { n - i } \\phi ^ { n - i } ( d _ i ) \\left ( \\phi ^ { n - i } ( d _ i ) \\backslash \\Delta \\right ) & ( \\ ; e _ i = \\phi ^ { n - i } ( d _ i ) \\backslash \\Delta ) \\\\ & = \\Delta ^ { n - i + 1 } & ( \\ ; \\phi ^ { n - i } ( d _ i ) \\left ( \\phi ^ { n - i } ( d _ i ) \\backslash \\Delta \\right ) = \\Delta ) . \\end{align*}"}
{"id": "6915.png", "formula": "\\begin{align*} \\frac { h ( u , v ) } { h ( v , u ) } = & \\frac { h _ 1 \\big ( \\frac { 1 } { \\sqrt { 2 } } ( u - v ) \\big ) h _ 2 \\big ( \\frac { 1 } { \\sqrt { 2 } } ( u + v ) \\big ) } { h _ 1 \\big ( \\frac { 1 } { \\sqrt { 2 } } ( v - u ) \\big ) h _ 2 \\big ( \\frac { 1 } { \\sqrt { 2 } } ( v + u ) \\big ) } \\\\ = & \\frac { h _ 1 \\big ( \\frac { 1 } { \\sqrt { 2 } } ( u - v ) \\big ) } { h _ 1 \\big ( \\frac { - 1 } { \\sqrt { 2 } } ( u - v ) \\big ) } \\ , \\end{align*}"}
{"id": "64.png", "formula": "\\begin{align*} G _ r ( k ) & = G _ 1 ( k ) + G _ 2 ( k ) + G _ 3 ( k ) + G _ 4 ( k ) \\\\ & = \\sum _ { h \\leq 0 } \\frac { 1 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } + \\sum _ { h = 1 } ^ { \\lfloor k / 2 \\rfloor } \\frac { 1 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } \\\\ & + \\sum _ { h = \\lfloor k / 2 \\rfloor + 1 } ^ { k - 1 } \\frac { 1 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } + \\sum _ { h \\geq k } \\frac { 1 } { ( 1 + | k - h | ) ^ { 2 s } ( 1 + | h | ) ^ { 2 r } } \\end{align*}"}
{"id": "5523.png", "formula": "\\begin{align*} \\abs { f ' ( A ( z ) ) } = \\abs { \\frac { b } { 2 \\lambda } } \\frac { 1 } { \\abs { g ( z ) } ^ 2 } \\abs { g ' ( z ) } \\abs { z + \\lambda i } ^ 2 . \\end{align*}"}
{"id": "157.png", "formula": "\\begin{align*} \\frac { 1 } { \\gamma } \\leq \\int _ 0 ^ \\infty \\exp \\Big [ - 2 \\int _ 0 ^ t \\dot \\kappa ^ { \\epsilon , L } _ s \\ , d s \\Big ] \\ , d t , \\dot \\kappa ^ { \\epsilon , L } _ t : = \\frac { 1 } { t } - \\frac { \\chi _ t ^ { \\epsilon , L } } { t ^ 2 } . \\end{align*}"}
{"id": "5824.png", "formula": "\\begin{align*} x _ n = \\prod _ { k \\leq n } \\left ( p _ k ^ 2 - \\chi ( p _ k ) \\right ) , y _ { n } = \\prod _ { k \\leq n } p _ k ^ 2 . \\end{align*}"}
{"id": "5778.png", "formula": "\\begin{align*} E ( G / H ) = \\dfrac { E ( G / G ^ 0 ) } { E ( H / H ^ 0 ) } \\equiv 0 \\mod p , \\end{align*}"}
{"id": "5605.png", "formula": "\\begin{align*} 1 = \\left ( \\frac { - 2 ^ { 3 r + 1 } P _ 1 ^ 2 P _ 2 x z _ 1 ^ 3 } { 2 ^ { r + 1 } P _ 1 P _ 2 P _ 3 x z _ 1 - y ^ 2 } \\right ) = \\left ( \\frac { - 2 ^ { 3 r + 1 } P _ 2 x z _ 1 } { 2 ^ { r + 1 } P _ 1 P _ 2 P _ 3 x z _ 1 - y ^ 2 } \\right ) . \\end{align*}"}
{"id": "7504.png", "formula": "\\begin{align*} E _ { d } ( x , t ) = t ^ { - \\frac { N } { 2 d } } E _ d ( t ^ { - \\frac { 1 } { 2 d } } x , 1 ) . \\end{align*}"}
{"id": "7715.png", "formula": "\\begin{align*} \\sum _ { \\nu = 1 } ^ { \\infty } \\frac { \\nu ^ t } { p ^ { \\nu k } } \\ll \\frac 1 { p ^ k } . \\end{align*}"}
{"id": "3472.png", "formula": "\\begin{align*} \\mathrm { T C E } _ { p } ( X ) = \\mu + \\sigma \\frac { c _ { 1 } \\overline { G } _ { 1 } \\left ( \\frac { 1 } { 2 } \\xi _ { p } ^ { 2 } \\right ) } { \\overline { F } _ { Y } ( \\xi _ { p } ) } , \\end{align*}"}
{"id": "1858.png", "formula": "\\begin{align*} W _ { j } ( z ) = \\frac { 1 } { z ^ { j + 1 } } + O \\left ( \\frac { 1 } { z ^ { j + 2 } } \\right ) . \\end{align*}"}
{"id": "4817.png", "formula": "\\begin{align*} \\sum _ { p \\in \\pi _ j \\cap J } s _ { k _ j } ( p , e ( \\pi _ j ) ) = 1 \\end{align*}"}
{"id": "9008.png", "formula": "\\begin{align*} 2 ^ a \\Phi ( - 1 , a , 2 b ) & = \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { \\left ( k + b \\right ) ^ a } - \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { \\left ( k + b + \\frac { 1 } { 2 } \\right ) ^ a } = \\zeta ( a , b ) - \\zeta \\left ( a , b + \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
{"id": "5740.png", "formula": "\\begin{align*} \\pi _ b \\cdot \\pi _ { [ a , b ] } = ( b - a + 1 ) \\pi _ { [ a , b ] } y _ { b + 1 } + ( b - a + 2 ) \\pi _ { [ a - 1 , b ] } \\end{align*}"}
{"id": "1893.png", "formula": "\\begin{align*} \\mathcal { P } _ { [ n , 0 ] } = \\bigcup _ { j = - 1 } ^ { p } \\mathcal { U } _ { [ n , j ] } , n \\geq 1 , \\end{align*}"}
{"id": "4152.png", "formula": "\\begin{align*} L ^ \\mu = - ( ( X _ 1 ^ \\mu ) ^ 2 + \\dots + ( X _ { d _ 1 } ^ \\mu ) ^ 2 ) . \\end{align*}"}
{"id": "8082.png", "formula": "\\begin{align*} g ( \\nabla _ { \\partial _ { i } } X , \\partial _ { j } ) = - g ( \\nabla _ { \\partial _ { j } } X , \\partial _ { i } ) \\end{align*}"}
{"id": "8140.png", "formula": "\\begin{align*} \\| V e ^ { - i t H _ 0 } \\psi \\| ^ 2 & \\leq C \\int \\limits _ { B _ { r _ 0 } } ( 1 + \\| x \\| + t ) ^ { - \\ell } \\ , d x \\leq C ( 1 + t ) ^ { - \\ell + d } \\end{align*}"}
{"id": "3701.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } \\left < \\frac { i } h [ - h ^ 2 \\Delta + V ( x ) - E ( h ) , \\chi _ \\alpha ( x _ n ) h D _ n ] u _ h , \\ , u _ h \\right > _ { L ^ 2 ( \\Omega _ \\Gamma ) } = o ( 1 ) . \\end{align*}"}
{"id": "191.png", "formula": "\\begin{align*} & \\P ( x ^ 1 _ t , x ^ 2 _ t | x ^ 1 _ { 1 : t - 1 } , x ^ 2 _ { 1 : t - 1 } , y _ { 1 : t - 1 } ) = \\\\ & q ^ 1 _ t ( x ^ 1 _ t | x ^ 1 _ { 1 : t - 1 } , y _ { 1 : t - 1 } ) q ^ 2 _ t ( x ^ 2 _ t | x ^ 2 _ { 1 : t - 1 } , Y _ { 1 : t - 1 } ) \\end{align*}"}
{"id": "7443.png", "formula": "\\begin{align*} \\Lambda ^ m _ { x x } = - c ^ { - 1 } \\left ( f ^ 5 + \\delta f ^ 1 _ x \\right ) , \\end{align*}"}
{"id": "5061.png", "formula": "\\begin{align*} L ^ { n , 1 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) ( s - \\eta _ n ( s ) ) ^ \\alpha ( W _ s - W _ { \\eta _ n ( s ) } ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) d s . \\end{align*}"}
{"id": "1293.png", "formula": "\\begin{align*} \\mu ( T _ 0 , \\lambda ) \\ge \\sum _ { j = 1 } ^ { k } \\mu ( T _ j , \\lambda ) ( \\forall \\lambda \\in \\mathbb { R } ) . \\end{align*}"}
{"id": "8077.png", "formula": "\\begin{align*} \\operatorname { g r a d } ( f ^ { 2 } ) = F \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) \\end{align*}"}
{"id": "7820.png", "formula": "\\begin{align*} A d _ { \\Theta _ n ( \\phi ) ^ { - 1 } } ( d \\Theta _ n ( e _ { \\phi ( \\lambda ) } ) ) \\stackrel { ( \\ref { e q - d i f f e r e n t i a l - o f - T h e t a - n - s e n d s - e - l a m b d a - t o } ) } { = } \\Theta _ n ( \\phi ) ^ { - 1 } e _ { \\theta ( \\phi ( \\lambda ) ) } \\Theta _ n ( \\phi ) = \\frac { t ( \\lambda , \\lambda ) } { 2 } e _ { \\theta ( \\lambda ) } ^ \\vee . \\end{align*}"}
{"id": "1009.png", "formula": "\\begin{align*} \\mu _ P ( [ B ] ) = p _ { b _ 1 } p _ { b _ 2 } \\cdots p _ { b _ k } . \\end{align*}"}
{"id": "3170.png", "formula": "\\begin{align*} e _ { \\Psi } \\left ( \\rho \\right ) = \\rho \\left ( \\mathfrak { e } _ { \\Psi } \\right ) \\mathfrak { e } _ { \\Psi } \\doteq \\sum \\limits _ { \\mathcal { Z } \\in \\mathcal { P } _ { \\mathrm { f } } , \\ ; \\mathcal { Z } \\ni 0 } \\frac { \\Psi _ { \\mathcal { Z } } } { \\left \\vert \\mathcal { Z } \\right \\vert } \\in \\mathcal { U } \\end{align*}"}
{"id": "3203.png", "formula": "\\begin{align*} g _ p ( x ) = \\sum _ { k = 1 } ^ \\infty \\left ( \\sum _ { i = n _ { k } } ^ { { n _ { k + 1 } } } \\frac { ( 2 ^ i ) ^ { \\frac { 1 } { p - 1 } } \\mu ^ { \\frac { 1 } { 1 - p } } ( A _ { 2 ^ i } ) } { \\sum _ { i = n _ k } ^ { n _ { k + 1 } } ( 2 ^ i ) ^ { \\frac { p } { p - 1 } } \\mu ^ { \\frac { 1 } { 1 - p } } ( A _ { 2 ^ i } ) } \\chi _ { A _ { 2 ^ i } } ( x ) \\right ) \\big ( { \\rm o r \\ \\ } g _ 1 ( x ) = \\sum _ { k = 1 } ^ \\infty 2 ^ { - i _ k } \\chi _ { A _ { 2 ^ { i _ k } } } ( x ) \\big ) . \\end{align*}"}
{"id": "1847.png", "formula": "\\begin{align*} \\mathcal { E } : = \\mathcal { E } _ { u } \\cup \\mathcal { E } _ { \\ell } \\cup \\mathcal { E } _ { d } , \\end{align*}"}
{"id": "1500.png", "formula": "\\begin{align*} \\phi ^ { \\prime } _ { n , \\lambda } ( x ) & = \\frac { d } { d x } \\phi _ { n , \\lambda } ( x ) = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ( 1 ) _ { n - k , \\lambda } \\phi _ { k , \\lambda } ( x ) - \\phi _ { n , \\lambda } ( x ) \\\\ & = \\sum _ { k = 0 } ^ { n - 1 } \\binom { n } { k } ( 1 ) _ { n - k , \\lambda } \\phi _ { k , \\lambda } ( x ) , ( n \\ge 1 ) . \\end{align*}"}
{"id": "2067.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : L i n f t y d e f } \\mathcal { L } _ \\infty \\Phi ( \\nu ) = \\frac { \\dd } { \\dd t } T _ { \\infty , t } \\Phi ( \\nu ) \\Big | _ { t = 0 } = \\frac { \\dd } { \\dd t } \\Phi \\big ( \\overline { S } _ t ( \\nu ) \\big ) \\Big | _ { t = 0 } & = { \\left \\langle \\dd \\Phi [ \\nu ] , \\frac { \\dd } { \\dd t } \\overline { S } _ t ( \\nu ) \\Big | _ { t = 0 } \\right \\rangle } \\\\ & = \\langle \\dd \\Phi [ \\nu ] , Q ( \\nu ) \\rangle . \\end{align*}"}
{"id": "4840.png", "formula": "\\begin{align*} \\delta : R ( G ) & \\rightarrow H ^ { \\mathrm { e v e n } \\geqslant 2 } ( S ) \\\\ \\rho & \\mapsto \\tau ^ \\mathrm { a n } ( M / S , \\rho ) - \\tau ^ \\mathrm { t o p } ( M / S , \\rho ) \\ ; . \\end{align*}"}
{"id": "3515.png", "formula": "\\begin{align*} \\overline { G } _ { ( 1 ) } ( u ) = ( 1 + \\sqrt { 2 u } ) \\exp ( - \\sqrt { 2 u } ) \\end{align*}"}
{"id": "301.png", "formula": "\\begin{align*} p _ j = P ( a ' ) > P ( ( a ' ) ^ * ) ^ { 1 / v } > P ( c ' ) ^ { 1 / \\sqrt { v } } \\ge P ( a ^ * ) ^ { 1 / \\sqrt { v } } . \\end{align*}"}
{"id": "620.png", "formula": "\\begin{align*} { \\alpha + j - 1 \\choose j } \\Big ( \\sum _ { k = 0 } ^ \\infty d _ k \\frac { ( x ^ { 2 k + 4 } + y ^ { 2 k + 4 } ) } { ( x ^ 2 + y ^ 2 + z ^ 2 ) } n ^ { - 2 k - 2 } \\Big ) ^ { j } = \\sum _ { k = 0 } ^ \\infty \\frac { F ' _ { k , j } ( x , y ) } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ j } n ^ { - 2 ( k + j ) } , \\end{align*}"}
{"id": "6335.png", "formula": "\\begin{align*} \\cos ( z ; q ) & : = \\frac { ( q ; q ) _ \\infty } { ( q ^ { 1 / 2 } ; q ) _ \\infty } ( z q ^ { - 1 / 2 } ) ^ { 1 / 2 } J ^ { ( 3 ) } _ { - 1 / 2 } ( z ( 1 - q ) / \\sqrt { q } ; q ^ 2 ) = \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n \\frac { q ^ { n ^ 2 } z ^ { 2 n } } { \\Gamma _ q ( 2 n + 1 ) } . \\end{align*}"}
{"id": "894.png", "formula": "\\begin{align*} r h = r h _ 0 + r h _ 1 + r h _ 2 + \\cdots \\end{align*}"}
{"id": "3656.png", "formula": "\\begin{align*} \\exists m _ \\ast \\geq 0 , \\begin{cases} H [ 0 , m _ \\ast ] , \\\\ H ( m _ \\ast , + \\infty ) . \\end{cases} \\end{align*}"}
{"id": "2834.png", "formula": "\\begin{align*} \\frac { q } { S _ * } \\| \\nabla f ( x _ N ) \\| ^ 2 + \\sum _ { i = 0 } ^ { N - 1 } \\frac { p _ i } { S _ * } \\min \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\ , , \\ , \\| \\nabla f ( x _ { i + 1 } ) \\| ^ 2 \\} { } \\leq { } \\frac { f ( x _ 0 ) - f _ * } { S _ * } . \\end{align*}"}
{"id": "5933.png", "formula": "\\begin{align*} \\lim \\limits _ { T \\to \\infty } \\frac 1 T \\ln \\langle { x ^ \\eta ( T ) } \\rangle = \\mathop \\sum \\limits _ n \\frac { w ^ { ( n ) } } { { n ! } } \\eta ^ n \\end{align*}"}
{"id": "3147.png", "formula": "\\begin{align*} \\Phi _ { \\Lambda + x } = \\alpha _ { x } \\left ( \\Phi _ { \\Lambda } \\right ) \\ , x \\in \\mathbb { Z } ^ { d } , \\ \\Lambda \\in \\mathcal { P } _ { \\mathrm { f } } , \\end{align*}"}
{"id": "2775.png", "formula": "\\begin{align*} \\begin{aligned} T _ 2 ( h _ i { } \\leq { } 1 ) { } = { } h _ i + h _ { i - 1 } - { h _ i } - { h _ { i - 1 } } \\left [ ( 1 + \\kappa ) \\tfrac { \\alpha _ { i - 1 } } { B } - \\kappa \\right ] { } = { } h _ { i - 1 } ( 1 + \\kappa ) \\left ( 1 - \\tfrac { \\alpha _ { i - 1 } } { B } \\right ) \\end{aligned} \\end{align*}"}
{"id": "7434.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\int _ 0 ^ L \\abs { \\omega } ^ 2 d x + c ( 1 - m ) \\int _ 0 ^ L \\abs { \\omega _ x } ^ 2 d x + \\Re \\left ( c m \\int _ 0 ^ L \\int _ 0 ^ { \\infty } \\sigma ( s ) \\eta _ x ( s ) \\overline { \\omega } _ x d s d x \\right ) - \\delta \\int _ 0 ^ L u _ t \\overline { w } _ x d x = 0 . \\end{align*}"}
{"id": "4074.png", "formula": "\\begin{align*} P _ 0 ( D ) = \\sum \\limits _ { j = 0 } ^ k P _ 0 ( D _ j ) \\end{align*}"}
{"id": "279.png", "formula": "\\begin{align*} ( B ( \\Lambda ' _ i - 1 ) ) _ { k j } - ( B ( \\Lambda ' _ j - 1 ) ) _ { k i } = \\pi L _ { i j k } ( \\Lambda ' ) . \\end{align*}"}
{"id": "897.png", "formula": "\\begin{align*} B ^ n = B ^ n ( a , r ) = B ^ { n _ 1 } ( a _ 1 , r _ 1 ) \\times \\cdots \\times B ^ { n _ k } ( a _ k , r _ k ) \\end{align*}"}
{"id": "2946.png", "formula": "\\begin{align*} \\int _ 1 ^ T c _ p ( t ) d t \\leq & \\langle B ( t ) : \\phi ( t ) \\rangle _ t \\vert _ { t = T } - \\langle B ( t ) : \\phi ( t ) \\rangle _ t \\vert _ { t = 1 } + \\| g ( t ) \\| _ { L ^ 1 _ t [ 1 , \\infty ) } \\\\ \\leq & \\left ( \\sup \\limits _ { t \\in \\R } \\| \\psi ( t ) \\| _ { \\mathcal { H } _ x ^ a } \\right ) ^ 2 + \\| g ( t ) \\| _ { L ^ 1 _ t [ 1 , \\infty ) } . \\end{align*}"}
{"id": "6568.png", "formula": "\\begin{align*} d g ( x , y ) = \\beta ( d f _ 1 ( x _ 1 , y _ 1 ) , \\ , f _ 2 ( x ) ) + \\beta ( f _ 1 ( x _ 1 ) , \\ , d f _ 2 ( x , y ) ) \\ , . \\end{align*}"}
{"id": "161.png", "formula": "\\begin{align*} \\kappa _ t ^ { \\epsilon , L } \\geq \\log ( m ^ 2 t + 1 ) - c , c : = \\int _ 0 ^ \\infty \\frac { \\tilde p _ { d , t , m ^ 2 } ( \\lambda ) } { t ^ 2 } . \\end{align*}"}
{"id": "4979.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\in [ 0 , T ] } E [ | \\widetilde { Y } ^ n _ t - Y ^ n _ t | ^ 2 ] = 0 . \\end{align*}"}
{"id": "2012.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : w e a k p d e m e a n f i e l d } \\forall \\varphi \\in \\mathcal { F } , \\quad \\frac { \\dd } { \\dd t } \\langle f _ t , \\varphi \\rangle = \\langle f _ t , L _ { f _ t } \\varphi \\rangle . \\end{align*}"}
{"id": "4266.png", "formula": "\\begin{align*} \\begin{aligned} & \\int \\limits _ { \\mathbb R ^ 2 } \\ ( \\mathcal E - \\mathcal L ( \\Phi ) + \\mathcal N ( \\Phi ) \\ ) \\partial _ \\rho W _ \\rho \\\\ & = \\mathfrak v _ \\infty \\mathfrak A \\frac { k } { \\rho ^ { \\nu + 1 } } ( 1 + o ( 1 ) ) - \\mathfrak B \\sqrt { \\frac k \\rho } e ^ { - 2 \\rho \\frac \\pi k } ( 1 + o ( 1 ) ) - \\beta \\mathfrak C \\sqrt { \\frac k \\rho } e ^ { - 4 \\rho \\frac { \\pi } { d k } } ( 1 + o ( 1 ) ) . \\end{aligned} \\end{align*}"}
{"id": "1938.png", "formula": "\\begin{align*} R _ { j } ( z ) = R _ { i } ( z ) \\ , S _ { j - i - 1 } ^ { ( i + 1 ) } ( z ) 0 \\leq i < j \\leq p . \\end{align*}"}
{"id": "352.png", "formula": "\\begin{align*} \\P _ \\pi ( T _ x < t ^ * , T _ y < t ^ * ) & \\le \\P _ \\pi ( T _ { \\{ x , y \\} } < t ^ * ) ( 1 - \\beta ) \\\\ & = \\P ( E ' _ 1 ) ( 1 - \\beta ) \\end{align*}"}
{"id": "999.png", "formula": "\\begin{align*} \\mathbf j ( z _ 1 + z _ 2 \\mathbf j ) = & \\mathbf j ( x _ 0 + x _ 1 \\mathbf i ) + \\mathbf j ( x _ 2 + x _ 3 \\mathbf i ) \\mathbf j \\\\ = & - ( x _ 2 - x _ 3 \\mathbf i ) + ( x _ 0 - x _ 1 \\mathbf i ) \\mathbf j \\\\ = & - \\overline z _ 2 + \\overline z _ 1 \\mathbf j \\end{align*}"}
{"id": "2702.png", "formula": "\\begin{align*} H _ { 2 \\mathrm { - m o d e } } = \\ ; & E _ 0 + E ^ w _ N + \\mathcal { T } \\big ( a ^ \\dagger _ 1 a _ 2 + a ^ \\dagger _ 2 a _ 1 \\big ) - \\mu \\mathcal { N } _ \\perp + \\frac { \\lambda U } { N - 1 } \\left ( \\mathcal { N } _ 1 - \\mathcal { N } _ 2 \\right ) ^ 2 \\\\ & + \\frac { 2 \\lambda } { N - 1 } w _ { 1 1 2 2 } \\mathcal { N } ^ 2 _ - + \\frac { \\lambda } { 4 ( N - 1 ) } ( w _ { 1 1 1 1 } - 2 w _ { 1 1 2 2 } + w _ { 1 2 1 2 } ) \\mathcal { N } _ \\perp ^ 2 . \\end{align*}"}
{"id": "4190.png", "formula": "\\begin{align*} \\tilde B _ { m } ^ { ( \\ell ) } : = B _ { 2 ^ { \\gamma \\iota + 1 } C R _ \\ell } ( x _ { m } ^ { ( \\ell ) } ) \\times B _ { 3 C R ^ 2 } ( 0 ) , 1 \\le m \\le M _ { \\ell } \\end{align*}"}
{"id": "2423.png", "formula": "\\begin{align*} \\| \\phi _ n ( a b ) - \\phi _ n ( a ) \\phi _ n ( b ) \\| & = \\| P _ n a b P _ n - P _ n a P _ n b P _ n \\| \\\\ & \\leq \\| P _ n a b - P _ n a P _ n b \\| \\\\ & \\leq \\| P _ n a - P _ n a P _ n \\| \\| b \\| \\approx 0 , \\end{align*}"}
{"id": "4249.png", "formula": "\\begin{align*} D ( n - 1 ) = \\frac { \\mathbb { G } ( n - 1 ) } { \\mathbb { G } ( n - 1 ) - \\mathbb { G } ( n ) } \\end{align*}"}
{"id": "3520.png", "formula": "\\begin{align*} \\mathrm { D T E } _ { ( p , q ) } ( X ) = \\mu + \\sigma \\frac { ( 1 + | \\xi _ { p } | ) \\exp ( - | \\xi _ { p } | ) - ( 1 + | \\xi _ { q } | ) \\exp ( - | \\xi _ { q } | ) } { 2 F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "6535.png", "formula": "\\begin{align*} \\P \\{ \\ell ^ { ( A ) } _ i > m - i \\} \\leq \\sum \\limits _ { k = 0 } ^ \\infty \\mu ( k ) \\left ( 1 \\wedge k \\P \\left \\{ \\exists t > 0 : t \\leq \\sum \\limits ^ { i + S _ { t } } _ { z = i + 1 } \\frac { 1 } { A ( z ) } S _ { t } \\geq m - i \\right \\} \\right ) . \\end{align*}"}
{"id": "5363.png", "formula": "\\begin{align*} \\langle m _ f ( u ) , v \\rangle \\vcentcolon = \\lim _ { i \\to \\infty } \\langle f , u _ i v _ i \\rangle \\mbox { f o r a l l } ( u , v ) \\in H ^ r ( \\R ^ n ) \\times H ^ { - t } ( \\R ^ n ) , \\end{align*}"}
{"id": "3953.png", "formula": "\\begin{align*} \\xi _ { ( w _ i ) _ i , \\phi } ^ { \\mathbf { G } } ( y _ { \\sigma ( 1 ) } ^ 1 , \\dots , y _ { \\sigma ( d ) } ^ d ) = \\widehat { \\xi ^ \\mathbf { G } } ( \\sqrt { w _ 2 - w _ 1 } , \\dots , \\sqrt { w _ d - w _ { d - 1 } } ) . \\end{align*}"}
{"id": "1341.png", "formula": "\\begin{align*} \\phi ( x ) = e ^ { \\omega \\langle x \\rangle } . \\end{align*}"}
{"id": "883.png", "formula": "\\begin{align*} R = R _ 0 \\supseteq R _ 1 \\supseteq R _ 2 \\supseteq \\cdots \\end{align*}"}
{"id": "5456.png", "formula": "\\begin{align*} r ^ C _ A ( b ) = l ^ B _ A ( r ^ C _ A ( b ) ) r ^ B _ A ( r ^ C _ A ( b ) ) . \\end{align*}"}
{"id": "5395.png", "formula": "\\begin{align*} \\Theta _ { \\gamma } \\colon \\R ^ { 2 n } \\to \\R ^ { n \\times n } , \\Theta _ { \\gamma } ( x , y ) \\vcentcolon = \\gamma ^ { 1 / 2 } ( x ) \\gamma ^ { 1 / 2 } ( y ) \\mathbf { 1 } _ { n \\times n } \\end{align*}"}
{"id": "5009.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) ) ^ 2 \\Psi ^ { n , 5 } _ s d s \\\\ & = \\kappa _ 5 \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "2883.png", "formula": "\\begin{align*} F ( x _ 1 , x _ 2 , \\dotsc , x _ { n } ) = \\big ( f ( x _ 1 , x _ 2 , \\dotsc , x _ { k } ) , f ( x _ 2 , x _ 3 , \\dotsc , x _ { k + 1 } ) , \\dotsc , f ( x _ k , x _ 1 , \\dotsc , x _ { k - 1 } ) \\big ) . \\end{align*}"}
{"id": "586.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i \\leq n } \\sum _ { i + 1 < j \\leq n } \\mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) = & V _ k ( n ) + O _ { \\alpha } \\Big ( \\sum _ { 1 \\leq i \\leq n } \\sum _ { i + 1 < j \\leq n } \\frac { \\tau _ 3 ( i ) \\tau _ 3 ( i + 1 ) \\tau _ 3 ( j ) \\tau _ 3 ( j + 1 ) } { \\sqrt { i } } \\Big ) \\\\ & + O _ { \\alpha } \\Big ( \\sum _ { 1 \\leq i \\leq n } \\sum _ { i + 1 < j \\leq n } \\frac { \\tau _ 3 ( j ) \\tau _ 3 ( j + 1 ) } { \\sqrt { j - i - 1 } } \\Big ) , \\end{align*}"}
{"id": "6711.png", "formula": "\\begin{align*} q _ { l , n } ( \\lambda _ 1 , \\lambda _ 2 ) : = q _ { l _ 1 , n _ 1 } ( \\lambda _ 1 ) \\cdot q _ { l _ 2 , n _ 2 } ( \\lambda _ 2 ) , \\ ; \\ ; \\ ; \\ ; ( \\lambda _ 1 , \\lambda _ 2 ) \\in \\mathbb { C } \\times \\mathbb { C } \\cong \\mathfrak { a } ^ * _ \\mathbb { C } , \\end{align*}"}
{"id": "6145.png", "formula": "\\begin{align*} \\begin{aligned} | b _ h ( u , v , w ) | & \\leq \\| \\nabla \\times u \\| _ { 2 } \\| v \\| _ { 6 } \\| w \\| _ { 3 } \\\\ & \\leq C \\| \\nabla u \\| _ { 2 } \\| \\nabla v \\| _ { 2 } \\| w \\| _ { 2 } ^ { 1 / 2 } \\| \\nabla w \\| _ { 2 } ^ { 1 / 2 } , \\end{aligned} \\end{align*}"}
{"id": "1638.png", "formula": "\\begin{align*} \\mathcal { F } | M = \\{ F \\in \\mathcal { F } : F \\in \\mathcal { A D } ( M ) \\} . \\end{align*}"}
{"id": "4124.png", "formula": "\\begin{align*} T ^ { i , j } \\circ S ^ { i - 1 , j + 1 } \\circ T ^ { i , j } = T ^ { i , j } , \\end{align*}"}
{"id": "4224.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty \\sum _ { n = 1 } ^ { d _ i } f ( b _ { i } + 2 ^ i n ) = \\sum _ { n = 1 } ^ \\infty f ( a ( n ) ) = \\infty , \\end{align*}"}
{"id": "7167.png", "formula": "\\begin{align*} g ( t ^ m \\otimes u ) = e ^ { - 2 \\pi i m } ( t ^ m \\otimes g u ) . \\end{align*}"}
{"id": "4609.png", "formula": "\\begin{align*} & A _ i = \\bar { x } _ i + ( \\bar { f } + k _ { i - 1 } ) ( \\bar { x } _ i + 1 ) , \\\\ & A _ { i , j } = \\bar { f } + ( k _ { i - 1 } + 1 ) ( \\bar { x } _ i + 1 ) + ( k _ { j - 1 } + \\bar { x } _ i + 1 ) ( \\bar { x } _ j + 1 ) , \\end{align*}"}
{"id": "3611.png", "formula": "\\begin{align*} \\gamma _ i = ( \\underbrace { 0 , \\ldots , 0 } _ { | \\mathfrak { A } | } , \\underbrace { 2 , \\ldots , 2 } _ { | N _ G ( \\mathfrak { A } ) | } , { 1 , \\ldots , 1 } ) \\end{align*}"}
{"id": "1793.png", "formula": "\\begin{align*} y ^ * - w ^ * - \\sum _ { j = 1 } ^ m \\lambda _ j \\widetilde { u } _ j \\in U ^ \\perp \\subseteq \\overline { K ^ { ( \\alpha ) } } . \\end{align*}"}
{"id": "7385.png", "formula": "\\begin{align*} \\psi _ n ( y ) : = \\psi _ n ( T _ i y ) \\quad \\textrm { o n } \\ \\ { \\C } _ i , \\ i = 2 , 3 , 4 . \\end{align*}"}
{"id": "8663.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\overline f > \\tau \\} } ( r ) r ^ p g ( r ) \\ , d r & = \\int _ 0 ^ { b _ \\tau } r ^ p g ( r ) \\ , d r \\leq b _ \\tau \\sup _ { \\beta > 0 } \\beta ^ { - 1 } \\int _ 0 ^ \\beta r ^ p g ( r ) \\ , d r \\\\ & = \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\overline f > \\tau \\} } ( r ) \\ , d r \\ \\underline \\nu _ p ( g ) \\ , . \\end{align*}"}
{"id": "4250.png", "formula": "\\begin{align*} p _ n = \\frac { 1 } { 2 } \\frac { \\mathbb { G } ( n - 1 ) + \\mathbb { G } ( n + 1 ) - 2 \\mathbb { G } ( n ) } { \\mathbb { G } ( n - 1 ) - \\mathbb { G } ( n + 1 ) } . \\end{align*}"}
{"id": "6286.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) - y ( x ) = 0 ( x \\in \\mathbb { R } ) \\end{align*}"}
{"id": "6211.png", "formula": "\\begin{align*} u ' ( x ) + p ( x ) u ( x ) + r ( x ) = 0 . \\end{align*}"}
{"id": "885.png", "formula": "\\begin{align*} a _ m z ^ m = a _ { m _ 1 } z ^ { m _ 1 } \\cdots a _ { m _ n } z ^ { m _ n } \\end{align*}"}
{"id": "7672.png", "formula": "\\begin{align*} \\lambda _ { m a x } = \\frac { g } { 2 } + ( e ^ 2 + f ^ 2 ) ^ { 1 / 2 } , \\lambda _ { m i n } = \\frac { g } { 2 } - ( e ^ 2 + f ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "7986.png", "formula": "\\begin{align*} H _ { ( Y _ + , D _ { Y , + } ) } ( y ) = e ^ { \\frac { t _ + } { 2 \\pi i } } \\sum _ { d \\in \\mathbb K _ { + } } y ^ d \\left ( \\frac { \\prod _ { i = 1 } ^ k \\Gamma ( 1 + \\frac { v _ i } { z } + v _ i \\cdot d ) } { \\prod _ { i \\in M _ 0 } \\Gamma ( 1 + \\frac { \\bar D _ i } { z } + D _ i \\cdot d ) } \\right ) \\textbf { 1 } _ { [ d ] } [ \\textbf { 1 } ] _ { ( D _ i \\cdot d ) _ { i \\in I _ + } } . \\end{align*}"}
{"id": "4796.png", "formula": "\\begin{align*} { \\rm k } _ { m , n } ( z , z ) = \\sum _ { k = 0 } ^ { n - 1 } | P _ { m , k } ( z ) | ^ 2 , \\end{align*}"}
{"id": "1637.png", "formula": "\\begin{align*} \\mathcal { F } _ B = \\{ F \\in \\mathcal { F } : B \\sqsubset F \\} . \\end{align*}"}
{"id": "4784.png", "formula": "\\begin{align*} \\left . \\left ( \\prod _ { x \\in X / S } \\left ( 1 + a _ x t + b _ x t + a _ x b _ x t \\right ) \\right ) \\right \\rvert _ { t = 1 } . \\end{align*}"}
{"id": "8689.png", "formula": "\\begin{align*} \\frac { \\partial h } { \\partial x ^ i } - \\frac { d } { d t } \\left ( \\frac { \\partial h } { \\partial \\dot { x } ^ i } \\right ) = 0 , ~ i = 1 , 2 . \\end{align*}"}
{"id": "5979.png", "formula": "\\begin{align*} u ( t ) = u ^ 0 + \\int _ 0 ^ t k _ 1 ( t - \\tau ) ( f ( \\tau ) - A u ^ 0 ) d \\tau , \\end{align*}"}
{"id": "66.png", "formula": "\\begin{align*} G _ 2 ( k ) & = \\sum _ { h = 1 } ^ { \\lfloor k / 2 \\rfloor } \\frac { 1 } { ( 1 + k - h ) ^ { 2 s } ( 1 + h ) ^ { 2 r } } \\lesssim \\frac { 1 } { k ^ { 2 r } } \\sum _ { h = 1 } ^ { \\lfloor k / 2 \\rfloor } \\frac { 1 } { ( 1 + k - h ) ^ { 2 ( s - r ) } ( 1 + h ) ^ { 2 r } } \\lesssim \\frac { 1 } { k ^ { 2 r } } \\sum _ { h = 1 } ^ { \\lfloor k / 2 \\rfloor } \\frac { 1 } { h ^ { 2 s } } \\end{align*}"}
{"id": "2115.png", "formula": "\\begin{align*} y _ i ( v ) = \\begin{cases} 1 & v = u _ 0 , \\\\ - 1 & v = u _ i , \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "5197.png", "formula": "\\begin{align*} \\sum _ { i \\in I } 2 d _ i + \\frac { 2 \\sum _ { i \\in I ( \\Gamma ) } a _ i + ( k _ 1 ( \\Gamma ) - 1 ) ( r - 2 ) } { r } & + \\frac { 2 \\sum _ { i \\in I ( \\Gamma ) } b _ i + ( k _ 2 ( \\Gamma ) - 1 ) ( s - 2 ) } { s } \\\\ & = 2 | I | + k _ 1 ( \\Gamma ) + k _ 2 ( \\Gamma ) - 2 . \\end{align*}"}
{"id": "5769.png", "formula": "\\begin{align*} E ( G / T ) = E ( G / H ) E ( H / T ) = E ( N ) E ( H / T ) = \\pm E ( H / T ) \\neq 0 , \\end{align*}"}
{"id": "4899.png", "formula": "\\begin{align*} C = h _ 1 h _ { 2 3 } + h _ 2 h _ { 3 1 } + h _ 3 h _ { 1 2 } \\end{align*}"}
{"id": "8958.png", "formula": "\\begin{align*} \\| u \\| _ { U ^ p ( E ) } = \\sup _ { v \\in V ^ { p ' } ( E ' ) : \\| v \\| _ { V ^ { p ' } ( E ' ) } = 1 } | B ( u , v ) | . \\end{align*}"}
{"id": "7584.png", "formula": "\\begin{align*} \\mbox { d e x p } ^ { - 1 } _ { \\Omega } ( H ) = \\sum _ { j = 0 } ^ { \\infty } \\frac { B _ j } { j ! } \\operatorname { a d } _ { \\Omega } ^ j ( H ) , \\ , \\end{align*}"}
{"id": "3595.png", "formula": "\\begin{align*} \\overline { I ^ n } = ( \\{ t ^ a \\mid a / n \\in { \\rm N P } ( I ) \\} ) = ( \\{ t ^ a \\mid \\langle a , u _ i \\rangle \\geq n \\mbox { f o r } i = 1 , \\ldots , p \\} ) \\end{align*}"}
{"id": "2514.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\{ g ( \\nabla _ U \\xi , V ) + g ( \\nabla _ V \\xi , U ) \\} + R i c ( U , V ) + \\mu g ( U , V ) = 0 . \\end{align*}"}
{"id": "6758.png", "formula": "\\begin{align*} \\binom { z } { k } : = \\frac { z ( z - 1 ) \\cdots ( z - k + 1 ) } { k ! } , \\binom { - z } { k } = ( - 1 ) ^ k \\binom { z + k - 1 } { k } . \\end{align*}"}
{"id": "4324.png", "formula": "\\begin{align*} \\int _ { \\mathcal { M } _ 3 } \\mathcal { I } = \\int _ { \\mathcal { M } _ 1 } \\mathcal { I } + \\int _ { \\mathcal { M } _ 2 } \\mathcal { I } . \\end{align*}"}
{"id": "1251.png", "formula": "\\begin{align*} \\mathcal { G } _ j = \\dfrac { W _ { l ( T ) + 1 - j } } { W _ { l ( T ) - j } } \\end{align*}"}
{"id": "2141.png", "formula": "\\begin{align*} d \\sigma _ { \\mu _ { n } } ( s ) = n \\ , \\frac { b _ { n } ^ { 2 } + s ^ { 2 } } { 1 + s ^ { 2 } } \\ , d \\sigma _ { \\mu } ( s / b _ { n } ) , \\end{align*}"}
{"id": "4311.png", "formula": "\\begin{align*} = 2 \\sum ( ) + \\sum ( ) + \\sum ( ) + n . \\end{align*}"}
{"id": "2402.png", "formula": "\\begin{align*} & \\| ( - A _ n ) ^ { \\frac { 1 } { 2 } } \\exp ( \\ ! - A _ n ^ 2 ( t \\ ! - \\ ! s ) ) \\Sigma _ n ( U ( s ) ) \\| ^ 2 _ { \\mathrm F } \\ ! = \\ ! \\sum _ { k , l = 1 } ^ { n - 1 } \\langle ( - A _ n ) ^ { \\frac { 1 } { 2 } } \\exp ( - A _ n ^ 2 ( t \\ ! - \\ ! s ) ) \\Sigma _ n ( U ( s ) ) e _ k , e _ l \\rangle ^ 2 \\\\ & = \\sum _ { l = 1 } ^ { n - 1 } ( - \\lambda _ { l , n } ) \\exp ( - 2 \\lambda ^ 2 _ { l , n } ( t - s ) ) \\| \\Sigma _ n ( U ( s ) ) e _ l \\| ^ 2 \\le C _ \\epsilon \\sum _ { l = 1 } ^ { n - 1 } ( - \\lambda _ { l , n } ) ^ { - \\frac { 1 } { 2 } - 2 \\epsilon } ( t - s ) ^ { - \\frac { 3 } { 4 } - \\epsilon } , \\end{align*}"}
{"id": "3375.png", "formula": "\\begin{align*} \\bigcup _ { k = 1 } ^ { \\infty } \\alpha ^ 1 _ k ( A _ { 1 , k } ) = \\bigcup _ { m = 1 } ^ { \\infty } \\alpha ^ m _ 1 ( A _ { m , 1 } ) \\end{align*}"}
{"id": "530.png", "formula": "\\begin{align*} L _ { n s } ( \\dot { x } , x , t ) = \\frac { 1 } { C _ 1 f ^ 2 ( t ) [ f ( t ) \\dot { x } - a _ o x + C _ 2 ] } \\ , \\end{align*}"}
{"id": "5902.png", "formula": "\\begin{align*} B _ r ( \\nu ) : = \\{ \\mu \\in \\mathcal { M } _ 1 ( \\mathbb { R } ^ k ) \\colon \\rho _ \\mathrm { L P } ( \\mu , \\nu ) < r \\} \\end{align*}"}
{"id": "8675.png", "formula": "\\begin{align*} z \\cdot W ( \\sigma ) = R _ { \\varphi ( \\sigma ) } ( u ) \\wedge v + u \\wedge R _ { \\varphi ( \\sigma ) } ( v ) . \\end{align*}"}
{"id": "3858.png", "formula": "\\begin{align*} N _ c ^ R : & = \\big \\{ y \\in N : \\exists ( y _ i ) _ { i = 1 , . . . , n } \\in N , y _ 1 = y , y _ n = x , ( y _ i , y _ { i + 1 } ) \\in E \\backslash c \\big \\} , \\\\ E _ c ^ R : & = \\big \\{ ( y , z ) \\in E : y , z \\in N _ c ^ R \\big \\} , H _ c ^ R : = \\big \\{ h \\cap N _ c ^ R : h \\in H \\big \\} \\backslash \\{ \\emptyset \\} , \\\\ L _ c ^ R : & N _ c ^ R \\to \\{ 1 , . . . , d \\} , L _ c ^ R = L | _ { N _ c ^ R } \\end{align*}"}
{"id": "5702.png", "formula": "\\begin{align*} k ! ( x _ i - x _ { i + 1 } ) e _ k ( x _ 1 , \\ldots , x _ i ) = 0 . \\end{align*}"}
{"id": "6166.png", "formula": "\\begin{align*} d i s t ^ { 2 } ( y , \\mathbf { S } ^ { 3 } ) = ( \\frac { | l _ { 1 } | ^ { 2 k } + | l _ { 2 } | ^ { 2 k } } { | l _ { i } | ^ { 2 k } } ) | v _ { i } | ^ { 2 } : = | s | _ { h } ^ { 2 } \\end{align*}"}
{"id": "6659.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t = \\Delta u - b | \\nabla u | ^ { q } + | u | ^ { p - 1 } u , \\ & x \\in \\Omega , t > 0 \\\\ u = 0 , \\ & x \\in \\partial \\Omega , t > 0 \\\\ u ( x , 0 ) = \\phi ( x ) \\geq 0 , \\ & x \\in \\Omega , \\end{array} \\right . \\end{align*}"}
{"id": "2269.png", "formula": "\\begin{align*} & \\int _ 0 ^ r \\left [ ( \\phi '' + \\frac { z } { 2 } \\phi ' - \\frac { z } { 2 } e _ \\sigma \\psi ' ) \\cdot \\phi '' + ( \\psi '' + \\frac { z } { 2 } \\psi ' - \\frac { z } { 2 } e _ \\sigma \\phi ' ) \\cdot \\psi '' \\right ] { \\rm { d } } z \\\\ & = \\int _ 0 ^ r | ( \\phi '' , \\psi '' ) | ^ 2 + \\frac { 1 } { 2 } \\int _ 0 ^ r z ( \\phi ' \\phi '' + \\psi ' \\psi '' ) - \\frac { 1 } { 2 } \\int _ 0 ^ r z e _ \\sigma ( \\psi ' \\phi '' + \\phi ' \\psi '' ) . \\end{align*}"}
{"id": "602.png", "formula": "\\begin{align*} \\xi _ Q ( s ) = ( \\det Q ) ^ { - 1 / 2 } \\xi _ { Q ^ { - 1 } } ( \\alpha / 2 - s ) , \\end{align*}"}
{"id": "655.png", "formula": "\\begin{align*} D L _ { ( g , P ( g ) ) } ( 0 , h ) = 2 \\Delta ^ { P ( g ) } _ { g } \\left ( 2 \\Delta ^ { P ( g ) } _ { g } h + h \\right ) - ( 4 \\pi ) ^ { - \\frac { n } { 2 } } \\int _ { M } h e ^ { - P ( g ) } \\ , d v _ { g } , \\quad g \\in \\mathcal { U } . \\end{align*}"}
{"id": "4698.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ 0 ( n ) \\ , q ^ n = \\frac { - 1 } { ( q ; q ) ^ 2 _ \\infty } \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ { n } q ^ { n ( n + 1 ) / 2 } \\end{align*}"}
{"id": "7329.png", "formula": "\\begin{align*} P _ 1 \\theta & = \\Delta \\theta - q { \\sf U } _ \\infty ^ { q - 1 } \\theta \\\\ & = - f ( { \\sf U } _ \\infty ) - \\sum _ { i = 1 } ^ N \\tfrac { f ^ { ( i ) } ( { \\sf U } _ \\infty ) } { i ! } ( \\theta - \\theta _ L ) ^ i + \\sum _ { i = 2 } ^ N \\tfrac { f _ 2 ^ { ( i ) } ( { \\sf U } _ \\infty ) } { i ! } ( \\theta - \\theta _ L ) ^ i . \\end{align*}"}
{"id": "7967.png", "formula": "\\begin{align*} I _ { \\mathbb P ^ 2 } ( y , z ) = z e ^ { H \\log y / z } \\sum _ { d \\geq 0 } y ^ { d } \\left ( \\frac { 1 } { \\prod _ { a = 1 } ^ d ( H + a z ) ^ 3 } \\right ) , \\end{align*}"}
{"id": "2272.png", "formula": "\\begin{align*} J _ 1 & = \\int _ 0 ^ r \\left [ \\frac { z } { 2 } ( \\bar \\phi + e _ \\delta - \\delta ) ( \\bar \\phi ' , \\bar \\psi ' ) \\cdot z ^ { 2 } ( \\phi , \\psi ) \\right ] { \\rm { d } } z \\\\ & \\leq \\Vert z ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert z ( \\bar \\phi ' , \\bar \\psi ' ) \\Vert _ { L ^ 2 } \\cdot \\frac { 1 } { 2 } \\left [ \\Vert z \\bar \\phi \\Vert _ { L ^ \\infty } + \\Vert z ( e _ \\delta - \\delta ) \\Vert _ { L ^ \\infty } \\right ] , \\end{align*}"}
{"id": "7535.png", "formula": "\\begin{align*} I _ 1 = & \\bigg \\| 1 _ { \\{ | T _ { n } - 1 | \\le 1 \\} } \\sup _ { t \\in [ 0 , 1 ] } | X _ { n } ( t ) - B ( T _ { k } ) | \\bigg \\| _ { L ^ { p } } \\\\ \\le & \\bigg \\| 1 _ { \\{ | T _ { n } - 1 | \\le 1 \\} } \\max _ { 0 \\le k \\le n - 1 } | \\zeta _ { k + 1 } | \\bigg \\| _ { L ^ { p } } \\\\ \\le & \\bigg \\| \\max _ { 0 \\le k \\le n - 1 } | \\zeta _ { k + 1 } | \\bigg \\| _ { L ^ { p } } \\\\ \\le & C n ^ { - \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { p } \\right ) } . \\end{align*}"}
{"id": "5704.png", "formula": "\\begin{align*} X _ { \\{ i \\} } \\cap \\Omega _ { J } = \\emptyset \\end{align*}"}
{"id": "4182.png", "formula": "\\begin{align*} F ^ { ( \\iota ) } ( \\sqrt L ) f _ j = \\frac { 1 } { 2 \\pi } \\int _ { 2 ^ { \\iota - 1 } \\le | \\tau | \\le 2 ^ { \\iota + 1 } } \\chi _ \\iota ( \\tau ) \\hat F ( \\tau ) \\cos ( \\tau \\sqrt L ) f _ j \\ , d \\tau . \\end{align*}"}
{"id": "2577.png", "formula": "\\begin{align*} \\mathbf S | _ { \\mathcal H ^ 0 _ { 1 2 3 } } = \\mathbf S ^ 0 _ { 1 2 3 } \\ , \\end{align*}"}
{"id": "7451.png", "formula": "\\begin{align*} \\| U ^ n \\| _ { \\mathcal { H } } = \\| \\left ( u ^ n , v ^ n , y ^ n , z ^ n , w ^ n , \\eta ^ n \\right ) ^ { \\top } \\| _ { \\mathcal { H } } = 1 , \\end{align*}"}
{"id": "3010.png", "formula": "\\begin{align*} & L _ { x y } ^ 2 = - \\frac { 4 n - 1 } { 2 n ( 4 n + 1 ) } , \\ \\ R _ 0 ^ 2 = R _ 1 ^ 2 = - \\frac { 2 n + 1 } { 2 ( 4 n + 1 ) } , \\\\ & L _ { x y } \\cdot R _ { 0 } = L _ { x y } \\cdot R _ { 1 } = \\frac { 1 } { 4 n + 1 } , \\ \\ R _ 0 \\cdot R _ { 1 } = \\frac { n } { 4 n + 1 } . \\\\ \\end{align*}"}
{"id": "1819.png", "formula": "\\begin{align*} s _ { n } = \\sum _ { \\gamma \\in \\mathcal { M } _ { n } } w ( \\gamma ) . \\end{align*}"}
{"id": "6299.png", "formula": "\\begin{align*} & \\int \\frac { x ^ 2 } { ( x ^ 2 q ^ { - 1 } ( 1 - q ) ; q ^ 2 ) _ \\infty } \\sin ( x ; q ) d _ q x = \\frac { q } { ( \\frac { x ^ 2 } { q } ( 1 - q ) ; q ^ 2 ) _ \\infty } \\Big ( x \\sin ( \\frac { x } { q } ; q ) - \\cos ( q ^ \\frac { - 1 } { 2 } x ; q ) \\Big ) , \\end{align*}"}
{"id": "5640.png", "formula": "\\begin{align*} ( \\psi ^ 2 + \\psi F ) = ( \\psi + \\frac { F } { 2 } ) ^ 2 - \\frac { F ^ 2 } { 4 } \\geq ( - \\frac { F } { 2 } + \\frac { \\alpha p - \\beta } { k - 1 } ) ^ 2 + \\frac { 1 } { ( k - 1 ) ^ 2 ( t - \\tau ) ^ 2 } - \\frac { F ^ 2 } { 4 } \\end{align*}"}
{"id": "915.png", "formula": "\\begin{align*} P ( u ) \\cdot v = \\begin{bmatrix} \\nabla u \\\\ \\nabla \\nabla u \\end{bmatrix} v = \\begin{bmatrix} f \\\\ h \\end{bmatrix} , v ( x ) \\perp \\mathrm { K e r } P ( u ) ( x ) . \\end{align*}"}
{"id": "4524.png", "formula": "\\begin{align*} a = a _ 0 \\cos \\theta - b _ 0 \\rho ^ { - 1 } \\sin \\theta , \\ ; \\ ; b = a _ 0 \\sin \\theta + b _ 0 \\rho ^ { - 1 } \\cos \\theta \\end{align*}"}
{"id": "8039.png", "formula": "\\begin{align*} s = - 1 - l , l \\geq 0 s = - \\frac { H } { z } - \\frac { m } { 3 } , m \\geq 1 . \\end{align*}"}
{"id": "1575.png", "formula": "\\begin{align*} d V _ { B H } = \\frac { \\int \\limits _ { 0 } ^ { \\pi } d t } { \\int \\limits _ { 0 } ^ { \\pi } ( b ' \\cos t ) ^ 2 d t } \\sqrt { \\det A } d x = \\frac { 2 } { b '^ 2 } \\sqrt { \\det A } d x . \\end{align*}"}
{"id": "6585.png", "formula": "\\begin{align*} f ( y , v _ 1 , \\ldots , v _ k ) - f ( x , v _ 1 , \\ldots , v _ k ) = \\int _ 0 ^ 1 d f ( x + t ( y - x ) , v _ 1 , \\ldots , v _ k , y - x , 0 , \\ldots , 0 ) \\ , d t \\end{align*}"}
{"id": "3921.png", "formula": "\\begin{align*} { } \\bar { \\pi } _ { h , { T _ n } } ( x ) & = \\frac { 1 } { T _ n \\prod _ { l = 1 } ^ d h _ l } \\int _ 0 ^ { T _ n } \\prod _ { m = 1 , 2 } K ( \\frac { x _ m - X _ u ^ m } { h _ m } ) \\prod _ { l = 3 } ^ d K ( \\frac { x _ l - X _ { \\varphi _ { n , l } ( u ) } ^ l } { h _ l } ) d u \\\\ & = : \\frac { 1 } { T _ n } \\int _ 0 ^ { T _ n } \\prod _ { m = 1 , 2 } K _ { h _ m } ( x _ m - X _ u ^ m ) \\prod _ { l = 3 } ^ d K _ { h _ l } ( x _ l - X _ { \\varphi _ { n , l } ( u ) } ^ l ) d u . \\end{align*}"}
{"id": "1300.png", "formula": "\\begin{align*} \\mathcal { G } _ 1 ( r ( T ) , x ) & = x \\displaystyle \\prod _ { j = 1 } ^ { k } \\mathcal { G } _ 1 ( r ( T _ j ) , x ) - \\displaystyle \\sum _ { j = 1 } ^ { k } \\left ( \\alpha _ j \\ \\mathcal { G } _ 2 ( r ( T _ j ) , x ) \\displaystyle \\prod _ { i = 1 , i \\neq j } ^ { k } \\mathcal { G } _ 1 ( r ( T _ i ) , x ) \\right ) . \\end{align*}"}
{"id": "4879.png", "formula": "\\begin{align*} Q = \\left \\{ \\begin{array} { l l l } x ^ { 3 / 2 } , & i f \\ k = 2 , \\\\ x ^ { 1 2 / 7 } , & i f \\ k = 3 . \\end{array} \\right . \\end{align*}"}
{"id": "5013.png", "formula": "\\begin{align*} R ^ { n , 2 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int _ 0 ^ \\tau \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) \\left [ \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) ^ 2 - \\int _ 0 ^ s \\psi ^ 2 _ { n , 1 } ( u , s ) d u \\right ] d s , \\end{align*}"}
{"id": "1236.png", "formula": "\\begin{align*} \\mathcal { G } ( u _ 1 , x ) & = \\mathcal { G } ( u _ 2 , x ) = \\mathcal { G } ( u _ 3 , x ) = \\mathcal { G } ( u _ 4 , x ) = \\mathcal { G } ( u _ 5 , x ) = x , \\\\ \\mathcal { G } ( u _ 6 , x ) & = x - \\frac { 2 } { x } = \\dfrac { x ^ 2 - 2 } { x } , \\\\ \\mathcal { G } ( u _ 7 , x ) & = x - \\frac { 3 } { x } = \\dfrac { x ^ 2 - 3 } { x } , \\\\ \\mathcal { G } ( u _ 8 , x ) & = x - \\frac { x } { x ^ 2 - 2 } - \\frac { x } { x ^ 2 - 3 } = \\frac { x ^ 5 - 7 x ^ 3 + 1 1 x } { ( x ^ 2 - 2 ) ( x ^ 2 - 3 ) } . \\end{align*}"}
{"id": "3804.png", "formula": "\\begin{align*} H y p ^ 4 ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta + \\sigma + \\kappa ) + i s ( \\mu _ 3 | \\kappa | + \\mu _ 2 | \\sigma | + \\mu | \\xi | + \\mu _ 1 | \\eta | ) } \\end{align*}"}
{"id": "6115.png", "formula": "\\begin{align*} \\abs { w } _ { X \\cup \\mathcal { P } } = \\abs { v } \\geq \\frac { \\abs { \\iota ( w ) } } { N } - ( N - 2 \\mu ) \\end{align*}"}
{"id": "4399.png", "formula": "\\begin{align*} & ( 1 ) \\ , \\ , \\left ( s + \\frac { s '^ { 2 } } { u '' s - s '' } \\right ) e ^ { u - t } = 1 , \\\\ & ( 2 ) \\ , \\ , s ' - s u ' = 1 , \\end{align*}"}
{"id": "3958.png", "formula": "\\begin{align*} j ( a * b ) ( \\gamma ) = \\sum _ { \\eta \\in G ^ { r ( \\gamma ) } } j ( a ) ( \\eta ) \\ , j ( b ) ( \\eta ^ { - 1 } \\gamma ) \\end{align*}"}
{"id": "7045.png", "formula": "\\begin{align*} \\sum _ { M \\geq 0 } g ( M ) \\ , z ^ M & = \\sum _ { M \\geq 0 } z ^ M \\sum _ { 0 \\leq 2 n \\leq M - 1 } ( M - 2 n ) \\binom { M - n - 1 } n \\\\ & = \\sum _ { M , n \\geq 0 } z ^ { M + 2 n + 1 } ( M + 1 ) \\binom { M + n } n \\\\ & = \\sum _ { M \\geq 0 } { z ^ { M + 1 } ( M + 1 ) } { ( 1 - z ^ 2 ) ^ { - ( M + 1 ) } } = \\sum _ { M \\geq 1 } { z ^ M M } { ( 1 - z ^ 2 ) ^ { - M } } \\\\ & = \\frac { \\frac { z } { 1 - z ^ 2 } } { \\left ( 1 - \\frac { z } { 1 - z ^ 2 } \\right ) ^ 2 } = \\frac { z - z ^ 3 } { ( 1 - z - z ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "5847.png", "formula": "\\begin{align*} V _ r ( a , b ) = \\sup _ { f \\in \\mathcal { M } ^ + ( a , b ) } \\frac { \\bigg ( \\int _ { a } ^ { b } f ^ r v \\bigg ) ^ { \\frac { 1 } { r } } } { \\int _ { a } ^ { b } f } . \\end{align*}"}
{"id": "8919.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\int _ { B _ R ( \\bar x _ n ^ i ) } ( v _ n ^ i ) ^ 2 \\ , d x \\to \\int _ { B _ R ( \\bar x ^ i ) } ( V ^ i ) ^ 2 \\ , d x , \\end{align*}"}
{"id": "6655.png", "formula": "\\begin{align*} f ( x ) = e ^ { - c x } ( 1 - x ) ^ c \\ ; , \\ , \\end{align*}"}
{"id": "336.png", "formula": "\\begin{align*} d _ k = g c _ i ^ { ( k ) } = h d _ { k - 1 } , \\end{align*}"}
{"id": "1828.png", "formula": "\\begin{align*} m _ { 1 } ( z ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { A _ { n } ^ { ( 1 ) } } { z ^ { n + 1 } } . \\end{align*}"}
{"id": "690.png", "formula": "\\begin{align*} u \\leq e ^ { C _ 1 s _ 0 ^ { - \\theta / 2 } ( s - s _ 0 ) } \\hat { u } + C s _ 0 ^ { - \\theta / 4 } e ^ { C _ 1 s _ 0 ^ { - \\theta / 2 } ( s - s _ 0 ) } & \\leq e ^ { C _ 1 s _ 0 ^ { - \\theta / 2 } \\log A } \\left ( \\hat { u } + C s _ 0 ^ { - \\theta / 4 } \\right ) \\\\ & \\leq \\left ( 1 + 2 C _ 1 s _ 0 ^ { - \\theta / 2 } \\log A \\right ) \\left ( \\hat { u } + C s _ 0 ^ { - \\theta / 4 } \\right ) ; \\end{align*}"}
{"id": "8175.png", "formula": "\\begin{align*} U _ 0 ( t ) f = e ^ { i \\Xi ( y , t ) } ( 2 i t ) ^ { - \\frac { 1 } { 2 } } \\hat { f } ( \\frac { y } { 2 t } ) \\end{align*}"}
{"id": "2534.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( L _ { Z _ 1 } g ) ( X _ 1 , Y _ 1 ) = \\frac { 1 } { 2 } \\{ e ^ { - 2 x _ 2 } a _ 1 a _ 4 a _ 5 + e ^ { - 2 x _ 2 } a _ 2 a _ 4 a _ 5 - 2 e ^ { - 2 x _ 2 } a _ 1 a _ 3 a _ 6 \\} . \\end{align*}"}
{"id": "321.png", "formula": "\\begin{align*} A = \\bigcup _ { j \\ge 1 } \\bigg ( S _ j \\setminus \\bigcup _ { 1 \\le i < j } { \\rm M } _ { I _ i } \\bigg ) . \\end{align*}"}
{"id": "7236.png", "formula": "\\begin{align*} & \\dot { x } ( t , s ) = \\lambda x ( t , s ) + \\partial _ s ^ 2 x ( t , s ) , s \\in [ 0 , 1 ] , \\\\ & \\dot { \\hat x } ( t , s ) = \\lambda \\hat x ( t , s ) + \\partial _ s ^ 2 \\hat x ( t , s ) + l ( s ) \\left ( \\partial _ s x ( t , 1 ) - \\partial _ s \\hat x ( t , 1 ) \\right ) , \\\\ & \\partial _ s x ( t , 0 ) = 0 , x ( t , 1 ) = 0 , \\\\ & \\partial _ s \\hat x ( t , 0 ) = 0 , \\hat x ( t , 1 ) = 0 . \\end{align*}"}
{"id": "6786.png", "formula": "\\begin{align*} f ( P ) - f ( P - i ' 0 ) = a _ { i ' 0 } \\left ( x ^ * _ { i ' 0 } + \\sum _ { i j \\in \\bar P } ( 1 - x ^ * _ { i j } ) - 1 \\right ) . \\end{align*}"}
{"id": "5090.png", "formula": "\\begin{align*} P ^ n _ \\tau & : = n ^ { \\alpha + \\frac 1 2 } \\int ^ { \\tau } _ 0 ( t - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\ , \\left ( \\int ^ s _ { \\eta _ n ( s ) } ( s - \\eta _ n ( u ) ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) d s \\\\ & = n ^ { \\alpha + \\frac 1 2 } \\int ^ { \\tau } _ 0 ( t - s ) ^ { \\alpha } \\sigma ' ( X _ { s } ) \\sigma ( X _ { \\eta _ n ( s ) } ) \\Xi ^ { n , 2 } _ s d s , \\end{align*}"}
{"id": "6120.png", "formula": "\\begin{align*} r _ k = \\max \\left \\{ r \\left | \\frac { j } { n } \\right | ^ { \\frac { 1 } { n - j } } , \\ , j = 1 , 2 , \\dots , k \\right \\} \\end{align*}"}
{"id": "921.png", "formula": "\\begin{align*} \\begin{aligned} r _ { i j } ^ n w _ n d x ^ i \\otimes d x ^ j : = & \\{ 2 R _ i { } ^ k { } _ j { } ^ n \\nabla _ k w _ n + R _ { i } { } ^ k { } _ j { } ^ m w _ n ( - \\Gamma _ { k m } ^ n ) + \\nabla ^ k ( { R _ { i k j } } ^ n ) w _ n + g ^ { k l } { R _ { i m j } } ^ n w _ n ( - \\Gamma _ { l k } ^ m ) \\\\ & + R _ i { } ^ m w _ n ( - \\Gamma _ { m j } ^ n ) \\} d x ^ i \\otimes d x ^ j . \\end{aligned} \\end{align*}"}
{"id": "5160.png", "formula": "\\begin{align*} B _ T = \\begin{pmatrix} A _ 1 & \\ 0 & \\cdots & \\ 0 \\\\ \\ 0 & A _ 2 & \\cdots & \\ 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\ 0 & \\ 0 & \\cdots & A _ { \\ell } \\\\ \\end{pmatrix} \\end{align*}"}
{"id": "2002.png", "formula": "\\begin{align*} \\nu _ p ( l ! ) = \\frac { l - \\sigma _ p ( l ) } { p - 1 } \\end{align*}"}
{"id": "3837.png", "formula": "\\begin{align*} E r r U ^ { b ; 3 } _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\sum _ { \\begin{subarray} { c } k _ 3 \\in \\Z _ + , n _ 3 \\in [ - M _ t , 2 ] \\cap \\Z \\\\ \\mu _ 3 \\in \\{ + , - \\} , i _ 3 \\in \\{ 0 , 1 , 2 , 3 , 4 \\} \\end{subarray} } \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { ( \\R ^ 3 ) ^ 3 } e ^ { i \\widetilde { \\Phi } ^ b _ { \\mu _ 1 , \\mu _ 2 } ( \\xi , \\eta , \\sigma ; s , X ( s ) , v ) } \\big ( \\sum _ { l _ 3 \\in [ - j _ 3 , 2 ] \\cap \\Z } E l l ^ { \\mu _ 3 , i _ 3 ; l _ 3 } _ { k _ 3 , j _ 3 ; n _ 3 } ( s , X ( s ) , V ( s ) ) \\end{align*}"}
{"id": "566.png", "formula": "\\begin{align*} 2 \\sum _ { 1 \\leq i < j \\leq n } f _ k ( i ) f _ k ( j ) = \\big ( \\sum _ { 1 \\leq i \\leq n } f _ k ( i ) \\big ) ^ 2 - \\sum _ { 1 \\leq i \\leq n } f _ k ^ 2 ( i ) = \\big ( \\sum _ { 1 \\leq i \\leq n } f _ k ( i ) \\big ) ^ 2 + O _ { \\varepsilon } ( n ^ { 1 + \\varepsilon } ) , \\end{align*}"}
{"id": "5787.png", "formula": "\\begin{align*} E ( G / A ) = E ( G / H ) E ( H / A ) \\neq 0 , \\end{align*}"}
{"id": "5177.png", "formula": "\\begin{align*} W = x ^ r + y ^ s . \\end{align*}"}
{"id": "7496.png", "formula": "\\begin{align*} t N _ n ( t , 1 ) = N _ n ( t , t ) . \\end{align*}"}
{"id": "853.png", "formula": "\\begin{align*} \\textrm { r a n k } & \\left ( \\mathbf { V } \\right ) = \\lim _ { \\varepsilon \\rightarrow 0 } \\frac { M \\log \\left ( \\frac { 1 } { \\varepsilon } \\right ) + \\log \\left | \\mathbf { V } + \\varepsilon \\mathbf { I } \\right | } { \\log \\left ( 1 + \\frac { 1 } { \\varepsilon } \\right ) } . \\end{align*}"}
{"id": "4009.png", "formula": "\\begin{align*} \\abs * { \\sum _ { t = 1 } ^ { T } ( f _ t ( M ) - F ( M ) ) } \\le C \\sqrt { 2 T H d \\log \\frac { 3 T ^ 2 } { \\delta } } \\end{align*}"}
{"id": "720.png", "formula": "\\begin{align*} & Q : C ^ { k , \\alpha } ( M \\times [ 0 , T ] ) \\rightarrow C ^ { k + 2 , \\alpha } ( M \\times [ 0 , T ] ) , \\\\ & Q : C ^ { k + 2 , \\alpha } ( M \\times [ 0 , T ] ) \\rightarrow s \\ C ^ { k + 2 , \\alpha } ( M \\times [ 0 , T ] ) . \\end{align*}"}
{"id": "5053.png", "formula": "\\begin{align*} K ^ { n , 4 } _ \\tau & : = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ { ( \\delta + \\frac 1 n ) \\wedge \\tau } \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) \\\\ & \\times \\left [ \\left ( \\int _ { \\eta _ n ( s ) - \\delta } ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) d W _ u \\right ) ^ 2 - \\left ( \\int _ 0 ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) d W _ u \\right ) ^ 2 \\right ] d s \\end{align*}"}
{"id": "3279.png", "formula": "\\begin{align*} \\int _ B e ^ { - i \\varphi } q ( x ) u _ 1 u _ 2 d x = & \\int e ^ { - i \\varphi } ( A + \\nabla \\vartheta ) \\cdot ( A _ 1 + A _ 2 ) u _ 1 u _ 2 d x \\cr & + \\int _ B i e ^ { - i \\varphi } ( A + \\nabla \\vartheta ) \\cdot ( u _ 1 \\nabla u _ 2 - u _ 2 \\nabla u _ 1 ) d x \\cr & + \\int _ B e ^ { - i \\varphi } ( A + \\nabla \\vartheta ) \\cdot \\nabla \\varphi u _ 1 u _ 2 d x + \\mathcal { R } \\cr : = & \\mathcal { J } _ 1 + \\mathcal { J } _ 2 + \\mathcal { J } _ 3 + \\mathcal { R } , \\end{align*}"}
{"id": "3323.png", "formula": "\\begin{align*} u ( x , t ) = \\frac 1 4 \\int _ 0 ^ \\infty \\exp ( - \\frac { k ^ 2 } { 8 } ) J _ 0 ( k | x | ) k \\cos ( k t ) d k , x \\in \\Omega ^ - , \\end{align*}"}
{"id": "2077.png", "formula": "\\begin{align*} \\overline { S } _ { t + s } ( \\nu ) = \\int _ E P ^ { \\overline { S } _ t ( \\nu ) } _ { s } ( y , \\cdot ) { \\left ( \\int _ E P ^ { \\nu } _ { t } ( x , \\dd y ) \\nu ( \\dd x ) \\right ) } = \\overline { S } _ { s } ( \\overline { S } _ { t } ( \\nu ) ) . \\end{align*}"}
{"id": "8524.png", "formula": "\\begin{align*} P _ k ^ * ( x ) & = \\int _ { - 1 } ^ 1 \\frac { P _ k ( x ) - P _ k ( t ) } { x - t } \\frac { 1 } { 2 } d t . \\end{align*}"}
{"id": "6007.png", "formula": "\\begin{align*} M _ { f _ 1 } ( 0 ) = M _ { f _ 1 } ( \\lambda _ 1 ) = ( \\mu _ { 1 , 0 } , \\mu _ { 1 , 1 } , \\mu _ { 1 , 2 } , \\mu _ { 1 , 3 } , \\mu _ { 1 , 4 } \\ ) = ( 3 , 2 , 1 , 0 , 0 ) \\end{align*}"}
{"id": "4177.png", "formula": "\\begin{align*} \\sum _ { \\ell = - 1 } ^ \\infty F _ \\ell ( L , U ) f = F ( \\sqrt L ) f . \\end{align*}"}
{"id": "146.png", "formula": "\\begin{align*} & \\eta _ t \\leq c \\log \\Big ( 1 + \\frac { 1 } { m ^ 2 t } \\Big ) { \\bf 1 } _ { d = 2 } + c m \\Big ( \\sqrt { 1 + \\frac { 1 } { t m ^ 2 } } - 1 \\Big ) { \\bf 1 } _ { d = 3 } , \\\\ & \\gamma _ t \\leq \\frac { c } { m ^ 2 ( m ^ 2 t + 1 ) } { \\bf 1 } _ { d = 2 } + c \\log \\Big ( 1 + \\frac { 1 } { m ^ 2 t } \\Big ) { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "1922.png", "formula": "\\begin{align*} S _ { 0 } ( z ) & = \\frac { 1 } { z - a _ { 0 } \\ , S _ { p - 1 } ^ { ( 1 ) } ( z ) } \\\\ S _ { j } ( z ) & = S _ { 0 } ( z ) \\ , S ^ { ( 1 ) } _ { j - 1 } ( z ) 1 \\leq j \\leq p . \\end{align*}"}
{"id": "926.png", "formula": "\\begin{align*} P _ c ( \\Psi _ t ) v ' = P _ c ( \\Psi _ t ) P ^ T ( \\Psi _ t ) [ P ( \\Psi _ t ) P ^ T ( \\Psi _ t ) ] ^ { - 1 } \\begin{bmatrix} 0 \\\\ h \\end{bmatrix} = ( I _ { \\frac { n ( n + 3 ) } { 2 } } - \\frac { 1 } { n } \\begin{bmatrix} 0 & 0 \\\\ 0 & J _ n \\end{bmatrix} ) \\begin{bmatrix} 0 \\\\ h \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ h \\end{bmatrix} , \\end{align*}"}
{"id": "744.png", "formula": "\\begin{align*} \\lim _ { z _ 2 \\to z _ 1 } \\frac { K _ { \\mathcal A _ 0 } ( \\boldsymbol z , \\boldsymbol z ) } { | z _ 1 - z _ 2 | ^ 2 } = \\frac { 1 } { 2 } \\frac { 1 } { ( 1 - | z _ 1 | ^ 2 ) ^ 4 } , ( z _ 1 , z _ 2 ) \\in \\mathbb D ^ 2 . \\end{align*}"}
{"id": "1767.png", "formula": "\\begin{align*} | f _ k ( y _ n ^ * ) - \\sum _ { j = 1 } ^ { m _ 1 } y _ n ^ * ( x ^ k _ j ) | & \\leq | f _ k ( y _ n ^ * ) - \\sum _ { j = 1 } ^ { r _ n } y _ n ^ * ( x ^ k _ j ) | + | \\sum _ { j = m _ 1 + 1 } ^ { r _ n } y _ n ^ * ( x ^ k _ j ) | \\\\ & \\leq \\epsilon / 8 + \\sum _ { j = m _ 1 + 1 } ^ \\infty | y _ n ^ * ( x ^ k _ j ) | < \\epsilon / 8 + \\epsilon / 8 < \\epsilon / 4 . \\end{align*}"}
{"id": "708.png", "formula": "\\begin{align*} v ^ 2 | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 = f ( u ) ^ 2 | \\nabla u | ^ 2 _ g . \\end{align*}"}
{"id": "219.png", "formula": "\\begin{align*} \\mathcal { K } _ { \\mathcal { S } } \\coloneqq \\left \\{ ( y ^ { n } _ { \\mathcal { T } } , \\tilde { y } ^ { n } _ { \\mathcal { T } } ) \\in \\mathcal { Y } ^ { n } _ { \\mathcal { T } } \\times \\mathcal { Y } ^ { n } _ { \\mathcal { T } } \\ : y ^ { n } _ { \\mathcal { S } } = \\tilde { y } ^ { n } _ { \\mathcal { S } } \\ \\wedge \\ y ^ { n } _ { i } \\neq \\tilde { y } ^ { n } _ { i } , \\ \\forall i \\in \\mathcal { S } ^ { c } \\right \\} . \\end{align*}"}
{"id": "5048.png", "formula": "\\begin{align*} N ^ { n , 1 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) d W _ u \\right ) ^ 2 \\ , d s \\end{align*}"}
{"id": "331.png", "formula": "\\begin{align*} \\frac { 1 } { x } \\sum _ { \\substack { n \\le x \\\\ n \\in { \\rm L } _ { A \\cap ( y , \\infty ) } } } 1 \\le \\frac { 1 } { x } \\sum _ { a \\in A , a > y } \\left \\lfloor \\frac { x } { a } \\right \\rfloor \\le \\sum _ { a \\in A , a > y } \\frac { 1 } { a } = o _ y ( 1 ) . \\end{align*}"}
{"id": "128.png", "formula": "\\begin{align*} T _ { \\lambda ^ 2 } : = 6 \\lambda ^ 2 \\Big [ C \\star \\big ( S ^ 3 - C ^ 3 \\big ) \\star S + C \\star \\big ( { \\bf 1 } ^ \\epsilon _ 0 \\| C ^ 3 \\| _ { L ^ 1 } - { \\bf 1 } ^ \\epsilon _ 0 \\| C _ \\infty ^ 3 \\| _ { L ^ 1 } \\big ) \\star S + C \\star \\psi \\star S \\Big ] , \\end{align*}"}
{"id": "6578.png", "formula": "\\begin{align*} \\beta ^ * ( g , \\lambda ) : = ( \\beta ( g ^ { - 1 } , \\cdot ) | _ { E _ { \\alpha ( g , x ) } } ^ { E _ x } ) ' ( \\lambda ) \\end{align*}"}
{"id": "5914.png", "formula": "\\begin{align*} P _ j ( \\hat \\phi _ j ) = 0 , 1 \\leq j \\leq n . \\end{align*}"}
{"id": "6014.png", "formula": "\\begin{align*} M _ f ( 0 ) & = ( 3 , 2 , 1 , 0 , 0 , 0 ) \\\\ M _ f ( 1 ) & = ( 1 , 0 , 1 , 0 , 0 , 0 ) . \\end{align*}"}
{"id": "7197.png", "formula": "\\begin{align*} \\langle f , u _ { p + \\frac { [ j _ 1 + j _ 2 ] } { T } } I ^ { \\circ } ( w _ 1 , z ) w _ 2 \\rangle = 0 \\end{align*}"}
{"id": "737.png", "formula": "\\begin{align*} \\partial \\bar { \\partial } \\log K ( z , z ) \\geq \\sup \\Big \\{ \\tfrac { | f ^ \\prime ( z ) | ^ 2 } { ( 1 - | f ( z ) | ^ 2 ) ^ 2 } : f \\in { \\rm R a t } ( \\Omega ) , \\| f \\| _ { \\Omega , \\infty } \\leq 1 \\Big \\} = \\mathbb S _ \\Omega ( z , z ) ^ 2 , \\end{align*}"}
{"id": "496.png", "formula": "\\begin{align*} \\langle \\alpha _ i ^ \\vee , v \\lambda \\rangle = 1 + \\langle \\alpha _ i ^ \\vee , \\alpha _ { j _ { p + 1 } } \\rangle + \\dots + \\langle \\alpha _ i ^ \\vee , \\alpha _ { j _ r } \\rangle \\le 1 \\end{align*}"}
{"id": "7653.png", "formula": "\\begin{align*} \\begin{pmatrix} \\lambda \\\\ \\mu \\end{pmatrix} \\in \\mathbb { C } ^ 2 \\end{align*}"}
{"id": "2744.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\sigma } _ { p } u ( x ) - \\lim _ { i \\rightarrow \\infty } ( - \\Delta ) ^ { \\sigma } _ { p } u _ { i } ( x ) = \\lim _ { R \\rightarrow \\infty } \\lim _ { i \\rightarrow \\infty } \\Psi _ { i } ( x , R ) . \\end{align*}"}
{"id": "4004.png", "formula": "\\begin{align*} \\omega ( a * b * c ) & = \\omega ( b * c * \\sigma _ i ^ R ( \\nabla ^ { - 1 } a ) ) = \\omega ( c * \\sigma _ i ^ R ( \\nabla ^ { - 1 } a ) * \\sigma _ i ^ R ( \\nabla ^ { - 1 } b ) ) \\\\ & = \\omega ( \\nabla \\sigma _ { - i } ^ R ( \\sigma _ i ^ R ( \\nabla ^ { - 1 } a ) * \\sigma _ i ^ R ( \\nabla ^ { - 1 } b ) ) * c ) . \\end{align*}"}
{"id": "4598.png", "formula": "\\begin{align*} W _ { c T } \\stackrel { \\mathrm { d } } { = } \\widehat { W } _ T . \\end{align*}"}
{"id": "4918.png", "formula": "\\begin{align*} R ( e _ i , e _ j ) e _ k = { R _ { i j k } } ^ { l } e _ l , \\end{align*}"}
{"id": "4584.png", "formula": "\\begin{align*} C _ R ( Q ) R _ { Q } = C _ { R _ { Q } } ( Q ) = C _ { R ' _ { Q } } ( Q ) = C _ { R ' } ( Q ) R ' _ { Q } . \\end{align*}"}
{"id": "1843.png", "formula": "\\begin{align*} \\phi _ { j } ^ { ( k ) } ( z ) : = \\langle ( z I - H ^ { ( k ) } ) ^ { - 1 } e _ { j } , e _ { 0 } \\rangle = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\langle ( H ^ { ( k ) } ) ^ { n } e _ { j } , e _ { 0 } \\rangle } { z ^ { n + 1 } } , 0 \\leq j \\leq p - 1 . \\end{align*}"}
{"id": "8890.png", "formula": "\\begin{align*} \\frac { n - 1 } { ( n + 1 ) \\sigma ^ 2 - \\| A \\| _ F ^ 2 } \\not = 0 , \\end{align*}"}
{"id": "8375.png", "formula": "\\begin{align*} a \\ell _ 1 + ( N / 8 - 2 - a ) \\ell _ 2 = N / 1 6 \\cdot ( k - 3 ) . \\end{align*}"}
{"id": "7018.png", "formula": "\\begin{align*} \\hat { \\tau } = \\hat { \\tau } ^ * K _ 0 ( \\hat { \\tau } ^ * ) + ( \\bar { \\tau } - \\hat { \\tau } ^ * ) K _ { \\bar { \\tau } } ( \\hat { \\tau } ^ * ) , \\end{align*}"}
{"id": "1455.png", "formula": "\\begin{align*} R _ { \\ell , i , s } ( z ) = \\sum _ { k = n } ^ { \\infty } \\dfrac { \\psi _ { { i , s } } ( t ^ k P _ { \\ell } ( t ) ) } { z ^ { k + 1 } } \\ \\ 0 \\le \\ell \\le r m , \\ 1 \\le i \\le m \\ \\ 0 \\le s \\le r - 1 \\enspace , \\end{align*}"}
{"id": "5612.png", "formula": "\\begin{align*} \\mathsf { L } ( b ) = \\mathsf { L } _ M ( b ) = \\{ | z | \\mid z \\in \\mathsf { Z } ( b ) \\} . \\end{align*}"}
{"id": "109.png", "formula": "\\begin{align*} \\forall t \\leq 1 / ( 2 | \\nu | + 1 ) , \\dot \\kappa _ t \\geq \\frac { 1 } { t } \\Big ( 1 - \\frac { 1 } { 1 + t \\nu } \\Big ) = \\frac { \\nu } { 1 + t \\nu } . \\end{align*}"}
{"id": "812.png", "formula": "\\begin{align*} W ^ { s , p } ( E ) : = \\left \\lbrace u \\in L ^ p ( E ) : [ u ] _ { W ^ { s , p } ( E ) } < \\infty \\right \\rbrace \\end{align*}"}
{"id": "8955.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } i \\partial _ t u + \\Delta u & = \\pm | u | ^ 4 u , ( t , x ) \\in \\R \\times \\R , \\\\ u ( 0 ) & = u _ 0 \\in M ^ s _ { 6 , 2 } ( \\R ) + L ^ 2 ( \\R ) \\end{array} \\right . \\end{align*}"}
{"id": "2552.png", "formula": "\\begin{align*} \\omega _ 1 = \\dfrac { 1 } { 3 } ( 2 \\alpha _ 1 + \\alpha _ 2 ) \\ \\ , \\ \\ \\ \\omega _ 2 = \\dfrac { 1 } { 3 } ( \\alpha _ 1 + 2 \\alpha _ 2 ) \\ \\ . \\end{align*}"}
{"id": "6815.png", "formula": "\\begin{align*} \\mathcal { W } ( 0 , 1 ) : = \\Bigl \\{ u \\in H ^ 2 _ { \\frac { 1 } { a } } ( 0 , 1 ) : a u '''' \\in L ^ 2 _ { \\frac { 1 } { a } } ( 0 , 1 ) \\Bigr \\} . \\end{align*}"}
{"id": "6954.png", "formula": "\\begin{align*} F ( z ) : = ( ( W - z I ) ^ { - 1 } p , p ) = \\int _ \\R \\frac { d \\rho ( s ) } { s - z } \\forall z \\in \\C \\setminus \\R . \\end{align*}"}
{"id": "6267.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } u ( x ) + \\frac { 1 } { q } u ( x ) u ( x / q ) + C ( x ) u ( x / q ) = 0 , \\end{align*}"}
{"id": "3185.png", "formula": "\\begin{align*} \\lim _ { \\gamma _ { - } \\rightarrow 0 ^ { + } } \\inf h _ { \\gamma _ { - } } \\left ( E _ { 1 } \\right ) = \\inf f _ { ( \\Phi , \\mathfrak { a } _ { \\mathfrak { b } _ { + } } - \\mathfrak { a } _ { \\mathfrak { b } _ { - } } ) } ^ { \\flat } \\left ( E _ { 1 } \\right ) = - \\mathrm { P } _ { ( \\Phi , \\mathfrak { a } _ { \\mathfrak { b } _ { + } } - \\mathfrak { a } _ { \\mathfrak { b } _ { - } } ) } ^ { \\flat } \\ . \\end{align*}"}
{"id": "1694.png", "formula": "\\begin{align*} y ( z x ) = - ( - 1 ) ^ { \\vert y \\vert \\vert z \\vert } z ( y x ) . \\end{align*}"}
{"id": "7887.png", "formula": "\\begin{align*} \\left \\langle [ \\gamma _ 1 ] _ { \\vec s ^ 1 } \\bar { \\psi } ^ { a _ 1 } , \\ldots , [ \\gamma _ m ] _ { \\vec s ^ m } \\bar { \\psi } ^ { a _ m } \\right \\rangle _ { g , \\{ \\vec s ^ j \\} _ { j = 1 } ^ m , \\beta } ^ { X _ { D , \\infty } } : = \\left [ \\left ( \\prod _ { i = 1 } ^ n r _ i ^ { s _ { i , - } } \\right ) \\left \\langle \\gamma _ 1 \\bar { \\psi } ^ { a _ 1 } , \\ldots , \\gamma _ m \\bar { \\psi } ^ { a _ m } \\right \\rangle _ { g , \\{ \\vec s ^ j \\} _ { j = 1 } ^ m , \\beta } ^ { X _ { D , \\vec r } } \\right ] _ { \\prod _ { i = 1 } ^ n r _ i ^ 0 } \\end{align*}"}
{"id": "1882.png", "formula": "\\begin{align*} \\mathcal { D } _ { [ n , 0 ] } = \\bigcup _ { j = 0 } ^ { p } \\mathcal { L } _ { [ n , j ] } \\end{align*}"}
{"id": "8002.png", "formula": "\\begin{align*} i ^ * H _ { ( X , K 3 ) , ( d , 0 ) } ( y , 0 ) = H _ { ( \\mathbb P ^ 3 , K 3 ) , 4 d } ( y ) , \\end{align*}"}
{"id": "4846.png", "formula": "\\begin{align*} ( X , \\dot { \\delta _ X } D f \\wedge \\delta _ X D g ) & = - ( X , \\delta _ X D g \\wedge \\dot { \\delta _ X } D f ) = - ( \\rho _ X \\delta _ X D g , \\dot \\delta _ X f ) \\\\ & = - ( \\dot { d _ X } \\rho _ X \\delta _ X D g , D f ) = - ( [ \\dot X , \\rho _ X \\delta _ X D g ] , D f ) \\\\ & = ( \\dot X , - O _ { X , D g } D f ) . \\end{align*}"}
{"id": "8539.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } \\frac { ( 1 - x ^ 2 ) ^ { k - i } - ( 1 - t ^ 2 ) ^ { k - i } } { x - t } d t & = - x \\sum _ { j = 0 } ^ { k - i - 1 } ( 1 - x ^ 2 ) ^ { k - i - j - 1 } 2 ^ { 2 j + 1 } \\frac { ( j ! ) ^ 2 } { ( 2 j + 1 ) ! } . \\end{align*}"}
{"id": "5127.png", "formula": "\\begin{align*} \\phi \\frac { \\partial } { \\partial x _ i } \\phi ^ { - 1 } ( x _ j ) = \\delta _ { i j } \\end{align*}"}
{"id": "7239.png", "formula": "\\begin{align*} & \\dot { \\hat x } ( t , s ) = \\lambda \\hat x ( t , s ) + \\partial _ s ^ 2 \\hat x ( t , s ) \\\\ & + \\int _ 0 ^ 1 l ( s ) w ( \\theta ) \\left ( \\partial _ s x ( t , \\theta ) - \\partial _ s \\hat x ( t , \\theta ) \\right ) d \\theta . \\end{align*}"}
{"id": "5225.png", "formula": "\\begin{align*} \\left [ \\left ( \\prod _ i x _ i ^ { n _ i } \\right ) \\partial _ { x _ j } , \\left ( \\prod _ i x _ i ^ { n _ i ' } \\right ) \\partial _ { x _ k } \\right ] = \\left ( n _ j ' \\prod _ { i = 1 } ^ n x _ i ^ { n _ i + n _ i ' - \\delta _ { i j } } \\right ) \\partial _ { x _ k } - \\left ( n _ k \\prod _ { i = 1 } ^ n x _ i ^ { n _ i + n _ i ' - \\delta _ { i k } } \\right ) \\partial _ { x _ j } . \\end{align*}"}
{"id": "3987.png", "formula": "\\begin{align*} E = \\{ e _ 0 \\} \\cup \\{ e _ 1 , \\cdots , e _ r \\} \\cup \\{ e _ { r + 1 } , \\overline e _ { r + 1 } , \\ , \\cdots , \\ , e _ { r + s } , \\overline e _ { r + s } \\} , \\end{align*}"}
{"id": "5496.png", "formula": "\\begin{align*} \\theta _ { t + 1 } & = \\theta _ t - \\beta _ t \\left ( \\phi ( s _ t , a _ t ) ^ T \\omega _ t \\right ) \\frac { d } { d \\theta } \\log \\pi _ { \\theta } ( a _ t | s _ t ) \\\\ \\omega _ { t + 1 } & = P _ { \\Omega } \\left [ \\omega _ t + \\alpha _ t \\left ( c _ t ' + \\gamma \\phi ( s _ t '' , a _ t '' ) ^ T \\omega _ t - \\phi ( s _ t ' , a _ t ' ) ^ T \\omega _ t \\right ) \\phi ( s _ t ' , a _ t ' ) \\right ] \\end{align*}"}
{"id": "3717.png", "formula": "\\begin{align*} \\bar { \\psi } _ { C _ 1 , C _ 2 } : = \\phi _ { C _ 1 , C _ 2 } . \\end{align*}"}
{"id": "4420.png", "formula": "\\begin{align*} \\begin{array} { l } T _ { e , j , 1 } = \\{ 4 k d + j , 4 k d + e + j , \\ldots , 4 k d + ( 4 k - 1 ) e + j \\} , \\\\ T _ { e , j , 2 } = \\{ 4 k N + j , 4 k N + 2 e + j \\} , \\\\ U _ { e , \\ell , 1 } = \\{ 4 k d + 4 k e + \\ell , 4 k d + ( 4 k + 1 ) e + \\ell , \\ldots , 4 k d + ( 8 k - 1 ) e + \\ell \\} , \\\\ U _ { e , \\ell , 2 } = \\{ 4 k N + e + \\ell , 4 k N + 3 e + \\ell \\} , \\end{array} \\end{align*}"}
{"id": "3432.png", "formula": "\\begin{align*} j ( g , z ) = A + B z . \\end{align*}"}
{"id": "3890.png", "formula": "\\begin{align*} | \\nabla H _ \\lambda | = O ( | \\nabla \\Gamma | ) \\hbox { i n } \\Omega \\end{align*}"}
{"id": "7168.png", "formula": "\\begin{align*} \\mathcal { L } ( V , g ) = \\oplus _ { r = 0 } ^ { r = T - 1 } t ^ { \\frac { r } { T } } \\mathbb { C } [ t , t ^ { - 1 } ] \\otimes V ^ r . \\end{align*}"}
{"id": "6038.png", "formula": "\\begin{align*} f _ { \\sigma } ^ { ( j ) } ( x ) = \\sum _ { k = j } ^ n ( k ) _ j \\sigma ( c _ k ) x ^ { k - j } . \\end{align*}"}
{"id": "3924.png", "formula": "\\begin{align*} \\sum _ { j = j _ { \\delta _ 1 } + 1 } ^ { j _ { \\delta _ 2 } } \\frac { 1 } { t _ j } \\le \\frac { c } { \\Delta _ n } \\log ( \\frac { j _ { \\delta _ 2 } } { j _ { \\delta _ 1 } } ) = \\frac { c } { \\Delta _ n } \\log ( \\frac { \\Delta _ n j _ { \\delta _ 2 } } { \\Delta _ n j _ { \\delta _ 1 } } ) = \\frac { c } { \\Delta _ n } \\log ( \\frac { \\delta _ 2 } { \\delta _ 1 } ) . \\end{align*}"}
{"id": "6365.png", "formula": "\\begin{align*} R _ { \\kappa } ( x ) = N _ { \\kappa } \\sum _ { m = 0 } ^ n c _ m x ^ m \\ \\ , \\end{align*}"}
{"id": "1617.png", "formula": "\\begin{align*} G ^ 1 = \\frac { - x ^ 1 ( y ^ 1 ) ^ 2 ( y ^ 2 ) ^ 2 } { \\left [ 2 ( y ^ 1 ) ^ 2 + ( x ^ 1 ) ^ 2 ( y ^ 2 ) ^ 2 \\right ] } , G ^ 2 = \\frac { 2 ( y ^ 1 ) ^ 3 ( y ^ 2 ) } { x ^ 1 \\left [ 2 ( y ^ 1 ) ^ 2 + ( x ^ 1 ) ^ 2 ( y ^ 2 ) ^ 2 \\right ] } . \\end{align*}"}
{"id": "387.png", "formula": "\\begin{align*} G ( x , y ) = \\int _ { 0 } ^ { \\infty } h _ t ( x , y ) \\mathrm { d } t . \\end{align*}"}
{"id": "1156.png", "formula": "\\begin{align*} | f ^ \\dagger f | = | f | ^ 2 \\end{align*}"}
{"id": "6851.png", "formula": "\\begin{align*} f ( V _ F ) p ( t , V _ F , g ) - f ( 0 ) p ( t , 0 , g ) = 0 , g \\in \\mathbb { R } , t > 0 , N ( t ) : = \\int _ { 0 } ^ { \\infty } g f ( V _ F ) p ( t , V _ F , g ) d g . \\end{align*}"}
{"id": "455.png", "formula": "\\begin{align*} K = \\{ r e ^ { i \\theta } : | r - r _ { 0 } | \\le \\varepsilon , | \\theta - h _ { \\nu } ( r _ { 0 } ) | \\le \\varepsilon \\} \\end{align*}"}
{"id": "5641.png", "formula": "\\begin{align*} = \\left ( \\frac { \\alpha p - \\beta } { k - 1 } \\right ) ^ 2 - F \\frac { ( \\alpha p - \\beta ) } { k - 1 } + \\frac { 1 } { ( k - 1 ) ^ 2 ( t - \\tau ) ^ 2 } . \\end{align*}"}
{"id": "7214.png", "formula": "\\begin{align*} C _ 0 : = \\| f \\| _ { L ^ { \\infty } ( \\R ) } + \\frac { 1 } { 2 } \\| g \\| _ { L ^ 1 ( \\R ) } > 0 . \\end{align*}"}
{"id": "3216.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\left ( \\sum _ { t = 1 } ^ { T } \\frac { \\gamma ^ { t } } { t } \\right ) & \\geq \\int _ { 1 } ^ { \\infty } \\frac { \\gamma ^ { t } } { t } d t \\\\ & = \\mathrm { E i } \\left ( \\infty \\ln \\gamma \\right ) - \\mathrm { E i } \\left ( \\ln \\gamma \\right ) = - \\mathrm { E i } \\left ( \\ln \\gamma \\right ) \\\\ & = \\mathrm { E _ { 1 } } \\left ( \\ln \\frac { 1 } { \\gamma } \\right ) > 0 , \\end{align*}"}
{"id": "5302.png", "formula": "\\begin{align*} \\sum _ { Q ' \\in f ^ { - 1 } ( Q ) } \\mathrm { C o n t } ( Q ' ) | _ { b _ l = 0 } = ( l - 2 ) \\mathrm { C o n t } ( Q ) . \\end{align*}"}
{"id": "4802.png", "formula": "\\begin{align*} \\frac { | \\pi ( w ) | ^ 2 } { z - w } = \\overline { \\pi ( w ) } \\frac { \\pi ( w ) - \\pi ( z ) } { z - w } + \\frac { \\pi ( z ) \\overline { \\pi ( w ) } } { z - w } . \\end{align*}"}
{"id": "7350.png", "formula": "\\begin{align*} [ x \\ast [ y , [ z , u ] ] , v ] = - [ x \\ast [ y , [ u , z ] ] , v ] \\overset { \\eqref { c h a n - 3 5 } } { \\equiv } [ x \\ast [ y , [ v , z ] ] , u ] \\equiv - [ x \\ast [ y , [ z , v ] ] , u ] . \\end{align*}"}
{"id": "1025.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { | I | + 1 } \\mu ( [ B _ j ^ { \\bar r _ j } ] ) . \\end{align*}"}
{"id": "2497.png", "formula": "\\begin{align*} g ( R ( X , Y ) Z , U ) = g ( ( \\nabla _ X A ) _ Y Z , U ) - g ( ( \\nabla _ Y A ) _ X Z , U ) - g ( T _ U Z , \\nu [ X , Y ] ) , \\end{align*}"}
{"id": "2314.png", "formula": "\\begin{align*} G : \\begin{cases} 0 \\to 0 \\\\ 1 \\to 0 1 \\end{cases} \\widetilde { G } : \\begin{cases} 0 \\to 0 \\\\ 1 \\to 1 0 \\end{cases} D : \\begin{cases} 0 \\to 1 0 \\\\ 1 \\to 1 \\end{cases} \\widetilde { D } : \\begin{cases} 0 \\to 0 1 \\\\ 1 \\to 1 \\end{cases} . \\end{align*}"}
{"id": "3896.png", "formula": "\\begin{align*} - \\Delta _ p ( \\mathit \\Gamma + \\mathcal { H } ) + \\Delta _ p \\mathit \\Gamma = \\mathcal G B _ 2 ( 0 ) \\setminus \\{ 0 \\} \\end{align*}"}
{"id": "1473.png", "formula": "\\begin{align*} \\Delta \\circ { { \\tilde { \\psi } } } _ { \\alpha , s } \\bigcirc _ { w = 1 } ^ { k _ s } ( \\theta _ { X _ s } + \\xi _ { s , w } ) ( V ( { \\hat { P } } ) ) = \\Delta \\circ { { \\tilde { \\psi } } } _ { \\alpha , s } \\bigcirc _ { w = 1 } ^ { s + 1 } ( \\theta _ { X _ s } + \\gamma _ { r - s + w - 1 } ) \\circ \\theta _ { X _ s } ^ { k _ s - s - 1 } ( V ( { \\hat { P } } ) ) \\enspace , \\end{align*}"}
{"id": "8446.png", "formula": "\\begin{align*} W _ { \\beta } ( f , \\varphi ) = \\bigcap _ { N = 1 } ^ { \\infty } \\bigcup _ { n = N } ^ { \\infty } W _ n . \\end{align*}"}
{"id": "3239.png", "formula": "\\begin{align*} \\nu _ p ( \\{ ( a , r ) \\in \\omega \\times \\R _ + : \\ c _ d ' r | \\nabla f ( a ) | > \\kappa \\} ) & = \\int _ { \\omega } \\int _ { \\kappa / ( c _ d ' | \\nabla f ( a ) | ) } ^ \\infty \\frac { d r } { r ^ { p + 1 } } \\ , d a \\\\ & = p ^ { - 1 } \\int _ \\omega \\left ( \\frac { c _ d ' | \\nabla f ( a ) | } { \\kappa } \\right ) ^ p d a \\\\ & = c _ { d , p } \\kappa ^ { - p } \\int _ \\omega | \\nabla f ( a ) | ^ p \\ , d a \\ , . \\end{align*}"}
{"id": "831.png", "formula": "\\begin{align*} \\mathbf { h } ^ { c } = \\sum _ { l = 1 } ^ { L } x _ { l } ^ { c } \\boldsymbol { a } \\left ( \\theta _ { l } ^ { c } \\right ) , \\end{align*}"}
{"id": "5653.png", "formula": "\\begin{align*} \\tilde { a } _ { i j } ( x ) D _ { i j } \\widetilde { w } = F ( A + D ^ 2 \\widetilde { w } ) - F ( A ) = g ( x ) - 1 \\quad \\ | x | > r _ 3 , \\end{align*}"}
{"id": "5950.png", "formula": "\\begin{align*} \\langle A _ { i j } ( t ) g [ \\mathbf { A } ] \\rangle = \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { n ! } \\int _ { \\mathbb { R } ^ n } \\bigl \\langle A _ { i j } ( t ) A _ { i _ 1 j _ 1 } ( t _ 1 ) \\dots A _ { i _ n j _ n } ( t _ n ) \\bigr \\rangle _ c \\ , \\biggl \\langle \\frac { \\delta ^ n \\ , g [ \\mathbf { A } ] } { \\delta A _ { i _ 1 j _ 1 } ( t _ 1 ) \\ , \\dots \\ , \\delta A _ { i _ n j _ n } ( t _ n ) } \\biggr \\rangle \\mathrm { d } t _ 1 \\dots \\mathrm { d } t _ n \\end{align*}"}
{"id": "3549.png", "formula": "\\begin{align*} w ^ { C _ 2 ^ n } ( \\pi ) & = 1 + P ( v _ 1 ^ 2 , \\ldots , v _ n ^ 2 ) \\\\ & = ( 1 + P ( v _ 1 , \\ldots , v _ n ) ) ^ 2 . \\end{align*}"}
{"id": "5022.png", "formula": "\\begin{align*} R ^ { n , M , 2 } _ \\tau = n ^ { 2 \\alpha + 1 } \\sum _ { i = 0 } ^ { M - 1 } \\int _ { \\tau _ i } ^ { \\tau _ { i + 1 } } \\gamma _ s \\left [ ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) - ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\tau _ i } ) \\right ] \\Xi ^ { n , 1 } _ s d s \\end{align*}"}
{"id": "2237.png", "formula": "\\begin{align*} \\sqrt { e ^ { - 2 \\pi t } \\sin ^ 2 { \\pi \\sigma } + ( 1 \\pm e ^ { - \\pi t } \\cos { \\pi \\sigma } ) ^ 2 } = \\sqrt { 1 + e ^ { - 2 \\pi t } \\pm 2 e ^ { - \\pi t } \\cos { \\pi \\sigma } } & \\geq \\sqrt { 1 + e ^ { - 2 \\pi t } - 2 e ^ { - \\pi t } } \\\\ & = \\sqrt { ( 1 - e ^ { - \\pi t } ) ^ 2 } \\\\ & = 1 - e ^ { - \\pi t } . \\end{align*}"}
{"id": "7724.png", "formula": "\\begin{align*} p ^ * \\sum _ { t = k _ 0 } ^ k e ^ T A _ { S \\bar { S } } ( t ) e & \\leq \\left ( \\pi _ { \\bar { S } } ^ T ( k _ 0 ) - \\pi _ { \\bar { S } } ^ T ( k ) \\right ) e + \\sum _ { t = k _ 0 } ^ k e ^ T A _ { \\bar { S } S } ( t ) e \\cr & \\leq 1 + \\sum _ { t = k _ 0 } ^ k e ^ T A _ { \\bar { S } S } ( t ) e \\end{align*}"}
{"id": "726.png", "formula": "\\begin{align*} ( \\partial _ s + \\Delta ) u = \\mathcal { H } v + \\frac { f ' ( u ) } { f ( u ) } ( m + v ^ 2 - 1 ) . \\end{align*}"}
{"id": "3385.png", "formula": "\\begin{align*} \\left ( \\zeta _ { t \\wedge t _ k } - \\zeta _ { t \\wedge s _ k } \\right ) \\ 1 { s _ k < \\rho _ r } & = \\left ( \\zeta _ { t \\wedge t _ k } - \\zeta _ { s _ k } \\right ) \\ 1 { s _ k < \\rho _ r \\wedge t } \\\\ & = \\left ( \\zeta ^ { ( f , \\theta ) } _ { t - s _ k } - \\zeta ^ { ( f , \\theta ) } _ 0 \\right ) \\ 1 { s _ k < \\rho _ r \\wedge t } , \\end{align*}"}
{"id": "1420.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } Q _ { N } \\Big [ \\varepsilon _ 0 \\Big ( \\frac { \\gamma } { \\beta + \\gamma } \\Big ) ^ { 1 / D } N ^ { \\frac { 1 } { D } \\left ( d - \\frac { 2 ( d - D ) } { D + 2 } \\right ) } & \\leq R _ { N , d , D } \\\\ & \\leq K _ { 0 } \\Big ( \\frac { \\beta + \\gamma } { \\beta } \\Big ) ^ { 1 / 2 } N ^ { \\frac { d } { 2 } + \\frac { d - D } { D + 2 } } \\Big ] = 1 . \\end{align*}"}
{"id": "2967.png", "formula": "\\begin{align*} T _ { p ^ 2 } \\psi = a _ \\varphi ( p ) \\psi . \\end{align*}"}
{"id": "3816.png", "formula": "\\begin{align*} \\sum _ { a = 0 , 1 , 2 , 3 } | { } _ { 0 } ^ 1 E l l H ^ { \\mu , a ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) | + \\sum _ { \\kappa \\in ( \\bar { \\kappa } , 2 ] \\cap \\Z } | { } _ { 0 } ^ \\kappa E l l H ^ { \\mu , 4 ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) | \\end{align*}"}
{"id": "3680.png", "formula": "\\begin{align*} K _ { i j } = \\frac 1 { \\sin ^ 2 r } \\left ( 1 - \\alpha \\cos ^ 2 r \\right ) \\geq 0 . \\end{align*}"}
{"id": "6421.png", "formula": "\\begin{align*} \\Lambda ( x , y ) = \\Phi _ 0 ( [ x , y ] _ \\mathfrak { g } ) - [ \\Phi _ 0 ( x ) , \\Phi _ 0 ( y ) ] . \\end{align*}"}
{"id": "4872.png", "formula": "\\begin{align*} \\begin{aligned} B ( n , q ; \\chi _ { 1 } , \\ldots , \\chi _ { 5 } ) = \\sum \\limits _ { a = 1 \\atop ( a , q ) = 1 } ^ { q } C _ { 1 } ( \\chi _ { 1 } , a ) C _ { 3 } ( \\chi _ { 2 } , a ) \\ldots C _ { 3 } ( \\chi _ { 5 } , a ) e \\left ( - \\frac { a n } { q } \\right ) \\end{aligned} \\end{align*}"}
{"id": "674.png", "formula": "\\begin{align*} f ^ \\infty _ s = f _ o \\circ \\Psi _ { s - s _ 0 } . \\end{align*}"}
{"id": "1209.png", "formula": "\\begin{align*} \\mathbb { M } _ n ( \\hat t _ n ) & = \\sqrt { n } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\hat \\theta _ n } ) + o _ { \\mathbb { P } } ( 1 ) \\\\ & \\leq \\inf _ { \\theta \\in \\Theta } \\sqrt { n } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ \\theta ) + o _ { \\mathbb { P } } ( 1 ) \\\\ & = \\inf _ { t \\in \\R ^ { d _ 0 } } \\mathbb { M } _ n ( t ) + o _ { \\mathbb { P } } ( 1 ) . \\end{align*}"}
{"id": "5172.png", "formula": "\\begin{align*} \\Delta _ h ^ { j _ \\ell } \\dots \\Delta _ h ^ { j _ 2 } \\Delta _ h ^ { j _ 1 } G ( x _ 0 , \\dots , x _ { m - 1 } ) = \\begin{cases} 1 & \\{ j _ 1 , \\dots , j _ \\ell \\} = A \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "7165.png", "formula": "\\begin{align*} M = \\mbox { s p a n } \\{ u _ n a | u \\in V , n \\in \\frac { 1 } { T } \\mathbf { N } \\} \\end{align*}"}
{"id": "3378.png", "formula": "\\begin{align*} \\frac { \\sqrt { 2 x + 2 } } { \\sqrt { x } + 1 } \\cdot \\frac { x + 1 } { 2 } \\leq \\max \\{ 1 , x \\} = \\frac { x + 1 + | x - 1 | } { 2 } . \\end{align*}"}
{"id": "3217.png", "formula": "\\begin{align*} \\operatorname { p a r d e g } ( E _ * ) \\coloneqq \\deg ( E ) + \\sum \\limits _ { p \\in D } \\sum \\limits _ { i = 1 } ^ { r _ p } \\alpha _ i ( p ) \\cdot \\dim ( E _ { p , i } / E _ { p , i + 1 } ) \\end{align*}"}
{"id": "7983.png", "formula": "\\begin{align*} I _ { ( Y _ - , D _ { Y , - } ) } ( y , z ) = z e ^ { t _ - / z } \\sum _ { d \\in \\mathbb K _ { - } } \\tilde y ^ { d } \\left ( \\prod _ { i \\in M _ 0 } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\\\ \\left ( \\prod _ { i = 1 } ^ k \\prod _ { 0 < a \\leq v _ i \\cdot d } ( v _ i + a z ) \\right ) \\textbf { 1 } _ { [ - d ] } I _ { D _ { Y , - } , d } , \\end{align*}"}
{"id": "1622.png", "formula": "\\begin{align*} P _ { o u t } ( i ) = \\textrm { P r } \\bigg [ \\bar { \\gamma _ i } < 1 \\bigg ] . \\end{align*}"}
{"id": "2006.png", "formula": "\\begin{align*} n = p D \\ \\ \\gcd ( D , p ) = 1 \\end{align*}"}
{"id": "273.png", "formula": "\\begin{align*} F ^ { \\nu } _ i + F ^ { \\nu t } _ i = 0 . \\end{align*}"}
{"id": "46.png", "formula": "\\begin{align*} f ^ { \\tilde { \\phi } ^ { - n } } ( v _ m ) = \\beta _ m \\end{align*}"}
{"id": "7444.png", "formula": "\\begin{align*} \\Lambda ^ m ( x ) = - c ^ { - 1 } \\int _ 0 ^ x \\int _ 0 ^ { x _ 1 } \\left ( f ^ 5 + \\delta f ^ 1 _ { x _ 2 } \\right ) d x _ 2 d x _ 1 + c ^ { - 1 } \\frac { x } { L } \\int _ 0 ^ L \\int _ 0 ^ { x _ 1 } \\left ( f ^ 5 + \\delta f ^ 1 _ { x _ 2 } \\right ) d x _ 2 d x _ 1 . \\end{align*}"}
{"id": "8358.png", "formula": "\\begin{align*} u \\times v \\mapsto \\langle u | v \\rangle _ t = e ^ { t ( ( \\tilde { f } ^ { | u \\rangle } , \\tilde { f } ^ { | v \\rangle } ) _ { { } _ { \\mathfrak { J } } } + 4 \\pi ) } = e ^ { - t 4 \\pi ( \\lambda \\textrm { c o t h } \\lambda - 1 ) } \\end{align*}"}
{"id": "2169.png", "formula": "\\begin{align*} I _ { 4 } = \\int _ { \\mathbb { R } } \\frac { 1 + s ^ { 2 } } { ( 1 - s G _ { \\nu } ^ { * } ( 0 ) ) ^ { 2 } } \\ , d \\sigma _ { \\mu } ( s ) \\leq \\frac { 1 } { I _ { 1 } } , \\end{align*}"}
{"id": "735.png", "formula": "\\begin{align*} { \\mathcal G } _ { K ^ { - 1 } } ( z , w ) : = \\Big ( \\ ! \\ ! \\Big ( \\ , K ( z , w ) \\partial _ i \\bar { \\partial } _ j K ( z , w ) - \\partial _ i K ( z , w ) \\bar { \\partial } _ j K ( z , w ) \\Big ) \\ ! \\ ! \\Big ) _ { i , j = 1 } ^ m , \\ ; \\ ; z , w \\in \\Omega , \\end{align*}"}
{"id": "4828.png", "formula": "\\begin{align*} \\tau ^ \\mathrm { a n } ( M / S , \\rho ) = \\tau ^ \\mathrm { t o p } ( M / S , \\rho ) \\ ; . \\end{align*}"}
{"id": "4720.png", "formula": "\\begin{align*} ( q - q ^ { - 1 } ) e _ { ( 2 ) } H _ 3 H _ 2 H _ { 3 } ^ { - 1 } + ( q ^ { - 1 } - q ) e _ { ( 2 ) } H _ 1 H _ { 2 } ^ { - 1 } H _ 1 ^ { - 1 } - ( q - q ^ { - 1 } ) ^ { 2 } e _ { ( 2 ) } = 0 . \\end{align*}"}
{"id": "5486.png", "formula": "\\begin{align*} \\overline { H } ( \\rho , t ) & \\approx r ^ { - n } e ^ { - \\frac { \\rho ^ 2 } { 4 t } - \\lambda _ 1 ( M ) t } = \\max \\left ( 1 , \\frac { \\rho } { t } \\right ) ^ n e ^ { - \\frac { \\rho ^ 2 } { 4 t } - \\lambda _ 1 ( M ) t } \\\\ & \\approx \\left ( \\frac { \\rho + t } { t } \\right ) ^ n e ^ { - \\frac { \\rho ^ 2 } { 4 t } - \\lambda _ 1 ( M ) t } \\lesssim ( 1 + \\rho ^ n ) e ^ { - \\frac { \\rho ^ 2 } { 4 t } - \\frac { ( n - 1 ) ^ 2 a ^ 2 } { 4 } t } , \\end{align*}"}
{"id": "3048.png", "formula": "\\begin{align*} c _ { n } = \\frac { \\Gamma ( n / 2 ) } { ( 2 \\pi ) ^ { n / 2 } } \\left [ \\int _ { 0 } ^ { \\infty } s ^ { n / 2 - 1 } g _ { n } ( s ) \\mathrm { d } s \\right ] ^ { - 1 } \\end{align*}"}
{"id": "5431.png", "formula": "\\begin{align*} s = \\theta s _ 1 + ( 1 - \\theta ) s _ 2 \\quad \\frac { 1 } { p } = \\frac { \\theta } { p _ 1 } + \\frac { 1 - \\theta } { p _ 2 } , \\end{align*}"}
{"id": "6419.png", "formula": "\\begin{align*} [ Q , \\Bar { \\Psi } ^ { ( n + 1 ) } ] = \\Bar \\Psi ^ { ( n ) } \\circ Q _ \\mathfrak { g } ^ { ( 1 ) } - \\Bar { Q } ^ { ( 1 ) } \\circ \\Bar \\Psi ^ { ( n ) } . \\end{align*}"}
{"id": "151.png", "formula": "\\begin{align*} \\sup _ { t > 0 } \\frac { c \\gamma _ t } { m _ t } \\leq c ' ( \\mu ) \\lim _ { \\mu \\rightarrow \\infty } c ' ( \\mu ) = 0 . \\end{align*}"}
{"id": "7436.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } c m \\int _ 0 ^ { L } \\int _ 0 ^ { \\infty } \\sigma ( s ) \\abs { \\eta _ x } ^ 2 d s d x - \\frac { c m } { 2 } \\int _ 0 ^ { L } \\int _ 0 ^ { \\infty } \\sigma ' ( s ) \\abs { \\eta _ x } ^ 2 d s d x = \\Re \\left ( c m \\int _ 0 ^ L \\int _ 0 ^ { \\infty } \\sigma ( s ) \\eta _ x \\overline { w } _ x d s d x \\right ) \\end{align*}"}
{"id": "5752.png", "formula": "\\begin{align*} \\pi _ { i _ 2 } \\cdots \\pi _ { i _ d } = \\sum _ { J \\subseteq [ n - 1 ] } c _ { J } \\pi _ J \\end{align*}"}
{"id": "343.png", "formula": "\\begin{align*} \\Phi ( S ) = \\frac { Q ( S , S ^ c ) } { \\pi ( S ) } , \\end{align*}"}
{"id": "3874.png", "formula": "\\begin{align*} \\nabla ( G _ \\lambda - \\Gamma ) \\in L ^ { \\bar q } ( \\Omega ) , \\bar q = \\frac { N ( p - 1 ) } { N - 1 } , \\end{align*}"}
{"id": "8322.png", "formula": "\\begin{align*} \\| e ^ { i c } - I \\| _ { 0 , N } \\leq \\int _ 0 ^ 1 \\| e ^ { i t c } \\| \\| c \\| _ { 0 , N } \\ , d t = \\| c \\| _ { 0 , N } \\ , . \\end{align*}"}
{"id": "6143.png", "formula": "\\begin{align*} b _ h ( u , v , w ) : = \\langle n l _ h ( u , v ) , w \\rangle _ { H ^ { - 1 } \\times H _ { \\# } ^ 1 } , \\end{align*}"}
{"id": "8597.png", "formula": "\\begin{align*} \\left \\{ \\alpha \\oplus \\frac { \\varepsilon _ 1 } { 2 ^ { k _ 1 } } \\oplus \\ldots \\oplus \\frac { \\varepsilon _ l } { 2 ^ { k _ l } } : \\ , \\varepsilon _ j = 0 , 1 \\right \\} \\subset E . \\end{align*}"}
{"id": "4100.png", "formula": "\\begin{align*} B \\overline { \\overline { u } } \\ , ' ( t ) + ( A + B Q ) \\overline { \\overline { u } } ( t ) = - H _ 1 ( { u } _ 2 ' ( t ) + Q { u } _ 2 ( t ) ) - H _ 2 ( { u } _ 1 ' ( t ) + Q { u } _ 1 ( t ) ) , \\overline { \\overline { u } } ( 0 ) = 0 . \\end{align*}"}
{"id": "3179.png", "formula": "\\begin{align*} \\mathrm { P } _ { ( \\Phi , \\hat { f } \\left ( 0 \\right ) \\delta _ { \\Psi } ) } = - \\inf _ { \\rho \\in E _ { 1 } } \\{ f _ { \\Phi } \\left ( \\rho \\right ) + \\hat { f } \\left ( 0 \\right ) \\Delta _ { \\mathfrak { e } _ { \\Psi } } \\left ( \\rho \\right ) \\} \\ . \\end{align*}"}
{"id": "2500.png", "formula": "\\begin{align*} d i v ( X ) = \\sum _ { i = 1 } ^ { m } g ( \\nabla _ { e _ i } X , e _ i ) , ~ \\forall X \\in \\Gamma ( T M ) . \\end{align*}"}
{"id": "4640.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\| B ^ n \\| } ^ { 1 / n } = \\lim _ { n \\to \\infty } { \\left | w \\right | } ^ { - \\frac { n + 1 } { 2 } } = 0 , \\end{align*}"}
{"id": "5730.png", "formula": "\\begin{align*} \\pi _ { [ a - 1 , i - 1 ] } \\pi _ { [ i , b - 1 ] } = \\binom { b - a + 1 } { i - a + 1 } \\pi _ { [ a - 1 , b - 1 ] } \\end{align*}"}
{"id": "3825.png", "formula": "\\begin{align*} { } _ { } ^ { 2 } E r r H y p ^ { \\mu , i } _ { k , j ; n } ( t _ 1 , t _ 2 ) : = \\sum _ { l \\in [ - M _ t , 2 ] \\cap \\Z } { } _ { } ^ { 2 } E r r H y p ^ { \\mu , i ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) , { } _ { } ^ { 2 } E r r H y p ^ { \\mu , i ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) : = \\sum _ { { ( \\tilde { m } , \\tilde { k } , \\tilde { j } , \\tilde { l } ) \\in \\mathcal { S } _ 1 ( t ) \\cup \\mathcal { S } _ 2 ( t ) } } H ^ { \\mu , i ; l ; \\tilde { m } } _ { k , n ; \\tilde { k } , \\tilde { j } , \\tilde { l } } ( t _ 1 , t _ 2 ) , \\end{align*}"}
{"id": "3332.png", "formula": "\\begin{align*} s i g n ( \\widehat { \\alpha . \\beta } ) = s i g n ( \\hat { \\alpha } ) + s i g n ( \\hat { \\beta } ) - M e y e r ( { \\mathcal B } _ { - 1 } ( \\alpha ) , { \\mathcal B } _ { - 1 } ( \\beta ) ) \\end{align*}"}
{"id": "876.png", "formula": "\\begin{align*} s * ^ { \\mu _ n } f _ { n } ( t , s ) * ^ { \\mu _ { n - 1 } } f _ { n - 1 } ( t , s ) * ^ { \\mu _ { n - 2 } } \\cdots * ^ { \\mu _ 1 } f _ 1 ( t , s ) = t , \\end{align*}"}
{"id": "3018.png", "formula": "\\begin{align*} \\phi ^ * ( D ) - t F \\equiv \\phi ^ \\ast \\left ( \\frac { 3 } { 2 } C _ x \\right ) - t F = \\frac { 3 } { 2 } \\left ( \\hat { L } _ { x y } + \\hat { R } _ 0 + \\hat { R } _ 1 \\right ) + \\left ( \\frac { 3 ( 2 n + 1 ) } { 2 ( 4 n + 1 ) } - t \\right ) F , \\end{align*}"}
{"id": "6158.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } ( \\| u _ h ^ m \\| _ { 2 } ^ 2 - \\| u _ h ^ { m - 1 } \\| _ { 2 } ^ 2 ) + \\nu \\Delta t \\| \\nabla u _ { h } ^ { m , \\frac { 1 } { 2 } } \\| _ { 2 } ^ 2 = 0 . \\end{align*}"}
{"id": "5373.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + \\sum _ { | \\alpha | \\leq m } a _ \\alpha ( D ^ \\alpha u ) & = F \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u & = f \\mbox { i n } \\ ; \\ ; \\Omega _ e \\end{align*}"}
{"id": "4434.png", "formula": "\\begin{align*} ( i _ u f ) _ { i _ 1 \\dots i _ { m + k } } & = \\sigma ( i _ 1 \\dots i _ { m + k } ) u _ { i _ 1 \\dots i _ k } f _ { i _ { k + 1 } \\dots i _ { k + m } } \\\\ ( j _ u g ) _ { i _ 1 \\dots i _ { m } } & = g _ { i _ 1 \\dots i _ { m + k } } u ^ { i _ { m + 1 } \\dots i _ { m + k } } . \\end{align*}"}
{"id": "5024.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\sup _ { n \\ge M } \\sup _ { \\tau \\in [ 0 , T ] } \\| R ^ { n , M , 2 } _ \\tau \\| _ 2 = 0 . \\end{align*}"}
{"id": "4047.png", "formula": "\\begin{align*} \\mu _ { V _ { 2 } , \\alpha ^ { ( 2 ) } } = \\frac { n } { n + 1 } \\mu _ { V _ { 1 } , \\alpha ^ { ( 1 ) } } + \\frac { 1 } { 2 ( n + 1 ) } ( \\delta _ { z } + \\delta _ { - z } ) , \\end{align*}"}
{"id": "1729.png", "formula": "\\begin{align*} \\frac { G ( z + \\omega _ 1 \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) } { G ( z \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) } = & F ( z + \\overline \\omega _ 1 \\ , | \\ , \\widetilde \\omega _ 1 , \\omega _ 2 ) ^ { - 1 } , \\\\ \\frac { G ( z + \\widetilde \\omega _ 1 \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) } { G ( z \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) } = & F ( z + \\overline \\omega _ 1 \\ , | \\ , \\omega _ 1 , \\omega _ 2 ) ^ { - 1 } . \\end{align*}"}
{"id": "3237.png", "formula": "\\begin{align*} \\lim \\kappa ^ p \\ , \\tilde \\nu _ \\gamma ( \\{ ( x , y ) \\in \\mathcal X : \\ | f ( x ) - f ( y ) | / | x - y | ^ { 1 + \\gamma / p } > \\kappa \\} ) = | \\gamma | ^ { - 1 } \\tilde c _ { d , p } \\ , \\| \\nabla f \\| _ { L ^ p ( \\R ^ d ) } ^ p \\ , , \\end{align*}"}
{"id": "5633.png", "formula": "\\begin{align*} \\mathcal { I } _ 2 = \\underbrace { \\iint _ { Q _ T } \\nabla \\cdot ( m \\nabla p ) | \\nabla p | ^ 2 } _ { \\mathcal { I } _ { 2 , 1 } } + \\iint _ { Q _ T } m F | \\nabla p | ^ 2 , \\end{align*}"}
{"id": "3741.png", "formula": "\\begin{align*} J ^ { e s s ; 0 } _ { k , m ' } ( t , x ) = \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } e ^ { i x \\cdot \\xi + i ( t - s ) \\omega \\cdot \\xi } \\frac { \\varphi ^ { e s s } ( \\omega , \\xi ) } { i ( \\omega + \\hat { v } ) \\cdot \\xi } \\hat { f } ( s , \\xi , v ) \\varphi _ k ( \\xi ) \\varphi _ j ( v ) \\varphi _ { m ' ; - k } ( t - s ) \\varphi _ { m ; - 1 0 M _ t } ( t - s ) d v d \\omega d \\xi \\big | _ { s = 0 } ^ t \\end{align*}"}
{"id": "8592.png", "formula": "\\begin{align*} S _ 2 = \\sum _ { u , v , z } b _ { u \\vee z } b _ { v \\vee z } \\int _ E w _ u ( x ) w _ v ( x ) d x , \\end{align*}"}
{"id": "8907.png", "formula": "\\begin{align*} \\lambda _ n u _ n ( x _ n ) - \\rho _ n u _ n ^ { p - 1 } ( x _ n ) = u _ n '' ( x _ n ) \\le 0 , \\end{align*}"}
{"id": "741.png", "formula": "\\begin{align*} & \\big ( \\big ( K _ 1 ^ 2 \\partial _ i \\bar { \\partial } _ j \\log K _ 1 \\big ) ( z , w ) \\big ) _ { i , j = 1 } ^ m \\\\ & \\quad \\quad = \\big ( \\big ( K _ 2 ^ 2 \\partial _ i \\bar { \\partial } _ j \\log K _ 2 \\big ) ( z , w ) \\big ) _ { i , j = 1 } ^ m + \\big ( \\big ( K _ 3 ^ 2 \\partial _ i \\bar { \\partial } _ j \\log K _ 3 \\big ) ( z , w ) \\big ) _ { i , j = 1 } ^ m + \\big ( \\left \\langle \\gamma _ j ( w ) , \\gamma _ i ( z ) \\right \\rangle \\big ) _ { i , j = 1 } ^ m . \\end{align*}"}
{"id": "30.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { L , n } = T _ { L | K } ( \\lambda _ { \\rho } ( \\alpha ) \\psi _ { L , n } ( \\beta ) ) \\cdot _ { \\rho } v _ n \\end{align*}"}
{"id": "6012.png", "formula": "\\begin{align*} f ( x ) & = x ^ 3 - 3 x ^ 2 = x ^ 2 ( x - 3 ) \\\\ f ' ( x ) & = 3 x ^ 2 - 6 x = 3 x ( x - 2 ) \\\\ f '' ( x ) & = 6 x - 6 = 6 ( x - 1 ) \\\\ f ^ { ( 3 ) } ( x ) & = 6 \\end{align*}"}
{"id": "605.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\abs { \\frac { H _ n ( 1 - s ) } { H _ n ( s ) } } = 1 . \\end{align*}"}
{"id": "8787.png", "formula": "\\begin{align*} H ( p , G ) = \\sup \\{ \\| | l _ \\phi | ^ { p ' } \\| _ { L ^ 1 ( G ) \\to L ^ \\infty ( G ) } ^ { 1 / p ' } \\mid \\| \\phi \\| _ p = 1 \\} . \\end{align*}"}
{"id": "931.png", "formula": "\\begin{align*} v _ { l + 1 } : = E ( \\Psi _ t ) ( 0 , - \\frac { 1 } { 2 } h ) + Q ( v _ l , v _ l ) , \\end{align*}"}
{"id": "3396.png", "formula": "\\begin{align*} \\sum _ { 1 = K _ s \\subsetneq K _ { s - 1 } \\subsetneq \\dots \\subsetneq K _ 0 = F } ( - 1 ) ^ s \\beta _ { K _ 0 , \\dots , K _ s } = ( - 1 ) ^ { \\sum _ i e _ i + \\sum _ i f _ i } \\prod _ i Q _ i ( e _ i ) \\end{align*}"}
{"id": "2205.png", "formula": "\\begin{align*} C ( J f ( x ) ) = C ^ * ( J f ( x ) ) . \\end{align*}"}
{"id": "1231.png", "formula": "\\begin{align*} [ Z _ j ] _ { \\alpha , \\beta } & = \\sum _ { i = 1 } ^ { n ( T , j + 1 ) } \\sum _ { h = 1 } ^ { n ( T , j + 1 ) } [ D _ { j + 1 } ^ T ] _ { \\alpha , i } \\ [ R _ { j + 1 } ^ { - 1 } ] _ { i , h } \\ [ D _ { j + 1 } ] _ { h , \\beta } \\\\ & = \\sum _ { i , h = 1 } ^ { n ( T , j + 1 ) } [ D _ { j + 1 } ^ T ] _ { \\alpha , i } \\ [ R _ { j + 1 } ^ { - 1 } ] _ { i , h } \\ [ D _ { j + 1 } ] _ { h , \\beta } \\ , . \\end{align*}"}
{"id": "1933.png", "formula": "\\begin{align*} C _ { [ m , 0 ] } & = [ w ^ { m ( p + 1 ) + 1 } ] \\ , h ( w ) \\\\ & = \\frac { 1 } { m ( p + 1 ) + 1 } \\ , [ t ^ { m ( p + 1 ) } ] \\ , ( t ^ { p + 1 } + 1 ) ^ { m ( p + 1 ) + 1 } \\\\ & = \\frac { 1 } { m ( p + 1 ) + 1 } \\binom { m ( p + 1 ) + 1 } { m } \\\\ & = \\frac { 1 } { m p + 1 } \\binom { m ( p + 1 ) } { m } . \\end{align*}"}
{"id": "2432.png", "formula": "\\begin{align*} \\widetilde { M } _ f : = \\frac { 1 } { [ | W _ f | ] } M _ f S _ 0 = \\sum _ { \\tau \\in W ^ f } q ^ { \\ell ( ^ f w _ 0 ) - \\ell ( \\tau ) } M _ { f \\cdot \\tau } \\end{align*}"}
{"id": "7435.png", "formula": "\\begin{align*} \\eta _ { x t } + \\eta _ { x s } - w _ x = 0 . \\end{align*}"}
{"id": "2740.png", "formula": "\\begin{align*} \\left | \\Phi _ { i } ( x , R ) \\right | \\leq C ( p , n , \\sigma , \\alpha , \\mathcal { M } ) \\begin{cases} \\varepsilon ^ { \\min \\{ ( 1 - \\sigma ) p , ( \\sigma p + \\alpha ) ( p - 2 ) + \\alpha \\} } , & , \\\\ \\varepsilon ^ { \\min \\{ ( 1 - \\sigma ) p , p + \\alpha - 2 \\} } , & , \\end{cases} \\end{align*}"}
{"id": "5938.png", "formula": "\\begin{align*} w ( \\eta ) = \\sup \\limits _ \\xi \\left ( \\eta \\xi - J ( \\xi ) \\right ) \\end{align*}"}
{"id": "6530.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ \\infty \\mu [ a _ { i + \\kappa _ n } , \\infty ) = \\sum \\limits _ { i = \\kappa _ n + 1 } ^ \\infty \\mu [ a _ i , \\infty ) < \\frac { 1 } { 2 ^ n } . \\end{align*}"}
{"id": "2407.png", "formula": "\\begin{align*} J _ 1 ( t ) & = \\langle E ( t ) , F _ n ( U ( t ) ) - F _ n ( \\tilde U ( t ) ) \\rangle + \\langle E ( t ) , F _ n ( \\tilde U ( t ) ) - F _ n ( \\mathbb U ( t ) ) \\rangle \\\\ & \\le \\frac { 3 } { 2 } \\| E ( t ) \\| ^ 2 + \\frac { 1 } { 2 } \\| F _ n ( \\mathbb U ( t ) ) - F _ n ( \\tilde U ( t ) ) \\| ^ 2 , \\end{align*}"}
{"id": "1759.png", "formula": "\\begin{align*} \\Omega ( \\alpha , A , z + a f _ k ) = \\{ z + a f _ k \\} \\cup \\bigcup _ { n = 1 } ^ \\infty \\Omega ( \\alpha - 1 , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \\end{align*}"}
{"id": "2715.png", "formula": "\\begin{align*} \\big \\langle \\Phi , ( \\mathcal { N } _ \\perp ^ 2 + 1 ) \\Phi \\big \\rangle \\le \\ ; & C \\\\ \\langle \\Phi , \\mathfrak { D } ^ 2 \\Phi \\rangle \\le \\ ; & C N \\\\ \\big \\langle U _ N ^ * \\Phi , \\ , \\mathcal { N } _ - \\ , U _ N ^ * \\Phi \\big \\rangle \\le \\ ; & C \\min \\{ N , \\frac { 1 } { T } \\} \\end{align*}"}
{"id": "5904.png", "formula": "\\begin{align*} \\sigma _ { p , \\alpha } ^ 2 = \\Big ( \\frac { p } { 1 + \\alpha } \\Big ) ^ { 2 / p } \\frac { \\Gamma ( 3 / p ) } { \\Gamma ( 1 / p ) } . \\end{align*}"}
{"id": "7072.png", "formula": "\\begin{align*} \\mathcal { S } _ 8 = \\{ W _ 2 = W _ 3 = W _ 4 = W _ 5 = W _ 6 = W _ 7 = W _ 8 = 0 \\} , \\end{align*}"}
{"id": "5382.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u _ f + \\sum _ { | \\alpha | \\leq m } a _ \\alpha D ^ \\alpha u _ f & = 0 \\mbox { i n } \\quad \\Omega , u _ f = f \\quad \\mbox { i n } \\quad \\Omega _ e \\end{align*}"}
{"id": "7174.png", "formula": "\\begin{align*} M ' = \\oplus _ { n \\in \\mathbb { N } } M ( \\frac { n } { T } ) ^ * , \\end{align*}"}
{"id": "6282.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } g ( x ) = A ( x ) g ( x ) , g ( 0 ) = 1 , \\end{align*}"}
{"id": "6586.png", "formula": "\\begin{align*} \\beta \\circ f | _ K = \\beta | _ { E _ 1 \\times \\cdots \\times E _ { j - 1 } \\times M } \\circ f | _ K \\end{align*}"}
{"id": "7153.png", "formula": "\\begin{align*} H ^ * ( X _ { D } ) & = \\mathbb { Z } [ x _ 1 , x _ 2 , x _ 3 ] / \\langle x _ 1 ^ { b } , x _ 2 ( x _ 1 + x _ 2 ) ^ { a - 1 } , x _ 3 ( x _ 1 + x _ 2 + x _ 3 ) ^ { c - 1 } \\rangle , \\\\ H ^ * ( X _ { \\widetilde { D } } ) & = \\mathbb { Z } [ y _ 1 , y _ 2 , y _ 3 ] / \\langle y _ 1 ^ { b } , y _ 2 ( y _ 1 + y _ 2 ) ^ { c - 1 } , y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } \\rangle . \\end{align*}"}
{"id": "2990.png", "formula": "\\begin{align*} \\sum _ { Q \\in \\Gamma \\backslash \\mathcal Q _ D } \\chi _ d ( Q ) \\int _ { C _ Q } \\varphi ( z ) y ^ { s - 1 } \\ , | d z | = \\pi ^ { - s - \\frac 1 2 } \\sqrt { d } \\ , \\Gamma ( \\tfrac s 2 + \\tfrac { i r } 2 + \\tfrac 1 4 ) \\Gamma ( \\tfrac s 2 - \\tfrac { i r } 2 + \\tfrac 1 4 ) L ( s + \\tfrac 1 2 , \\varphi \\times \\chi _ d ) . \\end{align*}"}
{"id": "4213.png", "formula": "\\begin{align*} ( \\mathrm { c u r l } \\ \\boldsymbol { u } , \\mathrm { c u r l } \\ \\boldsymbol { v } ) + ( \\boldsymbol { u } , \\boldsymbol { v } ) = ( \\boldsymbol { f } , \\boldsymbol { v } ) , \\forall \\boldsymbol { v } \\in \\boldsymbol { H } _ 0 ( \\mathrm { c u r l } ; \\Omega ) . \\end{align*}"}
{"id": "6860.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = g _ 0 + g _ 1 N ( b ^ * , c ^ * ) - b ^ * , \\\\ 0 & = 2 a _ 0 + 2 a _ 1 N ( b ^ * , c ^ * ) - 2 c ^ * . \\end{aligned} \\end{align*}"}
{"id": "5554.png", "formula": "\\begin{align*} \\lim \\limits _ { \\ell \\rightarrow \\infty } \\dfrac { \\mathbb { E } [ \\ell , k ] } { \\ell } = \\dfrac { b } { a } \\end{align*}"}
{"id": "8901.png", "formula": "\\begin{align*} H _ { \\mu } ^ 1 ( \\mathcal { G } ) : = \\left \\{ u \\in H ^ 1 ( \\mathcal { G } ) : \\int _ { \\mathcal { G } } | u | ^ 2 \\ , d x = \\mu \\right \\} . \\end{align*}"}
{"id": "1086.png", "formula": "\\begin{align*} B _ x & : = \\inf \\{ t \\ge 0 : x \\in \\eta _ n ( t ) \\} , \\\\ D _ x & : = \\sup \\{ t \\ge 0 : x \\in \\eta _ n ( t ) \\} , \\\\ L _ x & : = D _ x - B _ x . \\end{align*}"}
{"id": "1864.png", "formula": "\\begin{align*} W _ { j } ( z ) & = W _ { i } ( z ) \\ , A _ { j - i - 1 } ^ { ( i + 1 ) } ( z ) 0 \\leq i < j \\leq p . \\end{align*}"}
{"id": "1458.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ n A ( X - j ) = \\sum _ { k = 0 } ^ { r n } a _ { k , s } \\prod _ { { { w } } = 1 } ^ k ( X + \\gamma _ { r - s - 1 + { { w } } } ) \\enspace , \\end{align*}"}
{"id": "2820.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\big \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\big \\} { } \\leq { } \\frac { 2 L \\big [ f ( x _ 0 ) - f _ * \\big ] } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } p ( h _ i , \\kappa ) } \\end{align*}"}
{"id": "4823.png", "formula": "\\begin{align*} \\tau ^ \\mathrm { a n } ( M / S , F ) & = \\sum _ k \\bigg \\{ \\frac { 2 ^ { 2 k } \\big ( k ! \\big ) ^ 2 } { ( 2 k + 1 ) ! } \\tau ^ \\mathrm { B L } ( M / S , F ) \\bigg \\} ^ { [ 2 k ] } \\ ; , \\\\ \\tau ^ \\mathrm { t o p } ( M / S , F ) & = \\sum _ k \\bigg \\{ \\ ! - \\frac { k ! } { ( 2 \\pi ) ^ k } \\tau ^ \\mathrm { I K } ( M / S , F ) + \\frac { \\zeta ' ( - k ) \\mathrm { r k } F } { 2 } \\int _ Z \\mathrm { e } ( T Z ) \\mathrm { c h } ( T Z ) \\bigg \\} ^ { [ 2 k ] } \\ ; , \\end{align*}"}
{"id": "6630.png", "formula": "\\begin{align*} \\phi _ w ( g _ \\infty , g _ p ) : = \\sum _ { \\gamma \\in \\Gamma _ { p , \\ell } } w ( g _ \\infty \\gamma ) \\chi ( g _ p \\gamma ) \\hbox { f o r $ ( g _ \\infty , g _ p ) \\in G _ \\infty \\times G _ p . $ } \\end{align*}"}
{"id": "7021.png", "formula": "\\begin{align*} \\dot { K } _ { \\delta } ( \\hat \\tau ^ * ) = \\dot { \\hat { \\tau } } ^ * \\frac { k } { 2 } \\textrm { s e c h } ^ 2 \\left ( k ( \\hat \\tau ^ * - \\delta ) \\right ) \\end{align*}"}
{"id": "6808.png", "formula": "\\begin{align*} \\begin{cases} u _ t ( t , x ) + ( a u _ { x x } ) _ { x x } ( t , x ) = h ( t , x ) , & ( t , x ) \\in ( 0 , T ) \\times ( 0 , 1 ) , \\\\ u ( 0 , x ) = u _ 0 ( x ) , & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
{"id": "750.png", "formula": "\\begin{align*} ( 1 \\otimes x _ 2 \\cdot 1 \\otimes x _ 3 ) _ { \\lambda } = 1 \\otimes x _ 2 x _ 3 + \\sum _ i m _ i \\otimes y _ i \\end{align*}"}
{"id": "4183.png", "formula": "\\begin{align*} \\norm { F ^ { ( \\iota ) } ( \\sqrt L ) f _ j } _ p ^ p \\lesssim \\sum _ { j = 0 } ^ \\infty \\Vert F ^ { ( \\iota ) } ( \\sqrt L ) f _ j \\Vert _ p ^ p . \\end{align*}"}
{"id": "8019.png", "formula": "\\begin{align*} 0 \\longrightarrow \\mathbb { L } : = ( \\hat { \\beta } ) \\longrightarrow \\mathbb { Z } ^ { m + 1 } \\stackrel { \\hat { \\beta } } { \\longrightarrow } N \\oplus \\mathbb Z \\longrightarrow 0 , \\end{align*}"}
{"id": "4405.png", "formula": "\\begin{align*} \\Vert P _ n f \\Vert _ p \\leq c _ p \\Vert f \\Vert _ p \\mbox { f o r a l l } f \\in A _ { \\mu } ^ p \\mbox { a n d a l l } n \\mbox { a n d } \\lim _ { n \\rightarrow \\infty } \\Vert f - P _ n f \\Vert _ p = 0 . \\end{align*}"}
{"id": "8654.png", "formula": "\\begin{align*} \\overline B ' : = \\sup _ { s > 0 } \\left ( \\int _ s ^ \\infty V ( t ) ^ { - \\frac 1 { p - 1 } } \\ , d t \\right ) ^ { p - 1 } \\left ( \\int _ 0 ^ s W ( t ) \\ , d t \\right ) \\end{align*}"}
{"id": "1105.png", "formula": "\\begin{align*} G ( V , V ) G ( \\mbox { g r a d } b , P _ U U ) = G ( U , U ) G ( \\mbox { g r a d } b , P _ V V ) , \\end{align*}"}
{"id": "2906.png", "formula": "\\begin{align*} F = ( f , f \\circ S ^ { - m } , f \\circ S ^ { - 2 m } , \\dotsc , f \\circ S ^ { - ( n - 1 ) m } ) , \\end{align*}"}
{"id": "6676.png", "formula": "\\begin{align*} \\begin{aligned} \\| w \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } & \\leq C _ 1 ( \\| F _ 1 \\| _ { L ^ p ( D ) } + \\| z \\| _ { L ^ p ( D ) } + \\| w _ 0 \\| _ { W ^ { 2 , 0 } _ { p } ( [ 0 , 1 ] ) } ) \\\\ & \\leq C _ 1 ( \\| F _ 1 \\| _ { C ^ 0 ( D ) } + \\| z \\| _ { C ^ 0 ( D ) } + \\| w _ 0 \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } ) \\\\ \\end{aligned} \\end{align*}"}
{"id": "664.png", "formula": "\\begin{align*} \\left . \\int u _ { 1 , t } d \\nu ^ 2 _ { t } \\ , \\right | _ { t = s } ^ { t = s ' } = \\left . \\int u _ { 1 , t } d \\nu ^ 2 _ { t } \\ , \\right | _ { t = \\bar s } ^ { t = s ' } + \\left . \\int u _ { 1 , t } d \\nu ^ 2 _ { t } \\ , \\right | ^ { t = \\bar s } _ { t = s } . \\end{align*}"}
{"id": "4569.png", "formula": "\\begin{align*} \\Gamma ( \\mathbf v , d ) & = \\{ \\mathbf a \\in \\mathbb Z ^ 3 \\mid \\mathbf a \\times \\mathbf v / d \\in M ( \\mathbf v , d ) \\} \\\\ & = \\{ \\mathbf a \\in \\mathbb Z ^ 3 \\mid \\mathbf a \\times \\mathbf v ( \\mathbf a \\times \\mathbf v ) \\times \\mathbf v \\} . \\end{align*}"}
{"id": "7217.png", "formula": "\\begin{align*} \\| L ( | U ^ 0 | ^ { p - j } | U | ^ j ) \\| _ 2 \\le C _ 2 ( \\| U \\| _ 2 D ( T ) ) ^ j \\mbox { f o r } \\ j = 0 , 1 , \\end{align*}"}
{"id": "5449.png", "formula": "\\begin{align*} k = q ( r ^ B _ A ( k ) ) ^ { - 1 } r ^ B _ A ( k ) . \\end{align*}"}
{"id": "3337.png", "formula": "\\begin{align*} I m ( s _ 1 ^ { \\pm 1 } - I d ) = \\mathbb { R } e _ 2 \\ \\ \\ \\ \\ \\ \\ I m ( s _ 2 ^ { \\pm 1 } - I d ) = \\mathbb { R } e _ 1 \\end{align*}"}
{"id": "4172.png", "formula": "\\begin{align*} \\norm { ( x , u ) } : = ( | x | ^ 4 + | u | ^ 2 ) ^ { 1 / 4 } , ( x , u ) \\in G \\end{align*}"}
{"id": "7600.png", "formula": "\\begin{align*} X _ 1 = \\partial _ x , X _ 2 = \\partial _ y , & X _ 3 = x \\partial _ x , X _ 4 = y \\partial _ y , \\\\ X _ 5 = y \\partial _ x , X _ 6 = x \\partial _ y , X _ 7 & = x ^ 2 \\partial _ x + x y \\partial _ y , X _ 8 = x y \\partial _ x + y ^ 2 \\partial _ y . \\end{align*}"}
{"id": "2341.png", "formula": "\\begin{align*} B = \\lim _ { Q \\in \\Psi _ n } \\nu ( Q ) \\mbox { a n d } \\overline B = \\lim _ { Q \\in \\Psi _ n } \\nu _ Q ( F ) . \\end{align*}"}
{"id": "1396.png", "formula": "\\begin{align*} \\| N _ t u _ 0 \\| _ { H ^ { 1 + \\varepsilon } ( \\mathbb { T } ) } & = \\left \\| \\left ( L _ { t } [ S _ t u _ 0 ] S _ t - W _ t ^ \\gamma \\right ) u _ 0 \\right \\| _ { H _ x ^ { 1 + \\varepsilon } ( \\mathbb { T } ) } \\\\ & \\leq \\tilde { C } ( \\| f \\| _ { H ^ 1 _ x ( \\mathbb { T } ) } , \\gamma , \\varepsilon ) , \\end{align*}"}
{"id": "6686.png", "formula": "\\begin{align*} & 0 < m _ i < \\min \\big \\{ \\dfrac { 1 } { 2 } , \\dfrac { p - \\alpha } { \\alpha ( p - 1 ) } \\} , \\ C > \\dfrac { h } { m _ i } \\ \\ 0 < \\varepsilon < \\dfrac { d ^ { 2 ( p - 1 ) } a _ 1 \\gamma _ { B L } } { h ( 1 + C / 2 ) } , \\ i = 1 , 2 , \\end{align*}"}
{"id": "8577.png", "formula": "\\begin{align*} \\sup _ n \\left | \\sum _ { k = 1 } ^ \\infty t _ { n , k } a _ k w _ { m _ k } ( x ) \\right | < \\infty \\end{align*}"}
{"id": "3698.png", "formula": "\\begin{align*} | ( \\xi ' , \\xi _ n ) | _ x ^ 2 = \\xi _ n ^ 2 + R ( x ' , x _ n , \\xi ' ) , \\end{align*}"}
{"id": "6681.png", "formula": "\\begin{align*} \\begin{aligned} & \\sup \\limits _ { t \\in [ 0 , T _ 0 ] } \\big ( \\| w _ 1 ( t , \\cdot ) - w _ 2 ( t , \\cdot ) \\| _ { C ^ { 0 } ( [ 0 , 1 ] } + \\| z _ 1 ( t , \\cdot ) - z _ 2 ( t , \\cdot ) \\| _ { C ^ { 0 } ( [ 0 , 1 ] ) } \\big ) + \\| h _ 1 - h _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + \\| g _ 1 - g _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } \\\\ & \\leq ( C _ 3 + 1 ) ( \\| h _ 1 - h _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + \\| g _ 1 - g _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } ) + C _ 3 ( \\| z _ { 0 1 } - z _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } + \\| w _ { 0 1 } - w _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } ) \\\\ \\end{aligned} \\end{align*}"}
{"id": "7424.png", "formula": "\\begin{align*} \\bold { q } ( t ) = - \\int _ 0 ^ { \\infty } g ( s ) \\theta _ x ( t - s ) d s , \\end{align*}"}
{"id": "7426.png", "formula": "\\begin{align*} \\bold { q } _ t ( t ) + k \\bold { q } = - \\theta _ x ( t ) . \\end{align*}"}
{"id": "6683.png", "formula": "\\begin{align*} \\begin{aligned} & \\| h ' _ 1 - h ' _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } \\leq \\dfrac { 2 \\mu } { h _ { 0 1 } } T _ 0 ^ { \\beta _ 0 / 2 } \\| w _ { 1 } - w _ { 2 } \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } + 2 C _ 0 | h _ { 0 1 } - h _ { 0 2 } | , \\\\ & \\| g ' _ 1 - g ' _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } \\leq \\dfrac { 2 \\eta } { g _ { 0 1 } } T _ 0 ^ { \\beta _ 0 / 2 } \\| z _ { 1 } - z _ { 2 } \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } + 2 C _ 0 | g _ { 0 1 } - g _ { 0 2 } | , \\end{aligned} \\end{align*}"}
{"id": "6100.png", "formula": "\\begin{align*} \\theta _ { n + 1 \\ , \\ , n } = 0 \\theta _ { n + 1 \\ , \\ , n + 2 } = a \\ , \\omega _ { n + 1 \\ , \\ , n + 2 } \\end{align*}"}
{"id": "7946.png", "formula": "\\begin{align*} \\tilde y _ i = \\left \\{ \\begin{array} { c c } y _ i y _ \\mathrm { r } ^ { c _ i } & 1 \\leq i \\leq \\mathrm { r } - 1 \\\\ y _ \\mathrm { r } ^ { - c } & i = \\mathrm { r } \\end{array} \\right . \\end{align*}"}
{"id": "1633.png", "formula": "\\begin{align*} r _ n [ B , M ] ^ * = \\{ r _ n ( N ) : N \\in [ B , M ] ^ * \\} , \\end{align*}"}
{"id": "2215.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial \\varphi } { \\partial t } = \\varphi ^ { \\prime } ( t ) \\leq C ( 1 + t ) e ^ { - t } , \\end{array} \\end{align*}"}
{"id": "1504.png", "formula": "\\begin{align*} Y _ { \\lambda } ( x , p ) = \\sum _ { k = 1 } ^ { \\infty } ( k ) _ { p , \\lambda } \\sum _ { n = k } ^ { \\infty } \\frac { x ^ { n } } { n ! } = \\sum _ { k = 1 } ^ { \\infty } ( k ) _ { p , \\lambda } \\bigg ( e ^ { x } - \\sum _ { l = 0 } ^ { k - 1 } \\frac { x ^ { l } } { l ! } \\bigg ) . \\end{align*}"}
{"id": "3344.png", "formula": "\\begin{align*} { \\mathcal Q } = { \\mathcal P } ^ t { \\mathcal P } \\end{align*}"}
{"id": "7756.png", "formula": "\\begin{align*} { \\bf U } ^ k ( \\alpha _ i ) \\cong \\displaystyle \\sum _ { j = 1 } ^ { L } T _ j ^ k ( \\alpha _ i ) { \\boldsymbol \\xi } _ j k = 1 , \\ldots , m , \\ i = 1 , \\ldots , p , \\end{align*}"}
{"id": "4911.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial X } { \\partial t } ( x , t ) = u ^ \\alpha \\rho ^ \\delta f ^ { - \\beta } ( \\kappa ) \\nu ( x , t ) , \\\\ & X ( \\cdot , 0 ) = X _ 0 . \\end{cases} \\end{align*}"}
{"id": "5976.png", "formula": "\\begin{align*} \\| ( A + \\lambda I ) ^ { - 1 } \\| { = \\frac 1 { \\operatorname { d i s t } ( \\lambda , \\sigma ( A ) ) } } \\leq \\frac { 1 } { \\sin \\theta | \\lambda | } | \\arg \\lambda | < \\pi - \\theta \\end{align*}"}
{"id": "2795.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\| \\nabla f ( x _ i ) \\| ^ 2 { } \\leq { } \\frac { 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left ( 2 h _ i - h _ i ^ 2 \\right ) } \\ , , \\ , \\forall h _ i \\in ( 0 , 2 ) \\end{align*}"}
{"id": "4511.png", "formula": "\\begin{align*} R _ { \\alpha \\beta 3 \\alpha } = & | \\nabla u | ^ { - 1 } ( \\nabla _ \\alpha \\nabla _ \\beta - \\nabla _ \\beta \\nabla _ \\alpha ) \\nabla _ \\alpha u \\\\ = & - | \\nabla u | ^ { - 1 } \\nabla _ \\alpha ( k _ { \\alpha \\beta } | \\nabla u | ) + | \\nabla u | ^ { - 1 } \\nabla _ \\beta ( k _ { \\alpha \\alpha } | \\nabla u | ) \\\\ = & - \\nabla _ \\alpha k _ { \\alpha \\beta } + \\nabla _ { \\beta } k _ { \\alpha \\beta } + k _ { \\alpha 3 } k _ { \\alpha \\beta } - k _ { \\beta 3 } k _ { \\alpha \\alpha } , \\end{align*}"}
{"id": "2020.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dd X ^ { i , N } _ t & = & V ^ { i , N } _ t \\dd t \\\\ \\dd V ^ { i , N } _ t & = & F \\big ( X ^ { i , N } _ t , V ^ { i , N } _ t , \\mu _ { \\mathcal { X } ^ N _ t } \\big ) \\dd t + \\sigma \\big ( X ^ i _ t , V ^ { i , N } _ t , \\mu _ { \\mathcal { X } ^ N _ t } \\big ) \\dd B ^ { i } _ t , \\end{array} \\right . \\end{align*}"}
{"id": "5381.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u _ 2 ^ * + \\sum _ { | \\alpha | \\leq m } ( - 1 ) ^ { | \\alpha | } D ^ \\alpha ( a _ { 2 , \\alpha } u _ 2 ^ * ) & = 0 \\mbox { i n } \\ ; \\ ; \\Omega , u _ 2 ^ * - f _ 2 \\in \\widetilde H ^ s ( \\Omega ) , \\end{align*}"}
{"id": "3050.png", "formula": "\\begin{align*} \\overline { G } _ { n } ( u ) = \\int _ { u } ^ { \\infty } { g } _ { n } ( v ) \\mathrm { d } v \\end{align*}"}
{"id": "2717.png", "formula": "\\begin{align*} \\mathrm { d } \\Gamma _ \\perp ( A ) : = \\sum _ { m , n \\ge 3 } \\langle u _ m , A u _ n \\rangle a ^ * _ m a _ n \\end{align*}"}
{"id": "8190.png", "formula": "\\begin{align*} \\| P _ \\delta ( E ) \\| _ { \\mathrm { o p } } = \\| P ^ \\parallel _ \\delta ( E ^ \\parallel ) \\| _ { \\mathrm { o p } } \\| P ^ \\perp _ \\delta ( E ^ \\perp ) \\| _ { \\mathrm { o p } } \\end{align*}"}
{"id": "1285.png", "formula": "\\begin{align*} n ( T , k + 1 - j ) - n ( T , k - j ) & = \\frac { ( j - 1 ) ( k - 1 ) ! } { j ! } , \\end{align*}"}
{"id": "5564.png", "formula": "\\begin{align*} \\frac { 1 } { n + 2 } \\sum _ { j = 1 } ^ { n + 1 } ( 1 - e ^ { - y } ) ^ j - \\frac { y } { n + 2 } & = \\sum _ { l = 0 } ^ { n - 1 } \\frac { 1 } { l + 1 } \\dbinom { n + 1 } { l } e ^ { - l y } \\left ( 1 - e ^ { - y } \\right ) ^ { n - l + 1 } \\\\ & + \\frac { \\left ( e ^ { - ( n + 1 ) y } - 1 + ( n + 1 ) y \\right ) } { n + 2 } + \\left ( e ^ { - n y } - e ^ { - ( n + 1 ) y } - y \\right ) \\end{align*}"}
{"id": "7198.png", "formula": "\\begin{align*} \\langle f , I ^ { \\circ } ( w _ 1 , z ) u _ { p + \\frac { j _ 2 } { T } } w _ 2 \\rangle = 0 \\end{align*}"}
{"id": "5159.png", "formula": "\\begin{align*} d _ i & = \\alpha _ i d _ { i - 1 } \\\\ d _ { i + 1 } & = \\alpha _ { i + 1 } d _ i + d _ { i - 1 } = ( \\alpha _ { i + 1 } \\alpha _ i + 1 ) d _ { i - 1 } \\\\ \\end{align*}"}
{"id": "3757.png", "formula": "\\begin{align*} \\gamma _ 2 : = \\inf \\{ k : k \\in \\R _ + , | V ( x , v , \\tau _ { \\ast } , 0 ) | | \\leq 2 ^ { k M _ t } \\} . \\end{align*}"}
{"id": "6074.png", "formula": "\\begin{align*} \\Lambda _ 1 = ( 0 , 1 , 2 , 3 ) \\end{align*}"}
{"id": "1478.png", "formula": "\\begin{align*} s + 1 = r _ 1 + \\cdots + r _ { w - 1 } + s _ w \\enspace . \\end{align*}"}
{"id": "6971.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 1 } ^ N \\frac { a _ n ^ N } { \\lambda _ n ^ 2 - z } \\to \\sum \\limits _ { n \\ge 1 } \\frac { a _ n } { \\lambda _ n ^ 2 - z } N \\to \\infty \\end{align*}"}
{"id": "269.png", "formula": "\\begin{align*} ( D ^ { \\nu } _ i ) _ { j k } = ( D ^ { \\nu } _ j ) _ { i k } \\end{align*}"}
{"id": "5754.png", "formula": "\\begin{align*} \\pi _ { i _ 1 } \\pi _ J = \\pi _ { i _ 1 } \\cdot \\pi _ { J _ 1 } \\cdots \\pi _ { J _ m } \\end{align*}"}
{"id": "1980.png", "formula": "\\begin{align*} a _ s ( y ) & = a ( x _ 0 + s y ) \\\\ f _ s ( y ) & = s f ( x _ 0 + s y ) . \\end{align*}"}
{"id": "8219.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { r r r r } 0 & - 1 & 0 & 0 \\\\ - 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 \\\\ 0 & 0 & - 1 & 0 \\end{array} \\right ) , \\end{align*}"}
{"id": "4890.png", "formula": "\\begin{gather*} T _ { 2 , 2 } ( m ) = \\left \\lfloor \\frac m 3 \\left \\lfloor \\frac { m - 1 } 2 \\right \\rfloor \\right \\rfloor - [ m \\equiv 5 \\bmod 6 ] \\\\ T _ { 2 , b } ( m ) = \\begin{cases} \\left \\lfloor \\frac { b - 2 } 3 \\binom m 2 \\right \\rfloor + T _ { 2 , 2 } ( m ) & 2 \\mid m \\land 2 \\mid b \\\\ \\left \\lfloor \\frac { b - 1 } 3 \\binom m 2 \\right \\rfloor - [ m \\equiv 5 \\bmod 6 \\lor b \\equiv 5 \\bmod 6 ] & \\end{cases} \\end{gather*}"}
{"id": "5937.png", "formula": "\\begin{align*} { e ^ { W \\left [ \\eta _ T \\left ( t \\right ) \\right ] } } = { e ^ { W \\left [ \\eta \\left ( t / T \\right ) \\right ] } } \\sim e ^ { T \\int w ( \\eta ( \\tau ) ) d \\tau } \\end{align*}"}
{"id": "8563.png", "formula": "\\begin{align*} \\int \\exp _ 2 ^ * ( s _ p ( q ) ) \\ p \\ d \\mu = \\int \\frac q p \\log \\frac q p p \\ d \\mu - \\int \\frac q p \\ d \\mu + 1 = - \\int \\log \\frac p q \\ q \\ d \\mu \\ . \\end{align*}"}
{"id": "3271.png", "formula": "\\begin{align*} \\rho _ 2 = s \\Big ( - i \\omega _ 2 + \\big ( \\frac { \\xi } { 2 s } + \\sqrt { 1 - \\frac { \\vert \\xi \\vert ^ 2 } { 4 s ^ 2 } } \\omega _ 1 \\big ) \\Big ) : = s \\omega ^ { * } _ 2 ( s ) . \\end{align*}"}
{"id": "4221.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { 1 \\vee \\log \\left ( \\phi ^ { - 1 } ( k ) \\right ) } = \\infty . \\end{align*}"}
{"id": "6054.png", "formula": "\\begin{align*} \\lambda _ i = \\frac { \\kappa _ i - \\kappa _ 1 } { \\kappa _ 2 - \\kappa _ 1 } \\end{align*}"}
{"id": "3791.png", "formula": "\\begin{align*} \\sum _ { i = 3 , 4 } \\big | \\mathbf { P } \\big ( T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , E ) ( t , x , \\zeta ) + \\hat { \\zeta } \\times T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , B ) ( t , x , \\zeta ) \\big ) \\big | + 2 ^ { ( \\gamma _ 1 - \\gamma _ 2 ) M _ { t ^ { \\star } } } \\big | T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , E ) ( t , x , \\zeta ) + \\hat { \\zeta } \\times T _ { k , j ; n } ^ { \\mu , i } ( \\mathfrak { m } , B ) ( t , x , \\zeta ) \\big | \\end{align*}"}
{"id": "5793.png", "formula": "\\begin{align*} [ x , y z ] = [ x , z ] [ x , y ] ^ z \\mbox { a n d } [ x y , z ] = [ x , z ] ^ y [ y , z ] . \\end{align*}"}
{"id": "7954.png", "formula": "\\begin{align*} I _ { X _ + } ( y , z ) = z e ^ { t _ + / z } \\sum _ { d \\in \\mathbb K _ { + } } y ^ { d } \\left ( \\prod _ { i = 0 } ^ { m } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\textbf { 1 } _ { [ - d ] } , \\end{align*}"}
{"id": "2907.png", "formula": "\\begin{align*} f _ { \\ell k + 1 } \\circ S ^ \\ell = f _ { ( \\ell - 1 ) k + 1 } \\circ S ^ { \\ell - 1 } = \\dotsb f _ { k + 1 } \\circ S ^ { - 1 } = f _ 1 , \\end{align*}"}
{"id": "5952.png", "formula": "\\begin{align*} \\langle A _ { i j } ( t ) g [ \\mathbf { A } ] \\rangle = \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { n ! } w ^ { ( n + 1 ) } _ { i j i _ 1 j _ 1 . . . i _ n j _ n } \\biggl \\langle \\frac { \\delta ^ n \\ , g [ \\mathbf { A } ] } { \\delta A _ { i _ 1 j _ 1 } ( t _ 1 ) \\ , \\dots \\ , \\delta A _ { i _ n j _ n } ( t _ n ) } \\biggr \\rangle \\end{align*}"}
{"id": "7569.png", "formula": "\\begin{align*} X ( t , x ) = \\sum _ { \\alpha = 1 } ^ r b _ \\alpha ( t ) X _ \\alpha ( x ) , t \\in \\mathbb { R } , x \\in N , \\end{align*}"}
{"id": "2379.png", "formula": "\\begin{align*} \\nu \\left ( L ( h _ \\rho ) - L _ p ( h _ \\rho ) \\right ) > \\beta _ b + b \\gamma _ \\rho = \\nu ( L ( h _ \\rho ) ) . \\end{align*}"}
{"id": "6909.png", "formula": "\\begin{align*} p ( t , g ) & = e ^ { i \\mu g } \\mathcal { F } ^ { - 1 } _ { \\xi } [ \\hat { p } ( 0 , e ^ { - t } \\xi + \\mu ) ] * ( \\frac { 1 } { \\sqrt { 2 \\pi } } \\mathcal { F } ^ { - 1 } _ { \\xi } [ e ^ { - H _ t ( \\xi ) } ] ) \\\\ & = e ^ { i \\mu g } [ e ^ t p ( 0 , e ^ t \\cdot ) e ^ { - i \\mu e ^ t \\cdot } ] \\ast ( \\frac { 1 } { \\sqrt { 2 \\pi } } \\mathcal { F } ^ { - 1 } _ { \\xi } [ e ^ { - H _ t ( \\xi ) } ] ) . \\end{align*}"}
{"id": "1945.png", "formula": "\\begin{align*} \\phi _ { j } ^ { ( q ) } ( z ) & = A _ { j } ^ { ( q ) } ( z ) , \\\\ \\beta _ { j } ^ { ( q ) } ( z ) & = B _ { j } ^ { ( q ) } ( z ) , \\\\ \\psi _ { j } ( z ) & = W _ { j } ( z ) . \\end{align*}"}
{"id": "3059.png", "formula": "\\begin{align*} c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast } = \\frac { \\Gamma \\left ( ( n - 1 ) / 2 \\right ) } { ( 2 \\pi ) ^ { ( n - 1 ) / 2 } } \\left [ \\int _ { 0 } ^ { \\infty } x ^ { ( n - 3 ) / 2 } \\overline { G } _ { n } \\left ( \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } + x \\right ) \\mathrm { d } x \\right ] ^ { - 1 } , \\end{align*}"}
{"id": "5266.png", "formula": "\\begin{align*} n ( \\Gamma ) = \\sum _ { v \\in \\mathrm { V e r t } ( \\Lambda ) } n ( v ) - ( r + s ) | E | . \\end{align*}"}
{"id": "2863.png", "formula": "\\begin{align*} L _ { \\mathcal { L } } = \\| \\tfrac { 1 } { N } A ^ T B A \\| \\leq \\tfrac { 1 } { N } \\| A ^ T A \\| . \\end{align*}"}
{"id": "7873.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } T & = \\lambda ^ 3 + \\mu ^ 3 + \\nu ^ 3 ; \\\\ L & = \\lambda ^ 3 + \\mu ^ 3 + \\nu ^ 3 - \\left ( \\lambda ^ 2 \\mu + \\lambda \\mu ^ 2 + \\lambda \\nu ^ 2 + \\lambda ^ 2 \\nu + \\mu ^ 2 \\nu + \\mu \\nu ^ 2 \\right ) + 3 \\lambda \\mu \\nu . \\end{array} \\right . \\end{align*}"}
{"id": "2900.png", "formula": "\\begin{align*} a _ 1 x _ 2 + a _ 2 x _ 1 = b + a _ 1 + a _ 1 a _ 2 + a _ 3 , \\end{align*}"}
{"id": "3646.png", "formula": "\\begin{align*} m _ v \\coloneqq m - \\sum _ { 0 \\leq i < k } m _ i = \\lim _ { n \\to \\infty } \\int _ { B _ { r _ n } \\setminus \\cup _ { 0 \\leq i < k _ n } B ^ i _ n } u _ n . \\end{align*}"}
{"id": "5311.png", "formula": "\\begin{align*} \\lambda _ { b , k } = \\int _ 0 ^ 1 x ^ { k - 2 } ( 1 - x ) ^ { b - k } \\Lambda ( d x ) \\end{align*}"}
{"id": "1792.png", "formula": "\\begin{align*} y ^ * = \\lim _ { n \\rightarrow \\infty } \\left ( z ^ * + \\sum _ { m = 1 } ^ n c _ m \\widetilde { u } _ m \\right ) \\end{align*}"}
{"id": "2745.png", "formula": "\\begin{align*} \\mathcal { K } _ { 1 } : = & - u _ { i } ^ { p - 2 } ( y ) u _ { i } ( x ) , \\\\ \\mathcal { K } _ { 2 } : = & \\left ( u _ { i } ^ { p - 2 } ( y ) - | u _ { i } ( x ) - u _ { i } ( y ) | ^ { p - 2 } \\right ) u _ { i } ( x ) , \\\\ \\mathcal { K } _ { 3 } : = & - \\left ( u _ { i } ^ { p - 2 } ( y ) - | u _ { i } ( x ) - u _ { i } ( y ) | ^ { p - 2 } \\right ) u _ { i } ( y ) , \\\\ \\Theta : = & - | u _ { i } ( x ) - u _ { i } ( y ) | ^ { p - 2 } ( u _ { i } ( x ) - u _ { i } ( y ) ) . \\end{align*}"}
{"id": "3872.png", "formula": "\\begin{align*} \\nabla ( G _ 0 - \\Gamma ) \\in L ^ { \\bar q } ( \\Omega ) , \\bar q = \\frac { N ( p - 1 ) } { N - 1 } , \\end{align*}"}
{"id": "1368.png", "formula": "\\begin{align*} v = W _ t u _ 0 \\pm T [ W _ t u _ 0 , v , \\cdots , v ] + z . \\end{align*}"}
{"id": "1568.png", "formula": "\\begin{align*} S ( x , y ) = \\frac { \\partial G ^ m } { \\partial y ^ m } - y ^ m \\frac { \\partial \\left ( \\ln \\sigma _ F \\right ) } { \\partial x ^ m } , \\end{align*}"}
{"id": "8608.png", "formula": "\\begin{align*} \\mathcal { F } ( \\tilde A , \\tilde B ) = { \\| A - ( J - R ) Q \\| } _ F ^ 2 + { \\| B - ( R - J ) C ^ T \\| } _ F ^ 2 , \\end{align*}"}
{"id": "574.png", "formula": "\\begin{align*} M _ 2 = f _ k ( i ) f _ k ( i + 1 ) + O _ \\alpha \\Big ( \\frac { \\tau _ 3 ( i ) \\tau _ 3 ( i + 1 ) } { \\sqrt { i } } \\Big ) . \\end{align*}"}
{"id": "8832.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left \\| f _ t \\circ \\left ( 1 _ { A _ 1 ^ { ( n ) } } * 1 _ { A _ 2 ^ { ( n ) } } \\right ) \\right \\| = \\| f _ t \\circ ( \\phi _ 1 * \\phi _ 2 ) \\| , \\lim _ { n \\to \\infty } \\left \\| f _ t \\circ \\left ( 1 _ { A _ 1 ^ { ( n ) } } ^ * * 1 _ { A _ 2 ^ { ( n ) } } ^ * \\right ) \\right \\| = \\| f _ t \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) \\| \\end{align*}"}
{"id": "1049.png", "formula": "\\begin{align*} \\theta _ 0 ( t ) = \\sum _ { n = r + 1 } ^ \\infty \\psi _ 1 ^ { [ n * ] } ( t ) , \\end{align*}"}
{"id": "5287.png", "formula": "\\begin{align*} \\delta ( x _ 1 ^ { a _ 1 } \\cdots x _ { i - 1 } ^ { a _ { i - 1 } } x _ { i + 1 } ^ { a _ { i + 1 } } \\cdots x _ n ^ { a _ n } d x _ 1 \\wedge \\cdots \\wedge \\widehat { d x _ i } \\wedge \\cdots \\wedge d x _ n ) = ( - 1 ) ^ { i - 1 } \\hbar ^ { - 1 } r _ i x _ 1 ^ { a _ 1 } \\cdots x _ { i - 1 } ^ { a _ { i - 1 } } x _ i ^ { r _ i - 1 } x _ { i + 1 } ^ { a _ { i + 1 } } \\cdots x _ n ^ { a _ n } \\Omega . \\end{align*}"}
{"id": "2605.png", "formula": "\\begin{align*} E = \\left ( n _ 1 + n _ 2 + n _ 3 + \\dfrac { 3 } { 2 } \\right ) \\omega \\end{align*}"}
{"id": "8694.png", "formula": "\\begin{align*} A _ { B H } = \\frac { ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } } { 2 } \\int \\limits _ { t _ { 0 } } ^ { t _ { 1 } } ( x ^ { 1 } \\dot { x } ^ { 2 } - x ^ { 2 } \\dot { x } ^ { 1 } ) d t . \\end{align*}"}
{"id": "8804.png", "formula": "\\begin{align*} 1 _ { \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) } ^ * ( x ) = \\inf \\{ t ' > 0 \\mid \\mu ( 1 _ { \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) } ^ { - 1 } ( \\mathbb { R } _ { > t ' } ) ) \\leq 2 | x | \\} \\end{align*}"}
{"id": "8113.png", "formula": "\\begin{align*} \\P \\biggl ( \\sup _ { x \\in Q } Y ( t , x ) > R \\mathrel { \\Big | } A \\biggr ) & \\geq C \\P \\biggl ( \\sup _ { x \\in Q } g ( t - \\tau , x - \\eta ) > 4 R \\mathrel { \\Big | } A \\biggr ) \\\\ & = C \\P ( ( 2 \\pi ( t - \\tau ) ) ^ { - \\frac d 2 } > 4 R \\mid A ) . \\end{align*}"}
{"id": "4414.png", "formula": "\\begin{align*} T _ { e , i } = \\{ 4 k d + i , 4 k d + e + i , 4 k d + 2 e + i , \\ldots , 4 k d + ( 4 k - 1 ) e + i \\} . \\end{align*}"}
{"id": "2690.png", "formula": "\\begin{align*} H _ { 2 - \\mathrm { m o d e } } : = P ^ { \\otimes N } H _ N P ^ { \\otimes N } \\end{align*}"}
{"id": "6625.png", "formula": "\\begin{align*} \\left | \\sum _ { 0 < | c | \\le h ^ \\delta } \\sum _ { k = 1 } ^ \\infty k ^ { - 4 } S _ { k } ( c ; \\xi ) I _ { k , \\ell } ( { } ^ t g c ) \\right | \\ll _ { w , \\ell , \\theta , \\epsilon } \\| g \\| _ E h ^ { 7 / 2 + 3 \\delta + \\epsilon } . \\end{align*}"}
{"id": "8349.png", "formula": "\\begin{align*} \\sum \\limits _ { i } ^ { N } \\alpha _ i = 0 \\end{align*}"}
{"id": "2983.png", "formula": "\\begin{align*} z _ Q = - \\frac { b } { 2 a } + i \\frac { | d | } { 2 a } \\end{align*}"}
{"id": "5276.png", "formula": "\\begin{align*} \\begin{aligned} P ^ { C o n t , + r } & = \\{ ( I _ 0 , k _ 1 ( 0 ) , k _ 2 ( 0 ) ) , \\ldots , ( I _ h , k _ 1 ( h ) , k _ 2 ( h ) ) , ( \\emptyset , r , 0 ) \\} ; \\\\ P ^ { C o n t , + s } & = \\{ ( I _ 0 , k _ 1 ( 0 ) , k _ 2 ( 0 ) ) , \\ldots , ( I _ h , k _ 1 ( h ) , k _ 2 ( h ) ) , ( \\emptyset , 0 , s ) \\} . \\end{aligned} \\end{align*}"}
{"id": "6752.png", "formula": "\\begin{align*} K _ X ^ 2 = 1 6 n , p _ g \\left ( X \\right ) = 2 n , \\chi \\left ( \\mathcal { O } _ X \\right ) = 2 n , q \\left ( X \\right ) = 1 . \\end{align*}"}
{"id": "4366.png", "formula": "\\begin{align*} \\sum \\nolimits _ { k , \\beta } a _ { k , \\beta } \\ , \\zeta _ { k , \\beta } = 0 . \\end{align*}"}
{"id": "2366.png", "formula": "\\begin{align*} \\nu ( f - l ( h _ \\sigma ) ) \\geq \\min \\{ \\nu ( f ) , \\nu ( l ( h _ \\sigma ) ) \\} > \\nu _ \\sigma ( f ) = \\beta _ b + b \\gamma _ \\sigma , \\end{align*}"}
{"id": "764.png", "formula": "\\begin{align*} w _ m ^ * = \\frac { \\varrho | g _ m | ^ 2 } { \\sum _ { k = 1 } ^ { M } \\varrho | g _ k | ^ 2 } , m = 1 , 2 , \\ldots , M \\end{align*}"}
{"id": "1004.png", "formula": "\\begin{align*} \\mu ( S _ { N + 1 } \\in \\cdot \\ | \\ S _ N ) = \\frac { \\nu _ { N + 1 } ( S _ N ) } { \\nu _ { N } ( S _ N ) } \\delta _ { S _ N } ( \\cdot ) + \\left ( 1 - \\frac { \\nu _ { N + 1 } ( S _ N ) } { \\nu _ { N } ( S _ N ) } \\right ) \\delta _ { N + 1 } ( \\cdot ) \\end{align*}"}
{"id": "8908.png", "formula": "\\begin{align*} \\begin{cases} - V '' + V = V ^ { p - 1 } , U _ 0 > 0 & , \\\\ V ( 0 ) = \\max _ \\R V , \\\\ V ( x ) \\to 0 & . \\end{cases} \\end{align*}"}
{"id": "8988.png", "formula": "\\begin{align*} ( x - n + 1 ) _ n = \\sum _ { k = 0 } ^ n ( - 1 ) ^ { n - k } { n \\brack k } x ^ k , \\sum _ { n = k } ^ { \\infty } ( - 1 ) ^ { n - k } { n \\brack k } \\frac { z ^ n } { n ! } = \\frac { 1 } { k ! } \\ln ^ k ( 1 + z ) , \\end{align*}"}
{"id": "6050.png", "formula": "\\begin{align*} \\left ( \\Lambda _ { r , s } \\right ) _ { r ' , s ' } = \\Lambda \\end{align*}"}
{"id": "3805.png", "formula": "\\begin{align*} H y p ^ 4 ( t _ 1 , t _ 2 ) = \\sum _ { a = 0 , 1 } E r r ^ a _ { i , i _ 1 , i _ 2 , i _ 3 , i _ 4 } ( t _ 1 , t _ 2 ) + H y p ^ 5 ( t _ 1 , t _ 2 ) , H y p ^ 5 ( t _ 1 , t _ 2 ) : = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { ( \\R ^ 3 ) ^ 5 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta + \\sigma + \\kappa + \\chi ) } \\end{align*}"}
{"id": "2611.png", "formula": "\\begin{align*} \\begin{cases} r _ + = 2 a + 3 k \\ , \\ \\ r _ - = k \\\\ \\vdots \\\\ r _ + = a + 3 k \\ , \\ \\ r _ - = a + k \\end{cases} \\ , \\end{align*}"}
{"id": "1544.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } g '' ( r ) + \\frac { \\alpha } { 2 } g ' ( r ) - \\lambda g ( r ) = 0 \\end{align*}"}
{"id": "1535.png", "formula": "\\begin{align*} \\kappa \\sqrt { \\ss } = \\tfrac 1 2 C _ { \\widehat { r } } = ( \\pi - \\phi ( \\widehat { r } ) ) \\sinh ( \\beta \\widehat { r } ) . \\end{align*}"}
{"id": "4346.png", "formula": "\\begin{align*} & L _ { 1 } ( C ) = \\{ x \\in V ( C ) \\mid \\exists \\ , ( x , y ) \\in E ( D _ { G } ) \\ , \\ , \\ , \\ , y \\not \\in C \\} \\\\ & L _ { 2 } ( C ) = \\{ x \\in C \\mid \\mathcal { N } ( x ) \\subseteq C \\} \\\\ & L _ { 3 } ( C ) = C \\setminus ( L _ { 1 } ( C ) \\cup L _ { 2 } ( C ) ) \\end{align*}"}
{"id": "7163.png", "formula": "\\begin{align*} ( g _ { 3 1 } y _ 1 + g _ { 3 2 } y _ 2 + g _ { 3 3 } y _ 3 ) \\{ g _ { 3 1 } y _ 1 + g _ { 3 2 } y _ 2 + ( g _ { 1 3 } + g _ { 2 3 } + g _ { 3 3 } ) y _ 3 ) \\} ^ { b - 1 } \\\\ = g _ { 3 2 } g _ { 2 3 } ^ { b - 1 } y _ 2 ( y _ 1 + y _ 2 ) ^ { b - 1 } + \\alpha y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } , \\end{align*}"}
{"id": "6289.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } v ( x ) - q p ( x ) v ( x ) = 1 . \\end{align*}"}
{"id": "8235.png", "formula": "\\begin{align*} | b | = \\sqrt { \\frac { \\sqrt { 1 ^ { 2 } + ( 1 6 m \\beta \\lambda / 3 ) ^ { 2 } } - 1 } { 2 } } , \\end{align*}"}
{"id": "3518.png", "formula": "\\begin{align*} & \\mathrm { D T V } _ { ( p , q ) } ( X ) = - \\mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \\mu ^ { 2 } + 2 \\mu \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sigma ^ { 2 } \\left ( L _ { 1 } + 2 L _ { 2 } \\right ) , \\end{align*}"}
{"id": "93.png", "formula": "\\begin{align*} \\chi _ t : = \\max _ { x \\in \\Lambda } \\sum _ { y \\in \\Lambda } S _ t ( x , y ) . \\end{align*}"}
{"id": "427.png", "formula": "\\begin{align*} \\Im H ( x + i y ) \\begin{cases} > 0 , & y > f ( x ) , \\\\ < 0 , & y < f ( x ) . \\end{cases} \\end{align*}"}
{"id": "6297.png", "formula": "\\begin{align*} & \\int f ( x ) h ( x / q ) \\left ( \\frac { 1 } { q } u ( x ) u ( x / q ) + \\frac { 1 } { q } x r ( x ) ( q - 1 ) u ( x / q ) \\right ) y ( x ) d _ q x \\\\ & = f ( x / q ) h ( x / q ) \\left ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \\right ) . \\end{align*}"}
{"id": "5556.png", "formula": "\\begin{align*} \\chi ( M ) = \\frac { 1 } { 8 \\pi ^ 2 } \\int _ M \\left ( \\frac { s ^ 2 } { 2 4 } + | W _ + | ^ 2 + | W _ - | ^ 2 - \\frac { | \\mathring { r } | ^ 2 } { 2 } \\right ) d \\mu _ g \\end{align*}"}
{"id": "2437.png", "formula": "\\begin{align*} N _ f : = \\frac { [ | W _ \\zeta | ] } { [ | W _ f | ] } \\widetilde { N } _ f . \\end{align*}"}
{"id": "7885.png", "formula": "\\begin{align*} s _ { i , - } : = \\# \\{ j : s _ i ^ j < 0 \\} , i = 1 , 2 , \\ldots , n . \\end{align*}"}
{"id": "5548.png", "formula": "\\begin{align*} \\mathbb { E } [ \\ell , k ] = \\left . \\left [ \\dfrac { \\ell } { 2 } \\dbinom { \\ell } { k } + \\sum \\limits _ { i = 0 } ^ { ( k - 1 ) / 2 } \\dbinom { \\ell } { k - 2 i - 1 } \\right ] \\middle / \\dbinom { \\ell } { k } \\right . \\end{align*}"}
{"id": "2631.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ { k + 1 } X _ i & = \\biggl ( \\bigcup _ { i = 1 } ^ { k } X _ i \\biggr ) \\bigcup X _ { k + 1 } \\\\ & = \\biggl ( \\bigsqcup _ { i = 1 } ^ { k } X _ i ' \\biggr ) \\bigcup X _ { k + 1 } \\\\ & = \\biggl ( \\bigsqcup _ { i = 1 } ^ { k } X _ i ' \\biggr ) \\bigsqcup \\Biggl ( X _ { k + 1 } \\backslash \\biggl ( \\bigsqcup _ { i = 1 } ^ { k } X _ i ' \\biggr ) \\Biggr ) \\end{align*}"}
{"id": "1920.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { I } } _ { [ m , j ] } : = \\{ ( i _ { 1 } , \\ldots , i _ { m } ) \\in \\mathbb { Z } ^ { m } : - p \\leq i _ { 1 } \\leq ( m - 1 ) p + j \\ , \\ , \\mbox { a n d } \\ , \\ , i _ { k - 1 } - p \\leq i _ { k } \\leq ( m - k ) p + j , \\ , \\ , 2 \\leq k \\leq m \\} . \\end{align*}"}
{"id": "1919.png", "formula": "\\begin{align*} ( j - i _ { 1 } ) + ( i _ { 1 } + p - i _ { 2 } ) + ( i _ { 2 } + p - i _ { 3 } ) + \\cdots + ( i _ { \\ell - 1 } + p - i _ { \\ell } ) = j + ( \\ell - 1 ) p - i _ { \\ell } . \\end{align*}"}
{"id": "8010.png", "formula": "\\begin{align*} H _ { ( Z _ + , D _ { Z , + } ) } ( y ) = e ^ { \\frac { t _ + } { 2 \\pi i } } \\sum _ { d \\in \\mathbb K _ { + } } y ^ d \\left ( \\frac { \\prod _ { i = 1 } ^ 2 \\Gamma ( 1 + \\frac { v _ i } { z } + v _ i \\cdot d ) } { \\prod _ { i \\in M _ 0 } \\Gamma ( 1 + \\frac { \\bar D _ i } { z } + D _ i \\cdot d ) } \\right ) \\textbf { 1 } _ { [ d ] } [ \\textbf { 1 } ] _ { ( D _ i \\cdot d ) _ { i \\in I _ + } , v _ 2 \\cdot d } . \\end{align*}"}
{"id": "4515.png", "formula": "\\begin{align*} \\partial _ 2 A _ { 1 1 } - \\partial _ 1 A _ { 1 2 } = 2 A _ { 1 2 } k _ { 1 3 } - 2 A _ { 1 1 } k _ { 2 3 } . \\end{align*}"}
{"id": "2108.png", "formula": "\\begin{align*} ( L y _ S , y _ S ) _ V & = \\frac { 1 } { 2 } \\sum \\limits _ { e \\in E } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum \\limits _ { u , v \\in e } ( y _ S ( u ) - y _ S ( v ) ) ^ 2 \\\\ & \\le \\frac { 1 } { 2 } \\sum \\limits _ { e \\in \\partial S } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } 2 | S \\cap e | | e - S | 4 \\\\ & \\le | \\partial S | \\delta _ E ( \\max ) = b ( G ) \\delta _ E ( \\max ) . \\end{align*}"}
{"id": "3675.png", "formula": "\\begin{align*} \\tilde { \\mathcal { H } } ^ { \\tilde { j } , j ; a } _ { k , \\tilde { k } ; m , l } ( t _ 1 , t _ 2 ) : = \\int _ 0 ^ { t _ a } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { 0 } ^ { 2 \\pi } \\int _ { 0 } ^ { \\pi } 2 ^ { k - j + \\gamma M _ t + 2 0 0 \\epsilon M _ t } \\big ( 2 ^ { m + k } + 1 \\big ) \\varphi _ { m ; - 1 0 M _ t } ( t _ a - \\tau ) \\| E ( \\tau , \\cdot ) \\| _ { L ^ \\infty _ x } ( 1 + 2 ^ { k - 3 0 \\epsilon M _ t } | y \\cdot \\tilde { v } | ) ^ { - N _ 0 ^ 3 } \\end{align*}"}
{"id": "1566.png", "formula": "\\begin{align*} K ( x , y , P ) : = \\frac { g _ y ( R _ y ( u ) , u ) } { g _ y ( y , y ) g _ y ( u , u ) - g _ y ( y , u ) g _ y ( y , u ) } , \\end{align*}"}
{"id": "6932.png", "formula": "\\begin{align*} | \\Gamma | ^ 2 - | \\Gamma _ 1 | ^ 2 = u u ^ * . \\end{align*}"}
{"id": "676.png", "formula": "\\begin{align*} D \\psi _ s \\left ( \\frac { \\partial } { \\partial { x ^ i } } \\right ) = \\frac { \\partial \\widehat \\psi _ s ^ a } { \\partial x ^ i } \\frac { \\partial } { \\partial { y ^ a } } . \\end{align*}"}
{"id": "5806.png", "formula": "\\begin{align*} \\frac { \\partial c ( v ) } { \\partial t } = - a ( v ) c ( v ) + \\sum _ { v ' > v } \\bar b _ { v | v ' } a ( v ' ) c ( v ' ) . \\end{align*}"}
{"id": "1240.png", "formula": "\\begin{align*} P ( T , x ) & = \\prod _ { j = 1 } ^ { 1 3 } \\mathcal { G } ( u _ j , x ) \\\\ & = x ^ 9 \\ \\dfrac { x ^ 2 - 2 } { x } \\ \\dfrac { ( x ^ 2 - 1 ) ( x ^ 2 - 4 ) } { x ( x ^ 2 - 2 ) } \\ \\dfrac { x ^ 2 - 4 } { x } \\ \\dfrac { x ^ 6 - 8 x ^ 4 + 1 2 x ^ 2 - 4 } { x ( x ^ 2 - 1 ) ( x ^ 2 - 4 ) } \\\\ & = x ^ 5 ( x ^ 2 - 4 ) ( x ^ 6 - 8 x ^ 4 + 1 2 x ^ 2 - 4 ) . \\end{align*}"}
{"id": "5705.png", "formula": "\\begin{align*} \\pi _ J \\coloneqq \\prod _ { k = 1 } ^ m e _ { | J _ k | } ( y _ 1 , y _ 2 , \\ldots , y _ { \\max J _ k } ) \\in M , \\end{align*}"}
{"id": "1491.png", "formula": "\\begin{align*} ( x ) _ { n , \\lambda } = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) ( x ) _ { k } , ( n \\ge 0 ) , ( \\mathrm { s e e } \\ [ 8 ] ) . \\end{align*}"}
{"id": "2555.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\le r _ - \\le q \\le r _ + & \\le p + q \\ , \\ r _ - \\le \\nu _ 3 \\le r _ + \\ , \\\\ \\nu _ 1 = p + 2 q - ( r _ + + r _ - ) \\ , & \\ \\nu _ 2 = ( r _ + + r _ - ) - \\nu _ 3 \\ , \\ J = \\dfrac { 1 } { 2 } ( r _ + - r _ - ) \\ , \\end{aligned} \\end{align*}"}
{"id": "5664.png", "formula": "\\begin{align*} h _ 2 ( j ) = j + 1 . \\end{align*}"}
{"id": "2490.png", "formula": "\\begin{align*} \\nabla _ X U = A _ X U + \\nu \\nabla _ X U , \\end{align*}"}
{"id": "6607.png", "formula": "\\begin{align*} \\norm { \\rho _ t ( \\beta ) f _ t } ^ 2 & = \\sum _ { \\pi : \\lambda _ { \\pi _ \\infty } \\in t \\Omega } \\norm { \\pi _ { p } ( \\beta ) f _ \\pi } ^ 2 = \\sum _ { \\pi : \\lambda _ { \\pi _ \\infty } \\in t \\Omega } \\abs { \\beta ( \\omega _ { \\pi _ { p } } ) } ^ 2 \\\\ & = \\sum _ { \\pi : \\lambda _ { \\pi _ \\infty } \\in t \\Omega } ( \\beta ^ \\ast \\ast \\beta ) ( \\omega _ { \\pi _ { p } } ) = \\sum _ { \\pi : \\lambda _ { \\pi _ \\infty } \\in t \\Omega } \\chi _ { \\pi _ p } ( \\phi _ { \\beta ^ \\ast \\ast \\beta } ) . \\end{align*}"}
{"id": "6181.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l l } \\frac { \\partial } { \\partial t } \\varphi ( x , t ) & = & \\log \\frac { ( \\widehat { \\omega } _ { t } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\varphi ) ^ { n } \\wedge \\eta _ { 0 } } { \\Omega \\wedge \\eta _ { 0 } } , \\\\ \\widehat { \\omega } _ { t } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\varphi & > & 0 , \\\\ \\varphi ( 0 ) & = & \\varphi _ { 0 } . \\end{array} \\right . \\end{align*}"}
{"id": "6182.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l c l } \\frac { \\partial } { \\partial t } \\omega _ { Z } = - R i c ( \\omega _ { Z } ) & \\mathrm { o n } & ( 0 , T _ { 0 } ) \\times Z _ { \\operatorname { r e g } } , \\\\ \\pi ^ { \\ast } ( \\omega _ { Z } ( 0 ) ) = \\omega _ { 0 } & \\mathrm { o n } & Z . \\end{array} \\right . \\end{align*}"}
{"id": "5812.png", "formula": "\\begin{align*} \\beta ( s ) = \\sum _ { n \\geq 1 } \\frac { \\chi ( n ) } { n ^ s } = \\prod _ { p \\geq 3 } \\left ( 1 - \\frac { \\chi ( p ) } { p ^ s } \\right ) ^ { - 1 } , \\end{align*}"}
{"id": "4417.png", "formula": "\\begin{align*} ( 8 k + 4 ) N + 1 - ( ( 4 k + 2 ) N - i + 1 ) = ( 4 k + 2 ) N + i . \\end{align*}"}
{"id": "4485.png", "formula": "\\begin{align*} \\Phi ( \\eta _ 1 , \\eta _ 2 , \\xi ) = p ( \\eta _ 1 ) + p ( \\eta _ 2 ) + p ( \\xi - \\eta _ 1 - \\eta _ 2 ) - p ( \\xi ) = - \\frac { \\eta _ 1 } { \\sqrt { 1 + \\eta _ 1 ^ 2 } } - \\frac { \\eta _ 2 } { \\sqrt { 1 + \\eta _ 2 ^ 2 } } - \\frac { \\xi - \\eta _ 1 - \\eta _ 2 } { \\sqrt { 1 + ( \\xi - \\eta _ 1 - \\eta _ 2 ) ^ 2 } } + \\frac { \\xi } { \\sqrt { 1 + \\xi ^ 2 } } . \\end{align*}"}
{"id": "8047.png", "formula": "\\begin{align*} s = - \\frac { H } { 2 \\pi i } - \\frac { m } { d } - \\bar { f } , m \\geq 1 \\end{align*}"}
{"id": "5586.png", "formula": "\\begin{align*} m ^ \\tau _ { U , \\ , \\eta } \\ = \\ \\sup _ { \\alpha \\subset U , \\ , \\alpha } m _ \\alpha \\ , . \\end{align*}"}
{"id": "5816.png", "formula": "\\begin{align*} \\beta ( s ) = \\begin{cases} \\displaystyle r _ n \\pi ^ { s } & , \\\\ \\displaystyle r _ n \\pi ^ { s } - u _ n & , \\end{cases} \\end{align*}"}
{"id": "7979.png", "formula": "\\begin{align*} \\Phi ( h _ + ) = \\xi _ - - h _ - , \\Phi ( \\xi _ + ) = \\xi _ - \\end{align*}"}
{"id": "5716.png", "formula": "\\begin{align*} \\pi _ { [ 2 , 5 ] } = e _ 4 ( y _ 1 , y _ 2 , \\ldots , y _ 5 ) , \\pi _ { [ 3 , 2 ] } = 1 , \\pi _ { [ 0 , 1 ] } = 0 \\end{align*}"}
{"id": "4814.png", "formula": "\\begin{align*} = 2 ^ { 2 k - 2 } ( i _ { k - 1 } + 2 i _ k + . . . + 2 ^ { l + 1 } i _ { k + l - 1 } ) ^ 2 + 2 ^ { 2 k - 2 } ( j _ { k - 1 } + 2 j _ k + . . . + 2 ^ { l + 1 } j _ { k + l - 1 } ) ^ 2 \\mod 2 ^ { k + l } \\end{align*}"}
{"id": "441.png", "formula": "\\begin{align*} z G _ { \\mu } ( z ) = \\frac { 1 } { 1 - \\eta _ { \\mu } \\left ( \\frac { 1 } { z } \\right ) } , z \\in \\mathbb { C } \\setminus \\mathbb { R } _ { + } , \\end{align*}"}
{"id": "4942.png", "formula": "\\begin{align*} - \\sigma _ k ^ { i i } \\theta _ { i i } ^ 2 = & - \\sum _ { i > 1 } \\sigma _ k ^ { i i } ( a _ { i i } - 1 ) ^ 2 \\\\ = & - \\sum _ { i > 1 } \\sigma _ k ^ { i i } a _ { i i } ^ 2 + 2 \\sum _ { i > 1 } \\sigma _ k ^ { i i } a _ { i i } - \\sum _ { i > 1 } \\sigma _ k ^ { i i } \\\\ \\le & - \\sum _ { i > 1 } \\sigma _ k ^ { i i } a _ { i i } ^ 2 + 2 k \\sigma _ k - \\sum _ { i > 1 } \\sigma _ k ^ { i i } , \\end{align*}"}
{"id": "1776.png", "formula": "\\begin{align*} A \\cup \\{ k \\} = \\{ k \\} \\cup \\bigcup _ { j = 1 } ^ \\infty ( A _ j \\cup \\{ p ( j ) \\} ) , \\end{align*}"}
{"id": "7958.png", "formula": "\\begin{align*} I _ { ( X _ + , D _ + ) } ( y , z ) = z e ^ { t _ + / z } \\sum _ { d \\in \\mathbb K _ { + } } y ^ { d } \\left ( \\prod _ { i \\in M _ 0 } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\textbf { 1 } _ { [ - d ] } I _ { D _ + , d } , \\end{align*}"}
{"id": "8759.png", "formula": "\\begin{align*} [ 2 k + 2 a - 2 l + 2 ] [ 2 a ] - [ 2 k ] [ 2 l - 2 ] = [ 2 a - 2 \\ell + 2 ] [ 2 k + 2 a ] . \\end{align*}"}
{"id": "535.png", "formula": "\\begin{align*} \\tau ^ * ( n ) : = \\sum \\limits _ { \\substack { n = n _ 1 n _ 2 \\\\ \\gcd ( n _ 1 , n _ 2 ) = 1 } } 1 . \\end{align*}"}
{"id": "4368.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | F _ { \\delta } - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f | ^ 2 e ^ { v _ { t _ 0 , B } ( \\Psi ) - \\Psi } \\\\ \\le & \\left ( \\frac { 1 } { \\delta } e ^ { - t _ 1 } + e ^ { - t _ 1 } - e ^ { - t _ 0 - B } \\right ) \\int _ D \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f | ^ 2 e ^ { - \\Psi } . \\end{align*}"}
{"id": "4594.png", "formula": "\\begin{align*} c _ { i j } ^ { \\mathcal { S } } ( t ) : = \\begin{cases} c _ { i j } ( t ) & \\mathrm { i f } \\ , \\ , i , j \\in \\mathcal { J } \\ , \\ , \\mathrm { a n d } \\ , \\ , | i - j | \\le \\ell \\\\ 0 & \\mathrm { o t h e r w i s e } , \\end{cases} \\end{align*}"}
{"id": "2461.png", "formula": "\\begin{align*} \\dim \\left [ \\gg , x \\right ] = \\sum _ { \\mu } \\dim \\left [ \\gg ^ { \\mu } , x \\right ] , \\end{align*}"}
{"id": "2992.png", "formula": "\\begin{align*} \\sum _ { Q \\in \\Gamma \\backslash \\mathcal Q _ D } \\chi _ d ( Q ) \\int _ { C _ Q } i \\partial _ z \\varphi ( z ) y ^ { s } \\ , d z = - 2 \\pi i \\sum _ { n \\neq 0 } n a _ \\varphi ( n ) G ( n , d ) \\int _ 0 ^ \\infty y ^ { s + \\frac 1 2 } K _ { i r } ( 2 \\pi | n | y ) \\ , d y \\end{align*}"}
{"id": "685.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } c _ j ^ 2 = \\left \\| e ^ { f _ o } v ( \\cdot , 0 ) \\right \\| ^ 2 _ { L ^ 2 ( d \\nu _ * ) } \\leq \\overline C , \\end{align*}"}
{"id": "3829.png", "formula": "\\begin{align*} E r r H ^ 2 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot \\xi + i s \\mu _ 1 | \\xi - \\eta | } \\mathcal { F } \\big [ ( \\hat { v } - \\hat { V } ( s ) ) \\times B f \\big ] \\cdot \\nabla _ v \\big [ ( \\hat { V } ( s ) \\cdot \\xi - \\hat { v } \\cdot \\eta + \\mu _ 1 | \\xi - \\eta | ) ^ { - 1 } \\end{align*}"}
{"id": "7712.png", "formula": "\\begin{align*} S _ { \\log , k } ( x ) : = \\sum _ { n _ 1 \\cdots n _ k \\le x } \\log ( n _ 1 , \\ldots , n _ k ) = \\sum _ { d ^ k \\delta \\le x } \\Lambda ( d ) \\tau _ k ( \\delta ) = \\sum _ { p ^ { \\nu k } \\delta \\le x } ( \\log p ) \\tau _ k ( \\delta ) . \\end{align*}"}
{"id": "3470.png", "formula": "\\begin{align*} \\mathrm { D T E } _ { ( p , q ) } ( X ) = \\mu + \\sigma \\frac { c _ { 1 } \\left [ \\overline { G } _ { 1 } \\left ( \\frac { 1 } { 2 } \\xi _ { p } ^ { 2 } \\right ) - \\overline { G } _ { 1 } \\left ( \\frac { 1 } { 2 } \\xi _ { q } ^ { 2 } \\right ) \\right ] } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "3096.png", "formula": "\\begin{align*} \\sum _ i \\sum _ { ( x ) , ( f ^ i ( x _ 0 ) ) } h ^ i \\otimes ( S x _ { - 1 } ) f ^ i ( x _ 0 ) _ { - 1 } \\otimes f ^ i ( x _ 0 ) _ 0 = \\sum _ { ( x ) , ( f ( x _ 0 ) ) } ( S x _ { - 1 } ) f ( x _ 0 ) _ { - 2 } \\otimes ( S x _ { - 2 } ) f ( x _ 0 ) _ { - 1 } \\otimes f ( x _ 0 ) _ 0 . \\end{align*}"}
{"id": "2480.png", "formula": "\\begin{align*} \\inf _ { 2 \\leq l \\leq s } \\inf _ { ( x , y ) \\in \\bar B } \\sum _ { i = 1 } ^ { n - 2 } ( \\varphi _ i ^ l ( x , y ) - \\Phi _ i ( x , y ) ) ^ 2 > \\frac { 4 \\varepsilon } { 9 } . \\end{align*}"}
{"id": "8873.png", "formula": "\\begin{align*} \\frac { f _ { \\lambda \\lambda } ( \\lambda _ i ) } { f _ \\lambda ( \\lambda _ i ) } = 2 \\sum _ { j ( \\neq i ) } \\frac { 1 } { \\lambda _ i - \\lambda _ j } . \\end{align*}"}
{"id": "8485.png", "formula": "\\begin{align*} f _ * ( x ) & = \\left ( x - \\frac { 1 } { 2 } \\right ) L & \\left \\| f _ * \\right \\| _ 2 ^ 2 & = \\frac { L ^ 2 } { 1 2 } + 1 & & L \\leq 2 \\\\ f _ * ( x ) & = L \\left ( x - 1 + \\sqrt { \\frac { 2 } { L } } \\right ) _ + & \\left \\| f _ * \\right \\| _ 2 ^ 2 & = \\frac { 2 \\sqrt { 2 } } { 3 } \\sqrt { L } & & L > 2 . \\end{align*}"}
{"id": "4451.png", "formula": "\\begin{align*} ( W ^ k f ) _ { p _ 1 \\dots p _ { m - k } q _ 1 \\dots q _ { m - k } i _ 1 \\dots i _ k } = \\sigma ( q _ 1 \\dots q _ { m - k } i _ 1 \\dots i _ k ) ( W f ^ { i _ 1 \\dots i _ k } ) _ { p _ 1 \\dots p _ { m - k } q _ 1 \\dots q _ { m - k } } . \\end{align*}"}
{"id": "3347.png", "formula": "\\begin{align*} \\lambda ( k ) = ( - 1 ) ^ { \\Big [ \\frac { 2 A p k } { q } \\Big ] } \\end{align*}"}
{"id": "446.png", "formula": "\\begin{align*} C = \\{ r e ^ { i h _ { \\mu } ( r ) } : r \\in ( 0 , + \\infty ) \\} , \\end{align*}"}
{"id": "6250.png", "formula": "\\begin{align*} f ( q ^ n x ) = ( - 1 ) ^ n q ^ n f ( x ) ( n \\in \\mathbb { N } _ 0 ) . \\end{align*}"}
{"id": "3204.png", "formula": "\\begin{align*} { \\rm C a p } _ p ( B _ { \\hat { d } } ( \\mathcal O , 1 ) , B _ { \\hat { d } } ( \\mathcal O , 2 ^ i ) ) : = \\inf \\int _ X g _ u ^ p d \\mu \\end{align*}"}
{"id": "2657.png", "formula": "\\begin{align*} g _ \\alpha ( p ) & = ( n - 1 ) p ^ { 2 n - k - 5 } + ( n - k - 1 ) ( n - 2 ) p ^ { n - 3 } ( p - 1 ) - ( n - k - 1 ) p ^ { n - 3 } \\\\ & + \\left ( ( n - k - 2 ) + \\binom { n - k - 2 } { 2 } \\right ) p ^ { n - 3 } ( p - 2 ) ( p - 3 ) . \\end{align*}"}
{"id": "159.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\dot \\kappa ^ { \\epsilon , L } _ s \\ , d s \\geq \\log ( t + 1 ) - c _ d ( \\lambda , \\mu , m ^ 2 = 1 ) . \\end{align*}"}
{"id": "7776.png", "formula": "\\begin{align*} \\tilde { \\bf u } _ m ( \\alpha ) = { \\bf u } _ m ( \\alpha ) + \\tilde { \\boldsymbol { \\eta } } _ m ( \\alpha ) . \\end{align*}"}
{"id": "3272.png", "formula": "\\begin{align*} \\omega _ 1 ^ { * } \\cdot \\nabla \\varphi _ 1 = \\omega _ 1 ^ { * } \\cdot A _ 1 , \\omega _ 2 ^ { * } \\cdot \\nabla \\varphi _ 2 = - \\omega _ 2 ^ { * } \\cdot A _ 2 . \\end{align*}"}
{"id": "6799.png", "formula": "\\begin{align*} \\begin{cases} u '' ( 0 ) = u ''' ( 0 ) = 0 , \\\\ u '' ( 1 ) = u ''' ( 1 ) = 0 . \\end{cases} \\end{align*}"}
{"id": "6097.png", "formula": "\\begin{align*} 0 = d \\omega _ { i A } = { \\bar \\Omega } _ { i A } , 1 \\leq i \\leq n , 1 \\le A \\le n + 2 . \\end{align*}"}
{"id": "3660.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\hat { v } \\times \\nabla _ x f ( t , x , v ) d v = 0 , \\end{align*}"}
{"id": "8766.png", "formula": "\\begin{align*} B ( p ) : = \\frac { p ^ { 1 / 2 p } } { | p ' | ^ { 1 / 2 p ' } } , \\frac { 1 } { p } + \\frac { 1 } { p ' } = 1 \\end{align*}"}
{"id": "5014.png", "formula": "\\begin{align*} R ^ { n , 3 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int _ 0 ^ \\tau \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) \\int _ 0 ^ s \\psi ^ 2 _ { n , 1 } ( u , s ) d u d s . \\end{align*}"}
{"id": "1875.png", "formula": "\\begin{align*} A _ { [ n , j ] } = \\sum _ { k \\in \\mathbb { Z } } A _ { [ k , 0 ] } A _ { [ n - k - 1 , j - 1 ] } ^ { ( 1 ) } , n \\in \\mathbb { Z } . \\end{align*}"}
{"id": "4020.png", "formula": "\\begin{align*} V _ { \\epsilon } \\triangleq \\begin{cases} V _ { \\min } & \\epsilon < \\frac 1 2 \\\\ V _ { \\max } & \\epsilon \\geq \\frac 1 2 . \\end{cases} \\end{align*}"}
{"id": "2123.png", "formula": "\\begin{align*} \\varphi _ { \\mu } ( z ) = \\gamma + \\int _ { \\mathbb { R } } \\frac { 1 + s z } { z - s } \\ , d \\sigma ( s ) , z \\in \\mathbb { C } ^ { + } , \\end{align*}"}
{"id": "1513.png", "formula": "\\begin{align*} \\Big ( x \\frac { d } { d x } \\Big ) _ { p , \\lambda } \\frac { 1 } { 1 - x } = \\sum _ { n = 0 } ^ { \\infty } \\Big ( x \\frac { d } { d x } \\Big ) _ { p , \\lambda } x ^ { n } = \\sum _ { n = 0 } ^ { \\infty } ( n ) _ { p , \\lambda } x ^ { n } = \\frac { 1 } { 1 - x } F _ { p , \\lambda } \\Big ( \\frac { x } { 1 - x } \\Big ) . \\end{align*}"}
{"id": "1871.png", "formula": "\\begin{align*} \\mathcal { D } _ { [ n , j ] } = \\bigcup _ { k = 0 } ^ { n - j } \\mathcal { D } _ { [ n , j , k ] } , \\end{align*}"}
{"id": "2996.png", "formula": "\\begin{align*} \\log _ q N _ e > \\log _ q ( q ^ e - 2 q ^ { e / 2 } g ) = e / 2 + \\log _ q ( q ^ { e / 2 } - 2 g ) \\geq e / 2 \\geq \\log _ q N \\ . \\end{align*}"}
{"id": "5988.png", "formula": "\\begin{align*} E ^ 2 ( t ) = | u ( t ) - \\hat U ( t ) | \\end{align*}"}
{"id": "7407.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\frac { \\sum _ { k = 1 } ^ N f ( S _ k p / q ) - E ( f ) N } { \\sqrt { 2 N \\log \\log N } } = \\sigma \\mathrm { a . s . } \\end{align*}"}
{"id": "5119.png", "formula": "\\begin{align*} 0 = V _ 0 \\subset V _ 1 \\subset \\cdots \\subset V _ n = H _ * ( X ; \\Q ) \\end{align*}"}
{"id": "8637.png", "formula": "\\begin{align*} U ( t ) = \\exp \\left ( { \\rm i } f _ 1 ( t ) \\widehat { H } _ 1 \\right ) \\times \\ldots \\times \\exp \\left ( { \\rm i } f _ r ( t ) \\widehat { H } _ r \\right ) { , } \\end{align*}"}
{"id": "6964.png", "formula": "\\begin{align*} \\Phi ( \\infty ) : = \\lim _ { z \\to \\infty } \\Phi ( z ) = 1 \\end{align*}"}
{"id": "3374.png", "formula": "\\begin{align*} \\psi _ m : = \\pi ^ m _ 1 \\circ \\varphi ^ m _ { p ( m , 1 ) , 1 } . \\end{align*}"}
{"id": "3714.png", "formula": "\\begin{align*} \\{ [ v ' ] \\mid v ' \\in { \\downarrow _ 1 } ( v _ 1 ) \\} = \\{ [ v ' ] \\mid v ' \\in { \\downarrow _ 1 } ( v _ 2 ) \\} . \\end{align*}"}
{"id": "4487.png", "formula": "\\begin{align*} \\partial _ t [ \\partial _ \\xi \\hat h + \\rm { V I } ] = \\rm { I } + \\rm { I I } + \\rm { I I I } _ 1 + \\rm { I I I } _ { 2 1 } + \\rm { I I I } _ { 2 2 } + \\rm { I I I } _ { 2 3 } + \\rm { I I I } _ { 2 4 1 } + \\rm { I I I } _ { 2 4 2 } + \\rm { I V } + \\rm { V } . \\end{align*}"}
{"id": "2463.png", "formula": "\\begin{align*} \\dim \\gg ^ { \\mu } = 2 \\dim \\gg _ { 1 } ^ { \\mu } . \\end{align*}"}
{"id": "5057.png", "formula": "\\begin{align*} \\kappa _ 3 = \\sum _ { k = 1 } ^ \\infty \\int _ 0 ^ 1 [ ( x + k ) ^ \\alpha - k ^ \\alpha ] ^ 2 d x . \\end{align*}"}
{"id": "8031.png", "formula": "\\begin{align*} D _ + = \\sum _ { i \\in ( M _ + \\cup \\{ j \\} ) } \\bar D _ i \\subset X _ + D _ - = \\sum _ { i \\in M _ + } \\bar D _ i \\subset X _ - . \\end{align*}"}
{"id": "4684.png", "formula": "\\begin{align*} n = \\lambda _ 1 + \\lambda _ 2 + \\cdots + \\lambda _ k . \\end{align*}"}
{"id": "8560.png", "formula": "\\begin{align*} \\exp _ 2 ^ * ( a y ) = \\int _ 0 ^ { a y } \\frac { a y - s } { 1 + s } \\ d s = a \\int _ 0 ^ y \\frac { a ( y - t ) } { 1 + a t } \\ d t \\leq \\max ( a , a ^ 2 ) \\int _ 0 ^ y \\frac { y - t } { 1 + t } \\ d t \\ , \\end{align*}"}
{"id": "1560.png", "formula": "\\begin{align*} D ^ n _ x ( 1 ) = \\left \\lbrace ( y ^ 1 , y ^ 2 , . . . , y ^ n ) \\in T _ x M ^ n : F ( x , y ) < 1 \\right \\rbrace , y = y ^ { \\epsilon } z ^ i _ { \\epsilon } \\textnormal { a n d } z = \\left ( z ^ i _ { \\epsilon } \\right ) = \\left ( \\frac { \\partial \\varphi ^ i } { \\partial x ^ { \\epsilon } } \\right ) . \\end{align*}"}
{"id": "7931.png", "formula": "\\begin{align*} \\Sigma _ \\omega = \\{ \\sigma _ I : \\bar { I } \\in \\mathcal A _ \\omega \\} . \\end{align*}"}
{"id": "8595.png", "formula": "\\begin{align*} w _ m \\left ( \\alpha \\oplus \\frac { \\varepsilon _ 1 } { 2 ^ { k _ 1 } } \\oplus \\ldots \\oplus \\frac { \\varepsilon _ l } { 2 ^ { k _ l } } \\right ) = \\prod _ { j = 1 } ^ l r _ { k _ j } \\left ( \\alpha \\oplus \\frac { \\varepsilon _ j } { 2 ^ { k _ j } } \\right ) \\end{align*}"}
{"id": "8071.png", "formula": "\\begin{align*} \\phi ( N \\times \\{ 0 \\} ) = S ^ { 1 } \\cdot x . \\end{align*}"}
{"id": "1287.png", "formula": "\\begin{align*} n ( T , k + 1 - j ) - n ( T , k - j ) & = \\dfrac { ( k - 1 ) ! } { ( j - 1 ) ! } - \\dfrac { ( k - 1 ) ! } { j ! } \\\\ & = j \\dfrac { ( k - 1 ) ! } { j ! } - \\dfrac { ( k - 1 ) ! } { j ! } \\\\ & = \\dfrac { ( j - 1 ) ( k - 1 ) ! } { ( k - j ) ! } , \\end{align*}"}
{"id": "4532.png", "formula": "\\begin{align*} \\sigma _ { m , m ; n , n } = 4 \\sigma _ { 2 m - 1 , 2 n - 1 } - 2 \\sigma _ { 2 m - 1 , n - 1 } - 2 \\sigma _ { m - 1 , 2 n - 1 } + \\sigma _ { m - 1 , n - 1 } \\end{align*}"}
{"id": "8783.png", "formula": "\\begin{align*} \\| \\phi _ 1 * \\phi _ 2 * \\cdots * \\phi _ N \\| _ { q ( P ) } \\leq Y \\prod _ { k = 1 } ^ N \\| \\phi _ k \\| _ { p _ k } \\end{align*}"}
{"id": "6009.png", "formula": "\\begin{align*} g ( x ) = x ^ 5 - \\frac { 2 5 } { 8 } x ^ 4 + \\frac { 5 } { 2 } x ^ 3 = ( x - 1 ) ^ 5 + \\frac { 1 5 } { 8 } ( x - 1 ) ^ 4 - \\frac { 5 } { 4 } ( x - 1 ) ^ 2 + \\frac { 3 } { 8 } \\end{align*}"}
{"id": "1520.png", "formula": "\\begin{align*} \\frac { 1 } { r ! } \\Big ( \\frac { d } { d x } \\Big ) ^ { r } \\bigg [ \\frac { x ^ { r } } { ( 1 - x ) ^ { 2 } } F _ { p , \\lambda } \\Big ( \\frac { x } { 1 - x } \\Big ) \\bigg ] & = \\frac { 1 } { r ! } \\Big ( \\frac { d } { d x } \\Big ) ^ { r } \\bigg [ \\sum _ { n = 1 } ^ { \\infty } \\Big ( ( 1 ) _ { p , \\lambda } + ( 2 ) _ { p , \\lambda } + \\cdots + ( n ) _ { p , \\lambda } \\Big ) \\bigg ] x ^ { n + r } \\\\ & = \\sum _ { n = 1 } ^ { \\infty } \\Big ( ( 1 ) _ { p , \\lambda } + ( 2 ) _ { p , \\lambda } + \\cdots + ( n ) _ { p , \\lambda } \\Big ) \\binom { n + r } { r } x ^ { n } . \\end{align*}"}
{"id": "1944.png", "formula": "\\begin{align*} \\psi _ { j } ( z ) : = \\langle ( z I - \\mathcal { W } ) ^ { - 1 } \\hat { e } _ { j } , \\hat { e } _ { 0 } \\rangle = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\langle \\mathcal { W } ^ { n } \\ , \\hat { e } _ { j } , \\hat { e } _ { 0 } \\rangle } { z ^ { n + 1 } } , 0 \\leq j \\leq p , \\end{align*}"}
{"id": "7405.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\frac { \\sum _ { k = 1 } ^ N f ( S _ k \\alpha ) - E ( f ) N } { \\sqrt { 2 N \\log \\log N } } = \\sigma \\textrm { a . s . } \\end{align*}"}
{"id": "798.png", "formula": "\\begin{align*} | g ( z ) | = | f ( \\omega ( z ) ) | \\leq \\max _ { | z | = r } | f ( | z | \\leq r ) | \\leq \\int _ { 0 } ^ { r } \\frac { \\psi ( t ) } { 1 - t ^ 2 } d t = f _ 0 ( r ) , \\end{align*}"}
{"id": "4541.png", "formula": "\\begin{align*} J _ { 2 2 } = - \\frac { 4 } { m n \\pi ^ { 2 } } \\int _ { 0 } ^ { \\pi - h _ { 1 } } \\int _ { 0 } ^ { h _ { 2 } } H _ { x , z _ { 1 } , y , z _ { 2 } } ( t _ { 1 } + h _ { 1 } , t _ { 2 } ) \\frac { S ( t _ { 1 } , t _ { 2 } ) } { ( t _ { 1 } + h _ { 1 } ) ^ { 2 } \\left ( 2 \\sin \\frac { t _ { 2 } } { 2 } \\right ) ^ { 2 } } d t _ { 1 } d t _ { 2 } , \\end{align*}"}
{"id": "4529.png", "formula": "\\begin{align*} a _ { x _ 2 x _ 2 } = b _ { x _ 1 x _ 2 } \\quad a _ { x _ 1 x _ 2 } = b _ { x _ 1 x _ 1 } . \\end{align*}"}
{"id": "385.png", "formula": "\\begin{align*} \\min ( f ( n ) t , t ^ { 3 / 2 } ) = \\min ( a n , a ^ { 3 / 2 } D ^ 3 ) \\geq a ^ { 3 / 2 } n . \\end{align*}"}
{"id": "4836.png", "formula": "\\begin{align*} \\Big ( \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } g . \\varphi _ k \\Big ) _ { k = 0 , \\cdots , p ^ r - 1 } \\ ; . \\end{align*}"}
{"id": "1931.png", "formula": "\\begin{align*} w = \\frac { h ( w ) } { h ( w ) ^ { p + 1 } + 1 } . \\end{align*}"}
{"id": "6652.png", "formula": "\\begin{align*} G _ Z ( \\alpha ) = e ^ { \\lambda _ Z ( e ^ { \\alpha } - 1 ) } \\ , . \\ , \\end{align*}"}
{"id": "4241.png", "formula": "\\begin{align*} \\mathbb { G } ( X _ n , X _ 0 ) = \\sum _ { m = 0 } ^ \\infty p _ m ( X _ n , X _ 0 ) \\leq \\gamma ( N + 1 ) ^ 3 ( n + 1 ) ^ 2 p _ n ( X _ n , X _ 0 ) + \\sum _ { m = N + 1 } ^ \\infty p _ m ( X _ n , X _ 0 ) . \\end{align*}"}
{"id": "6962.png", "formula": "\\begin{align*} 1 - F ( z ) = \\prod _ { k \\ge 1 } \\left ( \\frac { z - \\mu _ k ^ 2 } { z - \\lambda _ k ^ 2 } \\right ) \\ , . \\end{align*}"}
{"id": "1481.png", "formula": "\\begin{align*} \\log \\ , \\left | \\dfrac { k ! } { ( \\beta ) _ k } \\right | _ p \\le \\begin{cases} C _ { p , k } \\log ( p ) & \\ \\ \\ \\ p \\mid N _ k \\\\ 0 & \\ \\ \\enspace , \\end{cases} \\end{align*}"}
{"id": "6345.png", "formula": "\\begin{align*} \\varPsi ( u ) = \\displaystyle \\int _ { \\Omega } \\int _ { \\Omega } \\varPhi _ { x , y } \\left ( \\dfrac { | u ( x ) - u ( y ) | } { | x - y | ^ s } \\right ) \\dfrac { d x d y } { | x - y | ^ N } + \\int _ { \\Omega } \\widehat { \\varPhi } _ { x } \\left ( | u ( x ) | \\right ) d x . \\end{align*}"}
{"id": "3098.png", "formula": "\\begin{align*} & \\rho ( d ^ X \\circ g ) ( x ) = \\sum _ { ( x ) } ( d ^ X \\circ g ) ( x _ 0 ) _ { - 1 } S ^ { - 1 } x _ { - 1 } \\otimes ( d ^ X \\circ g ) ( x _ 0 ) _ 0 \\\\ = & \\sum _ { ( x ) } g ( x _ 0 ) _ { - 1 } S ^ { - 1 } x _ { - 1 } \\otimes d ^ X ( g ( x _ 0 ) _ 0 ) \\\\ = & \\sum _ { ( x ) } g _ { - 1 } \\otimes d ^ X ( g _ 0 ( x ) ) , \\end{align*}"}
{"id": "715.png", "formula": "\\begin{align*} \\| u \\| _ { \\alpha } : = \\| u \\| _ { \\infty } + [ u ] ' _ { \\alpha } , [ u ] ' _ { \\alpha } : = \\sup _ { M ^ 2 _ { T , \\delta } } \\left \\{ \\dfrac { | u ( p , t ) - u ( p ' , t ' ) | } { d ( p , p ' ) ^ { \\alpha } + | t - t ' | ^ { \\alpha / 2 } } \\right \\} , \\end{align*}"}
{"id": "941.png", "formula": "\\begin{align*} P _ c ( \\Psi _ t ) ( x ) = [ \\nabla _ i \\Psi _ t ( x ) \\nabla _ i \\nabla _ j \\Psi _ t ( x ) - \\nabla _ l \\Psi _ t \\Gamma _ { i j } ^ l ( x ) \\nabla _ k \\nabla _ k \\Psi _ t ( x ) - \\nabla _ l \\Psi _ t \\Gamma _ { k k } ^ l ( x ) - \\frac { \\sum _ k ( \\nabla _ k \\nabla _ k \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { k k } ^ l ) } { n } ( x ) ] ^ T \\end{align*}"}
{"id": "5926.png", "formula": "\\begin{align*} \\langle { x ^ \\eta ( T ) } \\rangle = \\int e ^ { T \\eta a } \\rho _ { \\bar { \\xi } } ( a ) d a \\sim e ^ { T \\ , w ( \\eta ) } \\end{align*}"}
{"id": "6813.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\int _ { x _ 0 - \\delta } ^ { x _ 0 + \\delta } ( a u '' ) ' v ' d x = 0 . \\end{align*}"}
{"id": "2213.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { d } { d t } \\varphi = \\log \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } + \\varphi _ { \\alpha \\overline { \\beta } } ) - \\log \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } ) + \\kappa \\varphi - F . \\end{array} \\end{align*}"}
{"id": "3621.png", "formula": "\\begin{align*} v ( \\Box \\phi , w ) & = \\inf \\{ v ( \\phi , w ' ) : w R w ' \\} , & v ( \\lozenge \\phi , w ) & = \\sup \\{ v ( \\phi , w ' ) : w R w ' \\} . \\end{align*}"}
{"id": "990.png", "formula": "\\begin{align*} d ( L ) = \\prod _ { i = 1 } ^ { n } b _ { i i } . \\end{align*}"}
{"id": "5679.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ i ^ 2 & = ( \\varpi _ 1 x _ { 2 } - \\varpi _ { 0 } x _ 1 ) + ( \\varpi _ 2 x _ 3 - \\varpi _ 1 x _ 2 ) + \\cdots + ( \\varpi _ i x _ { i + 1 } - \\varpi _ { i - 1 } x _ i ) \\\\ & = - \\varpi _ { 0 } x _ 1 + \\varpi _ i x _ { i + 1 } \\\\ & = ( x _ 1 + x _ 2 + \\cdots + x _ i ) x _ { i + 1 } \\end{align*}"}
{"id": "6677.png", "formula": "\\begin{align*} \\begin{aligned} \\| z \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } & \\leq C _ 2 ( \\| F _ 2 \\| _ { L ^ p ( D ) } + \\| w \\| _ { L ^ p ( D ) } + \\| z _ 0 \\| _ { W ^ { 2 , 0 } _ { p } ( [ 0 , 1 ] ) } ) \\\\ & \\leq C _ 2 ( \\| F _ 2 \\| _ { C ^ 0 ( D ) } + \\| w \\| _ { C ^ 0 ( D ) } + \\| z _ 0 \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } ) , \\end{aligned} \\end{align*}"}
{"id": "2865.png", "formula": "\\begin{align*} \\begin{aligned} \\relax [ \\mu , L ] = [ 0 , L _ { \\mathcal { L } } ] + \\beta [ - \\tfrac { 1 } { \\sigma } , \\tfrac { 1 } { \\lambda - \\sigma } ] \\approx [ - \\tfrac { \\beta } { \\sigma } , \\tfrac { 1 } { N } \\| A ^ T A \\| + \\tfrac { \\beta } { \\lambda - \\sigma } ] \\end{aligned} \\end{align*}"}
{"id": "1529.png", "formula": "\\begin{align*} \\mu ( B _ Q ( R ) \\cap B _ O ( R ) ) = n C _ { \\alpha } e ^ { - \\frac { r } { 2 } } ( 1 + O ( e ^ { - ( \\alpha - \\frac 1 2 ) r } + e ^ { - r } ) ) . \\end{align*}"}
{"id": "1551.png", "formula": "\\begin{align*} \\prod _ { b \\in \\lambda } \\dfrac { 1 - t ^ { n + c ( b ) } } { 1 - t ^ { h ( b ) } } = \\prod _ { 1 \\leq i < j \\leq n } \\dfrac { 1 - t ^ { \\lambda _ i - \\lambda _ j + j - i } } { 1 - t ^ { j - i } } . \\end{align*}"}
{"id": "7703.png", "formula": "\\begin{align*} K _ { \\log , k } = \\sum _ p \\frac { \\log p } { p ^ k - 1 } , K _ { \\omega , k } = \\sum _ p \\frac 1 { p ^ k } , K _ { \\Omega , k } = \\sum _ p \\frac 1 { p ^ k - 1 } . \\end{align*}"}
{"id": "8792.png", "formula": "\\begin{align*} \\phi _ 1 : = \\frac { 1 _ { ( - 1 , 1 ) } } { 2 } , \\phi _ 2 : = y _ 1 1 _ { ( - 5 , - 3 ) \\cup ( - 1 - 2 \\lambda , - 1 ) \\cup ( 1 , 1 + 2 \\lambda ) \\cup ( 3 , 5 ) } + y _ 2 1 _ { ( - 3 , - 1 - 2 \\lambda ) \\cup ( - 1 , 1 ) \\cup ( 1 + 2 \\lambda , 3 ) } \\end{align*}"}
{"id": "5097.png", "formula": "\\begin{align*} F ^ n _ \\tau = n ^ { \\alpha + \\frac 1 2 } \\int ^ { \\tau } _ 0 ( t - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\ , \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) d s , \\end{align*}"}
{"id": "7675.png", "formula": "\\begin{align*} p _ { k } : = \\xi _ { k } , \\overline { y } _ { k + 1 } : = y _ { k } - \\frac { 1 } { \\sigma _ { k } } \\left ( g ( x _ { k } ) + \\nabla g ( x _ { k } ) ^ { \\top } \\xi _ { k } \\right ) , \\overline { Z } _ { k + 1 } : = \\Sigma _ { k } , \\end{align*}"}
{"id": "3605.png", "formula": "\\begin{align*} & y _ { i } \\leq \\langle ( y _ 1 , \\ldots y _ s ) , \\gamma _ j \\rangle \\leq d _ j - y _ { s + 3 } \\leq d _ j , \\ i = 1 , \\ldots , s \\\\ & y _ { s + 3 } \\leq y _ { s + 1 } \\leq \\langle ( y _ 1 , \\ldots , y _ s ) , u \\rangle \\leq \\langle ( y _ 1 , \\ldots y _ s ) , \\gamma _ j \\rangle \\leq d _ j - y _ { s + 3 } \\leq d _ j , \\end{align*}"}
{"id": "1555.png", "formula": "\\begin{align*} \\varphi ( x ^ 1 , x ^ 2 ) = ( f ( x ^ 1 ) \\cos x ^ 2 , f ( x ^ 1 ) \\sin x ^ 2 , x ^ 1 ) , \\end{align*}"}
{"id": "6328.png", "formula": "\\begin{align*} J _ { \\nu } ^ { ( 1 ) } ( z ; q ) : = \\frac { ( q ^ { v + 1 } ; q ) _ \\infty } { ( q ; q ) _ \\infty } \\sum _ { n = 0 } ^ { \\infty } \\dfrac { ( - 1 ) ^ n } { ( q , q ^ { v + 1 } ; q ) _ n } ( z / 2 ) ^ { 2 n + \\nu } , | z | < 2 , \\end{align*}"}
{"id": "3014.png", "formula": "\\begin{align*} S _ { S _ n , \\frac { 1 } { 2 } W } ( R ) = \\frac { 1 6 n ^ 3 + 1 6 n ^ 2 + 7 n + 1 } { 2 ( 2 n + 1 ) ( 4 n + 1 ) ^ 2 } . \\end{align*}"}
{"id": "175.png", "formula": "\\begin{align*} L ^ \\sigma [ \\psi ] ( x ) : = \\textup { t r } \\big ( \\sigma \\sigma ^ T D ^ 2 \\psi ( x ) \\big ) \\end{align*}"}
{"id": "1538.png", "formula": "\\begin{align*} \\sum _ { k = \\overline { k } } ^ { R } e ^ { - 2 \\beta k } I _ k \\geq \\sum _ { k = \\overline { k } } ^ { R } e ^ { - 2 \\beta k } \\widetilde { c } \\ss e ^ { - \\alpha ( R - k ) } Z _ k = \\widetilde { c } \\ss e ^ { - 2 \\beta R } \\sum _ { k = \\overline { k } } ^ R Z _ k > \\widetilde { c } \\ss e ^ { - 2 \\beta R } \\frac { \\widetilde { \\eta } } { 2 } ( R - \\overline { k } + 1 ) . \\end{align*}"}
{"id": "7041.png", "formula": "\\begin{align*} \\sum _ { M \\geq 1 } ( g _ 1 ( M ) - g _ 1 ( M - 1 ) ) \\ , x ^ M & = \\frac { x ^ 2 - x ^ 3 } { ( 1 - x - x ^ 2 ) ^ 2 } = \\sum _ { M \\geq 1 } ( h _ 1 ( M ) - h _ 1 ( M - 1 ) ) \\ , x ^ M , \\\\ \\sum _ { M \\geq 1 } ( g ( M ) - h ( M ) ) \\ , x ^ M & = \\frac { x ^ 2 - x ^ 3 } { ( 1 - x - x ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "3923.png", "formula": "\\begin{align*} | I _ 1 | \\le \\frac { \\Delta _ n ^ 2 \\ , n } { T _ n ^ 2 } \\sum _ { j = 0 } ^ { j _ { \\delta _ 1 } } \\frac { c } { \\prod _ { l = 1 } ^ d h _ l } = c \\frac { \\Delta _ n ^ 2 \\ , n } { T _ n ^ 2 } \\frac { 1 } { \\prod _ { l = 1 } ^ d h _ l } ( j _ { \\delta _ 1 } + 1 ) . \\end{align*}"}
{"id": "2121.png", "formula": "\\begin{align*} \\dot { x } _ t = f ( x _ t ) + \\epsilon ( L _ G ( g ( x _ t ) ) ) , \\end{align*}"}
{"id": "1094.png", "formula": "\\begin{align*} g ( P _ X Y , Z ) = g ( Y , P _ X Z ) , \\end{align*}"}
{"id": "7308.png", "formula": "\\begin{align*} \\begin{cases} w _ t = \\Delta _ x w - q { \\sf U } _ \\infty ^ { q - 1 } w + ( F _ 1 [ \\tilde v _ 1 , \\tilde w _ 1 ] - F _ 2 [ \\tilde w _ 1 ] ) ( 1 - \\chi _ { { \\sf s q } } ) \\\\ + k [ v ] + ( g + N [ \\epsilon , \\tilde v _ 1 , \\tilde w _ 1 ] ) { \\bf 1 } _ { | \\xi | > 1 } & x \\in \\R ^ n , \\ t \\in ( 0 , T ) , \\\\ w = w _ 0 & x \\in \\R ^ n , \\ t = 0 . \\end{cases} \\end{align*}"}
{"id": "3993.png", "formula": "\\begin{align*} \\lim \\limits _ { i \\to \\infty } \\textstyle \\frac { \\log _ q n _ i } { \\mu _ q ( n _ i ) } = 0 . \\end{align*}"}
{"id": "8615.png", "formula": "\\begin{align*} L Q G ( s ) = \\frac { - 1 . 5 9 3 s ^ 3 + 9 . 8 4 s ^ 2 - 1 2 . 5 8 s + 9 3 . 7 6 } { s ^ 4 + 3 . 8 4 7 s ^ 3 + 2 6 . 6 6 s ^ 2 + 4 6 . 8 6 s + 1 2 5 . 1 } . \\end{align*}"}
{"id": "7527.png", "formula": "\\begin{align*} \\mathcal { W } _ { q } ( \\mu , \\nu ) & : = { \\bigg ( \\inf _ { \\pi \\in \\Pi ( \\mu , \\nu ) } \\int _ \\mathcal { X } { d ( x , y ) } ^ q \\mathrm { d } \\pi ( x , y ) \\bigg ) } ^ { 1 / q } \\\\ & = \\inf \\{ [ \\mathbf { E } { d ( X , Y ) } ^ q ] ^ { 1 / q } ; \\hbox { l a w } ( X ) = \\mu , \\hbox { l a w } ( Y ) = \\nu \\} , \\end{align*}"}
{"id": "1229.png", "formula": "\\begin{align*} \\mathcal { F } _ 1 ( v , x ) & = ( x - \\beta ( v ) ) \\displaystyle \\prod _ { w \\in c ( v ) } \\mathcal { F } _ 1 ( w , x ) - \\displaystyle \\sum _ { w \\in c ( v ) } \\left ( \\mathcal { F } _ 2 ( w , x ) \\displaystyle \\prod _ { t \\in c ( v ) , t \\neq w } \\mathcal { F } _ 1 ( t , x ) \\right ) , \\\\ \\mathcal { F } _ 2 ( v , x ) & = \\displaystyle \\prod _ { w \\in c ( v ) } \\mathcal { F } _ 1 ( w , x ) , \\end{align*}"}
{"id": "6161.png", "formula": "\\begin{align*} w ^ 3 + x ^ 3 + y ^ 3 + z ^ 3 = 0 , \\end{align*}"}
{"id": "152.png", "formula": "\\begin{align*} \\| E \\| _ { L ^ 1 } \\leq G _ { d , t , \\mu , m ^ 2 } ( \\lambda ) + \\sum _ { i = 1 } ^ 3 \\lambda ^ i P ^ { ( i ) } _ { d , t , \\mu , m ^ 2 , c \\lambda } \\big ( \\| E \\| _ { L ^ 1 } \\big ) . \\end{align*}"}
{"id": "8738.png", "formula": "\\begin{align*} f ( k ) = \\frac { k - 1 } { 2 ^ { \\tau ^ * } } \\Bigl ( 2 ^ { 2 ( \\rho - \\tau ^ * ) } - 1 \\Bigr ) - \\frac { \\epsilon ^ 2 } { 8 } \\Bigl ( \\frac { 2 ^ { 2 ( \\rho - 1 ) } } { k ^ 2 } - 1 \\Bigr ) \\end{align*}"}
{"id": "3707.png", "formula": "\\begin{align*} \\frac { 1 } { 2 T } \\mu ( \\Lambda _ { S ^ * _ \\Gamma \\mathbb { T } ^ 2 \\backslash S ^ * \\Gamma } ) = 0 . \\end{align*}"}
{"id": "7309.png", "formula": "\\begin{align*} k [ v ] = - k _ 1 [ v ] + k _ 1 ' [ v ] \\lesssim { \\sf R } _ { \\sf m i d } ^ { - { \\sf c } _ 2 } { \\sf R } _ 1 ^ { - 1 } ( T - t ) ^ { { \\sf d } _ 1 } \\eta ^ \\frac { 2 q } { 1 - q } | \\xi | ^ { \\gamma - 2 - 2 { \\sf c } _ 2 } { \\bf 1 } _ { { \\sf R } _ { \\sf m i d } < | \\xi | < 2 { \\sf R } _ { \\sf m i d } } . \\end{align*}"}
{"id": "2896.png", "formula": "\\begin{align*} & f ( x _ 1 , \\ldots , x _ k ) \\\\ & = x _ 1 + x _ 2 + x _ 1 x _ 2 + x _ 2 x _ 3 + x _ 3 x _ 4 + \\cdots + x _ { k - 3 } x _ { k - 2 } + x _ { k - 2 } x _ { k - 1 } + x _ { k - 1 } + x _ k \\\\ & = x _ 1 + x _ 2 + x _ 1 x _ 2 + x _ 3 ( x _ 2 + x _ 4 ) + x _ 5 ( x _ 4 + x _ 6 ) + \\cdots + x _ { k - 2 } ( x _ { k - 3 } + x _ { k - 1 } ) + x _ { k - 1 } + x _ k \\\\ & = x _ 1 + x _ 1 x _ 2 + x _ 3 y _ 4 + \\cdots + x _ { k - 2 } y _ { k - 1 } + y _ 2 + y _ 4 + \\cdots + y _ { k - 1 } + x _ k , \\end{align*}"}
{"id": "8887.png", "formula": "\\begin{align*} \\sigma _ 1 \\leqslant a _ { m + 2 } = \\left ( a _ { m + 1 } ^ 2 - \\left | \\operatorname { d e t } \\left ( a _ { m + 1 } ^ 2 - A ^ { H } A \\right ) \\right | \\left ( \\frac { n - 1 } { ( n + 1 ) a _ { m + 1 } ^ 2 - \\| A \\| _ { F } ^ { 2 } } \\right ) ^ { n - 1 } \\right ) ^ { 1 / 2 } \\leqslant a _ { m + 1 } \\end{align*}"}
{"id": "640.png", "formula": "\\begin{align*} & \\zeta ( \\Delta _ n , s ) = V _ \\alpha ( s ) \\Bigg ( a ( s ) \\ , n ^ { \\alpha - 2 s } + \\sum _ { m = 0 } ^ { M - 1 } b _ m ( s ) \\ , n ^ { - 2 m } \\Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) , \\\\ & \\zeta ( \\widetilde { \\Delta } _ n , s ) = V _ \\alpha ( s ) \\Bigg ( \\widetilde { a } ( s ) \\ , n ^ { \\alpha - 2 s } + \\sum _ { m = 0 } ^ { M - 1 } \\widetilde { b } _ m ( s ) \\ , n ^ { - 2 m } \\Bigg ) + O ( n ^ { - 2 M - 2 s + 2 } ) . \\end{align*}"}
{"id": "4007.png", "formula": "\\begin{align*} \\abs * { \\sum _ { t = 1 } ^ { T } ( f _ t ( M ) - F ( M ) ) } \\le C \\sqrt { 2 T H d \\log \\frac { 3 T ^ 2 } { \\delta } } . \\end{align*}"}
{"id": "6451.png", "formula": "\\begin{align*} d t _ 1 \\left ( C _ 1 ( [ x , y ] _ \\mathfrak g , z ) - [ C _ 1 ( x , y ) , \\varrho ( z ) ^ 2 ] + \\circlearrowleft ( x , y , z ) \\right ) = 0 . \\end{align*}"}
{"id": "933.png", "formula": "\\begin{align*} \\nabla \\Psi _ { t , \\eta _ i } ^ { q ( t ) } \\cdot \\nabla v _ k - \\frac { \\mathrm { t r } _ g ( \\nabla \\Psi _ { t , \\eta _ i } ^ { q ( t ) } \\cdot \\nabla v _ k ) } { n } g + \\nabla v _ k \\cdot \\nabla \\Psi _ { t , \\eta _ i } ^ { q ( t ) } - \\frac { \\mathrm { t r } _ g ( \\nabla v _ k \\cdot \\nabla \\Psi _ { t , \\eta _ i } ^ { q ( t ) } ) } { n } g + \\nabla v _ k \\cdot \\nabla v _ k - \\frac { \\mathrm { t r } _ g ( \\nabla v _ k \\cdot \\nabla v _ k ) } { n } g = h . \\end{align*}"}
{"id": "480.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ N B ( x _ i , \\frac { 1 } { 2 } \\sin \\theta ) \\subseteq B ( 0 , 1 + \\frac { 1 } { 2 } \\sin \\theta ) . \\end{align*}"}
{"id": "1978.png", "formula": "\\begin{align*} \\mu \\int _ { B _ s } | \\nabla u | ^ 2 \\ , d x & \\le \\frac { 1 } { \\mu } \\L { 2 } { \\nabla u } { B _ t \\setminus B _ s } \\L { 2 } { \\nabla \\eta } { B _ 1 } + \\Big | \\int _ { B _ t } f ( x ) ( \\eta u ) \\ , d x \\Big | \\\\ & \\le \\frac { C ( \\mu ) } { 2 \\tau ^ { 2 } } \\L { 2 } { \\nabla u } { B _ t \\setminus B _ s } ^ { 2 } + 2 \\tau ^ { 2 } \\L { 2 } { \\nabla \\eta } { 2 } ^ 2 + C ( N ) . \\end{align*}"}
{"id": "6208.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d x ^ 2 } + p ( x ) \\frac { d y } { d x } + r ( x ) y ( x ) = 0 , \\end{align*}"}
{"id": "8931.png", "formula": "\\begin{align*} \\int _ \\Omega f = \\int _ { \\Omega ^ * } f ^ * . \\end{align*}"}
{"id": "7019.png", "formula": "\\begin{align*} \\hat { \\tau } ^ * = - \\frac { \\xi _ x \\nu _ x + \\xi _ y \\nu _ y } { \\nu _ x ^ 2 + \\nu _ y ^ 2 + \\varepsilon } , \\end{align*}"}
{"id": "2024.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m e a n f i e l d j u m p s d e } & X ^ i _ t = X ^ i _ 0 + \\int _ 0 ^ t a ( X ^ i _ s ) \\dd s \\\\ & + \\int _ { 0 } ^ t \\int _ { 0 } ^ { + \\infty } \\int _ \\Theta { \\left \\{ \\psi { \\left ( X ^ i _ { s ^ - } , \\mu _ { \\mathcal { X } ^ N _ { s ^ - } } , \\theta \\right ) } - X ^ { i } _ { s ^ - } \\right \\} } \\ 1 _ { \\big ( 0 , \\lambda { \\big ( X ^ i _ { s ^ - } , \\mu _ { \\mathcal { X } ^ N _ { s ^ - } } \\big ) } \\big ] } ( u ) \\ , \\ , \\mathcal { N } ^ i ( \\dd s , \\dd u , \\dd \\theta ) . \\end{align*}"}
{"id": "4325.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\partial _ i F _ i = \\Omega ( \\frac { \\partial } { \\partial y _ 1 } , \\ldots , \\frac { \\partial } { \\partial y _ n } ) . \\end{align*}"}
{"id": "306.png", "formula": "\\begin{align*} \\sum _ { i \\le k } c _ i d _ i \\le \\sum _ { i \\le k - 1 } ( c _ i - c _ { i + 1 } ) D _ i \\ + \\ c _ k D _ k = \\sum _ { i \\le k } c _ i ( D _ i - D _ { i - 1 } ) . \\end{align*}"}
{"id": "6918.png", "formula": "\\begin{align*} f - q | _ g = & \\ - L ^ 2 ( p | _ { L ^ 2 ( x y ) } ) z - p | _ { L ^ 2 ( x y ) } L ^ 2 ( z ) + L ^ 2 ( p | _ { L ^ 2 ( x ) y } z ) + L ^ 2 ( p | _ { x L ^ 2 ( y ) } z ) \\\\ \\equiv & \\ - L ^ 2 ( p | _ { L ^ 2 ( x ) y } ) z - L ^ 2 ( p | _ { x L ^ 2 ( y ) } ) z - p | _ { L ^ 2 ( x ) y } L ^ 2 ( z ) - p | _ { x L ^ 2 ( y ) } L ^ 2 ( z ) \\\\ & \\ + L ^ 2 ( p | _ { L ^ 2 ( x ) y } ) z + p | _ { L ^ 2 ( x ) y } L ^ 2 ( z ) + L ^ 2 ( p | _ { x L ^ 2 ( y ) } ) z + p | _ { x L ^ 2 ( y ) } L ^ 2 ( z ) \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "2136.png", "formula": "\\begin{align*} - \\lim _ { y \\rightarrow \\infty } y \\Im \\varphi _ \\mu ( i y ) = \\int _ \\mathbb { R } 1 + s ^ 2 \\ , d \\sigma _ \\mu ( s ) = + \\infty . \\end{align*}"}
{"id": "6080.png", "formula": "\\begin{align*} P _ 1 = M _ f ( \\Lambda _ 1 ) & \\\\ P _ 2 = M _ f ( \\Lambda _ 2 ) & \\\\ P _ 3 = M _ f ( \\Lambda _ 3 ) & , \\end{align*}"}
{"id": "7857.png", "formula": "\\begin{align*} \\tilde \\varphi ( \\cdot ) = \\lim _ { i \\to \\omega } \\varphi ( A _ i ^ { 1 / 2 } \\cdot A _ i ^ { 1 / 2 } ) , \\end{align*}"}
{"id": "5392.png", "formula": "\\begin{align*} C _ { n , s } = \\left ( \\int _ { \\R ^ n } \\frac { 1 - \\cos ( x _ 1 ) } { | x | ^ { n + 2 s } } \\ , d x \\right ) ^ { - 1 } < \\infty , \\end{align*}"}
{"id": "1428.png", "formula": "\\begin{align*} \\begin{aligned} C _ { N , \\beta , d , D } & = \\int _ { \\mathbb { R } ^ { D \\cdot N ( d ) } } \\exp \\left ( - \\sum _ { i = 1 } ^ { D } \\sum _ { k = 1 } ^ { N ( d ) } \\frac { ( x ^ { ( i ) } _ k ) ^ 2 } { 2 ( 2 \\beta \\lambda _ k ) ^ { - 1 } } \\right ) \\prod _ { i = 1 } ^ { D } \\prod _ { k = 1 } ^ { N ( d ) } d x ^ { ( i ) } _ k \\\\ & = \\frac { 1 } { ( 2 \\beta ) ^ { D N ( d ) / 2 } } \\prod _ { k = 1 } ^ { N ( d ) } \\frac { 1 } { \\lambda _ k ^ { D / 2 } } . \\end{aligned} \\end{align*}"}
{"id": "5116.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E [ \\Lambda _ n ] & = \\exp \\left ( - \\frac { \\lambda ^ 2 } 2 \\kappa _ 1 \\right ) \\lim _ { n \\rightarrow \\infty } E \\left [ \\exp \\left ( i \\mu Y ^ { n , 1 } _ t \\right ) \\right ] \\\\ & = E \\left [ \\exp \\left ( i \\lambda Z _ t \\right ) \\right ] E \\left [ \\exp \\left ( i \\mu Y ^ { \\infty , 1 } _ t \\right ) \\right ] , \\end{align*}"}
{"id": "7267.png", "formula": "\\begin{align*} T _ 1 ( y ) & = { \\sf A } _ 1 + O ( | y | ^ { - ( n - 4 ) } ) + O ( | y | ^ { - 2 } ) | y | \\to \\infty ( n \\geq 5 ) , \\\\ | \\nabla _ y T _ 1 ( y ) | & = O ( | y | ^ { - ( n - 3 ) } ) + O ( | y | ^ { - 3 } ) | y | \\to \\infty ( n \\geq 5 ) . \\end{align*}"}
{"id": "3172.png", "formula": "\\begin{align*} U _ { z } = \\int _ { \\mathrm { T } _ { d } } x _ { 1 } ^ { z _ { 1 } } \\cdots x _ { d } ^ { z _ { d } } \\mathrm { d } \\mathrm { P } \\left ( x \\right ) = \\int _ { \\Theta _ { d } } \\mathrm { e } ^ { i \\theta \\cdot z } \\mathrm { d } \\left ( \\mathrm { F } _ { \\ast } \\mathrm { P } \\right ) \\left ( \\theta \\right ) \\ , z \\in \\mathbb { Z } ^ { d } \\ , \\end{align*}"}
{"id": "2262.png", "formula": "\\begin{align*} B [ \\Phi , \\Phi _ 1 ] : = \\int _ 0 ^ r \\left [ \\phi ' \\phi ' _ 1 - \\frac { z } { 2 } \\phi ' \\phi _ 1 + \\frac { z } { 2 } e _ \\sigma \\psi ' \\phi _ 1 \\right ] + \\int _ 0 ^ r \\left [ \\psi ' \\psi ' _ 1 - \\frac { z } { 2 } \\psi ' \\psi _ 1 + \\frac { z } { 2 } e _ \\sigma \\phi ' \\psi _ 1 \\right ] ~ \\end{align*}"}
{"id": "6584.png", "formula": "\\begin{align*} f ( y , v _ 1 , \\ldots , v _ k ) = \\int _ 0 ^ 1 d f ( y , t v _ 1 , \\ldots , t v _ k , 0 , v _ 1 , \\ldots , v _ k ) \\ , d t \\ , . \\end{align*}"}
{"id": "6991.png", "formula": "\\begin{align*} ( f _ 1 + f _ 2 ) ( x ) = f _ 1 ( x ) + f _ 2 ( x ) , ( \\lambda f ) ( x ) = \\lambda f ( x ) . \\end{align*}"}
{"id": "7673.png", "formula": "\\begin{align*} \\mbox { S p a n } \\Bigl \\{ \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} e ^ { 2 \\pi i n \\cdot x } , \\frac { 1 } { \\sqrt 2 } \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} e ^ { 2 \\pi i n \\cdot x } , \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} e ^ { 2 \\pi i n \\cdot x } \\Bigr \\} _ { | n | \\leq k } \\end{align*}"}
{"id": "199.png", "formula": "\\begin{align*} x ( \\{ v _ 1 , v _ 3 \\} ) = x ( \\{ v _ 2 , v _ 3 \\} ) = 1 . \\end{align*}"}
{"id": "3015.png", "formula": "\\begin{align*} D = a W + b L _ { x y } + b _ 0 R _ 0 + b _ 1 R _ 1 + \\Delta , \\end{align*}"}
{"id": "8648.png", "formula": "\\begin{align*} c ( t ) e ^ { \\frac { k \\alpha ( t ) } { 2 } } = e ^ { - \\alpha ( t ) } l \\Longrightarrow c ( t ) e ^ { \\frac { \\alpha ( t ) ( k + 2 ) } { 2 } } = l , \\end{align*}"}
{"id": "754.png", "formula": "\\begin{align*} \\small \\frac { 1 - 1 4 \\delta } { 2 + 2 \\gamma } = \\frac { 1 } { 4 + 4 \\gamma } , \\varsigma \\in \\left ( 0 , \\frac { 1 - 1 4 \\delta } { 2 + 2 \\gamma } \\right ) , \\quad \\varsigma + \\varsigma \\gamma + 7 \\delta - \\frac 1 2 = \\frac { 1 } { 8 } + \\frac { 1 } { 4 } - \\frac 1 2 = - \\frac { 1 } { 8 } , \\end{align*}"}
{"id": "5613.png", "formula": "\\begin{align*} \\mathsf { D } ( r ) : = \\{ a D ^ \\times \\mid a \\in \\mathcal { A } ( D ) a \\mid _ D r \\} . \\end{align*}"}
{"id": "6773.png", "formula": "\\begin{align*} I _ { \\alpha } ( n ) = \\frac { \\sqrt { 2 \\pi } u } { \\sqrt { ( 1 + u ) n } } \\left ( \\frac { u } { e ^ { 1 / u } } \\right ) ^ n \\left ( 1 + \\sum _ { r = 1 } ^ { R - 1 } \\frac { a _ r ( u ) } { n ^ r } + O \\left ( \\frac { ( \\log n ) ^ R } { n ^ R } \\right ) \\right ) \\end{align*}"}
{"id": "6741.png", "formula": "\\begin{align*} j _ \\delta ' ( u _ \\delta ) h = ( p _ \\delta , h ) + \\alpha ( u , h ) + ( u - \\overline { u } , h ) , \\forall h \\in U . \\end{align*}"}
{"id": "3750.png", "formula": "\\begin{align*} \\big | d e t \\big ( \\begin{bmatrix} \\sin \\theta \\cos \\phi - \\sin \\tilde { \\theta } \\cos \\tilde { \\phi } & \\sin \\theta \\sin \\phi - \\sin \\tilde { \\theta } \\sin \\tilde { \\phi } & \\cos \\theta - \\cos \\tilde { \\theta } \\\\ - s \\sin \\theta \\sin \\phi & s \\sin \\theta \\cos \\phi & 0 \\\\ s \\cos \\theta \\cos \\phi & s \\cos \\theta \\sin \\phi & - s \\sin \\theta \\\\ \\end{bmatrix} \\big ) \\big | \\end{align*}"}
{"id": "389.png", "formula": "\\begin{align*} G ( o , x ) & = \\int _ { 0 } ^ { \\varepsilon D ^ 2 } h _ t ( o , x ) d t + \\int _ { \\varepsilon D ^ 2 } ^ { b D ^ 2 } h _ t ( o , x ) d t + \\int _ { b D ^ 2 } ^ { \\infty } h _ t ( o , x ) d t . \\end{align*}"}
{"id": "552.png", "formula": "\\begin{align*} \\sum \\limits _ { 1 \\leq n \\leq N } f _ k ( n ) f _ k ( n + 1 ) = \\sum _ { \\substack { w _ 1 \\leq N , w _ 2 \\leq N + 1 \\\\ ( w _ 1 , w _ 2 ) = 1 } } \\frac { g _ k ( w _ 1 ) g _ k ( w _ 2 ) } { w _ 1 w _ 2 } \\Big ( \\frac { N } { w _ 1 w _ 2 } + O ( 1 ) \\Big ) , \\end{align*}"}
{"id": "703.png", "formula": "\\begin{align*} v : = - \\overline { g } ( T , \\mu ) . \\end{align*}"}
{"id": "6425.png", "formula": "\\begin{align*} \\vartheta ( x , y , z ) = [ Q , \\Phi _ 2 ( x , y , z ) ] . \\end{align*}"}
{"id": "2525.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { 1 } { 2 } ( L _ Z g ) ( X , Y ) + \\frac { 1 } { \\lambda ^ 2 } R i c ^ N ( \\tilde { X } , \\tilde { Y } ) + \\frac { \\mu } { \\lambda ^ 2 } h ( \\tilde { X } , \\tilde { Y } ) = 0 . \\end{array} \\end{align*}"}
{"id": "5780.png", "formula": "\\begin{align*} G / N = ( G / K ) / ( N / K ) = H / f ( [ G , G ] ) \\end{align*}"}
{"id": "2334.png", "formula": "\\begin{align*} S _ n = \\{ f \\in K [ x ] \\mid \\nu _ Q ( f ) < \\nu ( f ) \\mbox { f o r e v e r y } Q \\in \\Psi _ n \\} . \\end{align*}"}
{"id": "7915.png", "formula": "\\begin{align*} D _ + = D _ { + , 1 } + \\cdots + D _ { + , n } + E , \\end{align*}"}
{"id": "1322.png", "formula": "\\begin{align*} X _ t ^ \\varepsilon ( x _ 0 , v _ 0 ) = X _ { \\varepsilon t } \\left ( x _ 0 , v _ 0 / \\varepsilon \\right ) \\mbox { a n d } V _ t ^ \\varepsilon ( x _ 0 , v _ 0 ) = \\varepsilon V _ { \\varepsilon t } \\left ( x _ 0 , v _ 0 / \\varepsilon \\right ) . \\end{align*}"}
{"id": "6360.png", "formula": "\\begin{align*} N _ { \\kappa } = \\frac { 1 } { ( n - 1 ) ! } \\prod _ { m = 0 } ^ n [ 1 + ( 2 m - n ) \\kappa ] \\ , \\ , \\ \\ . \\end{align*}"}
{"id": "5533.png", "formula": "\\begin{align*} f ^ { - 1 } \\circ X \\circ f = \\lambda \\frac { z - a } { 1 - \\bar { a } z } . \\end{align*}"}
{"id": "1496.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - x ( e _ { \\lambda } ( t ) - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } F _ { n , \\lambda } ( x ) \\frac { t ^ { n } } { n ! } . \\end{align*}"}
{"id": "3199.png", "formula": "\\begin{align*} \\mathcal R _ { p } ( \\mathcal O ) < \\infty \\implies \\begin{cases} 1 \\leq p < Q & 1 < Q < \\infty , \\\\ p = 1 & Q = 1 . \\end{cases} \\implies R _ { p } ( h , \\mathcal O ) < \\infty \\end{align*}"}
{"id": "5412.png", "formula": "\\begin{align*} ( ( - \\Delta ) ^ s + q ) v & = 0 \\quad \\quad \\Omega , \\\\ v & = f \\quad \\quad \\Omega _ e . \\end{align*}"}
{"id": "8037.png", "formula": "\\begin{align*} I _ { K _ S } ( q ) = ( 3 H ) q ^ { H / z } \\sum _ { d \\geq 0 } q ^ d \\frac { \\prod _ { k = 1 } ^ { 3 d } ( 3 H + k z ) ( - 1 ) ^ d \\prod _ { k = 0 } ^ { d - 1 } ( H + k z ) } { \\prod _ { k = 1 } ^ d ( H + k z ) ^ 4 } . \\end{align*}"}
{"id": "4843.png", "formula": "\\begin{align*} \\tau ^ \\mathrm { a n } ( M ' / S , \\mathbb { 1 } ) = \\tau ^ \\mathrm { t o p } ( M ' / S , \\mathbb { 1 } ) \\ ; . \\end{align*}"}
{"id": "7792.png", "formula": "\\begin{align*} [ e , f ] = h , \\ [ h , e ] = 2 e , \\ [ h , f ] = - 2 f . \\end{align*}"}
{"id": "1911.png", "formula": "\\begin{align*} z \\ , W _ { 0 } ( z ) - 1 = a _ { 0 } ^ { ( 0 ) } \\ , W _ { 0 } ( z ) + \\sum _ { j = 1 } ^ { p } \\sum _ { s = 0 } ^ { j } a _ { - s } ^ { ( j ) } \\ , A _ { j - s - 1 } ^ { ( 1 ) } ( z ) \\ , B _ { s - 1 } ^ { ( 1 ) } ( z ) \\ , W _ { 0 } ( z ) \\end{align*}"}
{"id": "5092.png", "formula": "\\begin{align*} P ^ { n , 1 } _ \\tau = n ^ { \\alpha + \\frac 1 2 } \\int ^ { \\tau } _ 0 ( t - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ( X _ { \\eta _ n ( s ) } ) \\Xi ^ { n , 2 } _ s d s \\end{align*}"}
{"id": "2802.png", "formula": "\\begin{align*} \\begin{aligned} - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\big [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\big ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 = 0 \\end{aligned} \\end{align*}"}
{"id": "2285.png", "formula": "\\begin{align*} - \\Delta v _ e ^ 1 = 0 , v _ e ^ 1 ( x , 0 ) = - v _ p ^ 0 ( x , 0 ) , v _ e ^ 1 ( x , \\infty ) = 0 , u _ e ^ 1 = \\int _ x ^ \\infty v _ { e Y } ^ 1 ( \\theta , Y ) { \\rm d } \\theta , \\end{align*}"}
{"id": "4735.png", "formula": "\\begin{align*} & \\sum _ { i , j = 1 } ^ { m } \\tau _ { a _ 1 } \\tau _ k \\tau _ i ^ { - 1 } \\tau _ j ^ { - 1 } q ^ { 2 m - 2 i - 2 j + 2 } v _ { j } \\otimes v _ { j } \\otimes v _ { i } \\otimes v _ { i } \\cdot H _ { 2 } H _ { 1 } \\\\ = & \\sum _ { i , j = 1 } ^ { m } \\tau _ { a _ 1 } \\tau _ k \\tau _ i ^ { - 1 } \\tau _ j ^ { - 1 } q ^ { 2 m - 2 i - 2 j + 2 } v _ { j } \\otimes v _ { j } \\otimes v _ { i } \\otimes v _ { i } \\cdot H _ { 2 } H _ { 3 } . \\end{align*}"}
{"id": "3507.png", "formula": "\\begin{align*} & \\mathrm { D T V } _ { ( p , q ) } ( X ) \\\\ & = - \\mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \\mu ^ { 2 } + 2 \\mu \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sigma ^ { 2 } \\left [ L _ { 1 } + \\frac { \\Psi _ { 1 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) } { \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) } L _ { 2 } \\right ] , \\end{align*}"}
{"id": "5940.png", "formula": "\\begin{align*} w ( \\eta _ 1 , . . . , \\eta _ d ) \\equiv w _ { \\xi } ( \\eta _ 1 , . . . , \\eta _ d ) = \\lim \\limits _ { T \\to \\infty } \\frac 1 T \\ln \\left \\langle e ^ { \\int \\limits _ 0 ^ T ( \\xi _ 1 \\eta _ 1 + . . . + \\xi _ d \\eta _ d ) d t } \\right \\rangle \\end{align*}"}
{"id": "5103.png", "formula": "\\begin{align*} \\Lambda _ n : = \\exp \\left ( i \\lambda N ^ { n , 1 } _ { \\eta _ n ( t ) } + i \\mu Y ^ { n , 1 } _ { t } \\right ) . \\end{align*}"}
{"id": "5416.png", "formula": "\\begin{align*} \\langle \\Lambda _ { \\gamma } f , g \\rangle & = B _ { \\gamma } ( u _ f , g ) = B _ q ( \\gamma ^ { 1 / 2 } u _ f , \\gamma ^ { 1 / 2 } g ) = B _ q ( v _ f , g ) = \\langle \\Lambda _ { q } f , g \\rangle . \\end{align*}"}
{"id": "5507.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\mathbb E \\| u ( t , s , \\zeta _ { s } ) - u ( t ) \\| ^ { 2 } = 0 . \\end{align*}"}
{"id": "3779.png", "formula": "\\begin{align*} \\lesssim 2 ^ { 4 \\epsilon M _ t } \\| \\mathfrak { m } ( \\cdot , \\zeta ) \\| _ { \\mathcal { S } ^ \\infty } \\big ( \\min \\{ 2 ^ { n + 4 j / 3 } , 2 ^ { k + n } \\} \\mathbf { 1 } _ { l > - j } + \\min \\{ 1 + 2 ^ { n + j } , 2 ^ { k / 2 + n } \\} \\mathbf { 1 } _ { l = - j } \\big ) \\end{align*}"}
{"id": "6781.png", "formula": "\\begin{align*} \\Big \\{ x \\in \\R ^ d \\mid & \\ x _ { i j } = 0 \\ \\forall i \\notin M _ P j \\in N _ i , \\sum _ { i \\in \\bar M } x _ { i n _ i } = | \\bar M | - 1 , \\\\ & \\sum _ { i \\in M _ P \\cap M _ 0 } a _ { i 1 } x _ { i 1 } + \\sum _ { j \\in N _ { i ' } - n _ { i ' } } a _ { i ' j } x _ { i ' j } = b - \\sum _ { i \\in \\bar M - i ' } a _ { i n _ i } \\Big \\} , \\end{align*}"}
{"id": "5949.png", "formula": "\\begin{align*} \\left | \\frac { \\d A _ { \\alpha } } { \\d X _ { \\gamma } } \\right | = 1 + \\left ( \\frac { \\d M _ { \\beta } } { \\d X _ { \\gamma } } \\right ) \\delta _ { \\beta \\gamma } \\Delta t + O ( \\Delta t ^ 2 ) \\end{align*}"}
{"id": "3508.png", "formula": "\\begin{align*} \\mathrm { D T E } _ { ( p , q ) } ( X ) = \\mu + \\sigma \\frac { \\phi ( \\xi _ { p } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) - \\phi ( \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) } { \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) } , \\end{align*}"}
{"id": "8806.png", "formula": "\\begin{align*} \\phi ( g ) = \\int _ { 0 } ^ { \\phi ( g ) } d t = \\int _ { 0 } ^ { \\infty } 1 _ { \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) } ( g ) d t . \\end{align*}"}
{"id": "2670.png", "formula": "\\begin{align*} & ( b _ i ^ 2 - b _ i ) = 0 k < i < \\ell \\\\ & b _ i b _ j = 0 k < i < \\ell , j > i . \\end{align*}"}
{"id": "147.png", "formula": "\\begin{align*} \\sup _ { t > 0 } \\frac { c \\eta _ t } { m _ t } \\leq { \\bf 1 } _ { d = 2 } c ( m ^ 2 ) + c _ 0 { \\bf 1 } _ { d = 3 } \\lim _ { m \\rightarrow \\infty } c ( m ^ 2 ) = 0 , \\end{align*}"}
{"id": "2600.png", "formula": "\\begin{align*} \\tilde { \\mathbf S } = \\dfrac { 1 } { 2 } \\begin{pmatrix} 7 & - 2 \\\\ 2 & - 7 \\end{pmatrix} \\ , \\end{align*}"}
{"id": "2971.png", "formula": "\\begin{align*} F _ 0 ( z , s ) = y ^ s + \\frac { \\Lambda ( 2 s - 1 ) } { \\Lambda ( 2 s ) } y ^ { 1 - s } + 2 \\sqrt y \\sum _ { n \\neq 0 } \\frac { | n | ^ { s - \\frac 1 2 } \\sigma _ { 1 - 2 s } ( | n | ) } { \\Lambda ( 2 s ) } K _ { s - \\frac 1 2 } ( 2 \\pi | n | y ) e ( n x ) , \\end{align*}"}
{"id": "3877.png", "formula": "\\begin{align*} H _ 0 = G _ 0 - \\Gamma \\in L ^ \\infty ( \\Omega ) . \\end{align*}"}
{"id": "7296.png", "formula": "\\begin{align*} w _ t & = \\Delta _ x w - q { \\sf U } _ \\infty ^ { q - 1 } w + ( F _ 1 [ \\tilde v _ 1 , \\tilde w _ 1 ] - F _ 2 [ \\tilde w _ 1 ] ) ( 1 - \\chi _ { { \\sf s q } } ) \\\\ & + k [ v ] + ( g + N [ \\epsilon , \\tilde v _ 1 , \\tilde w _ 1 ] ) { \\bf 1 } _ { | \\xi | > 1 } x \\in \\R ^ n , \\ t \\in ( 0 , T ) . \\end{align*}"}
{"id": "5239.png", "formula": "\\begin{align*} \\mathfrak { g } _ { A _ I , k } ^ { r , s } = \\bigoplus _ { \\substack { k _ 1 , k _ 2 \\ge 0 \\\\ 0 < k _ 1 + k _ 2 \\le k } } \\mathfrak { g } ^ { r , s } _ { ( k _ 1 , k _ 2 ) } . \\end{align*}"}
{"id": "5736.png", "formula": "\\begin{align*} & \\pi _ i \\cdot \\pi _ { [ a , b ] } = ( b - i + 1 ) \\pi _ { [ a - 1 , b ] } + ( i - a + 1 ) \\pi _ { [ a , b + 1 ] } \\end{align*}"}
{"id": "1929.png", "formula": "\\begin{align*} S _ { j } ( z ) = S _ { 0 } ( z ) ^ { j + 1 } , 1 \\leq j \\leq p . \\end{align*}"}
{"id": "7229.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left \\{ \\frac { ( t - x - R ) ^ { 1 - b } } { 1 + t - x } \\right \\} ^ { a _ n } \\left \\{ \\frac { ( t - x - R ) ^ { 1 - a } } { 1 + t - x } \\right \\} ^ { b _ n } \\quad \\mbox { i n } \\ D \\end{align*}"}
{"id": "7553.png", "formula": "\\begin{align*} \\chi _ 2 ( x ) & = { \\hat { T } _ 2 } ( s _ 2 ( x ) ) - s _ 2 ( T ( x ) ) = { \\hat { T } _ 2 } ( \\zeta ( s _ 1 ( x ) ) ) - \\zeta ( s _ 1 ( T ( x ) ) ) \\\\ & = \\zeta ( { \\hat { T } _ 1 } ( s _ 1 ( x ) ) - s _ 1 ( T ( x ) ) ) = \\zeta ( \\chi _ 1 ( x ) ) \\\\ & = \\chi _ 1 ( x ) . \\end{align*}"}
{"id": "4782.png", "formula": "\\begin{align*} F ( \\mathbf { B } _ U ) & \\ ! = \\ ! 3 ( \\kappa \\ ! - \\ ! \\nu ) ( \\kappa \\ ! - \\ ! \\nu \\ ! - \\ ! 1 ) ( \\kappa \\ ! - \\ ! 2 ) \\ ! + \\ ! \\kappa ( \\kappa \\ ! - \\ ! 1 ) ( \\kappa \\ ! - \\ ! 2 ) . \\end{align*}"}
{"id": "2712.png", "formula": "\\begin{align*} \\mathfrak { D } : = U _ N ( \\mathcal { N } _ 1 - \\mathcal { N } _ 2 ) U _ N ^ * \\qquad ( \\mathfrak { D } \\Phi ) _ { s , d } = d \\Phi _ { s , d } . \\end{align*}"}
{"id": "2893.png", "formula": "\\begin{align*} g _ 0 = g _ { k + 1 } ( 0 , x _ 2 , \\ldots , x _ { k + 1 } ) \\end{align*}"}
{"id": "7617.png", "formula": "\\begin{align*} S ( r ) < \\dfrac { e ^ { \\frac { r - 1 } { 2 } } } { 2 ^ r } \\sum _ { k = 2 r } ^ { \\infty } \\dfrac { 1 } { \\pi ^ k } = \\dfrac { \\pi } { \\sqrt { e } ( \\pi - 1 ) } \\Bigl ( \\dfrac { \\sqrt { e } } { 2 \\ r ^ 2 } \\Bigr ) ^ r < \\dfrac { 1 } { 1 0 ^ r } . \\end{align*}"}
{"id": "8523.png", "formula": "\\begin{align*} f _ k ( z ) & = \\sum _ { i = 0 } ^ n 4 ^ i c _ i z ^ i ; \\\\ f _ k ^ * ( z ) & = \\sum _ { i = 0 } ^ { n - 1 } 4 ^ i z ^ i \\int _ { - 1 } ^ 1 \\bigg ( \\sum _ { j \\geq i + 1 } c _ j ( t ^ 2 - 1 ) ^ { j - i - 1 } \\bigg ) \\frac { t ^ 2 } { 2 } d t . \\end{align*}"}
{"id": "7332.png", "formula": "\\begin{align*} a , W _ \\ell , b _ \\ell \\sim U ( - M ^ { - 1 / 2 } , M ^ { - 1 / 2 } ) , \\ell = 1 , \\cdots , L - 1 , \\end{align*}"}
{"id": "6155.png", "formula": "\\begin{align*} \\begin{aligned} b _ h ( u ^ { \\Delta t } _ { h } , u ^ { \\Delta t } _ { h } , P _ h ( u ^ { \\Delta t } _ { h } \\phi ) ) & = b _ h ( u ^ { \\Delta t } _ { h } , u ^ { \\Delta t } _ { h } , u ^ { \\Delta t } _ { h } \\phi ) + R _ { n l } . \\end{aligned} \\end{align*}"}
{"id": "3888.png", "formula": "\\begin{align*} - \\Delta _ p ( \\mathit \\Gamma + u ) + \\Delta _ p \\mathit \\Gamma = f \\Omega \\setminus \\{ 0 \\} \\end{align*}"}
{"id": "7611.png", "formula": "\\begin{align*} \\overline { p } ( n ) = \\frac { 1 } { 2 \\pi } \\underset { 2 \\nmid k } { \\sum _ { k = 1 } ^ { \\infty } } \\sqrt { k } \\underset { ( h , k ) = 1 } { \\sum _ { h = 0 } ^ { k - 1 } } \\dfrac { \\omega ( h , k ) ^ 2 } { \\omega ( 2 h , k ) } e ^ { - \\frac { 2 \\pi i n h } { k } } \\dfrac { d } { d n } \\biggl ( \\dfrac { \\sinh \\frac { \\pi \\sqrt { n } } { k } } { \\sqrt { n } } \\biggr ) , \\end{align*}"}
{"id": "5391.png", "formula": "\\begin{align*} \\nabla ^ s u ( x , y ) = \\sqrt { \\frac { C _ { n , s } } { 2 } } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { n / 2 + s + 1 } } ( x - y ) , \\end{align*}"}
{"id": "5529.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } \\setminus \\mathcal { H } } \\frac { 1 } { \\abs { f ' ( z ) } ^ q } d x d y = \\sum _ { \\gamma \\in \\Gamma } \\int _ { \\gamma \\Phi ^ * } \\frac { 1 } { \\abs { f ' ( z ) } ^ q } d x d y \\approx _ { \\Gamma , q } \\sum _ { \\gamma \\in \\Gamma } \\frac { \\abs { \\gamma ' ( 0 ) } ^ { q + 2 } } { \\abs { \\rho ( \\gamma ) ' ( 0 ) } ^ q } . \\end{align*}"}
{"id": "3848.png", "formula": "\\begin{align*} A _ { 1 , 1 } = & \\Big \\{ ( 0 , 1 ) , ( 1 , 0 ) \\Big \\} , A _ { 1 , 2 } = \\Big \\{ ( 0 , 1 , 1 ) , ( 0 , 1 , 2 ) , ( 1 , 0 , 1 ) , ( 1 , 0 , 2 ) , ( 1 , 1 , 0 ) , ( 1 , 2 , 0 ) \\Big \\} , \\\\ A _ { 2 , 1 } = & \\Big \\{ ( 0 , 0 , 1 ) , ( 0 , 1 , 0 ) , ( 1 , 0 , 0 ) \\Big \\} , A _ { 0 , 3 } = \\Big \\{ ( 1 , 1 , 1 ) , ( 1 , 1 , 2 ) , ( 1 , 2 , 1 ) , ( 1 , 2 , 2 ) , ( 1 , 2 , 3 ) \\Big \\} \\end{align*}"}
{"id": "6112.png", "formula": "\\begin{align*} O _ j ^ h \\subseteq ( H \\cap P _ i ^ { g _ j } ) ^ h = H ^ h \\cap P _ i ^ { p g _ k } = H \\cap P _ i ^ { g _ k } \\end{align*}"}
{"id": "2568.png", "formula": "\\begin{align*} C _ 3 ^ { x y z } = C _ 3 ^ { y z x } = C _ 3 ^ { z x y } \\ . \\end{align*}"}
{"id": "7521.png", "formula": "\\begin{align*} ( A _ i ) _ { x y } = \\begin{cases} 1 , \\\\ 0 . \\end{cases} \\end{align*}"}
{"id": "5407.png", "formula": "\\begin{align*} \\langle \\Theta \\nabla ^ s u , \\nabla ^ s \\phi \\rangle _ { L ^ 2 ( \\R ^ { 2 n } ) } = \\langle ( - \\Delta ) ^ { s / 2 } ( \\gamma ^ { 1 / 2 } u ) , ( - \\Delta ) ^ { s / 2 } ( \\gamma ^ { 1 / 2 } \\phi ) \\rangle _ { L ^ 2 ( \\R ^ n ) } + \\langle q \\gamma ^ { 1 / 2 } u , \\gamma ^ { 1 / 2 } \\phi \\rangle _ { L ^ 2 ( \\R ^ n ) } \\end{align*}"}
{"id": "4226.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { 1 \\vee \\log \\max \\{ \\phi ^ { - 1 } ( k ) , k ^ 2 \\} } = \\infty \\sum _ { k = 1 } ^ \\infty \\frac { 2 ^ k } { 1 \\vee \\max \\{ \\log \\phi ^ { - 1 } ( 2 ^ k ) , k \\log 4 \\} } = \\infty . \\end{align*}"}
{"id": "4660.png", "formula": "\\begin{align*} \\| \\Psi _ t \\| _ { 1 , h } = \\| \\Psi _ 1 \\| _ { 1 , h } = 1 , \\end{align*}"}
{"id": "2889.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d s _ i ^ { t _ i } \\ , ( t _ i ! ) . \\end{align*}"}
{"id": "1021.png", "formula": "\\begin{align*} \\binom x { \\ 1 _ S } = \\begin{pmatrix} a _ 1 , & a _ 2 , & a _ 3 , & \\dots \\\\ \\ 1 _ S ( 1 ) , & \\ 1 _ S ( 2 ) , & \\ 1 _ S ( 3 ) , & \\dots \\end{pmatrix} , \\end{align*}"}
{"id": "1054.png", "formula": "\\begin{align*} \\widetilde { B _ 0 } S _ n ( t ) \\in \\mathcal { C } _ { r _ n } r _ n = \\frac { p q } { n p + q } , n = 0 , 1 , \\ldots , l . \\end{align*}"}
{"id": "5620.png", "formula": "\\begin{align*} P = ( ( A ^ { - 1 } ) f ( X ) D [ X ] ) _ t = f ( X ) A ^ { - 1 } . \\end{align*}"}
{"id": "6762.png", "formula": "\\begin{align*} \\frac k { \\gcd ( k , j _ 1 , j _ 2 , \\dots , j _ m ) } \\binom { k } { j _ 1 , j _ 2 , \\dots , j _ m } . \\end{align*}"}
{"id": "800.png", "formula": "\\begin{align*} f _ 0 ( z ) = z \\exp { \\int _ { 0 } ^ { z } \\frac { \\psi ( t ) - 1 } { t } d t } . \\end{align*}"}
{"id": "4169.png", "formula": "\\begin{align*} ( F ( L , \\mathbf U ) f ) ^ \\mu ( x ) = F ( L ^ \\mu , \\mu ) f ^ \\mu ( x ) \\end{align*}"}
{"id": "1526.png", "formula": "\\begin{align*} \\Delta _ { a n g } = \\frac { 1 } { 2 \\sinh ^ 2 ( \\beta r ) } \\frac { \\partial ^ 2 } { \\partial \\theta ^ 2 } \\end{align*}"}
{"id": "8605.png", "formula": "\\begin{align*} Q = Q ^ T > 0 , \\ \\ \\ J = - J ^ T , \\ \\ \\ R = R ^ T \\geq 0 . \\end{align*}"}
{"id": "4633.png", "formula": "\\begin{align*} \\| B \\| = { | w | } ^ { - 1 } < 1 \\end{align*}"}
{"id": "7433.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\left ( \\int _ 0 ^ L ( \\rho \\abs { u _ t } ^ 2 + \\alpha _ 1 \\abs { u _ x } ^ 2 + \\mu \\abs { y _ t } ^ 2 + \\beta \\abs { \\gamma u _ x - y _ x } ^ 2 ) d x \\right ) + \\delta \\int _ 0 ^ L \\omega _ x \\overline { u _ t } d x = 0 . \\end{align*}"}
{"id": "3400.png", "formula": "\\begin{align*} \\dim H ^ 1 ( G , V ) = \\dim H ^ 1 ( H , V ) . \\end{align*}"}
{"id": "5309.png", "formula": "\\begin{align*} A \\phi ( x ) = a \\cdot \\nabla \\phi ( x ) + \\frac { 1 } { 2 } \\nabla \\cdot ( Q \\nabla \\phi ) ( x ) + \\int _ { \\R ^ d } \\left ( \\phi ( x + y ) - \\phi ( x ) - \\nabla \\phi ( x ) y 1 _ { \\{ | y | < 1 \\} } \\right ) \\nu ( d y ) \\end{align*}"}
{"id": "6403.png", "formula": "\\begin{align*} \\pi _ { \\tau } ( 1 + 1 ) = j _ 1 + 1 , \\ , \\pi _ { \\tau } ( 2 + 1 ) = j _ 2 + 1 , \\ldots , \\pi _ { \\tau } ( \\ell + 1 ) = j _ { \\ell } + 1 . \\end{align*}"}
{"id": "2654.png", "formula": "\\begin{align*} D _ { i - 2 } = \\frac { a _ { ( i - 2 ) i } ^ 2 - p ^ { e _ { i - 1 } - 1 } a _ { ( i - 2 ) i } - p ^ { \\ell - 1 } a _ { ( i - 2 ) ( i - 1 ) } d } { p ^ { e _ { i - 2 } + m - 2 } } \\in \\Z . \\end{align*}"}
{"id": "8295.png", "formula": "\\begin{align*} \\frac { \\sin ( k a ) } { k a } = \\mp \\frac { \\sinh ( k ' a ) } { k ' a } , \\end{align*}"}
{"id": "1947.png", "formula": "\\begin{align*} \\langle \\mathcal { H } _ { q } ^ { n } \\ , e _ { r } , e _ { 0 } \\rangle = \\langle e _ { r } , e _ { 0 } \\rangle = \\delta _ { r , 0 } = \\sum _ { \\gamma \\in \\mathcal { S } _ { 1 } ( 0 , r ) } w ( \\gamma ) . \\end{align*}"}
{"id": "5145.png", "formula": "\\begin{align*} \\mathcal { G } ( O _ { n + 1 } ) & = \\mathcal { G } ( h _ 1 ) \\oplus \\dots \\oplus \\mathcal { G } ( h _ i ) \\oplus \\dots \\oplus \\mathcal { G } ( h _ p ) & \\\\ & = \\mathcal { G } ( h _ 1 ) \\oplus \\dots \\oplus [ \\mathcal { G } ( h _ i - 1 ) \\oplus 1 ] \\oplus \\dots \\oplus \\mathcal { G } ( h _ p ) & \\eqref { o b 1 } \\\\ & = \\mathcal { G } ( O ' _ n ) \\oplus 1 & \\end{align*}"}
{"id": "2486.png", "formula": "\\begin{align*} h ( F _ \\ast X , F _ \\ast Y ) = \\lambda ^ 2 ( p ) g ( X , Y ) , ~ \\forall X , Y \\in \\Gamma ( K e r F _ \\ast ) ^ \\bot ~ ~ p \\in M . \\end{align*}"}
{"id": "3563.png", "formula": "\\begin{align*} \\binom { \\sum _ { i = 1 } ^ { n } c _ i N _ i } { 2 } & = \\sum _ { i = 1 } ^ { n } \\binom { c _ i N _ i } { 2 } + \\sum _ { 1 \\leq l < k \\leq n } c _ l c _ k N _ l N _ k \\\\ & = \\sum _ { i = 1 } ^ { n } \\left ( \\sum _ { k _ 1 + \\cdots + k _ { N _ i } = 2 } \\binom { c _ i } { k _ 1 } \\cdots \\binom { c _ i } { k _ { N _ i } } \\right ) + \\sum _ { 1 \\leq l < k \\leq n } c _ l c _ k N _ l N _ k \\\\ & = \\sum _ { i = 1 } ^ { n } N _ i \\binom { c _ i } { 2 } + \\sum _ { i = 1 } ^ { n } c _ i ^ 2 \\binom { N _ i } { 2 } + \\sum _ { 1 \\leq l < k \\leq n } c _ l c _ k N _ l N _ k . \\end{align*}"}
{"id": "5120.png", "formula": "\\begin{align*} 0 = V _ 0 \\subset \\cdots \\subset V _ r = V \\end{align*}"}
{"id": "4027.png", "formula": "\\begin{align*} \\log \\delta _ n = - \\frac { Q ^ { - 1 } ( \\epsilon _ n ) ^ 2 } { 2 } - \\frac { 1 } { 2 } \\log n . \\end{align*}"}
{"id": "5037.png", "formula": "\\begin{align*} Q ^ { n , 5 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( s - \\eta _ n ( s ) ) ^ { 2 \\alpha + 1 } [ ( \\sigma ' ( X _ s ) ) ^ 2 - ( \\sigma ' ( X _ { \\eta _ n ( s ) } ) ) ^ 2 ] \\ , \\sigma ^ 2 ( X _ { \\eta _ n ( s ) } ) d s , \\end{align*}"}
{"id": "260.png", "formula": "\\begin{align*} \\Lambda _ { i } ^ { ( s ) t } A _ i ^ { ( s ) } \\Lambda _ { i } ^ { ( s ) } = B ^ { ( s ) } . \\end{align*}"}
{"id": "1378.png", "formula": "\\begin{align*} \\Gamma [ z ] & = \\chi ( t / \\tilde { T } ) W _ t T [ u _ 0 ] \\pm i \\chi ( t / \\tilde { T } ) \\int _ 0 ^ t W _ { t - s } \\left ( \\mathfrak { F } ( v ) + \\mathcal { H L } _ B [ z , v , \\cdots , v ] \\right ) \\ , d s . \\end{align*}"}
{"id": "5544.png", "formula": "\\begin{align*} M _ { n , k } ( m ) = \\dbinom { m + n } { n - ( m - k ) + 1 } - \\dbinom { m + n } { n - ( m - k ) - 1 } . \\end{align*}"}
{"id": "7947.png", "formula": "\\begin{align*} D _ \\pm = \\sum _ { i \\in ( M _ + \\cup M _ - ) \\setminus S _ \\pm } \\bar { D } _ i . \\end{align*}"}
{"id": "8408.png", "formula": "\\begin{gather*} { \\varrho _ m ( \\gamma , \\upsilon ) } \\hat h | _ { k - \\frac 1 2 } ( \\gamma , \\upsilon ) = \\hat h \\end{gather*}"}
{"id": "3655.png", "formula": "\\begin{align*} f ( x , u , \\xi ) = \\bigl ( 1 + u ^ { p ( \\frac { 1 } { p ^ \\star } - 1 ) } \\bigr ) \\abs { \\xi } ^ { p ( x ) } , \\end{align*}"}
{"id": "6434.png", "formula": "\\begin{align*} Q ' : = \\dd ^ { } + Q + \\sum _ { k \\geq 1 , i _ 1 , \\ldots , i _ k = 1 , \\ldots , \\mathrm { d i m } ( \\mathfrak { g } ) } \\frac { 1 } { k ! } \\xi ^ { i _ 1 } \\odot \\cdots \\odot \\xi ^ { i _ k } \\Phi _ { k - 1 } ( \\xi _ { i _ 1 } , \\ldots , \\xi _ { i _ { k } } ) , \\end{align*}"}
{"id": "3327.png", "formula": "\\begin{align*} \\chi _ 4 ( \\hat { \\beta } ) = N - k \\end{align*}"}
{"id": "5206.png", "formula": "\\begin{align*} | I | - 3 = { \\sum _ { i \\in I } a _ i - ( r - 2 ) \\over r } + { \\sum _ { i \\in I } b _ i - ( s - 2 ) \\over s } + \\sum _ { i \\in I } d _ i , \\end{align*}"}
{"id": "2543.png", "formula": "\\begin{align*} g ( X _ 1 , Y _ 1 ) = g ( a _ 1 e _ 1 + a _ 2 e _ 2 , a _ 3 e _ 1 + a _ 4 e _ 2 ) = ( a _ 1 a _ 3 e ^ { 2 x _ 1 } + a _ 2 a _ 4 ) , \\end{align*}"}
{"id": "6458.png", "formula": "\\begin{align*} Q _ \\mathfrak { g } ( x _ 1 \\wedge \\cdots \\wedge x _ k ) : = \\sum _ { 1 \\leq i < j \\leq k } ( - 1 ) ^ { i + j - 1 } [ x _ i , x _ j ] _ \\mathfrak { g } \\wedge x _ 1 \\wedge \\cdots \\widehat { x } _ i \\cdots \\widehat { x } _ j \\cdots \\wedge x _ k , \\end{align*}"}
{"id": "490.png", "formula": "\\begin{align*} s _ { i _ { k } } \\cdots s _ { i _ \\ell } \\lambda = \\lambda - \\alpha _ { i _ \\ell } - \\alpha _ { i _ { \\ell - 1 } } - \\cdots - \\alpha _ { i _ k } \\end{align*}"}
{"id": "7049.png", "formula": "\\begin{align*} \\sum _ { M \\geq 0 } g _ 1 ( M ) \\ , q ^ M & = \\sum _ { M \\geq 1 } \\frac { z ^ { 2 M } } { ( 1 - z ) ^ { M + 1 } } + \\sum _ { M \\geq 0 } \\frac { z ^ { 2 M + 2 } M } { ( 1 - z ) ^ { M + 2 } } \\\\ & = \\frac { z ^ 2 } { ( 1 - z ) ( 1 - z - z ^ 2 ) } + \\frac { z ^ 4 } { ( 1 - z ) ^ 3 \\left ( 1 - \\frac { z ^ 2 } { 1 - z } \\right ) ^ 2 } = \\frac { z ^ 2 } { ( 1 - z - z ^ 2 ) ^ 2 } . \\end{align*}"}
{"id": "5306.png", "formula": "\\begin{align*} \\langle \\phi , \\mu \\rangle : = \\int _ { \\R ^ d } \\phi ( x ) \\mu ( d x ) = : \\mu ( \\phi ) . \\end{align*}"}
{"id": "1646.png", "formula": "\\begin{align*} ( y _ 1 , \\dots , y _ { g - 1 } ) \\longmapsto \\left ( \\ , \\prod _ { i = 1 } ^ { g - 1 } \\frac { 1 - \\kappa _ 1 y _ i } { 1 - \\kappa _ 2 y _ i } , \\ , \\ , \\prod _ { i = 1 } ^ { g - 1 } \\frac { 1 - \\kappa _ 3 y _ i } { 1 - \\kappa _ 4 y _ i } , \\dots , \\prod _ { i = 1 } ^ { g - 1 } \\frac { 1 - \\kappa _ { 2 g - 1 } y _ i } { 1 - \\kappa _ { 2 g } y _ i } \\ , \\right ) . \\end{align*}"}
{"id": "7028.png", "formula": "\\begin{align*} \\frac { \\prod _ { j \\in \\deg G } ( 1 + t ^ j ) } { \\prod _ { k \\in \\deg H } ( 1 + t ^ k ) } = \\frac { p ( G ) } { p ( H ) } = \\frac { p ( G ' ) } { p ( H ' ) } = ( 1 + t ^ n ) \\frac { 1 + t ^ { 2 m - 1 } } { 1 + t ^ { m - 1 } } \\mathrlap , \\end{align*}"}
{"id": "507.png", "formula": "\\begin{align*} \\smash { \\sum _ { \\substack { a \\in \\overline { E } \\\\ \\mathrm { h e a d } ( a ) = i } } \\epsilon ( a ) a a ^ * } ( x ) & = ( \\pi ( z _ 1 ) \\to i ) ( i \\to \\pi ( z _ 2 ) ) ( x ) + ( \\pi ( z _ 2 ) \\to i ) ( i \\to \\pi ( z _ 1 ) ) ( x ) \\\\ & = ( \\pi ( z _ 1 ) \\to i ) ( z _ 1 ) + ( \\pi ( z _ 2 ) \\to i ) ( z _ 2 ) \\\\ & = - x + x \\\\ & = 0 . \\end{align*}"}
{"id": "4489.png", "formula": "\\begin{align*} \\frac { p ' ( \\eta _ 3 ) - p ' ( \\eta _ 1 + \\eta _ 2 + \\eta _ 3 ) } { p ' ( \\eta _ 1 ) - p ' ( \\eta _ 3 ) } - \\frac { p ' ( \\eta _ 1 ) - p ' ( \\eta _ 3 ) } { p ' ( \\eta _ 3 ) - p ' ( \\eta _ 1 + \\eta _ 2 + \\eta _ 3 ) } = : \\frac { \\mathcal { A } } { \\mathcal { B } } , \\end{align*}"}
{"id": "8155.png", "formula": "\\begin{align*} & \\limsup \\limits _ { \\delta \\rightarrow 0 } \\limsup _ { n \\rightarrow \\infty } \\limsup _ { t \\rightarrow \\infty } \\| P _ \\delta ( W _ { n , m ; \\textrm { f a r } } ) e ^ { - i t H _ 0 } \\varphi \\| \\\\ & \\leq \\limsup \\limits _ { \\delta \\rightarrow 0 } \\limsup _ { n \\rightarrow \\infty } \\limsup _ { t \\rightarrow \\infty } \\| P _ \\delta ( W _ { n , m ; \\textrm { f a r } } ) e ^ { - i t H } \\psi \\| = 0 \\end{align*}"}
{"id": "6351.png", "formula": "\\begin{align*} | Z _ 2 | = 0 ~ ~ E _ { u _ n } ( x , y ) \\longrightarrow E _ u ( x , y ) ~ ~ ( x , y ) \\in \\R ^ { 2 N } \\setminus Z _ 2 . \\end{align*}"}
{"id": "2503.png", "formula": "\\begin{align*} 2 A _ X Y = \\nu [ X , Y ] - \\lambda ^ 2 g ( X , Y ) \\left ( \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) , \\end{align*}"}
{"id": "236.png", "formula": "\\begin{align*} R \\rightarrow W ( R ) = R ^ { \\mathbb N } \\end{align*}"}
{"id": "8661.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\overline f > \\tau \\} } ( r ) g ( r ) \\ , d r & = \\int _ 0 ^ { b _ \\tau } g ( r ) \\ , d r \\leq b _ \\tau ^ \\beta \\sup _ { s > 0 } s ^ { - \\beta } \\int _ 0 ^ s g ( r ) \\ , d r \\\\ & = \\beta \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\overline f > \\tau \\} } ( r ) \\ , \\frac { d r } { r ^ { 1 - \\beta } } \\ \\underline \\nu _ \\beta ( g ) \\ , . \\end{align*}"}
{"id": "7234.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left \\{ \\log \\left ( \\frac { 1 + t - x } { 1 + R } \\right ) \\right \\} ^ { a _ n } \\quad \\mbox { i n } \\ D \\end{align*}"}
{"id": "1346.png", "formula": "\\begin{align*} A ( x , v ) = a x - b v , \\mbox { a n d } B ( x , v ) = x + v ( v - 1 ) ( v - c ) , \\end{align*}"}
{"id": "3851.png", "formula": "\\begin{align*} & A _ { \\tilde { i } , \\tilde { j } | a } : = \\Big \\{ \\overline { a } \\in A _ { \\tilde { i } , \\tilde { j } } : \\forall p = 1 , . . . , i + j , \\overline { a } _ p = a _ p \\Big \\} . \\end{align*}"}
{"id": "7294.png", "formula": "\\begin{align*} { \\cal V } ( \\xi , t ) & = ( T - t ) ^ { { \\sf d } _ 1 } ( 1 + | \\xi | ^ 2 ) ^ \\frac { \\gamma } { 2 } , \\xi = \\eta ( t ) ^ { - 1 } x . \\end{align*}"}
{"id": "6134.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\end{gather*}"}
{"id": "8880.png", "formula": "\\begin{align*} \\det ( C ( \\alpha , \\beta ) ) = \\sum _ { \\gamma } \\det ( A ( \\alpha , \\gamma ) ) \\det ( B ( \\gamma , \\beta ) ) , \\end{align*}"}
{"id": "2493.png", "formula": "\\begin{align*} H ' = - \\frac { \\lambda ^ 2 } { 2 } \\left ( \\nabla _ \\nu \\frac { 1 } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
{"id": "8365.png", "formula": "\\begin{align*} \\varphi _ { { } _ { \\nu } } ( g ) = { \\textstyle \\frac { 1 } { 4 \\pi } } \\int \\limits _ { \\mathbb { S } ^ 2 } \\ , d ^ 2 p \\ , \\overline { ( p \\cdot u ) ^ { i \\nu - 1 } } ( p \\cdot v ) ^ { i \\nu - 1 } , \\ , \\ , \\ , v = ( 1 , 0 , 0 , 0 ) , \\ , u = g v = \\Lambda ( g ^ { - 1 } ) v \\end{align*}"}
{"id": "1827.png", "formula": "\\begin{align*} A _ { n } ^ { ( 1 ) } = \\sum _ { \\gamma \\in \\mathcal { M } _ { n } ^ { ( 1 ) } } w ( \\gamma ) , \\end{align*}"}
{"id": "3314.png", "formula": "\\begin{align*} L \\tilde u = \\left \\{ \\begin{array} { c c } - \\div \\kappa \\nabla \\tilde u & \\Omega ^ - \\\\ - \\Delta \\tilde u & \\Omega ^ + \\end{array} \\right . \\end{align*}"}
{"id": "1943.png", "formula": "\\begin{align*} \\mathcal { W } \\ , \\hat { e } _ { n } = \\hat { e } _ { n - 1 } + \\sum _ { k = 0 } ^ { p } a _ { n } ^ { ( k ) } \\hat { e } _ { n + k } , n \\in \\mathbb { Z } , \\end{align*}"}
{"id": "645.png", "formula": "\\begin{align*} \\xi _ { 2 } ( s ) = \\xi _ { 2 } ( 1 - s ) , \\end{align*}"}
{"id": "2350.png", "formula": "\\begin{align*} f = a _ { \\rho 0 } ( f ) + a _ { \\rho 1 } ( f ) Q _ \\rho + \\ldots + a _ { \\rho r } ( f ) Q _ \\rho ^ r \\end{align*}"}
{"id": "3394.png", "formula": "\\begin{align*} \\Delta _ { \\Sigma } g ( z ) = \\gamma \\Sigma _ { 0 0 } ( z ) \\left ( \\frac { 2 b ' ( x ) ^ 2 } { b ( x ) ^ 3 } - \\frac { b '' ( x ) } { b ( x ) ^ 2 } \\right ) \\| y \\| ^ 2 _ d + \\frac { 2 \\gamma } { b ( x ) } \\sum _ { i = 1 } ^ d \\Sigma _ { i i } ( z ) - 4 \\frac { b ' ( x ) } { b ( x ) ^ 2 } \\sum _ { i = 1 } ^ d y _ i \\Sigma _ { 0 i } ( z ) . \\end{align*}"}
{"id": "6986.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\prod \\limits _ { k = 1 } ^ { N - 1 } \\Bigg ( \\frac { \\lambda _ N ^ 2 + \\mu _ { k } ^ { 2 } } { \\lambda _ N ^ 2 + \\lambda _ { k + 1 } ^ { 2 } } \\Bigg ) = \\infty , \\end{align*}"}
{"id": "4349.png", "formula": "\\begin{align*} I ^ { \\vee } ( \\mathrm { p o l } ) ^ { \\vee } & = \\big < \\{ x _ { i , 1 } x _ { j , 1 } , \\ldots , x _ { i , w ( i ) } x _ { j , 1 } \\mid ( x _ { i } , x _ { j } ) \\in E ( D _ { G } ) \\} \\big > \\\\ & = I ( G ^ { D } ) \\end{align*}"}
{"id": "6812.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\int _ 0 ^ { x _ 0 - \\delta } a u '' v '' d x = \\int _ 0 ^ { x _ 0 } a u '' v '' d x , \\lim _ { \\delta \\rightarrow 0 } \\int _ { x _ 0 + \\delta } ^ 1 a u '' v '' d x = \\int _ { x _ 0 } ^ 1 a u '' v '' d x \\end{align*}"}
{"id": "8839.png", "formula": "\\begin{align*} \\| \\phi _ 1 * \\phi _ 2 \\| = \\| \\phi _ 1 ^ * * \\phi _ 2 ^ * \\| , \\end{align*}"}
{"id": "4337.png", "formula": "\\begin{align*} \\int _ \\Sigma u ^ * \\omega = \\int _ { \\partial \\Sigma } \\lambda = \\int _ { - \\infty } ^ \\infty d f _ L - \\int _ { - \\infty } ^ \\infty d f _ { L ' } = ( f _ L - f _ { L ' } ) ( q ) - ( f _ L - f _ { L ' } ) ( p ) . \\end{align*}"}
{"id": "7461.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\abs { \\omega } ^ 2 d x = o ( 1 ) \\int _ 0 ^ L \\abs { \\Lambda _ x } ^ 2 d x = o ( 1 ) . \\end{align*}"}
{"id": "2947.png", "formula": "\\begin{align*} \\psi _ r ( t ) = & e ^ { - i t H _ 0 } ( \\psi ( 0 ) - \\Omega _ \\alpha ^ * \\psi ( 0 ) ) + ( - i ) \\int _ 0 ^ t d s e ^ { - i ( t - s ) H _ 0 } \\mathcal { N } ( x , t , \\psi ( s ) ) \\\\ = & ( - i ) \\int _ 0 ^ t d s e ^ { - i ( t - s ) H _ 0 } \\mathcal { N } ( x , t , \\psi _ r ( s ) ) \\\\ & + ( - i ) \\int _ 0 ^ t d s e ^ { - i ( t - s ) H _ 0 } \\left ( \\mathcal { N } ( x , t , \\psi ( s ) ) - \\mathcal { N } ( x , t , \\psi _ r ( s ) ) \\right ) \\\\ = : & S _ 0 ( x , t ) + S _ 1 ( x , t ) + S _ 2 ( x , t ) . \\end{align*}"}
{"id": "2689.png", "formula": "\\begin{align*} h _ { m n } : = \\ ; & \\big \\langle u _ m , ( - \\Delta + V _ \\mathrm { D W } ) u _ n \\big \\rangle \\\\ w _ { m n p q } : = \\ ; & \\big \\langle u _ m \\otimes u _ n , w , u _ p \\otimes u _ q \\big \\rangle . \\end{align*}"}
{"id": "4761.png", "formula": "\\begin{align*} \\mathcal S _ { \\kappa , \\gamma } = \\left \\{ \\left [ n _ 0 , n _ 1 , \\ldots , n _ { 2 ^ { \\gamma } - 1 } \\right ] \\colon n _ i \\geq 0 , \\ ; \\sum _ { i = 0 } ^ { 2 ^ \\gamma - 1 } n _ i = \\kappa \\right \\} . \\end{align*}"}
{"id": "7365.png", "formula": "\\begin{align*} \\begin{cases} t _ 1 = ( ~ 1 , ~ 1 , ~ 1 , ~ 0 , \\cdots , 0 ) , \\\\ t _ 2 = ( 1 , - 1 , - 1 , 0 , \\cdots , 0 ) , \\\\ t _ 3 = ( - 1 , 1 , - 1 , 0 , \\cdots , 0 ) , \\\\ t _ 4 = ( - 1 , - 1 , 1 , 0 , \\cdots , 0 ) , \\end{cases} \\end{align*}"}
{"id": "2435.png", "formula": "\\begin{align*} \\mathbb T ^ { m | n } _ \\zeta : = \\mathbb T ^ { m | n } S _ \\zeta . \\end{align*}"}
{"id": "6675.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } z _ { t } - \\dfrac { \\Delta z } { g _ 1 ^ 2 } + B _ 1 ( t , s ) z _ { s } = B _ 3 ( t , s ) w + F _ 2 ( t , s ) , \\ \\ 0 < t < T _ 0 , \\ 0 < s < 1 , \\\\ z _ { s } ( t , 0 ) = z ( t , 1 ) = 0 , \\ \\ 0 < t < T _ 0 , \\\\ z ( 0 , s ) = v _ { 0 1 } - v _ { 0 2 } = z _ { 0 } ( s ) , \\ \\ 0 \\leq s \\leq 1 , \\end{array} \\right . \\end{align*}"}
{"id": "107.png", "formula": "\\begin{align*} \\chi ^ \\Lambda ( g , \\nu ) = \\sum _ { x \\in \\Lambda } \\avg { \\varphi _ 0 \\varphi _ x } _ { g , \\nu } . \\end{align*}"}
{"id": "195.png", "formula": "\\begin{align*} \\hat { \\pi } ^ i _ t ( m ^ i ) = \\frac { 1 _ { e ^ i _ t ( m ^ i ) } ( x ^ i _ t ) \\hat { \\pi } ^ i _ { t - 1 } ( m ^ i ) } { \\sum _ { \\tilde { m } ^ i } 1 _ { e ^ i _ t ( \\tilde { m } ^ i ) } ( x ^ i _ t ) \\hat { \\pi } ^ i _ { t - 1 } ( \\tilde { m } ^ i ) } , \\end{align*}"}
{"id": "5446.png", "formula": "\\begin{align*} ( a _ 1 \\star ( a _ 2 \\star q ) ) ( m ) & = ( a _ 2 \\star q ) ( m a _ 1 ) \\cdot a _ 1 ^ { - 1 } \\\\ & = q ( m a _ 1 a _ 2 ) \\cdot a _ 2 ^ { - 1 } \\cdot a _ 1 ^ { - 1 } \\\\ & = q ( m a _ 1 a _ 2 ) \\cdot ( a _ 1 a _ 2 ) ^ { - 1 } \\\\ & = ( ( a _ 1 a _ 2 ) \\star q ) ( m ) . \\end{align*}"}
{"id": "7154.png", "formula": "\\begin{align*} ( g _ { 1 1 } y _ 1 + g _ { 1 2 } y _ 2 + g _ { 1 3 } y _ 3 ) ^ b = g _ { 1 1 } ^ b y _ 1 ^ b + \\gamma y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } , \\end{align*}"}
{"id": "8460.png", "formula": "\\begin{align*} p \\left ( x _ 1 , x _ 2 , \\ldots , x _ d \\right ) = \\sum _ { i _ 1 = 1 } ^ k \\cdots \\sum _ { i _ d = 1 } ^ k W _ { i _ 1 , \\ldots , i _ d } p _ { 1 , i _ 1 } ( x _ 1 ) p _ { 2 , i _ 2 } ( x _ 2 ) \\cdots p _ { d , i _ d } ( x _ d ) . \\end{align*}"}
{"id": "2836.png", "formula": "\\begin{align*} \\min \\{ \\| \\nabla f ( x _ 0 ) \\| ^ 2 \\ , , \\ , \\| \\nabla f ( x _ { 1 } ) \\| ^ 2 \\} \\leq \\frac { f ( x _ 0 ) - f ( x _ { 1 } ) } { P ( h ) } \\end{align*}"}
{"id": "5666.png", "formula": "\\begin{align*} J _ 1 = \\{ 1 , 2 \\} , \\ J _ 2 = \\{ 4 , 5 , 6 \\} , \\ J _ 3 = \\{ 9 \\} \\end{align*}"}
{"id": "8262.png", "formula": "\\begin{align*} \\phi ( x ) = A e ^ { i k x } + B e ^ { - i k x } + C e ^ { k ' x } + D e ^ { - k ' x } . \\end{align*}"}
{"id": "8834.png", "formula": "\\begin{align*} \\| f _ t \\circ ( \\phi _ 1 * \\phi _ 2 ) \\| \\leq \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\| f _ { T ( t _ 1 , t _ 2 ) } \\circ ( 1 _ { ( - J _ 1 ( t _ 1 ) , J _ 1 ( t _ 1 ) ) } * 1 _ { ( - J _ 2 ( t _ 2 ) , J _ 2 ( t _ 2 ) ) } ) \\| d t _ 1 d t _ 2 = \\| f _ t \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) \\| \\end{align*}"}
{"id": "8447.png", "formula": "\\begin{align*} W _ n = \\bigcup _ { w \\in \\Sigma _ { \\beta } ^ n } \\bigcup _ { 0 \\leq k \\leq \\left \\lfloor \\frac { \\beta ^ n } { \\varphi ( n ) } \\right \\rfloor } \\left \\{ ( x , y ) \\in I _ { n , \\beta } ( w ) \\times J _ { n , \\beta } ( k ) : | T _ { \\beta } ^ n x - f ( x , y ) | < \\varphi ( n ) \\right \\} . \\end{align*}"}
{"id": "6560.png", "formula": "\\begin{align*} d ^ j f ( x , y _ 1 , \\ldots , y _ j ) ( v ) = d ^ j ( f ^ \\wedge ) ( ( x , v ) , ( y _ 1 , 0 ) , \\ldots , ( y _ j , 0 ) ) \\end{align*}"}
{"id": "5236.png", "formula": "\\begin{align*} \\psi ( x ) = x ( 1 + { ( b + 1 ) C \\over b - a } x ^ a y ^ b + \\cdots ) . \\end{align*}"}
{"id": "3613.png", "formula": "\\begin{align*} I = \\bigcap _ { 1 \\leq j _ 1 < \\cdots < j _ { s - d + 1 } \\leq s } ( t _ { j _ 1 } , \\ldots , t _ { j _ { s - d + 1 } } ) . \\end{align*}"}
{"id": "3803.png", "formula": "\\begin{align*} \\Phi _ 4 ( \\xi , \\eta , \\sigma , \\kappa , \\zeta ) : = \\hat { \\zeta } \\cdot ( \\xi + \\eta + \\sigma + \\kappa ) + \\mu _ 3 | \\kappa | + \\mu _ 2 | \\sigma | + \\mu | \\xi | + \\mu _ 1 | \\eta | . \\end{align*}"}
{"id": "5592.png", "formula": "\\begin{align*} K e r ( M ) \\cap K e r ( \\tau ) = K e r ( \\tau ^ M ) . \\end{align*}"}
{"id": "7797.png", "formula": "\\begin{align*} P ( X _ 2 , \\eta _ 2 ) : = \\eta _ 2 ( H ^ { 2 , 0 } ( X _ 2 ) ) = [ \\eta _ 2 \\circ ( \\eta _ 2 ^ { - 1 } \\psi \\eta _ 1 ) ] ( H ^ { 2 , 0 } ( X _ 1 ) ) = \\psi ( P ( X _ 1 , \\eta _ 1 ) ) . \\end{align*}"}
{"id": "3043.png", "formula": "\\begin{align*} H _ { \\lambda } ( q ) : = \\prod _ { x \\in \\lambda } ( 1 - q ^ { h ( x ) } ) , \\end{align*}"}
{"id": "6456.png", "formula": "\\begin{align*} \\Phi = \\sum _ { k \\geq 0 } \\Phi ^ { ( k ) } \\end{align*}"}
{"id": "7431.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\left ( \\rho \\int _ 0 ^ L \\abs { u _ t } ^ 2 d x + \\alpha \\int _ 0 ^ L \\abs { u _ x } ^ 2 d x \\right ) - \\gamma \\beta \\Re \\left ( \\int _ 0 ^ L y _ x \\overline { u _ { x t } } \\right ) + \\delta \\Re \\left ( \\int _ 0 ^ L \\omega _ x \\overline { u _ t } d x \\right ) = 0 \\end{align*}"}
{"id": "8325.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ { l + 1 } } = \\| f \\| _ { C ^ l } + \\left \\| \\frac { 1 } { 2 \\pi i } \\frac { d } { d x } \\ , f \\right \\| _ { C ^ l } . \\end{align*}"}
{"id": "3973.png", "formula": "\\begin{align*} h _ q ( \\delta ) = \\delta \\log _ q ( q - 1 ) - \\delta \\log _ q \\delta - ( 1 - \\delta ) \\log _ q ( 1 - \\delta ) , \\delta \\in [ 0 , 1 - q ^ { - 1 } ] , \\end{align*}"}
{"id": "1498.png", "formula": "\\begin{align*} \\phi _ { n , \\lambda } ( x ) & = \\sum _ { k = 0 } ^ { n } S _ { 2 , \\lambda } ( n , k ) x ^ { k } , ( n \\ge 0 ) , \\\\ & = e ^ { - x } \\sum _ { k = 0 } ^ { \\infty } \\frac { ( k ) _ { n , \\lambda } } { k ! } x ^ { k } , ( \\mathrm { s e e } \\ [ 9 , 1 0 , 1 4 , 1 5 ] ) . \\end{align*}"}
{"id": "4774.png", "formula": "\\begin{align*} | \\mathcal { K } _ { \\kappa , 3 } | = | S _ { \\kappa , 3 } ^ A | + | S _ { \\kappa , 3 } ^ B | / 3 + | S _ { \\kappa , 3 } ^ C | / 6 . \\end{align*}"}
{"id": "7977.png", "formula": "\\begin{align*} & H _ { ( X _ - , D _ - ) } ( y _ 1 , y _ 2 ) \\\\ = & e ^ { \\frac { \\xi _ - \\log y _ 1 + h _ - \\log y _ 2 } { 2 \\pi i } } \\sum _ { d _ 1 , d _ 2 \\geq 0 } \\frac { y _ 1 ^ { d _ 1 } y _ 2 ^ { d _ 2 } [ \\textbf { 1 } ] _ { ( d _ 1 - d _ 2 , \\cdots , d _ 1 - d _ 2 ) } } { \\Gamma ( 1 + \\frac { h _ - } { 2 \\pi i } + d _ 2 ) ^ { r ^ \\prime + 1 } \\Gamma ( 1 + \\frac { \\xi _ - - h _ - } { 2 \\pi i } + d _ 1 - d _ 2 ) ^ { r ^ \\prime + 1 } \\Gamma ( 1 + \\frac { \\xi _ - } { 2 \\pi i } + d _ 1 ) } . \\end{align*}"}
{"id": "7106.png", "formula": "\\begin{align*} \\mathcal { C } \\colon = C ^ * . \\end{align*}"}
{"id": "5671.png", "formula": "\\begin{align*} \\langle e _ 1 \\rangle \\subset \\langle e _ 1 , e _ 2 \\rangle \\subset \\cdots \\subset \\langle e _ 1 , e _ 2 , \\ldots , e _ n \\rangle = \\C ^ n \\end{align*}"}
{"id": "7499.png", "formula": "\\begin{align*} N _ n ( t , u ) = t ^ n N _ n ( 1 / t , u / t ) . \\end{align*}"}
{"id": "8129.png", "formula": "\\begin{align*} \\liminf \\limits _ { t \\rightarrow \\infty } \\| \\chi _ { S _ { v t } } e ^ { - i t H } \\psi \\| = 0 \\end{align*}"}
{"id": "1493.png", "formula": "\\begin{align*} \\mathrm { L i } _ { k , \\lambda } ( t ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - \\lambda ) ^ { n - 1 } ( 1 ) _ { n , 1 / \\lambda } } { ( n - 1 ) ! n ^ { k } } t ^ { n } , ( \\mathrm { s e e } \\ [ 8 ] ) . \\end{align*}"}
{"id": "2128.png", "formula": "\\begin{align*} \\liminf _ { z \\rightarrow \\alpha } \\int _ \\mathbb { R } \\frac { d \\nu ( s ) } { | z - s | ^ 2 } = \\int _ \\mathbb { R } \\frac { d \\nu ( s ) } { ( \\alpha - s ) ^ 2 } = \\lim _ { z \\rightarrow _ { \\sphericalangle } \\alpha } \\int _ \\mathbb { R } \\frac { d \\nu ( s ) } { | z - s | ^ 2 } , \\end{align*}"}
{"id": "4351.png", "formula": "\\begin{align*} \\nu _ { r - 1 } ( a ) = \\ell ' _ { r - 1 } s _ { r - 1 } ( a ) - \\ell _ { r - 1 } u _ { r - 1 } ( a ) \\in \\Z , \\end{align*}"}
{"id": "7060.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } = - y + d x + l \\ , x ^ 2 + m \\ , x \\ , y + n \\ , y ^ 2 , \\\\ \\dot { y } = x + b x \\ , y , \\end{cases} \\end{align*}"}
{"id": "3164.png", "formula": "\\begin{align*} \\mathit { \\Omega } _ { \\mathfrak { m } } \\left ( c _ { - } , c _ { + } \\right ) \\doteq \\left \\{ \\omega \\in \\mathit { M } _ { \\Phi _ { \\mathfrak { m } } ( c _ { + } + c _ { - } ) } : e _ { ( \\cdot ) } \\left ( \\omega \\right ) = c _ { + } + c _ { - } \\left \\vert \\mathfrak { a } \\right \\vert \\right \\} \\subseteq E _ { 1 } \\end{align*}"}
{"id": "2176.png", "formula": "\\begin{align*} g ( x ) = \\int _ { \\mathbb { R } } \\frac { 1 + s ^ { 2 } } { ( x - s ) ^ { 2 } } \\ , d \\rho ( s ) , x \\in \\mathbb { R } . \\end{align*}"}
{"id": "489.png", "formula": "\\begin{align*} \\langle \\alpha _ { i _ k } ^ \\vee , s _ { i _ { k + 1 } } \\cdots s _ { i _ { \\ell - 1 } } s _ { i _ \\ell } \\lambda \\rangle = 1 \\end{align*}"}
{"id": "4962.png", "formula": "\\begin{align*} \\kappa _ 3 = \\sum _ { k = 1 } ^ \\infty \\int _ { 0 } ^ 1 \\left [ ( x + k ) ^ { \\alpha } - k ^ { \\alpha } \\right ] ^ 2 d x , \\end{align*}"}
{"id": "3483.png", "formula": "\\begin{align*} & \\mathrm { D T S } _ { ( p , q ) } ( X ) \\\\ & = \\mathrm { D T V } _ { ( p , q ) } ^ { - 3 / 2 } ( X ) \\bigg \\{ \\sum _ { k = 2 } ^ { 3 } \\binom { 3 } { k } [ - \\mathrm { D T E } _ { ( p , q ) } ( X ) ] ^ { 3 - k } [ \\mu ^ { k } + k \\mu ^ { k - 1 } \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) ] \\\\ & ~ ~ + 2 \\mathrm { D T E } _ { ( p , q ) } ^ { 3 } ( X ) + 3 [ \\mu - \\mathrm { D T E } _ { ( p , q ) } ( X ) ] \\sigma ^ { 2 } \\left ( L _ { 1 } + 1 \\right ) + \\sigma ^ { 3 } \\left ( L _ { 1 } ^ { \\ast } + 2 L _ { 2 } ^ { \\ast } \\right ) \\bigg \\} , \\end{align*}"}
{"id": "8918.png", "formula": "\\begin{align*} \\psi _ n ( x ) = \\lambda _ n ^ { \\frac 1 { p - 2 } } e ^ { - \\gamma \\ , \\lambda _ n ^ { \\frac { 1 } { 2 } } | x - { \\rm v } _ 1 | } . \\end{align*}"}
{"id": "2062.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : L i n f t y T i n f t y } \\frac { \\dd } { \\dd t } T _ { \\infty , t } \\Phi = \\mathcal { L } _ \\infty [ T _ { \\infty , t } \\Phi ] = T _ { \\infty , t } \\mathcal { L } _ \\infty \\Phi . \\end{align*}"}
{"id": "7690.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma _ { k - 1 } } \\sum ^ { m } _ { j = 1 } \\lambda _ { j } ( \\left [ - X ( x _ k ) \\right ] _ { + } ) ^ 2 = \\frac { 1 } { \\sigma _ { k - 1 } } \\norm { \\left [ - X ( x _ k ) \\right ] _ { + } } _ { \\mathrm { F } } ^ 2 \\to 0 \\end{align*}"}
{"id": "4243.png", "formula": "\\begin{align*} \\mathbb { P } _ x \\bigl ( X _ n = o d ( o , X _ m ) \\leq r \\bigr ) & \\leq \\mathbb { P } _ x \\bigl ( \\tau _ \\dagger > n d ( o , X _ m ) \\leq r \\bigr ) \\\\ & \\leq \\left ( 1 - \\frac { c _ 1 ( \\log r ) ^ \\gamma } { r ^ 2 } \\right ) ^ n \\leq \\exp \\left [ - \\frac { c _ 1 ( \\log r ) ^ \\gamma n } { r ^ 2 } \\right ] , \\end{align*}"}
{"id": "6238.png", "formula": "\\begin{align*} k ( x ) = \\left \\{ \\begin{array} { l c } 1 + \\frac { x } { c } , & \\\\ \\\\ x , & \\end{array} \\right . \\end{align*}"}
{"id": "6019.png", "formula": "\\begin{align*} c _ j = \\frac { f ^ { ( j ) } ( \\lambda ) } { j ! } \\end{align*}"}
{"id": "1416.png", "formula": "\\begin{align*} s e p ( \\Lambda _ 1 , \\Lambda _ 2 ) & = \\inf _ { \\| T \\| = 1 } \\| T \\Lambda _ 1 - \\Lambda _ 2 T \\| \\\\ & = \\inf _ { \\| T \\| = 1 } \\| T ( \\Lambda _ 1 - t _ 0 I ) - ( \\Lambda _ 2 - t _ 0 I ) T \\| \\\\ & \\geq \\inf _ { \\| T \\| = 1 } \\left \\{ \\min _ { \\lambda \\in S ( \\Lambda _ 1 ) } | \\lambda - t _ 0 | \\| T \\| - \\max _ { \\mu \\in S ( \\Lambda _ 2 ) } | \\mu - t _ 0 | \\| T \\| \\right \\} \\\\ & = \\min _ { \\lambda \\in S ( \\Lambda _ 1 ) } | \\lambda - t _ 0 | - \\max _ { \\mu \\in S ( \\Lambda _ 2 ) } | \\mu - t _ 0 | . \\end{align*}"}
{"id": "7008.png", "formula": "\\begin{align*} S ( h + ( t _ 1 \\ldots , t _ n ) ) = \\left \\langle ( ( h _ 1 + t _ 1 ) [ D _ 1 ] + \\ldots + ( h _ m + t _ m ) [ D _ m ] ) ^ m , [ X _ { \\Sigma , \\Lambda } ] \\right \\rangle = t _ { i _ 1 } \\ldots t _ { i _ n } \\cdot n ! \\cdot \\left \\langle D _ { i _ 1 } \\ldots D _ { i _ n } , [ X _ { \\Sigma , \\Lambda } ] \\right \\rangle + \\ldots \\end{align*}"}
{"id": "8603.png", "formula": "\\begin{align*} \\begin{bmatrix} P A + A ^ { T } P & P B - A ^ { T } C ^ { T } \\\\ B ^ { T } P - C A & - ( C B + B ^ { T } C ^ { T } ) \\end{bmatrix} \\leq 0 . \\end{align*}"}
{"id": "6494.png", "formula": "\\begin{align*} \\int _ { b _ 1 } ^ { b _ 2 } \\left | v _ { x } \\right | ^ 2 d x = o \\left ( \\lambda ^ { - \\frac { 1 } { 2 } } \\right ) \\int _ { d _ 1 } ^ { d _ 2 } \\left | z _ { x } \\right | ^ 2 d x = o \\left ( \\lambda ^ { - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
{"id": "3002.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\mu - 1 } a _ { i , j } y _ { i , j } = \\sum _ { i = 0 } ^ { \\mu - 1 } a _ i ( x ( E _ j ) ) y ^ { ( A ) } _ i ( E _ j ) = \\sum _ { i = 0 } ^ { \\mu - 1 } ( a _ i y ^ { ( A ) } _ i ) ( E _ j ) = a ( E _ j ) \\ . \\end{align*}"}
{"id": "6782.png", "formula": "\\begin{align*} & \\sum _ { i \\in M _ P - i ^ * } \\sum _ { j \\in N _ i } a _ { i j } x _ { i j } + \\sum _ { \\substack { i j \\in P , \\\\ i \\in M _ P - M _ 0 - i ^ * } } ( b - s ) x _ { i j } + a _ { i ^ * j ^ * } x _ { i ^ * j ^ * } \\\\ + & \\sum _ { j \\in N _ { i ^ * } - j ^ * } a _ { i ^ * j ^ * } \\max \\left \\{ 1 , \\frac { a _ { i ^ * j } } { a _ { i ^ * j ^ * } + b - s } \\right \\} x _ { i ^ * j } \\leq b + ( | M _ P - M _ 0 | - 2 ) ( b - s ) \\end{align*}"}
{"id": "4585.png", "formula": "\\begin{align*} ( C _ { R } ( Q ) ) ^ h = C _ { R [ t ] } ( Q ^ h ) = \\langle x _ 1 , \\ldots , x _ c \\rangle . \\end{align*}"}
{"id": "6410.png", "formula": "\\begin{align*} \\Phi _ 0 ( x ) [ f ] = \\varrho ( x ) [ f ] , \\ ; \\ , \\ ; [ Q , \\Phi _ 0 ( x ) ] = 0 , \\forall x \\in \\mathfrak { g } [ 1 ] , f \\in \\mathcal { O } . \\end{align*}"}
{"id": "2713.png", "formula": "\\begin{align*} \\Theta : \\ell ^ 2 ( \\mathcal { F } _ \\perp ) \\to \\ell ^ 2 ( \\mathcal { F } _ \\perp ) \\qquad ( \\Theta \\Phi ) _ { s , d } = \\Phi _ { s , d - 1 } . \\end{align*}"}
{"id": "6863.png", "formula": "\\begin{align*} a _ 0 + a _ 1 N ( \\beta ( c ^ * ) , c ^ * ) - c ^ * = 0 . \\end{align*}"}
{"id": "7123.png", "formula": "\\begin{align*} { \\mathcal F } ( F _ 6 ) & = [ 0 , 1 , 1 , 2 , 3 , 5 , 0 , 5 , 5 , 2 , 7 , 1 ] , \\\\ { \\mathcal F } ^ 2 ( F _ 6 ) & = [ 0 , 1 , 1 , 4 , 1 , 1 ] , \\\\ { \\mathcal F } ^ 3 ( F _ 6 ) & = [ 0 , 1 , 1 , 0 , 3 , 5 , 0 , 5 , 5 , 0 , 7 , 1 ] , \\\\ { \\mathcal F } ^ 4 ( F _ 6 ) & = [ 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1 , 1 ] \\\\ & = [ 0 , 1 , 1 ] . \\end{align*}"}
{"id": "7666.png", "formula": "\\begin{align*} \\left ( 1 - q \\right ) ^ { - f ( u , v ) } = \\prod _ { i , j } \\left ( 1 - u ^ i v ^ j q \\right ) ^ { - p _ { i j } } \\end{align*}"}
{"id": "2430.png", "formula": "\\begin{align*} S _ 0 H _ \\sigma = q ^ { - \\ell ( \\sigma ) } S _ 0 = H _ \\sigma S _ 0 . \\end{align*}"}
{"id": "5102.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E \\left ( \\exp \\left ( i \\lambda N ^ { n , 1 } _ { \\eta _ n ( t ) } + i \\mu Y ^ { n , 1 } _ { t } \\right ) \\right ) = E \\left ( \\exp \\left ( i \\lambda Z _ { t } \\right ) \\right ) E \\left ( \\exp \\left ( i \\mu Y ^ { \\infty , 1 } _ { t } \\right ) \\right ) , \\end{align*}"}
{"id": "8952.png", "formula": "\\begin{align*} \\beta _ 0 \\cdot D w ( x _ 0 ) = e ^ { - \\kappa \\phi ( x _ 0 ) } [ \\beta _ 0 \\cdot D ( u - u _ 0 ) ( x _ 0 ) - \\kappa \\beta _ 0 \\cdot \\gamma ( x _ 0 ) ( u - u _ 0 ) ( x _ 0 ) ] \\ge 0 , \\end{align*}"}
{"id": "7131.png", "formula": "\\begin{align*} H ^ * ( X _ { D } ) = \\mathbb { Z } [ x _ { a _ 1 b _ 1 } , x _ { a _ 2 b _ 2 } , \\dots , x _ { a _ k b _ k } ] / \\langle p _ 1 , \\dots , p _ k \\rangle , \\end{align*}"}
{"id": "6912.png", "formula": "\\begin{align*} T = \\inf \\Big \\{ t \\in \\Phi : \\widehat { \\mathsf { F D P } } ( t ) \\leq q \\Big \\} , \\end{align*}"}
{"id": "193.png", "formula": "\\begin{align*} \\hat { \\pi } ^ i _ t ( m ^ i ) \\triangleq \\P ^ { g } ( M ^ i = m ^ i | x ^ i _ { 1 : t } , y _ { 1 : t } ) , i = 1 , 2 . \\end{align*}"}
{"id": "1001.png", "formula": "\\begin{align*} \\frac { \\P ( S _ { N + 1 } \\in \\cdot \\setminus \\{ 0 , 1 , \\ldots , N \\} \\ | \\ S _ N ) } { \\P ( S _ { N + 1 } \\notin \\{ 0 , 1 , \\ldots , N \\} \\ | \\ S _ N ) } = \\delta _ { N + 1 } ( \\cdot ) \\end{align*}"}
{"id": "4238.png", "formula": "\\begin{align*} P ( u , v ) = \\frac { c ( u , v ) } { c ( u ) + K ( u ) } P ( u , \\dagger ) = \\frac { K ( u ) } { c ( u ) + K ( u ) } , P ( \\dagger , \\dagger ) = 1 , \\end{align*}"}
{"id": "3840.png", "formula": "\\begin{align*} \\omega f ( x ) = f ( \\omega ^ { - 1 } x ) , \\omega \\in W . \\end{align*}"}
{"id": "8022.png", "formula": "\\begin{align*} \\tilde { D } _ i = ( D _ i , 0 ) , i \\in \\{ 1 , \\ldots , m \\} ; \\end{align*}"}
{"id": "337.png", "formula": "\\begin{align*} s ( g ) = \\min \\{ j \\in \\N \\ : \\ I ( g ) _ j \\ne 0 \\} \\leq d - 1 . \\end{align*}"}
{"id": "842.png", "formula": "\\begin{align*} \\boldsymbol { x } _ { A } ^ { p o s t } = \\boldsymbol { V } _ { A } ^ { p o s t } \\left ( ( \\boldsymbol { V } _ { A } ^ { p r i } ) ^ { - 1 } \\boldsymbol { x } _ { A } ^ { p r i } + \\frac { \\mathbf { F } ^ { H } \\boldsymbol { y } } { \\sigma _ { n } ^ { 2 } } \\right ) \\end{align*}"}
{"id": "1583.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial z ^ i _ { \\epsilon } } v ^ i = 0 , \\end{align*}"}
{"id": "1615.png", "formula": "\\begin{align*} g ^ { - 1 } = ( g ^ { i j } ) = \\frac { 1 } { \\left [ a ( y ^ 1 ) ^ 2 + \\tilde { b } ^ ( y ^ 2 ) ^ 2 \\right ] ^ 3 } \\begin{pmatrix} ( y ^ 1 ) ^ 4 \\left \\lbrace a ( y ^ 1 ) ^ 2 + 3 \\tilde { b } ( y ^ 2 ) ^ 2 \\right \\rbrace & 2 \\tilde { b } ( y ^ 1 ) ^ 3 ( y ^ 2 ) ^ 3 \\\\ 2 \\tilde { b } ( y ^ 1 ) ^ 3 ( y ^ 2 ) ^ 3 & \\frac { ( y ^ 1 ) ^ 2 } { 2 \\tilde { b } } \\left \\lbrace a ^ 2 ( y ^ 1 ) ^ 4 + 3 \\tilde { b } ^ 2 ( y ^ 2 ) ^ 4 \\right \\rbrace \\\\ \\end{pmatrix} . \\end{align*}"}
{"id": "2980.png", "formula": "\\begin{align*} \\phi _ { 2 , m } ( y , s ) = \\begin{cases} s y ^ { s - 1 } & m = 0 , \\\\ s m ^ { - 1 / 2 } ( 2 \\pi y ) ^ { - 1 } \\frac { \\Gamma ( s ) } { \\Gamma ( 2 s ) } M _ { 1 , s - \\frac 1 2 } ( 4 \\pi m y ) & m > 0 . \\end{cases} \\end{align*}"}
{"id": "6068.png", "formula": "\\begin{align*} f ^ { ( j ) } ( x ) & = \\sum _ { k = 0 } ^ j \\binom { j } { k } ( p ) _ k ( x - \\lambda ) ^ { p - k } g ^ { ( j - k ) } ( x ) \\\\ & = \\sum _ { k = 0 } ^ p \\binom { j } { k } ( p ) _ k ( x - \\lambda ) ^ { p - k } g ^ { ( j - k ) } ( x ) \\\\ & = ( x - \\lambda ) \\sum _ { k = 0 } ^ { p - 1 } \\binom { j } { k } ( p ) _ k ( x - \\lambda ) ^ { p - 1 - k } g ^ { ( j - k ) } ( x ) + \\binom { j } { p } p ! g ^ { ( j - p ) } ( x ) \\end{align*}"}
{"id": "5945.png", "formula": "\\begin{align*} w _ { \\xi } ( \\eta _ { 1 } , . . . , \\eta _ { d } ) = \\left ( { b + c } \\right ) \\mathop \\sum \\limits _ { k = 1 } ^ d \\left ( { \\frac { { d + 1 } } { 2 } - k } \\right ) { \\eta _ k } + \\frac { 1 } { 2 } \\mathop \\sum \\limits _ { k , p = 1 } ^ d \\left ( { \\left ( { b + c } \\right ) { \\delta _ { k p } } - a } \\right ) { \\eta _ k } { \\eta _ p } \\ , \\end{align*}"}
{"id": "5594.png", "formula": "\\begin{align*} \\bar { \\tau } ^ { M , 0 } = \\tau ^ { M , 0 } \\otimes I _ 3 \\in \\R ^ { 3 n \\times 3 n } . \\end{align*}"}
{"id": "178.png", "formula": "\\begin{align*} \\mathcal { X } _ R ( x ) : = \\mathcal { X } \\left ( \\frac { x } { R } \\right ) \\qquad R > 0 , \\end{align*}"}
{"id": "5400.png", "formula": "\\begin{align*} B _ q ( v , \\phi ) = F ( \\phi ) \\quad v - f \\in \\Tilde { H } ^ s ( \\Omega ) \\end{align*}"}
{"id": "852.png", "formula": "\\begin{align*} \\mathcal { P } _ { 1 } : & \\underset { \\left \\{ \\mathbf { V } _ { 2 , p } \\right \\} _ { p = 1 } ^ { P _ { 2 } } } { \\max } \\lambda \\\\ & \\textrm { t r } \\left ( \\mathbf { V } _ { 2 , p } \\right ) \\leq P _ { t } , \\\\ & \\mathbf { J } _ { \\textrm { e f f } } \\left ( \\left \\{ \\mathbf { V } _ { 2 , p } \\right \\} _ { p = 1 } ^ { P _ { 2 } } \\right ) \\succeq \\lambda \\mathbf { I } , \\\\ & \\textrm { r a n k } \\left ( \\mathbf { V } _ { 2 , p } \\right ) = 1 , p = 1 , 2 \\ldots , P _ { 2 } . \\end{align*}"}
{"id": "7721.png", "formula": "\\begin{align*} A _ { \\bar { S } } ( k ) e = e - A _ { \\bar { S } S } ( k ) e \\end{align*}"}
{"id": "3730.png", "formula": "\\begin{align*} K _ g ^ 2 : = | E \\cdot \\omega | ^ 2 + | B \\cdot \\omega | ^ 2 + | E - \\omega \\times B | ^ 2 + | B + \\omega \\times E | ^ 2 . \\end{align*}"}
{"id": "2982.png", "formula": "\\begin{align*} T _ m ( d ) = \\sum _ { \\substack { Q \\in \\Gamma _ \\infty \\backslash \\mathcal Q _ { d ^ 2 } \\\\ Q = [ a , b , c ] , a > 0 } } \\chi _ d ( Q ) \\int _ { C _ Q } e ( m x ) \\phi _ { 2 , m } ( y , s ) \\ , d z . \\end{align*}"}
{"id": "3573.png", "formula": "\\begin{align*} m _ { \\pi } & = \\frac { 1 } { 2 } ( \\dim \\pi - \\chi _ { \\pi } ( h _ 1 ) ) \\\\ & = \\frac { 1 } { 2 } \\left ( [ n ] _ q ! - [ n - 1 ] _ q ! \\left ( \\sum _ { i = 1 } ^ n \\chi _ i ( - 1 ) \\right ) \\right ) \\\\ & = \\frac { 1 } { 2 } [ n - 1 ] _ q ! \\left ( T _ { n - 1 } - \\sum _ { i = 1 } ^ n \\chi _ i ( - 1 ) \\right ) . \\end{align*}"}
{"id": "4227.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { 1 \\vee \\log \\phi ^ { - 1 } ( a ( k ) ) } = \\infty . \\end{align*}"}
{"id": "2016.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n v l a s o v - p d e } \\partial _ t f _ t ( x ) = - \\nabla _ x \\cdot \\{ b ( x , f _ t ) f _ t \\} + \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ d \\partial _ { x _ i } \\partial _ { x _ j } \\{ a _ { i j } ( x , f _ t ) f _ t \\} \\end{align*}"}
{"id": "2085.png", "formula": "\\begin{align*} \\widetilde { N _ H } ( n ) = C a r d ( \\{ v : | v | = n , u \\not \\sqsubset v w _ 1 v u \\in \\mathcal { L } _ H \\} ) . \\end{align*}"}
{"id": "6809.png", "formula": "\\begin{align*} \\begin{aligned} H ^ i _ a ( 0 , 1 ) : = \\{ u \\in H ^ { i - 1 } ( 0 , 1 ) : & \\ ; u ^ { ( i - 1 ) } \\\\ & [ 0 , 1 ] \\setminus \\{ x _ 0 \\} \\sqrt { a } u ^ { ( i ) } \\in L ^ 2 ( 0 , 1 ) \\} , \\end{aligned} \\end{align*}"}
{"id": "6533.png", "formula": "\\begin{align*} \\P \\{ m \\} = \\P \\{ Y _ m = 0 \\} = \\prod \\limits _ { i = 0 } ^ { m - 1 } \\P \\{ \\ell ^ { ( A ) } _ i \\leq m - i \\} = \\prod \\limits _ { i = 0 } ^ { m - 1 } \\left ( 1 - \\P \\{ \\ell ^ { ( A ) } _ i > m - i \\} . \\right ) \\end{align*}"}
{"id": "1963.png", "formula": "\\begin{align*} ( A _ { 0 } , A _ { 1 } , \\ldots , A _ { p - 1 } ) = \\cfrac { \\mathbf { 1 } } { \\mathbf { d } _ 1 + \\cfrac { \\mathbf { 1 } } { \\mathbf { d } _ 2 + \\cfrac { \\mathbf { 1 } } { \\mathbf { d } _ 3 + \\mathbf { v } _ 3 } } } . \\end{align*}"}
{"id": "7704.png", "formula": "\\begin{align*} \\sum _ { n _ 1 , \\ldots , n _ k \\le x } f ( [ n _ 1 , \\ldots , n _ k ] ) = C _ { f , k } \\frac { x ^ { k ( r + 1 ) } } { ( r + 1 ) ^ k } + O \\big ( x ^ { k ( r + 1 ) - \\frac 1 { 2 } + \\varepsilon } \\big ) , \\end{align*}"}
{"id": "5597.png", "formula": "\\begin{align*} \\Delta \\left ( a \\right ) & = \\int _ 0 ^ { \\infty } \\frac { \\ln \\left ( \\left | a \\right | - i x \\right ) } { \\cosh \\left ( \\pi x \\right ) } \\ , \\mathrm { d } x + \\int _ 0 ^ { \\infty } \\frac { \\ln \\left ( \\left | a \\right | + i x \\right ) } { \\cosh \\left ( \\pi x \\right ) } \\ , \\mathrm { d } x \\end{align*}"}
{"id": "2244.png", "formula": "\\begin{align*} & ( u _ p ^ 0 , h _ p ^ 0 ) ( x , 0 ) = ( - \\delta , - \\sigma ) , ( u _ p ^ 0 , h _ p ^ 0 ) ( x , \\infty ) = ( 0 , 0 ) , \\\\ & ( u _ p ^ 0 , h _ p ^ 0 ) ( 1 , y ) = ( u _ 0 ^ 0 , h _ 0 ^ 0 ) ( y ) . \\end{align*}"}
{"id": "8651.png", "formula": "\\begin{align*} \\frac { d \\beta } { d t } ( t ) = \\beta ^ 2 ( t ) + K ( t ) , \\frac { d \\alpha } { d t } ( t ) + 2 \\beta ( t ) = 0 . \\end{align*}"}
{"id": "6371.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\ ! \\ ! \\ ! x ^ { m } \\ , p \\ , ( n , x ) \\ , d x = - m \\frac { p ^ { ( n - 2 ) } ( 0 ) } { p ^ { ( n - 1 ) } ( 0 ) } \\int _ 0 ^ { \\infty } \\ ! \\ ! \\ ! x ^ { m - 1 } \\ , p \\ , ( n - 1 , x ) \\ , d x \\ \\ , \\end{align*}"}
{"id": "2561.png", "formula": "\\begin{align*} Q ( p _ 1 , q _ 1 ) \\otimes Q ( p _ 2 , q _ 2 ) = \\bigoplus _ { n = 0 } ^ { \\min ( p _ 1 , q _ 2 ) } \\bigoplus _ { m = 0 } ^ { \\min ( p _ 2 , q _ 1 ) } Q ( p _ 1 - n , p _ 2 - m ; q _ 1 - m , q _ 2 - n ) \\ , \\end{align*}"}
{"id": "4848.png", "formula": "\\begin{align*} & \\{ f , \\{ g , h \\} \\} + \\{ g , \\{ h , f \\} \\} + \\{ h , \\{ f , g \\} \\} = \\\\ & ( X , \\delta _ X D f \\wedge \\delta _ X ( D g , \\wedge \\delta _ X D h ) ) + \\circlearrowright \\\\ + & ( X , \\delta _ X D f \\wedge \\delta _ X ( O _ { X , D g } D h - O _ { X , D h } D g ) ) + \\circlearrowright \\\\ + & ( X , \\delta _ X f \\wedge \\delta _ X ( D ^ 2 g ( \\nu _ { X } D f , \\cdot ) - D ^ 2 f ( \\nu _ { X } D g , \\cdot ) ) ) + \\circlearrowright , \\end{align*}"}
{"id": "763.png", "formula": "\\begin{align*} w _ m ^ * = \\frac { | g _ m | ^ 2 } { \\sum _ { k = 1 } ^ { M } | g _ k | ^ 2 } , m = 1 , 2 , \\ldots , M . \\end{align*}"}
{"id": "4198.png", "formula": "\\begin{align*} \\norm { g _ { \\le \\iota } ^ { ( 2 ) } } _ 2 = \\bigg \\Vert \\sum _ { \\ell = - 1 } ^ \\iota \\sum _ { m = 1 } ^ { M _ { \\ell } } ( 1 - \\chi _ { m } ^ { ( \\ell ) } ) g _ { m } ^ { ( \\ell ) } \\bigg \\Vert _ 2 \\le \\sum _ { \\ell = - 1 } ^ \\iota \\sum _ { m = 1 } ^ { M _ { \\ell } } \\Vert g _ { m } ^ { ( \\ell ) } \\Vert _ 2 . \\end{align*}"}
{"id": "5887.png", "formula": "\\begin{align*} \\mu _ { V _ { n , k } X ^ { ( n ) } } ( A ) : = \\mathbb { P } ( V _ { n , k } X ^ { ( n ) } \\in A ) , \\end{align*}"}
{"id": "2418.png", "formula": "\\begin{align*} R _ 2 ^ { ( n ) } : & = \\sup _ { 0 \\le k , l \\le \\lfloor n ^ \\mu \\rfloor } \\sup _ { \\{ t : | t - t _ k ^ { ( \\mu ) } | \\le T / n ^ \\mu \\} } \\sup _ { \\{ x : | x - x _ l ^ { ( \\mu ) } | \\le \\pi / n ^ \\mu \\} } | u ^ n ( t _ k ^ { ( \\mu ) } , x _ l ^ { ( \\mu ) } ) - u ^ n ( t , x ) | ^ { 2 p } , \\\\ R _ 3 ^ { ( n ) } : & = \\sup _ { 0 \\le k , l \\le \\lfloor n ^ \\mu \\rfloor } \\sup _ { \\{ t : | t - t _ k ^ { ( \\mu ) } | \\le T / n ^ \\mu \\} } \\sup _ { \\{ x : | x - x _ l ^ { ( \\mu ) } | \\le \\pi / n ^ \\mu \\} } | u ( t _ k ^ { ( \\mu ) } , x _ l ^ { ( \\mu ) } ) - u ( t , x ) | ^ { 2 p } . \\end{align*}"}
{"id": "875.png", "formula": "\\begin{align*} s * ^ { \\alpha _ p } s _ { a _ p } * ^ { \\alpha _ { p - 1 } } s _ { a _ { p - 1 } } * ^ { \\alpha _ { p - 2 } } \\cdots * ^ { \\alpha _ 1 } s _ { a _ 1 } * ^ { - \\beta _ 1 } s _ { b _ 1 } * ^ { - \\beta _ 2 } s _ { b _ 2 } * ^ { - \\beta _ 3 } \\cdots * ^ { - \\beta _ q } s _ { b _ q } = t . \\end{align*}"}
{"id": "5054.png", "formula": "\\begin{align*} N ^ { n , 1 } _ \\tau = N ^ { n , 3 } _ \\tau + N ^ { n , 4 } _ \\tau , \\end{align*}"}
{"id": "8817.png", "formula": "\\begin{align*} \\psi ( x ' ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha _ { t _ 1 , t _ 2 } ( x ' ) d t _ 1 d t _ 2 \\end{align*}"}
{"id": "8990.png", "formula": "\\begin{align*} ( j + 1 ) _ m = \\sum _ { k = 0 } ^ m ( - 1 ) ^ k P _ k \\left ( m , x \\right ) \\left ( j + x \\right ) ^ { m - k } , \\end{align*}"}
{"id": "1502.png", "formula": "\\begin{align*} ( 1 - \\lambda ) _ { n - k , \\lambda } = ( 1 ) _ { n - k , \\lambda } \\big ( 1 - ( n - k ) \\lambda \\big ) . \\end{align*}"}
{"id": "471.png", "formula": "\\begin{align*} L _ K ( G ) : = \\mathop { \\oplus } \\limits _ { n = 1 } ^ \\infty ( \\gamma _ n ( G ) / \\gamma _ { n + 1 } ( G ) ) \\otimes _ { \\Z } K . \\end{align*}"}
{"id": "110.png", "formula": "\\begin{align*} \\chi ^ { \\epsilon , L } _ t ( \\lambda , \\mu , m ^ 2 ) : = \\epsilon ^ d \\sum _ { x \\in \\Lambda _ { \\epsilon , L } } \\big < \\varphi _ 0 \\varphi _ x \\big > ^ { \\epsilon , L } _ { \\lambda , \\mu + 1 / t , m ^ 2 } . \\end{align*}"}
{"id": "1986.png", "formula": "\\begin{align*} \\mathbf { T ^ 1 _ { k } } = \\int _ { B _ { 1 / 2 } } \\nabla ( U _ 0 - V ) \\cdot \\nabla ( ( 2 - \\eta _ r ) ( U _ k - U _ 0 ) ) \\ , d x & = \\int _ { B _ { 1 / 2 } } ( 2 - \\eta _ r ) \\nabla ( U _ 0 - V ) \\cdot \\nabla ( U _ k - U _ 0 ) \\ , d x \\\\ & - \\int _ { B _ { 1 / 2 } } \\big ( ( U _ k - U _ 0 ) \\nabla ( U _ 0 - V ) \\cdot \\nabla \\eta _ r \\big ) \\ , d x . \\end{align*}"}
{"id": "1447.png", "formula": "\\begin{align*} & \\psi _ { { i , s } } = \\psi _ { i , 0 } \\circ ( \\theta _ t + \\gamma _ 1 ) \\circ \\cdots \\circ ( \\theta _ t + \\gamma _ s ) \\enspace , \\\\ & \\psi _ { i , 0 } \\circ \\mathcal { T } _ { \\bold { c } } = [ \\alpha _ i ] \\circ { \\rm { E v a l } } _ { \\alpha _ i } \\ \\ 1 \\le i \\le m \\enspace . \\end{align*}"}
{"id": "7554.png", "formula": "\\begin{align*} [ x , y , z ] _ { \\mathfrak { g } _ 0 } & = l _ 3 ( x , y , z ) , \\\\ [ \\alpha , \\beta , \\gamma ] _ { \\mathfrak { g } _ 1 } & = l _ 3 ( d ( \\alpha ) , d ( \\beta ) , \\gamma ) = l _ 3 ( d ( \\alpha ) , \\beta , d ( \\gamma ) ) = l _ 3 ( \\alpha , d ( \\beta ) , d ( \\gamma ) ) . \\end{align*}"}
{"id": "2741.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\Phi _ { i } ( x , R ) = 0 . \\end{align*}"}
{"id": "2054.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n v l a s o v c o n t i n u i t y } \\partial _ t f ^ N _ t = - \\nabla \\cdot \\left ( \\left ( \\mathbf { b } ^ N - \\frac { \\sigma ^ 2 } { 2 } \\nabla \\log f ^ N _ t \\right ) f ^ N _ t \\right ) . \\end{align*}"}
{"id": "3995.png", "formula": "\\begin{align*} ( C _ i \\beta _ i ) \\cap ( C _ i \\beta _ i ) ^ { \\bot _ { A _ i } } = \\begin{cases} C _ i \\beta _ i , & \\mbox { i f } C _ i \\beta _ i \\overline { C _ i \\beta _ i } = 0 ; \\\\ 0 , & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
{"id": "3146.png", "formula": "\\begin{align*} \\mathcal { E } ( E _ { 1 } ) = \\left \\{ \\hat { \\rho } \\in E _ { 1 } : \\hat { \\rho } \\right \\} = \\bigcap \\limits _ { A \\in \\mathcal { U } } \\left \\{ \\hat { \\rho } \\in E _ { 1 } : \\lim \\limits _ { L \\rightarrow \\infty } \\hat { \\rho } ( \\left \\vert A _ { L } \\right \\vert ^ { 2 } ) = | \\hat { \\rho } ( A ) | ^ { 2 } \\right \\} \\ . \\end{align*}"}
{"id": "7583.png", "formula": "\\begin{align*} \\frac { d \\Omega } { d t } = \\operatorname { d e x p } ^ { - 1 } _ { \\Omega ( t ) } ( A ( t ) ) , \\qquad \\Omega ( 0 ) = { \\bf 0 } \\ , , \\end{align*}"}
{"id": "2443.png", "formula": "\\begin{align*} L _ f S _ \\zeta & = M _ f S _ \\zeta + \\sum _ { g \\prec f } \\ell _ { g f } ( q ) M _ g S _ \\zeta \\\\ & = \\widetilde { N } _ f + \\sum _ { g \\prec f ; g \\in \\Z ^ { m | n } } \\ell _ { g f } ( q ) q ^ { - \\ell ( \\tau _ g ) } \\widetilde { N } _ { g \\cdot \\tau ( g ) } , \\end{align*}"}
{"id": "6229.png", "formula": "\\begin{align*} & \\int ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty h ( x / q ) \\Big ( \\frac { 1 } { q } D _ { q ^ { - 1 } } u ( x ) + \\frac { 1 } { q } u ( x ) u ( x / q ) - \\frac { q ^ { - n } x } { 1 - q } u ( x / q ) + \\frac { q ^ { 1 - n } [ n ] _ q } { ( 1 - q ) } \\Big ) y ( x ) d _ q x \\\\ & = ( x ^ 2 ; q ^ 2 ) _ \\infty h ( x / q ) \\Big ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \\Big ) . \\end{align*}"}
{"id": "8273.png", "formula": "\\begin{align*} \\tanh ( k ' a / 2 ) = 0 , \\end{align*}"}
{"id": "8192.png", "formula": "\\begin{align*} \\hat { H } = - \\frac { \\hslash ^ { 2 } } { 2 m } \\partial _ { x } ^ { 2 } + \\frac { \\beta \\hslash ^ { 4 } } { 3 m } \\partial _ { x } ^ { 4 } \\end{align*}"}
{"id": "581.png", "formula": "\\begin{align*} M _ 3 = f _ k ( j ) f _ k ( j + 1 ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) \\tau _ 3 ( j + 1 ) } { \\sqrt { j - i - 1 } } \\Big ) . \\end{align*}"}
{"id": "731.png", "formula": "\\begin{align*} | \\nabla u | _ g ^ 2 = v ^ 2 - 1 \\ge v ^ 2 - \\frac { v ^ 2 } { 4 } = \\frac { 3 } { 4 } v ^ 2 . \\end{align*}"}
{"id": "3233.png", "formula": "\\begin{align*} m ( 2 m + 1 ) ( g - 1 ) + m ^ 2 n & = \\dim ( \\mathcal { U } ) \\\\ & = \\dim ( \\mathcal { U } ' ) \\\\ & = m ' ( 2 m ' + 1 ) ( g ' - 1 ) + m '^ 2 n ' , \\end{align*}"}
{"id": "24.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { M , n } & = \\Phi _ M ( \\beta ) ( \\xi ) - \\xi \\\\ & = t _ { M | L } ( \\Phi _ L ( \\beta ) ) ( \\xi ) - \\xi \\\\ & = \\prod _ i h _ i ( \\xi ) - \\xi \\\\ & = \\sum _ i ( h _ i ( \\xi ) - \\xi ) \\end{align*}"}
{"id": "5246.png", "formula": "\\begin{align*} \\Gamma _ P : = \\bigsqcup _ { j = 1 } ^ h \\Gamma _ { P , j } : = \\bigsqcup _ { j = 1 } ^ h \\Gamma _ { 0 , k _ 1 ( j ) , k _ 2 ( j ) , 1 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I _ j } } . \\end{align*}"}
{"id": "8502.png", "formula": "\\begin{align*} \\left < \\hat { H } , H \\right > & = \\left < \\sum _ { A \\in \\left [ b \\right ] ^ d } \\hat { w } _ A h _ { d , b , A } , \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { B \\in \\left [ b \\right ] ^ d } h _ { d , b , B } \\ 1 \\left ( X _ i \\in \\Lambda _ { d , b , B } \\right ) \\right > \\\\ & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { A \\in \\left [ b \\right ] ^ d } \\hat { w } _ A \\ 1 \\left ( X _ i \\in \\Lambda _ B \\right ) b ^ d = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\hat { H } ( X _ i ) . \\end{align*}"}
{"id": "4312.png", "formula": "\\begin{align*} L _ + = \\R ^ n , L _ - = ( e ^ { i \\phi _ 1 } , \\ldots e ^ { i \\phi _ n } ) \\R ^ n , 0 < \\phi _ i < \\pi , T X = \\R ^ n \\otimes \\C , \\end{align*}"}
{"id": "5618.png", "formula": "\\begin{align*} S = \\{ f \\in D [ X ] \\mid ( A _ f ) _ v = D \\} . \\end{align*}"}
{"id": "4892.png", "formula": "\\begin{gather*} \\bigwedge _ { i = 1 } ^ { n - 2 } \\neg c _ i \\lor c _ { i + 1 } \\\\ \\bigwedge _ { i = 1 } ^ n c _ { i - 1 } \\lor \\neg a _ i \\lor b _ i \\qquad \\bigwedge _ { i = 1 } ^ n c _ { i - 1 } \\lor a _ i \\lor b _ i \\lor \\neg c _ i \\\\ \\bigwedge _ { i = 1 } ^ n c _ { i - 1 } \\lor \\neg a _ i \\lor \\neg b _ i \\lor \\neg c _ i \\qquad \\bigwedge _ { i = 1 } ^ n c _ { i - 1 } \\lor a _ i \\lor \\neg b _ i \\lor c _ i \\end{gather*}"}
{"id": "4944.png", "formula": "\\begin{align*} \\sigma _ { k } ^ { i i } = \\frac { C _ n ^ k \\times k } { C _ n ^ { k - 1 } } \\sigma _ { k - 1 } \\ge C ( n , k ) \\sigma _ { k } ^ { \\frac { k - 1 } { k } } . \\end{align*}"}
{"id": "6573.png", "formula": "\\begin{align*} \\phi ( \\beta ( g ^ { - 1 } , \\psi ^ { - 1 } ( \\alpha ( g , x ) , v ) ) ) = ( x , A ( g , x , v ) ) \\mbox { f o r a l l $ \\ , g \\in U $ , $ x \\in V $ , a n d $ v \\in F $ , } \\end{align*}"}
{"id": "802.png", "formula": "\\begin{align*} h _ { \\psi } ( r ^ m ) + R ^ N ( r ) + h _ { \\psi } ( - 1 ) = 0 , \\end{align*}"}
{"id": "7501.png", "formula": "\\begin{align*} \\begin{cases} u _ t + ( - \\Delta ) ^ { d } u = | x | ^ { \\alpha } | u | ^ { p } + \\zeta ( t ) { \\mathbf w } ( x ) \\ \\quad \\mbox { f o r } ( x , t ) \\in \\mathbb { R } ^ { N } \\times ( 0 , \\infty ) , \\\\ u ( x , 0 ) = u _ 0 ( x ) , x \\in \\mathbb { R } ^ { N } , \\end{cases} \\end{align*}"}
{"id": "314.png", "formula": "\\begin{align*} f ( A ) & = f ( 2 ^ K ) + \\sum _ { p \\in A } f ( p ) + \\sum _ { p \\notin A } f ( A _ { p } ) \\\\ & \\le f ( 2 ^ K ) + \\sum _ { \\substack { p > 2 \\\\ p \\in A } } f ( p ) + \\sum _ { \\substack { p > 2 \\\\ p \\notin A } } b _ p f ( p ) + \\sum _ { \\substack { p > 2 \\\\ p \\notin A } } \\sum _ { k = 1 } ^ { K - 1 } f ( ( A ^ k ) _ { 2 ^ k p } ) , \\end{align*}"}
{"id": "3238.png", "formula": "\\begin{align*} \\lim _ { \\kappa \\to 0 } \\kappa ^ p \\nu _ p ( \\{ ( a , r ) \\in \\omega \\times \\R _ + : \\ c _ d ' r | \\nabla f ( a ) | \\pm C A r ^ 2 > \\kappa \\} = c _ { d , p } \\int _ { \\omega } | \\nabla f ( a ) | ^ p \\ , d a \\ , . \\end{align*}"}
{"id": "3407.png", "formula": "\\begin{align*} ( g , n ) \\cdot ( g ' , n ' ) = ( g g ' , n + n ' + \\sigma ( g , g ' ) ) . \\end{align*}"}
{"id": "4800.png", "formula": "\\begin{align*} \\mathcal { N } = \\phi ^ { - 1 } \\big ( \\mathbb { A } \\big ( \\rho ( \\tfrac 1 3 \\sigma ^ * ) \\big ) \\big ) , \\end{align*}"}
{"id": "5487.png", "formula": "\\begin{align*} \\int _ { N _ d ( K ) } H ( x , y , t ) d \\mathrm { v o l } ( y ) \\lesssim \\sum _ { \\rho = 1 } ^ \\infty ( 1 + \\rho ^ n ) e ^ { - \\frac { \\rho ^ 2 } { 4 t } - \\frac { ( n - 1 ) ^ 2 } { 4 } t - \\frac { n - 1 } { 2 } \\rho } \\mathrm { v o l } ( \\mathrm { A n } ( \\rho ) \\cap N _ d ( K ) ) , \\end{align*}"}
{"id": "2137.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\frac { 1 + s ^ { 2 } } { ( x - s ) ^ { 2 } } \\ , d \\sigma _ { \\mu } ( s ) \\geq f ( | [ x ] | + 1 ) \\int _ { [ [ x ] , [ x ] + 1 ) } \\frac { d \\lambda ( s ) } { ( x - s ) ^ { 2 } } = + \\infty \\end{align*}"}
{"id": "5520.png", "formula": "\\begin{align*} f ' ( A ( z ) ) = B ' ( g ( z ) ) g ' ( z ) \\left ( A ^ { - 1 } \\right ) ' ( A ( z ) ) . \\end{align*}"}
{"id": "55.png", "formula": "\\begin{align*} \\rho _ { { v } ^ { - 1 } } \\circ \\theta ( \\omega ' ) = \\rho _ { { v } ^ { - 1 } } ( \\omega ) = \\Phi _ K ( w ) ( \\omega ) & = \\Phi _ K ( u ) ( \\theta ( \\omega ' ) ) \\\\ & = \\theta ( \\Phi _ K ( u ) ( \\omega ' ) ) \\\\ & = \\theta \\circ [ u ^ { - 1 } ] _ f ( \\omega ' ) . \\end{align*}"}
{"id": "8568.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty a _ n \\phi _ n ( x ) . \\end{align*}"}
{"id": "7788.png", "formula": "\\begin{align*} [ e _ \\lambda , e _ \\lambda ^ \\vee ] = h , \\ [ h , e _ \\lambda ^ \\vee ] = - 2 e _ \\lambda ^ \\vee . \\end{align*}"}
{"id": "1268.png", "formula": "\\begin{align*} \\mathcal { E } ( \\mathcal { B } _ { 2 , k } ) & = \\begin{cases} 2 \\left ( \\cot \\left ( \\dfrac { \\pi } { 2 k + 2 } \\right ) - 1 \\right ) , & 2 \\nmid k , \\\\ 2 \\left ( \\csc \\left ( \\dfrac { \\pi } { 2 k + 2 } \\right ) - 1 \\right ) , & 2 \\mid k . \\end{cases} \\end{align*}"}
{"id": "6868.png", "formula": "\\begin{align*} \\Phi ( \\lambda ) + \\lambda ( 1 - \\Phi ( \\lambda ) ) \\frac { \\Phi ( \\lambda ) } { { \\phi ( \\lambda ) } } & \\geq \\Phi ( \\lambda ) + \\lambda ( 1 - \\Phi ( \\lambda ) ) \\frac { \\lambda } { 1 + \\lambda ^ 2 } \\\\ & = \\frac { \\lambda ^ 2 } { 1 + \\lambda ^ 2 } + \\Phi ( \\lambda ) \\frac { 1 } { \\lambda ^ 2 + 1 } \\geq \\frac { \\lambda ^ 2 } { 1 + \\lambda ^ 2 } \\geq \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "988.png", "formula": "\\begin{align*} d ( L ) = | \\operatorname { d e t } ( B ) | . \\end{align*}"}
{"id": "6172.png", "formula": "\\begin{align*} ( M ^ { \\prime } , \\alpha ^ { \\prime } ) = ( M - \\overset { \\circ } { \\mathcal { N } } ( { \\mathsf { S _ { \\mathbf { p } } ^ { 1 } } } ) , \\alpha ) \\underset { \\Psi } { \\cup } ( { \\mathbf { C } } \\times { \\mathbf { S } ^ { 3 } } , \\delta _ { a } ) . \\end{align*}"}
{"id": "5465.png", "formula": "\\begin{align*} ( b _ 1 b _ 2 ) \\star a = l ( ( b _ 1 b _ 2 ) a ) = & l ( b _ 1 ( b _ 2 a ) ) \\stackrel { ( L 3 ) } = l ( b _ 1 l ( b _ 2 a ) ) \\\\ = & l ( b _ 1 ( b _ 2 \\star a ) ) = b _ 1 \\star ( b _ 2 \\star a ) , \\end{align*}"}
{"id": "766.png", "formula": "\\begin{align*} P _ { \\rm F A } ^ { } ( \\lambda ) = & { \\rm P r } \\left ( \\left . \\frac { \\sum _ { m = 1 } ^ { M } ( T _ { { m } } - \\alpha ) T _ { { m } } } { \\sum _ { k = 1 } ^ { M } ( T _ { k } - \\alpha ) } > \\lambda \\right | \\mathcal { H } _ 0 \\right ) \\end{align*}"}
{"id": "3681.png", "formula": "\\begin{align*} \\textstyle r \\leq \\epsilon _ 0 : = e ^ { - \\frac { 1 } { 2 \\delta } } \\frac { R } { 2 } , \\end{align*}"}
{"id": "8092.png", "formula": "\\begin{align*} X = p \\partial _ { 1 } + q \\partial _ { 2 } . \\end{align*}"}
{"id": "681.png", "formula": "\\begin{align*} d \\nu _ * : = ( 4 \\pi ) ^ { - n / 2 } e ^ { - f _ o } \\ , d g _ o . \\end{align*}"}
{"id": "5454.png", "formula": "\\begin{align*} a q ( r ^ B _ A ( k ) ) ^ { - 1 } r ^ B _ A ( k ) \\stackrel { ( \\ref { s p l } ) } = a k = a k ' \\stackrel { ( \\ref { s p l } ) } = a q ( r ^ B _ A ( k ' ) ) ^ { - 1 } r ^ B _ A ( k ' ) . \\end{align*}"}
{"id": "2101.png", "formula": "\\begin{align*} ( L ( x ) , x ) _ V = - \\sum _ { e \\in E } \\frac { 1 } { 2 } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum _ { u , v \\in e } ( x ( u ) - x ( v ) ) ^ 2 \\le 0 . \\end{align*}"}
{"id": "1888.png", "formula": "\\begin{align*} z A _ { 0 } ( z ) - 1 = \\sum _ { j = 0 } ^ { i } a _ { 0 } ^ { ( j ) } A _ { j } ( z ) + \\sum _ { j = i + 1 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { i } ( z ) A _ { j - i - 1 } ^ { ( i + 1 ) } ( z ) . \\end{align*}"}
{"id": "1889.png", "formula": "\\begin{align*} \\mathcal { P } _ { [ n , j ] } = \\bigcup _ { k = i } ^ { n - j + i } \\mathcal { P } _ { [ n , j , k ] } \\end{align*}"}
{"id": "5738.png", "formula": "\\begin{align*} \\pi _ b \\cdot \\pi _ { [ a , b ] } = ( b - a + 1 ) \\pi _ { [ a , b ] } y _ { b + 1 } + ( b - a + 2 ) \\pi _ { [ a - 1 , b ] } ( 1 \\le a \\le b \\le n ) . \\end{align*}"}
{"id": "3760.png", "formula": "\\begin{align*} \\widetilde { T } _ { k , j ; n } ^ { b i l ; \\mu , i } ( \\mathfrak { m } , U ) ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i x \\cdot \\xi + i \\mu ( t - s ) | \\xi | } | \\xi | ^ { - 1 } \\mathcal { F } \\big ( ( E + \\hat { v } \\times B ) f \\big ) ( s , \\xi , v ) \\cdot \\nabla _ v \\big [ \\big ( \\frac { m ^ 0 _ U ( \\xi , v ) } { i ( \\mu | \\xi | + \\hat { v } \\cdot \\xi ) } \\end{align*}"}
{"id": "1684.png", "formula": "\\begin{align*} \\| ( c ( x ) , s ( x ) ) \\| _ Z = 1 \\end{align*}"}
{"id": "1197.png", "formula": "\\begin{align*} \\big | \\| h _ 1 - h _ 2 \\| _ { \\dot { H } ^ { - 1 , p } ( \\mu * \\gamma _ { \\sigma } ) } - \\| \\tilde { h } _ 1 - \\tilde { h } _ 2 \\| _ { \\dot { H } ^ { - 1 , p } ( \\mu * \\gamma _ { \\sigma } ) } \\big | \\le \\sum _ { i = 1 } ^ 2 \\| h _ i - \\tilde { h } _ i \\| _ { \\dot { H } ^ { - 1 , p } ( \\mu * \\gamma _ { \\sigma } ) } < 2 \\epsilon . \\end{align*}"}
{"id": "1441.png", "formula": "\\begin{align*} \\Big ( B ( - \\theta _ z ) z - A ( - \\theta _ z ) \\Big ) f ( z ) = B ( 0 ) \\enspace . \\end{align*}"}
{"id": "8810.png", "formula": "\\begin{align*} \\int _ { G } ^ { } \\int _ { 0 } ^ { \\infty } 1 _ { L _ 1 ( t _ 1 ) } ( g ' ) d t _ 1 \\int _ { 0 } ^ { \\infty } 1 _ { L _ 2 ( t _ 2 ) } ( g '^ { - 1 } g ) d t _ 2 d g ' = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\int _ { G } ^ { } 1 _ { L _ 1 ( t _ 1 ) } ( g ' ) 1 _ { L _ 2 ( t _ 2 ) } ( g '^ { - 1 } g ) d g ' d t _ 1 d t _ 2 \\end{align*}"}
{"id": "5023.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } E [ | R ^ { n , M , 1 } _ \\tau | ^ 2 ] = 0 . \\end{align*}"}
{"id": "622.png", "formula": "\\begin{align*} \\Big | F ' _ { k , j } ( x , y ) \\Big | & \\leq C _ 1 j ^ { \\alpha - 1 } \\frac { k ^ { j - 1 } } { ( j - 1 ) ! } \\frac { 2 ^ { 2 k + 3 j } \\pi ^ { 2 k + 2 j } \\Bigl ( x ^ 2 + y ^ 2 \\Bigr ) ^ { k + 2 j } } { \\Gamma ( 2 k _ j + 5 ) \\prod \\limits _ { \\ell = 1 } ^ { j - 1 } \\Gamma ( 2 ( k _ { \\ell } - k _ { \\ell + 1 } ) + 5 ) } \\\\ & \\leq C _ 1 j ^ { \\alpha - 1 } \\frac { k ^ { j - 1 } } { ( j - 1 ) ! } \\frac { 2 ^ { 2 k + 3 j } \\pi ^ { 2 k + 2 j } \\Bigl ( x ^ 2 + y ^ 2 \\Bigr ) ^ { k + 2 j } } { \\Bigl ( \\Gamma \\left ( 2 k / j + 5 \\right ) \\Bigr ) ^ j } , \\end{align*}"}
{"id": "2810.png", "formula": "\\begin{align*} \\begin{aligned} h _ * = \\arg \\max _ { 1 { } \\leq { } h < 2 } 2 h + \\frac { \\kappa h ^ 3 } { 2 - h \\left ( 1 + \\kappa \\right ) } \\end{aligned} \\end{align*}"}
{"id": "28.png", "formula": "\\begin{align*} L ^ n = \\{ \\beta \\in L ^ { \\times } ; ~ ( \\alpha , \\beta ) _ { L , n } = 0 ~ \\forall \\alpha \\in W _ L \\} . \\end{align*}"}
{"id": "22.png", "formula": "\\begin{align*} ( ~ , ~ ) _ { \\rho , L , n } : \\mathfrak { p } _ L \\times L ^ { \\times } & \\rightarrow W _ { \\rho } ^ n \\\\ ( \\alpha , \\beta ) & \\mapsto \\Phi _ L ( \\beta ) ( \\xi ) - \\xi , \\end{align*}"}
{"id": "4590.png", "formula": "\\begin{align*} \\widetilde { \\Omega } = \\widetilde { \\Omega } _ \\xi : = \\Big \\{ \\sup _ { 0 \\le t \\le T } \\max _ { i \\in [ N ] } N ^ { 2 / 3 } \\widehat { i } ^ { 1 / 3 } | \\lambda _ i ( t ) - \\gamma _ i ( t ) | \\le N ^ \\xi \\Big \\} \\end{align*}"}
{"id": "3790.png", "formula": "\\begin{align*} H _ { k , j ; n , l , r ; p , q ; \\star } ^ { \\mu , m , i ; n o n , a } ( t , x , \\zeta ) = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } E B ^ i ( t , s , x - y + ( t - s ) \\omega , \\omega , v ) \\cdot \\nabla _ v \\big ( ( t - s ) \\mathcal { H } ^ { \\star ; \\mu , E , i } _ { k , j ; n , l , r } ( y , \\omega , v , \\zeta ) + \\mathcal { H } ^ { \\star ; e r r ; \\mu , E , i } _ { k , j ; n , l , r } ( y , v , \\omega , \\zeta ) \\big ) \\end{align*}"}
{"id": "2052.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : o p t i m a l c o u p l i n g j u m p p a r a m } \\overline { X } { } ^ i _ { T ^ i _ n } = \\psi \\big ( \\overline { X } { } ^ i _ { T ^ { i , - } _ n } , f _ { T ^ i _ n } , \\theta \\big ) , X ^ i _ { T ^ i _ n } = \\psi \\Big ( X ^ { i } _ { T ^ { i , - } _ n } , \\mu _ { X ^ i _ { T ^ { i , - } _ n } } , \\theta \\Big ) , \\end{align*}"}
{"id": "3800.png", "formula": "\\begin{align*} = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { ( \\R ^ 3 ) ^ 4 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta + \\sigma + \\kappa ) + i s ( \\mu _ 2 | \\sigma | + \\mu | \\xi | + \\mu _ 1 | \\eta | ) } \\clubsuit K _ { k _ 3 , j _ 3 ; n _ 3 } ^ { \\mu _ 3 , i _ 3 } ( s , \\kappa , V ( s ) ) \\cdot { } ^ 3 \\clubsuit K ( s , \\xi , \\eta , \\sigma , X ( s ) , V ( s ) ) d \\xi d \\eta d \\sigma d \\kappa d s \\end{align*}"}
{"id": "2107.png", "formula": "\\begin{align*} ( L y _ i ) ( v ) & = \\begin{cases} - \\frac { \\delta _ E ( e _ 0 ) } { c } \\frac { 1 } { | e _ 0 | } y _ i ( v ) & v \\in e _ u , \\\\ 0 & . \\end{cases} \\end{align*}"}
{"id": "6937.png", "formula": "\\begin{align*} \\| x \\| ^ 2 = \\sum _ { k = 0 } ^ { \\infty } \\| ( \\Sigma ^ * ) ^ k x \\| ^ 2 , \\end{align*}"}
{"id": "2512.png", "formula": "\\begin{align*} \\begin{array} { l l } R i c ( U , X ) & = ( m - n ) g ( \\nabla _ U H , X ) - \\sum \\limits _ { i = n + 1 } ^ { m } g ( ( \\nabla _ { U _ i } T ) _ U U _ i , X ) \\\\ & + \\sum \\limits _ { j = 1 } ^ { n } g ( ( \\nabla _ X A ) _ { X _ j } X _ j , U ) - \\sum \\limits _ { j = 1 } ^ { n } g ( ( \\nabla _ { X _ j } A ) _ X X _ j , U ) \\\\ & - \\sum \\limits _ { j = 1 } ^ { n } g ( T _ U X _ j , \\nu [ X , X _ j ] ) , \\end{array} \\end{align*}"}
{"id": "9011.png", "formula": "\\begin{align*} \\beta ^ { \\prime } ( 0 ) = - 2 \\ln \\left ( \\frac { 2 \\Gamma \\left ( \\frac { 3 } { 4 } \\right ) } { \\Gamma \\left ( \\frac { 1 } { 4 } \\right ) } \\right ) . \\end{align*}"}
{"id": "6949.png", "formula": "\\begin{align*} H _ u ( f ) = P _ + ( u \\bar f ) , f \\in H ^ 2 ; \\end{align*}"}
{"id": "6515.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\mu \\left ( \\left [ a _ i , \\infty \\right ) \\right ) < \\infty . \\end{align*}"}
{"id": "6290.png", "formula": "\\begin{align*} \\int \\frac { f ( x ) } { f ( x / q ) } \\left ( v ( x / q ) - \\frac { 1 } { q } x ( 1 - q ) \\right ) r ( x ) y ( x ) d _ q x = y ( x / q ) - v ( x / q ) D _ { q ^ { - 1 } } y ( x ) , \\end{align*}"}
{"id": "1521.png", "formula": "\\begin{align*} & \\frac { 1 } { r ! } \\Big ( \\frac { d } { d x } \\Big ) ^ { r } \\bigg [ \\frac { x ^ { r } } { ( 1 - x ) ^ { 2 } } F _ { p , \\lambda } \\Big ( \\frac { x } { 1 - x } \\Big ) \\bigg ] - \\Big ( x \\frac { d } { d x } \\Big ) _ { p , \\lambda } \\Big ( \\frac { 1 } { 1 - x } \\Big ) ^ { r + 1 } \\\\ & = \\sum _ { k = 1 } ^ { \\infty } ( k ) _ { p , \\lambda } \\bigg ( \\frac { 1 } { ( 1 - x ) ^ { r + 1 } } - \\sum _ { j = 0 } ^ { k } \\binom { r + j } { j } x ^ { j } \\bigg ) . \\end{align*}"}
{"id": "6777.png", "formula": "\\begin{align*} & \\sum _ { j \\in N _ { i ' } } \\max \\left \\{ a _ { i ' j } , b - \\sum _ { k \\in M _ C - i ' } a _ { k t _ k } \\right \\} x _ { i ' j } \\\\ & + \\sum _ { i \\in M _ C - i ' } \\left ( \\sum _ { j = 1 } ^ { t _ i } a _ { i t _ i } \\max \\left \\{ 1 , \\frac { a _ { i j } } { b - \\sum _ { k \\in M _ C - { i , i ' } } a _ { k t _ k } - a _ { i ' n _ { i ' } } } \\right \\} x _ { i j } + \\sum _ { j = t _ i + 1 } ^ { n _ i } a _ { i j } x _ { i j } \\right ) > b ? \\end{align*}"}
{"id": "2794.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\| \\nabla f ( x _ i ) \\| ^ 2 { } \\leq { } \\frac { 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left [ 2 h _ i - \\frac { h _ i ^ 2 } { 2 } \\max ( 1 , h _ i ) \\right ] } \\end{align*}"}
{"id": "8127.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } \\| e ^ { - i t H _ 0 } \\psi - e ^ { - i t H } \\varphi \\| = 0 \\end{align*}"}
{"id": "7925.png", "formula": "\\begin{align*} N E ( X _ \\omega ) = C _ \\omega ^ { \\prime , \\vee } = \\left \\{ d \\in H _ 2 ( X _ \\omega ; \\mathbb R ) : \\eta \\cdot d \\geq 0 \\eta \\in C _ \\omega ^ \\prime \\right \\} . \\end{align*}"}
{"id": "5645.png", "formula": "\\begin{align*} ( D ^ 2 u ) = f \\quad \\ \\mathbb { R } ^ n , \\end{align*}"}
{"id": "5419.png", "formula": "\\begin{align*} & \\langle \\Lambda _ { \\gamma _ 1 } f , g \\rangle = \\langle \\Lambda _ { \\gamma _ 2 } f , g \\rangle \\Leftrightarrow \\langle \\Lambda _ { q _ 1 } f , g \\rangle = \\langle \\Lambda _ { q _ 2 } f , g \\rangle \\end{align*}"}
{"id": "3691.png", "formula": "\\begin{align*} \\Lambda _ { A } : = \\bigcup _ { T } ^ { \\infty } \\Lambda _ { A , T } \\ , , \\qquad \\Lambda _ { A , T } : = \\bigcup _ { t = - T } ^ { T } \\phi _ t ( A ) \\end{align*}"}
{"id": "4769.png", "formula": "\\begin{align*} \\mathcal T _ { \\kappa , 2 } = \\left \\{ [ n _ 0 , n _ 1 , n _ 2 , n _ 3 ] \\in \\mathcal S _ { \\kappa , 2 } \\colon n _ 1 = n _ 2 \\right \\} . \\end{align*}"}
{"id": "8809.png", "formula": "\\begin{align*} \\int _ { G } ^ { } \\phi _ 1 ( g ' ) \\phi _ 2 ( g '^ { - 1 } g ) d g ' = \\int _ { G } ^ { } \\int _ { 0 } ^ { \\infty } 1 _ { L _ 1 ( t _ 1 ) } ( g ' ) d t _ 1 \\int _ { 0 } ^ { \\infty } 1 _ { L _ 2 ( t _ 2 ) } ( g '^ { - 1 } g ) d t _ 2 d g ' . \\end{align*}"}
{"id": "1048.png", "formula": "\\begin{align*} \\theta ( t ) = \\sum _ { n = r + 1 } ^ \\infty \\phi * \\psi _ 1 ^ { [ n * ] } ( t ) < \\infty , \\end{align*}"}
{"id": "7079.png", "formula": "\\begin{align*} K \\phi \\varphi _ { i } ( \\gamma ) + K \\phi \\gamma \\varphi ^ { \\prime } _ { i } ( \\gamma ) = \\theta . \\end{align*}"}
{"id": "5772.png", "formula": "\\begin{align*} E ( G / A ) = E ( N _ G ( A ) / A ) = \\pm | W ( K ) | \\end{align*}"}
{"id": "4336.png", "formula": "\\begin{align*} \\mu _ { L , L ' } ( p ) = \\frac { 1 } { \\pi } ( \\sum \\phi _ i + \\theta _ L ( p ) - \\theta _ { L ' } ( p ) ) \\end{align*}"}
{"id": "3412.png", "formula": "\\begin{align*} r \\cdot w _ 1 \\cdots w _ s = h _ s \\cdots h _ 1 \\cdot r ' . \\end{align*}"}
{"id": "1558.png", "formula": "\\begin{align*} d V _ { B H } = \\frac { \\int \\limits _ { 0 } ^ { \\pi } \\sin ^ { n - 2 } ( t ) d t } { \\int \\limits _ { 0 } ^ { \\pi } \\frac { \\sin ^ { n - 2 } ( t ) } { \\phi ( b \\cos ( t ) ) ^ n } d t } d V _ { \\alpha } , \\end{align*}"}
{"id": "5143.png", "formula": "\\begin{align*} \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( n + 1 ) \\oplus \\mathcal { G } ( 2 a c + 1 ) = \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( n ) \\oplus \\mathcal { G } ( 2 a c + 3 ) , \\ ; a \\neq 2 \\bmod 6 \\end{align*}"}
{"id": "6938.png", "formula": "\\begin{align*} I - S S ^ * = e _ 0 e _ 0 ^ * , I - \\Sigma \\Sigma ^ * = q q ^ * , \\end{align*}"}
{"id": "1764.png", "formula": "\\begin{align*} \\sum _ { j = m _ 0 } ^ \\infty | y _ n ^ * ( x _ j ^ k ) | < \\epsilon / 8 . \\end{align*}"}
{"id": "8094.png", "formula": "\\begin{align*} Z = f X . \\end{align*}"}
{"id": "239.png", "formula": "\\begin{align*} ( \\delta ^ { ( s ) } ) ^ 2 u = f ^ { ( 2 ) } ( u , u ^ { \\phi ^ s } , u ^ { \\phi ^ { 2 s } } ) , \\end{align*}"}
{"id": "7880.png", "formula": "\\begin{align*} D _ I : = \\cap _ { i \\in I } D _ i . \\end{align*}"}
{"id": "187.png", "formula": "\\begin{align*} X ^ i _ t & = \\tilde { f } _ t ^ i ( M ^ i , X ^ i _ { 1 : t - 1 } , Y _ { 1 : t - 1 } ) = f _ t ^ i ( M ^ i , Y _ { 1 : t - 1 } ) , i = 1 , 2 . \\end{align*}"}
{"id": "139.png", "formula": "\\begin{align*} \\| C ( C ^ 2 \\star C ^ 2 ) \\| _ { L ^ 1 } \\leq \\frac { c } { m ^ 4 _ t } { \\bf 1 } _ { d = 2 } + \\frac { c } { m _ t } { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "3968.png", "formula": "\\begin{align*} j ( a ^ { * } ) ( e ) = \\overline { j ( a ) ( e ^ { - 1 } ) } \\quad j ( a * b ) ( e ' ) = \\sum _ { q ( e ) \\in G ^ { r ( e ' ) } } j ( a ) ( e ) \\ , j ( b ) ( e ^ { - 1 } e ' ) . \\end{align*}"}
{"id": "3182.png", "formula": "\\begin{align*} \\mathrm { P } _ { ( \\Phi , - \\hat { f } \\left ( 0 \\right ) \\delta _ { \\Psi } ) } ^ { \\sharp } = \\mathrm { P } _ { ( \\Phi , - \\hat { f } \\left ( 0 \\right ) \\delta _ { \\Psi } ) } ^ { \\flat } \\leq \\inf _ { \\gamma \\in \\left ( 0 , 1 \\right ) } \\mathrm { P } _ { \\Phi + \\mathcal { K } _ { \\gamma } \\left ( \\Psi , - f \\right ) } ^ { \\sharp } \\ , \\Psi \\in \\mathbb { S } , \\ \\Phi \\in \\mathcal { W } _ { 1 } , \\ f \\in \\mathfrak { D } _ { 0 , + } \\ , \\end{align*}"}
{"id": "6959.png", "formula": "\\begin{align*} F ( z ) = \\int _ \\R \\frac { d \\rho ( s ) } { s - z } \\forall z \\in \\C \\setminus \\R \\end{align*}"}
{"id": "4290.png", "formula": "\\begin{align*} \\frac { d } { d t } \\langle X \\rangle _ t = - \\frac { 1 } { 2 } \\sum \\limits _ { j } \\langle [ C _ { j } , [ C _ { j } , X ] ] ) \\rangle _ t . \\end{align*}"}
{"id": "7233.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left \\{ \\frac { t - x - R } { ( 1 + t - x ) ^ { 1 + b } } \\right \\} ^ { a _ n } \\left \\{ \\log \\left ( \\frac { 1 + t + x } { 1 + t - x } \\right ) \\right \\} ^ { b _ n } \\quad \\mbox { i n } \\ D \\end{align*}"}
{"id": "6329.png", "formula": "\\begin{align*} J _ { \\nu } ^ { ( 2 ) } ( z ; q ) : = \\frac { ( q ^ { v + 1 } ; q ) _ \\infty } { ( q ; q ) _ \\infty } \\sum _ { n = 0 } ^ { \\infty } \\dfrac { ( - 1 ) ^ n q ^ { n ( n + \\nu ) } } { ( q , q ^ { v + 1 } ; q ) _ n } ( z / 2 ) ^ { 2 n + \\nu } , z \\in \\mathbb C , \\end{align*}"}
{"id": "4573.png", "formula": "\\begin{align*} ( \\phi \\cdot \\psi ) ( F _ * ^ { e + d } r ) = \\phi \\left ( F _ * ^ e ( \\psi ( F _ * ^ d r ) ) \\right ) . \\end{align*}"}
{"id": "5361.png", "formula": "\\begin{align*} s _ { \\theta } = ( 1 - \\theta ) s _ 0 + \\theta s _ 1 = \\frac { t - r } { t - s } s + \\frac { r - s } { t - s } t = r . \\end{align*}"}
{"id": "3906.png", "formula": "\\begin{align*} a _ i = W _ c ( \\mu , \\nu _ { i / I } ) - W _ c ( \\mu , \\nu _ { ( i - 1 ) / I } ) \\forall i \\in \\mathcal I . \\end{align*}"}
{"id": "4740.png", "formula": "\\begin{align*} \\begin{cases} \\Psi _ { \\ell , 1 } ( j _ 1 , j _ 2 , j _ 3 , j _ 1 ' ) = 0 , \\\\ \\partial _ X \\Psi _ { \\ell , 1 } ( j _ 1 , j _ 2 , j _ 3 , j _ 1 ' ) \\cdot j _ k ' - \\Psi _ { \\ell , k } ( j _ 1 , j _ 2 , j _ 3 , j _ 1 ' ) = 0 . \\end{cases} \\end{align*}"}
{"id": "6251.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } A i _ q ( x ) = \\frac { 1 } { 1 - q ^ 2 } \\ , _ 1 \\phi _ 1 ( 0 ; - q ^ 2 ; q , - x ) , \\end{align*}"}
{"id": "609.png", "formula": "\\begin{align*} f ( x , y , n , z ) : = \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi x } { n } \\Big ) + \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi y } { n } \\Big ) + z ^ 2 . \\end{align*}"}
{"id": "697.png", "formula": "\\begin{align*} ( Z + \\zeta ) ( s ) \\le ( Z + \\zeta ) ( s _ 1 ) e ^ { - \\frac { 1 } { 4 } ( s - s _ 1 ) } = 2 \\zeta ( s _ 1 ) e ^ { - \\frac { 1 } { 4 } ( s - s _ 1 ) } \\le C s _ 1 ^ { - \\theta } e ^ { - \\frac { 1 } { 4 } ( s - s _ 1 ) } , \\end{align*}"}
{"id": "7268.png", "formula": "\\begin{align*} u ( x , t ) = \\lambda ^ { - \\frac { n - 2 } { 2 } } { \\sf Q } ( y ) + \\lambda ^ { - \\frac { n - 2 } { 2 } } \\sigma { \\sf A } _ 1 ( 1 + o ) | y | \\to \\infty . \\end{align*}"}
{"id": "5890.png", "formula": "\\begin{align*} \\nu = \\mathcal { D } \\Big ( \\sum _ { j = 1 } ^ \\infty A _ { \\bullet , j } Z _ j + \\sigma ( I _ k - A A ^ T ) ^ { 1 / 2 } N _ k \\Big ) \\end{align*}"}
{"id": "8484.png", "formula": "\\begin{align*} y ^ * _ i & = ( i - 1 ) L + \\left [ \\frac { U } { n } - \\frac { ( n - 1 ) L } { 2 } \\right ] \\\\ & = ( i - 1 ) L + \\left [ \\frac { ( n - 1 ) L } { 2 } - \\frac { ( n - 1 ) L } { 2 } \\right ] \\\\ & = ( i - 1 ) L . \\end{align*}"}
{"id": "8081.png", "formula": "\\begin{align*} \\Gamma ^ { 1 } _ { 2 2 } = - c _ { 1 } \\partial _ r c _ { 1 } , \\Gamma ^ { 2 } _ { 1 2 } = \\Gamma ^ { 2 } _ { 2 1 } = \\frac { \\partial _ r c _ { 1 } } { c _ { 1 } } . \\end{align*}"}
{"id": "7602.png", "formula": "\\begin{align*} \\begin{gathered} a _ { 1 1 } = 1 , a _ { 1 2 } = - \\lambda _ 7 , a _ { 1 3 } = - \\lambda _ 8 , a _ { 2 1 } = \\lambda _ 1 , \\\\ a _ { 2 2 } = ( 1 + \\lambda _ 5 \\lambda _ 6 ) e ^ { \\lambda _ 3 } - \\lambda _ 1 \\lambda _ 7 , a _ { 2 3 } = \\lambda _ 5 e ^ { \\lambda _ 3 } - \\lambda _ 1 \\lambda _ 8 , a _ { 3 1 } = \\lambda _ 2 , \\\\ a _ { 3 2 } = \\lambda _ 6 e ^ { \\lambda _ 4 } - \\lambda _ 2 \\lambda _ 7 , a _ { 3 3 } = e ^ { \\lambda _ 4 } - \\lambda _ 2 \\lambda _ 8 . \\end{gathered} \\end{align*}"}
{"id": "7640.png", "formula": "\\begin{align*} \\chi ^ K = \\prod _ i ( f ^ g ) ^ { \\chi _ i ^ { K _ i } } & = \\prod _ i ( \\chi _ i ^ { - 1 } ) ^ { K _ i } \\circ g ^ { - 1 } \\circ f \\circ g \\circ \\chi _ i ^ { K _ i } \\\\ & = \\prod _ i { g ^ { - 1 } } \\circ ( ( \\chi _ i ^ { - 1 } ) ^ { K _ i } ) ^ { g ^ { - 1 } } \\circ f \\circ ( \\chi _ i ^ { K _ i } ) ^ { g ^ { - 1 } } \\circ g \\\\ & = ( \\prod _ i ( \\chi _ i ^ { - 1 } ) ^ { K _ i g ^ { - 1 } } \\circ f \\circ \\chi _ i ^ { K _ i g ^ { - 1 } } ) ^ g . \\end{align*}"}
{"id": "2456.png", "formula": "\\begin{align*} T _ { A , B } : = \\left ( \\begin{array} { c | r } A & B \\\\ \\hline B & A \\end{array} \\right ) , \\end{align*}"}
{"id": "3767.png", "formula": "\\begin{align*} \\widetilde { T } _ { k , j ; n , l , r } ^ { S ; \\mu , m , i } ( \\mathfrak { m } , U ) ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big [ ( t - s ) K _ { k ; n } ^ { \\mu } ( \\mathfrak { m } ) ( y , \\zeta , \\omega ) + \\widetilde { K } _ { k ; n } ^ { \\mu } ( \\mathfrak { m } ) ( y , \\zeta ) \\big ] f ( s , x - y + ( t - s ) \\omega , v ) \\big ( E ( s , x - y + ( t - s ) \\omega ) \\end{align*}"}
{"id": "2148.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } ( x - s ) ^ { - 2 } \\ , d \\nu ( s ) = + \\infty , x \\in \\mathbb { R } , \\end{align*}"}
{"id": "4120.png", "formula": "\\begin{align*} F ^ { i } : = \\left ( \\begin{array} { c c c c c c } I & K ^ { i , 1 } & \\frac { 1 } { 2 } K ^ { i , 1 } K ^ { i , 2 } & \\frac { 1 } { 6 } K ^ { i , 1 } K ^ { i , 2 } K ^ { i , 3 } & \\cdots & \\frac { 1 } { N ! } K ^ { i , 1 } K ^ { i , 2 } \\cdots K ^ { i , N } \\\\ 0 & I & K ^ { i , 2 } & \\frac { 1 } { 2 } K ^ { i , 2 } K ^ { i , 3 } & \\cdots & \\frac { 1 } { ( N - 1 ) ! } K ^ { i , 2 } K ^ { i , 3 } \\cdots K ^ { i , N } \\\\ & & \\cdots & \\cdots & & \\\\ 0 & 0 & 0 & 0 & \\cdots & I \\end{array} \\right ) . \\end{align*}"}
{"id": "1459.png", "formula": "\\begin{align*} \\Theta = \\dfrac { \\prod _ { i = 1 } ^ m \\alpha ^ r _ i \\prod _ { s = 0 } ^ { r - 1 } a _ { 0 , s } ^ m } { ( n - 1 ) ! ^ { r ^ 2 m } } \\cdot { \\rm { d e t } } \\left ( { \\rm { E v a l } } _ { \\alpha _ i } \\bigcirc _ { { { w } } = 0 } ^ s ( \\theta _ t + \\gamma _ { r - s + { { w } } } ) ^ { - 1 } ( t ^ n H _ { \\ell } ( t ) ) \\right ) _ { \\substack { 0 \\le \\ell \\le r m - 1 \\\\ 1 \\le i \\le m , 0 \\le s \\le r - 1 } } . \\end{align*}"}
{"id": "4820.png", "formula": "\\begin{align*} w = \\sum _ { p \\in J } S ( p ) ^ { d } w \\geq \\sum _ { p \\in J } \\prod _ { j = 1 } ^ d \\frac { \\tilde S ( p , \\pi _ j ( p ) , \\alpha , \\lambda ) } { { \\lambda + k _ j \\choose k _ j } } \\geq \\frac { 1 } { \\prod _ { j = 1 } ^ d { \\lambda + k _ j \\choose k _ j } } { \\lambda + n \\choose n } \\gtrsim _ { n } 1 , \\end{align*}"}
{"id": "3017.png", "formula": "\\begin{align*} \\Delta \\cdot L _ { x y } = \\left ( D - a W - b L _ { x y } - b _ 0 R _ 0 - b _ 1 R _ 1 \\right ) \\cdot L _ { x y } \\leq \\left ( D - b L _ { x y } \\right ) \\cdot L _ { x y } = \\frac { 3 + 2 b ( 4 n - 1 ) } { 4 n ( 4 n + 1 ) } < \\frac { 1 } { \\lambda } . \\end{align*}"}
{"id": "5501.png", "formula": "\\begin{align*} \\frac { \\partial q _ { \\theta } } { \\partial \\theta _ i } = u ^ T ( I - \\gamma P _ { \\theta } ) ^ { - 1 } \\left [ - \\gamma \\frac { \\partial P _ { \\theta } } { \\partial \\theta _ i } \\right ] ( I - \\gamma P _ { \\theta } ) ^ { - 1 } v \\end{align*}"}
{"id": "3074.png", "formula": "\\begin{align*} c _ { n } ^ { \\ast } = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } \\Psi _ { 1 } ^ { \\ast } ( - 1 , \\frac { n } { 2 } , 1 ) } \\end{align*}"}
{"id": "7891.png", "formula": "\\begin{align*} \\tilde { T } _ { \\vec s , k } = [ T _ { I _ { \\vec s } , k } ] _ { \\vec s } . \\end{align*}"}
{"id": "3016.png", "formula": "\\begin{align*} D = a W + b L _ { x y } + c R + \\Omega , \\end{align*}"}
{"id": "4111.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\langle V ' ( \\vec p ) \\widehat { \\vec p } _ 1 ( { u } _ 2 ^ \\prime ( t ) + Q { u } _ 2 ( t ) ) , w ( t ) \\big \\rangle _ X \\d t & = - \\int _ 0 ^ T \\langle u _ 2 , V ' ( \\vec p ) \\widehat { \\vec p } _ 1 ( w ^ \\prime ( t ) + Q w ( t ) ) \\big \\rangle _ X \\d t \\\\ & = \\int _ 0 ^ T \\langle V ' ( \\vec p ) \\widehat { \\vec p } _ 2 ( { u } ^ \\prime ( t ) + Q { u } ( t ) ) , z ( t ) \\big \\rangle _ X \\d t . \\end{align*}"}
{"id": "2817.png", "formula": "\\begin{align*} \\min \\limits _ { 0 \\leq i \\leq N } \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\} & \\leq \\frac { f ( x _ 0 ) - f _ * } { q + \\sum \\limits _ { i = 0 } ^ { N - 1 } p _ i } \\end{align*}"}
{"id": "8490.png", "formula": "\\begin{align*} \\int _ a ^ b \\left | \\alpha - f ( x ) \\right | d x & = \\int _ a ^ b \\alpha - f ( x ) d x \\\\ & = \\int _ a ^ b \\alpha - \\max \\left ( f ( a ) , f ( b ) \\right ) + \\max ( f ( a ) , f ( b ) ) - f ( x ) d x \\\\ & = \\int _ a ^ b \\left | \\alpha - \\max ( f ( a ) , f ( b ) ) \\right | + \\left | \\max ( f ( a ) , f ( b ) ) - f ( x ) \\right | d x \\\\ & > \\int _ a ^ b \\left | \\max ( f ( a ) , f ( b ) ) - f ( x ) \\right | d x . \\end{align*}"}
{"id": "8575.png", "formula": "\\begin{align*} m = 2 ^ { k _ 1 } + 2 ^ { k _ 2 } + \\ldots + 2 ^ { k _ s } , 1 \\le s \\le l . \\end{align*}"}
{"id": "8181.png", "formula": "\\begin{align*} H ( \\omega ) = H _ 0 + V _ \\omega \\end{align*}"}
{"id": "4601.png", "formula": "\\begin{align*} [ ( x _ 1 \\phi _ 1 ) _ \\lambda ( x _ 2 \\phi _ 2 ) ] = [ ( x _ 1 \\phi _ 1 ) _ { \\lambda } x _ 2 ] \\phi _ 2 + ( - 1 ) ^ { ( \\bar { x } _ 1 + \\bar { \\phi } _ 1 ) \\bar { x } _ 2 } x _ 2 [ ( x _ 1 \\phi _ 1 ) _ { \\lambda } \\phi _ 2 ] + \\int ^ \\lambda _ 0 [ [ ( x _ 1 \\phi _ 1 ) _ { \\lambda } x _ 2 ] _ \\mu \\phi _ 2 ] d \\mu , \\end{align*}"}
{"id": "6153.png", "formula": "\\begin{align*} \\begin{aligned} & v ^ { \\Delta t } _ { h } , \\ , u ^ { \\Delta t } _ { h } \\to u \\textrm { s t r o n g l y i n } L ^ { 2 } ( 0 , T ; L ^ { 2 } _ { \\# } ) \\quad \\textrm { a s } ( \\Delta t , h ) \\to ( 0 , 0 ) . \\end{aligned} \\end{align*}"}
{"id": "6984.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\ln 2 \\left ( \\frac { \\lambda _ { N } ^ { 2 } + \\mu _ { k } ^ { 2 } } { \\lambda _ { N } ^ { 2 } + \\lambda _ { k } ^ { 2 } } - 1 \\right ) = \\sum _ { k = N } ^ \\infty - \\ln 2 \\frac { \\lambda _ { k } ^ { 2 } - \\mu _ { k } ^ { 2 } } { \\lambda _ { N } ^ { 2 } + \\lambda _ { k } ^ { 2 } } \\ge - \\ln 2 \\sum _ { k = N } ^ { \\infty } \\frac { \\lambda _ k ^ 2 - \\mu _ k ^ 2 } { \\lambda _ N ^ 2 } \\ge - \\ln 2 \\end{align*}"}
{"id": "6678.png", "formula": "\\begin{align*} \\begin{aligned} \\| w _ 1 - w _ 2 \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } & \\leq C _ 1 ( C _ { 0 1 } \\| h _ 1 - h _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + T _ 0 ^ { \\frac { \\beta _ 0 + 1 } { 2 } } \\| z _ 1 - z _ 2 \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } \\\\ & + \\| z _ { 0 1 } - z _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } + \\| w _ { 0 1 } - w _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } ) \\\\ \\end{aligned} \\end{align*}"}
{"id": "5295.png", "formula": "\\begin{align*} \\frac { ( r - 2 ) k + 2 ( r ( I ) + p r ) } { r } = \\frac { ( r - 2 ) r ( I ) + 2 r ( I ) + 2 p r } { r } = r ( I ) + 2 p . \\end{align*}"}
{"id": "4544.png", "formula": "\\begin{align*} \\Vert J _ { 2 } \\Vert _ { p } = \\mathcal { O } \\left ( \\left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \\right ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) \\right ) . \\end{align*}"}
{"id": "5615.png", "formula": "\\begin{align*} A \\mapsto A _ t : = \\bigcup \\big \\{ J _ v \\mid J \\in F ( D ) J \\subseteq A \\big \\} . \\end{align*}"}
{"id": "4339.png", "formula": "\\begin{align*} \\int _ { \\Sigma } u ^ * \\omega = ( f _ { L _ 0 } - f _ { L _ k } ) ( q ) - \\sum _ 1 ^ k ( f _ { L _ { i - 1 } } - f _ { L _ { i } } ) ( p _ i ) . \\end{align*}"}
{"id": "3076.png", "formula": "\\begin{align*} c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast \\ast } & = \\frac { \\Gamma ( ( n - 1 ) / 2 ) } { ( 2 \\pi ) ^ { ( n - 1 ) / 2 } } \\left \\{ \\int _ { 0 } ^ { \\infty } t ^ { ( n - 3 ) / 2 } \\ln \\left [ 1 + \\exp \\left ( - \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } , k } ^ { 2 } \\right ) \\exp ( - t ) \\right ] \\mathrm { d } t \\right \\} ^ { - 1 } , \\end{align*}"}
{"id": "6884.png", "formula": "\\begin{align*} | N _ 0 ( t ) - N ( B ( t ) , C ( t ) ) | & = \\left | \\int _ { \\mathbb { R } } ( N ( B ( t ) + y , C ( t ) ) - N ( B ( t ) , C ( t ) ) p _ { t , 0 } ( y ) d y \\right | \\\\ & \\leq \\int _ { \\mathbb { R } } | N ( B ( t ) + y , C ( t ) ) - N ( B ( t ) , C ( t ) | p _ { t , 0 } ( y ) d y . \\end{align*}"}
{"id": "8943.png", "formula": "\\begin{align*} | \\det D T u | = e ^ { [ \\tau ( 1 - t ) + \\epsilon ] ( u - u _ 0 ) } [ t f + ( 1 - t ) f ^ * \\circ T u _ 0 | \\det ( D T u _ 0 ) | ] / f ^ * \\circ T u , { \\rm i n } \\ \\Omega , \\\\ \\end{align*}"}
{"id": "8041.png", "formula": "\\begin{align*} J _ W : = \\left ( \\exp \\left ( \\frac { 2 \\pi i } { 3 } \\right ) , \\exp \\left ( \\frac { 2 \\pi i } { 3 } \\right ) , \\exp \\left ( \\frac { 2 \\pi i } { 3 } \\right ) , \\exp \\left ( \\frac { 2 \\pi i } { 3 } \\right ) \\right ) \\in ( \\mathbb C ^ \\times ) ^ 4 . \\end{align*}"}
{"id": "7034.png", "formula": "\\begin{align*} P = c \\big ( u _ 1 \\otimes u _ 1 + u _ 2 \\otimes u _ 2 \\big ) \\end{align*}"}
{"id": "3346.png", "formula": "\\begin{align*} 6 A + B q = 1 \\end{align*}"}
{"id": "2887.png", "formula": "\\begin{align*} S ( F ( c ( x ) ) ) = F ( S ( c ( x ) ) ) = F ( c ( x ) ) , \\end{align*}"}
{"id": "4067.png", "formula": "\\begin{align*} 2 g ( Y _ t ) - 2 - K _ { X _ t } \\cdot h _ t ( Y _ t ) = \\deg N _ { h _ t / X _ t } . \\end{align*}"}
{"id": "4501.png", "formula": "\\begin{align*} E = \\lim _ { r \\rightarrow \\infty } \\frac { 1 } { 1 6 \\pi } \\int _ { S _ { r } } \\sum _ i \\left ( g _ { i j , i } - g _ { i i , j } \\right ) \\upsilon ^ j d A , \\quad P _ i = \\lim _ { r \\rightarrow \\infty } \\frac { 1 } { 8 \\pi } \\int _ { S _ { r } } \\left ( k _ { i j } - ( \\mathrm { T r } _ g k ) g _ { i j } \\right ) \\upsilon ^ j d A \\end{align*}"}
{"id": "7685.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma _ { k - 1 } } \\lambda _ { i } ( \\sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \\left [ \\lambda _ { i } ( \\sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \\right ] _ { + } \\to 0 , \\ i = 1 , \\dots , m , \\end{align*}"}
{"id": "5435.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { p } \\vcentcolon = \\frac { 1 } { p _ 1 } + \\frac { 1 } { r _ 2 } = \\frac { 1 } { p _ 2 } + \\frac { 1 } { r _ 1 } < 1 , \\end{align*}"}
{"id": "4072.png", "formula": "\\begin{align*} D = \\bigcup \\limits _ { j = 0 } ^ k D _ j . \\end{align*}"}
{"id": "707.png", "formula": "\\begin{align*} | \\nabla u | _ g ^ 2 = \\frac { f ( u ) ^ 2 } { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 } - 1 = v ^ 2 - 1 . \\end{align*}"}
{"id": "2042.png", "formula": "\\begin{align*} \\big \\langle \\mu _ { \\mathbf { x } ^ N } ^ { \\otimes k } , \\varphi _ k \\big \\rangle & = \\frac { 1 } { N ^ k } \\sum _ { i _ 1 , \\ldots , i _ k } \\varphi _ k { \\left ( x ^ { i _ 1 } , \\ldots , x ^ { i _ k } \\right ) } \\\\ & = \\frac { 1 } { N ^ k } \\sum _ { \\substack { i _ 1 , \\ldots , i _ k \\\\ } } \\varphi _ k { \\left ( x ^ { i _ 1 } , \\ldots , x ^ { i _ k } \\right ) } + R _ { k , N } , \\end{align*}"}
{"id": "813.png", "formula": "\\begin{align*} [ u ] _ { W ^ { s , p } ( E ) } : = \\left ( \\int _ { E } \\int _ { E } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + s p } } ~ d x d y \\right ) ^ { 1 / p } . \\end{align*}"}
{"id": "6631.png", "formula": "\\begin{align*} \\Gamma _ { p , \\ell } \\cap B _ h & = \\big \\{ x \\in \\Lambda [ 1 / p ] : \\ , \\ , N ( x ) = 1 , \\ , x = I \\ , ( \\hbox { m o d } \\ , \\ell ) , \\ , \\| x \\| _ p \\le h \\big \\} \\\\ & = \\big \\{ h ^ { - 1 } y : \\ , y \\in \\Lambda , \\ , N ( y ) = h ^ 2 , \\ , y = h I \\ , ( \\hbox { m o d } \\ , \\ell ) \\big \\} , \\end{align*}"}
{"id": "6501.png", "formula": "\\begin{align*} \\int _ { c _ 1 + \\delta } ^ { c _ 2 - \\delta } \\abs { y _ x } ^ 2 d x = o ( 1 ) . \\end{align*}"}
{"id": "3025.png", "formula": "\\begin{align*} \\frac { 2 0 n + 4 } { 2 0 n + 5 } = \\frac { 1 } { \\lambda } < b + \\mu \\leq \\frac { 1 6 n + 3 } { 1 6 n + 4 } + \\epsilon . \\end{align*}"}
{"id": "5962.png", "formula": "\\begin{align*} \\omega _ j ( t ) = 0 t \\leq t _ j . \\end{align*}"}
{"id": "577.png", "formula": "\\begin{align*} \\mathbb { V } ( \\overline { T } _ n ) = \\frac { 1 } { n ^ 2 } \\sum _ { 1 \\leq i \\leq n } \\mathbb { E } ( X _ i ^ 2 X _ { i + 1 } ^ 2 ) + \\frac { 2 } { n ^ 2 } \\sum _ { 1 \\leq i < j \\leq n } \\mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) - \\frac { 1 } { n ^ 2 } \\mathbb { E } ^ 2 \\Big ( \\sum _ { 1 \\leq i \\leq n } X _ i X _ { i + 1 } \\Big ) . \\end{align*}"}
{"id": "1181.png", "formula": "\\begin{align*} ( \\tilde { \\varphi } _ x * \\phi _ \\sigma ) ( x ) = \\frac { 1 } { 1 + | x | } \\| g _ x \\| _ { L ^ p ( \\gamma _ { \\sigma } ) } = \\frac { 1 } { 1 + | x | } e ^ { ( p - 1 ) | x | ^ 2 / ( 2 \\sigma ^ 2 ) } . \\end{align*}"}
{"id": "165.png", "formula": "\\begin{align*} \\theta ( k ) : = \\sum _ { k = 1 } ^ 3 \\theta ( k _ i ) , \\theta ( k _ i ) : = \\frac { 4 } { \\epsilon ^ 2 } \\sin ^ 2 \\Big ( \\frac { k _ i \\epsilon } { 2 } \\Big ) , k \\in \\Lambda ^ * . \\end{align*}"}
{"id": "2763.png", "formula": "\\begin{align*} \\bar { u } _ { j } ( x ) : = \\varepsilon _ { 0 } ^ { s } u _ { j } ( \\varepsilon _ { 0 } ( x + 4 e _ { 1 } ) ) , \\quad \\mathrm { a n d } \\ ; \\ , \\bar { K } _ { j } ( x ) : = \\varepsilon _ { 0 } ^ { \\sigma p - s ( p - 1 ) ( q - 1 ) } K _ { j } ( \\varepsilon _ { 0 } ( x + 4 e _ { 1 } ) ) . \\end{align*}"}
{"id": "7032.png", "formula": "\\begin{align*} e ^ { ( t + t _ 1 + t _ 2 ) A } = e ^ { t _ 1 A } e ^ { t A } e ^ { t _ 2 A } \\end{align*}"}
{"id": "6062.png", "formula": "\\begin{align*} f ( x ) & = x ^ 3 + a x ^ 2 = ( x - 1 ) ^ 3 + b ( x - 1 ) \\\\ & = x ^ 3 - 3 x ^ 2 + ( b + 3 ) x - ( b + 1 ) \\end{align*}"}
{"id": "7173.png", "formula": "\\begin{align*} M = \\mbox { s p a n } \\{ u ( n ) a | u ( n ) \\in V [ g ] \\} \\end{align*}"}
{"id": "8540.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } & \\frac { x ( 1 - x ^ 2 ) ^ { k - i } - t ( 1 - t ^ 2 ) ^ { k - i } } { x - t } d t \\cr & = x \\int _ { - 1 } ^ { 1 } \\frac { ( 1 - x ^ 2 ) ^ { k - i } - ( 1 - t ^ 2 ) ^ { k - i } } { x - t } d t + \\int _ { - 1 } ^ { 1 } ( 1 - t ^ 2 ) ^ { k - i } . \\end{align*}"}
{"id": "2761.png", "formula": "\\begin{align*} \\partial ^ { 2 } _ { x _ { k } x _ { l } } K _ { j } ( 0 ) = 0 , \\quad \\end{align*}"}
{"id": "7301.png", "formula": "\\begin{align*} e ^ { \\mu _ 1 s } ( \\epsilon _ 3 , \\psi _ 1 ) _ { L _ y ^ 2 ( B _ { 4 { \\sf R } _ { \\sf i n } } ) } - ( \\epsilon _ 0 , \\psi _ 1 ) _ { L _ y ^ 2 ( B _ { 4 { \\sf R } _ { \\sf i n } } ) } & = \\int _ 0 ^ s e ^ { \\mu _ 1 s _ 1 } ( \\mathcal { R } _ 1 + \\mathcal { R } _ 2 , \\psi _ 1 ) _ { L _ y ^ 2 ( B _ { 4 { \\sf R } _ { \\sf i n } } ) } d s _ 1 . \\end{align*}"}
{"id": "6516.png", "formula": "\\begin{align*} \\sum _ { { z = 1 } } ^ \\infty \\frac { 1 } { A ( z ) } < \\infty \\end{align*}"}
{"id": "5764.png", "formula": "\\begin{align*} [ G , G ] = N _ 1 S \\end{align*}"}
{"id": "1512.png", "formula": "\\begin{align*} S _ { 2 , \\lambda } ( n + 1 , k ) = S _ { 2 , \\lambda } ( n , k - 1 ) + ( k - n \\lambda ) S _ { 2 , \\lambda } ( n , k ) , \\end{align*}"}
{"id": "607.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } u ( i ) & = \\int _ 0 ^ n u ( x ) d x + \\frac { 1 } { 2 } \\big ( u ( 0 ) - u ( n ) \\big ) + \\sum _ { j = 1 } ^ { M } \\frac { B _ { 2 j } } { ( 2 j ) ! } \\big ( u ^ { ( 2 j - 1 ) } ( n ) - u ^ { ( 2 j - 1 ) } ( 0 ) \\big ) \\\\ & + \\frac { 1 } { ( 2 M + 1 ) ! } \\int _ 0 ^ n B _ { 2 M + 1 } ( x - [ x ] ) u ^ { ( 2 M + 1 ) } ( x ) d x . \\end{align*}"}
{"id": "8802.png", "formula": "\\begin{align*} 1 _ { ( \\phi ^ * ) ^ { - 1 } ( \\mathbb { R } _ { > t } ) } = 1 _ { ( - \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) / 2 , \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) / 2 ) } \\end{align*}"}
{"id": "7084.png", "formula": "\\begin{align*} 1 + \\frac { \\sum _ { i = 1 } ^ { n } \\alpha _ { i } } { \\sum _ { i = 1 } ^ { n } \\frac { 1 + \\beta _ { i } } { 1 - \\beta _ { i } } \\alpha _ { i } } = \\phi . \\end{align*}"}
{"id": "8080.png", "formula": "\\begin{align*} \\Gamma ^ { k } _ { i j } = \\frac { 1 } { 2 } \\left ( \\sum _ { a = 1 } ^ { 2 } g ^ { k a } \\left ( \\partial _ { i } g _ { j a } + \\partial _ { j } g _ { i a } - \\partial _ { a } g _ { i j } \\right ) \\right ) . \\end{align*}"}
{"id": "5440.png", "formula": "\\begin{align*} \\| f g \\| _ { H ^ { \\theta s _ 1 , p } ( \\R ^ n ) } & \\leq C ( \\| f \\| _ { H ^ { \\theta s _ 1 , p _ 1 / \\theta } ( \\R ^ n ) } \\| g \\| _ { L ^ { r _ 2 } ( \\R ^ n ) } + \\| f \\| _ { L ^ { \\infty } ( \\R ^ n ) } \\| g \\| _ { H ^ { \\theta s _ 1 , p _ 2 } ( \\R ^ n ) } ) \\\\ & \\leq C ( \\| f \\| _ { H ^ { s _ 1 , p _ 1 } ( \\R ^ n ) } ^ { \\theta } \\| f \\| _ { L ^ { \\infty } ( \\R ^ n ) } ^ { 1 - \\theta } \\| g \\| _ { L ^ { r _ 2 } ( \\R ^ n ) } + \\| f \\| _ { L ^ { \\infty } ( \\R ^ n ) } \\| g \\| _ { H ^ { \\theta s _ 1 , p _ 2 } ( \\R ^ n ) } ) . \\end{align*}"}
{"id": "902.png", "formula": "\\begin{align*} \\left ( r \\tilde { \\rho } \\right ) '' - \\dfrac { f ^ { 2 } } { 2 r } \\tilde { \\rho } = - \\dfrac { 1 } { 4 } r \\left ( A - B \\right ) ^ { 2 } \\tilde { \\rho } + \\dfrac { \\lambda } { 2 } r \\tilde { \\rho } \\left ( \\tilde { \\rho } ^ { 2 } - \\dfrac { 2 \\mu ^ { 2 } } { \\lambda } \\right ) \\end{align*}"}
{"id": "4065.png", "formula": "\\begin{align*} & i _ { 2 p - 1 } = - i _ { 2 p } , & & i _ { \\sigma ( 2 p - 1 ) } = - i _ { \\sigma ( 2 p ) } , & & p = 1 , \\ldots , t . \\end{align*}"}
{"id": "7080.png", "formula": "\\begin{align*} t _ { k + 1 } & = \\min \\{ t > t _ { k } : \\mathsf { G } ( u ( t _ { k } ) , u ( t ) ) = \\lambda \\} , \\end{align*}"}
{"id": "8050.png", "formula": "\\begin{align*} { \\mathfrak m } { \\mathfrak c } _ { u , v } : = - \\int _ { M } g ( [ u , v ] , [ u , v ] ) \\mu - \\int _ { M } g ( u , [ [ u , v ] , v ] ) \\mu , \\end{align*}"}
{"id": "6324.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ { q } y ( x ) - \\dfrac { 1 + q } { q x ( 1 - q ) } D _ { q ^ { - 1 } } y ( x ) + \\frac { 1 } { q x ( 1 - q ) ^ 2 } y ( x ) = 0 . \\end{align*}"}
{"id": "1384.png", "formula": "\\begin{align*} \\partial _ t \\frac { 1 } { 2 } \\| u \\| _ { L ^ 2 ( \\mathbb { T } ) } ^ 2 & = \\frac { 1 } { 2 } \\int _ \\mathbb { T } \\overline { u } \\left ( i u _ x x - i | u | ^ { p - 1 } u - \\gamma u + i f \\right ) + u \\left ( - i \\overline { u } _ x x + i | u | ^ { p - 1 } \\overline { u } - \\gamma \\overline { u } - i \\overline { f } \\right ) \\ , d x \\\\ & = - \\gamma \\| u \\| _ { L ^ 2 _ x ( \\mathbb { T } ) } ^ 2 + \\int _ \\mathbb { T } \\Im ( u \\overline { f } ) \\ , d x \\end{align*}"}
{"id": "6602.png", "formula": "\\begin{align*} \\left < \\tau ( \\beta ^ S _ h ) \\phi , \\phi \\right > = \\frac { 1 } { m _ { G _ S } ( B ^ S _ h ) } \\int _ { U } m _ S \\big ( B ^ S _ h \\cap x p ^ { - 1 } ( U ) ^ { - 1 } \\big ) \\ , d \\nu ( x L _ S ) . \\end{align*}"}
{"id": "1921.png", "formula": "\\begin{align*} S _ { [ m ( p + 1 ) + j , j ] } ^ { ( q ) } & = \\sum _ { i _ { 1 } = q } ^ { j + q } \\ , \\ , \\sum _ { i _ { 2 } = q } ^ { i _ { 1 } + p } \\ , \\ , \\sum _ { i _ { 3 } = q } ^ { i _ { 2 } + p } \\ , \\ , \\cdots \\ , \\ , \\sum _ { i _ { m } = q } ^ { i _ { m - 1 } + p } \\ , \\ , \\prod _ { k = 1 } ^ { m } a _ { i _ { k } } , \\\\ T _ { [ m ( p + 1 ) + j , j ] } ^ { ( q ) } & = \\sum _ { i _ { 1 } = - j - p - q } ^ { - p - q } \\ , \\ , \\sum _ { i _ { 2 } = i _ { 1 } - p } ^ { - p - q } \\ , \\ , \\sum _ { i _ { 3 } = i _ { 2 } - p } ^ { - p - q } \\ , \\ , \\cdots \\ , \\ , \\sum _ { i _ { m } = i _ { m - 1 } - p } ^ { - p - q } \\ , \\ , \\prod _ { k = 1 } ^ { m } a _ { i _ { k } } . \\end{align*}"}
{"id": "6996.png", "formula": "\\begin{align*} H _ I = \\bigcap _ { j \\in I } H _ j . \\end{align*}"}
{"id": "6760.png", "formula": "\\begin{align*} \\binom { k } { j _ 1 , j _ 2 , \\dots , j _ m } 1 j _ 1 + 2 j _ 2 + \\dots + m j _ m = n . \\end{align*}"}
{"id": "7318.png", "formula": "\\begin{align*} | w _ 2 | < \\bar { w } _ 2 = 2 T ^ { \\frac { 1 } { 4 } { \\sf d } _ 1 } e ^ { - 2 ( T - t ) ^ \\frac { 1 } { 2 } } { \\sf U } _ \\infty ( x ) | z | > { \\sf l } _ { \\sf o u t } , \\ t \\in ( 0 , T ) . \\end{align*}"}
{"id": "3468.png", "formula": "\\begin{align*} \\mathrm { E } [ X ^ { n } | x _ { p } < X < x _ { q } ] & = \\frac { \\int _ { \\xi _ { p } } ^ { \\xi _ { q } } ( \\sigma y + \\mu ) ^ { n } c _ { 1 } g _ { 1 } \\left ( \\frac { 1 } { 2 } y ^ { 2 } \\right ) \\mathrm { d } y } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } \\\\ & = \\frac { \\sum _ { i = 0 } ^ { n } \\binom { n } { i } \\mu ^ { n - i } \\sigma ^ { i } \\int _ { \\xi _ { p } } ^ { \\xi _ { q } } y ^ { i } c _ { 1 } g _ { 1 } \\left ( \\frac { 1 } { 2 } y ^ { 2 } \\right ) \\mathrm { d } y } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } . \\end{align*}"}
{"id": "325.png", "formula": "\\begin{align*} \\lim _ { y \\to \\infty } \\sup _ { \\substack { A \\subset [ y , \\infty ) \\\\ \\textnormal { L - p r i m i t i v e } A \\not \\ni q } } f ( A _ q ) \\ = \\ \\sup _ { \\textnormal { L - p r i m i t i v e } A \\not \\ni q } f ( A _ q ) \\ = \\ e ^ \\gamma { \\rm d } ( { \\rm L } _ q ) . \\end{align*}"}
{"id": "4748.png", "formula": "\\begin{align*} \\theta _ { a , b } ( z , \\tau ) = \\sum _ { m \\in \\Z ^ g } \\exp \\left ( i \\pi ( m + \\tfrac a 2 ) ^ t \\tau ( m + \\tfrac a 2 ) + 2 i \\pi ( m + \\tfrac a 2 ) ^ t ( z + \\tfrac b 2 ) \\right ) . \\end{align*}"}
{"id": "906.png", "formula": "\\begin{align*} \\Tilde { \\Psi } _ t ^ * g _ { \\mathrm { c a n } } = g + O ( t ^ l ) \\end{align*}"}
{"id": "4548.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 2 \\right ) } \\Vert _ { p } = \\mathcal { O } ( 1 ) \\left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \\right ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "5402.png", "formula": "\\begin{align*} \\langle \\Theta \\nabla ^ s u , \\nabla ^ s \\phi \\rangle _ { L ^ 2 ( \\R ^ { 2 n } ) } = \\langle ( - \\Delta ) ^ { s / 2 } ( \\gamma ^ { 1 / 2 } u ) , ( - \\Delta ) ^ { s / 2 } ( \\gamma ^ { 1 / 2 } \\phi ) \\rangle _ { L ^ 2 ( \\R ^ n ) } + \\langle q \\gamma ^ { 1 / 2 } u , \\gamma ^ { 1 / 2 } \\phi \\rangle _ { L ^ 2 ( \\R ^ n ) } \\end{align*}"}
{"id": "2914.png", "formula": "\\begin{align*} m ^ { ( i ) } ( x ) = \\prod _ { j \\in C _ i } ( x - \\alpha ^ j ) , \\end{align*}"}
{"id": "7860.png", "formula": "\\begin{align*} [ T ^ 0 _ g , J x _ m J ] = & \\sum _ n E _ n T ^ { i ( n ) } _ g [ E _ n , J x _ m J ] + \\sum _ n [ E _ n , J x _ m J ] T ^ { i ( n ) } _ g E _ n \\\\ & + \\sum _ n E _ n [ T ^ { i ( n ) } _ g , J x _ m J ] E _ n . \\end{align*}"}
{"id": "4229.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { 1 \\vee \\log ( \\psi ^ { - 1 } ( a ( k ) ) ) } = \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { 1 \\vee ( \\log 8 + \\log ( \\phi ^ { - 1 } ( a ( 8 k ) ) ) ) } = \\infty \\end{align*}"}
{"id": "1663.png", "formula": "\\begin{align*} f = \\sum _ { D \\in F } \\left ( \\sum _ { \\gamma \\in I _ D } ( f _ D ) | _ { D _ { \\gamma , i _ { \\gamma } } } \\right ) = \\sum _ { D \\in F } \\left ( \\sum _ { i = 1 } ^ k \\sum _ { \\gamma \\in I _ D } ( f _ D ) | _ { D _ { \\gamma , i } } \\right ) = \\sum _ { i = 1 } ^ k \\left ( \\sum _ { D \\in F } \\sum _ { \\gamma \\in I _ D } ( f _ D ) | _ { D _ { \\gamma , i } } \\right ) . \\end{align*}"}
{"id": "588.png", "formula": "\\begin{align*} \\mathbb { V } ( \\overline { T } _ n ) = O _ { k , \\alpha } ( n ^ { - 1 / 2 + \\varepsilon } ) \\end{align*}"}
{"id": "6217.png", "formula": "\\begin{align*} \\int f ( x ) h ( x / q ) \\Big [ \\frac { 1 } { q } \\frac { D _ { q ^ { - 1 } } D _ q h ( x ) } { h ( x / q ) } + p ( x ) \\frac { D _ { q ^ { - 1 } } h ( x ) } { h ( x / q ) } + \\frac { r ( x ) h ( x ) } { h ( x / q ) } \\Big ] y ( x ) d _ q x \\\\ = f ( x / q ) h ( x / q ) \\Big [ y ( x / q ) \\frac { D _ { q ^ { - 1 } } h ( x ) } { h ( x / q ) } - D _ { q ^ { - 1 } } y ( x ) \\Big ] . \\end{align*}"}
{"id": "1998.png", "formula": "\\begin{align*} g ( x ) : = g ( x , n , s ) = ( - 1 ) ^ n L _ n ^ { ( - n - s - 1 ) } = \\displaystyle \\sum _ { j = 0 } ^ { n } \\binom { n + s - j } { n - j } \\frac { x ^ j } { j ! } = \\displaystyle \\sum _ { j = 0 } ^ { n } b _ j \\frac { x ^ j } { j ! } \\end{align*}"}
{"id": "1277.png", "formula": "\\begin{align*} \\mathcal { E } ( \\mathcal { B } _ { d , k } ) & = \\sum _ { j = 1 } ^ { k - 1 } ( d - 1 ) ^ { k - 1 - j } \\Psi ( E _ { j + 1 } ( x , d - 1 ) ) - \\sum _ { j = 1 } ^ { k - 1 } ( d - 1 ) ^ { k - 1 - j } \\Psi ( E _ j ( x , d - 1 ) ) \\\\ & = \\sum _ { j = 1 } ^ { k - 1 } ( d - 1 ) ^ { k - 1 - j } \\left ( \\Psi ( E _ { j + 1 } ( x , d - 1 ) ) - \\Psi ( E _ j ( x , d - 1 ) ) \\right ) . \\end{align*}"}
{"id": "4937.png", "formula": "\\begin{align*} 1 + p ' u = \\frac { - \\frac { 1 } { 2 } \\min u } { u - \\frac { 1 } { 2 } \\min u } < 0 . \\end{align*}"}
{"id": "2629.png", "formula": "\\begin{align*} \\max _ { I \\in \\mathcal { B } ( \\bigcap _ { i = 1 } ^ m \\mathcal { M } _ i ) } | I | \\leqslant \\min \\sum _ { i = 1 } ^ m r _ i ( X _ i ) \\end{align*}"}
{"id": "5683.png", "formula": "\\begin{align*} - \\sum _ { j = 1 } ^ i \\varpi _ { j - 1 } x _ j ^ d = - \\sum _ { j = 0 } ^ { i - 1 } \\varpi _ { j } x _ { j + 1 } ^ d = - \\sum _ { j = 1 } ^ { i } \\varpi _ j x _ { j + 1 } ^ d + \\varpi _ i x _ { i + 1 } ^ d \\end{align*}"}
{"id": "2781.png", "formula": "\\begin{align*} \\begin{aligned} \\tfrac { 2 L \\sigma _ N } { B } = 1 + h _ { N - 1 } \\tfrac { 1 } { 2 - ( 1 + \\kappa ) h _ { N - 1 } } \\ , \\ , , \\tfrac { \\alpha _ { N - 1 } } { B } = \\tfrac { 1 - \\kappa h _ { N - 1 } } { 2 - ( 1 + \\kappa ) h _ { N - 1 } } \\end{aligned} \\end{align*}"}
{"id": "7354.png", "formula": "\\begin{align*} \\left \\vert \\frac { k _ { \\phi } ( z ) } { k ' _ { \\phi } ( z ) } \\right \\vert \\leq \\frac { r } { m ( r ) } , \\vert z \\vert = r . \\end{align*}"}
{"id": "7091.png", "formula": "\\begin{align*} \\begin{pmatrix} s _ { 2 , 1 } & s _ { 3 , 1 } & s _ { 4 , 1 } \\\\ s _ { 2 , 2 } & s _ { 3 , 2 } & s _ { 4 , 2 } \\end{pmatrix} = Q R \\end{align*}"}
{"id": "2723.png", "formula": "\\begin{align*} c _ { d , s } = \\begin{cases} \\frac { 1 } { Z _ N } e ^ { - d ^ 2 / 4 \\sigma _ N ^ 2 } & \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "2041.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : c l o s u r e f 2 } f ^ { 2 , N } _ t = f ^ { 1 , N } _ t \\otimes f ^ { 1 , N } _ t , \\end{align*}"}
{"id": "6053.png", "formula": "\\begin{align*} \\tilde { \\lambda _ i } = r \\sigma ( \\lambda _ i ) + s \\end{align*}"}
{"id": "113.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p } : = \\Big ( \\epsilon ^ d \\sum _ { x \\in \\Lambda _ { \\epsilon , L } } | f ( x ) | ^ p \\Big ) ^ { 1 / p } , \\| f \\| _ { L ^ \\infty } : = \\max _ { x \\in \\Lambda _ { \\epsilon , L } } | f ( x ) | , f \\in \\R ^ { \\Lambda _ { \\epsilon , L } } . \\end{align*}"}
{"id": "810.png", "formula": "\\begin{align*} f ( z ) = \\int _ { 0 } ^ { z } \\frac { 1 } { y } \\int _ { 0 } ^ { y } G ' ( t ) \\psi ( \\omega ( t ) ) d t d y . \\end{align*}"}
{"id": "45.png", "formula": "\\begin{align*} \\eta \\delta _ n ( \\beta ) = \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { f ' ( v _ { n + 1 } + w ) } { f ( v _ { n + 1 } + w ) } = \\operatorname { T } _ { n + 1 , n } ( \\delta _ { n + 1 } ( \\beta ) ) . \\end{align*}"}
{"id": "8171.png", "formula": "\\begin{align*} h ( y ) = - \\frac { d ^ 2 } { d x ^ 2 } + V ( x , y ) \\end{align*}"}
{"id": "5004.png", "formula": "\\begin{align*} \\Psi ^ { n , 6 } _ s = 2 \\left ( \\int ^ s _ { \\eta _ n ( s ) } \\left ( s - \\eta _ n ( u ) \\right ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\ , \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) . \\end{align*}"}
{"id": "4738.png", "formula": "\\begin{align*} \\xi ( A ) = X ^ 2 - s _ 1 X + s _ 2 . \\end{align*}"}
{"id": "6872.png", "formula": "\\begin{align*} p ( t , g ) = ( p _ { t , 0 } * G _ t ) ( g ) . \\end{align*}"}
{"id": "6415.png", "formula": "\\begin{align*} D = D ^ h + \\sum _ { s \\geq 0 } D ^ { v _ s } \\end{align*}"}
{"id": "5354.png", "formula": "\\begin{align*} P _ { n , s , p } ( \\Omega ) = P _ { k , s , p } ( \\omega ) . \\end{align*}"}
{"id": "254.png", "formula": "\\begin{align*} \\delta ^ { ( s ) } w = \\beta w ^ { ( p ^ s ) } , \\end{align*}"}
{"id": "3554.png", "formula": "\\begin{align*} j _ n ^ * ( s _ l ) = \\begin{cases} s _ l ' , & \\\\ 0 , & \\end{cases} \\end{align*}"}
{"id": "774.png", "formula": "\\begin{align*} \\beta f ' _ { 0 } ( r ^ m ) + ( 1 - \\beta ) f _ { 0 } ( r ^ m ) + \\sum _ { n = 1 } ^ { \\infty } M ( n ) \\phi _ { n } ( r ) = - f _ { 0 } ( - 1 ) , \\end{align*}"}
{"id": "7251.png", "formula": "\\begin{align*} & \\angle _ { f ( \\gamma _ j \\gamma ) } ( \\xi , f ( e ) ) = \\angle _ { f ( \\gamma _ j \\gamma ) } ( \\xi , q _ j ) + \\angle _ { f ( \\gamma _ j \\gamma ) } ( q _ j , f ( e ) ) , \\\\ & \\angle _ { f ( \\gamma _ j \\gamma ) } ( \\xi , q _ j ) = \\angle _ { q _ j } ( \\xi , f ( \\gamma _ j \\gamma ) ) = \\frac { \\pi } { 2 } . \\end{align*}"}
{"id": "7867.png", "formula": "\\begin{align*} \\delta ( x y ) = x \\delta ( y ) + \\delta ( x ) y \\ \\ \\ \\ \\ x , y \\in D ( \\delta ) . \\end{align*}"}
{"id": "6547.png", "formula": "\\begin{align*} \\P \\{ \\theta _ \\infty < \\infty \\} = 0 , \\end{align*}"}
{"id": "7937.png", "formula": "\\begin{align*} K _ { \\tilde X } = f _ - ^ * K _ { X _ - } + \\left ( 1 + \\sum _ { i \\in M _ - } D _ i \\cdot e \\right ) E = f _ + ^ * K _ { X _ + } + \\left ( 1 - \\sum _ { i \\in M _ + } D _ i \\cdot e \\right ) E . \\end{align*}"}
{"id": "6952.png", "formula": "\\begin{align*} ( ( W - z I ) ^ { - 1 } p , p ) = \\int _ \\R \\frac { d \\rho ( s ) } { s - z } \\forall z \\in \\C \\setminus \\R . \\end{align*}"}
{"id": "4055.png", "formula": "\\begin{align*} & q _ { i j } = q _ { j i } ^ { - 1 } , & & q _ { i i } = 1 . \\end{align*}"}
{"id": "7902.png", "formula": "\\begin{align*} \\nabla _ { \\vec s , k } L ( t , z ) \\alpha = 0 , \\end{align*}"}
{"id": "4651.png", "formula": "\\begin{align*} p _ t ( x ) = t ^ { - d / \\alpha } p _ 1 ( t ^ { - { 1 / \\alpha } } x ) \\ , . \\end{align*}"}
{"id": "6520.png", "formula": "\\begin{align*} \\ell ^ { ( A ) } _ x = \\max \\Big \\{ k \\in \\Z _ + : \\exists t > 0 , j \\in \\overline { 1 , \\eta ( x ) } t \\leq \\sum \\limits ^ { x + S _ { t } ^ { ( x , j ) } } _ { z = x + 1 } \\frac { 1 } { A ( z ) } S _ { t } ^ { ( x , j ) } \\geq k \\Big \\} \\vee 0 . \\end{align*}"}
{"id": "4508.png", "formula": "\\begin{align*} \\nabla _ 3 k _ { \\alpha 3 } - \\nabla _ \\alpha k _ { 3 3 } = - \\nabla _ 3 \\frac { \\nabla ^ 2 _ { \\alpha 3 } u } { | \\nabla u | } + \\nabla _ \\alpha \\frac { \\nabla ^ 2 _ { 3 3 } u } { | \\nabla u | } = R _ { \\alpha 3 3 3 } = 0 \\end{align*}"}
{"id": "4943.png", "formula": "\\begin{align*} \\frac { \\beta \\sigma _ k ^ { 1 1 } } { k \\sigma _ k } \\theta _ 1 ^ 2 \\le \\frac { \\beta \\sigma _ k ^ { 1 1 } } { k \\sigma _ k } a _ { 1 1 } \\le \\beta . \\end{align*}"}
{"id": "3740.png", "formula": "\\begin{align*} E B ^ 2 ( t , s , x , \\omega , v ) = \\big ( \\sum _ { i = 1 , 2 , 3 } - ( \\hat { v } + \\omega ) _ 2 ( \\omega \\times \\mathbf { e } _ 3 ) \\cdot \\mathbf { e } _ i ( B \\cdot ( \\omega \\times \\mathbf { e } _ i ) ) - ( \\hat { v } + \\omega ) _ 3 B _ 2 , \\sum _ { i = 1 , 2 , 3 } ( \\hat { v } + \\omega ) _ 1 ( \\omega \\times \\mathbf { e } _ 3 ) \\cdot \\mathbf { e } _ i ( B \\cdot ( \\omega \\times \\mathbf { e } _ i ) ) \\end{align*}"}
{"id": "6933.png", "formula": "\\begin{align*} R ^ 2 - R _ 1 ^ 2 = p p ^ * . \\end{align*}"}
{"id": "1272.png", "formula": "\\begin{align*} \\Psi ( E _ j ( x , a ) ) & = 4 \\sqrt { a } \\sum _ { h = 1 } ^ { \\lfloor \\frac { j } { 2 } \\rfloor } \\frac { \\zeta ^ h + \\zeta ^ { - h } } { 2 } \\\\ & = 2 \\sqrt { a } \\sum _ { h = 1 } ^ { \\lfloor \\frac { j } { 2 } \\rfloor } \\left ( \\zeta ^ h + \\zeta ^ { - h } \\right ) \\\\ & = 2 \\sqrt { a } \\left ( \\left ( \\sum _ { h = - \\lfloor \\frac { j } { 2 } \\rfloor } ^ { \\lfloor \\frac { j } { 2 } \\rfloor } \\zeta ^ h \\right ) - 1 \\right ) . \\end{align*}"}
{"id": "6831.png", "formula": "\\begin{align*} p _ k ( t , g ) = e ^ { i k ( \\frac { 2 \\pi } { V _ F } ) g } ( p _ { t , k } \\ast G _ { t , k } ) ( g ) . \\end{align*}"}
{"id": "5895.png", "formula": "\\begin{align*} \\nu = \\mathcal { D } \\Big ( \\Big ( \\frac { 1 } { \\beta } \\Big ) ^ { 1 / p } \\sum _ { j = 1 } ^ \\infty A _ { \\bullet , j } Z _ j + \\sigma _ { p , \\beta } ( I _ k - A A ^ T ) ^ { 1 / 2 } N _ k \\Big ) \\end{align*}"}
{"id": "4819.png", "formula": "\\begin{align*} \\begin{aligned} & S ( p ) ^ d \\ ( \\prod _ { j = 1 } ^ d { \\lambda + k _ j \\choose k _ j } \\ ) \\abs { W ( ( \\pi _ j ) _ j , \\alpha ) - W ( ( \\pi _ j ) _ j , \\alpha ^ \\prime ) } \\\\ & = \\abs { \\prod _ { j = 1 } ^ d \\ ( \\tilde S ( p , \\pi _ j , \\alpha ^ \\prime , \\lambda ) - ( \\tilde S ( p , \\pi _ j , \\alpha ^ \\prime , \\lambda ) - \\tilde S ( p , \\pi _ j , \\alpha , \\lambda ) ) \\ ) - \\prod _ { j = 1 } ^ d \\tilde S ( p , \\pi _ j , \\alpha ^ \\prime , \\lambda ) } . \\end{aligned} \\end{align*}"}
{"id": "5610.png", "formula": "\\begin{align*} ( - 1 ) ^ { ( P _ 2 z - 1 ) / 2 + ( x - 1 ) / 2 + ( y - 1 ) / 2 } \\left ( \\frac { x y } { P _ 2 z } \\right ) \\left ( \\frac { P _ 2 z x } { y } \\right ) \\left ( \\frac { P _ 2 z y } { x } \\right ) = 1 . \\end{align*}"}
{"id": "8422.png", "formula": "\\begin{gather*} \\frac { | C _ { \\varphi } ( D ) | ^ 2 } { \\langle \\varphi , \\varphi \\rangle } = \\frac { \\sqrt { | D | } } { 2 \\pi } \\frac { L ( f \\otimes D , 1 ) } { \\langle f , f \\rangle } \\end{gather*}"}
{"id": "6322.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) - \\dfrac { x } { 1 - q } D _ q y ( x ) + \\frac { [ n ] _ q } { 1 - q } y ( x ) = 0 . \\end{align*}"}
{"id": "426.png", "formula": "\\begin{align*} D = \\{ x + i y : x \\in ( \\alpha - \\varepsilon , \\alpha + \\varepsilon ) | y - f ( x ) | < \\delta \\} , \\end{align*}"}
{"id": "6387.png", "formula": "\\begin{align*} \\pi \\wedge \\rho : = \\{ V \\cap W \\mid V \\in \\pi , \\ , W \\in \\rho , \\ , V \\cap W \\neq \\emptyset \\} . \\end{align*}"}
{"id": "8585.png", "formula": "\\begin{align*} g ( x ) \\ge 0 , \\int _ 0 ^ 1 g ( x ) d x = 1 , \\end{align*}"}
{"id": "1367.png", "formula": "\\begin{align*} \\widehat { T } [ W _ t u _ 0 , v , \\cdots , v ] _ k = \\sum _ { \\ell \\ , \\ , o d d } \\widehat { T _ \\ell } [ v , \\cdots , v , \\underset { \\ell ' t h } { \\underbrace { W _ t u _ 0 } } , v , \\cdots , v ] _ k = \\frac { p + 1 } { 2 } \\widehat { T _ 1 } [ W _ t u _ 0 , v , \\cdots , v ] _ k , \\end{align*}"}
{"id": "2841.png", "formula": "\\begin{align*} \\begin{aligned} \\big [ 2 - h ( 1 + \\kappa ) \\big ] \\big ( f _ { 0 } - f _ { 1 } \\big ) { } \\geq { } \\frac { h } { 2 L } \\big [ \\kappa h ^ 2 - 2 h ( 1 + \\kappa ) + 3 \\big ] \\big \\| g _ { 0 } \\big \\| ^ 2 + \\frac { h } { 2 L } \\| g _ 1 \\| ^ 2 \\end{aligned} \\end{align*}"}
{"id": "2383.png", "formula": "\\begin{align*} \\int _ G f ( x ) \\ , d \\mu _ G ( x ) = \\Delta _ G ( y ) \\int _ G f ( x y ) \\ , d \\mu _ G ( x ) , \\end{align*}"}
{"id": "3126.png", "formula": "\\begin{align*} \\eta ^ 2 ( \\mathcal { T } , \\mathcal { M } ) : = \\big ( \\eta ( \\mathcal { T } , \\mathcal { M } ) \\big ) ^ 2 : = \\sum _ { T \\in \\mathcal { M } } \\eta ^ 2 ( \\mathcal { T } , T ) \\quad \\eta ^ 2 ( \\mathcal { T } ) : = \\eta ( \\mathcal { T } , \\mathcal { T } ) . \\end{align*}"}
{"id": "1682.png", "formula": "\\begin{align*} \\| ( 1 , 0 ) \\| _ Z = \\| ( 0 , 1 ) \\| _ Z = 1 \\hbox { a n d } \\| ( a , b ) \\| _ Z = \\| ( \\pm a , \\pm b ) \\| _ Z . \\end{align*}"}
{"id": "5684.png", "formula": "\\begin{align*} x _ 1 ^ { d + 1 } + x _ 2 ^ { d + 1 } + \\cdots + x _ i ^ { d + 1 } = \\sum _ { j = 1 } ^ i \\varpi _ j ( x _ j ^ { d - 1 } - x _ { j + 1 } ^ { d - 1 } ) x _ { j + 1 } + \\varpi _ i x _ { i + 1 } ^ d . \\end{align*}"}
{"id": "3562.png", "formula": "\\begin{align*} w _ 2 ( \\pi ) & = \\left ( \\sum _ { 1 \\leq i < j \\leq n } N _ i N _ j c _ i c _ j + \\sum _ { i = 1 } ^ { n } N _ i \\binom { c _ i } { 2 } + \\sum _ { i = 1 } ^ { n } c _ i ^ 2 \\binom { N _ i } { 2 } \\right ) \\left ( \\sum _ { t = 1 } ^ { n } v _ t ^ 2 \\right ) \\\\ & + \\left ( \\sum _ { i = 1 } ^ { n } c _ i N _ i '' + 2 \\sum \\limits _ { 1 \\leq i < j \\leq n } N _ i N _ j c _ i c _ j \\right ) \\cdot \\left ( \\sum _ { k < l } v _ k v _ l \\right ) . \\end{align*}"}
{"id": "3577.png", "formula": "\\begin{align*} m _ { \\pi _ 1 } & = \\frac { 1 } { 2 } \\left ( [ 5 ] _ q ! - [ 4 ] _ q ! ( 1 + 2 ( - 1 ) ^ i + 2 ( - 1 ) ^ j ) \\right ) \\\\ & = \\frac { 1 } { 2 } ( q + 1 ) ^ 2 ( q ^ 2 + 1 ) ( q ^ 2 + q + 1 ) \\left ( q ( q + 1 ) ( q ^ 2 + 1 ) - 2 ( - 1 ) ^ i - 2 ( - 1 ) ^ j \\right ) . \\end{align*}"}
{"id": "8163.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } \\| \\chi _ { S _ { v t } ^ c } e ^ { - i t H } \\psi \\| = 0 \\end{align*}"}
{"id": "1034.png", "formula": "\\begin{align*} \\widetilde { ( B + C ) } T ( t , A ) x = ( \\tilde { B } + \\tilde { C } ) T ( t , A ) x \\forall x \\in \\mathcal { H } , \\end{align*}"}
{"id": "841.png", "formula": "\\begin{align*} \\boldsymbol { \\theta } ^ { * } & = \\underset { \\boldsymbol { \\theta } } { } \\ln p ( \\boldsymbol { y } | \\boldsymbol { \\theta } ) . \\end{align*}"}
{"id": "8238.png", "formula": "\\begin{align*} \\phi _ { \\pm } = A _ { \\pm } e ^ { + k _ { \\pm } x } + B _ { \\pm } e ^ { - k _ { \\pm } x } + C _ { \\pm } e ^ { + k ' _ { \\pm } x } + D _ { \\pm } e ^ { - k ' _ { \\pm } x } , \\end{align*}"}
{"id": "3661.png", "formula": "\\begin{align*} \\psi _ { k } ( x ) : = \\tilde { \\psi } ( | x | / 2 ^ k ) - \\tilde { \\psi } ( | x | / 2 ^ { k - 1 } ) , \\varphi _ 0 ( x ) : = \\tilde { \\psi } ( | x | ) , \\quad \\forall j \\in ( 0 , \\infty ) \\cap \\Z , \\varphi _ j ( x ) : = \\psi _ j ( x ) . \\end{align*}"}
{"id": "1758.png", "formula": "\\begin{align*} \\mathcal { W } = \\{ x _ n \\} _ { n \\in \\mathbb { N } } \\cup \\{ u _ n \\} _ { n \\in \\mathbb { N } } \\end{align*}"}
{"id": "4319.png", "formula": "\\begin{align*} \\int _ L \\Omega = \\int _ { \\mathcal { M } } \\tilde { \\Omega } _ { L } , \\int _ { L _ i } \\Omega = \\int _ { \\mathcal { M } } \\tilde { \\Omega } _ { L _ i } , i = 1 , 2 . \\end{align*}"}
{"id": "1795.png", "formula": "\\begin{align*} 0 \\in \\begin{bmatrix} 0 & A ^ T \\\\ - A & 0 \\end{bmatrix} \\begin{bmatrix} x \\\\ z \\end{bmatrix} + \\begin{bmatrix} \\partial f ( x ) + \\nabla h ( x ) \\\\ \\partial g ^ \\ast ( z ) \\end{bmatrix} . \\end{align*}"}
{"id": "5998.png", "formula": "\\begin{align*} e ^ { \\max } = \\max _ { j = 1 , \\dots , N } \\| P _ h u ( t _ j ) - U _ j \\| _ { L ^ 2 ( \\Omega ) } , \\end{align*}"}
{"id": "6309.png", "formula": "\\begin{align*} & \\int x ^ { n - 2 } ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty h _ n ( x ; q ) d _ q x = \\frac { x ^ { n } ( x ^ 2 ; q ^ 2 ) _ \\infty } { [ n - 1 ] _ q } \\left ( \\frac { h _ n ( \\frac { x } { q } ; q ) } { x } - \\frac { 1 } { q } h _ { n - 1 } ( \\frac { x } { q } ; q ) \\right ) . \\end{align*}"}
{"id": "5567.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { 1 } { x } \\int _ 1 ^ \\infty [ f ( e ^ { - y } x ) - f ( x ) + ( e ^ { - y } - 1 ) x f ' ( x ) ] \\Pi ( d y ) = 0 . \\end{align*}"}
{"id": "2588.png", "formula": "\\begin{align*} \\alpha _ { x , y } : = \\psi _ { x , y } ^ { - 1 } \\circ \\psi _ { y , x } \\circ \\iota : \\mathcal O _ { x , y } \\to \\mathcal O _ { x , y } \\ , \\end{align*}"}
{"id": "2884.png", "formula": "\\begin{align*} g ( f ( x _ 1 , \\ldots , x _ k ) , f ( x _ 2 , \\ldots , x _ { k + 1 } ) , \\ldots ) = x _ 1 . \\end{align*}"}
{"id": "890.png", "formula": "\\begin{align*} V = \\{ a _ 0 + a _ 1 z + a _ 2 z ^ 2 B ( z ) + a _ 3 z ^ 3 B ( z ) + \\cdots \\ : : \\ : \\sum \\abs { a _ j } ^ 2 < \\infty \\} \\end{align*}"}
{"id": "5404.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) = - \\frac { C _ { n , s } } { 2 } \\int _ { \\R ^ n } \\frac { \\delta u ( x , y ) } { | y | ^ { n + 2 s } } \\ , d y \\quad x \\in \\R ^ n . \\end{align*}"}
{"id": "8079.png", "formula": "\\begin{align*} g ^ { - 1 } = ( g ^ { i j } ) = \\begin{pmatrix} 1 & 0 \\\\ 0 & c _ { 1 } ^ { - 2 } \\end{pmatrix} . \\end{align*}"}
{"id": "4539.png", "formula": "\\begin{align*} \\Vert D _ { m , n } \\Vert _ { p } ^ { ( v , v ) } : = \\Vert D _ { m , n } \\Vert _ { p } + \\sup _ { z _ { 1 } \\neq 0 , \\ , \\ , z _ { 2 } \\neq 0 } \\frac { \\Vert D _ { m , n } ( x + z _ { 1 } , y + z _ { 2 } ) - D _ { m , n } ( x , y ) \\Vert _ { p } } { v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) } . \\end{align*}"}
{"id": "1375.png", "formula": "\\begin{align*} s > \\frac { 2 p - 6 } { 2 ( 2 p - 2 ) } = \\frac { p - 3 } { 2 ( p - 1 ) } . \\end{align*}"}
{"id": "4402.png", "formula": "\\begin{align*} s ( A ) = \\{ f \\in A : g \\in A \\mbox { f o r a l l h o l o m o r p h i c } g \\mbox { w i t h } | \\hat { g } ( k ) | \\leq | \\hat { f } ( k ) | \\mbox { f o r a l l } k \\} \\end{align*}"}
{"id": "6981.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { \\lambda _ N ^ 2 } \\prod \\limits _ { k = 1 } ^ { \\infty } \\Bigg ( \\frac { \\lambda _ N ^ 2 + \\mu _ { k } ^ { 2 } } { \\lambda _ N ^ 2 + \\lambda _ { k } ^ { 2 } } \\Bigg ) = \\infty \\end{align*}"}
{"id": "2014.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : F P g e n e r a t o r } \\forall \\varphi \\in C ^ 2 _ b ( \\R ^ d ) , L _ \\mu \\varphi ( x ) : = b ( x , \\mu ) \\cdot \\nabla \\varphi + \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ d a _ { i j } ( x , \\mu ) \\partial _ { x _ i } \\partial _ { x _ j } \\varphi , \\end{align*}"}
{"id": "1390.png", "formula": "\\begin{align*} G ( v , x ) = \\frac { F } { \\bigtriangleup + i \\gamma } . \\end{align*}"}
{"id": "7082.png", "formula": "\\begin{align*} \\begin{aligned} \\mathsf { T } _ { 2 } ( t _ { k } ) & = \\frac { \\phi \\lambda - \\sum _ { i = 1 } ^ { n } \\beta _ { i } u _ { i } ( t _ { k } ) } { \\sum _ { i = 1 } ^ { n } \\alpha _ { i } } , \\\\ u _ i ( t _ { k + 1 } ) & = \\beta _ { i } u _ i ( t _ { k } ) + \\alpha _ { i } \\mathsf { T } _ { 2 } ( t _ { k } ) . \\end{aligned} \\end{align*}"}
{"id": "8001.png", "formula": "\\begin{align*} H _ { ( X , K 3 ) } ( y _ 1 , y _ 2 ) = e ^ { \\frac { H \\log y _ 1 + P \\log y _ 2 } { 2 \\pi i } } \\sum _ { d _ 1 , d _ 2 \\geq 0 } y ^ d \\frac { \\Gamma ( 1 + \\frac { 4 H } { 2 \\pi i } + 4 d _ 1 ) \\Gamma ( 1 + \\frac { H + P } { 2 \\pi i } + d _ 1 + d _ 2 ) } { \\Gamma ( 1 + \\frac { H } { 2 \\pi i } + d _ 1 ) ^ 5 \\Gamma ( 1 + \\frac { P } { 2 \\pi i } + d _ 2 ) } [ \\textbf { 1 } ] _ { ( d _ 1 , 0 ) } , \\end{align*}"}
{"id": "851.png", "formula": "\\begin{align*} \\mathcal { P } : & \\underset { \\left \\{ \\boldsymbol { v } _ { 2 , p } \\right \\} _ { p = 1 } ^ { P _ { 2 } } } { \\max } \\lambda \\\\ & \\textrm { t r } \\left ( \\boldsymbol { v } _ { 2 , p } \\boldsymbol { v } _ { 2 , p } ^ { H } \\right ) \\leq P _ { t } , \\\\ & \\mathbf { J } _ { \\textrm { e f f } } \\left ( \\left \\{ \\boldsymbol { v } _ { 2 , p } \\right \\} _ { p = 1 } ^ { P _ { 2 } } \\right ) \\succeq \\lambda \\mathbf { I } , \\end{align*}"}
{"id": "7311.png", "formula": "\\begin{align*} w _ 2 = \\sum _ { j = 0 } ^ J a _ j ( \\tau ) e _ j ( z ) + w _ 2 ^ \\bot a _ j ( \\tau ) = ( w _ 2 ( \\cdot , \\tau ) , e _ j ) _ \\rho . \\end{align*}"}
{"id": "4583.png", "formula": "\\begin{align*} ( C _ R ( J ) ) ^ h = C _ { R [ y ] } ( J ^ h ) . \\end{align*}"}
{"id": "6096.png", "formula": "\\begin{align*} { \\bar \\Omega } _ { A B } - k _ A k _ B \\theta _ A \\wedge \\theta _ B = \\Omega _ { A B } \\end{align*}"}
{"id": "2046.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : O m e g a k } \\Omega _ k \\left ( f ^ N , f \\right ) : = W _ p { \\left ( f ^ { k , N } , f ^ { \\otimes k } \\right ) } \\underset { N \\to + \\infty } { \\longrightarrow } 0 . \\end{align*}"}
{"id": "7844.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in M , \\| x \\| \\leq 1 } | \\langle x \\xi u _ n , \\eta \\rangle | = 0 . \\end{align*}"}
{"id": "5999.png", "formula": "\\begin{align*} e ^ { \\max } _ { } = \\max _ { j = 1 , \\dots , N } E ^ 2 _ { } ( t _ j ) , \\end{align*}"}
{"id": "5943.png", "formula": "\\begin{align*} { w _ \\alpha } \\left ( { { \\eta _ 1 } , . . . , { \\eta _ d } } \\right ) = \\mathop \\sum \\limits _ { n = 1 } ^ \\infty \\mathop \\sum \\limits _ { { i _ 1 } . . . { i _ n } } \\frac { w _ { { i _ 1 } . . . . { i _ n } } ^ { \\left ( n \\right ) } } { { n ! } } { \\eta _ { { i _ 1 } } } . . . { \\eta _ { { i _ n } } } \\end{align*}"}
{"id": "1243.png", "formula": "\\begin{align*} \\mathcal { G } _ { l ( T ) } & = x , \\\\ \\mathcal { G } _ j & = x - \\dfrac { c _ j } { \\mathcal { G } _ { j + 1 } } ( \\forall j \\in \\overline { 1 , l ( T ) - 1 } ) , \\end{align*}"}
{"id": "2720.png", "formula": "\\begin{align*} [ b _ { r , \\alpha } , b ^ * _ { r , \\beta } ] = [ b _ { r , \\alpha } , c _ { \\ell , \\beta } ] = [ b _ { r , \\alpha } , c ^ * _ { \\ell , \\beta } ] = \\ ; & 0 \\\\ [ b _ { r , \\alpha } , b ^ * _ { r , \\beta } ] = [ c _ { \\ell , \\alpha } , c ^ * _ { \\ell , \\beta } ] = \\ ; & \\delta _ { \\alpha , \\beta } . \\end{align*}"}
{"id": "886.png", "formula": "\\begin{align*} \\abs { m } = \\abs { m _ 1 } + \\cdots + \\abs { m _ n } , \\end{align*}"}
{"id": "4507.png", "formula": "\\begin{align*} \\nabla _ \\beta k _ { \\alpha \\beta } - \\nabla _ \\alpha k _ { \\beta \\beta } + \\nabla _ 3 k _ { \\alpha 3 } - \\nabla _ \\alpha k _ { 3 3 } = 0 . \\end{align*}"}
{"id": "7945.png", "formula": "\\begin{align*} y ^ d \\mapsto \\prod _ { i = 1 } ^ { \\mathrm { r } } \\tilde y _ i ^ { p _ i ^ - \\cdot d } . \\end{align*}"}
{"id": "8825.png", "formula": "\\begin{align*} t : = \\phi _ 1 ^ * * \\phi _ 2 ^ * ( x _ 0 ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } T ( t _ 1 , t _ 2 ) d t _ 1 d t _ 2 \\end{align*}"}
{"id": "166.png", "formula": "\\begin{align*} \\psi = C _ { ( m ) } ^ 3 - { \\bf 1 } ^ \\epsilon _ 0 \\| C _ { ( m ) } ^ 3 \\| _ { L ^ 1 } , \\end{align*}"}
{"id": "96.png", "formula": "\\begin{align*} \\forall \\varphi \\in \\R ^ \\Lambda , V _ t ( \\varphi ) = - \\log { \\bf E } _ { C _ t } \\Big [ e ^ { - V _ 0 ( \\varphi + \\cdot ) } \\Big ] . \\end{align*}"}
{"id": "6070.png", "formula": "\\begin{align*} \\emph { c o l s u m } _ 0 ( M ) = \\mu _ { 1 , 0 } + \\mu _ { 2 , 0 } = n . \\end{align*}"}
{"id": "6271.png", "formula": "\\begin{align*} \\int ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty h _ n ( x ; q ) d _ q x = - q ^ { n - 1 } ( 1 - q ) ( x ^ 2 ; q ^ 2 ) _ \\infty h _ { n - 1 } ( \\frac { x } { q } ; q ) . \\end{align*}"}
{"id": "5399.png", "formula": "\\begin{align*} ( ( - \\Delta ) ^ s + q ) v & = F \\quad \\quad \\Omega , \\\\ v & = f \\quad \\quad \\Omega _ e , \\end{align*}"}
{"id": "2138.png", "formula": "\\begin{align*} d \\sigma _ { \\mu } ^ { + } ( s ) = p c \\ , s ^ { - ( a + 1 ) } \\ , d \\lambda ( s ) , \\quad 1 \\leq s < + \\infty , \\end{align*}"}
{"id": "3748.png", "formula": "\\begin{align*} U _ { { j } ; { l } } ^ { { m } ; \\tilde { p } , \\tilde { q } } ( t , x ) : = 2 ^ { { m } - { j } - 2 { l } } \\big ( \\int _ { 0 } ^ { t } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\varphi _ { { m } ; - 1 0 M _ t } ( t - s ) \\big [ | E ( s , x + ( t - s ) \\omega ) - \\omega \\times B ( s , x + ( t - s ) \\omega ) | ^ 2 + | \\omega \\cdot B ( s , x + ( t - s ) \\omega ) | ^ 2 \\big ] \\end{align*}"}
{"id": "2855.png", "formula": "\\begin{align*} \\begin{aligned} h _ * = \\arg \\max _ { 1 { } \\leq { } h < 2 } 2 h + \\frac { \\kappa h ^ 3 } { 2 - h \\left ( 1 + \\kappa \\right ) } \\end{aligned} , \\end{align*}"}
{"id": "5104.png", "formula": "\\begin{align*} Y ^ { n , 1 } _ t = n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { t } ( t - s ) ^ \\alpha \\xi ^ n _ s d W _ s , \\end{align*}"}
{"id": "616.png", "formula": "\\begin{align*} f ( x , y , n , z ) & : = \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi x } { n } \\Big ) + \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi y } { n } \\Big ) + z ^ 2 , \\\\ g ( x , y , n , z ) & : = \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi x } { n } \\Big ) + \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi y } { n } \\Big ) + z ^ 2 \\\\ & - \\frac { 2 n ^ 2 } { 3 \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi x } { n } \\Big ) \\sin ^ 2 \\Big ( \\frac { \\pi y } { n } \\Big ) . \\end{align*}"}
{"id": "6879.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d B ( t ) } { d t } & = g _ 0 + g _ 1 N ( B , C ) - B ( t ) + g _ 1 \\epsilon ( t ) , \\\\ \\frac { d C ( t ) } { d t } & = 2 a _ 0 + 2 a _ 1 N ( B , C ) - 2 C ( t ) + 2 a _ 1 \\epsilon ( t ) . \\end{aligned} \\end{align*}"}
{"id": "2626.png", "formula": "\\begin{align*} \\frac { d F } { d t } = [ F , H ] + \\frac { \\partial F } { \\partial t } \\ , \\end{align*}"}
{"id": "2056.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : L N h a t } \\widehat { \\mathcal { L } } _ N \\Phi ( \\mu _ { \\mathbf { x } ^ N } ) = \\mathcal { L } _ N [ \\Phi \\circ \\boldsymbol { \\mu } _ N ] ( \\mathbf { x } ^ N ) . \\end{align*}"}
{"id": "2234.png", "formula": "\\begin{align*} \\sqrt { e ^ { - 2 \\pi t } \\sin ^ 2 { \\pi \\sigma } + ( 1 \\pm e ^ { - \\pi t } \\cos { \\pi \\sigma } ) ^ 2 } = \\sqrt { 1 + e ^ { - 2 \\pi t } \\pm 2 e ^ { - \\pi t } \\cos { \\pi \\sigma } } & \\geq \\sqrt { 1 + e ^ { - 2 \\pi t } - 2 e ^ { - \\pi t } } \\\\ & = \\sqrt { ( 1 - e ^ { - \\pi t } ) ^ 2 } \\\\ & = 1 - e ^ { - \\pi t } . \\end{align*}"}
{"id": "5623.png", "formula": "\\begin{align*} X ^ { 2 \\frac { 1 } { 2 ^ k } } + X ^ { \\frac { 1 } { 2 ^ k } } + 1 = \\big ( X ^ { 2 \\frac { 1 } { 2 ^ { k + 1 } } } + X ^ { \\frac { 1 } { 2 ^ { k + 1 } } } + 1 \\big ) ^ 2 \\end{align*}"}
{"id": "1732.png", "formula": "\\begin{align*} G ( z + \\omega _ 2 \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) \\cdot G ( z \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , - \\omega _ 2 ) = \\prod _ { k _ 1 \\geq 0 , k _ 2 \\geq 0 } \\big ( 1 - x _ 2 q _ 2 ^ { k _ 1 + \\frac { 1 } { 2 } } \\widetilde { q } _ 2 ^ { k _ 2 + \\frac { 1 } { 2 } } \\big ) \\cdot \\prod _ { k _ 1 \\geq 0 , k _ 2 \\geq 0 } \\big ( 1 - x _ 2 ^ { - 1 } q _ 2 ^ { k _ 1 + \\frac { 1 } { 2 } } \\widetilde { q } _ 2 ^ { k _ 2 + \\frac { 1 } { 2 } } \\big ) . \\end{align*}"}
{"id": "7961.png", "formula": "\\begin{align*} I _ { D _ - , d } = \\frac { 1 } { \\prod _ { i \\in I _ - , D _ i \\cdot d > 0 } ( \\bar D _ i + ( D _ i \\cdot d ) z ) } [ \\textbf { 1 } ] _ { ( - D _ i \\cdot d ) _ { i \\in I _ - } } . \\end{align*}"}
{"id": "2072.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : r e c u r s i v e j u m p i n g t i m e s } t _ { k + 1 } = \\max _ { \\ell , p , n } \\big \\{ T ^ { i _ \\ell , p } _ n \\ , | \\ , T ^ { i _ \\ell , p } _ n < t _ k , \\ , \\ell \\leq k \\big \\} . \\end{align*}"}
{"id": "1332.png", "formula": "\\begin{align*} v ' = v - ( u \\cdot \\omega ) \\omega , \\mbox { a n d } v _ * ' = v _ * - ( u \\cdot \\omega ) \\omega . \\end{align*}"}
{"id": "8960.png", "formula": "\\begin{align*} & \\big \\| \\int _ 0 ^ t e ^ { i ( t - s ) \\Delta } [ F ( u + w ) ( s ) - F ( u ( s ) ) ] d s \\big \\| _ { X ^ s ( [ 0 , T ] ) } \\\\ & \\lesssim \\| w \\| _ { X ^ s ( [ 0 , T ] ) } \\big ( \\| u \\| _ { X ^ s ( [ 0 , T ] ) } + \\| w \\| _ { X ^ s ( [ 0 , T ] ) } \\big ) ^ { \\frac { 4 } { d - 2 } } . \\end{align*}"}
{"id": "6197.png", "formula": "\\begin{align*} L _ { Z } \\cdot C = \\int _ { C } R _ { h _ { Z } } \\geq 0 . \\end{align*}"}
{"id": "1872.png", "formula": "\\begin{align*} \\mathcal { D } _ { [ n , j , k ] } : = \\{ \\gamma \\in \\mathcal { D } _ { [ n , j ] } : \\kappa _ { 0 } ( \\gamma ) = k \\} , 0 \\leq k \\leq n - j . \\end{align*}"}
{"id": "4902.png", "formula": "\\begin{align*} \\lvert v _ { i , j } \\rvert ^ 2 \\prod _ { k = 1 ; k \\ne i } ^ { n - 1 } ( { \\lambda _ i } ( A ) - { \\lambda _ k } ( A ) ) = \\prod _ { k = 1 } ^ { n - 1 } ( { \\lambda _ i } ( A ) - { \\lambda _ k } ( M _ j ) ) . \\end{align*}"}
{"id": "891.png", "formula": "\\begin{align*} \\norm { f _ m } ^ 2 = \\norm { g _ 0 } ^ 2 + \\norm { g _ 1 } ^ 2 + \\cdots + \\norm { g _ m } ^ 2 \\leq \\norm { r h } ^ 2 \\end{align*}"}
{"id": "2.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & - 1 & 1 & 1 \\\\ 1 & 1 & - 1 & 1 \\\\ 1 & 1 & 1 & - 1 \\end{pmatrix} \\end{align*}"}
{"id": "8892.png", "formula": "\\begin{align*} a _ { k + 1 } = \\left ( a _ k ^ 2 - | \\det ( a _ k ^ 2 I _ n - A ^ H A ) | \\left ( \\frac { n - 1 } { ( n + 1 ) a _ k ^ 2 - \\| A \\| _ F ^ 2 } \\right ) ^ { n - 1 } \\right ) ^ { 1 / 2 } , k = 1 , 2 , \\cdots , \\end{align*}"}
{"id": "1599.png", "formula": "\\begin{align*} - f ( x ^ 1 ) f '' ( x ^ 1 ) + f '^ 2 ( x ^ 1 ) + 1 = 0 . \\end{align*}"}
{"id": "7132.png", "formula": "\\begin{align*} p _ i : = x _ { a _ i b _ i } \\prod _ { j = 1 } ^ { \\ell _ i - 1 } \\left ( - \\sum _ { u \\in S ( w _ { i j } ) } x _ { \\phi ( u ) } + \\sum _ { u \\in S ( v _ i ) } x _ { \\phi ( u ) } \\right ) . \\end{align*}"}
{"id": "3892.png", "formula": "\\begin{align*} | \\nabla H _ { \\lambda } | \\leq \\frac { C ' } { | x | ^ \\frac { N - 1 } { p - 1 } } = C | \\nabla \\Gamma | \\hbox { o n } \\partial B _ R ( 0 ) \\end{align*}"}
{"id": "8236.png", "formula": "\\begin{align*} \\sqrt { 1 + 1 6 i m \\beta \\lambda / 3 } = \\pm \\left ( \\sqrt { \\frac { \\sqrt { 1 ^ { 2 } + ( 1 6 m \\beta \\lambda / 3 ) ^ { 2 } } + 1 } { 2 } } + i \\sqrt { \\frac { \\sqrt { 1 ^ { 2 } + ( 1 6 m \\beta \\lambda / 3 ) ^ { 2 } } - 1 } { 2 } } \\right ) \\end{align*}"}
{"id": "1114.png", "formula": "\\begin{align*} C ( v ) = \\frac { \\sigma _ Z ^ 2 v ^ 2 } { 2 } . \\end{align*}"}
{"id": "662.png", "formula": "\\begin{align*} d \\nu ^ i _ t : = d \\nu ^ i _ { x _ i , s _ i \\ , | \\ , t } = v _ { i , t } \\ , d g _ { i , t } , \\end{align*}"}
{"id": "5875.png", "formula": "\\begin{align*} d ^ \\gamma ( { \\mathcal K } , Y _ n ) _ X : = \\inf _ { \\Phi _ n } \\sup _ { f \\in { \\mathcal K } } \\inf _ { y \\in B _ { Y _ n } } \\| f - \\Phi _ n ( y ) \\| _ X , \\end{align*}"}
{"id": "8986.png", "formula": "\\begin{align*} \\langle \\nabla u _ { q , r } , \\nabla \\nu \\rangle = \\frac { 1 + 2 \\delta } { ( 1 + \\delta ) ^ 2 } \\xi ^ 2 _ { \\tilde K } \\langle \\nabla w , \\nabla w \\rangle + ( 1 + u _ { q , r } ) ^ { 2 \\delta + 1 } \\langle \\nabla u _ { q , r } , \\nabla ( \\xi _ { \\tilde K } ) ^ 2 \\rangle , \\end{align*}"}
{"id": "3749.png", "formula": "\\begin{align*} \\times \\varphi _ { \\tilde { j } , \\tilde { l } } ( u , \\tilde { \\omega } ) d \\omega d \\tilde { \\omega } d u d s d \\tau \\lesssim \\sum _ { \\kappa \\in [ - 1 0 M _ t , \\epsilon M _ t ] \\cap \\Z } H _ { \\kappa } ( t , x ) , H _ { \\kappa } ( t , x ) : = \\int _ 0 ^ t \\int _ { \\tau } ^ { t } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\int _ { \\mathbb { S } ^ 2 } f ( \\tau , x + ( t - s ) \\omega + ( s - \\tau ) \\tilde { \\omega } , u ) \\end{align*}"}
{"id": "4298.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } L _ t = L _ 1 + \\ldots L _ N . \\end{align*}"}
{"id": "4799.png", "formula": "\\begin{align*} \\sigma _ j = \\sigma ^ * - \\frac { \\sigma ^ * j } { 2 k } , j = 1 , \\ldots , k . \\end{align*}"}
{"id": "5673.png", "formula": "\\begin{align*} \\ i _ * = \\ j _ * . \\end{align*}"}
{"id": "5285.png", "formula": "\\begin{align*} ( X , W ) = ( \\C ^ n , W = \\sum _ i x _ i ^ { r _ i } ) . \\end{align*}"}
{"id": "6279.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } q ^ { \\frac { k ( k - 3 ) } { 2 } } x ^ k S _ n ( q ^ k x ; q ) = \\frac { 1 } { 1 - q ^ n } S _ { n - 1 } ( q x ; q ) . \\end{align*}"}
{"id": "2109.png", "formula": "\\begin{align*} ( L y ) ( v ) & = - \\sum _ { e \\in E _ V } \\frac { \\delta _ E ( e ) } { \\delta _ V ( v ) } \\frac { 1 } { | e | ^ 2 } \\sum _ { u \\in e \\cap w } | V _ 2 | \\\\ & = - k ( v ) | V _ 2 | \\end{align*}"}
{"id": "2324.png", "formula": "\\begin{align*} C _ 1 & : = \\{ ( x , y , z ) ^ \\top \\in \\mathbb { R } ^ 3 \\colon 0 \\leq x , 0 \\leq y , 0 \\leq z \\leq x + y \\} , \\\\ C _ 2 & : = \\{ ( x , y , z ) ^ \\top \\in \\mathbb { R } ^ 3 \\colon 0 \\leq x , 0 \\geq y , y \\leq z \\leq x \\} , \\\\ C _ 3 & : = \\{ ( 0 , 0 , z ) ^ \\top \\in \\mathbb { R } ^ 3 \\colon 0 \\leq z \\} \\end{align*}"}
{"id": "5055.png", "formula": "\\begin{align*} N ^ { n , 4 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ^ 2 ( u , s ) d u \\ , d s . \\end{align*}"}
{"id": "1059.png", "formula": "\\begin{align*} \\theta _ 1 ( t ) = \\sum _ { n = \\ell + 1 } ^ \\infty \\psi _ { 1 } ^ { [ * n ] } ( t ) , \\end{align*}"}
{"id": "3883.png", "formula": "\\begin{align*} - \\Delta _ p z _ n = a _ n | \\hat { u } _ n | ^ { p - 2 } \\hat { u } _ n + \\hat { f } _ n \\hbox { i n } \\Omega . \\end{align*}"}
{"id": "4580.png", "formula": "\\begin{align*} I S [ x ] ^ { [ q ] } : I S [ x ] = ( I ^ { [ q ] } : I ) S [ x ] . \\end{align*}"}
{"id": "82.png", "formula": "\\begin{align*} h ( \\R \\times \\{ - \\infty \\} ) & = W _ 1 , h ( \\{ + \\infty \\} \\times \\R ) = W _ 2 , \\\\ h ( \\{ - \\infty \\} \\times \\R ) & = W _ 1 ' , h ( \\R \\times \\{ + \\infty \\} ) = W _ 2 ' . \\end{align*}"}
{"id": "2871.png", "formula": "\\begin{align*} P ^ f _ { H , d } = \\{ ( x _ k , 0 , ( p _ x ) _ k , ( p _ y ) _ k , x _ { k + 1 } , 0 , ( p _ x ) _ { k + 1 } , ( p _ y ) _ { k + 1 } ) \\in { \\mathbb R } ^ 8 \\} \\equiv { \\mathbb R } ^ 3 \\times { \\mathbb R } ^ 3 \\end{align*}"}
{"id": "2224.png", "formula": "\\begin{align*} K _ { \\emph { Z } } ^ { o r b } = \\varphi ^ { \\ast } ( K _ { Z } + [ \\Delta ] ) . \\end{align*}"}
{"id": "4531.png", "formula": "\\begin{align*} \\sigma _ { m , k ; n , \\ell } : = & \\left ( 1 + \\frac { m } { k } \\right ) \\left ( 1 + \\frac { n } { \\ell } \\right ) \\sigma _ { m + k - 1 , n + \\ell - 1 } - \\left ( 1 + \\frac { m } { k } \\right ) \\frac { n } { \\ell } \\sigma _ { m + k - 1 , n - 1 } \\\\ & - \\frac { m } { k } \\left ( 1 + \\frac { n } { \\ell } \\right ) \\sigma _ { m - 1 , n + \\ell - 1 } + \\frac { m n } { k \\ell } \\sigma _ { m - 1 , n - 1 } , \\end{align*}"}
{"id": "6305.png", "formula": "\\begin{align*} \\int f ( x ) h ( x / q ) S _ q ( x ) y ( x ) d _ q x = f ( x / q ) h ( x / q ) \\left ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \\right ) , \\end{align*}"}
{"id": "2439.png", "formula": "\\begin{align*} \\overline { \\widetilde { N } } _ f & = \\overline { M _ f S _ \\zeta } = \\overline { M } _ f S _ \\zeta = M _ f S _ \\zeta + \\sum _ { g \\prec f } r _ { g f } M _ g S _ \\zeta = \\widetilde { N } _ f + \\sum _ { g \\prec f ; g \\in \\Z ^ { m | n } _ { \\zeta - } } r ' _ { g f } \\widetilde { N } _ g . \\end{align*}"}
{"id": "5421.png", "formula": "\\begin{align*} 0 = & \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) u ^ { ( 1 ) } _ f u ^ { ( 2 ) } _ g \\ , d x \\\\ = & \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) ( u ^ { ( 1 ) } _ f - f ) ( u ^ { ( 2 ) } _ g - g ) \\ , d x + \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) ( u ^ { ( 1 ) } _ f - f ) g \\ , d x \\\\ & + \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) f ( u ^ { ( 2 ) } _ g - g ) \\ , d x + \\int _ { \\R ^ n } ( q _ 1 - q _ 2 ) f g \\ , d x . \\end{align*}"}
{"id": "5578.png", "formula": "\\begin{align*} L u = \\textrm { d i v } ( A \\nabla u ) , A = ( a _ { i j } ( x ) ) _ { i , j = 1 } ^ n , a _ { i j } = a _ { j i } \\in L ^ \\infty ( \\Omega ) , \\end{align*}"}
{"id": "4700.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ { k - 1 } } { ( q ; q ) _ \\infty } \\sum _ { n = 1 - k } ^ { k } ( - 1 ) ^ { n } q ^ { n ( 3 n - 1 ) / 2 } = ( - 1 ) ^ { k - 1 } + \\sum _ { n = k } ^ \\infty \\frac { q ^ { { k \\choose 2 } + ( k + 1 ) n } } { ( q ; q ) _ n } \\begin{bmatrix} n - 1 \\\\ k - 1 \\end{bmatrix} , \\end{align*}"}
{"id": "8917.png", "formula": "\\begin{align*} \\psi _ n ( x ) = e ^ { - \\gamma R } \\lambda _ n ^ { \\frac 1 { p - 2 } } e ^ { - \\gamma \\ , \\lambda _ n ^ { \\frac { 1 } { 2 } } | x - P ^ i _ n | } , \\end{align*}"}
{"id": "8794.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } f \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) ( x ) d x - \\int _ \\mathbb { R } f \\circ ( \\phi _ 1 * \\phi _ 2 ) ( x ) d x = 4 ( \\lambda f ( y _ 1 ) + ( 1 - \\lambda ) f ( y _ 2 ) - f ( \\lambda y _ 1 + ( 1 - \\lambda ) y _ 2 ) ) . \\end{align*}"}
{"id": "8368.png", "formula": "\\begin{align*} F ( k , s , t ) & = \\min \\{ N \\mid \\exists \\ , \\mathit { O A } ( N , k , s , t ) \\} , \\\\ F ^ * ( k , s , t ) & = \\min \\{ N \\mid \\exists \\mathit { O A } ( N , k , s , t ) \\} . \\\\ \\end{align*}"}
{"id": "2112.png", "formula": "\\begin{align*} & \\left ( 4 \\sum _ { e \\in E } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum \\limits _ { \\{ u , v \\} \\subset e \\cap V _ 1 } y ( u ) y ( v ) + \\alpha \\right ) \\\\ & = \\left ( 2 \\sum _ { e \\in E } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum \\limits _ { \\{ u , v \\} \\subset e \\cap V _ 1 } y ( u ) y ( v ) + \\sum _ { e \\in \\partial V _ 1 } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum \\limits _ { u \\in e \\cap V _ 2 ; v \\in e \\cap V _ 1 } z _ 2 ( u ) z _ 2 ( v ) \\right ) \\ge 0 . \\end{align*}"}
{"id": "4056.png", "formula": "\\begin{align*} & v _ a = \\frac { 1 + \\sigma _ a } 2 - \\frac { \\epsilon _ a } { \\omega } , & & a = 1 , \\ldots , k - 1 , \\end{align*}"}
{"id": "7743.png", "formula": "\\begin{align*} \\lambda _ * H ^ { ( 1 ) } ( \\mu _ N ) - \\lambda _ * \\Psi ( \\zeta ^ { \\binom { N } { 2 } } ) \\cong \\Psi _ * \\binom { N } { 2 } , \\end{align*}"}
{"id": "1026.png", "formula": "\\begin{align*} \\binom x { \\ 1 _ S } = \\begin{pmatrix} x _ 1 , & x _ 2 , & x _ 3 , & \\dots \\\\ \\ 1 _ S ( 1 ) , & \\ 1 _ S ( 2 ) , & \\ 1 _ S ( 3 ) , & \\dots \\end{pmatrix} . \\end{align*}"}
{"id": "2609.png", "formula": "\\begin{align*} \\nu = a + 2 k \\end{align*}"}
{"id": "794.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { r ^ m } \\frac { \\psi ( t ) } { 1 - t ^ 2 } + R ^ N ( r ) = \\int _ { 0 } ^ { 1 } \\dfrac { \\psi ( - t ) } { 1 + t ^ 2 } d t , \\end{align*}"}
{"id": "6806.png", "formula": "\\begin{align*} ( a u '' ) ' ( 1 ) = ( a u '' ) ' ( 0 ) = 0 ( a u '' ) ( 1 ) = ( a u '' ) ( 0 ) = 0 . \\end{align*}"}
{"id": "3744.png", "formula": "\\begin{align*} J ^ { e s s ; 1 } _ { k , m ' } ( t , x ) : = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big ( E ( s , x - y + ( t - s ) \\omega ) + \\hat { v } \\times B ( s , x - y + ( t - s ) \\omega ) \\big ) \\cdot \\nabla _ v \\mathcal { K } _ { k , j } ^ { e s s } ( y , \\omega , v ) \\end{align*}"}
{"id": "6672.png", "formula": "\\begin{align*} K ( i , t ) = & \\ \\ ( A _ 1 ( i , h _ 1 ) - A _ 1 ( i , h _ 2 ) ) \\overline { \\varphi } _ { 1 , i i } \\\\ & + [ ( B _ 1 ( i , h _ 1 ) - B _ 1 ( i , h _ 2 ) ) + ( h ' _ { 1 } C _ 1 ( i , h _ 1 ) - h ' _ 2 C ( i , h _ 2 ) ) + ( D _ 1 ( i , h _ 1 ) - D _ 1 ( i , h _ 2 ) ) ] \\overline { \\varphi } _ { 1 , i } \\\\ & + a _ { 0 1 } ( \\chi _ 1 ^ { p } - \\chi _ 2 ^ { p } ) + \\lambda _ { 0 1 } \\big [ \\sqrt { A _ 1 ( i , h _ 1 ) } ^ { \\alpha } | \\varphi _ { 1 , i } | ^ { \\alpha } - \\sqrt { A _ 1 ( i , h _ 2 ) } ^ { \\alpha } | \\varphi _ { 2 , i } | ^ { \\alpha } \\big ] . \\end{align*}"}
{"id": "6503.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { 2 } \\int _ 0 ^ L \\left ( \\abs { u _ t } ^ 2 + \\abs { \\nabla u } ^ 2 + \\abs { y _ t } ^ 2 + \\abs { \\nabla y } ^ 2 \\right ) d x . \\end{align*}"}
{"id": "2300.png", "formula": "\\begin{align*} \\begin{cases} ( u , v , h , g ) | _ { y = 0 } = ( u , v , h , g ) | _ { y = \\beta } = ( 0 , 0 , 0 , 0 ) , \\\\ ( u , v , h , g ) | _ { x = 1 } = ( u , v , h , g ) | _ { x \\rightarrow \\infty } = ( 0 , 0 , 0 , 0 ) . \\end{cases} \\end{align*}"}
{"id": "6091.png", "formula": "\\begin{align*} { \\bar \\Omega } _ { A B } = d \\theta _ { A B } - \\sum \\theta _ { A C } \\wedge \\theta _ { C B } \\end{align*}"}
{"id": "2307.png", "formula": "\\begin{align*} ( \\widehat { u } , \\widehat { v } , \\widehat { h } , \\widehat { g } , \\widehat { p } ) = ( u _ 1 - u _ 2 , v _ 1 - v _ 2 , h _ 1 - h _ 2 , g _ 1 - g _ 2 , p _ 1 - p _ 2 ) , \\end{align*}"}
{"id": "2892.png", "formula": "\\begin{align*} f _ 1 = f _ { k + 1 } ( x _ 1 , \\ldots , x _ { i - 1 } , 1 , x _ { i + 1 } , \\ldots , x _ { k + 1 } ) \\end{align*}"}
{"id": "3782.png", "formula": "\\begin{align*} \\widetilde { K } ^ { m ; p , q ; \\tilde { m } } _ { k , j ; n , l ; \\tilde { k } ; \\tilde { j } , \\tilde { l } } ( t , x , \\zeta ) : = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ { \\pi } 2 ^ { m + 3 k + 2 n - j + \\max \\{ p , l \\} + 3 \\epsilon M _ t } f ( s , x - y + ( t - s ) \\omega , v ) \\big | B ^ { \\tilde { m } } _ { \\tilde { k } ; \\tilde { j } , \\tilde { l } } ( s , x - y + ( t - s ) \\omega ) \\big | \\end{align*}"}
{"id": "7807.png", "formula": "\\begin{align*} \\widetilde { H } ( \\widetilde { \\Theta } _ n ( \\phi _ 1 ) ) = \\det ( \\phi _ 1 ) ^ { n + 1 } ( B _ { - \\delta / 2 } \\circ \\tilde { \\iota } _ { \\phi _ 1 } \\circ B _ { \\delta / 2 } ) . \\end{align*}"}
{"id": "4125.png", "formula": "\\begin{align*} S ^ { i , j } \\circ T ^ { i + 1 , j - 1 } \\circ S ^ { i , j } = S ^ { i , j } . \\end{align*}"}
{"id": "646.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\bigg | \\frac { H _ n ( 1 - s ) } { H _ n ( s ) } \\bigg | = 1 . \\end{align*}"}
{"id": "3362.png", "formula": "\\begin{align*} \\Phi : = \\begin{pmatrix} ( a _ { 1 , 1 } , b _ { 1 , 1 } ) & ( a _ { 1 , 2 } , b _ { 1 , 2 } ) & \\ldots & ( a _ { 1 , K } , b _ { 1 , K } ) \\\\ ( a _ { 2 , 1 } , b _ { 2 , 1 } ) & ( a _ { 2 , 2 } , b _ { 2 , 2 } ) & \\ldots & ( a _ { 2 , K } , b _ { 2 , K } ) \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ ( a _ { L , 1 } , b _ { L , 1 } ) & ( a _ { L , 2 } , b _ { L , 2 } ) & \\ldots & ( a _ { L , K } , b _ { L , K } ) \\end{pmatrix} \\end{align*}"}
{"id": "2771.png", "formula": "\\begin{align*} U = \\frac { 2 L \\Delta } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left [ 2 h _ i - h _ i ^ 2 \\frac { - \\kappa } { 2 \\min \\left ( 1 , \\frac { 1 } { h _ i } \\right ) - ( 1 + \\kappa ) } \\right ] } \\end{align*}"}
{"id": "1822.png", "formula": "\\begin{align*} \\pi _ { n } ( z ) = \\frac { Q _ { n } ( z ) } { P _ { n } ( z ) } \\end{align*}"}
{"id": "355.png", "formula": "\\begin{align*} N ( \\ell ) \\lesssim n ^ \\ell ( d + 1 ) ^ { k ^ 2 \\delta _ { } } = n ^ { \\ell + o ( 1 ) } . \\end{align*}"}
{"id": "6588.png", "formula": "\\begin{align*} d ^ k ( g \\circ f ) ( x , y ) = \\sum _ { j = 1 } ^ k \\sum _ { P \\in P _ { k , j } } d ^ j g ( f ( x ) , d ^ { | I _ 1 | } ( x , y _ { I _ 1 } ) , \\ldots , d ^ { | I _ j | } ( x , y _ { I _ j } ) ) \\end{align*}"}
{"id": "4783.png", "formula": "\\begin{align*} \\prod _ { x \\in X / S } \\left ( 1 + a _ x \\right ) \\cdot \\prod _ { x \\in X / S } \\left ( 1 + b _ x \\right ) = \\prod _ { x \\in X / S } \\left ( \\left ( 1 + a _ x \\right ) \\left ( 1 + b _ x \\right ) \\right ) \\end{align*}"}
{"id": "1774.png", "formula": "\\begin{align*} b \\widetilde { z } + \\sum _ { j = 1 } ^ m Q _ { c _ j } ( \\alpha _ j , A _ j , x ^ k _ j + a _ j f _ { p ( j ) } ) \\subseteq Q _ b ( \\alpha , A , z + a f _ k ) . \\end{align*}"}
{"id": "2331.png", "formula": "\\begin{align*} \\Psi _ n = \\{ Q \\in K [ x ] \\mid Q \\mbox { i s a k e y p o l y n o m i a l f o r } \\nu \\mbox { a n d } \\deg ( Q ) = n \\} . \\end{align*}"}
{"id": "6802.png", "formula": "\\begin{align*} A _ 1 u : & = ( a u '' ) '' , \\\\ \\forall \\ ; u \\in D ( A _ 1 ) : = \\{ u \\in { \\mathcal Z } ( 0 , 1 ) : \\ , u '' ( 0 ) & = u '' ( 1 ) = 0 , u ''' ( 0 ) = u ''' ( 1 ) = 0 \\} , \\end{align*}"}
{"id": "2431.png", "formula": "\\begin{align*} M _ f S _ 0 & = M _ f \\sum _ { \\sigma \\in W _ f } q ^ { \\ell ( w _ 0 ^ f ) - \\ell ( \\sigma ) } H _ \\sigma \\sum _ { \\tau \\in W ^ f } q ^ { \\ell ( ^ f w _ 0 ) - \\ell ( \\tau ) } H _ \\tau \\\\ & = M _ f \\sum _ { \\sigma \\in W _ f } q ^ { \\ell ( w _ 0 ^ f ) - 2 \\ell ( \\sigma ) } \\sum _ { \\tau \\in W ^ f } q ^ { \\ell ( ^ f w _ 0 ) - \\ell ( \\tau ) } H _ \\tau \\\\ & = [ | W _ f | ] M _ f \\sum _ { \\tau \\in W ^ f } q ^ { \\ell ( ^ f w _ 0 ) - \\ell ( \\tau ) } H _ \\tau \\\\ & = [ | W _ f | ] \\sum _ { \\tau \\in W ^ f } q ^ { \\ell ( ^ f w _ 0 ) - \\ell ( \\tau ) } M _ { f \\cdot \\tau } . \\end{align*}"}
{"id": "5164.png", "formula": "\\begin{align*} \\begin{pmatrix} - \\frac { | R _ 1 | - 2 } { 3 } & 0 & 0 & \\cdots & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "3664.png", "formula": "\\begin{align*} \\tilde { M } _ { n } ( t ) = \\sup _ { s \\in [ 0 , t ] } M _ { n } ( s ) + ( 1 + t ) ^ { n ^ 3 } \\lesssim \\big ( \\tilde { M } _ { n } ( t ) \\big ) ^ { 1 - \\epsilon } + ( 1 + t ) ^ { n ^ 3 } , \\Longrightarrow \\tilde { M } _ { n } ( t ) \\lesssim ( 1 + t ) ^ { n ^ 3 } . \\end{align*}"}
{"id": "898.png", "formula": "\\begin{align*} z _ { n _ j } \\in B ^ { n _ j } , \\ , \\ , j = 1 , \\ldots , k \\end{align*}"}
{"id": "4624.png", "formula": "\\begin{align*} h ^ { \\ast } [ \\alpha ] \\psi [ \\alpha ] ^ { - 1 } \\psi ^ { - 1 } { h ^ { \\ast } } ^ { - 1 } = [ \\alpha ] \\phi [ \\alpha ] ^ { - 1 } \\phi ^ { - 1 } \\end{align*}"}
{"id": "5741.png", "formula": "\\begin{align*} \\pi _ b \\cdot \\pi _ { [ a , b ] } & = ( b - a + 1 ) \\pi _ { [ a , b ] } y _ { b + 1 } + ( b - a + 2 ) \\pi _ { [ a - 1 , b ] } \\\\ & = \\pi _ { [ a - 1 , b ] } + ( b - a + 1 ) \\Big ( \\pi _ { [ a - 1 , b ] } + \\pi _ { [ a , b ] } y _ { b + 1 } \\Big ) . \\end{align*}"}
{"id": "6691.png", "formula": "\\begin{align*} \\overline { U } ( t , r ) = c e ^ { - k t } w ( \\dfrac { r } { \\overline { s } _ { 1 } ( t ) } ) , \\ 0 \\leq r < \\overline { s } _ { 1 } ( t ) , t \\geq 0 \\ \\ \\overline { V } ( t , r ) = d e ^ { - h t } w ( \\dfrac { r } { \\overline { s } _ { 2 } ( t ) } ) , \\ 0 \\leq r < \\overline { s } _ { 2 } ( t ) , t \\geq 0 , \\end{align*}"}
{"id": "1841.png", "formula": "\\begin{align*} z y _ { n } = y _ { n + 1 } + a _ { n } ^ { ( 0 ) } \\ , y _ { n } + a _ { n - 1 } ^ { ( 1 ) } \\ , y _ { n - 1 } + \\cdots + a _ { n - p } ^ { ( p ) } \\ , y _ { n - p } , n \\geq p . \\end{align*}"}
{"id": "992.png", "formula": "\\begin{align*} L _ { \\alpha , \\beta } = L \\left ( H ^ { * } ( \\overline { \\alpha } \\otimes \\overline { \\beta } ) \\right ) . \\end{align*}"}
{"id": "7177.png", "formula": "\\begin{align*} \\theta ( Y _ t ( u , z ) ) = Y _ t ( e ^ { z L ( 1 ) } ( - z ^ { - 2 } ) ^ { L ( 0 ) } u , z ^ { - 1 } ) . \\end{align*}"}
{"id": "3484.png", "formula": "\\begin{align*} \\mathrm { D T E } _ { ( p , q ) } ( X ) = \\mu + \\sigma \\frac { \\phi ( \\xi _ { p } ) - \\phi ( \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "1801.png", "formula": "\\begin{align*} \\begin{array} { l l l l } d = d _ - & \\tilde d = d _ \\mathrm { p c v } & \\tilde \\phi = \\phi _ \\mathrm { p c v } & \\mbox { f o r B r e g m a n p r i m a l C o n d a t - - V \\ ~ u ~ \\eqref { e - b c v } , } \\\\ d = d _ + & \\tilde d = d _ \\mathrm { d c v } & \\tilde \\phi = \\phi _ \\mathrm { d c v } & \\mbox { f o r B r e g m a n d u a l C o n d a t - - V \\ ~ u ~ \\eqref { e - b c v - d u a l } . } \\end{array} \\end{align*}"}
{"id": "177.png", "formula": "\\begin{align*} \\Delta _ h \\psi ( x ) = \\sum _ { i = 1 } ^ { N } \\frac { \\psi ( x + h e _ i ) + \\psi ( x - h e _ i ) - 2 \\psi ( x ) } { h ^ 2 } . \\end{align*}"}
{"id": "5707.png", "formula": "\\begin{align*} \\pi _ J = 0 . \\end{align*}"}
{"id": "4136.png", "formula": "\\begin{align*} | A | = q ^ { n - 1 - \\lfloor \\frac { n - 1 } { k } \\rfloor } , \\end{align*}"}
{"id": "7116.png", "formula": "\\begin{align*} 0 = \\frac { \\partial h ( \\tau , p ( \\tau ; \\cdot ) ) } { \\partial x } \\left [ \\frac { \\partial p ( \\tau ; \\cdot ) } { \\partial t } + \\frac { \\partial p ( \\tau ; \\cdot ) } { \\partial x } \\frac { \\partial p ( t ; \\cdot ) } { \\partial \\tau } \\right ] \\ , , \\end{align*}"}
{"id": "5255.png", "formula": "\\begin{align*} A ^ 1 _ Q = \\bigcup _ { j = 1 } ^ h \\bigcup _ { R _ 0 = 0 } ^ { k _ 1 ( j ) - 1 } A ^ 1 _ { Q , j , R _ 0 } ; A ^ 2 _ Q = \\bigcup _ { j = 1 } ^ h \\bigcup _ { S _ 0 = 0 } ^ { k _ 2 ( j ) - 1 } A ^ 2 _ { Q , j , S _ 0 } . \\end{align*}"}
{"id": "1400.png", "formula": "\\begin{align*} X _ 1 = \\frac { 1 } { \\sqrt { 1 + \\epsilon } } \\begin{pmatrix} 1 & 1 \\\\ \\epsilon ^ { \\frac { 1 } { 2 } } & - \\epsilon ^ { \\frac { 1 } { 2 } } \\\\ 0 & 0 \\end{pmatrix} , X _ 2 = \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} . \\end{align*}"}
{"id": "2933.png", "formula": "\\begin{align*} G ( y ) : = \\sum _ { j = 1 } ^ m \\max \\{ F ( x ) : Z ( y ) = h ^ 1 , h ^ 2 , \\ldots , h ^ m \\land x \\leq h ^ j \\land x \\in P ( \\mathcal { F } ) \\} . \\end{align*}"}
{"id": "3990.png", "formula": "\\begin{align*} \\textstyle a b = a _ 0 b _ 0 + \\sum _ { i = 1 } ^ r a _ i b _ i + \\sum _ { j = 1 } ^ s a _ { r + j } b _ { r + j } . \\end{align*}"}
{"id": "1443.png", "formula": "\\begin{align*} [ t ^ k ] \\circ \\theta ^ { n } _ t ( t ^ m ) = m ^ n t ^ { m + k } \\enspace . \\end{align*}"}
{"id": "6225.png", "formula": "\\begin{align*} & \\int x ( q ^ 2 x ^ 2 ; q ^ 2 ) _ \\infty h _ n ( x ; q ) d _ q x = \\frac { q ^ { n - 1 } ( x ^ 2 ; q ^ 2 ) _ \\infty } { [ n - 1 ] _ q } \\left ( ( 1 - q ) h _ n ( \\frac { x } { q } ; q ) - \\frac { 1 - q ^ n } { q } x h _ { n - 1 } ( \\frac { x } { q } ; q ) \\right ) , \\end{align*}"}
{"id": "7848.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in M , \\| x \\| \\leq 1 } | \\langle x \\xi y _ n , \\xi \\rangle | = 0 , \\end{align*}"}
{"id": "5497.png", "formula": "\\begin{align*} \\omega _ { t + 1 } = P _ { \\Omega } \\left ( ( I + \\alpha _ t A _ t ) \\omega _ t - \\alpha _ t b _ t ) \\right ) . \\end{align*}"}
{"id": "594.png", "formula": "\\begin{align*} \\sigma ( \\widetilde \\Delta _ n ) = \\Bigg \\{ \\left . \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi k _ 1 } { n } \\Big ) + \\frac { n ^ 2 } { \\pi ^ 2 } \\sin ^ 2 \\Big ( \\frac { \\pi k _ 2 } { n } \\Big ) \\right | \\forall _ { j = 1 , 2 } : k _ j = 0 \\ , , \\cdots , n - 1 \\Bigg \\} . \\end{align*}"}
{"id": "946.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ T \\int \\limits _ { V } \\eta p ( \\rho _ m ) ( \\psi \\rho _ m - \\psi \\alpha _ m ) \\ d x \\ d t = J _ 1 + J _ 2 , \\end{align*}"}
{"id": "4893.png", "formula": "\\begin{align*} ( x , y ) \\cdot ( x ' , y ' ) = ( x + x ' , y + y ' + ( x x '^ T - x ' x ^ T ) ) \\end{align*}"}
{"id": "7327.png", "formula": "\\begin{align*} \\Theta _ { \\sf a } & = \\Theta _ J - w , \\\\ P _ 1 \\theta & = \\Delta _ x \\theta - q { \\sf U } _ \\infty ^ { q - 1 } \\theta . \\end{align*}"}
{"id": "6736.png", "formula": "\\begin{align*} \\mu \\int _ { \\Omega } \\varepsilon y _ \\delta : \\varepsilon v + \\nu \\int _ { \\Omega } \\mathsf { m } _ \\delta ( \\varepsilon y _ \\delta ) : \\varepsilon v = \\int _ { \\Omega } u \\cdot v , \\forall v \\in Y . \\end{align*}"}
{"id": "4240.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log p _ n ( X _ n , X _ 0 ) } { \\log \\mathbb { G } ( X _ n , X _ 0 ) } = 1 \\end{align*}"}
{"id": "6774.png", "formula": "\\begin{align*} \\sum _ { i \\in M _ C } \\sum _ { j = 1 } ^ { r _ i - 1 } a _ { i r _ i } x _ { i j } + \\sum _ { i \\in M _ C } \\sum _ { j = r _ i } ^ { n _ i } \\max \\left \\{ a _ { i j } , b - \\sum _ { k \\in M _ C - i } a _ { k r _ k } \\right \\} x _ { i j } \\leq b \\end{align*}"}
{"id": "1713.png", "formula": "\\begin{align*} \\mathbb { E } _ q ( x ) = \\prod _ { k \\geq 0 } ( 1 - x q ^ k ) , \\end{align*}"}
{"id": "8930.png", "formula": "\\begin{align*} | \\psi | ( x , u , p ) = \\frac { f ( x ) } { f ^ * \\circ Y ( x , u , p ) } , \\end{align*}"}
{"id": "2134.png", "formula": "\\begin{align*} S = \\left \\{ s \\in \\mathbb { R } : ( - \\Im G _ { \\mu \\boxplus \\nu } ) ^ { * } ( s ) = + \\infty \\right \\} \\end{align*}"}
{"id": "624.png", "formula": "\\begin{align*} | G | \\leqslant \\sum _ { m = N } ^ { \\infty } C _ 2 ^ m n ^ { - 2 ( m - N ) } \\sum _ { j = 0 } ^ m \\frac { ( x ^ 2 + y ^ 2 ) ^ j } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { j } } \\leqslant C , \\end{align*}"}
{"id": "7190.png", "formula": "\\begin{align*} & o _ I \\big ( ( u \\ast _ { g _ 1 g _ 2 } v ) \\ast _ { g _ 1 g _ 2 , g _ 2 } w _ 1 - u \\ast _ { g _ 1 g _ 2 , g _ 2 } ( v \\ast _ { g _ 1 g _ 2 , g _ 2 } w _ 1 \\big ) ) \\mid _ { M ^ 2 ( 0 ) } \\\\ = & o _ { M ^ 3 } ( u \\ast _ { g _ 1 g _ 2 } v ) o _ I ( w _ 1 ) \\mid _ { M ^ 2 ( 0 ) } - o _ { M ^ 3 } ( u ) o _ I ( v \\ast _ { g _ 1 g _ 2 , g _ 2 } w _ 1 ) \\mid _ { M ^ 2 ( 0 ) } \\\\ = & o _ { M ^ 3 } ( u ) o _ { M ^ 3 } ( v ) o _ I ( w _ 1 ) \\mid _ { M ^ 2 ( 0 ) } - o _ { M ^ 3 } ( u ) o _ { M ^ 3 } ( v ) o _ I ( w _ 1 ) \\mid _ { M ^ 2 ( 0 ) } \\\\ = & 0 \\end{align*}"}
{"id": "532.png", "formula": "\\begin{align*} P _ { i + 1 } = P _ i + \\begin{cases} ( 1 , 0 ) & , \\\\ ( 0 , 1 ) & , \\end{cases} \\end{align*}"}
{"id": "7645.png", "formula": "\\begin{align*} 1 - g ( S _ n ) = 1 - n \\binom { n } { 3 } + ( n + 1 ) \\binom { n - 1 } { 3 } . \\end{align*}"}
{"id": "5512.png", "formula": "\\begin{align*} K ( \\Omega ) & = \\inf \\{ K > 0 : K f \\in \\mathcal { F } \\} , \\\\ K _ ( \\Omega ) & = \\inf \\{ K > 0 : K f \\in \\mathcal { F } _ \\} . \\end{align*}"}
{"id": "2487.png", "formula": "\\begin{align*} A _ E E ' = \\mathcal { H } \\nabla _ { \\mathcal { H } E } \\nu E ' + \\nu \\nabla _ { \\mathcal { H } E } \\mathcal { H } E ' , \\end{align*}"}
{"id": "6400.png", "formula": "\\begin{align*} B _ { \\pi } : = \\{ b _ { \\ell } , \\ldots , b _ 1 , b _ 0 \\} \\subseteq \\{ 1 , \\ldots , k + 1 \\} . \\end{align*}"}
{"id": "7621.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( H ) \\colon i \\in e } t ' _ { e } = \\big | V ^ * _ i \\big | t ' _ { \\hat e } \\ge z \\ge \\frac { n } { 5 m } \\end{align*}"}
{"id": "3310.png", "formula": "\\begin{align*} \\begin{pmatrix} - \\gamma ^ - \\partial _ t u \\\\ 0 \\end{pmatrix} + \\begin{pmatrix} 0 & \\frac 1 2 I \\\\ - \\frac 1 2 I & 0 \\end{pmatrix} \\begin{pmatrix} \\varphi \\\\ \\psi \\end{pmatrix} + B ( \\partial _ t ) \\begin{pmatrix} \\varphi \\\\ \\psi \\end{pmatrix} = \\begin{pmatrix} - \\partial _ t \\beta _ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
{"id": "2682.png", "formula": "\\begin{align*} \\mathcal { E } ^ \\mathrm { H } [ u ] : = \\ ; & \\int _ { \\mathbb { R } ^ d } | \\nabla u ( x ) | ^ 2 d x + \\int _ { \\mathbb { R } ^ d } V _ \\mathrm { D W } ( x ) | u ( x ) | ^ 2 d x \\\\ & + \\frac { \\lambda } { 2 } \\iint | u ( x ) | ^ 2 w ( x - y ) | u ( y ) | ^ 2 d x d y , \\end{align*}"}
{"id": "1953.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { S } _ { 2 } ( n + 1 , r ) } w ( \\gamma ) & = \\sum _ { \\gamma \\in \\mathcal { S } _ { 2 } ( n , r - 1 ) } w ( \\gamma ) + \\sum _ { k = 0 } ^ { p } a _ { - k - q - r } ^ { ( k ) } \\Big ( \\sum _ { \\gamma \\in \\mathcal { S } _ { 2 } ( n , r + k ) } w ( \\gamma ) \\Big ) \\\\ & = \\langle \\mathcal { B } _ { q } ^ { n } e _ { r - 1 } , e _ { 0 } \\rangle + \\sum _ { k = 0 } ^ { p } a _ { - k - q - r } ^ { ( k ) } \\langle \\mathcal { B } _ { q } ^ { n } e _ { r + k } , e _ { 0 } \\rangle \\\\ & = \\langle \\mathcal { B } _ { q } ^ { n + 1 } e _ { r } , e _ { 0 } \\rangle \\end{align*}"}
{"id": "8889.png", "formula": "\\begin{align*} | \\det ( \\sigma ^ 2 I _ n - A ^ H A ) | \\left ( \\frac { n - 1 } { ( n + 1 ) \\sigma ^ 2 - \\| A \\| _ F ^ 2 } \\right ) ^ { n - 1 } = 0 . \\end{align*}"}
{"id": "694.png", "formula": "\\begin{align*} u ( x , s ) = e ^ { \\frac { n } { 2 } ( s - s _ 0 ) } v ^ { \\lambda } ( x , - e ^ { s - s _ 0 } ) = | t | ^ { n / 2 } v ^ { \\lambda } ( x , t ) . \\end{align*}"}
{"id": "7700.png", "formula": "\\begin{align*} \\sum _ { n _ 1 \\cdots n _ k \\le x } ( n _ 1 , \\ldots , n _ k ) = x \\ , Q _ { k - 1 } ( \\log x ) + O ( x ^ { \\theta _ k + \\varepsilon } ) , \\end{align*}"}
{"id": "1024.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\frac { C _ N } { D _ N } \\le \\frac 1 { \\nu ( [ 1 ] ) } \\sum _ { n = 0 } ^ \\infty \\lambda ( [ 1 , * ^ n , 1 ] ) \\cdot \\nu ( [ 1 , 0 ^ n , 1 ] ) = \\sum _ { n = 0 } ^ \\infty \\lambda ( [ 1 , * ^ n , 1 ] ) \\cdot c _ n , \\end{align*}"}
{"id": "6748.png", "formula": "\\begin{align*} \\chi _ { j _ 1 j _ 2 j _ 3 } \\left ( a _ 1 , a _ 2 , a _ 3 \\right ) : = e ^ { \\left ( \\pi a _ 1 j _ 1 \\right ) \\sqrt { - 1 } } e ^ { \\left ( \\pi a _ 2 j _ 2 \\right ) \\sqrt { - 1 } } e ^ { \\left ( \\pi a _ 3 j _ 3 \\right ) \\sqrt { - 1 } } \\end{align*}"}
{"id": "6048.png", "formula": "\\begin{align*} g _ { r , s , j } ( \\kappa _ i ) = r ^ { j + \\sum _ { i = 1 } ^ { m } \\mu _ { i , j } } g _ j \\left ( r \\kappa _ i + s \\right ) = r ^ { j + \\sum _ { i = 1 } ^ { m } \\mu _ { i , j } } g _ j \\left ( \\lambda _ i \\right ) \\neq 0 \\end{align*}"}
{"id": "5518.png", "formula": "\\begin{align*} \\int _ H \\frac { 1 } { \\abs { f ' ( z ) } ^ q } d x d y & \\leq \\frac { 1 } { C ^ q \\abs { f ' ( z _ 0 ) } ^ q } \\int _ H d x d y = \\frac { 1 } { C ^ q \\abs { f ' ( z _ 0 ) } ^ q } \\frac { \\pi ( 1 - \\abs { z _ 0 } ) ^ 2 } { 4 } \\\\ & \\leq \\frac { \\pi } { 4 C ^ q } \\frac { \\left ( 1 - \\abs { z _ 0 } ^ 2 \\right ) ^ 2 } { \\abs { f ' ( z _ 0 ) } ^ q } . \\end{align*}"}
{"id": "4989.png", "formula": "\\begin{align*} \\langle M ^ { ( n ) } , W \\rangle _ \\tau = n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { \\tau } \\sum _ { i = 1 } ^ Q \\rho _ i \\mathbf { 1 } _ { [ 0 , t _ i ] } ( \\tau ) ( t _ i - s ) ^ { \\alpha } \\sigma ' ( X _ s ) \\Theta ^ n _ s d s \\end{align*}"}
{"id": "3035.png", "formula": "\\begin{align*} \\tilde { H } _ { \\lambda } ( \\Z ; q ) = \\sum _ { \\tau } \\tilde { K } _ { \\tau , \\lambda } ( q ) s _ { \\tau } ( \\Z ) . \\end{align*}"}
{"id": "8404.png", "formula": "\\begin{gather*} \\phi ( \\tau , z ) = h ( \\tau ) ^ { \\rm t } \\theta _ m ( \\tau , z ) \\end{gather*}"}
{"id": "3153.png", "formula": "\\begin{align*} \\mathbb { S } \\doteq \\left \\{ \\Phi \\in \\mathcal { W } _ { 1 } : \\left \\Vert \\Phi \\right \\Vert _ { \\mathcal { U } } = 1 \\right \\} \\end{align*}"}
{"id": "6081.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 - a x = x \\left ( x ^ 3 - a \\right ) \\end{align*}"}
{"id": "1913.png", "formula": "\\begin{align*} R _ { j } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { R _ { [ n , j ] } } { z ^ { n + 1 } } = \\sum _ { m = 0 } ^ { \\infty } \\frac { R _ { [ m ( p + 1 ) + j , j ] } } { z ^ { m ( p + 1 ) + j + 1 } } , \\\\ S _ { j } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { S _ { [ n , j ] } } { z ^ { n + 1 } } = \\sum _ { m = 0 } ^ { \\infty } \\frac { S _ { [ m ( p + 1 ) + j , j ] } } { z ^ { m ( p + 1 ) + j + 1 } } , \\\\ T _ { j } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { T _ { [ n , j ] } } { z ^ { n + 1 } } = \\sum _ { m = 0 } ^ { \\infty } \\frac { T _ { [ m ( p + 1 ) + j , j ] } } { z ^ { m ( p + 1 ) + j + 1 } } . \\end{align*}"}
{"id": "1855.png", "formula": "\\begin{align*} A _ { [ n , j ] } = B _ { [ n , j ] } = W _ { [ n , j ] } & = 0 n < 0 , 0 \\leq j \\leq p , \\\\ A _ { [ n , j ] } ^ { ( q ) } = B _ { [ n , j ] } ^ { ( q ) } & = 0 n < 0 , 0 \\leq j \\leq p , q \\geq 0 . \\end{align*}"}
{"id": "8520.png", "formula": "\\begin{align*} \\Delta _ 1 ( k , m + 1 ) & = z \\Delta _ 1 ( k , m ) + a _ { m + 1 } \\Delta _ 2 ( k , m ) + a _ { m + 1 } ^ 2 \\Delta _ 1 ( k , m - 1 ) ; \\\\ \\Delta _ 2 ( k , m + 1 ) & = ( 2 z + 1 ) \\Delta _ 1 ( k , m ) + a _ { m + 1 } \\Delta _ 2 ( k , m ) + a _ { m + 1 } \\Delta _ 3 ( k , m ) ; \\\\ \\Delta _ 3 ( k , m + 1 ) & = - \\Delta _ 1 ( k , m ) - a _ { m + 1 } \\Delta _ 3 ( k , m ) ; \\\\ \\Delta _ 4 ( k , m + 1 ) & = 2 z \\Delta _ 1 ( k , m ) + a _ { m + 1 } \\Delta _ 2 ( k , m ) . \\end{align*}"}
{"id": "7710.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\tau _ k ( n ) = \\frac 1 { ( k - 1 ) ! } x ( \\log x ) ^ { k - 1 } + O ( x ( \\log x ) ^ { k - 2 } ) , \\end{align*}"}
{"id": "1952.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { S } _ { 2 } ( n + 1 , 0 ) } w ( \\gamma ) = \\sum _ { k = 0 } ^ { p } a _ { - k - q } ^ { ( k ) } \\Big ( \\sum _ { \\gamma \\in \\mathcal { S } _ { 2 } ( n , k ) } w ( \\gamma ) \\Big ) = \\sum _ { k = 0 } ^ { p } a _ { - k - q } ^ { ( k ) } \\langle \\mathcal { B } _ { q } ^ { n } e _ { k } , e _ { 0 } \\rangle = \\langle \\mathcal { B } _ { q } ^ { n + 1 } e _ { 0 } , e _ { 0 } \\rangle \\end{align*}"}
{"id": "4348.png", "formula": "\\begin{align*} \\mathrm { d e p t h } \\ , ( R / J ) & = 1 + \\mathrm { d e p t h } \\ , ( K [ x _ { k + 1 } , \\ldots , x _ { n } ] / I ( D _ { F } ) ) \\\\ & = 1 + \\mathrm { d i m } \\ , ( K [ x _ { k + 1 } , \\ldots , x _ { n } ] / I ( D _ { F } ) ) \\\\ & = 1 + ( n - k ) - ( g - k ) \\\\ & = 1 + n - g . \\end{align*}"}
{"id": "3490.png", "formula": "\\begin{align*} c _ { 1 } = \\frac { \\Gamma \\left ( ( m + 1 ) / 2 \\right ) } { \\Gamma ( m / 2 ) ( m \\pi ) ^ { \\frac { 1 } { 2 } } } , \\end{align*}"}
{"id": "3908.png", "formula": "\\begin{align*} \\overline \\varepsilon & = \\frac { 1 } { 4 } \\min _ { i \\in \\mathcal I } \\left \\{ | a _ i | : a _ i \\neq 0 \\right \\} = \\frac { 1 } { 4 } \\min _ { i \\in \\mathcal I } | a _ i | , \\end{align*}"}
{"id": "8321.png", "formula": "\\begin{align*} S _ \\mathcal { E } = \\left \\{ x \\in H : x = \\sum _ { k = 0 } ^ \\infty x _ k E _ k , \\{ x _ k \\} _ { k = 0 } ^ \\infty \\textrm { i s R D } \\right \\} . \\end{align*}"}
{"id": "5994.png", "formula": "\\begin{align*} u ( t , x ) = ( 1 + t ^ \\beta + H ( t - r ) t ^ \\beta ) \\sin x \\end{align*}"}
{"id": "3895.png", "formula": "\\begin{align*} - \\Delta _ p u _ i - a u _ i ^ { p - 1 } = f _ i \\Omega \\end{align*}"}
{"id": "3028.png", "formula": "\\begin{align*} \\tilde { N } _ { \\lambda } ( z , w ) = \\ ! \\ ! \\ ! \\prod _ { \\substack { x \\in \\lambda \\\\ h ( x ) \\equiv 0 \\ ! \\ ! \\ ! \\mod 2 } } \\ ! \\ ! \\ ! ( z ^ { a ( x ) + 1 } - w ^ { l ( x ) } ) ( z ^ { a ( x ) } - w ^ { l ( x ) + 1 } ) , \\end{align*}"}
{"id": "2972.png", "formula": "\\begin{align*} F _ m ( z , s ) = f _ m ( z , s ) + \\frac { 2 | m | ^ { 1 / 2 - s } \\sigma _ { 2 s - 1 } ( | m | ) } { ( 2 s - 1 ) \\Lambda ( 2 s ) } y ^ { 1 - s } + 2 \\sqrt y \\sum _ { n \\neq 0 } \\Phi ( m , n ; s ) K _ { s - \\frac 1 2 } ( 2 \\pi | n | y ) e ( n x ) , \\end{align*}"}
{"id": "1286.png", "formula": "\\begin{align*} n ( T , j ) & = ( k - 1 ) ( k - 2 ) \\cdots ( k + 1 - j ) = \\dfrac { ( k - 1 ) ! } { ( k - j ) ! } , \\end{align*}"}
{"id": "2757.png", "formula": "\\begin{align*} 0 \\geq J _ { 1 } \\geq - \\int _ { B _ { 1 } ( x ) ^ { c } } \\frac { d y } { | x - y | ^ { n + \\sigma p } } = - \\gamma , \\end{align*}"}
{"id": "6455.png", "formula": "\\begin{align*} \\sum _ { i _ 1 + \\cdot + i _ r = k } \\sum _ { \\sigma \\in \\mathfrak { S } ( i _ 1 , \\ldots , i _ r ) } \\epsilon ( \\sigma ) \\frac { 1 } { r ! } \\prod _ { j = 1 } ^ r \\Phi _ { i _ j - 1 } ( x _ { \\sigma ( i _ { 1 } + \\cdots + i _ { j - 1 } + 1 ) } , \\ldots , x _ { \\sigma ( i _ { 1 } + \\cdots + i _ j ) } ) . \\end{align*}"}
{"id": "2873.png", "formula": "\\begin{align*} \\hbox { \\rm G r a p h } f = \\{ ( x , f ( x ) ) \\ , | \\ , x \\in M \\} \\end{align*}"}
{"id": "1756.png", "formula": "\\begin{align*} q _ A ( x ) = \\sup \\{ | f ( x ) | : \\ ; f \\in A \\cap B _ { X ^ * } \\} \\end{align*}"}
{"id": "7966.png", "formula": "\\begin{align*} H _ { ( X _ + , D _ + ) , f _ + , \\vec d _ + } = H _ { ( X _ - , D _ - ) , f _ - , \\vec d _ - } \\end{align*}"}
{"id": "8618.png", "formula": "\\begin{align*} \\vartheta _ p \\cdot \\vartheta _ q = \\sum _ { r \\in B ( \\mathbb { Z } ) } \\sum _ { \\textbf { A } \\in N E ( X ) } \\alpha _ { p , q , \\textbf { A } } ^ r t ^ \\textbf { A } \\vartheta _ r . \\end{align*}"}
{"id": "9013.png", "formula": "\\begin{align*} \\beta ( s ) = \\left ( \\frac { \\pi } { 2 } \\right ) ^ { s - 1 } \\cos \\left ( \\frac { \\pi s } { 2 } \\right ) \\Gamma ( 1 - s ) \\beta ( 1 - s ) . \\end{align*}"}
{"id": "4374.png", "formula": "\\begin{align*} \\int _ { U } | F | ^ { 2 ( 1 - p ' _ 0 ) \\frac { p _ 0 } { p _ 0 - p ' _ 0 } } = \\int _ { U } e ^ { - 2 ( p ' _ 0 - 1 ) \\frac { p _ 0 } { p _ 0 - p ' _ 0 } \\log | F | } < + \\infty . \\end{align*}"}
{"id": "2816.png", "formula": "\\begin{align*} \\min \\limits _ { 0 \\leq i \\leq N } \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\} & \\leq \\frac { f ( x _ 0 ) - f ( x _ { N } ) } { \\sum \\limits _ { i = 0 } ^ { N - 1 } p _ i } \\end{align*}"}
{"id": "5383.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u _ f ^ * + \\sum _ { | \\alpha | \\leq m } ( - 1 ) ^ { | \\alpha | } D ^ \\alpha ( a _ \\alpha u _ f ^ * ) & = 0 \\mbox { i n } \\quad \\Omega , u _ f ^ * = f \\mbox { i n } \\quad \\Omega _ e , \\end{align*}"}
{"id": "5241.png", "formula": "\\begin{align*} \\mathfrak { g } _ { A _ { I , \\mathrm { s y m } } , k } ^ { r , s } : = \\widetilde \\psi _ I ^ { - 1 } ( \\mathfrak { g } ^ { r , s } _ { A _ I , k } ) . \\end{align*}"}
{"id": "1706.png", "formula": "\\begin{align*} & \\frac { e ^ { t ( | \\xi | ^ 2 - | \\eta | ^ 2 - | \\xi - \\eta | ^ 2 ) } - 1 } { | \\xi | ^ 2 - | \\eta | ^ 2 - | \\xi - \\eta | ^ 2 } - \\frac { e ^ { t ( | \\xi | ^ 2 - | \\eta | ^ 2 ) } - 1 } { | \\xi | ^ 2 - | \\eta | ^ 2 } \\\\ = & \\sum _ { k = 1 } ^ \\infty \\frac { t ^ { k + 1 } ( | \\xi | ^ 2 - | \\eta | ^ 2 - | \\xi - \\eta | ^ 2 ) ^ { k } } { ( k + 1 ) ! } - \\sum _ { k = 1 } ^ \\infty \\frac { t ^ { k + 1 } ( | \\xi | ^ 2 - | \\eta | ^ 2 ) ^ { k } } { ( k + 1 ) ! } . \\end{align*}"}
{"id": "537.png", "formula": "\\begin{align*} | \\tau ^ * ( n ) | \\leq \\tau _ 2 ( n ) = O _ { \\varepsilon } ( n ^ { \\varepsilon } ) . \\end{align*}"}
{"id": "3794.png", "formula": "\\begin{align*} E r r ^ 2 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\sum _ { \\begin{subarray} { c } k _ 2 \\in \\Z _ + , j _ 2 \\in [ 0 , ( 1 + 2 \\epsilon ) M _ t ] \\cap \\Z , i _ 2 \\in \\{ 0 , 1 , 2 , 3 , 4 \\} \\\\ n _ 2 \\in [ - M _ t , 2 ] \\cap \\Z , \\mu _ 2 \\in \\{ + , - \\} , l _ 2 \\in [ - j _ 2 , 2 ] \\cap \\Z , a _ 2 \\in \\{ 0 , 1 , 2 , 3 \\} \\\\ \\end{subarray} } \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta ) + i \\mu s | \\xi | + i \\mu _ 1 s | \\eta | } \\end{align*}"}
{"id": "3101.png", "formula": "\\begin{align*} \\min \\{ \\lambda _ h ( { k } ) , \\lambda _ k \\} \\kappa ^ 2 _ m h _ { \\max } ^ { 2 { m } } \\le 1 \\quad \\quad \\lambda _ h ( { k } ) \\le \\lambda _ { k } \\quad k = 1 , \\dots , M . \\end{align*}"}
{"id": "4110.png", "formula": "\\begin{align*} \\big ( F '' ( V ( \\vec p ) ) [ V ' ( \\vec p ) \\widehat { \\vec p } _ 1 , V ' ( \\vec p ) \\ , \\cdot ] ^ * \\vec g \\big ) \\widehat { \\vec p } _ 2 & = \\int _ 0 ^ T \\langle V ' ( \\vec p ) \\widehat { \\vec p } _ 1 ( { u } _ 2 ^ \\prime ( t ) + Q { u } _ 2 ( t ) ) , w ( t ) \\big \\rangle _ X \\d t \\\\ & \\qquad \\qquad + \\int _ 0 ^ T \\langle V ' ( \\vec p ) \\widehat { \\vec p } _ 2 ( { u } _ 1 ^ \\prime ( t ) + Q { u } _ 1 ( t ) ) , w ( t ) \\big \\rangle _ X \\d t . \\end{align*}"}
{"id": "32.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { L , n } = T _ { L | K } ( \\lambda _ { \\rho } ( \\alpha ) \\Psi _ { L , n } ( \\beta ) ) \\cdot _ { \\rho } v _ n \\end{align*}"}
{"id": "4255.png", "formula": "\\begin{align*} - \\Delta U + U = U ^ 3 \\ \\hbox { i n } \\ \\mathbb R ^ n . \\end{align*}"}
{"id": "5019.png", "formula": "\\begin{align*} \\Xi ^ { n , 1 } _ s = \\left [ \\left ( \\int ^ { s } _ 0 \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) ^ 2 - \\int ^ { s } _ 0 \\psi _ { n , 1 } ^ 2 ( u , s ) d u \\ , \\right ] , \\end{align*}"}
{"id": "1951.png", "formula": "\\begin{align*} \\langle \\mathcal { B } _ { q } ^ { n } \\ , e _ { r } , e _ { 0 } \\rangle = \\sum _ { \\gamma \\in \\mathcal { S } _ { 2 } ( n , r ) } w ( \\gamma ) , \\end{align*}"}
{"id": "8481.png", "formula": "\\begin{align*} y _ k & = \\frac { U } { m } - \\frac { L ( m - 1 ) } { 2 } \\\\ & = \\frac { U } { m } - \\frac { L ( m + 1 ) } { 2 } + L . \\end{align*}"}
{"id": "4605.png", "formula": "\\begin{align*} r _ X ( k + h ^ \\vee _ { X ^ + } ) ( \\ell + h ^ \\vee _ { Y ^ - } ) = 1 \\end{align*}"}
{"id": "89.png", "formula": "\\begin{align*} d \\varphi _ t = ( \\Delta ^ \\epsilon \\varphi _ t - \\lambda \\varphi _ t ^ 3 - ( \\mu + a ^ \\epsilon ( \\lambda ) ) \\varphi _ t ) \\ , d t + \\sqrt { 2 } d W _ t ^ { \\epsilon , L } \\end{align*}"}
{"id": "227.png", "formula": "\\begin{align*} a b + c + 1 & = \\frac { \\gamma ^ 2 + \\gamma + 1 } { \\gamma + 1 } \\cdot \\frac { \\gamma ^ 2 + 1 } { \\gamma ^ 3 + \\gamma ^ 2 + \\gamma } + \\frac { \\gamma + 1 } { \\gamma ^ 3 + \\gamma ^ 2 + \\gamma } + 1 = \\frac { ( \\gamma ^ 3 + 1 ) + ( \\gamma + 1 ) + ( \\gamma ^ 3 + \\gamma ^ 2 + \\gamma ) } { \\gamma ^ 3 + \\gamma ^ 2 + \\gamma } \\\\ & = \\frac { \\gamma ^ 2 } { \\gamma ^ 3 + \\gamma ^ 2 + \\gamma } \\ne 0 \\end{align*}"}
{"id": "3036.png", "formula": "\\begin{align*} \\mathcal { Q } ^ { \\tau } _ { \\lambda } ( q ) = \\sum _ { \\nu } \\chi ^ { \\nu } _ { \\lambda } \\tilde { K } _ { \\nu \\tau } ( q ) . \\end{align*}"}
{"id": "1429.png", "formula": "\\begin{align*} \\begin{aligned} & \\log Q _ N ( R _ { N , 2 , 1 } > \\alpha N ^ { 4 / 3 } ( \\log N ) ^ { 4 / 3 } ) \\\\ & \\leq \\log P ( R _ { N , 2 , 1 } > \\alpha N ^ { 4 / 3 } ( \\log N ) ^ { 4 / 3 } ) - \\log Z _ { N , 2 , 1 } . \\end{aligned} \\end{align*}"}
{"id": "65.png", "formula": "\\begin{align*} G _ 3 ( k ) & \\lesssim \\frac { 1 } { k ^ { 2 r } } \\sum _ { h = \\lfloor k / 2 \\rfloor + 1 } ^ { k - 1 } \\frac { 1 } { ( 1 + k - h ) ^ { 2 s } } \\leq \\frac { 1 } { k ^ { 2 r } } \\sum _ { h = 1 } ^ { + \\infty } \\frac { 1 } { h ^ { 2 s } } = O ( \\frac { 1 } { k ^ { 2 r } } ) \\ k \\to + \\infty . \\end{align*}"}
{"id": "5862.png", "formula": "\\begin{align*} C _ 5 ^ { \\frac { p q } { p - q } } & \\lesssim \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p - q } } \\bigg ( \\int _ 0 ^ { { x _ { k + 1 } } } \\bigg ( \\int _ t ^ { { x _ { k + 1 } } } u \\bigg ) ^ { \\frac { 1 } { 1 - q } } d \\big [ V _ r ( 0 , t ) ^ { \\frac { q } { 1 - q } } \\big ] \\bigg ) ^ { \\frac { p ( 1 - q ) } { p - q } } = D _ 1 . \\end{align*}"}
{"id": "5314.png", "formula": "\\begin{align*} v ( t ) : = \\inf \\{ r > 0 : \\int _ r ^ \\infty \\psi _ \\Lambda ( u ) ^ { - 1 } d u > t \\} , \\end{align*}"}
{"id": "1138.png", "formula": "\\begin{align*} \\L _ 3 ( 1 8 , 2 , 1 ) & = \\{ 1 3 _ g + 4 _ r + 1 _ r , ~ 1 2 _ g + 5 _ r + 1 _ r , ~ 1 2 _ r + 5 _ g + 1 _ r , ~ 1 1 _ g + 6 _ r + 1 _ r , ~ 1 1 _ r + 6 _ g + 1 _ r , \\\\ & 1 0 _ r + 7 _ g + 1 _ r , ~ 1 1 _ g + 5 _ r + 2 _ r , ~ 1 0 _ g + 6 _ r + 2 _ r , ~ 1 0 _ r + 6 _ g + 2 _ r , ~ 9 _ r + 6 _ r + 3 _ g \\} , \\end{align*}"}
{"id": "8323.png", "formula": "\\begin{align*} \\| T ( f ) \\| \\le \\underset { x \\in \\R / \\Z } { \\textrm { s u p } } | f ( x ) | = \\| f \\| _ \\infty \\ , . \\end{align*}"}
{"id": "5873.png", "formula": "\\begin{align*} d _ 0 ( { \\mathcal K } ) _ X : = \\sup _ { f \\in { \\mathcal K } } \\| f \\| _ X , d _ n ( { \\mathcal K } ) _ X : = \\inf _ { \\dim ( X _ n ) = n } \\sup _ { f \\in { \\mathcal K } } { \\rm d i s t } ( f , X _ n ) _ X , n \\geq 1 , \\end{align*}"}
{"id": "6024.png", "formula": "\\begin{align*} f ( x ) & = ( x - 1 ) ^ 3 + b ( x - 1 ) ^ 2 + c \\\\ & = x ^ 3 + ( b - 3 ) x ^ 2 + ( 3 - 2 b ) x + ( b + c - 1 ) . \\end{align*}"}
{"id": "2833.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N - 1 } \\frac { p _ i } { S _ * } \\min \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\ , , \\ , \\| \\nabla f ( x _ { i + 1 } ) \\| ^ 2 \\} { } \\leq { } \\frac { f ( x _ 0 ) - f ( x _ { N } ) } { S _ * } . \\end{align*}"}
{"id": "330.png", "formula": "\\begin{align*} f ( A ) = \\sum _ q f ( A _ q ) < \\sum _ q \\max \\{ f ( q ) , e ^ \\gamma { \\rm d } ( { \\rm L } _ q ) \\} = f ( \\mathcal Q ) + e ^ \\gamma \\big ( 1 - { \\rm d } ( { \\rm L } _ { \\mathcal Q } ) ) . \\end{align*}"}
{"id": "2529.png", "formula": "\\begin{align*} \\tau ( F ) = ( n - 2 ) \\frac { \\lambda ^ 2 } { 2 } F _ \\ast \\left ( \\nabla _ { \\mathcal { H } } \\frac { 1 } { \\lambda ^ 2 } \\right ) - ( m - n ) F _ \\ast ( H ) . \\end{align*}"}
{"id": "3224.png", "formula": "\\begin{align*} \\det ( \\lambda - \\Phi ) = \\lambda ^ { 2 m } + s _ 2 \\lambda ^ { 2 m - 2 } + \\cdots + s _ { 2 m } , \\end{align*}"}
{"id": "2277.png", "formula": "\\begin{align*} ( \\phi , \\psi ) = T ( \\bar \\phi , \\bar \\psi ) \\in B _ { R ( \\delta , \\sigma ) } \\subset H _ w ^ 2 , ~ { \\rm { f o r } } ~ { \\rm { a n y } } ~ ( \\bar \\phi , \\bar \\psi ) \\in B _ { R ( \\delta , \\sigma ) } \\subset H _ w ^ 2 , \\end{align*}"}
{"id": "4442.png", "formula": "\\begin{align*} ( N _ m f ) _ { i _ 1 \\dots i _ m } ( x ) = f _ { j _ 1 \\dots j _ m } * \\frac { x ^ { j _ 1 } \\dots x ^ { j _ m } x ^ { i _ 1 } \\dots x ^ { i _ m } } { | x | ^ { 2 m + n - 1 } } \\end{align*}"}
{"id": "181.png", "formula": "\\begin{align*} \\lambda _ { u _ 0 } ( \\zeta ) : = \\sup _ { | \\xi | \\leq \\zeta } \\| u _ 0 - u _ 0 ( \\cdot + \\xi ) \\| _ { L ^ 1 ( \\R ^ N ) } , \\ , \\lambda _ g ( \\zeta ) : = \\sup _ { | \\xi | \\leq \\zeta } \\| g - g ( \\cdot + \\xi , \\cdot + 0 ) \\| _ { L ^ 1 ( Q _ T ) } . \\end{align*}"}
{"id": "7819.png", "formula": "\\begin{align*} \\Theta _ n ( \\phi ) ^ { - 1 } e _ { ( \\widetilde { H } ( \\Theta _ n ( \\phi ) ) ) ( \\theta ( \\lambda ) ) } \\Theta _ n ( \\phi ) = \\frac { t ' ( \\lambda , \\lambda ) } { 2 } e _ { \\theta ( \\lambda ) } ^ \\vee . \\end{align*}"}
{"id": "3514.png", "formula": "\\begin{align*} g _ { 1 } ( u ) = \\exp ( - \\sqrt { 2 u } ) , \\end{align*}"}
{"id": "7503.png", "formula": "\\begin{align*} u ( x , t ) = e ^ { - t ( - \\Delta ) ^ { d } } u _ 0 + \\int _ { 0 } ^ { t } e ^ { - ( t - s ) ( - \\Delta ) ^ { d } } \\left ( | \\cdot | ^ { \\alpha } | u ( s ) | ^ p \\right ) \\ , d s + \\int _ { 0 } ^ { t } \\ , \\zeta ( s ) \\ , e ^ { - ( t - s ) ( - \\Delta ) ^ { d } } { \\mathbf w } \\ , d s , \\end{align*}"}
{"id": "7562.png", "formula": "\\begin{align*} S ^ m ( f _ 1 , \\dots , f _ m ) ( x ) & \\geq \\int _ { B ^ { k } ( 0 , 1 ) } \\prod _ { i = 1 } ^ k | f _ i ( x - x \\sqrt { k } y _ i ) | d y _ 1 \\dots d y _ k \\\\ & \\gtrsim | x | ^ { - k } \\end{align*}"}
{"id": "4327.png", "formula": "\\begin{align*} \\bar { \\mathcal { S } } ( L ) = \\left ( \\sum _ 1 ^ N ( \\sup _ { L _ i } f _ { L _ i } ) e ^ { - i \\hat { \\theta } } \\int _ { L _ i } \\Omega \\right ) - ( \\sup _ { L _ 0 } f _ { L _ 0 } ) ( e ^ { - i \\hat { \\theta } } \\int _ { L _ 0 } \\Omega ) . \\end{align*}"}
{"id": "2784.png", "formula": "\\begin{align*} \\begin{aligned} T _ 4 ( h _ i { } \\leq { } 1 ) = 0 \\end{aligned} \\end{align*}"}
{"id": "1787.png", "formula": "\\begin{align*} \\mathcal { W } = \\bigcup _ { i \\in \\mathbb { N } } \\{ u _ i \\} \\cup \\{ x ^ t _ s : \\ : s \\in \\mathbb { N } , \\ ; t \\in A _ i \\cup \\{ q ( i ) \\} \\} , \\end{align*}"}
{"id": "3033.png", "formula": "\\begin{align*} s _ { \\lambda } ( \\Z ) = \\sum _ { \\tau } K _ { \\lambda , \\tau } ( q ) P _ { \\tau } ( \\Z ; q ) , \\end{align*}"}
{"id": "3734.png", "formula": "\\begin{align*} = \\int _ { \\R ^ 3 } \\int _ { | y - x | \\leq t } \\frac { \\varphi _ { m ; - 1 0 M _ t } ( | y - x | ) } { | y - x | } ( E ( t - | y - x | , y ) + \\hat { v } \\times B ( t - | y - x | , y ) ) \\cdot \\nabla _ v \\big ( \\frac { ( \\hat { v } _ 2 \\omega _ 3 - \\hat { v } _ 3 \\omega _ 2 ) \\varphi _ { j , l } ( v , \\omega ) } { 1 + \\hat { v } \\cdot \\omega } \\big ) \\end{align*}"}
{"id": "6124.png", "formula": "\\begin{align*} | \\omega _ m | > r _ m = r \\max \\left \\{ \\left | \\frac { j } { n } \\right | ^ { \\frac { 1 } { n - j } } , \\ , \\ , j = 1 , 2 , \\dots , m \\right \\} . \\end{align*}"}
{"id": "4130.png", "formula": "\\begin{align*} d _ V ^ { i } A ^ { i } \\alpha = A ^ { i + 1 } D ^ { i } \\alpha . \\end{align*}"}
{"id": "6643.png", "formula": "\\begin{align*} A = \\min \\left ( P _ { \\ , } , P _ { \\ , } \\right ) \\ ; . \\ , \\end{align*}"}
{"id": "5426.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s m - \\frac { ( - \\Delta ) ^ s m _ 1 } { \\gamma _ 1 ^ { 1 / 2 } } m & = 0 \\quad \\Omega , \\\\ m & = m _ 0 \\quad \\Omega _ e . \\end{align*}"}
{"id": "328.png", "formula": "\\begin{align*} \\sum _ { b \\in B ( v ) _ q } { \\rm d } ( { \\rm L } _ b ) = { \\rm d } ( { \\rm L } _ { B ( v ) _ q } ) = { \\rm d } ( { \\rm L } _ q ) . \\end{align*}"}
{"id": "5863.png", "formula": "\\begin{align*} D _ 1 & \\lesssim \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p - q } } \\bigg ( \\int _ { x _ k } ^ { { x _ { k + 1 } } } \\bigg ( \\int _ t ^ { { x _ { k + 1 } } } u \\bigg ) ^ { \\frac { q } { 1 - q } } u ( t ) V _ r ( 0 , t ) ^ { \\frac { q } { 1 - q } } \\bigg ) ^ { \\frac { p ( 1 - q ) } { p - q } } \\\\ & + \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ { - k \\frac { q } { p - q } } \\bigg ( \\int _ { x _ { k } } ^ { { x _ { k + 1 } } } u \\bigg ) ^ { \\frac { p } { p - q } } V _ r ( 0 , x _ k ) ^ { \\frac { p q } { p - q } } . \\end{align*}"}
{"id": "3090.png", "formula": "\\begin{align*} a \\# b \\rightharpoonup m \\otimes x = \\sum _ { ( b ) } a ( b _ { - 1 } \\rightharpoonup m ) \\otimes b _ 0 x = \\sum _ { ( b ) } ( a \\# b _ { - 1 } ) m \\otimes b _ 0 x \\end{align*}"}
{"id": "1052.png", "formula": "\\begin{align*} T ( t , A + B ) x - T ( t , A ) x = \\int _ { 0 } ^ { t } T ( t - s , A + B ) \\tilde { B } T ( s , A ) x \\ , \\textnormal { d } s . \\end{align*}"}
{"id": "5289.png", "formula": "\\begin{align*} \\begin{aligned} x _ 1 ^ { a _ 1 } \\cdots x _ { i - 1 } ^ { a _ { i - 1 } } x _ i ^ { r _ i - 1 } x _ { i + 1 } ^ { a _ { i + 1 } } \\cdots x _ n ^ { a _ n } \\Omega & = 0 \\\\ x _ i ^ r x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } \\Omega & = - \\hbar \\tfrac { a _ i + 1 } { r _ i } x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } \\Omega . \\end{aligned} \\end{align*}"}
{"id": "7591.png", "formula": "\\begin{align*} [ X _ 0 , X _ 1 ] = X _ 0 , [ X _ 0 , X _ 2 ] = 2 X _ 1 , [ X _ 1 , X _ 2 ] = X _ 2 . \\end{align*}"}
{"id": "7639.png", "formula": "\\begin{align*} \\prod _ i f ^ { \\chi _ i ^ { K _ i h } } & = \\prod _ i ( \\chi _ i ^ { - 1 } ) ^ { K _ i h } \\circ f \\circ \\chi _ i ^ { K _ i h } \\\\ & = \\prod _ i ( ( \\chi _ i ^ { - 1 } ) ^ { K _ i } ) ^ h \\circ f ^ h \\circ ( \\chi _ i ^ { K _ i } ) ^ h \\\\ & = \\prod _ i ( f ^ { \\chi _ i ^ { K _ i } } ) ^ h \\\\ & = ( \\prod _ i f ^ { \\chi _ i ^ { K _ i } } ) ^ h \\\\ & = ( \\chi ^ K ) ^ h = \\chi ^ { K h } . \\end{align*}"}
{"id": "8948.png", "formula": "\\begin{align*} L : = F ^ { i j } ( M [ u _ 0 ] ) D _ { i j } , \\end{align*}"}
{"id": "5697.png", "formula": "\\begin{align*} & \\varpi _ { \\{ 3 , 4 , 5 \\} } = \\frac { 1 } { 6 } \\varpi _ 3 \\varpi _ 4 \\varpi _ 5 = e _ 3 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 ) , \\\\ & \\varpi _ { \\{ 2 , 4 , 5 \\} } = \\frac { 1 } { 2 } \\varpi _ 2 \\varpi _ 4 \\varpi _ 5 = e _ 1 ( x _ 1 , x _ 2 ) e _ 2 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 ) . \\end{align*}"}
{"id": "1971.png", "formula": "\\begin{align*} ( S _ { 0 } ( z ) , \\ldots , S _ { p - 1 } ( z ) ) = \\cfrac { \\mathbf { 1 } } { ( 0 , \\ldots , 0 , z ) + \\cfrac { ( 1 , \\ldots , 1 , - a _ { 0 } ) } { ( 0 , \\ldots , 0 , z ) + \\cfrac { ( 1 , \\ldots , 1 , - a _ { 1 } ) } { \\ddots } } } . \\end{align*}"}
{"id": "8786.png", "formula": "\\begin{align*} H ( p , G ) : = \\sup \\{ \\| \\gamma \\| _ 2 ^ { 2 / p ' } \\mid \\gamma \\in L ^ 2 ( G ) , \\ ; \\exists \\phi \\in L ^ 1 ( G ) , \\ ; \\| \\phi \\| _ p = 1 , \\ ; l _ \\gamma = | l _ \\phi | ^ { p ' / 2 } \\} \\end{align*}"}
{"id": "6876.png", "formula": "\\begin{align*} p ( t , g ) = \\int _ { \\mathbb { R } } \\frac { 1 } { \\sqrt { 2 \\pi C ( t ) } } \\exp \\left ( - \\frac { ( g - B ( t ) - y ) ^ 2 } { 2 C ( t ) } \\right ) p _ { t , 0 } ( y ) d y . \\end{align*}"}
{"id": "2489.png", "formula": "\\begin{align*} \\nabla _ U V = T _ U V + \\nu \\nabla _ U V , \\end{align*}"}
{"id": "7068.png", "formula": "\\begin{align*} \\begin{aligned} W _ 2 = & 2 ( m _ 1 - m _ 2 ) / 3 , \\\\ W _ 3 = & \\pi m _ 2 ( l _ 1 + l _ 2 + n _ 2 + n _ 1 ) / 8 , \\\\ W _ 4 = & - 2 m _ 2 ( l _ 1 + l _ 2 + 1 ) ( 3 l _ 1 + l _ 2 + 4 n _ 2 ) / 4 5 , \\\\ W _ 5 = & - \\pi m _ 2 ( l _ 1 + l _ 2 + 1 ) ( l _ 1 - l _ 2 ) ^ 2 / 1 5 3 6 , \\\\ W _ 6 = & - ` 2 m _ 2 ( l _ 2 + 2 ) ( l _ 2 - 1 ) ( l _ 1 + l _ 2 + 1 ) ( l _ 1 - l _ 2 ) / 4 7 2 5 . \\\\ \\end{aligned} \\end{align*}"}
{"id": "5513.png", "formula": "\\begin{align*} h ( z ) = z + \\sum _ { n = 2 } ^ \\infty a _ n z ^ n . \\end{align*}"}
{"id": "5192.png", "formula": "\\begin{align*} e _ i : = \\frac { 2 \\sum _ j a _ { j , i } + ( - 1 + \\sum _ { T : i \\in T } k _ T ) ( r _ i - 2 ) } { r _ i } \\in \\Z , ~ ~ i = 1 , \\ldots , a , \\end{align*}"}
{"id": "7663.png", "formula": "\\begin{align*} e _ { \\mathrm { t o p } } ( S _ n ) = n ^ 2 ( 1 0 - 1 0 n + 3 n ^ 2 ) + \\binom { n } { 2 } ( - 1 0 + 1 5 n - 6 n ^ 2 ) + \\binom { n } { 3 } ( 4 n - 5 ) - \\binom { n } { 4 } . \\end{align*}"}
{"id": "534.png", "formula": "\\begin{align*} \\tau _ l ( n ) = O _ { l , \\varepsilon } ( n ^ \\varepsilon ) \\end{align*}"}
{"id": "3865.png", "formula": "\\begin{align*} \\Big \\{ \\tilde { \\phi } ^ { T , \\Upsilon , Y } [ \\tilde { h } _ j ^ Y ] : \\tilde { h } _ j ^ Y \\in Z ^ a [ Y _ 1 , . . . , Y _ n ] , \\tilde { h } _ j ^ \\Upsilon \\in Z ^ a [ \\Upsilon _ 1 , . . . , \\Upsilon _ n ] \\Big \\} = Z ^ a [ T _ 1 , . . . , T _ n ] . \\end{align*}"}
{"id": "1360.png", "formula": "\\begin{align*} \\widehat { \\mathcal { R } ^ 1 } [ u , \\cdots , u ] _ k : = \\sum _ { \\substack { \\ell = 1 \\\\ o d d } } ^ { p } \\sum _ { \\substack { k = k _ 1 - k _ 2 + \\cdots - k _ { p - 1 } + k _ p \\\\ k _ \\ell = k } } \\prod _ { \\substack { i = 1 \\\\ o d d } } ^ { p } \\widehat { u } _ { k _ i } \\prod _ { \\substack { i = 2 \\\\ e v e n } } ^ p \\overline { \\widehat { u } _ { k _ i } } = \\frac { p + 1 } { 4 \\pi } \\widehat { u } _ k \\int _ \\mathbb { T } | u | ^ { p - 1 } \\ , d x . \\end{align*}"}
{"id": "33.png", "formula": "\\begin{align*} \\mu ( \\alpha ) = \\sum _ { w \\in W _ { \\rho } ^ n } \\min \\{ \\mu ( \\xi ) , \\mu ( w ) \\} . \\end{align*}"}
{"id": "7171.png", "formula": "\\begin{align*} \\mbox { d e g } ( t ^ m \\otimes u ) = \\mbox { w t } u - m - 1 \\end{align*}"}
{"id": "8165.png", "formula": "\\begin{align*} & H _ x = - \\frac { d ^ 2 } { d x ^ 2 } + V _ 0 ( x ) \\\\ & H _ y = - \\frac { d ^ 2 } { d y ^ 2 } \\end{align*}"}
{"id": "857.png", "formula": "\\begin{align*} \\gamma _ { m } ^ { f } = \\frac { \\rho _ { 0 , 1 } ( 1 - \\pi _ { m - 1 } ^ { i n } ) ( 1 - \\gamma _ { m - 1 } ^ { f } ) + \\rho _ { 1 , 1 } \\pi _ { m - 1 } ^ { i n } \\gamma _ { m - 1 } ^ { f } } { ( 1 - \\pi _ { m - 1 } ^ { i n } ) ( 1 - \\gamma _ { m - 1 } ^ { f } ) + \\pi _ { m - 1 } ^ { i n } \\gamma _ { m - 1 } ^ { f } } , \\end{align*}"}
{"id": "1457.png", "formula": "\\begin{align*} \\Theta = { \\rm { d e t } } { \\begin{pmatrix} \\vec { q } _ { 0 } \\ \\cdots \\ \\vec { q } _ { r m - 1 } \\end{pmatrix} } \\enspace . \\end{align*}"}
{"id": "2122.png", "formula": "\\begin{align*} F ( z ) = a + b z + \\int _ { \\mathbb { R } } \\frac { 1 + s z } { s - z } \\ , d \\rho ( s ) , z \\in \\mathbb { C } ^ { + } , \\end{align*}"}
{"id": "1740.png", "formula": "\\begin{align*} f ^ c _ { k - 2 } ( z , \\overline \\omega _ 1 ) = \\int _ { c \\cdot C } \\frac { e ^ { z s } \\ , s ^ { k - 2 } } { e ^ { \\overline \\omega _ 1 s } - 1 } \\ d s , \\end{align*}"}
{"id": "3268.png", "formula": "\\begin{align*} \\norm { A _ j } _ { W ^ { 2 , \\infty } } \\leq M , \\norm { q _ j } _ { L ^ \\infty } \\leq M , j = 1 , 2 , \\end{align*}"}
{"id": "2832.png", "formula": "\\begin{align*} \\min _ { 0 \\leq i \\leq N } \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\} { } = { } \\min _ { 0 \\leq i \\leq N } \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\} \\sum _ { i = 0 } ^ { N - 1 } \\frac { p _ i } { S } { } \\leq { } \\sum _ { i = 0 } ^ { N - 1 } \\frac { p _ i } { S } \\min \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\ , , \\ , \\| \\nabla f ( x _ { i + 1 } ) \\| ^ 2 \\} \\end{align*}"}
{"id": "7108.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\tau _ x \\left ( [ P ^ E _ t ] \\right ) = \\tau _ x \\left ( [ P ^ E _ 0 ] \\right ) . \\end{align*}"}
{"id": "5698.png", "formula": "\\begin{align*} & I \\coloneqq ( e _ k ( y _ 1 , y _ 2 , \\ldots , y _ n ) \\mid 1 \\le k \\le n ) , \\\\ & I ' \\coloneqq ( ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \\ldots , y _ i ) \\mid 1 \\le i \\le n - 1 , \\ 1 \\le k \\le \\min \\{ i , n - i \\} ) . \\end{align*}"}
{"id": "6186.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\mathrm { d i s t } ^ { 2 } ( p , S _ { 0 } ^ { 3 } ) = ( \\frac { | l _ { 1 } | ^ { 2 } + | l _ { 2 } | ^ { 2 } } { | l _ { i } | ^ { 2 } } ) | v _ { i } | ^ { 2 } : = | s | _ { h } ^ { 2 } \\end{array} \\end{align*}"}
{"id": "8939.png", "formula": "\\begin{align*} \\int _ \\Omega f = \\int _ { \\Omega ^ * } f ^ * . \\end{align*}"}
{"id": "3464.png", "formula": "\\begin{align*} c _ { ( k ) } ^ { \\ast } = \\frac { 1 } { \\sqrt { 2 } } \\left [ \\int _ { 0 } ^ { \\infty } s ^ { - 1 / 2 } \\overline { G } _ { ( k ) } ( s ) \\mathrm { d } s \\right ] ^ { - 1 } . \\end{align*}"}
{"id": "7004.png", "formula": "\\begin{align*} \\mathcal Z _ K = \\bigsqcup _ { i = 1 } ^ s \\widetilde U _ { \\tau _ i } . \\end{align*}"}
{"id": "4041.png", "formula": "\\begin{align*} E e = 0 \\end{align*}"}
{"id": "2579.png", "formula": "\\begin{align*} \\overline C = \\{ x \\ , \\omega _ 1 + y \\ , \\omega _ 2 : x , y \\ge 0 \\} \\end{align*}"}
{"id": "4220.png", "formula": "\\begin{align*} t _ n = \\inf \\left \\{ m > t _ { n - 1 } : Z _ m < a _ n Z _ { t _ { n - 1 } } Z _ m < ( 1 - n ^ { z ^ \\prime - 1 } ) Z _ m ^ * \\right \\} \\end{align*}"}
{"id": "7519.png", "formula": "\\begin{align*} p _ { s u c c } = \\sum _ { i = 1 } ^ { m i n ( m , ( 2 | E | - \\gamma _ { R S U } ) k ) } \\mbox { P r } ( C _ { i } ) \\sum _ { j = 0 } ^ { k } \\binom { m - i } { j } ( p _ { m a t c h , j } ) ^ { 2 } , \\end{align*}"}
{"id": "2260.png", "formula": "\\begin{align*} \\begin{cases} - \\phi '' - \\frac { z } { 2 } \\phi ' + \\frac { z } { 2 } e _ \\sigma \\psi ' = f _ 1 , \\\\ - \\psi '' - \\frac { z } { 2 } \\psi ' + \\frac { z } { 2 } e _ \\sigma \\phi ' = f _ 2 , \\\\ \\phi ( 0 ) = \\phi ( r ) = 0 , \\psi ( 0 ) = \\psi ( r ) = 0 . \\end{cases} \\end{align*}"}
{"id": "12.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { \\rho , L , n } = \\Phi _ L ( \\beta ) ( \\xi ) - \\xi ; ~ ~ \\rho _ { \\eta ^ n } ( \\xi ) = \\alpha . \\end{align*}"}
{"id": "8496.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } \\rho ( x ) = \\infty . \\end{align*}"}
{"id": "407.png", "formula": "\\begin{align*} \\int _ { \\R } e ^ { - i t s } \\int _ { \\sf T _ \\varphi \\cal Q _ { k } } e ^ { i \\psi ( u ) } \\big ( 1 - \\kappa ( \\psi ) \\big ) \\sum _ { n \\in \\N } \\big ( \\cal L _ { \\varphi + i \\psi } ^ n p _ F \\big ) ( x ) \\cal F v _ G ( \\psi ) d \\psi _ 0 d s = O ( t ^ { - N } ) \\end{align*}"}
{"id": "335.png", "formula": "\\begin{align*} C : = \\{ d _ { k - 1 } < c _ i ^ { ( k ) } \\le d _ k \\ : \\ k , i \\ge 0 \\} \\ ; = \\ ; \\bigcup _ { k \\ge 0 } \\{ c _ i ^ { ( k ) } \\ : \\ i \\in [ i _ k , r _ { J ( d _ k ) } ] \\} . \\end{align*}"}
{"id": "3603.png", "formula": "\\begin{align*} { \\rm R C } ( I ^ \\vee ) = \\left ( \\bigcap _ { i = 1 } ^ { s + 1 } H ^ + _ { e _ i } \\right ) \\bigcap \\left ( \\bigcap _ { i = 1 } ^ q H ^ + _ { ( \\delta _ i , - f _ i ) } \\right ) \\bigcap \\left ( \\bigcap _ { i = q + 1 } ^ { p _ 1 } H ^ + _ { ( \\delta _ i , - f _ i ) } \\right ) , \\end{align*}"}
{"id": "2513.png", "formula": "\\begin{align*} s = \\sum \\limits _ { i = n + 1 } ^ { m } R i c ( U _ i , V _ i ) + \\sum \\limits _ { j = 1 } ^ { n } R i c ( X _ j , X _ j ) , \\end{align*}"}
{"id": "5370.png", "formula": "\\begin{align*} P = \\sum _ { | \\alpha | \\leq m } a _ \\alpha D ^ \\alpha , \\end{align*}"}
{"id": "2931.png", "formula": "\\begin{align*} h _ { j + 1 } ^ { j + 1 } = h _ { j + 1 } ^ { 1 } + h _ { j + 1 } ^ { 2 } + \\cdots + h _ { j + 1 } ^ { j } + h _ { j + 1 } ^ { j + 1 } \\geq j + 1 - y _ { j + 1 } + 1 \\geq j + 1 - ( j + 1 ) + 1 = 1 , \\end{align*}"}
{"id": "1541.png", "formula": "\\begin{align*} \\phi ^ { ( \\ss ) } : = \\Big ( \\frac { \\ss ^ { \\frac { 1 } { \\alpha } } } { e ^ { R } } \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
{"id": "2637.png", "formula": "\\begin{align*} I ^ \\alpha = \\{ f \\in S ^ \\alpha \\ , \\vert \\ , \\Lambda ( f g ) = 0 \\ \\mbox { f o r a l l } \\ g \\in S ^ { N - \\alpha } \\} ; \\end{align*}"}
{"id": "7011.png", "formula": "\\begin{align*} \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } V o l ( h ) = - \\sum _ { l \\ne i _ j } \\langle \\chi , \\Lambda ( \\rho _ { l } ) \\rangle \\partial _ l \\partial _ { i _ 1 } ^ { k _ 1 - 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } V o l ( h ) \\end{align*}"}
{"id": "1468.png", "formula": "\\begin{align*} & { { \\tilde { \\psi } } } _ { \\alpha , s , \\boldsymbol { \\xi } _ s , \\ell _ s } = { { \\tilde { \\psi } } } _ { \\alpha , s } \\bigcirc _ { w = 1 } ^ { \\ell _ s } ( \\theta _ { X _ s } + \\xi _ { s , w } ) \\ \\ 0 \\le s \\le r - 1 \\enspace , \\\\ & \\Psi _ { \\alpha , \\boldsymbol { \\Xi } _ { \\boldsymbol { \\ell } } } = \\bigcirc _ { s = 0 } ^ { r - 1 } { { \\tilde { \\psi } } } _ { \\alpha , s , \\boldsymbol { \\xi } _ s , \\ell _ s } \\enspace , \\end{align*}"}
{"id": "8147.png", "formula": "\\begin{align*} & \\int \\limits _ 0 ^ \\infty \\| \\chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \\perp } P _ \\delta ^ \\perp ( W ^ \\perp _ { n ; \\textrm { o u t } } ) \\| _ \\textrm { o p } \\ , d t \\leq \\int \\limits _ 0 ^ \\frac { 1 } { n } 1 \\ , d t + C \\int \\limits _ { \\frac { 1 } { n } } ^ \\infty t ^ { - \\frac { 1 } { 2 } } ( n + t m ) ^ { - \\ell } \\ , d t \\\\ & \\leq \\frac { 1 } { n } + C \\sqrt { n } \\int \\limits _ { \\frac { 1 } { n } } ^ \\infty ( n + t m ) ^ { - \\ell } \\ , d t = \\frac { 1 } { n } + C \\frac { \\sqrt { n } } { m } ( n + \\frac { m } { n } ) ^ { - \\ell + 1 } \\end{align*}"}
{"id": "5153.png", "formula": "\\begin{align*} [ D / \\mathcal { W } \\vec { y } ] _ { v _ i } = \\sum _ { j = 1 , j \\neq i } ^ { k } d _ G ( w _ i , w _ j ) | W _ j | \\vec { x } _ { w _ j } + c _ i ( | W _ i | - 1 ) \\vec { x } _ { w _ i } , \\end{align*}"}
{"id": "6219.png", "formula": "\\begin{align*} r ( x ) \\frac { h ( x ) } { h ( x / q ) } & = \\frac { r ( x ) } { h ( x / q ) } \\left ( h ( x / q ) + ( 1 - \\frac { 1 } { q } ) x D _ { q ^ { - 1 } } h ( x ) \\right ) \\\\ & = r ( x ) \\left ( 1 + ( 1 - \\frac { 1 } { q } ) x u ( \\frac { x } { q } ) \\right ) . \\end{align*}"}
{"id": "2878.png", "formula": "\\begin{align*} A _ n = \\{ x \\in A : S ^ n ( x ) = x \\} = \\{ x \\in A : x _ i = x _ { i + n } , i \\in \\Z \\} . \\end{align*}"}
{"id": "7956.png", "formula": "\\begin{align*} t _ + = \\sum _ { a = 1 } ^ \\mathrm { r } \\bar p _ a ^ + \\log y _ a , y ^ d = y _ 1 ^ { p _ 1 ^ + \\cdot d } \\cdots y _ \\mathrm { r } ^ { p _ \\mathrm { r } ^ + \\cdot d } \\end{align*}"}
{"id": "1863.png", "formula": "\\begin{align*} W _ { 0 } ( z ) & = \\frac { 1 } { z - a _ { 0 } ^ { ( 0 ) } - \\sum _ { j = 1 } ^ { p } \\sum _ { k = 0 } ^ { j } a _ { - k } ^ { ( j ) } \\ , A _ { j - k - 1 } ^ { ( 1 ) } ( z ) \\ , B _ { k - 1 } ^ { ( 1 ) } ( z ) } \\\\ W _ { j } ( z ) & = W _ { 0 } ( z ) \\ , A _ { j - 1 } ^ { ( 1 ) } ( z ) 1 \\leq j \\leq p . \\end{align*}"}
{"id": "1270.png", "formula": "\\begin{align*} \\Psi ( E _ j ( x , a ) ) & = \\sum _ { h = 1 } ^ { j } \\left | 2 \\sqrt { a } \\cos \\left ( \\frac { h } { j + 1 } \\pi \\right ) \\right | = 2 \\sqrt { a } \\sum _ { h = 1 } ^ { j } \\left | \\cos \\left ( \\frac { h } { j + 1 } \\pi \\right ) \\right | . \\end{align*}"}
{"id": "1907.png", "formula": "\\begin{align*} & \\sum _ { j = 2 } ^ { p } \\sum _ { s = 1 } ^ { j - 1 } \\sum _ { k \\in \\mathbb { Z } } \\sum _ { \\ell \\in \\mathbb { Z } } a _ { - s } ^ { ( j ) } \\ , A _ { [ k - 1 , j - s - 1 ] } ^ { ( 1 ) } B _ { [ \\ell - k - 1 , s - 1 ] } ^ { ( 1 ) } W _ { [ n - \\ell - 1 , 0 ] } \\\\ = & \\sum _ { j = 2 } ^ { p } \\sum _ { s = 1 } ^ { j - 1 } \\sum _ { \\ell \\in \\mathbb { Z } } \\sum _ { k \\in \\mathbb { Z } } a _ { - s } ^ { ( j ) } \\ , A _ { [ \\ell - 1 , j - s - 1 ] } ^ { ( 1 ) } B _ { [ k - \\ell - 1 , s - 1 ] } ^ { ( 1 ) } W _ { [ n - k - 1 , 0 ] } \\end{align*}"}
{"id": "1244.png", "formula": "\\begin{align*} P ( T , x ) & = \\prod _ { j = 1 } ^ { l ( T ) } \\mathcal { G } _ j ^ { n ( T , j ) } . \\end{align*}"}
{"id": "7256.png", "formula": "\\begin{align*} \\mathbb { E } ( \\tau ) = \\sum _ { n = 1 } ^ { \\infty } i b _ i = \\sum _ { j = 1 } ^ { \\infty } \\left \\lceil { \\frac { j } { k } } \\right \\rceil a _ j ' \\leq \\frac { 1 } { k } \\left ( k + \\sum _ { j = 1 } ^ { \\infty } j a _ j ' \\right ) = 1 + \\sum _ { n = 1 } ^ { \\infty } n a _ n , \\end{align*}"}
{"id": "2068.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : a b s t r a c t t h e o r e m s p l i t t i n g } \\big \\langle f ^ { k , N } _ t - f _ t ^ { \\otimes k } , \\varphi _ k \\big \\rangle = \\big \\langle f ^ { k , N } _ t - F ^ { k , N } _ t , \\varphi _ k \\big \\rangle & + \\big \\langle F ^ { k , N } _ t - \\overline { F } { } ^ { k , N } _ t , \\varphi _ k \\big \\rangle + \\big \\langle \\overline { F } { } ^ { k , N } _ t - f _ t ^ { \\otimes k } , \\varphi _ k \\big \\rangle . \\end{align*}"}
{"id": "6391.png", "formula": "\\begin{align*} | \\pi | _ 1 + 2 \\ , | \\pi | _ 2 = k , \\mbox { t h e t o t a l n u m b e r o f p o i n t s p a r t i t i o n e d b y $ \\pi $ } . \\end{align*}"}
{"id": "1862.png", "formula": "\\begin{align*} A _ { 0 } ^ { ( k ) } ( z ) = \\frac { 1 } { z - a _ { k } ^ { ( 0 ) } - \\sum _ { j = 1 } ^ { p } a _ { k } ^ { ( j ) } \\prod _ { \\ell = 1 } ^ { j } A _ { 0 } ^ { ( k + \\ell ) } ( z ) } , k \\geq 0 . \\end{align*}"}
{"id": "3377.png", "formula": "\\begin{align*} \\sup \\left \\{ s \\in [ 0 , 1 ] : \\frac { x - 1 } { \\log x } \\le s \\sqrt { x } + ( 1 - s ) \\left ( \\frac { x + 1 } { 2 } \\right ) , ( x > 0 ) \\right \\} = \\frac { 2 } { 3 } . \\end{align*}"}
{"id": "4.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 + i & 1 & 0 \\\\ 2 + i & 1 + 4 i & 1 & 1 \\\\ 2 + i & 1 + 5 i & 1 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "508.png", "formula": "\\begin{align*} M = \\mathrm { s p a n } ( & u _ 1 ^ 1 + u _ 1 ^ 2 + u _ 1 ^ 3 , \\\\ & v _ 1 ^ 1 , v _ 1 ^ 2 , v _ 1 ^ 3 , v _ 2 ^ 1 + v _ 2 ^ 2 + v _ 2 ^ 3 , \\\\ & w _ 1 ^ 1 + w _ 1 ^ 2 + w _ 1 ^ 3 , 3 w _ 1 ^ 3 + 2 w _ 1 ^ 2 + w _ 1 ^ 3 ) . \\end{align*}"}
{"id": "8102.png", "formula": "\\begin{align*} \\operatorname { d i v } _ { \\mu _ { H } } ( Y ) = \\operatorname { d i v } _ { \\mu } ( H Y ) . \\end{align*}"}
{"id": "3032.png", "formula": "\\begin{align*} p _ { \\lambda } ( \\Z ) = & \\sum _ { \\tau \\in \\mathcal { P } } \\chi ^ { \\tau } _ { \\lambda } s _ { \\tau } ( \\Z ) , \\\\ s _ { \\lambda } ( \\Z ) = & \\sum _ { \\tau } z _ { \\tau } ^ { - 1 } \\chi ^ { \\lambda } _ { \\tau } p _ { \\tau } ( \\Z ) , \\end{align*}"}
{"id": "1413.png", "formula": "\\begin{align*} \\bar q ( z ) = \\left ( z + | \\hat \\lambda _ { 1 } | \\right ) \\left ( z + | \\hat \\lambda _ { 2 } | \\right ) \\cdots \\left ( z + | \\hat \\lambda _ { r } | \\right ) = z ^ r + \\bar \\sigma _ 1 z ^ { r - 1 } + \\bar \\sigma _ 2 z ^ { r - 2 } + \\cdots + \\bar \\sigma _ r . \\end{align*}"}
{"id": "4370.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t \\} \\cap D _ 0 } | \\tilde F | ^ 2 e ^ { - k \\varphi - a \\Psi } \\\\ \\le & 2 \\int _ { \\{ \\psi < - t \\} \\cap D _ 0 } | f | ^ 2 e ^ { - a \\Psi } + 2 \\int _ { \\{ \\Psi < - t \\} \\cap D _ 0 } | \\tilde F - f F ^ { 2 k } | ^ 2 e ^ { - k \\varphi - a \\Psi } \\\\ < & + \\infty . \\end{align*}"}
{"id": "6866.png", "formula": "\\begin{align*} \\frac { N } { \\sqrt { c ^ * } } = \\int _ { \\mathbb { R } } ( g + \\frac { b } { \\sqrt { c ^ * } } ) _ + \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - \\frac { 1 } { 2 } g ^ 2 } d g & = \\int _ { - \\lambda } ^ { + \\infty } ( g + \\lambda ) \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - \\frac { 1 } { 2 } g ^ 2 } d g \\\\ & = \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - \\lambda ^ 2 / 2 } + \\lambda \\int _ { - \\lambda } ^ { + \\infty } \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - g ^ 2 / 2 } d g . \\end{align*}"}
{"id": "1851.png", "formula": "\\begin{align*} w ( \\gamma ) = \\prod _ { e \\subset \\gamma } w ( e ) , \\end{align*}"}
{"id": "8581.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 e ^ { 2 \\pi i m ( x + u ) } & \\prod _ { k = 1 } ^ s \\cos ( 2 \\pi n _ { k _ j } u ) d u \\\\ & \\qquad = \\left \\{ \\begin{array} { l l } 2 ^ { - s } e ^ { 2 \\pi i m x } , & \\hbox { i f } m = \\varepsilon _ 1 n _ { k _ 1 } + \\varepsilon _ { 2 } n _ { k _ 2 } + \\ldots + \\varepsilon _ 1 n _ { k _ s } , \\\\ 0 , & \\hbox { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
{"id": "8899.png", "formula": "\\begin{align*} \\int _ { \\mathcal { G } } | u | ^ 2 \\ , d x = \\mu > 0 , \\end{align*}"}
{"id": "1594.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial z ^ i _ { \\epsilon } } v ^ i = 0 , \\end{align*}"}
{"id": "2114.png", "formula": "\\begin{align*} ( A _ G x ) ( v ) = \\sum \\limits _ { e \\in E _ v } \\frac { \\delta _ E ( e ) } { \\delta _ V ( v ) } \\frac { 1 } { | e | ^ 2 } \\sum \\limits _ { u \\in e ; u \\neq v } x ( u ) . \\end{align*}"}
{"id": "1334.png", "formula": "\\begin{align*} \\phi ( x , v ) = H ( x , v ) + \\alpha x \\cdot v + \\beta | x | ^ 2 , \\end{align*}"}
{"id": "3040.png", "formula": "\\begin{align*} \\mathbf { D } _ { L _ I } \\big ( \\prod _ { j = 1 } ^ { 2 k } g ^ { - 1 } t _ j \\sigma ( t _ j ) g \\big ) = \\mathbf { D } _ { L _ I } \\big ( \\prod _ { j = 1 } ^ { 2 k } ( w _ j s _ j w _ j ^ { - 1 } ) \\sigma ( w _ j s _ j w _ j ^ { - 1 } ) \\big ) , \\end{align*}"}
{"id": "5221.png", "formula": "\\begin{align*} \\psi ^ * ( d x _ 1 \\wedge \\cdots \\wedge d x _ n ) = d x _ 1 \\wedge \\cdots \\wedge d x _ n . \\end{align*}"}
{"id": "257.png", "formula": "\\begin{align*} \\Gamma ^ { k ( s ) } _ { i j } ( 1 ) = 0 \\ \\ \\textup { f o r a l l } \\ \\ i , j , k \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "1303.png", "formula": "\\begin{align*} \\mu ( T _ 0 , \\lambda ) \\ge \\sum _ { j = 1 } ^ { k } ( \\alpha _ j - 1 ) \\mu ( T _ j , \\lambda ) \\end{align*}"}
{"id": "6320.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + \\dfrac { 1 - q x } { q x ^ 2 ( 1 - q ) } D _ { q ^ { - 1 } } y ( x ) + \\frac { [ n ] _ q } { x ^ 2 ( 1 - q ) } y ( x ) = 0 . \\end{align*}"}
{"id": "2135.png", "formula": "\\begin{align*} - \\lim _ { \\epsilon \\rightarrow 0 ^ + } \\frac { \\Im \\varphi _ \\mu ( x + i \\epsilon ) } { \\epsilon } = \\int _ \\mathbb { R } \\frac { 1 + s ^ 2 } { ( x - s ) ^ 2 } \\ , d \\sigma _ { \\mu } ( s ) > 1 , x \\in \\mathbb { R } , \\end{align*}"}
{"id": "2255.png", "formula": "\\begin{align*} \\phi ( z ) = \\phi _ * ( z ) - e _ \\delta ( z ) , \\psi ( z ) = \\psi _ * ( z ) - e _ \\sigma ( z ) . \\end{align*}"}
{"id": "8106.png", "formula": "\\begin{align*} & \\partial _ t u + \\nabla _ { u } u = - \\operatorname { g r a d } p t \\in [ 0 , t _ { 0 } ] , \\\\ & \\operatorname { d i v } u = 0 , \\\\ & u | _ { t = 0 } = \\dot { \\eta } ( 0 ) , \\end{align*}"}
{"id": "1902.png", "formula": "\\begin{align*} \\mathcal { V } _ { [ n , \\ell , k ] } : = \\{ \\gamma \\in \\mathcal { V } _ { [ n , \\ell ] } : \\eta ( \\gamma ) = k \\} , \\ell \\leq k \\leq n - 1 . \\end{align*}"}
{"id": "3509.png", "formula": "\\begin{align*} L _ { 1 } = \\frac { \\xi _ { p } \\phi ( \\xi _ { p } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) - \\xi _ { q } \\phi ( \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) } { \\Psi _ { 2 } ^ { \\ast } ( - 1 , \\frac { 1 } { 2 } , 1 ) F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { p } ) ) ( 1 + \\sqrt { 2 \\pi } \\phi ( \\xi _ { q } ) ) } , ~ L _ { 2 } & = \\frac { F _ { Y _ { ( 1 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "3957.png", "formula": "\\begin{align*} j ( a ^ { * } ) ( \\gamma ) = j ( a ) ( \\gamma ^ { - 1 } ) ^ { * } \\end{align*}"}
{"id": "5590.png", "formula": "\\begin{align*} \\tau _ { i , j } = \\delta _ { i , j } \\sum _ { k = 1 } ^ { n } B _ { i , k } - B _ { i , j } , i , j = 1 , \\cdots , n . \\end{align*}"}
{"id": "1591.png", "formula": "\\begin{align*} v o l ( D ^ 2 _ x ( 1 ) ) : = \\frac { v o l ( B ^ 2 ( 1 ) ) b ^ 2 A ^ { \\gamma \\tau } \\left [ \\cos ^ 2 \\theta z ^ { 1 } _ { \\gamma } z ^ { 1 } _ { \\tau } + \\sin ^ 2 \\theta z ^ { 2 } _ { \\gamma } z ^ { 2 } _ { \\tau } + 2 \\sin \\theta \\cos \\theta z ^ { 1 } _ { \\gamma } z ^ { 2 } _ { \\tau } \\right ] } { 2 \\sqrt { \\det A } } . \\end{align*}"}
{"id": "1582.png", "formula": "\\begin{align*} E = b ^ 2 f ^ 2 ( x ^ 1 ) \\left [ f '^ 2 ( x ^ 1 ) + \\sin ^ { 2 } \\left ( x ^ 2 \\right ) \\right ] . \\end{align*}"}
{"id": "1115.png", "formula": "\\begin{align*} C ( v ) = - \\ln \\left ( 1 - \\frac { v ^ 2 } { q ^ 2 } \\right ) . \\end{align*}"}
{"id": "548.png", "formula": "\\begin{align*} | f _ k ( n ) | \\leq \\tau _ 3 ( n ) = O _ { \\varepsilon } ( n ^ { \\varepsilon } ) . \\end{align*}"}
{"id": "3416.png", "formula": "\\begin{align*} g _ i = x _ i ( h _ 1 , \\ldots , h _ s ) . \\end{align*}"}
{"id": "6216.png", "formula": "\\begin{align*} & \\int f ( x ) h ( x / q ) \\Big ( \\frac { 1 } { q } D _ { q ^ { - 1 } } u ( x ) + \\frac { 1 } { q } u ( x ) u ( x / q ) + A ( x ) u ( x / q ) + r ( x ) \\Big ) y ( x ) d _ q x \\\\ & = f ( x / q ) h ( x / q ) \\Big ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \\Big ) , \\end{align*}"}
{"id": "4344.png", "formula": "\\begin{align*} m _ 0 ^ b = m _ 0 + m _ 1 ( b ) + m _ 2 ( b , b ) + \\ldots = 0 \\in C F _ { s e l f } ^ 2 ( L , L ) . \\end{align*}"}
{"id": "1645.png", "formula": "\\begin{align*} \\tau ( x , y , t ) \\ , \\ , = \\ , \\ , \\sum _ { I \\in \\binom { [ n ] } { k } } p _ I \\prod _ { \\substack { i , j \\in I \\\\ i < j } } ( \\kappa _ i - \\kappa _ j ) \\cdot \\exp \\biggl [ x \\cdot \\sum _ { i \\in I } \\kappa _ i + y \\cdot \\sum _ { i \\in I } \\kappa _ i ^ 2 + t \\cdot \\sum _ { i \\in I } \\kappa _ i ^ 3 \\biggr ] , \\end{align*}"}
{"id": "1337.png", "formula": "\\begin{align*} C = \\frac { 1 } { 1 - \\beta \\kappa ^ 2 e ^ { - \\tau \\| \\sigma \\| _ \\infty } } , \\mbox { a n d } \\lambda = - \\frac { \\log ( 1 - \\beta \\kappa ^ 2 e ^ { - \\tau \\| \\sigma \\| _ \\infty } ) } { \\tau } . \\end{align*}"}
{"id": "1013.png", "formula": "\\begin{align*} h _ \\mu ( \\P ) = \\lim _ { n \\to \\infty } \\frac 1 n H _ \\mu \\ ! \\left ( \\bigvee _ { i = 0 } ^ { n - 1 } T ^ { - i } \\P \\right ) , \\end{align*}"}
{"id": "4860.png", "formula": "\\begin{align*} \\begin{cases} N _ { 1 } = p _ { 1 } + p _ { 2 } ^ { 2 } + p _ { 3 } ^ { 3 } + p _ { 4 } ^ { 3 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k _ { 2 } } } \\\\ N _ { 2 } = p _ { 5 } + p _ { 6 } ^ { 2 } + p _ { 7 } ^ { 3 } + p _ { 8 } ^ { 3 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k _ { 2 } } } \\end{cases} \\end{align*}"}
{"id": "8685.png", "formula": "\\begin{align*} d V _ { \\max } = ( 1 + \\| \\beta \\| _ { \\alpha } ) ^ { n + 1 } d V _ { \\alpha } \\end{align*}"}
{"id": "7951.png", "formula": "\\begin{align*} D _ + = \\sum _ { i \\in M _ - } \\bar { D } _ i \\subset X _ + D _ - = \\sum _ { i \\in ( M _ - \\cup \\{ j \\} ) } \\bar { D } _ i \\subset X _ - . \\end{align*}"}
{"id": "4030.png", "formula": "\\begin{align*} \\underline { \\gamma } & = n \\chi ( \\underline { \\epsilon } _ n ) \\\\ \\overline { \\gamma } & = n \\chi ( \\overline { \\epsilon } _ n ) . \\end{align*}"}
{"id": "2491.png", "formula": "\\begin{align*} \\nabla _ X Y = A _ X Y + \\mathcal { H } \\nabla _ X Y , \\end{align*}"}
{"id": "8536.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { n - 1 } \\epsilon _ p & = \\sum _ { p = 1 } ^ { \\infty } \\epsilon _ p - \\sum _ { p = n } ^ { \\infty } \\epsilon _ p \\\\ & = B - r _ n \\end{align*}"}
{"id": "1292.png", "formula": "\\begin{align*} \\mu ( T _ 0 , \\lambda ) \\ge \\sum _ { j = 1 } ^ { k } ( \\alpha _ j - 1 ) \\mu ( T _ j , \\lambda ) ( \\forall \\lambda \\in \\mathbb { R } ) . \\end{align*}"}
{"id": "7774.png", "formula": "\\begin{align*} A _ m ( \\alpha ) { \\tilde { \\bf u } } _ m ( \\alpha ) = { \\bf f } _ m ( \\alpha ) + \\boldsymbol { \\eta } _ m , \\end{align*}"}
{"id": "3853.png", "formula": "\\begin{align*} & \\partial _ { ( c , b _ { i + 1 } , b _ { n } ) } f ( \\mu ) ( v _ { 1 } , . . . , v _ { p } , v _ { p + 1 } ) \\cdot e _ { 1 } \\otimes . . . \\otimes e _ { i } \\otimes e _ { i + 1 } \\otimes e _ { { n } } \\\\ & = \\partial _ { ( c , b _ { n } , b _ { i + 1 } ) } f ( \\mu ) ( v _ { 1 } , . . . , v _ { p } , v _ { p + 1 } ) \\cdot e _ { 1 } \\otimes . . . \\otimes e _ { i } \\otimes e _ { { n } } \\otimes e _ { i + 1 } . \\end{align*}"}
{"id": "4284.png", "formula": "\\begin{align*} \\frac { d } { d t } \\rho _ t = \\mathcal { L } ( \\rho _ t ) , \\mathcal { L } ( \\rho ) = - \\frac 1 2 \\sum _ { j = 1 } ^ { N } [ C _ j , [ C _ j , \\rho ] ] , \\end{align*}"}
{"id": "378.png", "formula": "\\begin{align*} \\frac { 1 } { \\mathbb { E } _ { \\alpha } [ T _ { A } ] } = \\lambda = \\frac { \\langle ( I - P ) f _ { 1 } , f _ { 1 } \\rangle _ { \\pi } } { \\| f _ 1 \\| _ 2 ^ 2 } = \\mathrm { V a r } _ { \\pi } f _ 1 \\frac { \\langle ( I - P ) f _ { 1 } , f _ { 1 } \\rangle _ { \\pi } } { \\mathrm { V a r } _ { \\pi } f _ 1 } . \\end{align*}"}
{"id": "7769.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { Q } _ { j , } ^ k \\sim \\mathcal { G P } ( q _ { j , } ^ k , z _ { j , } ^ k ) & k = 1 , \\ldots , m , \\ j = 1 , \\ldots , L , \\end{aligned} \\end{align*}"}
{"id": "3232.png", "formula": "\\begin{align*} \\mathcal { C } = \\mathcal { C } _ X \\cup \\bigcup _ { x \\in D } \\mathcal { C } _ x . \\end{align*}"}
{"id": "6364.png", "formula": "\\begin{align*} Q _ { \\kappa } ( x ) = \\frac { d \\ , R _ { \\kappa } ( x ) } { d \\ , x } + N _ { \\kappa } \\ , x ^ { n - 1 } \\ \\ . \\end{align*}"}
{"id": "1876.png", "formula": "\\begin{align*} A _ { j } ( z ) = \\sum _ { n \\in \\mathbb { Z } } \\frac { A _ { [ n , j ] } } { z ^ { n + 1 } } = \\sum _ { n \\in \\mathbb { Z } } \\frac { 1 } { z ^ { n + 1 } } \\sum _ { k \\in \\mathbb { Z } } A _ { [ k , 0 ] } A _ { [ n - k - 1 , j - 1 ] } ^ { ( 1 ) } = A _ { 0 } ( z ) A _ { j - 1 } ^ { ( 1 ) } ( z ) . \\end{align*}"}
{"id": "8390.png", "formula": "\\begin{align*} B ^ C & = \\{ ( P _ f ) _ { f \\in G } \\in \\wp ( G ) ^ G \\ ; | \\ ; { \\rm t h e r e \\ ; e x i s t s } \\ ; g \\in P _ f \\ ; { \\rm a n d } \\ ; h \\in P _ { g f } \\ ; { \\rm s u c h \\ ; t h a t } \\ ; h g \\notin P _ { f } \\ ; { \\rm f o r \\ ; s o m e \\ ; } f \\in G \\} \\\\ & = \\bigcup _ { f , g , h \\in G } ( W _ { ( f , g ) } \\cap W _ { ( g f , h ) } \\cap W _ { ( f , h g ) } ^ C ) . \\end{align*}"}
{"id": "8803.png", "formula": "\\begin{align*} 1 _ { \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) } ^ * = 1 _ { ( - \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) / 2 , \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) / 2 ) } . \\end{align*}"}
{"id": "6784.png", "formula": "\\begin{align*} \\sum _ { \\substack { i \\in M _ P , \\\\ i \\neq i ^ * } } \\sum _ { j \\in N _ i } a _ { i j } \\tilde x _ { i j } + \\sum _ { \\substack { i j \\in P , \\\\ i \\notin M _ 0 , i \\neq i ^ * } } ( b - s ) \\tilde x _ { i j } & \\leq s + ( | M _ P - M _ 0 | - 1 ) ( b - s ) \\\\ & = b + ( | M _ P - M _ 0 | - 2 ) ( b - s ) . \\end{align*}"}
{"id": "2275.png", "formula": "\\begin{align*} J _ 5 & = \\int _ 0 ^ r z ^ { 2 } ( \\bar \\phi + e _ \\delta - \\delta ) ( 1 - \\delta + \\bar \\phi + e _ \\delta ) \\cdot ( e '' _ \\delta \\phi + e '' _ \\sigma \\psi ) { \\rm { d } } z \\\\ & \\leq \\Vert z ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert z ( e '' _ \\delta , e '' _ \\sigma ) \\Vert _ { L ^ 2 } \\cdot \\Vert ( \\bar \\phi + e _ \\delta - \\delta ) ( 1 - \\delta + \\bar \\phi + e _ \\delta ) \\Vert _ { L ^ \\infty } , \\end{align*}"}
{"id": "8282.png", "formula": "\\begin{align*} A = \\sinh ( k a / 2 ) \\cosh ( k a / 2 ) \\left [ - \\frac { a } { 2 } + \\cosh ^ { 2 } ( k a / 2 ) \\left ( \\frac { 2 a } { 3 } - \\frac { 1 } { 2 ( a / 2 ) k ^ { 2 } } + \\frac { \\sinh ( k a / 2 ) \\cosh ( k a / 2 ) } { 2 ( a / 2 ) ^ { 2 } k ^ { 3 } } \\right ) \\right ] ^ { - 1 / 2 } \\end{align*}"}
{"id": "464.png", "formula": "\\begin{align*} | \\Phi _ { n } ( r \\zeta ) | \\begin{cases} < 1 , & r < R _ { n } ( \\zeta ) , \\\\ = 1 , & r = R _ { n } ( \\zeta ) , \\\\ > 1 , & r > R _ { n } ( \\zeta ) . \\end{cases} \\end{align*}"}
{"id": "675.png", "formula": "\\begin{align*} V _ s = - X _ s + \\Phi _ { s * } X = - X _ s + \\widehat \\psi _ { s * } X \\end{align*}"}
{"id": "6295.png", "formula": "\\begin{align*} D _ { q } u ( x ) + p ( x ) u ( q x ) + r ( x ) = 0 , \\end{align*}"}
{"id": "8205.png", "formula": "\\begin{align*} U _ { 4 } = \\left ( \\begin{array} { r r r r } - 1 & 0 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 \\\\ 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 0 & - 1 \\end{array} \\right ) \\end{align*}"}
{"id": "4724.png", "formula": "\\begin{align*} \\overline { e _ { ( k + 1 ) } } = \\Big ( \\frac { q - q ^ { - 1 } } { z - z ^ { - 1 } } \\Big ) ^ { k - 1 } e _ { ( k ) } H _ { 2 k } ^ { - 1 } H _ { 2 k + 1 } ^ { - 1 } H _ { 2 k - 1 } H _ { 2 k } e _ { ( k ) } . \\end{align*}"}
{"id": "3451.png", "formula": "\\begin{align*} \\mathrm { T C E } _ { q } ( X ) = E ( X | X > x _ { q } ) \\end{align*}"}
{"id": "6314.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } q ^ { \\frac { k ( k - 5 ) } { 2 } + n k } x ^ k S _ n ( q ^ k x ; q ) = S _ n ( \\frac { x } { q } ; q ) + \\frac { x } { 1 - q ^ n } S _ { n - 1 } ( q x ; q ) , \\end{align*}"}
{"id": "333.png", "formula": "\\begin{align*} \\sum _ { n \\in L ^ x \\setminus { \\rm L } _ { A ^ y } } \\frac { 1 } { n } & = \\ \\sum _ { n \\in L ^ x } \\frac { 1 } { n } \\ - \\sum _ { n \\in L ^ x \\cap { \\rm L } _ { A ^ y } } \\frac { 1 } { n } \\\\ & = \\big ( { \\rm d } ( { \\rm L } _ { A ^ x } ) - { \\rm d } ( { \\rm L } _ { A ^ y } ) \\big ) \\prod _ { p \\le x } ( 1 - \\tfrac { 1 } { p } ) ^ { - 1 } \\ \\ll \\ ( \\log x ) \\big ( { \\rm d } ( { \\rm L } _ { A ^ x } ) - { \\rm d } ( { \\rm L } _ { A ^ y } ) \\big ) . \\end{align*}"}
{"id": "186.png", "formula": "\\begin{align*} \\P ( y _ t | x ^ 1 _ { 1 : t } , x ^ 2 _ { 1 : t } , y _ { 1 : t - 1 } ) = Q ( y _ t | x ^ 1 _ t , x ^ 2 _ t ) . \\end{align*}"}
{"id": "8857.png", "formula": "\\begin{align*} \\Delta \\lambda _ i = - \\frac { \\Delta f ( \\lambda _ i ) + 2 \\nabla f _ { \\lambda } ( \\lambda _ i ) \\cdot \\nabla \\lambda _ i + f _ { \\lambda \\lambda } ( \\lambda _ i ) \\nabla \\lambda _ i \\cdot \\nabla \\lambda _ i } { f _ \\lambda ( \\lambda _ i ) } , \\end{align*}"}
{"id": "6641.png", "formula": "\\begin{align*} 0 \\leq u _ { i , t } \\leq P _ { \\ , } \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } u _ { i , t } \\leq P _ { \\ , } \\ ; , \\ , \\end{align*}"}
{"id": "3246.png", "formula": "\\begin{align*} \\partial _ { x _ i } \\partial _ { x _ j } \\bigl ( a ^ { i j } \\varrho \\bigr ) - \\partial _ { x _ i } \\bigl ( b ^ i \\varrho \\bigr ) = 0 \\end{align*}"}
{"id": "2956.png", "formula": "\\begin{align*} X _ c \\cdot ( f , a ) = f ( c ) + X _ { c + a } . \\end{align*}"}
{"id": "4514.png", "formula": "\\begin{align*} \\partial _ 2 A _ { 1 1 } = & \\partial _ 2 ( \\nabla _ 3 k _ { 1 1 } - \\nabla _ 1 k _ { 1 3 } ) \\\\ = & \\nabla _ 2 ( \\nabla _ 3 k _ { 1 1 } - \\nabla _ 1 k _ { 1 3 } ) - k _ { 2 } ^ \\alpha \\nabla _ \\alpha k _ { 1 1 } + 2 k _ { 2 1 } \\nabla _ 3 k _ { 3 1 } \\\\ & - k _ { 2 1 } \\nabla _ 3 k _ { 1 3 } - k _ { 2 1 } \\nabla _ 1 k _ { 3 3 } + k _ { 2 } ^ \\alpha \\nabla _ 1 k _ { 1 \\alpha } \\\\ = & \\nabla _ 2 ( \\nabla _ 3 k _ { 1 1 } - \\nabla _ 1 k _ { 1 3 } ) . \\end{align*}"}
{"id": "1592.png", "formula": "\\begin{align*} \\mathcal { F } ( x , z ) = \\frac { 2 C ^ 3 } { E } . \\end{align*}"}
{"id": "354.png", "formula": "\\begin{align*} S ( \\delta _ { } ) = o ( 1 ) . \\end{align*}"}
{"id": "1075.png", "formula": "\\begin{align*} \\left ( I + ( \\mu - \\lambda ) R ( \\lambda , A _ 2 ) \\right ) & \\left ( R ( \\mu , A _ 2 ) - R ( \\mu , A _ 1 ) \\right ) \\\\ & = \\left ( R ( \\lambda , A _ 2 ) - R ( \\lambda , A _ 1 ) \\right ) \\left ( 1 + ( \\lambda - \\mu ) R ( \\mu , A _ 1 ) \\right ) , \\end{align*}"}
{"id": "7237.png", "formula": "\\begin{align*} l ( s ) = - \\sqrt { \\lambda } \\frac { I _ 1 \\left ( \\sqrt { \\lambda ( 1 - s ^ 2 ) } \\right ) } { \\sqrt { 1 - s ^ 2 } } . \\end{align*}"}
{"id": "8572.png", "formula": "\\begin{align*} \\sum _ { k = - m } ^ { m } c _ k e ^ { i n _ k x } , m = 1 , 2 , \\ldots , ( n _ { - k } = n _ k ) \\end{align*}"}
{"id": "163.png", "formula": "\\begin{align*} \\forall x \\in \\Lambda _ { \\epsilon , L } , C _ { ( m ) } ( x ) : = C _ { ( m ) } ( 0 , x ) = \\frac { 1 } { L ^ d } \\sum _ { k \\in \\Lambda ^ * } e ^ { i k \\cdot x } \\frac { 1 } { m ^ 2 + \\theta ( k ) } . \\end{align*}"}
{"id": "1175.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\frac { \\phi _ { \\sigma } ^ { p } ( x - y ) } { \\phi ^ { p - 1 } _ { \\sigma } ( y ) } d y = \\exp \\left ( \\frac { p ( p - 1 ) | x | ^ 2 } { 2 \\sigma ^ 2 } \\right ) , \\end{align*}"}
{"id": "6239.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } \\widetilde { h } _ n ( x ; q ) = q ^ { 1 - n } [ n ] _ q \\widetilde { h } _ { n - 1 } ( x ; q ) , \\end{align*}"}
{"id": "5542.png", "formula": "\\begin{align*} \\mathbb { E } [ \\ell ] = \\dfrac { \\ell } { 2 ^ { \\ell + 1 } } \\left ( 2 ^ \\ell + \\dbinom { \\ell } { \\ell / 2 } \\right ) . \\end{align*}"}
{"id": "2556.png", "formula": "\\begin{align*} t = ( \\nu _ 1 - \\nu _ 2 ) / 2 \\ , \\ \\ u = ( \\nu _ 2 - \\nu _ 3 ) / 2 \\ , \\ \\ v = ( \\nu _ 1 - \\nu _ 3 ) / 2 \\end{align*}"}
{"id": "7736.png", "formula": "\\begin{align*} A _ { \\bar S S } ( \\tau : t _ 0 ) & = \\sum _ { k = t _ 0 } ^ { \\tau - 2 } A _ { \\bar S } ( \\tau - 2 ) \\cdots A _ { \\bar S } ( k + 1 ) A _ { \\bar S S } ( k ) A _ S ( k : t _ 0 ) \\cr & { = } \\sum _ { k = t _ 0 } ^ { \\tau - 2 } \\tilde A ( \\tau - 1 : k + 1 ) A _ { \\bar S S } ( k ) A _ S ( k : t _ 0 ) \\cr & \\stackrel { ( a ) } { \\geq } e ^ { - M ( \\gamma ) \\lambda } \\sum _ { k = t _ 0 } ^ { \\tau - 2 } \\tilde A ( \\tau - 1 : k + 1 ) A _ { \\bar S S } ( k ) , \\end{align*}"}
{"id": "8862.png", "formula": "\\begin{align*} f ( \\lambda ) = \\lambda ^ N + \\sum _ { k = 1 } ^ { N } ( - 1 ) ^ k \\lambda ^ { N - k } \\sum _ { 1 \\le j _ 1 < \\cdots < j _ k \\le N } \\underset { 1 \\le \\ell , m \\le k } { { \\rm d e t } } ( H _ { j _ \\ell j _ m } ) , \\end{align*}"}
{"id": "5588.png", "formula": "\\begin{align*} B _ { i , j } : = b _ { i , j } \\rho _ i \\rho _ j \\end{align*}"}
{"id": "4276.png", "formula": "\\begin{align*} g ^ { - 1 } b _ 2 \\sigma ( g ) = b _ 1 . \\end{align*}"}
{"id": "5621.png", "formula": "\\begin{align*} f ( X ) D [ X ] = ( A P ) _ t = ( A _ t P ) _ t = ( ( A _ f ) _ t P ) _ t = ( A _ f P ) _ t , \\end{align*}"}
{"id": "5726.png", "formula": "\\begin{align*} \\pi _ { [ 1 , 1 ] } \\cdot \\pi _ { [ 2 , 2 ] } = 2 \\pi _ { [ 1 , 2 ] } . \\end{align*}"}
{"id": "1668.png", "formula": "\\begin{align*} f _ { i } ( \\alpha ) & = { f } _ { i } ^ 0 ( \\alpha ) + \\sum _ { j = 1 } ^ k \\ \\sum _ { \\{ p : j \\in S _ p \\} } ( { f } _ { j } ^ 0 ) | _ { C _ { p , S _ p } } ( \\alpha ) \\\\ & = { f } _ { i } ^ 0 ( \\alpha ) + \\sum _ { j \\in S _ q } ( { f } _ { j } ^ 0 ) ( \\alpha ) \\\\ & = f ( \\gamma _ p ) \\end{align*}"}
{"id": "5843.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ M c _ k b _ k = \\sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \\sum _ { i = k } ^ M c _ i + \\bigg ( \\sum _ { k = N } ^ M c _ k \\bigg ) b _ N . \\end{align*}"}
{"id": "2069.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : g i b b s m e a s u r e } \\mu ^ N \\big ( \\dd \\mathbf { x } ^ N \\big ) = \\frac { 1 } { Z _ N } \\exp { \\left [ N G ( \\mu _ { \\mathbf { x } ^ N } ) \\right ] } \\mu _ 0 ^ { \\otimes N } \\big ( \\dd \\mathbf { x } ^ N \\big ) , \\end{align*}"}
{"id": "6660.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t - u _ { x x } = u ^ { p } , & t > 0 , \\ 0 < x < s ( t ) , \\\\ u _ x ( t , 0 ) = u ( t , x ) = 0 , & t > 0 , \\\\ s ^ { \\prime } ( t ) = - u _ x ( t , s ( t ) ) , & t > 0 , \\\\ s ( 0 ) = s _ 0 > 0 , \\ u ( 0 , x ) = u _ 0 ( x ) , & 0 \\leq x \\leq s _ 0 , \\end{array} \\right . \\end{align*}"}
{"id": "5059.png", "formula": "\\begin{align*} \\| N ^ { n , 5 } _ \\tau \\| _ 2 \\le \\alpha ^ 2 \\sup _ { 0 \\le s \\le \\tau } \\| ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) \\| _ 2 \\frac 1 n \\sum _ { j = 0 } ^ { \\lfloor n \\tau \\rfloor } \\int _ { 0 } ^ { 1 \\wedge ( n \\tau - j ) } \\gamma _ { \\frac { x + j } n } d x \\sum _ { k = j + 1 } ^ { \\infty } k ^ { 2 \\alpha - 2 } . \\end{align*}"}
{"id": "2327.png", "formula": "\\begin{align*} - A < A F - B E \\leq A ( B - 1 ) - B A = - A , \\end{align*}"}
{"id": "3358.png", "formula": "\\begin{align*} \\widehat { \\varphi } _ 1 ( f ) = \\begin{pmatrix} \\varphi _ 1 ( f ) & 0 \\\\ 0 & 1 \\end{pmatrix} \\widehat { \\varphi } _ 2 ( f ) = \\begin{pmatrix} 1 & 0 \\\\ 0 & \\varphi _ 2 ( f ) \\end{pmatrix} . \\end{align*}"}
{"id": "5973.png", "formula": "\\begin{align*} k ( t ) = \\frac 1 { 2 \\pi i } \\int _ \\Gamma e ^ { s t } K ( s ) d s \\end{align*}"}
{"id": "4711.png", "formula": "\\begin{align*} s _ { i , j } = \\begin{cases} s _ i s _ { i + 1 } \\cdots s _ { j } & \\hbox { i f } i \\leq j , \\\\ s _ i s _ { i - 1 } \\cdots s _ { j } & \\hbox { i f } i > j . \\end{cases} \\end{align*}"}
{"id": "6754.png", "formula": "\\begin{align*} h ^ 0 \\left ( L _ { \\chi } + K _ Y \\right ) = 0 \\end{align*}"}
{"id": "985.png", "formula": "\\begin{align*} M _ { \\mathbb { Z } } ^ { * } = \\left \\{ H ^ { * } ( \\overline { f } ) \\mid \\overline { f } \\in \\mathbb { Z } ^ { n } \\right \\} . \\end{align*}"}
{"id": "2970.png", "formula": "\\begin{align*} f _ m ( z , s ) = \\frac { \\Gamma ( s ) } { 2 \\pi \\sqrt { | m | } \\Gamma ( 2 s ) } M _ { 0 , s - \\frac 1 2 } ( 4 \\pi | m | y ) e ( m x ) . \\end{align*}"}
{"id": "7105.png", "formula": "\\begin{align*} R ( K ) \\to K ^ * ( C ^ * _ r G ) \\colon [ E ] \\to [ P ^ E _ t ] - \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "2754.png", "formula": "\\begin{align*} u _ { j } ( x ) = & \\begin{cases} j + j ^ { q } \\phi ( x ) , & \\quad \\mathrm { i n } \\ ; B _ { R } , \\\\ ( 1 - \\varphi ( x ) ) ( j + j ^ { q } ) + \\varphi ( x ) | x | ^ { - s } , & \\quad \\mathrm { i n } \\ ; B _ { R } ^ { c } , \\end{cases} \\end{align*}"}
{"id": "3979.png", "formula": "\\begin{align*} \\textstyle C ^ { \\le \\delta } = \\big \\{ c \\ ; \\big | \\ ; c \\in C , ~ \\frac { { \\rm w } ( c ) } { n } \\le \\delta \\big \\} . \\end{align*}"}
{"id": "3401.png", "formula": "\\begin{align*} \\dim H ^ 1 ( G ^ { \\mathcal C } , V ) = \\dim H ^ 1 ( G ^ { \\mathcal C } , V ^ \\vee ) \\textrm { a n d } H ^ 2 ( G ^ { \\mathcal C } , V ) ^ { \\mathcal C , \\tau } = H ^ 2 ( G ^ { \\mathcal C } , V ^ \\vee ) ^ { \\mathcal C , \\tau } = 0 \\end{align*}"}
{"id": "4215.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\frac { 1 } { u ( 1 \\vee \\log \\Phi ^ { - 1 } ( u ) ) } \\ \\mathrm { d } u = \\infty \\limsup _ { n \\to \\infty } \\frac { \\mathbb { G } ( X _ n , X _ m ) } { \\Phi ( n ) } < \\infty \\end{align*}"}
{"id": "2025.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : s i m u l t a n e o u s j u m p s o p e r a t o r N } \\widetilde { L } ^ N _ \\mu \\varphi ( x ) : = N \\iint _ { E \\times E } \\lambda ( z , \\mu ) \\{ \\varphi ( y ) - \\varphi ( x ) \\} \\widetilde { P } _ \\mu ^ N ( x , z , \\dd y ) \\mu ( \\dd z ) . \\end{align*}"}
{"id": "1642.png", "formula": "\\begin{align*} \\tau \\tau _ { x x x x } \\ , \\ , - \\ , \\ , 4 \\tau _ { x x x } \\tau _ x \\ , \\ , + \\ , \\ , 3 \\tau _ { x x } ^ 2 \\ , \\ , + \\ , \\ , 4 \\tau _ x \\tau _ t \\ , \\ , - \\ , \\ , 4 \\tau \\tau _ { x t } \\ , \\ , + \\ , \\ , 3 \\tau \\tau _ { y y } \\ , \\ , - \\ , \\ , 3 \\tau _ y ^ 2 \\ , \\ , = \\ , \\ , 0 . \\end{align*}"}
{"id": "6482.png", "formula": "\\begin{align*} u ( c _ 2 ) = u _ x ( c _ 2 ) = 0 y ( c _ 1 ) = y _ x ( c _ 1 ) = 0 . \\end{align*}"}
{"id": "3159.png", "formula": "\\begin{align*} \\mathit { M } _ { \\mathfrak { m } } \\doteq \\left \\{ \\omega \\in E _ { 1 } : f _ { \\mathfrak { m } } ^ { \\sharp } \\left ( \\omega \\right ) = \\inf \\ , f _ { \\mathfrak { m } } ^ { \\sharp } ( E _ { 1 } ) = - \\mathrm { P } _ { \\mathfrak { m } } ^ { \\sharp } \\right \\} \\ . \\end{align*}"}
{"id": "3572.png", "formula": "\\begin{align*} \\chi _ { \\pi } ( h _ k ) = [ k ] _ q ! [ n - k ] _ q ! \\sum _ { 1 \\leq i _ 1 < i _ 2 < \\cdots < i _ k \\leq n } \\chi _ { i _ 1 } ( - 1 ) \\chi _ { i _ 2 } ( - 1 ) \\cdots \\chi _ { i _ k } ( - 1 ) . \\end{align*}"}
{"id": "1192.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\int _ { \\R ^ d } \\varphi d ( \\mu _ t - \\nu _ t ) = \\frac { 1 } { t } \\iint _ { \\R ^ d \\times \\R ^ d } \\{ \\varphi ( x ) - \\varphi ( y ) \\} d \\pi _ t ( x , y ) . \\end{align*}"}
{"id": "300.png", "formula": "\\begin{align*} p _ j = P ( d ) \\le P ( c ) < P ( a ^ * ) ^ { 1 / \\sqrt { v } } . \\end{align*}"}
{"id": "647.png", "formula": "\\begin{align*} \\bigg | \\frac { \\Omega ( 1 - s ) } { \\Omega ( s ) } \\bigg | = 1 . \\end{align*}"}
{"id": "4373.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi _ 1 < - t \\} \\cap V } | f | ^ 2 | F | ^ { 2 p _ 0 } e ^ { - 2 p _ 0 c _ o ^ { f F } ( \\psi ) \\psi } = \\int _ { \\{ \\Psi _ 1 < - t \\} \\cap V } | f | ^ 2 e ^ { - p _ 0 \\Psi } < + \\infty \\end{align*}"}
{"id": "7089.png", "formula": "\\begin{align*} \\delta ( t _ { k + 1 } ) & \\leq \\delta ( t _ { 0 } ) + 0 . 5 \\sum ^ { k } _ { l = 0 } | \\beta ^ { \\top } e ( t _ { l } ) | \\mathsf { T } _ { \\max } \\\\ & \\leq \\delta ( t _ { 0 } ) + 0 . 5 \\beta _ { \\max } \\sum ^ { k } _ { l = 0 } \\| e ( t _ { l } ) \\| \\mathsf { T } _ { \\max } . \\end{align*}"}
{"id": "7069.png", "formula": "\\begin{align*} \\begin{aligned} W _ 2 = & 4 ( l _ 1 - l _ 2 ) / 3 , \\\\ W _ 3 = & \\pi ( m _ 1 + m _ 2 ) ( l _ 2 + n ) / 8 . \\\\ \\end{aligned} \\end{align*}"}
{"id": "3891.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ p G _ { \\lambda , R } - \\lambda R ^ p G _ { \\lambda , R } ^ { p - 1 } = \\delta _ 0 & \\Omega _ R \\\\ G _ { \\lambda , R } \\geq 0 & \\Omega _ R \\\\ G _ { \\lambda , R } = 0 & \\partial \\Omega _ R . \\end{cases} \\end{align*}"}
{"id": "4426.png", "formula": "\\begin{align*} 4 k d + ( 8 k - 1 ) e + \\ell \\leq 4 k d + 8 k e = 4 k N < 4 k N + j , \\end{align*}"}
{"id": "2310.png", "formula": "\\begin{align*} \\left \\| \\frac { f } { x ^ { 1 - \\sigma ' } } \\right \\| _ { L ^ { 2 } } \\leq \\left \\| \\left \\| \\frac { f } { ( x - 1 ) ^ { 1 - \\sigma ' } } \\right \\| _ { L _ { x } ^ { 2 } } \\right \\| _ { L _ { y } ^ { 2 } } \\lesssim \\| \\| f _ { x } ( x - 1 ) ^ { \\sigma ' } \\left \\| _ { L _ { x } ^ { 2 } } \\right \\| _ { L _ { y } ^ { 2 } } \\lesssim \\left \\| f _ { x } x ^ { \\sigma ' } \\right \\| _ { L ^ { 2 } } \\end{align*}"}
{"id": "5041.png", "formula": "\\begin{align*} Q ^ { n , 7 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 [ \\gamma _ s - \\gamma _ { \\eta _ n ( s ) } ] ( s - \\eta _ n ( s ) ) ^ { 2 \\alpha + 1 } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) d s , \\end{align*}"}
{"id": "4463.png", "formula": "\\begin{align*} \\bigcup _ { k = 1 } ^ N \\overline { \\Omega _ k ( t ) } = \\R ^ 2 , \\end{align*}"}
{"id": "2822.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\nabla f ( x _ N ) \\| ^ 2 { } \\leq { } \\tfrac { 2 L \\big [ f ( x _ 0 ) - f ( x _ N ) \\big ] } { \\min \\big \\{ - 1 + ( 1 - h ) ^ { - 2 N } \\ , , \\ , { 2 N h } \\big \\} } \\| \\nabla f ( x _ N ) \\| ^ 2 { } \\leq { } \\tfrac { 2 L \\big [ f ( x _ 0 ) - f _ * \\big ] } { \\min \\big \\{ ( 1 - h ) ^ { - 2 N } \\ , , \\ , { 1 + 2 N h } \\big \\} } . \\end{aligned} \\end{align*}"}
{"id": "8782.png", "formula": "\\begin{align*} C ( p _ 1 , p _ 2 , p _ 3 ) = \\frac { B ( p _ 1 ) B ( p _ 2 ) B ( p _ 3 ) } { B \\circ q ( p _ 1 , p _ 2 , p _ 3 ) } = \\frac { B ( p _ 1 ) B ( p _ 2 ) } { B \\circ q ( p _ 1 , p _ 2 ) } \\cdot \\frac { B \\circ q ( p _ 1 , p _ 2 ) B ( p _ 3 ) } { B \\circ q ( q ( p _ 1 , p _ 2 ) , p _ 3 ) } = C ( p _ 1 , p _ 2 ) C ( q ( p _ 1 , p _ 2 ) , p _ 3 ) \\end{align*}"}
{"id": "5606.png", "formula": "\\begin{align*} \\left ( \\frac { P _ 2 x z _ 1 } { 2 ^ { r + 1 } P _ 1 P _ 2 P _ 3 x z _ 1 - y ^ 2 } \\right ) = - 1 . \\end{align*}"}
{"id": "6473.png", "formula": "\\begin{align*} u ( \\cdot , 0 ) = u _ 0 ( \\cdot ) , \\ u _ t ( \\cdot , 0 ) = u _ 1 ( \\cdot ) , \\ y ( \\cdot , 0 ) = y _ 0 ( \\cdot ) \\ \\ y _ t ( \\cdot , 0 ) = y _ 1 ( \\cdot ) \\ \\ \\Omega , \\end{align*}"}
{"id": "7083.png", "formula": "\\begin{align*} u ( t _ { k + 1 } ) = \\left \\{ \\begin{aligned} & A _ { 1 } u ( t _ { k } ) + B _ { 1 } & & \\beta ^ { \\top } u ( t _ { k } ) \\geq ( 2 - \\phi ) \\lambda \\\\ & A _ { 2 } u ( t _ { k } ) + B _ { 2 } & & \\beta ^ { \\top } u ( t _ { k } ) < ( 2 - \\phi ) \\lambda \\end{aligned} \\right . \\end{align*}"}
{"id": "2183.png", "formula": "\\begin{align*} \\alpha = s _ { 0 } - \\varphi _ { \\mu } \\left ( F _ { \\nu } ^ { * } ( \\alpha ) \\right ) = s _ { 0 } - \\gamma _ { \\mu } + \\int _ { \\mathbb { R } } \\frac { 1 + s F _ { \\nu } ^ { * } ( \\alpha ) } { s - F _ { \\nu } ^ { * } ( \\alpha ) } \\ , d \\sigma _ { \\mu } ( s ) , \\end{align*}"}
{"id": "7833.png", "formula": "\\begin{align*} \\varphi ( e , f ) = \\langle z ( e a \\oplus b e ) , ( f c \\oplus d f ) \\rangle . \\end{align*}"}
{"id": "434.png", "formula": "\\begin{align*} K = \\left \\{ s + i t : s \\in [ A , B ] , \\delta \\le t \\le \\frac { 1 } { \\delta } \\right \\} \\end{align*}"}
{"id": "1064.png", "formula": "\\begin{align*} \\lim _ { | y | \\to \\infty } \\| B R ( x + i y , A ) \\| _ q = 0 \\forall x < - \\omega _ 0 ( A ) . \\end{align*}"}
{"id": "8187.png", "formula": "\\begin{align*} \\eta ( x ) = \\eta ^ \\parallel ( x ^ \\parallel ) \\eta ^ \\perp ( x ^ \\perp ) \\end{align*}"}
{"id": "6154.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { T } \\left ( \\partial _ t v ^ { \\Delta t } _ { h } , P _ h ( u ^ { \\Delta t } _ { h } \\phi ) \\right ) \\ , d t = \\int _ { 0 } ^ { T } \\left ( \\partial _ t v ^ { \\Delta t } _ { h } , u ^ { \\Delta t } _ { h } \\phi \\right ) \\ , d t + \\int _ { 0 } ^ { T } \\left ( \\partial _ t v ^ { \\Delta t } _ { h } , P _ h ( u ^ { \\Delta t } _ { h } \\phi ) - u ^ { \\Delta t } _ { h } \\phi \\right ) \\ , d t = : I _ 1 + I _ 2 . \\end{align*}"}
{"id": "1754.png", "formula": "\\begin{align*} g _ s = 2 \\pi i / \\overline \\delta , Q = \\exp ( - 2 \\pi i \\overline \\mu ) , \\end{align*}"}
{"id": "8306.png", "formula": "\\begin{align*} a _ n = - \\frac { N ! } { \\pi n } \\sum _ { k = 0 } ^ { N - 1 } \\frac { ( - 1 ) ^ k F _ k ( n ) } { ( N - k ) ! } \\cos \\left ( \\frac { \\pi } { 2 } k \\right ) , \\end{align*}"}
{"id": "5931.png", "formula": "\\begin{align*} W \\left [ { \\eta \\left ( t \\right ) } \\right ] = \\mathop \\sum \\limits _ n \\frac { 1 } { { n ! } } \\mathop \\int { W ^ { \\left ( n \\right ) } } \\left ( { { t _ 1 } - { t _ 2 } , \\ldots , { t _ 1 } - { t _ n } } \\right ) \\eta \\left ( { { t _ 1 } } \\right ) . . . \\eta \\left ( { { t _ n } } \\right ) d { t _ 1 } . . . d { t _ n } \\end{align*}"}
{"id": "2351.png", "formula": "\\begin{align*} \\nu \\left ( a _ { \\rho 0 } ( f ) \\right ) = \\nu _ \\rho ( f ) = \\nu ( l ( h _ \\rho ) ) . \\end{align*}"}
{"id": "5322.png", "formula": "\\begin{align*} \\Pi ^ t ( s ) = ( \\pi ^ t _ 1 ( s ) , \\pi ^ t _ 2 ( s ) , \\dots , \\pi ^ t _ { N ^ t _ s } ( s ) ) , \\end{align*}"}
{"id": "136.png", "formula": "\\begin{align*} \\big \\| C \\star \\psi \\star S \\big \\| _ { L ^ 1 \\cap L ^ \\infty } & \\leq c \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } \\Big [ \\frac { 1 } { m ^ 4 _ t } { \\bf 1 } _ { d = 2 } + \\Big ( \\frac { 1 } { m _ t ^ { 1 / 2 } } + \\frac { 1 } { m _ t ^ { 5 / 2 } } \\Big ) { \\bf 1 } _ { d = 3 } \\Big ] \\\\ & + c \\Big ( \\frac { 1 } { m ^ 4 _ t } + \\frac { 1 } { m _ t ^ { 6 } } \\Big ) { \\bf 1 } _ { d = 2 } + c \\Big ( \\frac { 1 } { m _ t } + \\frac { 1 } { m _ t ^ { 9 / 2 } } \\Big ) { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "8772.png", "formula": "\\begin{align*} \\tilde { \\mathcal { B } } : = \\{ ( \\phi _ 1 , \\phi _ 2 , \\cdots , \\phi _ N ) \\mid \\phi _ 1 , \\phi _ 2 , \\cdots , \\phi _ N \\colon G \\to \\mathbb { R } _ { \\geq 0 } , \\ ; \\| \\phi _ 1 \\| _ { p _ 1 } = \\| \\phi _ 2 \\| _ { p _ 2 } = \\cdots = \\| \\phi _ N \\| _ { p _ N } = 1 \\} \\end{align*}"}
{"id": "5799.png", "formula": "\\begin{align*} F _ { i , j } = F _ { j , i } , \\end{align*}"}
{"id": "7007.png", "formula": "\\begin{align*} \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } S ( h ) = n ! \\cdot \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } V o l ( h ) , \\end{align*}"}
{"id": "4447.png", "formula": "\\begin{align*} c _ { l , s } = \\prod \\limits _ { w = 0 } ^ { ( s - l - 1 ) } ( n - 1 + 2 w ) \\frac { ( - 1 ) ^ l \\ , s ! } { 2 ^ l \\ , l ! \\ , ( s - 2 l ) ! } . \\end{align*}"}
{"id": "6967.png", "formula": "\\begin{align*} \\rho = \\sum _ { n \\ge 1 } a _ n \\delta _ { \\lambda _ n ^ 2 } , a _ n > 0 . \\end{align*}"}
{"id": "4931.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": "5724.png", "formula": "\\begin{align*} \\varpi _ { [ a , i ] } \\cdot \\varpi _ { [ i + 1 , b ] } = \\frac { 1 } { ( i - a + 1 ) ! } \\cdot \\frac { 1 } { ( b - i ) ! } \\cdot \\varpi _ { a } \\varpi _ { a + 1 } \\cdots \\varpi _ { b } = \\binom { b - a + 1 } { i - a + 1 } \\varpi _ { [ a , b ] } \\end{align*}"}
{"id": "8256.png", "formula": "\\begin{align*} \\lambda _ { 1 } = + \\lambda _ { + } , \\lambda _ { 2 } = + \\lambda _ { - } , \\lambda _ { 3 } = - \\lambda _ { + } , \\lambda _ { 4 } = - \\lambda _ { - } , \\end{align*}"}
{"id": "7371.png", "formula": "\\begin{align*} f ( r ) = - \\frac { U _ 0 ( r ) ^ { p - 1 } U _ 0 ' ( r ) } { r } \\end{align*}"}
{"id": "8985.png", "formula": "\\begin{align*} \\| \\nabla ( \\xi _ { \\tilde K } w ) \\| ^ 2 _ { L ^ 2 ( M ) } = \\| \\xi _ { \\tilde K } \\nabla w + w \\nabla \\xi _ { \\tilde K } \\| ^ 2 _ { L ^ 2 ( M ) } . \\end{align*}"}
{"id": "7528.png", "formula": "\\begin{align*} W _ { n } ( t ) : = \\frac { 1 } { \\sqrt { n } } \\bigg [ \\sum _ { j = 0 } ^ { [ n t ] - 1 } v \\circ T ^ j + ( n t - [ n t ] ) v \\circ T ^ { [ n t ] } \\bigg ] , t \\in [ 0 , 1 ] , \\end{align*}"}
{"id": "4069.png", "formula": "\\begin{align*} k = 0 , 1 , 2 , \\ldots , m , i _ k = \\tilde { \\omega } _ k = j _ k . \\end{align*}"}
{"id": "639.png", "formula": "\\begin{align*} V _ 2 ^ { - 1 } ( s - 1 ) - 2 V ^ { - 1 } _ 3 ( s - 1 ) + V ^ { - 1 } _ 4 ( s - 1 ) = \\frac { \\pi ( 2 - s ) } { 2 \\sin ( \\pi s ) } \\frac { s ( 1 - s ) } { 6 } = \\frac { s ( 2 - s ) } { 6 } V ^ { - 1 } _ 2 ( s ) . \\end{align*}"}
{"id": "6346.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\dfrac { \\widehat { \\varPhi } _ { x } ^ { - 1 } ( \\tau ) } { \\tau ^ { \\frac { N + s } { N } } } d \\tau = \\infty ~ ~ x \\in \\overline { \\Omega } . \\end{align*}"}
{"id": "8796.png", "formula": "\\begin{align*} 1 _ A ^ { - 1 } ( \\mathbb { R } _ { > t } ) = \\left \\{ \\begin{aligned} & A & & \\ ; t < 1 \\\\ & \\emptyset & & \\ ; t \\geq 1 \\end{aligned} \\right . \\end{align*}"}
{"id": "4019.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\frac { 1 } { \\epsilon _ n } = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\frac { 1 } { 1 - \\epsilon _ n } = 0 , \\end{align*}"}
{"id": "3113.png", "formula": "\\begin{align*} \\alpha _ { k } = ( v _ { \\mathrm { n c } } , \\phi _ { k } ) _ { 1 + \\delta } = ( z _ { \\mathrm { n c } } , \\phi _ { k } ) _ { 1 + \\delta } - t ( u _ { \\mathrm { n c } } , \\phi _ { k } ) _ { 1 + \\delta } = 0 . \\end{align*}"}
{"id": "8142.png", "formula": "\\begin{align*} & \\lim \\limits _ { n \\rightarrow \\infty } \\| \\varphi _ { n ; \\mathrm { i n } } \\| = 0 \\\\ \\end{align*}"}
{"id": "8218.png", "formula": "\\begin{align*} \\phi _ { 0 } ( x ) = \\frac { 1 } { \\sqrt { a } } , \\end{align*}"}
{"id": "4436.png", "formula": "\\begin{align*} I _ m f ( x , \\xi ) = \\int \\limits _ { - \\infty } ^ \\infty \\L f ( x + t \\xi ) , \\xi ^ m \\R \\ , d t \\end{align*}"}
{"id": "8364.png", "formula": "\\begin{align*} p \\rightarrow \\left ( { \\textstyle \\frac { ( 1 - z ) 2 ^ z } { 1 6 \\sqrt { 2 } \\pi ^ { 5 / 2 } } } \\right ) ^ { 1 / 2 } ( p \\cdot v ) ^ { z - 2 } , \\ , \\ , \\ , \\ , v = ( 1 , 0 , 0 , 0 ) , \\end{align*}"}
{"id": "192.png", "formula": "\\begin{align*} \\pi _ t & ( m ^ 1 , m ^ 2 ) = \\frac { Q ( y _ t | e ^ 1 _ t ( m ^ 1 ) , e ^ 2 _ t ( m ^ 2 ) ) \\pi _ { t - 1 } ( m ^ 1 , m ^ 2 ) } { \\sum \\limits _ { m ^ 1 , m ^ 2 } Q ( y _ t | e ^ 1 _ t ( m ^ 1 ) , e ^ 2 _ t ( m ^ 2 ) ) \\pi _ { t - 1 } ( m ^ 1 , m ^ 2 ) } \\end{align*}"}
{"id": "5617.png", "formula": "\\begin{align*} A _ t = ( A , B C ) _ t = ( A , A C , B C ) _ t = ( A , ( A , B ) C ) _ t = ( A , ( A , B ) _ t C ) _ t = ( A , C ) _ t , \\end{align*}"}
{"id": "4320.png", "formula": "\\begin{align*} _ J ( L ) = \\int _ L e ^ { - i \\theta } \\Omega \\geq \\sum _ 1 ^ N | \\int _ { L _ i } \\Omega | . \\end{align*}"}
{"id": "7102.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\kappa _ t ^ s ( k \\exp ( t v ) ) = \\kappa _ 0 ^ s ( k , v ) . \\end{align*}"}
{"id": "4184.png", "formula": "\\begin{align*} \\psi : = \\sum _ { j = - 2 } ^ 2 \\chi _ j \\end{align*}"}
{"id": "5566.png", "formula": "\\begin{align*} G _ \\Pi f ( x ) & = \\frac { 1 } { x } \\int _ 0 ^ \\infty [ f ( x e ^ { - y } ) - f ( x ) + y x f ' ( x ) ] \\Pi ( d y ) \\\\ & = \\frac { 1 } { x } \\int _ 0 ^ \\infty [ f ( x e ^ { - y } ) - f ( x ) - ( e ^ { - y } - 1 ) x f ' ( x ) ] \\Pi ( d y ) \\\\ & + f ' ( x ) \\int _ 0 ^ \\infty ( e ^ { - y } - 1 + y ) \\Pi ( d y ) \\\\ & = ~ A _ 1 ( x ) + A _ 2 ( x ) . \\end{align*}"}
{"id": "6589.png", "formula": "\\begin{align*} d \\beta ( x , y ) = \\sum _ { \\nu = 1 } ^ k \\beta ( x _ 1 , \\ldots , x _ { \\nu - 1 } , y _ \\nu , x _ { \\nu + 1 } , \\ldots , x _ k ) , \\end{align*}"}
{"id": "8027.png", "formula": "\\begin{align*} V _ { + } : = \\mathcal O _ { Y _ + } ( - 1 ) , V _ { - } : = \\mathcal O _ { Y _ - } ( - 1 ) . \\end{align*}"}
{"id": "1014.png", "formula": "\\begin{align*} h _ \\mu ( T ) = \\sup _ \\P h _ \\mu ( \\P ) , \\end{align*}"}
{"id": "5445.png", "formula": "\\begin{align*} ( a _ 0 \\star q ) ( m _ 1 m _ 2 ) & = & \\ , \\ , \\star \\\\ & = q ( m _ 1 m _ 2 a _ 0 ) a ^ { - 1 } _ 0 & \\ , \\ , ( \\textsc { L } 3 ) \\\\ & = q ( m _ 1 q ( m _ 2 a _ 0 ) ) a ^ { - 1 } _ 0 \\\\ & = q ( m _ 1 q ( m _ 2 a _ 0 ) a ^ { - 1 } _ 0 a _ 0 ) a ^ { - 1 } _ 0 & \\ , \\ , \\star \\\\ & = ( a _ 0 \\star q ) ( m _ 1 q ( m _ 2 a _ 0 ) a ^ { - 1 } _ 0 ) & \\ , \\ , \\star \\\\ & = ( a _ 0 \\star q ) ( m _ 1 ( a _ 0 \\star q ) ( m _ 2 ) ) . \\end{align*}"}
{"id": "1359.png", "formula": "\\begin{align*} | \\Phi | \\geq | k ^ 2 - k _ 1 ^ 2 | - p ( k _ 2 ^ * ) ^ 2 = | k - k _ 1 | | k + k _ 1 | - p | k _ 2 ^ * | ^ 2 \\gtrsim | k _ 1 | . \\end{align*}"}
{"id": "7934.png", "formula": "\\begin{align*} \\bigcap _ { j \\in M _ - } \\{ z _ j = 0 \\} \\subset X _ + , \\bigcap _ { j \\in M _ + } \\{ z _ j = 0 \\} \\subset X _ - . \\end{align*}"}
{"id": "6085.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 + a x ^ 3 + b x = ( x - 1 ) ^ 4 + c ( x - 1 ) ^ 2 + d ( x - 1 ) \\end{align*}"}
{"id": "6859.png", "formula": "\\begin{align*} \\begin{aligned} b ^ * & \\geq \\frac { g _ 0 } { 1 - g _ 1 / V _ F } > 0 , \\\\ c ^ * & \\geq a _ 0 + \\frac { a _ 1 } { V _ F } \\frac { g _ 0 } { 1 - g _ 1 / V _ F } > 0 . \\end{aligned} \\end{align*}"}
{"id": "4147.png", "formula": "\\begin{align*} \\chi _ j ( \\lambda ) = \\chi ( \\lambda / 2 ^ j ) j \\in \\Z . \\end{align*}"}
{"id": "5220.png", "formula": "\\begin{align*} \\psi _ I : A _ { I , \\mathrm { s y m } } \\rightarrow & A _ I \\\\ t _ { \\alpha , \\beta , d } \\mapsto & \\sum _ { i \\in I : ( a _ i , b _ i ) = ( \\alpha , \\beta ) } u _ { i , d } . \\end{align*}"}
{"id": "5045.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } Q ^ { n , 2 } _ \\tau = \\frac 1 { 2 \\alpha + 2 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ \\alpha ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "5390.png", "formula": "\\begin{align*} P _ j = \\sum _ { \\abs { \\alpha } \\leq m } ( a _ { j , \\alpha } + b _ { j , \\alpha } ) D ^ \\alpha , a _ { j , \\alpha } \\in M _ 0 ( H ^ { s - | \\alpha | } \\rightarrow H ^ { - s } ) , P _ { j , s } \\vcentcolon = \\sum _ { \\abs { \\alpha } \\leq m } { b _ { j , \\alpha } } D ^ \\alpha \\in \\mathcal { P } _ { m , s } ( \\Omega ) \\end{align*}"}
{"id": "2156.png", "formula": "\\begin{align*} y = \\lim _ { n \\rightarrow \\infty } y _ { n } \\geq \\lim _ { n \\rightarrow \\infty } f ( x _ { n } ) = f ( x ) , \\end{align*}"}
{"id": "50.png", "formula": "\\begin{align*} \\dfrac { ( f ^ { \\tilde { \\phi } ^ { - n } } \\circ \\rho _ { \\eta } ( X ) ) ' } { f ^ { \\tilde { \\phi } ^ { - n } } \\circ \\rho _ { \\eta } ( X ) } = \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { ( f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ( X + w ) ) ' } { f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ( X + w ) } = \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { ( f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ) ' ( X + w ) } { f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ( X + w ) } . \\end{align*}"}
{"id": "2452.png", "formula": "\\begin{align*} & \\deg ( x _ i ) = \\deg ( y _ i ) = 1 + ( n - 1 ) \\delta _ { i , 0 } , \\end{align*}"}
{"id": "1125.png", "formula": "\\begin{align*} \\omega _ i = \\frac { \\lambda A [ ( 2 i + 1 ) / k - 1 ] } { ( \\sigma _ N ^ 2 + \\sigma _ Z ^ 2 ) \\lambda ^ 2 + \\rho } , \\end{align*}"}
{"id": "69.png", "formula": "\\begin{align*} F _ 3 ( k ) & = \\sum _ { h = 1 } ^ { + \\infty } \\frac { \\big ( k ^ { 2 s - 1 } + h ^ { 2 s - 1 } \\big ) ^ 2 } { ( 1 + k + h ) ^ { 2 s } ( 1 + h ) ^ { 2 r } } \\\\ & \\leq 2 \\sum _ { h = 1 } ^ { + \\infty } \\frac { k ^ { 2 ( 2 s - 1 ) } + h ^ { 2 ( 2 s - 1 ) } } { ( k + h ) ^ { 2 s } h ^ { 2 r } } \\\\ & = 2 k ^ { 2 ( 2 s - 1 ) } \\sum _ { h = 1 } ^ { + \\infty } \\frac { 1 } { ( k + h ) ^ { 2 s } h ^ { 2 r } } + 2 \\sum _ { h = 1 } ^ { + \\infty } \\frac { 1 } { ( k + h ) ^ { 2 s } h ^ { 2 ( r + 1 - 2 s ) } } . \\end{align*}"}
{"id": "5028.png", "formula": "\\begin{align*} Q ^ { n , 3 } _ \\tau & = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( s - \\eta _ n ( s ) ) ^ { 2 \\alpha } \\left ( ( \\sigma ' ( X _ s ) ) ^ 2 - ( \\sigma ' ( X _ { \\eta _ n ( s ) } ) ) ^ 2 \\right ) \\sigma ^ 2 ( X _ { \\eta _ n ( s ) } ) \\ , \\\\ & \\times \\left [ \\left ( W _ s - W _ { \\eta _ n ( s ) } \\right ) ^ 2 - ( s - \\eta _ n ( s ) ) \\right ] \\ , d s \\end{align*}"}
{"id": "3925.png", "formula": "\\begin{align*} | I _ 2 | \\le c \\frac { \\delta _ 2 ^ { \\frac { 1 } { 2 } } } { T _ n } \\frac { 1 } { \\prod _ { l \\ge 2 } h _ l } . \\end{align*}"}
{"id": "7516.png", "formula": "\\begin{align*} P ( S _ i ) = \\frac { { m _ x \\choose i } \\sum _ { j = 0 } ^ { i } ( - 1 ) ^ { j } { { i } \\choose { j } } ( i - j ) ^ { \\gamma _ x k _ x } } { m _ x ^ { \\gamma _ x k _ x } } . \\end{align*}"}
{"id": "8013.png", "formula": "\\begin{align*} f _ + ^ * ( D _ + ) = \\sum _ { i = 1 } ^ m a _ { i , + } \\tilde { D } _ i + \\left ( \\sum _ { i = 1 } ^ m a _ { i , + } D _ i \\cdot e \\right ) E , \\end{align*}"}
{"id": "3420.png", "formula": "\\begin{align*} n ( z , x _ 0 ) \\cdot g = \\begin{pmatrix} a ' & * & * \\\\ b ' & * & * \\\\ c & * & * \\end{pmatrix} , b ' = b + \\sqrt { - 3 } \\bar z c . \\end{align*}"}
{"id": "5003.png", "formula": "\\begin{align*} \\Psi ^ { n , 5 } _ s = 2 \\sigma ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\ , \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) \\end{align*}"}
{"id": "5085.png", "formula": "\\begin{align*} S ^ { n , 1 } _ \\tau = S ^ { n , M , 1 } _ \\tau + S ^ { n , M , 2 } _ \\tau , \\end{align*}"}
{"id": "952.png", "formula": "\\begin{align*} \\varphi ( A ) = N ( A ) \\prod _ { P | A } ( 1 - \\frac { 1 } { N ( P ) } ) \\end{align*}"}
{"id": "3433.png", "formula": "\\begin{align*} \\mathfrak { q } _ 1 & = ( X _ 1 + X _ 2 , X _ 4 + ( \\zeta + 1 ) X _ 6 , X _ 3 ^ 3 , X _ 2 ^ 3 + X _ 5 ^ 3 - X _ 6 ^ 3 ) , \\\\ \\mathfrak { q } _ 2 & = ( X _ 1 + X _ 2 , X _ 4 - \\zeta X _ 6 , X _ 3 ^ 3 , X _ 2 ^ 3 + X _ 5 ^ 3 - X _ 6 ^ 3 ) , \\\\ \\mathfrak { q } _ 3 & = ( X _ 1 + X _ 2 , X _ 4 - X _ 6 , X _ 3 ^ 3 , X _ 2 ^ 3 + X _ 5 ^ 3 - X _ 6 ^ 3 ) . \\end{align*}"}
{"id": "6654.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ n y _ t \\leq n \\left ( \\rho P _ { \\max } + \\lambda + \\delta \\right ) \\ , , \\ , \\end{align*}"}
{"id": "5464.png", "formula": "\\begin{align*} r ( m m ' ) = & r ( l ( m ) \\cdot a \\cdot r ( m ) \\cdot r ( m ' ) ) \\\\ \\stackrel { ( R 2 ) } = & r ( l ( m ) a ) \\cdot r ( m ) \\cdot r ( m ' ) \\\\ = & \\textsf { 1 } _ B \\cdot r ( m ) \\cdot r ( m ' ) = r ( m ) \\cdot r ( m ' ) . \\end{align*}"}
{"id": "8860.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ( \\lambda _ j ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { k | k } \\right ) & = \\sum _ { k = 1 } ^ N \\prod _ { p ( \\neq k ) } ( \\lambda _ i ( t ) - \\lambda _ p ( t ) ) ( \\lambda _ j ( t ) - \\lambda _ p ( t ) ) \\\\ & = \\begin{cases} f _ { \\lambda } ( \\lambda _ i ( t ) ) ^ 2 & i = j \\\\ 0 & i \\neq j \\end{cases} . \\end{align*}"}
{"id": "5043.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } \\| Q ^ { n , 7 } _ \\tau \\| _ 2 = 0 . \\end{align*}"}
{"id": "1205.png", "formula": "\\begin{align*} \\inf _ { \\theta \\in \\Theta } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\theta } ) = \\inf _ { \\theta \\in N } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\theta } ) \\end{align*}"}
{"id": "540.png", "formula": "\\begin{align*} S = S _ 1 + S _ 2 , \\end{align*}"}
{"id": "2665.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 2 } - p ^ { n - 3 } ( n - k - 1 ) & = - p ^ { n - 3 } \\binom { n - 1 } { 2 } . \\end{align*}"}
{"id": "6903.png", "formula": "\\begin{align*} \\frac { d \\xi ( t ) } { d t } = \\xi ( t ) - \\mu , \\end{align*}"}
{"id": "2635.png", "formula": "\\begin{align*} \\max \\{ | S | : S \\in \\bigcap _ { i = 1 } ^ { k + 1 } \\mathcal { F } _ i \\} & = \\max \\{ | S | : S \\in ( \\bigcap _ { i = 1 } ^ { k } \\mathcal { F } _ i ) \\bigcap \\mathcal { F } _ { k + 1 } \\} \\\\ & = \\min \\{ \\Bar { r } _ k ( X ) + r _ { k + 1 } ( E \\backslash X ) : X \\subseteq E \\} \\end{align*}"}
{"id": "7046.png", "formula": "\\begin{align*} h ( M ) = \\sum _ { 0 \\leq 2 j \\leq M - 1 } ( j + 1 ) \\binom { M - j - 1 } j . \\end{align*}"}
{"id": "2539.png", "formula": "\\begin{align*} R i c ( e _ 1 , e _ 2 ) = 0 . \\end{align*}"}
{"id": "1611.png", "formula": "\\begin{align*} \\left ( 1 + 2 f '^ 2 ( x ^ 1 ) \\right ) \\left \\lbrace f '^ 2 ( x ^ 1 ) + 3 f ( x ^ 1 ) f '' ( x ^ 1 ) \\right \\rbrace - 1 = 0 . \\end{align*}"}
{"id": "3732.png", "formula": "\\begin{align*} B ^ m _ { j , l } ( t , x ) = \\int _ { \\R ^ 3 } \\int _ { | y - x | \\leq t } \\varphi _ { m ; - 1 0 M _ t } ( | y - x | ) \\frac { \\varphi _ { j , l } ( v , \\omega ) } { | y - x | } \\hat { v } \\times \\nabla _ x f ( t - | y - x | , y , v ) d y d v , \\omega : = \\frac { y - x } { | y - x | } . \\end{align*}"}
{"id": "6804.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } ( a u '' ) '' v \\ , d x = \\int _ { 0 } ^ { 1 } a u '' v '' d x ; \\end{align*}"}
{"id": "3542.png", "formula": "\\begin{align*} \\delta = \\begin{cases} 0 , & , \\\\ 1 , & . \\end{cases} \\end{align*}"}
{"id": "430.png", "formula": "\\begin{align*} \\frac { d } { d x } H ( x + i f ( x ) ) = a - b f ' ( x ) = a ( 1 + f ' ( x ) ^ { 2 } ) . \\end{align*}"}
{"id": "3462.png", "formula": "\\begin{align*} \\overline { G } _ { ( 1 ) } ( u ) = \\int _ { u } ^ { \\infty } g _ { 1 } ( s ) \\mathrm { d } s \\end{align*}"}
{"id": "8300.png", "formula": "\\begin{align*} \\int _ { x - \\Delta x / 2 } ^ { x + \\Delta x / 2 } | \\phi ( x ) | ^ { 2 } d x = \\frac { \\Delta x } { a } , \\end{align*}"}
{"id": "4315.png", "formula": "\\begin{align*} Y = \\frac { \\partial u } { \\partial s } d s , \\eta = \\omega ( \\cdot , \\frac { \\partial u } { \\partial s } ) d s . \\end{align*}"}
{"id": "503.png", "formula": "\\begin{align*} t _ x ( S ) = \\begin{cases} S \\cup \\{ x \\} , & x \\not \\in S S \\cup \\{ x \\} \\in J ( P ) \\\\ S \\setminus \\{ x \\} , & x \\in S S \\setminus \\{ x \\} \\in J ( P ) \\\\ S & \\end{cases} \\end{align*}"}
{"id": "8779.png", "formula": "\\begin{align*} Y _ O ( P , G ) \\leq C ( P ) < 1 = \\tilde { Y } _ O ( P , G ) \\end{align*}"}
{"id": "5240.png", "formula": "\\begin{align*} \\mathfrak { g } _ { A _ { I , \\mathrm { s y m } } , ( k _ 1 , k _ 2 ) } ^ { r , s } : = \\widetilde \\psi _ I ^ { - 1 } ( \\mathfrak { g } ^ { r , s } _ { A _ I , ( k _ 1 , k _ 2 ) } ) \\end{align*}"}
{"id": "6829.png", "formula": "\\begin{align*} N ( t ) : = \\int _ { 0 } ^ { \\infty } g p ( t , V _ F , g ) d g . \\end{align*}"}
{"id": "6599.png", "formula": "\\begin{align*} B _ h ^ S : = \\{ g \\in G _ S : \\ , \\hbox { H } _ S ( g ) \\le h \\} . \\end{align*}"}
{"id": "7036.png", "formula": "\\begin{align*} H : = L ^ 2 ( \\Omega ) \\times L ^ 2 ( \\partial \\Omega , \\beta ^ { - 1 } d S ) \\end{align*}"}
{"id": "4780.png", "formula": "\\begin{align*} F ( \\mathbf { B } _ U ) = ( \\kappa - \\nu ) ( \\kappa - \\nu - 1 ) ( \\kappa - 2 ) \\ , . \\end{align*}"}
{"id": "1711.png", "formula": "\\begin{align*} \\big ( f _ 1 ( \\tau , \\theta ) \\cdot y _ { \\gamma _ { m 1 } } \\big ) \\widehat { \\ast } \\big ( f _ 2 ( \\tau , \\theta ) \\cdot y _ { \\gamma _ { m 2 } } \\big ) = f _ 1 ( \\tau , \\theta - \\langle \\gamma _ { m 2 } , - \\rangle \\tau / 2 ) f _ 2 ( \\tau , \\theta + \\langle \\gamma _ { m 1 } , - \\rangle \\tau / 2 ) \\cdot y _ { \\gamma _ { m 1 } + \\gamma _ { m 2 } } , \\end{align*}"}
{"id": "295.png", "formula": "\\begin{align*} m _ q : = \\inf _ { \\substack { p \\ge q \\\\ p \\in \\mathcal P } } \\mu _ p , \\quad M _ x : = \\sup _ { \\substack { y \\ge x \\\\ y \\in \\R } } \\mu _ y . \\end{align*}"}
{"id": "5136.png", "formula": "\\begin{align*} \\left [ y \\frac { \\partial } { \\partial x } , x \\frac { \\partial } { \\partial u } \\right ] = y \\frac { \\partial } { \\partial u } = \\delta \\left ( \\frac { \\partial } { \\partial z } \\right ) \\end{align*}"}
{"id": "3024.png", "formula": "\\begin{align*} \\frac { b } { 4 n + 1 } + \\frac { n } { 4 n + 1 } ( 4 ( b + \\mu ) - 3 ) + \\frac { \\mu } { 4 n + 1 } = ( b + \\mu ) - \\frac { 3 n } { 4 n + 1 } < \\frac { 4 n + 3 } { 4 ( 4 n + 1 ) } + \\epsilon . \\end{align*}"}
{"id": "4614.png", "formula": "\\begin{align*} g [ \\alpha ] ^ { - 1 } g ^ { - 1 } = [ c '^ { - 1 } ( c ^ { - 1 } f ^ { - 1 } a ^ { - 1 } c '^ { - 1 } ) c ' ] \\end{align*}"}
{"id": "494.png", "formula": "\\begin{gather*} f _ i ( v ) = \\begin{cases} s _ i v & s _ i v > v s _ i v \\in W ^ J \\\\ 0 & \\end{cases} e _ i ( v ) = \\begin{cases} s _ i v & s _ i v < v \\\\ 0 & \\end{cases} \\end{gather*}"}
{"id": "4106.png", "formula": "\\begin{align*} \\big \\langle \\Phi ' ( \\vec p ) ^ * \\vec g , \\widehat { \\vec p } \\big \\rangle _ { ( L ^ \\infty ( D ) ^ 5 ) ' \\times L ^ \\infty ( D ) ^ 5 } & = \\big \\langle F ' ( V ( \\vec p ) ) ^ * \\vec g , V ' ( \\vec p ) \\widehat { \\vec p } \\big \\rangle _ { \\mathcal { L } ( X ) ' \\times \\mathcal { L } ( X ) } \\\\ [ 1 m m ] & = \\int _ 0 ^ T \\ ! \\big \\langle V ' ( \\vec p ) \\widehat { \\vec p } \\big ( u ' ( t ) + Q u ( t ) \\big ) , w ( t ) \\rangle _ X \\ , \\d t \\end{align*}"}
{"id": "20.png", "formula": "\\begin{align*} P _ { n m _ 0 } = Q _ { n m _ 0 } Q _ { n m _ 0 - 1 } \\cdots Q _ 1 \\end{align*}"}
{"id": "8737.png", "formula": "\\begin{align*} & ( 3 \\alpha _ 1 ^ 2 - 1 , \\ldots , 3 \\alpha _ k ^ 2 - 1 ) + \\sum _ { i = 1 } ^ { \\hat { k } } \\mu _ i ( - 1 , \\ldots , - 1 ) + \\lambda ( 1 , \\ldots , 1 ) = 0 , \\\\ & \\sum _ { i = 1 } ^ { \\hat { k } } \\alpha _ i = n , \\alpha _ 1 , \\ldots , \\alpha _ { \\hat { k } } \\ge 1 , \\mu _ 1 , \\ldots , \\mu _ { \\hat { k } } \\ge 0 , \\sum _ { i = 1 } ^ { \\hat { k } } \\mu _ i \\alpha _ i = 0 \\end{align*}"}
{"id": "3830.png", "formula": "\\begin{align*} E r r M ^ 2 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot \\xi - i s \\hat { v } \\cdot \\eta + i s \\mu _ 1 | \\xi - \\eta | } \\mathcal { F } \\big [ ( \\hat { v } - \\hat { V } ( s ) ) \\times B f \\big ] \\cdot \\nabla _ v \\big [ ( \\hat { V } ( s ) \\cdot \\xi - \\hat { v } \\cdot \\eta + \\mu _ 1 | \\xi - \\eta | ) ^ { - 1 } \\end{align*}"}
{"id": "2877.png", "formula": "\\begin{align*} f _ i ( x ) = f \\circ S ^ { 1 - i - w } ( x ) = f ( x _ { i + w } , \\dotsc , x _ { i + w + k - 1 } ) i \\in \\Z . \\end{align*}"}
{"id": "8188.png", "formula": "\\begin{align*} \\eta _ { x , p ; \\delta } ( y ) = \\eta ^ \\parallel _ { x ^ \\parallel , p ^ \\parallel ; \\delta } ( y ^ \\parallel ) \\eta _ { x ^ \\perp , p ^ \\perp ; \\delta } ^ \\perp ( y ^ \\perp ) \\end{align*}"}
{"id": "4805.png", "formula": "\\begin{align*} W _ { i / n } - \\widetilde W ^ { ( n ) } _ { i / n } - ( W _ { 1 } - \\widetilde W ^ { ( n ) } _ { 1 } ) i / n = \\sum _ { k = 1 } ^ i \\eta _ { n , k } . \\end{align*}"}
{"id": "1540.png", "formula": "\\begin{align*} \\widetilde { \\Delta } _ { r a d } ( f ) : = \\frac { 1 } { 2 } f '' + \\frac { \\alpha } { 2 } f ' , \\end{align*}"}
{"id": "8731.png", "formula": "\\begin{align*} U c _ 1 \\int \\limits _ { t _ 0 } ^ { t _ 1 } \\cos t d t + U c _ 2 \\int \\limits _ { t _ 0 } ^ { t _ 1 } \\sin t d t + U \\mu \\int \\limits _ { t _ 0 } ^ { t _ 1 } \\frac { a ^ 2 } { \\lambda } d t = 0 . \\end{align*}"}
{"id": "5247.png", "formula": "\\begin{align*} \\begin{aligned} 0 & \\le k _ 1 ( 0 ) \\le k _ 1 ( j ) , & 0 & \\le k _ 2 ( 0 ) \\le k _ 2 ( j ) , \\\\ 0 & \\le R _ 0 \\le k _ 1 ( j ) - k _ 1 ( 0 ) , & 0 & \\le S _ 0 \\le k _ 2 ( j ) - k _ 2 ( 0 ) . \\end{aligned} \\end{align*}"}
{"id": "5296.png", "formula": "\\begin{align*} c x ^ { \\sum _ { j = 1 } ^ h k _ { I _ j } } \\hbar ^ { - h } \\prod _ { j = 1 } ^ h \\langle \\prod _ { i \\in I _ j } \\tau ^ { a _ i } \\sigma ^ { k _ { I _ j } + 1 } \\rangle ^ { \\tfrac 1 r , o } \\end{align*}"}
{"id": "3151.png", "formula": "\\begin{align*} \\mathrm { P } _ { \\Phi } = - \\inf f _ { \\Phi } ( E _ { 1 } ) < \\infty , \\end{align*}"}
{"id": "6646.png", "formula": "\\begin{align*} \\delta _ n = c \\rho ^ 2 \\epsilon _ n = c \\rho ^ 2 a n ^ { \\frac { 1 } { 2 } ( b - 1 ) } \\ ; , \\ , \\end{align*}"}
{"id": "8217.png", "formula": "\\begin{align*} \\phi _ { n } ^ { \\pm } ( x ) = \\frac { e ^ { \\pm i \\left ( 2 n \\pi x / a - \\theta \\right ) } } { \\sqrt { a } } ; E _ { n } ^ { \\pm } = \\frac { \\hslash ^ { 2 } } { 2 m } \\left ( \\frac { 2 n \\pi } { a } \\right ) ^ { 2 } + \\frac { \\beta \\hslash ^ { 4 } } { 3 m } \\left ( \\frac { 2 n \\pi } { a } \\right ) ^ { 4 } , \\end{align*}"}
{"id": "7853.png", "formula": "\\begin{align*} \\lVert \\frac { 1 } { N } \\sum _ { n = 1 } ^ N u _ n ^ * T u _ n \\rVert _ { \\infty , 2 } ^ 2 & \\leq \\frac { 1 } { N ^ 2 } \\sum _ { n , m = 1 } ^ N \\lVert u _ m ^ * T u _ m u _ n ^ * T u _ n \\rVert _ { \\infty , 1 } \\\\ & \\leq \\frac { 2 } { N ^ 2 } \\sum _ { 1 \\leq m < n \\leq N } 2 ^ { - n } + \\frac { 1 } { N ^ 2 } \\sum _ { n = 1 } ^ N \\lVert u _ n ^ * T ^ 2 u _ n \\rVert _ { \\infty , 1 } \\\\ & \\leq \\frac { 2 } { N } + \\frac { \\| T \\| ^ 2 } { N } \\to 0 . \\end{align*}"}
{"id": "4612.png", "formula": "\\begin{align*} [ X , Y ] _ { \\pi _ 1 ( M ) } = \\sum _ { p \\in \\alpha \\cap \\beta } \\epsilon ( p ; \\alpha , \\beta ) \\{ \\alpha _ p \\beta _ p \\} \\end{align*}"}
{"id": "3191.png", "formula": "\\begin{align*} g \\left ( x \\right ) = \\sum _ { z \\in \\mathbb { Z } ^ { d } } u _ { z } \\mathrm { e } ^ { 2 \\pi \\gamma ^ { - 1 } i z \\cdot x } , x \\in \\mathbb { R } ^ { d } \\ , \\end{align*}"}
{"id": "8438.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } | T _ { \\beta } ^ n x - x _ 0 | = 0 \\end{align*}"}
{"id": "743.png", "formula": "\\begin{align*} \\mathcal R _ 1 ( f ) = \\frac { 1 } { \\sqrt { \\alpha \\beta ( \\alpha + \\beta ) } } \\begin{pmatrix} ( \\beta \\partial _ { 1 } f - \\alpha \\partial _ { m + 1 } f ) _ { | \\Delta } \\\\ \\vdots \\\\ ( \\beta \\partial _ { m } f - \\alpha \\partial _ { 2 m } f ) _ { | \\Delta } \\end{pmatrix} \\end{align*}"}
{"id": "4001.png", "formula": "\\begin{align*} \\textstyle \\lim \\limits _ { i \\to \\infty } \\frac { \\log _ q n _ i } { \\mu _ q ( n _ i ) } = 0 ; ~ ~ ~ \\mbox { a n d } ~ ~ ~ - 1 \\in \\langle q \\rangle _ { n _ i } , \\forall ~ i = 1 , 2 , \\cdots ; \\end{align*}"}
{"id": "406.png", "formula": "\\begin{align*} \\psi = s _ \\psi \\varphi + \\psi _ 0 \\end{align*}"}
{"id": "2862.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla _ x \\mathcal { L } ( x ) & = \\tfrac { 1 } { N } A ^ T \\big ( \\sigma ( A _ i x ) - y \\big ) \\\\ \\nabla ^ 2 _ x \\mathcal { L } ( x ) & = \\tfrac { 1 } { N } A ^ T B A \\end{aligned} \\end{align*}"}
{"id": "1418.png", "formula": "\\begin{align*} Q _ { N , d , D , \\beta , \\gamma } ( A ) = \\frac { 1 } { Z _ { N , d , D , \\beta , \\gamma } } E \\big [ \\mathcal { E } _ { N , d , D , \\gamma } \\mathbf { 1 } _ A \\big ] . \\end{align*}"}
{"id": "1495.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\big ( e _ { \\lambda } ( t ) - 1 \\big ) ^ { k } = \\sum _ { n = k } ^ { \\infty } S _ { 2 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } , ( k \\ge 0 ) , ( \\mathrm { s e e } \\ [ 8 ] ) . \\end{align*}"}
{"id": "4596.png", "formula": "\\begin{align*} a _ { i j } = a _ { i j } ( t ) : = \\frac { \\eta } { N ( ( \\lambda _ i ( t ) - \\lambda _ j ( t ) ) ^ 2 + \\eta ^ 2 ) } , \\end{align*}"}
{"id": "8751.png", "formula": "\\begin{align*} \\begin{array} { l l } A _ n ( t ) = \\psi ^ { - 1 } \\partial _ t ^ { n } \\psi = \\Pi _ { k = 1 } ^ n \\left [ ( \\partial _ t ^ { k - 1 } \\psi ) ^ { - 1 } \\partial _ t ^ { k } \\psi \\right ] . \\end{array} \\end{align*}"}
{"id": "6612.png", "formula": "\\begin{align*} I _ { k , \\ell } ( c ) & : = \\int _ { \\mathbb { R } ^ 4 } w ( h ^ { - 1 } x ) H \\big ( h ^ { - 1 } k , h ^ { - 2 } F _ h ( x ) \\big ) e _ { k \\ell } ( - c \\cdot x ) \\ , d x \\\\ & = h ^ 4 \\int _ { \\mathbb { R } ^ 4 } w ( x ) H \\big ( h ^ { - 1 } k , N ( x ) - 1 \\big ) e _ { k \\ell } ( - h c \\cdot x ) \\ , d x . \\end{align*}"}
{"id": "386.png", "formula": "\\begin{align*} n p _ t ( o , x ) \\leq \\frac { c _ 1 n } { t f ( n ) } e ^ { - c _ 2 / a } = \\frac { c _ 1 } { a } e ^ { - c _ 2 / a } . \\end{align*}"}
{"id": "3576.png", "formula": "\\begin{align*} n _ { \\pi } & = \\frac { 1 } { 2 } [ n - 2 ] _ q ! \\left ( T _ { n - 2 } T _ { n - 1 } - ( 1 + q ) \\sum _ { i < j } \\chi _ i ( - 1 ) \\chi _ j ( - 1 ) \\right ) . \\end{align*}"}
{"id": "123.png", "formula": "\\begin{align*} T _ { \\lambda ^ 0 } : = | \\mu - m ^ 2 | \\ , C \\star S = | \\mu - m ^ 2 | \\big ( C \\star C + C \\star E \\big ) . \\end{align*}"}
{"id": "84.png", "formula": "\\begin{align*} \\Psi _ \\kappa ^ { - 1 } x _ 1 & = x _ 1 , \\\\ \\Psi _ \\kappa ^ { - 1 } x _ \\ell & = x _ \\ell - \\sum _ { j = 1 } ^ { \\ell - 1 } \\Big ( \\# W ^ u ( x _ \\ell ; P ) \\cap W ^ s ( x _ j ; \\tilde P ) \\ \\ 2 \\Big ) \\cdot \\Phi _ \\kappa ^ { - 1 } x _ j , \\forall \\ell \\geq 2 . \\end{align*}"}
{"id": "6242.png", "formula": "\\begin{align*} & \\int \\dfrac { x ^ { \\alpha + 1 } } { ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( q + \\frac { 1 - q ^ { 1 - \\nu } [ \\nu ] ^ 2 } { q x ^ 2 } \\right ) J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) d _ q x = \\\\ & \\frac { x ^ { \\alpha + 1 } } { q ^ { \\alpha } ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\frac { q ^ { 1 - \\nu } ( 1 - q ) [ \\nu ] ^ 2 - 1 } { x } J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) - D _ { q ^ { - 1 } } J _ \\nu ^ { ( 2 ) } ( x | q ^ 2 ) \\right ) . \\end{align*}"}
{"id": "1445.png", "formula": "\\begin{align*} [ t ^ k ] \\circ \\mathcal { T } _ { \\bold { c } } ( t ^ m ) & = \\dfrac { t ^ { k + m } } { c _ m } = \\dfrac { 1 } { c _ { m + k } } \\dfrac { A ( m + k - 1 ) \\cdots A ( m ) } { B ( m + k ) \\cdots B ( m + 1 ) } t ^ { k + m } \\\\ & = \\mathcal { T } _ { \\bold { c } } \\circ A ( \\theta _ t - 1 ) \\circ \\cdots \\circ A ( \\theta _ t - k ) \\circ B ( \\theta _ t ) ^ { - 1 } \\circ \\cdots \\circ B ( \\theta _ t - k + 1 ) ^ { - 1 } \\circ [ t ^ k ] ( t ^ m ) \\enspace . \\end{align*}"}
{"id": "561.png", "formula": "\\begin{align*} \\mathbb { E } ^ 2 \\Big ( \\sum _ { 1 \\leq i \\leq n } { X _ i } \\Big ) = \\Big ( \\sum _ { 1 \\leq i \\leq n } f _ k ( i ) \\Big ) ^ 2 + O _ { \\varepsilon } ( n ^ { 3 / 2 + \\varepsilon } ) , \\end{align*}"}
{"id": "4851.png", "formula": "\\begin{align*} ( d _ { X , 0 } F ) ( a _ 1 , \\ldots , a _ { p + 2 } ) & = [ a _ 1 , F ( a _ 2 , \\ldots , a _ { p + 2 } ) ] _ X , \\\\ ( d _ { X , i } F ) ( a _ 1 , \\ldots , a _ { p + 2 } ) & = F ( a _ 1 , \\ldots , [ a _ i , a _ { i + 1 } ] _ X , \\ldots , a _ { p + 2 } ) , \\\\ ( d _ { X , n } F ) ( a _ 1 , \\ldots , a _ { n + 1 } ) & = [ F ( a _ 1 , \\ldots , a _ { p + 1 } ) , a _ { p + 2 } ] _ X \\end{align*}"}
{"id": "7588.png", "formula": "\\begin{align*} Y _ { k + 1 } & = e ^ { \\Omega _ k } Y _ k , \\\\ x _ { k + 1 } & = \\varphi ( Y _ { k + 1 } , x _ k ) , \\end{align*}"}
{"id": "437.png", "formula": "\\begin{align*} F _ { \\nu _ { n } } ( H _ { \\nu _ { n } } ( s + i t ) ) = s + i t , s + i t \\in Q , \\ , t > f _ { n } ( s ) , \\ , n \\ge N . \\end{align*}"}
{"id": "4359.png", "formula": "\\begin{align*} f = b _ 0 + b _ 1 \\phi + \\cdots + b _ s \\phi ^ s + \\cdots , b _ s \\in \\Z _ p [ x ] . \\end{align*}"}
{"id": "347.png", "formula": "\\begin{align*} 1 / g ( t ) = \\begin{cases} \\frac { C } { t ^ { 3 / 2 } } & 1 \\leq t \\leq f ( n ) ^ 2 \\\\ \\frac { C } { t f ( n ) } & f ( n ) ^ 2 \\leq t \\leq D ^ 2 . \\end{cases} \\end{align*}"}
{"id": "133.png", "formula": "\\begin{align*} 0 \\leq \\gamma _ t \\leq \\begin{cases} \\frac { 1 } { t m ^ 2 m _ t ^ 2 } & d = 2 , \\\\ c \\log \\Big ( 1 + \\frac { 1 } { m ^ 2 t } \\Big ) & d = 3 , \\end{cases} \\end{align*}"}
{"id": "7923.png", "formula": "\\begin{align*} \\Sigma _ \\omega : = \\{ \\sigma _ I | \\overline { I } \\in \\mathcal A _ \\omega \\} . \\end{align*}"}
{"id": "49.png", "formula": "\\begin{align*} f ^ { \\tilde { \\phi } ^ { - n } } \\circ \\rho _ { \\eta } ( X ) = \\Pi _ { w \\in W _ { \\rho } ^ { m _ 0 } } f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ( X + w ) . \\end{align*}"}
{"id": "6728.png", "formula": "\\begin{align*} \\mu \\int _ { \\Omega } \\varepsilon z : \\varepsilon v + \\nu \\int _ { \\Omega } \\mathsf { m } ' \\left ( \\varepsilon y ; \\varepsilon z \\right ) : \\varepsilon v = \\int _ { \\Omega } h \\cdot v , \\forall v \\in Y , \\end{align*}"}
{"id": "6338.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } \\cos ( z ; q ) = - q ^ { \\frac { 1 } { 2 } } \\sin ( q ^ { \\frac { - 1 } { 2 } } z ; q ) . \\end{align*}"}
{"id": "738.png", "formula": "\\begin{align*} K ^ 2 ( z , w ) B _ m ^ { - 2 } ( z , w ) \\Big ( \\ ! \\ ! \\Big ( \\ , \\partial _ i \\bar { \\partial } _ j \\log B _ m ( z , w ) \\Big ) \\ ! \\ ! \\Big ) _ { i , j = 1 } ^ m \\preceq B _ m ^ { - 2 } ( z , w ) \\mathcal G _ { K ^ { - 1 } } ( z , w ) . \\end{align*}"}
{"id": "6489.png", "formula": "\\begin{align*} u = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) u ( c _ 1 ) = u _ x ( c _ 1 ) = 0 . \\end{align*}"}
{"id": "3864.png", "formula": "\\begin{align*} E ^ { T , \\Upsilon , Y } \\cup Z ^ a \\big [ T _ 1 , . . . , T _ n \\big ] = \\bigg ( \\bigcup _ { i = 1 } ^ n E ^ { T _ i , \\Upsilon _ i , Y _ i } \\bigg ) \\cup \\Big \\{ \\tilde { \\phi } ^ { T , \\Upsilon , Y } \\big [ \\tilde { h } _ j ^ { Y } \\big ] : \\tilde { h } _ j ^ { Y } \\in Z ^ a \\big [ Y _ 1 , . . . , Y _ n \\big ] \\Big \\} . \\end{align*}"}
{"id": "3463.png", "formula": "\\begin{align*} \\overline { G } _ { ( k ) } ( u ) = \\int _ { u } ^ { \\infty } \\overline { G } _ { ( k - 1 ) } ( s ) \\mathrm { d } s , ~ k \\geq 2 . \\end{align*}"}
{"id": "1790.png", "formula": "\\begin{align*} w _ { n , i } ^ * & \\in \\left ( Q _ { a _ { n , i } } ( \\alpha - 1 , A _ i , u _ i + a _ i f _ { q ( i ) } ) \\right ) ^ { ( \\alpha - 1 ) } \\\\ & \\subseteq \\left ( \\bigcap _ { j = 1 } ^ \\infty K ( \\alpha - 1 , A _ j , u _ j + a _ j f _ { q ( j ) } ) \\right ) ^ { ( \\alpha - 1 ) } = K ^ { ( \\alpha - 1 ) } . \\end{align*}"}
{"id": "6897.png", "formula": "\\begin{align*} p ( t , v , g ) = \\rho ( t , v ) \\frac { 1 } { \\sqrt { 2 \\pi a ( t ) } } \\exp \\left ( - \\frac { ( g - g _ { } ( t ) ) ^ 2 } { 2 a ( t ) } \\right ) . \\end{align*}"}
{"id": "818.png", "formula": "\\begin{align*} H f ( x ) : = \\frac { 1 } { x } \\int _ 0 ^ x f ( t ) d t \\end{align*}"}
{"id": "5378.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + \\sum _ { | \\alpha | \\leq m } a _ \\alpha D ^ \\alpha u & = 0 \\mbox { i n } \\ ; \\ ; \\Omega , u - f \\in \\widetilde H ^ s ( \\Omega ) \\end{align*}"}
{"id": "1232.png", "formula": "\\begin{align*} [ Z _ j ] _ { \\alpha , \\beta } & = \\sum _ { i = 1 } ^ { n ( T , j + 1 ) } [ D _ { j + 1 } ^ T ] _ { \\alpha , i } \\ [ R _ { j + 1 } ^ { - 1 } ] _ { i , i } \\ [ D _ { j + 1 } ] _ { i , \\beta } \\\\ & = \\sum _ { i = 1 } ^ { n ( T , j + 1 ) } [ D _ { j + 1 } ] _ { i , \\alpha } \\ [ D _ { j + 1 } ] _ { i , \\beta } \\ [ R _ { j + 1 } ^ { - 1 } ] _ { i , i } \\ , . \\end{align*}"}
{"id": "7555.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { 1 } { | B _ r ( 0 ) | } \\int _ { B _ r ( x ) } f ( y ) \\ , d y = f ( x ) . \\end{align*}"}
{"id": "4939.png", "formula": "\\begin{align*} \\widehat { u } ( x , t ) = u ( \\widehat { x } , t ) , \\widehat { x } = x - 2 < x , \\vec { a } > \\vec { a } . \\end{align*}"}
{"id": "5002.png", "formula": "\\begin{align*} \\Psi ^ { n , 4 } _ s = 2 \\sigma ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) \\left ( \\int ^ s _ { \\eta _ n ( s ) } \\left ( s - \\eta _ n ( u ) \\right ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) , \\end{align*}"}
{"id": "8744.png", "formula": "\\begin{align*} A _ 1 ( t ) = \\psi ^ { - 1 } \\partial _ t \\psi = ( I - \\kappa e ^ { - a ( t , s ) } ) ^ { - 1 } \\partial _ t a ( t , s ) \\end{align*}"}
{"id": "7184.png", "formula": "\\begin{align*} & ( z _ 0 + z _ 2 ) ^ { k + \\frac { j _ 2 } { T } } Y _ { M ^ 3 } ( u , z _ 0 + z _ 2 ) I ( w _ 1 , z _ 2 ) w _ 2 \\\\ = & ( z _ 2 + z _ 0 ) ^ { k + \\frac { j _ 2 } { T } } I ( Y _ { M ^ 1 } ( u , z _ 0 ) w _ 1 , z _ 2 ) w _ 2 . \\end{align*}"}
{"id": "8137.png", "formula": "\\begin{align*} P _ \\delta ( E ^ \\parallel \\times E ^ \\perp ) = P ^ \\parallel _ \\delta ( E ^ \\parallel ) \\otimes P ^ \\perp _ \\delta ( E ^ \\perp ) \\end{align*}"}
{"id": "1829.png", "formula": "\\begin{align*} m ( z ) = \\frac { 1 } { z - b _ { 0 } - a _ { 0 } \\ , m _ { 1 } ( z ) } . \\end{align*}"}
{"id": "2876.png", "formula": "\\begin{align*} f _ i \\circ S ^ j = f _ { i - \\ell } i \\in \\Z . \\end{align*}"}
{"id": "7817.png", "formula": "\\begin{align*} d \\Theta _ n ( e _ \\lambda ) = e _ { \\theta ( \\lambda ) } . \\end{align*}"}
{"id": "1784.png", "formula": "\\begin{align*} | y ^ * ( u _ n ) | = a _ n | f _ { q ( n ) } ( y ^ * ) | \\leq C a _ n \\end{align*}"}
{"id": "3261.png", "formula": "\\begin{align*} \\begin{array} { c c c c c } \\mathcal { M } _ { A , q } & : & H ^ 1 ( D ) & \\rightarrow & H ^ 2 _ { \\mathrm { l o c } } ( \\R ^ 3 ) , \\\\ & & v & \\mapsto & \\mathcal { M } _ { A , q } v : = u _ { A , q } ^ s \\end{array} \\end{align*}"}
{"id": "5882.png", "formula": "\\begin{align*} d _ { n - 1 } ( { \\mathcal K } , N ) _ H \\geq C '' \\frac { 1 } { [ \\log _ 2 ( n + \\lceil \\log _ 2 N \\rceil ) ] ^ \\alpha } , n = 1 , 2 , \\ldots . \\end{align*}"}
{"id": "2026.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : b o l t z m a n n g e n e r a t o r } \\mathcal { L } _ N \\varphi _ N = \\sum _ { i = 1 } ^ N L ^ { ( 1 ) } \\diamond _ i \\varphi _ N + \\frac { 1 } { N } \\sum _ { i < j } L ^ { ( 2 ) } \\diamond _ { i j } \\varphi _ N , \\end{align*}"}
{"id": "8743.png", "formula": "\\begin{align*} A _ 1 ( t ) = \\psi ^ { - 1 } \\partial _ t \\psi = ( I + \\kappa U ( s , t ) ) \\partial _ t { \\rm L o g } ( U ( t , s ) + \\kappa I ) \\end{align*}"}
{"id": "1.png", "formula": "\\begin{align*} g _ i = t \\tilde { g } _ i . \\end{align*}"}
{"id": "6894.png", "formula": "\\begin{align*} c _ k : = \\int _ { \\mathbb { R } } \\int _ { 0 } ^ { V _ F } p _ { } ( v , g ) e ^ { - i k v \\frac { 2 \\pi } { V _ F } } d v , k \\in \\mathbb { Z } . \\end{align*}"}
{"id": "1128.png", "formula": "\\begin{align*} E _ { \\mbox { \\tiny M D } } ( \\theta ) = \\sup _ { \\lambda \\ge 0 } \\sup _ { P \\le P _ w } \\left \\{ \\lambda ( \\sqrt { P _ s P } - \\theta ) - \\tilde { C } _ \\lambda ( P ) - \\frac { \\lambda ^ 2 \\sigma _ N ^ 2 P } { 2 } \\right \\} . \\end{align*}"}
{"id": "1814.png", "formula": "\\begin{align*} x P _ { n } ( x ) = P _ { n + 1 } ( x ) + b _ { n } P _ { n } ( x ) + a _ { n - 1 } P _ { n - 1 } ( x ) , n \\geq 1 , \\end{align*}"}
{"id": "882.png", "formula": "\\begin{align*} \\tau _ j : = \\eta _ \\varepsilon \\rho _ j \\eta _ \\varepsilon : = ( 1 - \\varepsilon ) \\frac { \\alpha _ { \\min } - \\Lambda _ 0 } { 2 \\Lambda _ 1 ( 1 + \\delta _ { \\min } ^ { - 1 } ) } \\end{align*}"}
{"id": "8588.png", "formula": "\\begin{align*} m = n _ { j _ 1 } + \\ldots + n _ { j _ s } , 1 \\le s \\le l , \\end{align*}"}
{"id": "8454.png", "formula": "\\begin{align*} \\dim _ { \\rm H } B \\cap W ( \\mathbf { t } ) \\geq \\min _ { A \\in \\mathcal { A } } \\left \\{ \\# \\mathcal { K } _ 1 + \\# \\mathcal { K } _ 2 + \\frac { \\sum _ { k \\in \\mathcal { K } _ 3 } a _ k - \\sum _ { k \\in \\mathcal { K } _ 2 } t _ k } { A } \\right \\} : = s , \\end{align*}"}
{"id": "8684.png", "formula": "\\begin{align*} d V _ { H T } = d V _ { \\alpha } . \\end{align*}"}
{"id": "3594.png", "formula": "\\begin{align*} I ^ { ( n ) } = ( \\{ t ^ a \\mid a / n \\in \\mathcal { Q } ( I ^ \\vee ) \\} ) = ( \\{ t ^ a \\mid \\langle a , u _ i \\rangle \\geq n \\mbox { f o r } i = 1 , \\ldots , m \\} ) , \\end{align*}"}
{"id": "3778.png", "formula": "\\begin{align*} \\widetilde { K } ^ { m ; p , q , i ; a } _ { k , j ; n , l , r } ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ { \\pi } \\big [ ( t - s ) K _ { k ; n } ^ { \\mu } ( \\mathfrak { m } ) ( y , \\zeta , \\omega ) + \\widetilde { K } _ { k ; n } ^ { \\mu } ( \\mathfrak { m } ) ( y , \\zeta ) \\big ] f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "773.png", "formula": "\\begin{align*} \\beta f ' _ { 0 } ( r ^ m ) + ( 1 - \\beta ) f _ { 0 } ( r ^ m ) + \\sum _ { n = 1 } ^ { \\infty } M ( n ) \\phi _ { n } ( r ) < - f _ { 0 } ( - 1 ) . \\end{align*}"}
{"id": "3617.png", "formula": "\\begin{align*} V ( \\mathcal { Q } ( I ) ) = & \\{ ( 0 , \\ , 0 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 1 ) , \\ , ( 0 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 0 ) , \\ , ( 1 , \\ , 0 , \\ , 0 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 1 ) , \\ , \\\\ & ( 1 , \\ , 1 , \\ , 0 , \\ , 0 , \\ , 1 , \\ , 1 , \\ , 1 ) , \\ , ( 1 , \\ , 1 , \\ , 1 , \\ , 0 , \\ , 0 , \\ , 1 , \\ , 1 ) , \\ , ( 1 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 0 , \\ , 0 , \\ , 1 ) , \\ , \\\\ & ( 1 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 1 , \\ , 0 , \\ , 0 ) , \\ , ( { 1 } / { 2 } , \\ , { 1 } / { 2 } , \\ , { 1 } / { 2 } , \\ , { 1 } / { 2 } , \\ , { 1 } / { 2 } , \\ , { 1 } / { 2 } , \\ , { 1 } / { 2 } ) \\} , \\end{align*}"}
{"id": "1838.png", "formula": "\\begin{align*} \\phi _ { j } ( z ) : = \\langle ( z I - H ) ^ { - 1 } e _ { j } , e _ { 0 } \\rangle = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\langle H ^ { n } e _ { j } , e _ { 0 } \\rangle } { z ^ { n + 1 } } , 0 \\leq j \\leq p - 1 , \\end{align*}"}
{"id": "4475.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } c _ 1 ^ + + c _ 2 ^ + + q _ + = c _ 1 ^ - + c _ 2 ^ - + q _ - , \\\\ c _ 1 ^ + - c _ 2 ^ + = c _ 1 ^ - - c _ 2 ^ - . \\end{array} \\right . \\end{align*}"}
{"id": "7220.png", "formula": "\\begin{align*} M _ { n + 1 } = C \\eta ^ { - n } \\mu ^ { - n } M _ n ^ p . \\end{align*}"}
{"id": "8619.png", "formula": "\\begin{align*} d i m _ x ^ { l o g } f ^ { - 1 } ( f ( x ) ) : = \\end{align*}"}
{"id": "7884.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m s _ i ^ j = \\int _ \\beta [ D _ i ] , i \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "4737.png", "formula": "\\begin{align*} \\chi ( A ) = X ^ 4 - s _ 1 X ^ 3 + ( s _ 2 + 2 q ) X ^ 2 - q s _ 1 X + q ^ 2 , \\end{align*}"}
{"id": "4797.png", "formula": "\\begin{align*} \\tau \\log | \\phi _ \\tau | + \\Re \\mathcal { Q } _ \\tau = \\breve { Q } _ \\tau , \\end{align*}"}
{"id": "2951.png", "formula": "\\begin{align*} ( ( \\exp , 0 ) \\cdot X _ 0 - x _ 0 ) ( ( f , a ) ) = \\exp ( a ) . \\end{align*}"}
{"id": "7403.png", "formula": "\\begin{align*} \\psi _ { \\mathrm { d i s c } } ( k ) : = \\max _ { 0 \\le a < q } \\left | \\Pr ( \\{ S _ k p / q \\} \\le a / q ) - ( a + 1 ) / q \\right | , \\end{align*}"}
{"id": "4057.png", "formula": "\\begin{align*} & s _ { ( k ) } = \\frac 1 { k ! } \\prod _ { b = 2 } ^ k \\frac { ( y _ b + 1 ) ( y _ b + \\omega + b - 3 ) } { 2 b + \\omega - 4 } , \\end{align*}"}
{"id": "8215.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { c } \\phi ( + a / 2 ) = \\phi ( - a / 2 ) \\\\ \\partial _ { x } \\phi ( + a / 2 ) = \\partial _ { x } \\phi ( - a / 2 ) \\\\ \\partial _ { x } ^ { 2 } \\phi ( + a / 2 ) = \\partial _ { x } ^ { 2 } \\phi ( - a / 2 ) \\\\ \\partial _ { x } ^ { 3 } \\phi ( + a / 2 ) = \\partial _ { x } ^ { 3 } \\phi ( - a / 2 ) \\end{array} \\right . . \\end{align*}"}
{"id": "571.png", "formula": "\\begin{align*} M _ 1 = \\sum _ { \\substack { 0 \\leq l \\leq i \\\\ \\gcd ( l , i ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } \\\\ \\gcd ( l , i + 1 ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } C _ { \\alpha } ( i , l ) \\end{align*}"}
{"id": "4500.png", "formula": "\\begin{align*} | \\partial ^ l ( g _ { i j } - \\delta _ { i j } ) ( x ) | = O ( | x | ^ { - \\tau - l } ) , l = 0 , 1 , 2 , \\quad | \\partial ^ l k _ { i j } ( x ) | = O ( | x | ^ { - \\tau - 1 - l } ) , l = 0 , 1 , \\end{align*}"}
{"id": "6198.png", "formula": "\\begin{align*} C _ { M } ^ { B } = \\{ [ \\alpha ] _ { B } \\in H _ { B } ^ { 1 , 1 } ( M , \\mathbf { R } ) | \\exists \\omega > 0 [ \\omega ] _ { B } = [ \\alpha ] _ { B } \\} . \\end{align*}"}
{"id": "3003.png", "formula": "\\begin{align*} \\deg a _ i \\leq \\frac { 1 } { \\mu } ( \\delta _ A ( a ) + \\deg A ) i = 0 , \\dots , \\mu - 1 \\ , \\end{align*}"}
{"id": "817.png", "formula": "\\begin{align*} m _ i = p + i , R _ i = \\frac { 7 \\bar R _ 0 } { 8 } - 4 ( 2 i + 1 ) h _ 0 \\quad \\mbox { f o r a l l } i = 0 , \\dots , i _ \\infty . \\end{align*}"}
{"id": "3609.png", "formula": "\\begin{align*} \\rho _ { i c } ( I ) = \\max \\left \\{ ( d _ i f _ \\ell ) / \\langle \\gamma _ i , \\delta _ \\ell \\rangle \\right \\} _ { i , \\ell } . \\end{align*}"}
{"id": "2697.png", "formula": "\\begin{align*} E ^ \\mathrm { B o g } = \\ ; & - \\frac { 1 } { 2 } \\mathrm { T r } _ { \\perp , r } \\left [ D _ r + \\lambda Q _ r K _ { 1 1 } Q _ r - \\sqrt { D _ r ^ 2 + 2 \\lambda D _ r ^ { 1 / 2 } Q _ r K _ { 1 1 } Q _ r D _ r ^ { 1 / 2 } } \\right ] \\\\ & - \\frac { 1 } { 2 } \\mathrm { T r } _ { \\perp , \\ell } \\left [ D _ \\ell + \\lambda Q _ \\ell K _ { 2 2 } Q _ \\ell - \\sqrt { D _ \\ell ^ 2 + 2 \\lambda D _ \\ell ^ { 1 / 2 } Q _ \\ell K _ { 2 2 } Q _ \\ell D _ \\ell ^ { 1 / 2 } } \\right ] . \\end{align*}"}
{"id": "7000.png", "formula": "\\begin{align*} \\partial _ { i _ 1 } ^ { k _ 1 } \\cdots \\partial _ { i _ r } ^ { k _ r } \\left ( I _ Q \\right ) ( f ) = 0 \\end{align*}"}
{"id": "6427.png", "formula": "\\begin{align*} \\frac { \\dd \\Xi _ t ^ { ( 0 ) } } { \\dd t } & = \\Bar { Q } ^ { ( 0 ) } \\circ H _ t ^ { ( 0 ) } + H _ t ^ { ( 0 ) } \\circ Q _ \\mathfrak g ^ { ( 0 ) } \\\\ & = [ Q , H _ { 0 } ] \\\\ & = \\Psi _ { 0 } - \\Phi _ { 0 } = \\Bar { \\Psi } ^ { ( 0 ) } - \\Bar { \\Phi } ^ { ( 0 ) } . \\end{align*}"}
{"id": "6700.png", "formula": "\\begin{align*} l ^ \\gamma ( t \\circ \\varphi ( \\lambda _ 1 ) ) = l ^ \\tau ( t ) \\circ \\varphi ( \\lambda _ 2 ) . \\end{align*}"}
{"id": "456.png", "formula": "\\begin{align*} K ' = \\{ r e ^ { i \\theta } : | r - r _ { 0 } | \\le \\varepsilon , \\theta \\in [ h _ { \\nu } ( r ) + \\varepsilon , \\pi ] \\} . \\end{align*}"}
{"id": "3065.png", "formula": "\\begin{align*} \\mathrm { M D T E } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { X } ) & = \\mathrm { E } \\left [ \\mathbf { X } | \\boldsymbol { a } < \\mathbf { X } \\leq \\boldsymbol { b } \\right ] \\end{align*}"}
{"id": "8292.png", "formula": "\\begin{align*} \\psi _ { B _ { k } } = B _ { k } \\left [ \\frac { \\cos ( k x ) . \\sinh ( k ' x ) } { \\cos ( k a / 2 ) . \\sinh ( k ' a / 2 ) } - \\frac { \\sin ( k x ) . \\cosh ( k ' x ) } { \\sin ( k a / 2 ) . \\cosh ( k ' a / 2 ) } \\right ] , \\end{align*}"}
{"id": "2318.png", "formula": "\\begin{align*} R _ { \\widetilde { G } } = \\left ( \\begin{array} { l l l } 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right ) , \\ R _ G = \\left ( \\begin{array} { l l l } 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\ R _ { \\widetilde { D } } = \\left ( \\begin{array} { l l l } 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\ \\ R _ D = \\left ( \\begin{array} { l l l } 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 1 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
{"id": "3721.png", "formula": "\\begin{align*} \\phi ^ { [ \\lambda \\lambda _ k \\lambda '' v ] } & = ( \\lambda \\lambda _ k \\lambda '' ) \\cdot \\phi \\\\ & = \\lambda \\cdot ( ( \\lambda _ k \\lambda '' ) \\cdot \\phi ) \\\\ & = \\lambda \\cdot \\phi ^ { [ \\lambda _ k \\lambda '' v ] } . \\end{align*}"}
{"id": "4551.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 5 \\right ) } \\Vert _ { p } = \\mathcal { O } \\left ( \\frac { 1 } { n } \\right ) ( v ( | z _ { 1 } | ) + ( v ( | z _ { 2 } | ) ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "7117.png", "formula": "\\begin{align*} 0 = h ( \\boldsymbol { R } _ 1 ( \\tau ; t , x ) , p ( \\boldsymbol { R } _ 1 ( \\tau ; t , x ) ; t , x ) ) \\ , , \\end{align*}"}
{"id": "597.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { j = 1 } ^ N \\sum _ { k = 0 } ^ { n _ j } a _ { j k } x ^ { \\alpha _ j } \\log ^ k ( x ) + \\sum _ { k = 0 } ^ { n _ 0 } a _ { 0 k } \\log ^ k ( x ) + o ( x ^ { \\alpha _ N } \\log ^ { n _ N } ( x ) ) \\end{align*}"}
{"id": "8482.png", "formula": "\\begin{align*} y _ i & = \\left ( L ( i - k ) + y _ k \\right ) _ + \\\\ & = \\left ( L ( i - n + m - 1 ) + \\frac { U } { m } - \\frac { L ( m + 1 ) } { 2 } + L \\right ) _ + \\\\ & = \\left ( L ( i - n + m ) + \\frac { U } { m } - \\frac { L ( m + 1 ) } { 2 } \\right ) _ + \\end{align*}"}
{"id": "2940.png", "formula": "\\begin{align*} & \\| F _ c \\phi \\| _ { L ^ p _ x } = \\| e ^ { i x ^ 2 / 4 t } F _ c \\phi \\| _ { L ^ p _ x } \\lesssim \\| P e ^ { i x ^ 2 / 4 t } F _ c \\phi \\| _ { L ^ 2 _ x } ^ a \\| e ^ { i x ^ 2 / 4 t } F _ c \\phi \\| _ { L ^ 2 _ x } ^ { 1 - a } \\\\ & \\lesssim \\| ( 1 / t ) | 2 P t - x | F _ c \\phi \\| _ { L ^ 2 _ x } ^ a \\| F _ c \\phi \\| _ { L ^ 2 _ x } ^ { { 1 - a } } \\lesssim t ^ { ( - 1 + \\alpha ) a } \\| F _ c \\phi \\| _ { L ^ 2 _ x } = t ^ { ( - 1 + \\alpha ) a } \\| F ( \\frac { | x | } { t ^ { \\alpha } } \\leq 1 ) e ^ { i \\Delta t } \\phi \\| _ { L ^ 2 _ x } \\end{align*}"}
{"id": "4739.png", "formula": "\\begin{align*} \\begin{cases} V _ 0 - V _ 1 S _ 0 ( U _ 1 ) = 0 , \\\\ V _ 1 ^ 2 - S _ 1 ( U _ 1 ) = 0 , \\\\ U _ 0 - R _ 0 ( U _ 1 ) = 0 , \\\\ R _ 1 ( U _ 1 ) = 0 \\end{cases} \\end{align*}"}
{"id": "8318.png", "formula": "\\begin{align*} G ( t , x ) \\frac { \\left ( e ^ { t } - 1 \\right ) } { t - 2 \\pi i k } = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\int _ { 0 } ^ { 1 } e ^ { - 2 \\pi i k z } \\bar { P } _ { n } ( x , z ) d z \\right ) \\frac { t ^ n } { n ! } , \\end{align*}"}
{"id": "1477.png", "formula": "\\begin{align*} { { \\mathcal { L } } } _ m ( { { \\mathfrak { A } } } ( \\boldsymbol { t } ) ) = { \\rm { d e t } } ( \\psi _ s ( t ^ { u + \\ell } ( t - 1 ) ^ { r n } ) ) _ { \\substack { 0 \\le s \\le r - 1 \\\\ 0 \\le \\ell \\le r - 1 } } \\enspace . \\end{align*}"}
{"id": "424.png", "formula": "\\begin{align*} \\Im H _ { \\mu } ( s + i t ) = t \\left [ 1 - \\int _ { \\mathbb { R } } \\frac { 1 + x ^ { 2 } } { | s + i t - x | ^ { 2 } } d \\sigma ( x ) \\right ] . \\end{align*}"}
{"id": "8398.png", "formula": "\\begin{gather*} \\left ( Q \\left | \\left ( \\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix} \\right ) \\right . \\right ) ( x , y ) : = Q ( a x + b y , c x + d y ) , \\end{gather*}"}
{"id": "1152.png", "formula": "\\begin{align*} | f g | ^ 2 = ( f g , f g ) \\leq ( f ^ \\dagger f , f ^ \\dagger f ) ( g g ^ \\dagger , g g ^ \\dagger ) = | f | ^ 2 | g | ^ 2 . \\end{align*}"}
{"id": "4611.png", "formula": "\\begin{align*} \\Psi _ { n + 1 } ( d _ n f \\otimes P ) = \\Psi _ n ( f \\otimes \\partial _ { n + 1 } P ) , \\end{align*}"}
{"id": "8745.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t ^ 2 \\psi = A _ 2 ( t ) \\psi \\end{array} \\end{align*}"}
{"id": "3766.png", "formula": "\\begin{align*} c ^ { m , E } _ { j , l , r ; e r r } ( t - s , v , \\omega ) : = \\big ( \\frac { \\hat { v } + \\omega } { ( 1 + \\hat { v } \\cdot \\omega ) ^ 2 } \\big ( 1 - | \\hat { v } ^ 2 | \\big ) + \\omega - \\frac { ( \\omega + \\hat { v } ) \\hat { v } \\cdot \\omega } { 1 + \\hat { v } \\cdot \\omega } \\big ) \\varphi _ { l ; r } ( \\tilde { v } + \\omega ) \\varphi _ j ( v ) \\varphi _ { m ; - 1 0 M _ t } ( t - s ) \\end{align*}"}
{"id": "4481.png", "formula": "\\begin{align*} \\| f \\| _ Z : = \\| ( | \\xi | ^ { r } + | \\xi | ^ { w } ) \\hat f ( \\xi ) \\| _ { L ^ \\infty _ \\xi } . \\end{align*}"}
{"id": "1422.png", "formula": "\\begin{align*} N ( d ) = | S _ N ^ d | - 1 = ( 2 N + 1 ) ^ d - 1 . \\end{align*}"}
{"id": "4929.png", "formula": "\\begin{align*} \\nabla _ i u = & g ^ { k l } h _ { i k } \\nabla _ l \\Phi , \\\\ \\nabla _ i \\nabla _ j u = & g ^ { k l } \\nabla _ k h _ { i j } \\nabla _ l \\Phi + \\phi ' h _ { i j } - ( h ^ 2 ) _ { i j } u , \\end{align*}"}
{"id": "2913.png", "formula": "\\begin{align*} c _ n = ( 2 ^ { \\ell _ 1 } - 1 ) ( 2 ^ { \\ell _ 2 } - 1 ) \\cdots ( 2 ^ { \\ell _ t } - 1 ) . \\end{align*}"}
{"id": "6528.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 2 } ^ { \\infty } \\frac { 1 } { \\alpha _ i } \\sum \\limits _ { k : k \\geq 0 , k \\leq \\alpha _ i } \\mu ( k ) k \\leq c _ { \\alpha } \\frac { \\alpha _ 1 } { \\alpha _ 2 } + c _ { \\alpha } ( 1 - \\beta _ 1 ) . \\end{align*}"}
{"id": "6317.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) + \\dfrac { x - q ( a + b - q a b ) } { a b q ^ 2 ( 1 - q ) ( 1 - x ) } D _ { q ^ { - 1 } } y ( x ) - \\frac { q ^ { - n - 1 } [ n ] _ q } { a b ( 1 - q ) ( 1 - x ) } y ( x ) = 0 . \\end{align*}"}
{"id": "714.png", "formula": "\\begin{align*} \\| u \\| _ { \\alpha } : = \\| u \\| _ { \\infty } + [ u ] _ { \\alpha } . \\end{align*}"}
{"id": "8514.png", "formula": "\\begin{align*} \\Psi ( C _ 1 \\cap C _ 2 ) = \\left \\langle \\left \\{ \\gamma ^ { j } \\texttt { l c m } ( \\widehat { \\ , F _ { 1 , j } \\ ; } , \\widehat { \\ , F _ { 2 , j } \\ ; } ) \\ ; : \\ ; 0 \\leq j < t \\right \\} \\right \\rangle . \\end{align*}"}
{"id": "3875.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ p G - \\lambda G ^ { p - 1 } = \\delta _ 0 & \\Omega \\\\ G \\geq 0 & \\Omega \\\\ G = g & \\partial \\Omega \\end{cases} \\end{align*}"}
{"id": "6355.png", "formula": "\\begin{align*} c = I _ 1 ( u ) \\leqslant \\liminf _ { n \\rightarrow \\infty } I _ 1 \\left ( \\dfrac { u _ n + u } { 2 } \\right ) . \\end{align*}"}
{"id": "3919.png", "formula": "\\begin{align*} \\tilde { \\Sigma } : = \\{ ( a , b ) \\in \\Sigma ( \\beta , \\mathcal { L } , a _ { } , a _ 0 , a _ 1 , b _ 0 , b _ 1 , \\tilde { C } , \\tilde { \\rho } ) : \\ , a , b \\mbox { a r e } \\mathcal { C } ^ 3 \\} . \\end{align*}"}
{"id": "7601.png", "formula": "\\begin{align*} Y = \\begin{pmatrix} y _ { 1 1 } & y _ { 1 2 } & y _ { 1 3 } \\\\ y _ { 2 1 } & y _ { 2 2 } & y _ { 2 3 } \\\\ y _ { 3 1 } & y _ { 3 2 } & y _ { 3 3 } \\end{pmatrix} = \\prod _ { i = 1 } ^ 8 \\exp ( \\lambda _ i M _ i ) \\in \\mathrm { S L } ( 3 , \\mathbb { R } ) , \\end{align*}"}
{"id": "7379.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\int _ { \\mathbb { R } ^ N } \\sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \\frac { \\partial U _ { h _ n , i } } { \\partial h _ n } \\ , \\phi _ n ~ d y \\\\ & = \\sum _ { k = 1 } ^ 4 \\int _ { { \\C } _ k } \\left ( \\sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \\frac { \\partial U _ { h _ n , i } } { \\partial h _ n } \\ , \\phi _ n \\right ) d y \\\\ & = \\sum _ { k = 1 } ^ 4 \\int _ { { \\C } _ k } \\left ( \\sum _ { i = 1 } ^ 4 f ( | y - h _ n t _ i | ) ( y - h _ n t _ i ) \\cdot t _ i \\phi _ n ( y ) \\right ) d y , \\end{aligned} \\end{align*}"}
{"id": "8729.png", "formula": "\\begin{align*} c _ 1 \\cos t _ 0 + c _ 2 \\sin t _ 0 + \\frac { \\mu a ^ 2 } { \\lambda } = 0 , \\end{align*}"}
{"id": "5475.png", "formula": "\\begin{align*} \\tilde { r } _ * e _ \\alpha = \\sum _ i \\tilde { r } ^ i _ \\alpha e _ i . \\end{align*}"}
{"id": "4449.png", "formula": "\\begin{align*} \\sigma ( i _ 1 \\cdots i _ { r - 1 } ) \\{ \\delta _ { i _ 1 i _ 2 } \\cdots \\delta _ { i _ { r - 1 } i _ r } \\} = \\sigma ( i _ 1 \\cdots i _ r ) \\{ \\delta _ { i _ 1 i _ 2 } \\cdots \\delta _ { i _ { r - 1 } i _ r } \\} . \\end{align*}"}
{"id": "2595.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ 3 \\partial _ k ^ 1 { \\partial _ k ^ 1 } ^ \\ast \\psi = \\sum _ { k = 1 } ^ 3 \\partial _ k ^ 2 { \\partial _ k ^ 2 } ^ \\ast \\psi = 0 \\ , \\mbox { a n d } \\end{align*}"}
{"id": "7570.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] = X _ 1 , [ X _ 1 , X _ 3 ] = 2 X _ 2 , [ X _ 2 , X _ 3 ] = X _ 3 , \\end{align*}"}
{"id": "8008.png", "formula": "\\begin{align*} I _ { D _ { Z , + } , d } = \\frac { \\prod _ { 0 < a < v _ 2 \\cdot d } ( v _ 2 + a z ) } { \\prod _ { i \\in I _ + , D _ i \\cdot d > 0 } ( \\bar D _ i + ( D _ i \\cdot d ) z ) } [ \\textbf { 1 } ] _ { ( - D _ i \\cdot d ) _ { i \\in I _ + } , v _ 2 \\cdot d } , \\end{align*}"}
{"id": "2565.png", "formula": "\\begin{align*} \\begin{aligned} A _ { j k } ^ \\dagger = A _ { k j } \\ & , \\ \\ A _ { 1 1 } + A _ { 2 2 } + A _ { 3 3 } = 0 \\ , \\\\ [ A _ { j k } , A _ { l m } ] & = \\delta _ { l , k } A _ { j m } - \\delta _ { j , m } A _ { l k } \\ . \\end{aligned} \\end{align*}"}
{"id": "1141.png", "formula": "\\begin{align*} \\sum _ { n , k , \\ell \\ge 0 } L _ 1 ( n , k , \\ell ) x ^ k y ^ { \\ell } q ^ n & = \\sum _ { m \\ge 0 } \\frac { ( - y q / x ) _ m x ^ m q ^ { \\binom { m + 1 } { 2 } } } { ( q ) _ m } , \\\\ \\sum _ { n , k , \\ell \\ge 0 } A ( n , k , \\ell ) x ^ k y ^ { \\ell } q ^ n & = \\sum _ { k , \\ell \\ge 0 } \\frac { x ^ k y ^ { \\ell } q ^ { \\binom { k + \\ell + 1 } { 2 } + \\binom { \\ell + 1 } { 2 } } } { ( q ) _ k ( q ) _ { \\ell } } . \\end{align*}"}
{"id": "6083.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 + a x ^ 3 + b x \\end{align*}"}
{"id": "1749.png", "formula": "\\begin{align*} & B _ { n + 1 } ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) = B _ { n } ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) \\cdot ( 1 - x y ^ { n } ) ^ { - 1 } , \\\\ & D _ { n + 1 } ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) = D _ { n } ( v , w , t , q ^ { \\frac { 1 } { 2 } } ) \\cdot \\prod _ { k = 0 } ^ { n - 1 } ( 1 - ( q ^ { \\frac { 1 } { 2 } } ) ^ { 1 - n + 2 k } x y ^ { n } ) ^ { - 1 } , \\\\ & x = \\exp ( - 2 \\pi i v / t ) , y = \\exp ( - 2 \\pi i w / t ) . \\end{align*}"}
{"id": "7040.png", "formula": "\\begin{align*} \\sum _ { M \\geq 1 } g ( M ) \\ , x ^ M & = \\frac { x - x ^ 3 } { ( 1 - x - x ^ 2 ) ^ 2 } , \\sum _ { M \\geq 1 } h ( M ) \\ , x ^ M = \\frac { x - x ^ 2 } { ( 1 - x - x ^ 2 ) ^ 2 } , \\\\ \\sum _ { M \\geq 1 } g _ 1 ( M ) \\ , x ^ M & = \\frac { x ^ 2 } { ( 1 - x - x ^ 2 ) ^ 2 } = \\sum _ { M \\geq 1 } h _ 1 ( M ) \\ , x ^ M . \\end{align*}"}
{"id": "2082.png", "formula": "\\begin{align*} U _ k & : = \\ell ( \\gamma _ 2 \\gamma _ 1 \\phi _ k \\gamma _ 1 \\gamma _ 2 ) \\end{align*}"}
{"id": "4976.png", "formula": "\\begin{align*} L _ 1 : = \\sup _ { n \\ge 1 } \\sup _ { s \\in [ 0 , T ] } E ( \\sigma ^ 2 ( X ^ n _ { \\eta _ { n } ( s ) } ) ) < \\infty . \\end{align*}"}
{"id": "8498.png", "formula": "\\begin{align*} \\varepsilon & = \\sqrt { \\frac { 2 b d k \\log ( \\frac { 4 b d k } { \\varepsilon } ) } { n } + \\frac { \\log ( \\frac { 3 } { \\delta } ) } { 2 n } } \\\\ & \\le \\sqrt { \\frac { 2 b d k \\log ( 4 b d k n ) } { n } + \\frac { \\log ( \\frac { 3 } { \\delta } ) } { 2 n } } \\\\ & \\leq \\sqrt { \\frac { 2 b d k \\log ( 4 b d k n ) } { n } } + \\sqrt { \\frac { \\log ( \\frac { 3 } { \\delta } ) } { 2 n } } . \\end{align*}"}
{"id": "3428.png", "formula": "\\begin{align*} F ( n ( z , x ) ) = \\begin{pmatrix} z & \\frac { x } { 2 } & 0 & 0 \\end{pmatrix} , F ( n ( z , x ) ^ t ) = \\begin{pmatrix} 0 & 0 & z & \\frac { x } { 2 } \\end{pmatrix} . \\end{align*}"}
{"id": "2416.png", "formula": "\\begin{align*} u = u _ R ~ \\textrm { o n } ~ \\Omega _ R a . s . , \\textrm { a n d } ~ \\lim _ { R \\rightarrow \\infty } \\mathbb P ( \\Omega _ R ) = 1 . \\end{align*}"}
{"id": "3674.png", "formula": "\\begin{align*} \\tilde { \\mathcal { H } } ^ { j ; a } _ { k , \\tilde { k } ; m , l } ( t _ 1 , t _ 2 ) = \\int _ 0 ^ { t _ a } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } E ( \\tau , X ( t _ a ) - y + ( t _ a - \\tau ) \\omega ) \\cdot \\nabla _ v \\big [ \\big ( \\tilde { \\mathcal { K } } ^ { 0 } _ { k , \\tilde { k } ; l } ( y , v , V ( t _ a ) ) + ( t _ a - \\tau ) \\tilde { \\mathcal { K } } ^ { 1 } _ { k , \\tilde { k } ; l } ( y , v , V ( t _ a ) ) \\big ) \\varphi _ { j } ( v ) \\big ] \\end{align*}"}
{"id": "7495.png", "formula": "\\begin{align*} N ( t , u , z ) = 1 + z ( N ( t , t , z ) - 1 + u ) N ( t , u , z ) , \\end{align*}"}
{"id": "5088.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , t ] } E [ | S ^ { n , M , 1 } _ \\tau | ^ 2 ] = 0 . \\end{align*}"}
{"id": "7659.png", "formula": "\\begin{align*} \\deg ( \\mathsf { W } _ { m , 4 } ' ) & = \\deg ( N _ { m + 3 } ) = \\binom { m + 5 } { 2 } \\\\ g ( \\mathsf { W } _ { m , 4 } ' ) & = ( m + 4 ) \\binom { m + 4 } { 3 } - ( m + 5 ) \\binom { m + 3 } { 3 } , \\end{align*}"}
{"id": "3158.png", "formula": "\\begin{align*} \\mathrm { P } _ { \\Phi _ { \\mathfrak { m } } ( c ) } = - \\inf f _ { \\Phi _ { \\mathfrak { m } } ( c ) } \\left ( E _ { 1 } \\right ) , c \\in L ^ { 2 } ( \\mathbb { S } ; \\mathbb { C } ; | \\mathfrak { \\mathfrak { a } } | ) , \\end{align*}"}
{"id": "7836.png", "formula": "\\begin{align*} \\langle \\Psi _ \\varphi ( \\xi ) \\hat y , \\hat x \\rangle = \\varphi ( x ^ * \\xi y ) . \\end{align*}"}
{"id": "2372.png", "formula": "\\begin{align*} b _ { i j } h _ \\theta ^ j = a _ { i 0 j } + a _ { i 1 j } Q _ \\theta + \\ldots + a _ { i s j } Q _ \\theta ^ s . \\end{align*}"}
{"id": "5080.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { \\tau } ( t - s ) ^ \\alpha \\sigma ' ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) d s = 0 , \\end{align*}"}
{"id": "2390.png", "formula": "\\begin{align*} \\left ( U ^ { - 1 } \\xi ( L ) \\right ) ( \\nu ) = \\frac { \\| ( 1 , 0 ) L ^ { - 1 } \\| \\ , \\xi \\big ( ( 1 , 0 ) L ^ { - 1 } , \\nu ^ { - 1 } \\big ) } { | \\nu | ^ { 1 / 2 } } \\nu \\in \\R ^ * . \\end{align*}"}
{"id": "4333.png", "formula": "\\begin{align*} & \\partial _ t \\mathcal { S } = - \\int _ { L _ t } ( \\theta _ t - \\hat { \\theta } ) ( e ^ { - i \\hat { \\theta } } \\Omega ) = - \\int _ { L _ t } ( \\theta _ t - \\hat { \\theta } ) ( e ^ { i ( \\theta - \\hat { \\theta } ) } ) d v o l _ { L _ t } \\\\ = & - \\int _ { L _ t } ( \\theta _ t - \\hat { \\theta } ) \\sin ( \\theta _ t - \\hat { \\theta } ) d v o l _ { L _ t } . \\end{align*}"}
{"id": "880.png", "formula": "\\begin{align*} \\mathcal { D } _ 1 ^ { n s } \\cong \\langle T _ { a } , T _ { b } ~ \\mid ~ T _ { a } * T _ { b } * T _ { a } = T _ { b } , T _ { b } * T _ { a } * T _ { b } = T _ { a } \\rangle . \\end{align*}"}
{"id": "6820.png", "formula": "\\begin{align*} y '' ( \\delta ) = y '' ( 1 ) - \\int _ \\delta ^ 1 y ''' ( x ) d x , \\end{align*}"}
{"id": "6511.png", "formula": "\\begin{align*} \\int _ { \\partial \\omega _ c } ( m \\cdot \\nu ) \\abs { \\nabla u } ^ 2 d x = \\int _ { \\Gamma _ 0 } ( m \\cdot \\nu ) \\left | \\frac { \\partial u } { \\partial \\nu } \\right | ^ 2 d \\Gamma \\ \\ \\ \\ \\Re \\left ( \\int _ { \\Gamma _ 0 } \\frac { \\partial u } { \\partial \\nu } \\left ( m \\cdot \\nabla \\bar { u } \\right ) d \\Gamma \\right ) = \\int _ { \\Gamma _ 0 } ( m \\cdot \\nu ) \\left | \\frac { \\partial u } { \\partial \\nu } \\right | ^ 2 d \\Gamma . \\end{align*}"}
{"id": "3106.png", "formula": "\\begin{align*} a _ { \\mathrm { p w } } ( \\phi , v _ { \\mathrm { n c } } ) = \\mu ( \\phi , v _ { \\mathrm { n c } } ) _ { 1 + \\delta } \\quad v _ { \\mathrm { n c } } \\in V ( { \\mathcal { T } } ) . \\end{align*}"}
{"id": "2060.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : a b s t r a c t p d e } \\partial _ t f _ t = Q ( f _ t ) , \\end{align*}"}
{"id": "6798.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } + \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\Biggl ( a \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } \\Biggr ) = f . \\end{align*}"}
{"id": "6810.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } ( a u '' ) '' v \\ , d x = [ ( a u '' ) ' v ] ^ { x = 1 } _ { x = 0 } - \\int _ { 0 } ^ { 1 } ( a u '' ) ' v ' d x . \\end{align*}"}
{"id": "3235.png", "formula": "\\begin{align*} ( 4 m - 1 ) ( g - 1 ) + ( 2 m - 1 ) n & = h ^ 0 ( K _ { X } ^ { 2 m } ( D ^ { 2 m - 1 } ) ) \\\\ & = h ^ 0 ( K _ { X ' } ^ { 2 m } ( ( D ' ) ^ { 2 m - 1 } ) ) \\\\ & = ( 4 m - 1 ) ( g ' - 1 ) + ( 2 m - 1 ) n ' \\end{align*}"}
{"id": "7305.png", "formula": "\\begin{align*} L _ { \\xi } v = \\Delta _ \\xi v - q { \\sf U } ^ { q - 1 } ( 1 - \\chi _ 1 ) v - \\eta \\dot \\eta ( \\Lambda _ \\xi v ) , \\Lambda _ \\xi = \\tfrac { 2 } { 1 - q } - \\xi \\cdot \\nabla _ \\xi . \\end{align*}"}
{"id": "6648.png", "formula": "\\begin{align*} v _ { i , t } & = \\rho u _ { i , t } + \\lambda > \\lambda \\ ; , \\ , \\end{align*}"}
{"id": "5046.png", "formula": "\\begin{align*} N ^ n _ \\tau : = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s \\left ( \\sigma ' ( X _ s ) \\right ) ^ 2 \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) ^ 2 \\ , d s . \\end{align*}"}
{"id": "2102.png", "formula": "\\begin{align*} ( L ( x ) , x ) _ V = - \\sum _ { e \\in E } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum _ { \\{ u , v \\} \\subset e } ( x ( u ) - x ( v ) ) ^ 2 . \\end{align*}"}
{"id": "7317.png", "formula": "\\begin{align*} | w _ 2 | < \\bar { w } _ 2 = ( T - t ) ^ { \\frac { \\gamma } { 2 } + J + \\frac { 5 } { 4 } { \\sf d } _ 1 } | z | ^ { \\gamma + 2 J + 3 { \\sf d } _ 1 } | z | > { \\sf m } _ 3 , \\ t \\in ( 0 , T ) . \\end{align*}"}
{"id": "789.png", "formula": "\\begin{align*} & \\beta ( 1 + D r ^ m ) ( 1 + E r ^ m ) ^ { \\frac { D - 2 E } { E } } + ( 1 - \\beta ) r ^ m ( 1 + E r ^ m ) ^ { \\frac { D - E } { E } } \\\\ & + \\sum _ { n = N } ^ { \\infty } \\prod _ { k = 0 } ^ { N - 2 } \\frac { | E - D + E k | } { k + 1 } r ^ n = ( 1 - E ) ^ { \\frac { D - E } { E } } , \\end{align*}"}
{"id": "1227.png", "formula": "\\begin{align*} \\mathcal { F } ( w , x ) & = \\dfrac { \\mathcal { F } _ 1 ( w , x ) } { \\mathcal { F } _ 2 ( w , x ) } , \\end{align*}"}
{"id": "4630.png", "formula": "\\begin{align*} \\epsilon _ h = s i g n ( b _ h \\tilde c _ 1 ( \\alpha , h ) , \\tilde c _ 2 ( \\alpha , h ) ) \\end{align*}"}
{"id": "399.png", "formula": "\\begin{align*} k _ 1 ( x , t ) - k _ 2 ( x , t ) = U ( \\phi _ t x ) - U ( x ) . \\end{align*}"}
{"id": "8578.png", "formula": "\\begin{align*} m = n _ { j _ 1 } + \\ldots + n _ { j _ s } , \\end{align*}"}
{"id": "6854.png", "formula": "\\begin{align*} V _ F N ( b , c ) = \\int _ { \\mathbb { R } } g _ + \\frac { 1 } { \\sqrt { 2 \\pi c } } e ^ { - \\frac { ( g - b ) ^ 2 } { 2 c } } d g = \\int _ { \\mathbb { R } } ( g + b ) _ + \\frac { 1 } { \\sqrt { 2 \\pi c } } e ^ { - \\frac { g ^ 2 } { 2 c } } = \\int _ { \\mathbb { R } } ( \\sqrt { c } g + b ) _ + \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - \\frac { g ^ 2 } { 2 } } d g . \\end{align*}"}
{"id": "367.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ \\infty [ p ^ \\# _ m ( o , o ) - \\pi ( o ) ] & \\ge \\sum _ { m = 0 } ^ { \\delta ^ 2 } [ p ^ \\# _ m ( o , o ) - \\frac 1 n ] \\ge \\left ( \\sum _ { m = 0 } ^ { \\delta ^ 2 } p ^ \\# _ m ( o , o ) \\right ) - \\frac 1 { f ( n ) } , \\end{align*}"}
{"id": "2571.png", "formula": "\\begin{align*} \\mathbf S _ { 1 2 3 } : = \\dfrac { 1 } { 3 } ( \\mathbf S _ { 1 2 } + \\mathbf S _ { 2 3 } + \\mathbf S _ { 3 1 } ) = \\mathbf S _ { 1 2 } - \\dfrac { 1 } { 3 } C _ 3 ^ { ( 1 ) } + \\dfrac { 1 } { 3 } C _ 3 ^ { ( 2 ) } \\ . \\end{align*}"}
{"id": "5809.png", "formula": "\\begin{align*} n = x ^ 2 + 1 6 y ^ 2 + 4 8 z ^ 2 . \\end{align*}"}
{"id": "8271.png", "formula": "\\begin{align*} \\psi _ { A _ { 0 } } = A _ { 0 } \\left [ 1 - \\frac { \\cosh ( k ' x ) } { \\cosh ( k ' a / 2 ) } \\right ] \\end{align*}"}
{"id": "6439.png", "formula": "\\begin{align*} \\ell ' _ 2 ( \\ell _ 2 ' ( x , y ) , z ) _ { | _ m } + \\circlearrowleft ( x , y , z ) = 0 \\ , & \\Longrightarrow \\ , \\ell ' _ 2 ( [ x , y ] _ \\mathfrak { g } , z ) _ { | _ m } + \\ell ' _ 2 ( \\eta ( x , y ) , z ) _ { | _ m } + \\circlearrowleft ( x , y , z ) = 0 , \\\\ & \\Longrightarrow \\ , \\nu ( z ) \\left ( \\eta ( x , y ) _ { | _ m } \\right ) - \\eta ( [ x , y ] _ \\mathfrak { g } , z ) _ { | _ m } + \\circlearrowleft ( x , y , z ) = 0 . \\end{align*}"}
{"id": "80.png", "formula": "\\begin{align*} \\mathcal U _ n : = \\Big \\{ ( X , u ) \\in \\mathfrak X _ 1 \\times \\mathcal C \\ \\Big | \\ \\big ( u ( \\cdot ) , \\ , ( F ) \\setminus \\{ x , y \\} \\big ) > \\frac 1 n \\Big \\} , \\forall n \\in \\N . \\end{align*}"}
{"id": "305.png", "formula": "\\begin{align*} \\sum _ { i \\le k } c _ i d _ i = \\sum _ { i \\le k } c _ i \\Big ( \\sum _ { j \\le i } d _ j - \\sum _ { j \\le i - 1 } d _ j \\Big ) = \\sum _ { i \\le k - 1 } ( c _ i - c _ { i + 1 } ) \\sum _ { j \\le i } d _ j \\ + \\ c _ k \\sum _ { i \\le k } d _ i . \\end{align*}"}
{"id": "7322.png", "formula": "\\begin{align*} N _ 6 [ w ] & = \\{ \\dot { \\sf M } + f ( u ) - f _ 2 ( u ) \\} ( 1 - \\chi _ 4 ) \\\\ & = \\underbrace { \\{ \\dot { \\sf M } + f ( u ) - f _ 2 ( u ) \\} ( 1 - \\chi _ { 4 , { \\sf a } } ) } _ { = { \\sf t } _ 5 } + \\underbrace { \\{ \\dot { \\sf M } + f ( u ) - f _ 2 ( u ) \\} ( \\chi _ { 4 , { \\sf a } } - \\chi _ 4 ) } _ { = { \\sf t } _ 6 } . \\end{align*}"}
{"id": "5781.png", "formula": "\\begin{align*} G / Z ( G ) = \\dfrac { G / Z ( G ) ^ 0 } { Z ( G ) / Z ( G ) ^ 0 } . \\end{align*}"}
{"id": "8513.png", "formula": "\\begin{align*} \\dim ( C ^ \\perp \\cap D ^ \\perp ) = n \\omega - \\dim ( C ) - \\dim ( D ) + \\dim ( C \\cap D ) . \\end{align*}"}
{"id": "6235.png", "formula": "\\begin{align*} & \\int \\frac { x ^ { n - 2 } } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\widetilde { h } _ n ( x ; q ) d _ q x = \\frac { x ^ n } { [ n - 1 ] _ q ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\frac { \\widetilde { h } _ n ( x ; q ) } { x } - \\widetilde { h } _ { n - 1 } ( x ; q ) \\right ) . \\end{align*}"}
{"id": "6665.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t - \\Delta u = v ^ { p } , \\ & t > 0 , x \\in \\R ^ N \\\\ v _ t - \\Delta v = u ^ { q } , & t > 0 , x \\in \\R ^ N \\end{array} \\right . \\end{align*}"}
{"id": "2406.png", "formula": "\\begin{align*} \\| F _ n ( \\mathbb U ( t ) ) - F _ n ( \\tilde U ( t ) ) \\| ^ 2 & \\le C \\sum _ { k = 1 } ^ { n - 1 } \\big ( 1 + \\mathbb U _ k ( t ) ^ 2 + \\tilde U _ k ( t ) ^ 2 \\big ) ^ 2 | \\mathbb U _ k ( t ) - \\tilde U _ k ( t ) | ^ 2 \\\\ & \\le C n \\big ( 1 + \\| \\mathbb U ( t ) \\| _ { l ^ { 4 q ^ \\prime } _ n } ^ 4 + \\| \\tilde U ( t ) \\| _ { l _ n ^ { 4 q ^ \\prime } } ^ { 4 } \\big ) \\| \\mathbb U ( t ) - \\tilde U ( t ) \\| ^ 2 _ { l _ n ^ { 2 q } } , \\end{align*}"}
{"id": "8315.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\frac { c _ m ( n ) } { m } = 0 . \\end{align*}"}
{"id": "3708.png", "formula": "\\begin{align*} \\| h \\partial _ { x _ 2 } \\phi _ h ( x _ 1 , x _ 2 ) \\| _ { L ^ 2 ( \\Gamma ) } = 0 \\end{align*}"}
{"id": "3799.png", "formula": "\\begin{align*} \\Phi _ 3 ( \\xi , \\eta , \\sigma , \\zeta ) : = \\hat { \\zeta } \\cdot ( \\xi + \\eta + \\sigma ) + \\mu _ 2 | \\sigma | + \\mu | \\xi | + \\mu _ 1 | \\eta | . \\end{align*}"}
{"id": "7240.png", "formula": "\\begin{align*} \\dot { x } ( t , s ) & = - \\partial _ s x ( t , s ) + f ( s ) x ( t , s ) , s \\in [ 0 , 1 ] , \\\\ x ( t , 0 ) & = \\int _ 0 ^ 1 h ( s ) x ( t , s ) d s , \\end{align*}"}
{"id": "947.png", "formula": "\\begin{align*} b ( \\rho ) = \\begin{cases} \\log ( \\overline { \\rho } - \\rho ) & \\mbox { i f } \\rho \\in [ 0 , \\overline { \\rho } - \\delta ) , \\\\ \\log \\delta & \\mbox { i f } \\rho \\geq \\overline { \\rho } - \\delta , \\end{cases} \\end{align*}"}
{"id": "8573.png", "formula": "\\begin{align*} m = \\varepsilon _ 1 n _ { k _ 1 } + \\varepsilon _ 2 n _ { k _ 2 } + \\ldots + \\varepsilon _ { l } n _ { k _ l } , \\end{align*}"}
{"id": "5118.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t - \\delta } \\left [ ( t - \\eta _ n ( s ) ) ^ { \\alpha } - ( t - s ) ^ { \\alpha } \\right ] ^ 2 \\ , d s & \\leq \\int _ { 0 } ^ { t - \\delta } ( t - s ) ^ { 2 \\alpha - 2 } ( s - \\eta _ n ( s ) ) ^ 2 d s \\\\ & \\leq \\frac { 1 } { n ^ 2 } \\int _ { 0 } ^ { t - \\delta } ( t - s ) ^ { 2 \\alpha - 2 } d s \\\\ & = \\frac 1 { n ^ 2 } \\int _ \\delta ^ t s ^ { 2 \\alpha - 2 } d s \\\\ & \\leq \\frac { 2 \\delta ^ { 2 \\alpha - 1 } } { n ^ 2 ( 1 - 2 \\alpha ) } = C _ 2 { n ^ { - 2 } } \\delta ^ { 2 \\alpha - 1 } . \\end{align*}"}
{"id": "7111.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\tau } p ( \\tau ; \\cdot ) = f ( \\tau , p ( \\tau ; \\cdot ) ) + g ( \\tau , p ( \\tau ; \\cdot ) ) \\mu ( \\tau , p ( \\tau ; \\cdot ) ) \\end{align*}"}
{"id": "5108.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E [ \\Lambda ^ { ( 1 ) } _ { n , \\delta } ] = 0 . \\end{align*}"}
{"id": "3789.png", "formula": "\\begin{align*} H _ { k , j ; n , l , r ; p , q ; \\star } ^ { \\mu , m , i ; l i n } ( t , x , \\zeta ) : = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big [ ( t - s ) \\mathcal { K } _ { k , n , l , r } ^ { \\star ; \\mu , m } ( y , v , \\omega , \\zeta ) + \\mathcal { K } _ { k , n , l , r } ^ { \\star ; e r r , \\mu , m } ( y , v , \\omega , \\zeta ) \\big ] f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "900.png", "formula": "\\begin{align*} K _ n ( z , w ) = \\sum \\ , \\frac { z ^ k } { \\norm { z ^ k } } \\cdot \\frac { \\overline { w } ^ k } { \\norm { \\overline { w } ^ k } } = \\frac { n ! } { \\pi ^ n } \\cdot \\frac { 1 } { ( 1 - z \\cdot \\overline { w } ) ^ { n + 1 } } \\end{align*}"}
{"id": "6367.png", "formula": "\\begin{align*} F _ { \\ , \\kappa } \\ , ( x ) = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\ , \\ , { \\rm e r f } _ { \\kappa } \\ ! \\left ( \\sqrt { \\beta } \\ , x \\right ) \\ \\ , \\end{align*}"}
{"id": "5471.png", "formula": "\\begin{align*} \\nabla _ z g _ { r , x } & = \\nabla _ { \\chi _ r ( d ( z ) ) \\Phi _ x ( g ( z ) ) } \\Phi _ x ^ { - 1 } \\left ( \\chi _ r ' ( d ( z ) ) \\left ( \\nabla _ z d \\right ) \\Phi _ x ( g ( z ) ) + \\chi _ r ( d ( z ) ) \\nabla _ { g ( z ) } \\Phi _ x \\nabla _ z g \\right ) , \\end{align*}"}
{"id": "4710.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty p ( N , M , n ) q ^ n = \\frac { ( q ; q ) _ { N + M } } { ( q ; q ) _ N ( q ; q ) _ M } . \\end{align*}"}
{"id": "1018.png", "formula": "\\begin{align*} \\mathsf { F r } _ x ( a ) = \\mu ( [ a ] ) . \\end{align*}"}
{"id": "783.png", "formula": "\\begin{align*} f _ 0 ( z ) = \\left \\{ \\begin{array} { l r } z ( 1 + E z ) ^ { \\frac { D - E } { E } } , & E \\neq 0 ; \\\\ z e ^ { D z } , & E = 0 . \\end{array} \\right . \\end{align*}"}
{"id": "7259.png", "formula": "\\begin{align*} u _ t = \\Delta u + | u | ^ { p - 1 } u \\end{align*}"}
{"id": "4309.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { S } ( L _ t ) = \\int _ { L _ t } h _ t ( e ^ { - i \\hat { \\theta } } \\Omega ) \\end{align*}"}
{"id": "288.png", "formula": "\\begin{align*} \\sum _ p p ^ { - t } = 1 + \\Big ( 1 - \\sum _ p p ^ { - 2 t } \\Big ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "4364.png", "formula": "\\begin{align*} f = r _ s + q _ s g ^ s , r _ s = \\sum \\nolimits _ { 0 \\le t < s } a _ s g ^ t . \\end{align*}"}
{"id": "6179.png", "formula": "\\begin{align*} \\Omega \\wedge \\eta _ { 0 } = ( \\sqrt { - 1 } ) ^ { n } F ( z _ { 1 } , . . . , z _ { n } ) d z _ { 1 } \\wedge d \\overline { z } _ { 1 } \\wedge . . . \\wedge d z _ { n } \\wedge d \\overline { z } _ { n } \\wedge d x \\end{align*}"}
{"id": "1451.png", "formula": "\\begin{align*} \\Delta ( z ) = ( - 1 ) ^ { r m } \\left ( \\sum _ { \\ell = 0 } ^ { r m } P _ { \\ell } ( z ) \\Delta _ { 1 , \\ell + 1 } ( z ) \\right ) \\enspace . \\end{align*}"}
{"id": "8076.png", "formula": "\\begin{align*} { \\mathfrak m } { \\mathfrak c } _ { Z , Y } : = - | [ Z , Y ] | ^ { 2 } - \\langle Z , [ [ Z , Y ] , Y ] \\rangle \\end{align*}"}
{"id": "1220.png", "formula": "\\begin{align*} Q ( T , x ) & = \\prod _ { v \\in V ( T ) } \\mathcal { H } ( v , x ) . \\end{align*}"}
{"id": "6005.png", "formula": "\\begin{align*} f _ 1 ( x ) = x ^ 4 + x ^ 3 \\end{align*}"}
{"id": "3938.png", "formula": "\\begin{align*} V a r ( \\hat { \\pi } ^ a _ { h ^ * , T _ n } ( x ) ) & = \\frac { 2 } { T _ n ^ 2 } \\int _ 0 ^ { T _ n } \\int _ 0 ^ { t } k ( t , s ) 1 _ { s < t } \\ , \\big ( 1 _ { | t - s | \\le h _ 1 ^ * h _ 2 ^ * } \\\\ & + 1 _ { h _ 1 ^ * h _ 2 ^ * \\le | t - s | \\le ( \\prod _ { j \\ge 3 } h _ j ^ * ) ^ { \\frac { 2 } { d - 2 } } } + 1 _ { ( \\prod _ { j \\ge 3 } h _ j ^ * ) ^ { \\frac { 2 } { d - 2 } } \\le | t - s | \\le D } + 1 _ { D \\le | t - s | \\le T _ n } \\big ) d s d t \\\\ & = \\sum _ { j = 1 } ^ 4 \\tilde { I } _ j , \\end{align*}"}
{"id": "7031.png", "formula": "\\begin{align*} ( e ^ { t B } - e ^ { t A } ) f _ n ( 1 ) & = \\frac { 1 } { 3 n ^ 2 } ( 1 - e ^ { - t } ) - \\frac { e ^ { - t / 2 } } { 2 n } ( 1 - e ^ { - t } ) \\\\ & = \\left ( \\frac { 1 } { 3 n ^ 2 } - \\frac { e ^ { - t / 2 } } { 2 n } \\right ) ( 1 - e ^ { - t } ) \\end{align*}"}
{"id": "8128.png", "formula": "\\begin{align*} V ( x ) = O ( | x | ^ { - ( 1 + \\epsilon ) } ) x \\rightarrow \\infty \\end{align*}"}
{"id": "3386.png", "formula": "\\begin{align*} S _ { k } : = \\inf \\{ t \\geq T _ { k - 1 } : D _ t \\leq ( h ' ) ^ 2 \\} , T _ { k } : = \\inf \\{ t \\geq S _ k + 1 + C r ^ 2 : D _ t \\geq h ^ 2 \\} . \\end{align*}"}
{"id": "8024.png", "formula": "\\begin{align*} \\tilde { \\omega } _ + = ( \\omega _ + , \\epsilon ) , \\tilde { \\omega } _ - = ( \\omega _ - , \\epsilon ) , \\end{align*}"}
{"id": "3334.png", "formula": "\\begin{align*} q _ { \\gamma _ 1 , \\gamma _ 2 } = \\Omega ( e , v _ 1 + v _ 2 ) \\end{align*}"}
{"id": "2999.png", "formula": "\\begin{align*} Q = \\sum _ { \\substack { a , b \\geq 0 \\\\ a + b \\geq s } } c _ { a , b } \\phi ^ a ( z - r ) ^ b \\end{align*}"}
{"id": "1607.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } = \\frac { f ' ( x ^ 1 ) } { \\sqrt { 1 + f '^ 2 ( x ^ 1 ) } } \\delta _ { \\epsilon 1 } \\left [ f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \\right ] , \\end{align*}"}
{"id": "2908.png", "formula": "\\begin{align*} F ( A x + e ) = B G ( x ) + d . \\end{align*}"}
{"id": "3809.png", "formula": "\\begin{align*} \\sum _ { i = 0 , 1 , 2 } | \\nabla _ v \\mathcal { K } ^ { \\mu , i } _ { k , j ; n , l } ( y , v , V ( s ) ) | + \\sum _ { a = 3 , 4 } 2 ^ { - \\max \\{ n , ( \\gamma _ 1 - \\gamma _ 2 ) M _ { t ^ \\star } \\} - \\epsilon M _ t } | \\nabla _ v \\mathcal { K } ^ { \\mu , a } _ { k , j ; n , l } ( y , v , V ( s ) ) | \\end{align*}"}
{"id": "6438.png", "formula": "\\begin{align*} \\begin{array} { c } \\cdots \\stackrel { \\dd = \\ell _ 1 } { \\longrightarrow } E _ { - 3 } \\stackrel { \\dd = \\ell _ 1 } { \\longrightarrow } E _ { - 2 } \\stackrel { \\dd = \\ell _ 1 } { \\longrightarrow } \\mathfrak g [ 1 ] \\oplus E _ { - 1 } \\stackrel { { \\rho ' } } { \\longrightarrow } T M , \\end{array} \\end{align*}"}
{"id": "4881.png", "formula": "\\begin{align*} n = p _ { 1 } ^ { 3 } + \\cdots + p _ { 4 } ^ { 3 } - p _ { 5 } ^ { 3 } - \\cdots - p _ { 8 } ^ { 3 } , 0 \\leq | n | \\leq N _ { i } , \\end{align*}"}
{"id": "5889.png", "formula": "\\begin{align*} I ( \\nu ) = - \\frac { 1 } { 2 } \\log \\mathrm { d e t } ( I _ k - A A ^ T ) \\end{align*}"}
{"id": "4015.png", "formula": "\\begin{align*} \\ddot { \\vec { r } } = - \\frac { \\mu } { r ^ 3 } \\vec { r } + \\vec { a } _ O , \\end{align*}"}
{"id": "3467.png", "formula": "\\begin{align*} \\mathrm { E } [ X ^ { n } | x _ { p } < X < x _ { q } ] = \\frac { \\int _ { x _ { p } } ^ { x _ { q } } x ^ { n } \\frac { c _ { 1 } } { \\sigma } g _ { 1 } \\left ( \\frac { 1 } { 2 } \\left ( \\frac { x - \\mu } { \\sigma } \\right ) ^ { 2 } \\right ) \\mathrm { d } x } { F _ { X } ( x _ { p } , x _ { q } ) } . \\end{align*}"}
{"id": "2528.png", "formula": "\\begin{align*} \\begin{array} { l l } \\sum \\limits _ { i = n + 1 } ^ { m } g ( \\nabla _ { U _ i } U _ i , U _ i ) + \\sum \\limits _ { j = 1 } ^ { n } g ( \\nabla _ { X _ j } X _ j , X _ j ) \\\\ + s ^ { K e r F _ \\ast } + \\frac { 1 } { \\lambda ^ 2 } s ^ { N } + \\mu ( m - n + n ) = 0 , \\end{array} \\end{align*}"}
{"id": "8970.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { L ^ 2 _ \\delta ( \\Omega ) } = \\int _ \\Omega u ^ 2 \\rho ^ { - 2 \\delta - n } d V _ { g } \\leq \\left [ \\int _ { \\Omega } | u | ^ q d V _ { g } \\right ] ^ { \\frac { 2 } { q } } \\left [ \\int _ { \\Omega } \\rho ^ { - \\frac { 2 \\delta q } { q - 2 } - \\frac { n q } { q - 2 } } d V _ { g } \\right ] ^ { \\frac { q - 2 } { q } } \\end{align*}"}
{"id": "8656.png", "formula": "\\begin{align*} \\sup \\left \\{ \\int _ 0 ^ \\infty f g \\ , d r : \\ g \\geq 0 \\ , , \\ \\overline \\nu _ \\alpha ( g ) \\leq 1 \\right \\} = \\alpha \\ , \\underline \\mu _ \\alpha ( f ) \\end{align*}"}
{"id": "1241.png", "formula": "\\begin{align*} Q ( T , x ) & = \\prod _ { j = 1 } ^ { 1 3 } \\mathcal { H } ( u _ j , x ) \\\\ & = ( x - 1 ) ^ 9 \\ \\dfrac { x ^ 2 - 4 x + 1 } { x - 1 } \\ \\dfrac { x ^ 4 - 9 x ^ 3 + 2 2 x ^ 2 - 1 1 x + 1 } { ( x - 1 ) ( x ^ 2 - 4 x + 1 ) } \\ \\dfrac { x ^ 2 - 6 x + 1 } { x - 1 } \\\\ & \\qquad \\cdot \\dfrac { x ^ 8 - 1 9 x ^ 7 + 1 3 7 x ^ 6 - 4 6 7 x ^ 5 + 7 6 3 x ^ 4 - 5 4 1 x ^ 3 + 1 5 5 x ^ 2 - 1 3 x } { ( x - 1 ) ( x ^ 2 - 6 x + 1 ) ( x ^ 4 - 9 x ^ 3 + 2 2 x ^ 2 - 1 1 x + 1 ) } \\\\ & = x ( x - 1 ) ^ 5 ( x ^ 7 - 1 9 x ^ 6 + 1 3 7 x ^ 5 - 4 6 7 x ^ 4 + 7 6 3 x ^ 3 - 5 4 1 x ^ 2 + 1 5 5 x - 1 3 ) . \\end{align*}"}
{"id": "4162.png", "formula": "\\begin{align*} L _ 0 ^ \\lambda \\Phi _ { \\nu , \\nu ' } ^ \\lambda = ( 2 | \\nu | _ 1 + n ) \\lambda \\Phi _ { \\nu , \\nu ' } ^ \\lambda , \\end{align*}"}
{"id": "5552.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\rightarrow \\infty } \\left . \\sum \\limits _ { i = 0 } ^ { b r } \\dbinom { a r } { i } \\middle / r \\dbinom { a r } { b r } \\right . \\leq \\lim \\limits _ { r \\rightarrow \\infty } \\dfrac { a r - b r + 1 } { r ( a r - 2 b r + 1 ) } = 0 . \\end{align*}"}
{"id": "5504.png", "formula": "\\begin{align*} \\theta _ { t + 1 } = \\theta _ t - \\beta _ t \\nabla V ( \\theta _ t ) - \\beta _ t d _ t - \\beta _ t w _ { \\rm a } ( t ) , \\end{align*}"}
{"id": "7793.png", "formula": "\\begin{align*} \\Psi ( \\lambda _ 1 \\cdots \\lambda _ k ) = e _ { \\lambda _ 1 } \\cdots e _ { \\lambda _ k } ( \\alpha ^ n / n ! ) . \\end{align*}"}
{"id": "8574.png", "formula": "\\begin{align*} x ^ { l - 1 } = x ^ { l - 2 } + \\ldots + 1 , x > 1 , \\end{align*}"}
{"id": "3855.png", "formula": "\\begin{align*} \\mbox { F o r } x _ 0 \\notin N , \\tilde { N } = N \\cup \\{ x _ 0 \\} , \\quad & \\tilde { h _ 0 } = h _ 0 \\cup \\{ x _ 0 \\} , \\\\ \\mbox { F o r } \\{ x _ 1 , . . . , x _ n \\} \\subseteq N \\mbox { s u c h t h a t } \\quad & \\forall j = 1 , . . . , n \\mbox { a n d } \\forall y \\in N , x _ j \\leq y . \\quad \\mbox { T h e n } \\\\ \\tilde { E } = E \\cup & \\{ ( x _ 1 , x _ 0 ) , . . . , ( x _ n , x _ 0 ) \\} . \\end{align*}"}
{"id": "6551.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\prod _ { i = 0 } ^ m ( 1 - r _ { m - i } ^ i ) & = \\sum _ { m = 1 } ^ { m _ 0 - 1 } \\prod _ { i = 0 } ^ m \\underbrace { ( 1 - r _ { m - i } ^ i ) } _ { \\leq 1 } + \\sum _ { m = m _ 0 } ^ { \\infty } \\prod _ { i = 0 } ^ m ( 1 - r _ { m - i } ^ i ) \\\\ & \\leq m _ 0 - 1 + \\sum _ { m = m _ 0 } ^ \\infty \\prod _ { i = i _ 0 } ^ m ( 1 - r _ { m - i } ^ i ) \\end{align*}"}
{"id": "5710.png", "formula": "\\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \\ldots , y _ i ) = 0 ( 1 \\le k \\le \\min \\{ i , n - i \\} ) . \\end{align*}"}
{"id": "6919.png", "formula": "\\begin{align*} f - q | _ g = & \\ - L ^ 2 ( x ) p | _ { L ^ 2 ( y z ) } - x L ^ 2 ( p | _ { L ^ 2 ( y z ) } ) + L ^ 2 ( x p | _ { L ^ 2 ( y ) z } ) + L ^ 2 ( x p | _ { y L ^ 2 ( z ) } ) \\\\ \\equiv & \\ - L ^ 2 ( x ) p | _ { L ^ 2 ( y ) z } - L ^ 2 ( x ) p | _ { y L ^ 2 ( z ) } - x L ^ 2 ( p | _ { L ^ 2 ( y ) z } ) - x L ^ 2 ( p | _ { y L ^ 2 ( z ) } ) \\\\ & \\ + L ^ 2 ( x ) p | _ { L ^ 2 ( y ) z } + x L ^ 2 ( p | _ { L ^ 2 ( y ) z } ) + L ^ 2 ( x ) p | _ { y L ^ 2 ( z ) } + x L ^ 2 ( p | _ { y L ^ 2 ( z ) } ) \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "258.png", "formula": "\\begin{align*} \\Gamma ^ { k ( s ) } _ { i j } - \\Gamma ^ { k ( s ) } _ { j i } = L ^ { k ( s ) } _ { i j } ( \\Lambda ^ { ( s ) } ) . \\end{align*}"}
{"id": "3196.png", "formula": "\\begin{align*} D \\cdot h _ s ^ + = 0 , D \\cdot h _ s ^ - = 2 , D \\cdot l _ s = 1 . \\end{align*}"}
{"id": "1731.png", "formula": "\\begin{align*} F ( z + \\omega _ 2 \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) \\cdot F ( z \\ , | \\ , \\overline \\omega _ 1 , - \\omega _ 2 ) = \\prod _ { k \\geq 0 } \\big ( 1 - x _ 2 ( q _ 2 \\widetilde { q } _ 2 ) ^ { k / 2 } \\big ) \\cdot \\prod _ { k \\geq 1 } \\big ( 1 - x _ 2 ^ { - 1 } ( q _ 2 \\widetilde { q } _ 2 ) ^ { k / 2 } \\big ) ^ { - 1 } , \\end{align*}"}
{"id": "2168.png", "formula": "\\begin{align*} I _ { 3 } = \\int _ { \\mathbb { R } } \\frac { 1 + s ^ { 2 } } { s ^ { 2 } } \\ , d \\sigma _ { \\mu } ( s ) < + \\infty , \\end{align*}"}
{"id": "6002.png", "formula": "\\begin{align*} u ( t ) = \\frac 1 { 2 \\pi i } \\int _ { \\sigma \\pm i \\infty } K ( s ) \\int _ 0 ^ t e ^ { s ( t - \\tau ) } g ( t ) d s = \\frac 1 { 2 \\pi i } \\int _ { \\sigma \\pm i \\infty } K ( s ) y ( t ; s ) d s , \\end{align*}"}
{"id": "5184.png", "formula": "\\begin{align*} \\zeta \\cdot ( X , T ) = ( e ^ { \\frac { 2 \\pi i } { d } } X , e ^ { m \\frac { 2 \\pi i } { d } } T ) . \\end{align*}"}
{"id": "524.png", "formula": "\\begin{align*} L _ { n s } ( \\dot { x } , x , t ) = \\frac { 1 } { C _ 1 f ^ 2 ( t ) [ f ( t ) \\dot { x } - a _ o x + C _ 2 ] } \\ , \\end{align*}"}
{"id": "1789.png", "formula": "\\begin{align*} ( u _ i + a _ i f _ { q ( i ) } ) ( w _ { n , i } ^ * ) & = ( u _ i + a _ i f _ { q ( i ) } ) ( z _ n ^ * ) = ( u _ i + a _ i f _ { q ( i ) } ) ( y _ n ^ * ) - a _ i f _ { q ( i ) } ( y _ n ^ * ) \\\\ & = y _ n ^ * ( u _ i ) = 0 , \\end{align*}"}
{"id": "1327.png", "formula": "\\begin{align*} f ( t , i , v ) = \\tilde R _ i M _ i ( v ) , M _ i = \\frac { 1 } { T _ i } e ^ { - \\frac { v ^ 2 } { 2 T _ i } } \\end{align*}"}
{"id": "6039.png", "formula": "\\begin{align*} f _ { \\sigma } ^ { ( j ) } ( \\kappa _ i ) & = f _ { \\sigma } ^ { ( j ) } ( \\sigma ( \\lambda _ i ) ) = \\sum _ { k = j } ^ n ( k ) _ j \\sigma ( c _ k ) \\sigma ( \\lambda _ i ) ^ { k - j } \\\\ & = \\sigma \\left ( \\sum _ { k = j } ^ n ( k ) _ j c _ k \\lambda _ i ^ { k - j } \\right ) = \\sigma \\left ( f ^ { ( j ) } ( \\lambda _ i ) \\right ) . \\end{align*}"}
{"id": "8553.png", "formula": "\\begin{align*} \\int f \\ d \\xi = \\int ( f ( x ) - f ( y ) ) \\ a ( d x , d y ) , \\end{align*}"}
{"id": "442.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( z \\Sigma _ { \\mu } ( z ) ) = z , z \\in V . \\end{align*}"}
{"id": "4736.png", "formula": "\\begin{align*} T _ { i - 1 } & T _ { i + 1 } ( E _ { i } ) \\\\ & = E _ { i + 1 } E _ { i - 1 } E _ { i } - q ^ { - 1 } E _ { i - 1 } E _ { i } E _ { i + 1 } - q ^ { - 1 } E _ { i + 1 } E _ { i } E _ { i - 1 } + q ^ { - 2 } E _ { i } E _ { i - 1 } E _ { i + 1 } . \\end{align*}"}
{"id": "3243.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\ 1 ( c _ d ' r | \\nabla f ( a ) | - C A r ^ 2 > \\kappa ) \\ , \\frac { d r } { r ^ { p + 1 } } & = p ^ { - 1 } \\left ( R _ - ^ { - p } - R _ + ^ { - p } \\right ) . \\end{align*}"}
{"id": "2950.png", "formula": "\\begin{align*} Y \\cdot ( f , a ) = a + Y . \\end{align*}"}
{"id": "1271.png", "formula": "\\begin{align*} \\Psi ( E _ j ( x , a ) ) & = 4 \\sqrt { a } \\sum _ { h = 1 } ^ { \\lfloor \\frac { j } { 2 } \\rfloor } \\cos \\left ( \\frac { h } { j + 1 } \\pi \\right ) . \\end{align*}"}
{"id": "650.png", "formula": "\\begin{align*} \\frac { \\Omega ( 1 - s ) } { \\Omega ( s ) } = \\frac { \\xi _ 2 ( - s ) } { \\xi _ 2 ( s - 1 ) } = \\frac { \\xi _ 2 ( - s ) } { \\xi _ 2 ( 2 - s ) } = \\frac { \\pi ^ 2 } { s ( s - 1 ) } \\cdot \\frac { \\zeta ( \\Delta , - s ) } { \\zeta ( \\Delta , 2 - s ) } . \\end{align*}"}
{"id": "2917.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n b _ j ( g \\circ S ^ { 1 - j } ) ( x ) + d ' = f ( A x + e ) . \\end{align*}"}
{"id": "5685.png", "formula": "\\begin{align*} x _ 1 ^ { d + 1 } + x _ 2 ^ { d + 1 } + \\cdots + x _ i ^ { d + 1 } = ( x _ 1 + x _ 2 + \\cdots + x _ i ) x _ { i + 1 } ^ d . \\end{align*}"}
{"id": "601.png", "formula": "\\begin{align*} \\xi _ Q ( s ) : = \\pi ^ { - s } \\Gamma ( s ) \\zeta _ Q ( s ) . \\end{align*}"}
{"id": "3718.png", "formula": "\\begin{align*} \\bar { \\psi } _ { C _ 1 , C ' _ 1 } = \\phi ^ { [ y _ 1 ] } _ { C _ 1 , C ' _ 1 } \\quad \\quad \\bar { \\psi } _ { C _ 2 , C ' _ 2 } = \\phi ^ { [ y _ 1 ] } _ { C _ 2 , C ' _ 2 } . \\end{align*}"}
{"id": "4832.png", "formula": "\\begin{align*} \\rho = \\sum _ { k = 1 } ^ m \\mathrm { I n d } _ { H _ k } ^ G \\varphi _ k \\ ; . \\end{align*}"}
{"id": "4165.png", "formula": "\\begin{align*} \\varphi _ k ^ \\lambda ( z ) = \\lambda ^ n L _ k ^ { n - 1 } ( \\tfrac 1 2 \\lambda | z | ^ 2 ) e ^ { - \\frac 1 4 \\lambda | z | ^ 2 } , z \\in \\R ^ { 2 n } , \\end{align*}"}
{"id": "6620.png", "formula": "\\begin{align*} I _ { k } ( c ) : = h ^ 4 \\int _ { \\mathbb { R } ^ 4 } w ( x ) H \\big ( h ^ { - 1 } k , N ( x ) - 1 \\big ) e _ { k } ( - h c \\cdot x ) \\ , d x . \\end{align*}"}
{"id": "3801.png", "formula": "\\begin{align*} H y p _ { i , i _ 1 , i _ 2 , i _ 3 } ( t _ 1 , t _ 2 ) = \\sum _ { a = 0 , 1 } E r r ^ a _ { i , i _ 1 , i _ 2 , i _ 3 } ( t _ 1 , t _ 2 ) + \\widetilde { H y p } _ { i , i _ 1 , i _ 2 , i _ 3 } ( t _ 1 , t _ 2 ) , \\widetilde { H y p } _ { i , i _ 1 , i _ 2 , i _ 3 } ( t _ 1 , t _ 2 ) : = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { ( \\R ^ 3 ) ^ 4 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta + \\sigma + \\kappa ) } \\end{align*}"}
{"id": "4332.png", "formula": "\\begin{align*} \\int _ L \\lambda \\wedge \\chi = - \\int _ L f _ L d \\chi . \\end{align*}"}
{"id": "4810.png", "formula": "\\begin{align*} \\int _ a ^ b f ( v ) \\ , d v & = \\int _ a ^ { v _ 0 } \\cdots + \\int _ { v _ 0 } ^ b \\cdots \\ge f ( a ) ( v _ 0 - a ) + f ( v _ 0 ) ( b - v _ 0 ) \\\\ & = \\mbox { $ \\frac 1 2 $ } f ( a ) ( b - a ) + \\mbox { $ \\frac 1 2 $ } f ( v _ 0 ) ( b - a ) + ( f ( v _ 0 ) - f ( a ) ) ( \\mbox { $ \\frac 1 2 $ } ( a + b ) - v _ 0 ) , \\end{align*}"}
{"id": "892.png", "formula": "\\begin{align*} f _ m = g _ 0 + g _ 1 + \\cdots + g _ m . \\end{align*}"}
{"id": "5790.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & x & y & & \\\\ & 1 & z & & \\\\ & & 1 & & \\\\ & & & a & \\\\ & & & & b \\\\ \\end{pmatrix} \\mapsto \\begin{pmatrix} 1 & x & y & & \\\\ & 1 & z & & \\\\ & & 1 & & \\\\ & & & e ^ { \\alpha } & \\\\ & & & & e ^ { \\beta } \\\\ \\end{pmatrix} \\end{align*}"}
{"id": "6376.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\ , \\sum _ { i = 1 } ^ n ( U ( \\gamma _ i ) - A _ 0 ) \\end{align*}"}
{"id": "3817.png", "formula": "\\begin{align*} \\sum _ { a = 0 , 1 } | { } _ { 0 } ^ 1 E l l H ^ { \\mu , a ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) | + \\sum _ { \\kappa \\in ( \\bar { \\kappa } , 2 ] \\cap \\Z } | { } _ { 0 } ^ \\kappa E l l H ^ { \\mu , 2 ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) | \\lesssim \\sum _ { \\begin{subarray} { c } k _ 1 \\in \\Z _ { + } , j _ 1 \\in [ 0 , ( 1 + 2 \\epsilon ) M _ t ] \\cap \\Z , \\mu _ 1 \\in \\{ + , - \\} \\\\ n _ 1 \\in [ - M _ t , 2 ] \\cap \\Z , i , i _ 1 \\in \\{ 0 , 1 , 2 , 3 , 4 \\} \\end{subarray} } \\big [ H y p E l l _ { k _ 1 , j _ 1 ; n _ 1 } ^ { i , i _ 1 , \\mu _ 1 } ( t _ 1 , t _ 2 ) \\end{align*}"}
{"id": "7388.png", "formula": "\\begin{align*} \\psi _ n + c _ n \\left ( \\sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \\frac { \\partial U _ { h _ n , i } } { \\partial h _ n } \\right ) \\in E _ n . \\end{align*}"}
{"id": "7235.png", "formula": "\\begin{align*} \\dot { x } ( t , s ) & = - \\partial _ s x ( t , s ) + f ( s ) x ( t , s ) , s \\in [ 0 , 1 ] , \\\\ x ( t , 0 ) & = \\int _ 0 ^ 1 h ( s ) x ( t , s ) d s , \\end{align*}"}
{"id": "8075.png", "formula": "\\begin{align*} \\phi _ { x _ { 0 } } ( N \\times \\{ 0 \\} ) = S ^ { 1 } \\cdot x _ { 0 } \\subset U ^ { + } \\end{align*}"}
{"id": "8501.png", "formula": "\\begin{align*} & \\int _ { \\left [ 0 , 1 \\right ] ^ d } \\left ( p ( x ) - \\hat { p } \\left ( x \\right ) \\right ) ^ 2 d x \\\\ & = \\int _ { \\left [ 0 , 1 \\right ] ^ d } \\hat { p } \\left ( x \\right ) ^ 2 d x - 2 \\int _ { \\left [ 0 , 1 \\right ] ^ d } p ( y ) \\hat { p } ( y ) d y + \\int _ { \\left [ 0 , 1 \\right ] ^ d } p ( z ) ^ 2 d z . \\end{align*}"}
{"id": "1928.png", "formula": "\\begin{align*} S _ { 0 } ( z ) & = \\frac { 1 } { z - S _ { 0 } ^ { p } ( z ) } , \\\\ S _ { j } ( z ) & = S _ { 0 } ( z ) \\ , S _ { j - 1 } ( z ) , 1 \\leq j \\leq p , \\end{align*}"}
{"id": "2458.png", "formula": "\\begin{align*} x ' = \\exp \\left ( \\operatorname { a d } _ h \\right ) \\left ( x \\right ) . \\end{align*}"}
{"id": "2266.png", "formula": "\\begin{align*} I _ 3 & = \\int _ 0 ^ r \\left [ ( \\bar \\phi + e _ \\delta - \\delta ) ( 2 - \\delta + \\bar \\phi + e _ \\delta ) ( \\bar \\phi '' , \\bar \\psi '' ) \\cdot ( \\phi , \\psi ) \\right ] { \\rm { d } } z \\\\ & \\leq \\Vert ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert ( \\bar \\phi '' , \\bar \\psi '' ) \\Vert _ { L ^ 2 } \\cdot \\Vert ( \\bar \\phi + e _ \\delta - \\delta ) ( 2 - \\delta + \\bar \\phi + e _ \\delta ) \\Vert _ { L ^ \\infty } , \\end{align*}"}
{"id": "5304.png", "formula": "\\begin{align*} \\int _ { \\Psi _ j } x ^ k e ^ { x ^ r / \\hbar } d x & = \\int _ 0 ^ \\infty t ^ k e ^ { ( \\pi i + \\arg \\hbar i + 2 \\pi i j ) ( k + 1 ) / r } e ^ { t ^ r \\hbar ^ { - 1 } e ^ { \\pi i + \\arg \\hbar i } } d t \\\\ & = \\exp \\left ( ( \\pi i + \\arg \\hbar i + 2 \\pi i j ) \\frac { k + 1 } { r } \\right ) \\int _ 0 ^ \\infty t ^ k e ^ { - t ^ r / | \\hbar | } d t \\end{align*}"}
{"id": "6742.png", "formula": "\\begin{align*} \\min _ { u \\in B ( \\overline { u } , r ) } j _ \\delta ( u ) \\end{align*}"}
{"id": "8291.png", "formula": "\\begin{align*} \\psi _ { A _ { k } } = A _ { k } \\left [ \\frac { \\cos ( k x ) . \\cosh ( k ' x ) } { \\cos ( k a / 2 ) . \\cosh ( k ' a / 2 ) } - \\frac { \\sin ( k x ) . \\sinh ( k ' x ) } { \\sin ( k a / 2 ) . \\sinh ( k ' a / 2 ) } \\right ] \\end{align*}"}
{"id": "2037.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : s p a t i a l l y h o m o g e n e o u s b o l t z m a n n } \\partial _ t f ( t , v ) = \\int _ { \\R ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } B ( v - v _ * , \\sigma ) \\Big ( f _ t ( v _ * ' ) f _ t ( v ' ) - f _ t ( v _ * ) f _ t ( v ) \\Big ) \\dd v _ * \\dd \\sigma , \\end{align*}"}
{"id": "955.png", "formula": "\\begin{align*} N ( A ) / p ^ { n } q ^ { n } = p ^ { f _ { 1 } - n } q ^ { f _ { 2 } - n } , \\end{align*}"}
{"id": "8167.png", "formula": "\\begin{align*} e ^ { - i t H } \\varphi = e ^ { - i t H _ x } \\psi _ 0 \\otimes e ^ { - i t H _ y } \\psi _ 1 = e ^ { - i t E _ 0 } \\psi _ 0 \\otimes e ^ { - i t H _ y } \\psi _ 1 \\end{align*}"}
{"id": "8405.png", "formula": "\\begin{gather*} h _ { r } ( \\tau ) = ( - 1 ) ^ k h _ { - r } ( \\tau ) \\end{gather*}"}
{"id": "97.png", "formula": "\\begin{align*} \\Sigma _ t ( \\varphi ) = \\Big ( \\big < \\zeta _ x ; \\zeta _ y \\big > ^ { C _ t ^ { - 1 } \\varphi } _ { A , g , \\nu + 1 / t } \\Big ) _ { x , y \\in \\Lambda } , \\end{align*}"}
{"id": "779.png", "formula": "\\begin{align*} \\beta f ' _ { 0 } ( r ^ m ) + ( 1 - \\beta ) f _ { 0 } ( r ^ m ) + \\sum _ { n = 1 } ^ { \\infty } | a _ n ( f _ { 0 } ) | \\phi _ { n } ( r ) < - f _ { 0 } ( - 1 ) . \\end{align*}"}
{"id": "587.png", "formula": "\\begin{align*} 2 V _ k ( n ) & = \\Big ( \\sum _ { 1 \\leq i \\leq n } f _ k ( i ) f _ k ( i + 1 ) \\Big ) ^ 2 - \\sum _ { 1 \\leq i \\leq n } \\big ( f _ k ( i ) f _ k ( i + 1 ) \\big ) ^ 2 + O _ \\varepsilon ( n ^ { 1 + \\varepsilon } ) \\\\ & = \\Big ( \\sum _ { 1 \\leq i \\leq n } f _ k ( i ) f _ k ( i + 1 ) \\Big ) ^ 2 + O _ \\varepsilon ( n ^ { 1 + \\varepsilon } ) . \\end{align*}"}
{"id": "7537.png", "formula": "\\begin{align*} \\bigg \\| \\max _ { 0 \\le k \\le n - 1 } \\big | V _ { k + 1 } - V _ { k } \\big | \\bigg \\| _ { L ^ { \\gamma p } } ^ { \\gamma } = \\bigg \\| \\max _ { 1 \\le k \\le n } \\big | \\mathbf { E } \\big ( \\frac { m ^ 2 } { n \\sigma ^ 2 } | f ^ { - 1 } \\mathcal { M } \\big ) \\circ f ^ { n - k } \\big | \\bigg \\| _ { L ^ { \\gamma p } } ^ { \\gamma } \\le C n ^ { - \\left ( \\gamma - \\frac { 2 \\gamma } { p } \\right ) } , \\end{align*}"}
{"id": "5682.png", "formula": "\\begin{align*} x _ 1 ^ { d + 1 } + x _ 2 ^ { d + 1 } + \\cdots + x _ i ^ { d + 1 } = \\sum _ { j = 1 } ^ i \\varpi _ j x _ j ^ { d - 1 } x _ { j + 1 } - \\sum _ { j = 1 } ^ i \\varpi _ { j - 1 } x _ j ^ d . \\end{align*}"}
{"id": "5742.png", "formula": "\\begin{align*} \\pi _ { [ a - 1 , b ] } + \\pi _ { [ a , b ] } y _ { b + 1 } = \\pi _ { [ a , b + 1 ] } \\end{align*}"}
{"id": "628.png", "formula": "\\begin{align*} a _ 0 ( s ) = \\int _ 0 ^ 1 z ^ { 2 \\alpha - 2 s - 1 } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\Big ( \\frac { \\sin ^ 2 ( \\pi x ) } { \\pi ^ 2 } + \\frac { \\sin ^ 2 ( \\pi y ) } { \\pi ^ 2 } + z ^ 2 \\Big ) ^ { - \\alpha } d x d y d z \\end{align*}"}
{"id": "1719.png", "formula": "\\begin{align*} R _ { \\ell , \\gamma _ m } ( t , q ^ { \\frac { 1 } { 2 } } ) = R _ { \\ell , - \\gamma _ m } ( t , q ^ { \\frac { 1 } { 2 } } ) ^ { - 1 } , \\end{align*}"}
{"id": "7321.png", "formula": "\\begin{align*} N _ 2 [ \\epsilon , v , w ] & = \\{ f _ 2 ( u ) + f _ 2 ( \\eta ^ \\frac { 2 } { 1 - q } { \\sf U } ) - q \\eta ^ { - 2 } { \\sf U } ^ { q - 1 } ( u + \\eta ^ \\frac { 2 } { 1 - q } { \\sf U } ) \\} ( 1 - \\chi _ 1 ) \\chi _ 2 \\\\ & = \\tfrac { 1 } { 2 } f _ 2 '' ( u _ \\kappa ) ( u + \\eta ^ \\frac { 2 } { 1 - q } { \\sf U } ) ^ 2 ( 1 - \\chi _ 1 ) \\chi _ 2 , \\end{align*}"}
{"id": "7582.png", "formula": "\\begin{align*} Y ( t ) = \\exp ( \\Omega ( t ) ) . \\end{align*}"}
{"id": "1150.png", "formula": "\\begin{align*} \\frac { \\partial f _ 0 } { \\partial x ^ 1 } & = \\frac { \\partial f _ 2 } { \\partial x ^ 2 } & \\frac { \\partial f _ 0 } { \\partial x ^ 2 } & = - \\frac { \\partial f _ 2 } { \\partial x ^ 1 } \\end{align*}"}
{"id": "5480.png", "formula": "\\begin{align*} N _ d ( \\mathrm { C H } ( S ) ) \\cap \\mathrm { A n } ( R ) & \\subseteq \\mathrm { A n } ( R ) \\cap \\bigcup _ { i = 1 } ^ N N _ { C + d } ( \\mathrm { C o n e } ( x , B ( y _ i , \\varepsilon ) ) ) \\setminus B ( x , R ) \\\\ & \\subseteq \\mathrm { A n } ( R ) \\cap \\bigcup _ { i = 1 } ^ N \\mathrm { C o n e } ( x , B ( y _ i , \\varepsilon ( 1 + A C ' ) ) ) , \\end{align*}"}
{"id": "8006.png", "formula": "\\begin{align*} I _ { ( Z _ + , D _ { Z , + } ) } ( y , z ) = z e ^ { t _ + / z } \\sum _ { d \\in \\mathbb K _ { + } } y ^ { d } \\left ( \\prod _ { i \\in M _ 0 } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\\\ \\left ( \\prod _ { 0 < a \\leq v _ 1 \\cdot d } ( v _ 1 + a z ) \\right ) \\textbf { 1 } _ { [ - d ] } I _ { D _ { Z , + } , d } , \\end{align*}"}
{"id": "7204.png", "formula": "\\begin{align*} \\phi _ 1 ( s ) = s ^ { - a } \\log ( 2 + s ) , \\end{align*}"}
{"id": "6402.png", "formula": "\\begin{align*} B _ { \\pi } = \\{ 3 , 5 , 7 , 8 \\} \\cup \\{ 6 \\} , \\end{align*}"}
{"id": "2838.png", "formula": "\\begin{align*} P ( h ) = \\frac { 1 } { 2 L } \\frac { h ( \\kappa h - 2 \\kappa + 2 ) } { 1 - \\kappa } = \\frac { 1 } { 2 L } \\big [ 2 h - h ^ 2 \\frac { - \\kappa } { 1 - \\kappa } \\big ] . \\end{align*}"}
{"id": "2703.png", "formula": "\\begin{align*} \\Big | E _ { 2 - \\mathrm { m o d e } } - E _ 0 - E ^ w _ N - N \\frac { \\mu _ + - \\mu _ - } { 2 } \\Big | \\le C _ \\varepsilon \\max \\{ T ^ { 1 / 2 - \\varepsilon } , N ^ { - 1 + \\varepsilon \\delta } \\} . \\end{align*}"}
{"id": "6830.png", "formula": "\\begin{align*} \\begin{aligned} p ( t , v , g ) & = \\frac { 1 } { V _ F } \\sum _ { k = - \\infty } ^ { + \\infty } p _ k ( t , g ) e ^ { i k v \\frac { 2 \\pi } { V _ F } } , \\\\ p _ k ( t , g ) : & = \\int _ { 0 } ^ { V _ F } p ( t , v , g ) e ^ { - i k v \\frac { 2 \\pi } { V _ F } } d v , k \\in \\mathbb { Z } . \\end{aligned} \\end{align*}"}
{"id": "4726.png", "formula": "\\begin{align*} e _ { ( k ) } H _ { 2 k } ^ { - 1 } H _ { 2 k + 1 } ^ { - 1 } H _ { 2 k - 1 } H _ { 2 k } e _ { ( k ) } = e _ { ( k ) } H _ { 2 k } ^ { - 1 } H _ { 2 k + 1 } H _ { 2 k - 1 } ^ { - 1 } H _ { 2 k } e _ { ( k ) } . \\end{align*}"}
{"id": "3648.png", "formula": "\\begin{gather*} e ( \\partial B _ { R _ \\ell } ( x _ 0 ) ) = u ( \\partial B _ { r _ \\ell } ( x _ 0 ) ) = 0 , \\\\ \\shortintertext { a n d } \\lim _ { \\ell \\to \\infty } \\frac { e ( B _ { R _ \\ell } ( x _ 0 ) ) } { u ( B _ { r _ \\ell } ( x _ 0 ) ) } = \\limsup _ { R \\to 0 ^ + } \\frac { e ( B _ R ( x _ 0 ) ) } { u ( B _ R ( x _ 0 ) ) } . \\end{gather*}"}
{"id": "3012.png", "formula": "\\begin{align*} S ( W _ { \\bullet , \\bullet } ^ { L _ { x y } } ; p ) & = \\frac { 1 6 n ( 4 n + 1 ) } { 9 ( 2 n + 1 ) } \\int _ 0 ^ \\frac { 3 } { 2 } h _ { p , L } ( t ) d t \\\\ & = \\frac { 4 n + 1 } { 1 8 n ( 2 n + 1 ) } \\left \\lbrace \\int _ 0 ^ \\frac { 3 } { 4 } \\frac { 1 } { ( 4 n + 1 ) ^ 2 } ( 2 ( 4 n - 1 ) t + 3 ) ^ 2 d t + \\int _ \\frac { 3 } { 4 } ^ \\frac { 3 } { 2 } ( 3 - 2 t ) ^ 2 d t \\right \\rbrace \\\\ & = \\frac { 4 n ^ 2 + 3 n + 1 } { 4 n ( 2 n + 1 ) ( 4 n + 1 ) } . \\end{align*}"}
{"id": "5599.png", "formula": "\\begin{align*} X ^ 3 + Y ^ 3 + Z ^ 3 = n X Y Z , \\end{align*}"}
{"id": "780.png", "formula": "\\begin{align*} \\beta | f ' ( z ^ m ) | + ( 1 - \\beta ) | f ( z ^ m ) | + \\sum _ { n = 1 } ^ { \\infty } | a _ n | \\phi _ { n } ( r ) \\leq d ( 0 , \\partial { \\Omega } ) \\end{align*}"}
{"id": "3093.png", "formula": "\\begin{align*} ( a \\# b ) ( n \\otimes y ) = \\sum _ { ( b ) } ( S ^ { - 1 } ( b _ { - 1 } y _ { - 1 } ) \\rightharpoonup a ) n \\otimes b _ 0 y _ 0 . \\end{align*}"}
{"id": "2290.png", "formula": "\\begin{align*} J _ 3 = & \\int _ 0 ^ \\infty ( f _ u ^ { ( 1 ) } , f _ h ^ { ( 1 ) } ) \\cdot ( u _ { p x } ^ 1 , h _ { p x } ^ 1 ) x ^ { 1 - 2 \\sigma _ 1 } { \\rm d } y \\\\ \\leq & \\Vert ( f _ u ^ { ( 1 ) } , f _ h ^ { ( 1 ) } ) x ^ { \\frac { 1 } { 2 } - \\sigma _ 1 } \\Vert _ { L _ y ^ 2 } \\Vert ( u _ { p x } ^ 1 , h _ { p x } ^ 1 ) x ^ { \\frac { 1 } { 2 } - \\sigma _ 1 } \\Vert _ { L _ y ^ 2 } \\\\ \\leq & C x ^ { - \\frac { 3 } { 2 } } + + \\delta _ 0 \\Vert ( u _ { p x } ^ 1 , h _ { p x } ^ 1 ) x ^ { \\frac { 1 } { 2 } - \\sigma _ 1 } \\Vert _ { L _ y ^ 2 } ^ 2 , \\end{align*}"}
{"id": "7720.png", "formula": "\\begin{align*} \\pi _ { S } ^ T ( k + 1 ) A _ { S \\bar { S } } ( k ) e + \\pi _ { \\bar { S } } ^ T ( k + 1 ) A _ { \\bar { S } } ( k ) e = \\pi _ { \\bar { S } } ^ T ( k ) e \\ \\end{align*}"}
{"id": "7607.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\boldsymbol { x } ( t ) - \\boldsymbol { x } _ c & \\to \\hat { \\boldsymbol { x } } ( t ) , \\\\ \\boldsymbol { u } ( t ) - \\boldsymbol { u } _ c & \\to \\hat { \\boldsymbol { u } } ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "965.png", "formula": "\\begin{align*} H = H _ { \\phi } = \\left ( \\begin{array} { c c c | c } 0 & \\cdots & 0 & \\phi _ 0 \\\\ \\hline & & & \\phi _ 1 \\\\ & I _ { n - 1 } & & \\vdots \\\\ & & & \\phi _ { n - 1 } \\\\ \\end{array} \\right ) \\in \\mathbb { Z } ^ { n \\times n } , \\end{align*}"}
{"id": "414.png", "formula": "\\begin{align*} F _ { \\mu } ( z + \\varphi _ { \\mu } ( z ) ) = z \\end{align*}"}
{"id": "2344.png", "formula": "\\begin{align*} \\nu _ Q \\left ( f \\right ) = \\min _ { 0 \\leq i \\leq r } \\{ \\nu _ Q ( f _ i Q '^ i ) \\} . \\end{align*}"}
{"id": "8645.png", "formula": "\\begin{align*} U ( t ) : = \\exp \\left ( { \\rm i } \\alpha ( t ) \\widehat { H } _ 2 \\right ) \\exp \\left ( { \\rm i } \\beta ( t ) \\widehat { H } _ 3 \\right ) . \\end{align*}"}
{"id": "7895.png", "formula": "\\begin{align*} \\Omega ( f , g ) = _ { z = 0 } ( f ( - z ) , g ( z ) ) d z , f , g \\in \\mathcal H , \\end{align*}"}
{"id": "5935.png", "formula": "\\begin{align*} w \\left ( \\eta \\right ) = \\mathop \\sum \\limits _ n \\frac { w ^ { ( n ) } } { { n ! } } \\eta ^ n \\end{align*}"}
{"id": "3479.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast \\ast } & = \\frac { c _ { 1 } \\left [ \\xi _ { p } \\overline { G } _ { ( 2 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { p } ^ { 2 } \\right ) - \\xi _ { q } \\overline { G } _ { ( 2 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { q } ^ { 2 } \\right ) \\right ] } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } + \\frac { c _ { 1 } } { c _ { ( 2 ) } ^ { \\ast } } \\frac { F _ { Y _ { ( 2 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } . \\end{align*}"}
{"id": "4388.png", "formula": "\\begin{align*} \\int _ { U } | f | ^ 2 e ^ { - k \\varphi - r _ 1 \\psi } \\le \\left ( \\int _ { U } | f | ^ { 2 q _ 1 } e ^ { - q _ 1 k \\varphi } \\right ) ^ { \\frac { 1 } { q _ 1 } } \\cdot \\left ( \\int _ U e ^ { - q _ 2 r _ 1 \\psi } \\right ) ^ { \\frac { 1 } { q _ 2 } } < + \\infty , \\end{align*}"}
{"id": "493.png", "formula": "\\begin{align*} w t ( b _ 1 \\otimes b _ 2 ) & = w t ( b _ 1 ) + w t ( b _ 2 ) , \\\\ \\varepsilon _ i ( b _ 1 \\otimes b _ 2 ) & = \\max \\left ( \\varepsilon _ i ( b _ 1 ) , \\varepsilon _ i ( b _ 2 ) - \\langle \\alpha _ i ^ \\vee , w t ( b _ 1 ) \\rangle \\right ) , \\\\ \\varphi _ i ( b _ 1 \\otimes b _ 2 ) & = \\max \\left ( \\varphi _ i ( b _ 2 ) , \\varphi _ i ( b _ 1 ) + \\langle \\alpha _ i ^ \\vee , w t ( b _ 2 ) \\rangle \\right ) , \\end{align*}"}
{"id": "4095.png", "formula": "\\begin{align*} U = U _ { + } \\oplus U _ { - } , \\end{align*}"}
{"id": "7804.png", "formula": "\\begin{align*} \\widetilde { Q } _ { ( X _ 1 , \\eta _ 1 , X _ 2 , \\eta _ 2 ) , \\omega _ 1 } : = \\{ ( X _ { 1 , t } , \\eta _ { 1 , t } , X _ { 2 , t } , \\eta _ { 2 , \\psi ( t ) } ) \\ : \\ t \\in Q _ { ( X _ 1 , \\eta _ 1 , \\omega _ 1 ) } \\} . \\end{align*}"}
{"id": "2507.png", "formula": "\\begin{align*} - g ( A _ { \\mathcal { H } \\nabla _ X W } X , W ) = 0 . \\end{align*}"}
{"id": "630.png", "formula": "\\begin{align*} J _ \\alpha ^ 0 ( s ) = I ( s , n ) + 8 \\Bigg ( \\sum _ { m = 0 } ^ { N - 1 } n ^ { - 2 m } \\sum _ { j = 0 } ^ m W _ { m , j } ^ 1 ( s , n ) + n ^ { - 2 N } R _ N ^ 1 ( s , n ) \\Bigg ) \\\\ + 4 \\Bigg ( \\sum _ { m = 0 } ^ { N ' - 1 } n ^ { - 2 m } \\sum _ { j = 0 } ^ m W _ { m , j } ^ 1 ( s , n ) + n ^ { - 2 N ' } R _ { N ' } ^ 2 ( s , n ) \\Bigg ) . \\end{align*}"}
{"id": "5744.png", "formula": "\\begin{align*} \\pi _ i \\cdot \\pi _ { [ a , b ] } = \\pi _ i \\pi _ { [ a - 1 , b - 1 ] } + \\pi _ i \\pi _ { [ a , b - 1 ] } y _ { b } \\end{align*}"}
{"id": "4340.png", "formula": "\\begin{align*} \\mathcal { M } ( p _ 1 , \\ldots p _ k , q ) = \\deg q - \\sum _ 1 ^ k \\deg p _ i + k - 2 . \\end{align*}"}
{"id": "116.png", "formula": "\\begin{align*} ( f \\star g ) ( x ) = \\epsilon ^ d \\sum _ { y \\in \\Lambda _ { \\epsilon , L } } f ( x - y ) g ( y ) , f , g \\in \\R ^ { \\Lambda _ { \\epsilon , L } } . \\end{align*}"}
{"id": "4387.png", "formula": "\\begin{align*} \\frac { 1 } { r _ 2 ^ 2 } \\int _ { \\{ 2 a _ 0 ^ f ( \\Psi ) \\Psi \\le 2 \\log r _ 2 \\} } | f | ^ 2 & = \\lim _ { p \\rightarrow 2 a _ 0 ^ f ( \\Psi ) + 0 } \\frac { 1 } { r _ 2 ^ 2 } \\int _ { \\{ p \\Psi < 2 \\log r _ 2 \\} } | f | ^ 2 \\\\ & \\ge G ( 0 ; \\Psi , I _ + ( 2 a _ o ^ f ( \\Psi ) \\Psi ) _ o , f ) . \\end{align*}"}
{"id": "1979.png", "formula": "\\begin{align*} & \\tilde A ( y ) : = A ( r _ 0 ^ { k _ 0 } y ) \\\\ & \\tilde f ( y ) : = r _ 0 ^ { k _ 0 ( 2 - \\alpha ) } f ( r _ 0 ^ { k _ 0 } y ) . \\end{align*}"}
{"id": "1811.png", "formula": "\\begin{align*} \\| A \\| _ { 1 , 2 } = \\sup _ { v \\neq 0 } \\frac { \\| A v \\| } { \\| v \\| _ 1 } = \\max _ { i = 1 , \\ldots , n } \\| a _ i \\| = \\sqrt 2 , \\end{align*}"}
{"id": "2059.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : l i n e a r P m u } \\langle Q ( \\mu ) , \\tilde { \\varphi } \\rangle = \\langle \\mu , K \\star \\tilde { \\varphi } \\rangle - \\langle \\mu , \\tilde { \\varphi } \\rangle . \\end{align*}"}
{"id": "2527.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\{ g ( \\nabla _ X \\xi , Y ) + g ( \\nabla _ Y \\xi , X ) \\} + R i c ( X , Y ) + \\mu g ( X , Y ) = 0 , \\end{align*}"}
{"id": "5575.png", "formula": "\\begin{align*} \\varkappa ( B \\cap E , \\sigma ) = \\varkappa ( B , \\sigma _ E ) . \\end{align*}"}
{"id": "6537.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { \\infty } \\mu \\left ( \\left [ C \\alpha _ i , \\infty \\right ) \\right ) \\precsim \\sum \\limits _ { i = 1 } ^ { \\infty } \\mu \\left ( \\left [ \\alpha _ i , \\infty \\right ) \\right ) \\end{align*}"}
{"id": "5651.png", "formula": "\\begin{align*} F ( D ^ 2 u ) = g \\quad \\ \\mathbb { R } ^ n \\setminus \\overline { B } _ { r _ 0 } \\end{align*}"}
{"id": "3982.png", "formula": "\\begin{align*} \\Pr \\big ( \\Delta ( C _ A ) \\le \\delta \\big ) = \\Pr ( X \\ge 1 ) . \\end{align*}"}
{"id": "5970.png", "formula": "\\begin{align*} C ^ 2 _ { T , \\beta } = 2 \\frac { T ^ { 1 - \\beta } } { \\Gamma ( 2 - \\beta ) } \\max ( 2 ^ { 1 - \\beta } \\Gamma ( 1 - \\beta ) T ^ { \\beta } , 1 ) . \\end{align*}"}
{"id": "8387.png", "formula": "\\begin{align*} \\omega ( t , X ( t , x ) ) = \\omega _ 0 ( x ) , \\end{align*}"}
{"id": "8565.png", "formula": "\\begin{align*} m = 2 ^ { k _ 1 } + 2 ^ { k _ 2 } + \\ldots + 2 ^ { k _ l } , \\end{align*}"}
{"id": "1991.png", "formula": "\\begin{align*} & \\tilde A ( y , s ) : = A ( R _ 0 ^ { k _ 0 } y , s ) \\\\ & \\tilde f ( y , s ) : = R _ 0 ^ { ( 2 k _ 0 ( 1 - \\alpha ) + k _ 0 \\alpha ) } f ( R _ 0 ^ { k _ 0 } y , s ) \\\\ & \\tilde Q ( y ) : = R _ 0 ^ { 2 k _ 0 ( 1 - \\alpha ) } Q ( r _ 0 ^ { k _ 0 } y ) . \\end{align*}"}
{"id": "8439.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } | T _ { \\beta } ^ n x - x | = 0 . \\end{align*}"}
{"id": "2448.png", "formula": "\\begin{align*} & x \\cdot m = x m - m x , \\end{align*}"}
{"id": "2370.png", "formula": "\\begin{align*} \\nu \\left ( \\partial _ i L ( h _ \\sigma ) - \\partial _ i L ( h _ \\rho ) \\right ) = \\min _ { 1 \\leq k \\leq n - i } \\left \\{ \\nu \\left ( { i + k \\choose i } \\partial _ { i + k } L ( h _ \\rho ) ( h _ \\sigma - h _ \\rho ) ^ k \\right ) \\right \\} . \\end{align*}"}
{"id": "6333.png", "formula": "\\begin{align*} \\S _ q z & : = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( q ; q ^ 2 ) _ \\infty } ( z ) ^ { 1 / 2 } J ^ { ( 2 ) } _ { 1 / 2 } ( 2 z ; q ^ 2 ) = \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n \\frac { q ^ { 2 n ^ 2 + n } z ^ { 2 n + 1 } } { [ 2 n + 1 ] _ q ! } , \\end{align*}"}
{"id": "438.png", "formula": "\\begin{align*} \\frac { d \\nu _ { n } } { d x } \\left ( H _ { \\nu _ { n } } ( s + i f _ { n } ( s ) \\right ) = \\frac { 1 } { \\pi } \\frac { f _ { n } ( s ) } { s ^ { 2 } + f _ { n } ( s ) ^ { 2 } } , s \\in [ A , B ] . \\end{align*}"}
{"id": "5269.png", "formula": "\\begin{align*} \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { k _ 1 ( 0 ) + r } a ^ { \\widehat { Q } ^ { + r } } + \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { k _ 2 ( 0 ) + s } b ^ { \\widehat { Q } ^ { + s } } = 1 + \\sum _ { j = 1 } ^ h \\left ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { Q } _ j } + \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { Q } _ j } \\right ) . \\end{align*}"}
{"id": "8332.png", "formula": "\\begin{align*} \\delta ( a ) = \\left [ \\sum _ { n \\ge 0 } \\sum _ { j = 0 } ^ \\infty \\beta _ { n , j } P _ { j , j + n } + \\sum _ { n < 0 } \\sum _ { j = 0 } ^ \\infty \\beta _ { n , j } P _ { j - n , j } , a \\right ] \\end{align*}"}
{"id": "3971.png", "formula": "\\begin{align*} [ \\sigma , e ] [ \\tau , f ] = [ \\sigma \\tau , e \\alpha _ { \\sigma } ( f ) ] \\quad [ \\sigma , e ] ^ { * } = [ \\sigma ^ { - 1 } , \\alpha _ { \\sigma } ^ { - 1 } ( e ^ { * } ) ] . \\end{align*}"}
{"id": "3252.png", "formula": "\\begin{align*} \\Phi ( x , y ) : = \\frac { 1 } { 4 \\pi } \\frac { e ^ { i k \\abs { x - y } } } { \\abs { x - y } } , x \\neq y \\end{align*}"}
{"id": "8066.png", "formula": "\\begin{align*} \\operatorname { s g n } ( Z ) : = \\begin{cases} \\operatorname { s g n } ( F ) & U _ { 0 } : = \\{ x \\in M \\mid \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) \\neq 0 \\} \\\\ 0 & M \\backslash U _ { 0 } , \\end{cases} \\end{align*}"}
{"id": "4460.png", "formula": "\\begin{align*} { } n e _ n = \\sum _ { r = 1 } ^ n ( - 1 ) ^ { r - 1 } p _ r e _ { n - r } . \\end{align*}"}
{"id": "2369.png", "formula": "\\begin{align*} \\beta _ i : = \\nu ( \\partial _ i L ( Q ) ) = \\nu ( \\partial _ i L ( h _ { \\rho } ) ) \\end{align*}"}
{"id": "1967.png", "formula": "\\begin{align*} \\mathbf { v } _ { k } ( z ) : = \\begin{cases} ( A _ { 0 } ^ { ( k ) } , \\ldots , A _ { p - k - 1 } ^ { ( k ) } , - \\sum _ { j = k } ^ { p } a _ { 0 } ^ { ( j ) } A _ { j - k } ^ { ( k ) } , \\ldots , - \\sum _ { j = 1 } ^ { p } a _ { k - 1 } ^ { ( j ) } A _ { j - 1 } ^ { ( k ) } ) , & 0 \\leq k \\leq p , \\\\ [ 0 . 5 e m ] ( - a _ { k - p } ^ { ( p ) } A _ { 0 } ^ { ( k ) } , - \\sum _ { j = p - 1 } ^ { p } a _ { k - p + 1 } ^ { ( j ) } A _ { j - p + 1 } ^ { ( k ) } , \\ldots , - \\sum _ { j = 1 } ^ { p } a _ { k - 1 } ^ { ( j ) } A _ { j - 1 } ^ { ( k ) } ) , & k \\geq p + 1 , \\end{cases} \\end{align*}"}
{"id": "7463.png", "formula": "\\begin{align*} \\mathcal { V } _ { \\left \\{ C _ { i } \\right \\} _ { i = 1 } ^ { r } } : = \\mathcal { R } _ { C _ { r } } \\dots \\mathcal { R } _ { C _ { 1 } } , \\end{align*}"}
{"id": "6230.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } u ( x ) + u ( x ) u ( x / q ) = 0 , \\end{align*}"}
{"id": "711.png", "formula": "\\begin{align*} R ^ { N } _ { \\alpha \\beta \\gamma } { } ^ \\delta = \\overline { \\Gamma } _ { \\beta \\gamma , \\alpha } ^ \\delta - \\overline { \\Gamma } _ { \\alpha \\gamma , \\beta } ^ \\delta + \\sum _ \\eta \\overline { \\Gamma } _ { \\beta \\gamma } ^ \\eta \\overline { \\Gamma } _ { \\alpha \\eta } ^ \\delta - \\overline { \\Gamma } _ { \\alpha \\gamma } ^ \\eta \\overline { \\Gamma } _ { \\beta \\eta } ^ \\delta . \\end{align*}"}
{"id": "8854.png", "formula": "\\begin{align*} H : = \\begin{pmatrix} \\sqrt { 2 } x _ 1 & y _ { 1 2 } & 0 & \\cdots & \\cdots & 0 \\\\ y _ { 1 2 } & \\sqrt { 2 } x _ 2 & y _ { 2 3 } & & & \\vdots \\\\ 0 & y _ { 2 3 } & & \\ddots & & \\vdots \\\\ \\vdots & & \\ddots & \\ \\ddots & y _ { N - 2 N - 1 } & 0 \\\\ \\vdots & & & y _ { N - 2 N - 1 } & \\sqrt { 2 } x _ { N - 1 } & y _ { N - 1 N } \\\\ 0 & \\cdots & \\cdots & 0 & y _ { N - 1 N } & \\sqrt { 2 } x _ N \\end{pmatrix} , \\end{align*}"}
{"id": "8248.png", "formula": "\\begin{align*} A . a = \\lambda a \\Rightarrow \\left ( \\begin{array} { l r } G & O \\\\ O & - G \\end{array} \\right ) \\left ( \\begin{array} { l r } a _ { + } \\\\ a _ { - } \\end{array} \\right ) = \\lambda \\left ( \\begin{array} { l r } a _ { + } \\\\ a _ { - } \\end{array} \\right ) , \\end{align*}"}
{"id": "6690.png", "formula": "\\begin{align*} & \\langle \\xi _ 0 , \\xi _ 0 \\rangle + \\langle A _ r \\xi _ 0 , \\xi \\rangle + \\langle A _ 0 \\xi , \\xi \\rangle = ( \\xi _ 1 ^ 0 ) ^ 2 + \\dfrac { N - 1 } { r ^ 2 } ( \\xi _ 1 ) ^ 2 + \\alpha \\lambda | W | ^ { \\alpha - 1 } W e ^ { a ( \\alpha - 1 ) t } \\xi _ 1 ^ 0 \\xi _ 1 + \\\\ & + ( \\xi _ 2 ^ 0 ) ^ 2 + \\dfrac { N - 1 } { r ^ 2 } ( \\xi _ 2 ) ^ 2 + \\alpha \\lambda | Z | ^ { \\alpha - 1 } Z e ^ { a ( \\alpha - 1 ) t } \\xi _ 2 ^ 0 \\xi _ 2 + a ( \\xi _ 1 ) ^ { 2 } + a ( \\xi _ 2 ) ^ 2 - p ( u ^ { p - 1 } + v ^ { p - 1 } ) \\xi _ 1 \\xi _ 2 . \\end{align*}"}
{"id": "4753.png", "formula": "\\begin{align*} \\rho = \\min _ { 0 \\leq n \\leq n _ 0 } \\left ( \\frac { m _ n } { 2 } \\prod _ { j = 0 } ^ { n - 1 } \\sqrt { \\frac { m _ n } { 2 M _ 0 + m _ n } } \\right ) , \\end{align*}"}
{"id": "1856.png", "formula": "\\begin{align*} W _ { j } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { W _ { [ n , j ] } } { z ^ { n + 1 } } = \\sum _ { n \\in \\mathbb { Z } } \\frac { W _ { [ n , j ] } } { z ^ { n + 1 } } \\\\ A _ { j } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { A _ { [ n , j ] } } { z ^ { n + 1 } } = \\sum _ { n \\in \\mathbb { Z } } \\frac { A _ { [ n , j ] } } { z ^ { n + 1 } } \\\\ B _ { j } ( z ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { B _ { [ n , j ] } } { z ^ { n + 1 } } = \\sum _ { n \\in \\mathbb { Z } } \\frac { B _ { [ n , j ] } } { z ^ { n + 1 } } \\end{align*}"}
{"id": "1230.png", "formula": "\\begin{align*} R _ j & = x I - C _ j - D _ { j + 1 } ^ T R _ { j + 1 } ^ { - 1 } D _ { j + 1 } , \\end{align*}"}
{"id": "5368.png", "formula": "\\begin{align*} M _ \\Omega ( H ^ s \\to H ^ { - s } ) \\vcentcolon = \\{ \\ , a \\in M ( H ^ s \\to H ^ { - s } ) \\ , ; \\ , \\inf _ { a ' \\in M _ 0 ( H ^ s \\to H ^ { - s } ) } \\norm { a - a ' } _ { s , - s } < \\delta ( \\Omega ) \\ , \\} . \\end{align*}"}
{"id": "2008.png", "formula": "\\begin{align*} H _ { s , c , 1 } = \\{ n \\in H _ { s , c } \\ \\ P ( n ) \\leq [ s c ^ { - 1 } ] \\} \\end{align*}"}
{"id": "5352.png", "formula": "\\begin{align*} P _ { n , s , p } ( \\Omega ) \\vcentcolon = \\inf _ { u \\in W _ \\Omega ^ { s , p } ( \\R ^ n ) , u \\neq 0 } \\frac { [ u ] _ { s , p , \\R ^ n } ^ p } { \\norm { u } _ { L ^ p ( \\Omega ) } ^ p } , \\end{align*}"}
{"id": "6663.png", "formula": "\\begin{align*} u _ t - \\Delta u ^ m = \\dfrac { u ^ p } { ( 1 + | x | ) ^ \\gamma } \\quad \\R ^ N \\times ( 0 , T ) , \\textrm { f o r s o m e $ T > 0 $ , } \\end{align*}"}
{"id": "6771.png", "formula": "\\begin{align*} I ( n ) : = \\int _ { - 1 } ^ { 1 } e ^ { n \\cdot f ( z ) } g ( z ) \\ , d z \\end{align*}"}
{"id": "3592.png", "formula": "\\begin{align*} \\alpha _ { k , j } & = \\begin{cases} \\sum \\limits _ { i = 1 } ^ { n } a _ i \\cdot \\mathtt { b } ^ + ( \\phi _ i ) \\geqslant 1 - c & t _ { k , j } = f ^ \\sharp _ { k , j } \\\\ \\sum \\limits _ { i = 1 } ^ { n } a _ i \\cdot \\mathtt { b } ^ + ( \\phi _ i ) < - c & \\end{cases} \\end{align*}"}
{"id": "2932.png", "formula": "\\begin{align*} G ( y ) : = \\sum _ { j = 1 } ^ m \\max \\{ F ( x ) : Z ( y ) = h ^ 1 , h ^ 2 , \\ldots , h ^ m \\land x \\in P ( h ^ j ) \\} . \\end{align*}"}
{"id": "103.png", "formula": "\\begin{align*} \\Lambda = \\Lambda _ { \\epsilon , L } , A = \\epsilon ^ d \\big ( - \\Delta ^ \\epsilon + m ^ 2 \\big ) , g = \\epsilon ^ d \\lambda , \\nu = \\epsilon ^ d ( \\mu - m ^ 2 + a ^ \\epsilon ( \\lambda , m ^ 2 ) ) . \\end{align*}"}
{"id": "5770.png", "formula": "\\begin{align*} H / H ^ 0 = W ( K ) , \\end{align*}"}
{"id": "8868.png", "formula": "\\begin{align*} f _ { \\lambda } ( \\lambda ) ^ 2 - \\dfrac { 1 } { 2 } \\nabla f \\cdot \\nabla f ( \\lambda ) = - f ( \\lambda ) \\Delta f ( \\lambda ) + 2 \\sum _ { \\l - k > 1 } \\det \\left ( ( \\lambda I _ N - H ) _ { k | k } \\right ) \\det \\left ( ( \\lambda I _ N - H ) _ { \\l | \\l } \\right ) . \\end{align*}"}
{"id": "53.png", "formula": "\\begin{align*} \\eta \\delta _ n ( \\beta ) = \\sum _ { w \\in W _ { \\rho } ^ { m _ 0 } } \\dfrac { ( f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ) ' ( v _ { n + 1 } + w ) } { f ^ { \\tilde { \\phi } ^ { - ( n + 1 ) } } ( v _ { n + 1 } + w ) } = \\operatorname { T } _ { n + 1 , n } ( \\delta _ { n + 1 } ( \\beta ) ) . \\end{align*}"}
{"id": "8053.png", "formula": "\\begin{align*} 2 \\nabla _ { X } X = - \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) . \\end{align*}"}
{"id": "4518.png", "formula": "\\begin{align*} \\Delta ^ \\Sigma F = | \\nabla u | ^ { - 2 } ( A _ { 1 1 } + A _ { 2 2 } ) = - | \\nabla u | ^ { - 2 } J _ 3 = | \\nabla u | ^ { - 2 } \\mu \\ge 0 . \\end{align*}"}
{"id": "3928.png", "formula": "\\begin{align*} V a r ( \\hat { \\pi } _ { h , n } ( x ) ) & \\le \\frac { c } { T _ n } \\big [ \\frac { 1 } { \\prod _ { l = 1 } ^ d h _ l } ( \\delta _ 1 + \\Delta _ n ) + \\frac { 1 } { \\prod _ { l \\ge 3 } h _ l } \\log ( \\frac { \\delta _ 2 } { \\delta _ 1 } ) + \\delta _ 2 ^ { 1 - \\frac { d } { 2 } } + D + \\frac { 1 } { ( \\prod _ { l = 1 } ^ d h _ l ) ^ 2 } e ^ { - \\rho D } \\big ] . \\end{align*}"}
{"id": "3449.png", "formula": "\\begin{align*} \\Delta = \\langle \\begin{pmatrix} - 1 \\\\ - 1 \\\\ - 1 \\end{pmatrix} , \\begin{pmatrix} 5 \\\\ 1 \\\\ 3 \\end{pmatrix} , \\begin{pmatrix} - 1 \\\\ 1 0 \\\\ 0 \\end{pmatrix} , \\begin{pmatrix} - 1 \\\\ - 1 \\\\ 0 \\end{pmatrix} \\rangle \\end{align*}"}
{"id": "8210.png", "formula": "\\begin{align*} \\psi _ { B _ { k } } = B _ { k } \\left [ \\frac { \\sin ( k x ) } { \\sin ( k a / 2 ) } - \\frac { \\sinh ( k ' x ) } { \\sinh ( k ' a / 2 ) } \\right ] , \\end{align*}"}
{"id": "4865.png", "formula": "\\begin{align*} \\mathfrak { M } _ { i } = \\bigcup \\limits _ { q _ { i } \\leq P _ { i } } \\bigcup \\limits _ { \\substack { 1 \\leq a _ { i } \\leq q _ { i } \\\\ ( a _ { i } , q _ { i } ) = 1 } } \\mathfrak { M } _ { i } ( a _ { i } , q _ { i } ) , \\mathfrak { m } _ { i } = \\left [ \\frac { 1 } { Q _ { i } } , 1 + \\frac { 1 } { Q _ { i } } \\right ] \\backslash \\mathfrak { M } _ { i } , \\end{align*}"}
{"id": "8237.png", "formula": "\\begin{align*} \\sqrt { 1 - 1 6 i m \\beta \\lambda / 3 } = \\pm \\left ( \\sqrt { \\frac { \\sqrt { 1 ^ { 2 } + ( 1 6 m \\beta \\lambda / 3 ) ^ { 2 } } + 1 } { 2 } } - i \\sqrt { \\frac { \\sqrt { 1 ^ { 2 } + ( 1 6 m \\beta \\lambda / 3 ) ^ { 2 } } - 1 } { 2 } } \\right ) , \\end{align*}"}
{"id": "8090.png", "formula": "\\begin{align*} M _ { a } ^ { 3 } : = \\{ ( x , y , z , w ) \\in { \\mathbb R } ^ { 4 } \\mid x ^ { 2 } + y ^ { 2 } = a ^ { 2 } ( 1 - z ^ { 2 } - w ^ { 2 } ) \\} . \\end{align*}"}
{"id": "1616.png", "formula": "\\begin{align*} K ( x , y ) = \\frac { - 6 b ^ 2 ( y ^ 1 ) ^ 6 ( y ^ 2 ) ^ 2 } { \\left [ 2 ( y ^ 1 ) ^ 2 + ( x ^ 1 ) ^ 2 ( y ^ 2 ) ^ 2 \\right ] ^ 4 } . \\end{align*}"}
{"id": "8635.png", "formula": "\\begin{align*} \\frac { \\partial \\psi } { \\partial t } = - { \\rm i } \\widehat { H } ( t ) \\psi , \\psi \\in \\mathcal { H } , t \\in \\mathbb { R } , \\end{align*}"}
{"id": "706.png", "formula": "\\begin{align*} \\nabla u = \\frac { \\widetilde { \\nabla } u } { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 } , | \\nabla u | _ g ^ 2 = \\frac { | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 } { f ( u ) ^ 2 - | \\widetilde { \\nabla } u | _ { \\widetilde { g } } ^ 2 } , \\end{align*}"}
{"id": "5162.png", "formula": "\\begin{align*} h ( W _ 3 ) = & h ( W _ 4 ) = h ( W _ 7 ) = h ( W _ 8 ) = h ( W _ 9 ) = 0 , \\\\ h ( W _ 2 ) = & h ( W _ 6 ) = 1 , \\\\ h ( W _ 5 ) = & 2 , \\\\ h ( W _ 1 ) = & 3 . \\end{align*}"}
{"id": "6512.png", "formula": "\\begin{align*} 2 \\Re \\left ( \\int _ { \\omega _ c } \\Delta u ( m \\cdot \\nabla \\bar { u } ) d x \\right ) = ( d - 2 ) \\int _ { \\omega _ c } \\abs { \\nabla u } ^ 2 d x + \\int _ { \\Gamma _ 0 } ( m \\cdot \\nu ) \\left | \\frac { \\partial u } { \\partial \\nu } \\right | ^ 2 d \\Gamma . \\end{align*}"}
{"id": "6783.png", "formula": "\\begin{align*} \\sum _ { \\substack { i \\in M _ P , \\\\ i \\neq i ^ * } } \\sum _ { j \\in N _ i } a _ { i j } \\tilde x _ { i j } + \\sum _ { \\substack { i j \\in P , \\\\ i \\notin M _ 0 , i \\neq i ^ * } } ( b - s ) \\tilde x _ { i j } + a _ { i ^ * j ^ * } \\tilde x _ { i ^ * j ^ * } & \\leq s + ( b - s ) | \\bar P | \\\\ & = s + ( | M _ P - M _ 0 | - 1 ) ( b - s ) \\\\ & = b + ( | M _ P - M _ 0 | - 2 ) ( b - s ) . \\end{align*}"}
{"id": "8476.png", "formula": "\\begin{align*} y _ k = y _ 1 = \\frac { U } { n } - \\frac { L ( n - 1 ) } { 2 } , \\end{align*}"}
{"id": "7608.png", "formula": "\\begin{align*} \\frac { d P } { d t } = P B R ( t ) ^ { - 1 } B ^ \\textsf { T } P - P A - A ^ \\textsf { T } P - Q ( t ) , P ( t _ f ) = S . \\end{align*}"}
{"id": "6914.png", "formula": "\\begin{align*} \\Big ( ( \\gamma _ j , \\gamma ' _ j ) \\big | ( \\boldsymbol { \\gamma } _ { - j } , \\boldsymbol { \\gamma } ' _ { - j } ) = ( \\boldsymbol { a } , \\boldsymbol { b } ) \\Big ) \\sim \\mathcal { N } ( \\boldsymbol { \\eta } ^ { ( j ) } , \\sigma ^ 2 \\ , \\mathbf { R } ^ { ( j ) } ) \\ , \\end{align*}"}
{"id": "4681.png", "formula": "\\begin{align*} f _ { ( s , r ) } = \\tilde { f } _ { \\tilde { \\nu } ( s , r ) } \\circ f _ r : \\Big ( \\dot r U ^ - P ^ + _ K \\Big ) \\bigcap f _ r ^ { - 1 } \\Big ( \\dot { \\tilde { \\nu } } ( s , r ) \\tilde { U } ^ { - } \\tilde { B } ^ + \\Big ) \\rightarrow \\tilde { G } . \\end{align*}"}
{"id": "3430.png", "formula": "\\begin{align*} \\Upsilon _ { \\mathrm { i n d e x } \\ 3 } ( a , b , c , d ) = \\{ g \\in \\Upsilon : a \\cdot g _ { 1 , 2 } + b \\cdot g _ { 1 , 3 } + c \\cdot g _ { 2 , 1 } + d \\cdot g _ { 3 , 1 } \\equiv 0 \\bmod 3 \\} . \\end{align*}"}
{"id": "1927.png", "formula": "\\begin{align*} S _ { j } ( z ) = S _ { j } ^ { ( q ) } ( z ) = \\sum _ { m = 0 } ^ { \\infty } \\frac { C _ { [ m , j ] } } { z ^ { m ( p + 1 ) + j + 1 } } , \\mbox { f o r a l l } \\ , \\ , \\ , \\ , 0 \\leq j \\leq p , \\ , \\ , \\ , q \\geq 1 . \\end{align*}"}
{"id": "5056.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } E [ | N ^ { n , 3 } _ \\tau | ^ 2 ] = 0 . \\end{align*}"}
{"id": "4564.png", "formula": "\\begin{align*} \\textup { d c o v } ( P ) = \\sum _ { p \\ , \\in \\ , P } x ^ { \\textup { d c o v } ( p ) } = 1 + 7 x + 7 x ^ 2 + x ^ 3 \\ , , \\end{align*}"}
{"id": "5768.png", "formula": "\\begin{align*} G / A = ( G / A ) ^ 0 \\rtimes F ' \\end{align*}"}
{"id": "8414.png", "formula": "\\begin{gather*} \\phi ( \\tau , z ) = h _ 0 ( \\tau ) \\theta _ { 1 , 0 } ( \\tau , z ) + h _ 1 ( \\tau ) \\theta _ { 1 , 1 } ( \\tau , z ) \\end{gather*}"}
{"id": "4998.png", "formula": "\\begin{align*} ( \\widehat { \\Theta } ^ n _ s ) ^ 2 = \\sum _ { j = 1 } ^ 6 \\Psi ^ { n , j } _ s , \\end{align*}"}
{"id": "3056.png", "formula": "\\begin{align*} f _ { \\mathbf { M ^ { \\ast } } _ { - \\xi _ { \\boldsymbol { s } k } } } ( \\boldsymbol { w } ) = c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast } \\overline { G } _ { n } \\left \\{ \\frac { 1 } { 2 } \\boldsymbol { w } ^ { T } \\boldsymbol { w } + \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } \\right \\} , ~ \\boldsymbol { w } \\in \\mathbb { R } ^ { n - 1 } , \\end{align*}"}
{"id": "3985.png", "formula": "\\begin{align*} \\textstyle \\Pr \\big ( \\R ( C _ { a , a ' } ) \\ ! = \\ ! \\frac { 1 } { 2 } \\big ) \\ge ( 1 \\ ! - \\ ! q ^ { - 2 } ) \\prod _ { i = 1 } ^ m ( 1 \\ ! - \\ ! q ^ { - 2 \\mu _ q ( n ) } ) \\ge ( 1 \\ ! - \\ ! q ^ { - 2 } ) \\cdot ( 1 \\ ! - \\ ! q ^ { - 2 \\mu _ q ( n ) } ) ^ n . \\end{align*}"}
{"id": "5493.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } V _ { \\theta } = \\sum _ { s , t = 0 , 1 , \\ldots } \\gamma ^ t p _ { t , \\theta } ( s ) \\sum _ { a } \\pi _ { \\theta } ( a | s ) Q _ { \\theta } ^ * ( s , a ) \\frac { d } { d \\theta } \\log \\pi _ { \\theta } ( a | s ) . \\end{align*}"}
{"id": "8749.png", "formula": "\\begin{align*} \\partial _ t ^ n u ( t ) = A _ n ( t ) u ( t ) \\end{align*}"}
{"id": "4205.png", "formula": "\\begin{align*} A = - \\Delta _ z + \\tfrac 1 4 | z | ^ 2 - i N , \\end{align*}"}
{"id": "4924.png", "formula": "\\begin{align*} \\nabla _ W h ( Y , Z ) - \\nabla _ Z h ( Y , W ) = - \\bar { R } ( \\nu , Y , Z , W ) = 0 \\end{align*}"}
{"id": "2177.png", "formula": "\\begin{align*} \\overline { V } = \\bigcup _ { k \\in K } [ a _ { k } , b _ { k } ] \\cup ( - \\infty , a ] \\cup [ b , \\infty ) , \\end{align*}"}
{"id": "3699.png", "formula": "\\begin{align*} & \\left < \\frac { i } h [ - h ^ 2 \\Delta + V ( x ) - E ( h ) , \\chi _ \\alpha ( x _ n ) h D _ n ] u _ h , \\ , u _ h \\right > _ { L ^ 2 ( \\Omega _ \\Gamma ) } \\\\ = & \\int _ { \\Sigma _ { \\Omega _ \\Gamma } } \\left ( 2 \\chi _ \\alpha ' ( x _ n ) \\xi _ n ^ 2 - \\chi _ \\alpha ( x _ n ) \\partial _ { x _ n } ( R + V ) \\right ) \\ , d \\mu + o ( 1 ) \\\\ = & I _ 1 - I _ 2 + o ( 1 ) \\end{align*}"}
{"id": "6960.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { k \\ge 1 } \\frac { a _ k } { \\lambda _ k ^ 2 - x } . \\end{align*}"}
{"id": "1492.png", "formula": "\\begin{align*} \\log _ { \\lambda } ( 1 + t ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda ^ { n - 1 } ( 1 ) _ { n , 1 / \\lambda } } { n ! } t ^ { n } , ( \\mathrm { s e e } \\ [ 8 ] ) . \\end{align*}"}
{"id": "2574.png", "formula": "\\begin{align*} \\mathbf S _ { 2 3 } + \\mathbf S _ { 3 1 } = C _ 3 ^ { 1 2 2 } - C _ 3 ^ { 2 1 1 } - C _ 3 ^ { ( 1 ) } + C _ 3 ^ { ( 2 ) } = 2 \\ , \\mathbf S _ { 1 2 } - C _ 3 ^ { ( 1 ) } + C _ 3 ^ { ( 2 ) } \\ , \\end{align*}"}
{"id": "4334.png", "formula": "\\begin{align*} \\bar { \\mathcal { S } } = \\sum _ 1 ^ { N - 1 } ( \\sup _ { L _ i } f _ { L _ i } - \\sup _ { L _ { i + 1 } } f _ { L _ { i + 1 } } ) \\left ( e ^ { - i \\hat { \\theta } } \\int _ { \\mathcal { E } _ i } \\Omega \\right ) . \\end{align*}"}
{"id": "3839.png", "formula": "\\begin{align*} E r r U ^ { b ; 0 } _ { i , i _ 1 , i _ 2 , i _ 3 } ( t _ 1 , t _ 2 ) : = \\sum _ { a = 1 , 2 } ( - 1 ) ^ { a - 1 } \\int _ { ( \\R ^ 3 ) ^ 4 } e ^ { i \\Phi _ { \\mu _ 1 , \\mu _ 2 , \\mu _ 3 } ( \\xi , \\eta , \\sigma , \\kappa ; t _ a , X ( t _ a ) ) } \\big ( \\Phi ^ b _ { \\mu _ 1 , \\mu _ 2 , \\mu _ 3 } ( \\xi , \\eta , \\sigma , \\kappa ; V ( t _ a ) , v ) \\big ) ^ { - 1 } \\end{align*}"}
{"id": "2227.png", "formula": "\\begin{align*} \\partial _ t u ( t ) = A _ i ( t ) u ( t ) \\end{align*}"}
{"id": "6544.png", "formula": "\\begin{align*} \\P ( D _ n ) \\geq c , \\ \\ \\ n \\in \\N \\end{align*}"}
{"id": "4922.png", "formula": "\\begin{align*} \\bar \\nabla _ Y \\nu = A ( Y ) , \\end{align*}"}
{"id": "2649.png", "formula": "\\begin{align*} m _ { \\beta , \\gamma , 1 , \\eta } ( \\theta ) = \\frac { 1 } { K ^ * _ { \\beta , \\gamma , 1 , \\eta } } \\sum _ { k = 0 } ^ { \\gamma - 1 } { \\gamma - 1 \\choose k } ( - 1 ) ^ { k } \\theta ^ { - 1 - k + \\beta } , 1 < y < \\eta . \\end{align*}"}
{"id": "3671.png", "formula": "\\begin{align*} \\mathfrak { H i g h } ^ { j } _ { k , \\tilde { k } } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\langle V ( s ) \\rangle ^ { - 1 } E ( s , X ( s ) ) \\cdot \\widetilde { \\mathfrak { H i g h } } ^ { j } _ { k , \\tilde { k } } ( s ) d s , \\end{align*}"}
{"id": "2441.png", "formula": "\\begin{align*} \\texttt { t } ' _ { g f } = t _ { g \\cdot w _ 0 ^ \\zeta , f \\cdot w _ 0 ^ \\zeta } , f , g \\in \\Z ^ { m | n } _ { \\zeta - } . \\end{align*}"}
{"id": "828.png", "formula": "\\begin{align*} \\boldsymbol { y } _ { t , p } ^ { r } = \\mathbf { H } ^ { r } \\boldsymbol { v } _ { t , p } + \\boldsymbol { n } _ { t , p } ^ { r } , \\end{align*}"}
{"id": "1071.png", "formula": "\\begin{align*} \\lim _ { | y | \\to \\infty } \\operatorname { d i s t } \\left ( x + i y , \\operatorname { S p e c } A \\right ) = \\infty . \\end{align*}"}
{"id": "642.png", "formula": "\\begin{align*} \\xi _ 2 ( s ) : = \\pi ^ { - s } \\ , \\Gamma ( s ) \\ , \\zeta ( \\Delta , s ) , \\Omega ( s ) : = \\frac { 1 } { 3 } \\ , s \\ , \\pi ^ { 2 - s } \\ , \\Gamma ( s ) \\ , \\zeta ( \\Delta , s - 1 ) . \\end{align*}"}
{"id": "4480.png", "formula": "\\begin{align*} r = 0 . 4 , ~ ~ w = 1 1 , ~ ~ s = 1 3 0 , ~ ~ p _ 0 = 1 0 ^ { - 4 } . \\end{align*}"}
{"id": "3719.png", "formula": "\\begin{align*} \\lambda \\cdot \\bar { \\psi } = \\bar { \\psi } \\end{align*}"}
{"id": "2322.png", "formula": "\\begin{align*} \\R _ { \\geq 0 } ^ n = \\{ x \\in \\mathbb { R } ^ n : \\} \\end{align*}"}
{"id": "4682.png", "formula": "\\begin{align*} \\Delta _ { \\lambda } ( \\dot { t } ^ { \\sharp } \\dot { u } ^ { \\flat , - 1 } g f _ { ( s , r ) } ( h ) ) & = \\Delta _ { \\lambda } ( \\dot { t } ^ \\sharp h _ 5 ^ { \\sharp , - 1 } z _ 5 ^ \\sharp \\dot { r } ^ { - 1 , \\sharp } ) = \\Delta _ { \\lambda } ( \\iota ( \\dot { t } ^ \\sharp h _ 5 ^ { \\sharp , - 1 } z _ 5 ^ \\sharp \\dot { r } ^ { - 1 , \\sharp } ) ) \\\\ & = \\Delta _ { \\lambda } ( \\iota ( \\dot { t } ^ \\sharp ) \\iota ( h _ 5 ^ { \\sharp , - 1 } ) z _ 5 ^ \\sharp \\iota ( \\dot { r } ^ { - 1 , \\sharp } ) ) . \\end{align*}"}
{"id": "4381.png", "formula": "\\begin{align*} G ( 0 ; \\Psi _ 1 , I _ + ( \\Psi _ 1 ) _ o , f ) \\le \\int _ { \\{ \\Psi < 0 \\} } | \\tilde f _ 0 | ^ 2 \\le \\liminf _ { j \\rightarrow + \\infty } \\int _ { \\{ \\Psi _ 1 < 0 \\} \\cap D _ j } | \\tilde f _ j | ^ 2 = \\lim _ { j \\rightarrow + \\infty } C _ j . \\end{align*}"}
{"id": "8204.png", "formula": "\\begin{align*} - \\frac { \\hslash ^ { 2 } } { 2 m } \\partial _ { x } ^ { 2 } \\psi + \\frac { \\beta \\hslash ^ { 4 } } { 3 m } \\partial _ { x } ^ { 4 } \\psi = E \\psi , \\end{align*}"}
{"id": "3373.png", "formula": "\\begin{align*} \\iota _ j \\circ \\pi _ j \\circ \\varphi _ { n _ j , n _ { j - 1 } } ^ { m } = \\varphi _ { n _ j , n _ { j - 1 } } ^ { m } \\end{align*}"}
{"id": "4498.png", "formula": "\\begin{align*} \\lim \\limits _ { \\zeta \\to 0 } \\frac { ( 1 - e ^ { i 2 ^ { j _ k } \\xi _ k \\zeta } ) + 2 ^ { 2 j _ k } \\zeta ^ 2 e ^ { i 2 ^ { j _ k } \\xi _ k \\zeta } + i 2 ^ { j _ k } \\xi _ k \\zeta } { \\zeta ^ 2 } = 2 ^ { 2 j _ k } ( \\xi _ k ^ 2 + 1 ) , k = 2 , \\cdots , 2 \\mu + 1 . \\end{align*}"}
{"id": "5178.png", "formula": "\\begin{align*} \\int _ { \\Xi _ { \\mu } } x ^ { \\mu ' } e ^ { W / \\hbar } d x _ 1 \\wedge \\cdots \\wedge d x _ a = \\delta _ { \\mu \\mu ' } , \\end{align*}"}
{"id": "6858.png", "formula": "\\begin{align*} ( g _ 1 \\delta _ 1 + 2 a _ 1 \\delta _ 0 ) ^ 2 = 9 a _ 1 ^ 2 \\delta _ 0 ^ 2 < 4 ( 1 - g _ 1 ) \\delta _ 0 , \\end{align*}"}
{"id": "8457.png", "formula": "\\begin{align*} G \\supset \\bigcap _ { N = 1 } ^ { \\infty } \\bigcup _ { n = N } ^ { \\infty } \\bigcup _ { w \\in \\Sigma _ { \\beta _ 1 } ^ n , v \\in \\Sigma _ { \\beta _ 2 } ^ n } \\tilde { J } _ { n , 1 } ( w , v ) \\times \\tilde { J } _ { n , 2 } ( w , v ) . \\end{align*}"}
{"id": "8669.png", "formula": "\\begin{align*} \\begin{aligned} V _ 1 ( x ) & = x _ 3 R _ { f _ 2 } ( x ) - x _ 2 R _ { f _ 3 } ( x ) , \\\\ V _ 2 ( x ) & = x _ 1 R _ { f _ 3 } ( x ) - x _ 3 R _ { f _ 1 } ( x ) , \\\\ V _ 3 ( x ) & = x _ 2 R _ { f _ 1 } ( x ) - x _ 1 R _ { f _ 2 } ( x ) . \\end{aligned} \\end{align*}"}
{"id": "467.png", "formula": "\\begin{align*} P ( Q ( \\infty ) \\geq k ) = \\rho ^ k , k \\geq 0 . \\end{align*}"}
{"id": "2424.png", "formula": "\\begin{align*} \\delta ( a b ) - \\delta ( a ) \\delta ( b ) & = ( \\delta ( a b ) - a b ) - ( \\delta ( a ) - a ) \\delta ( b ) - \\delta ( a ) ( \\delta ( b ) - b ) \\\\ & + ( \\delta ( a ) - a ) ( \\delta ( b ) - b ) \\in \\mathbb K ( H ) , \\end{align*}"}
{"id": "927.png", "formula": "\\begin{align*} ( P ( \\Psi _ t ) P ^ T ( \\Psi _ t ) ( x ) ) ^ { - 1 } = \\begin{bmatrix} I _ n + O ( t ) & O ( t ^ 2 ) \\\\ O ( t ^ 2 ) & 2 t \\cdot \\begin{pmatrix} \\begin{bmatrix} I _ { \\frac { n ( n - 1 ) } { 2 } } & 0 \\\\ 0 & ( 3 \\cdot \\Xi ( \\frac { 1 } { 3 } ) ) ^ { - 1 } \\end{bmatrix} + O ( t ) \\end{pmatrix} \\end{bmatrix} _ . \\end{align*}"}
{"id": "5591.png", "formula": "\\begin{align*} \\tau ^ M : = \\tau + M ; M = \\mathrm { d i a g } ( M _ 1 \\rho _ 1 , \\dots , M _ n \\rho _ n ) . \\end{align*}"}
{"id": "6131.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\\\ \\end{gather*}"}
{"id": "4268.png", "formula": "\\begin{align*} \\lim \\limits _ { | \\zeta | \\to \\infty } f ( \\zeta ) = \\mathfrak c , \\end{align*}"}
{"id": "7615.png", "formula": "\\begin{align*} \\log P ( r , k ) = \\sum _ { i = 0 } ^ { r - 1 } \\log \\Bigl ( 1 + \\dfrac { 2 i } { k } \\Bigr ) < \\sum _ { i = 0 } ^ { r - 1 } \\dfrac { 2 i } { k } = \\dfrac { r ( r - 1 ) } { k } \\ \\implies P ( r , k ) < e ^ { \\frac { r ( r - 1 ) } { k } } . \\end{align*}"}
{"id": "457.png", "formula": "\\begin{align*} \\psi _ { \\mu } ( z ) = \\int _ { \\mathbb { T } } \\frac { z \\zeta } { 1 - z \\zeta } \\ , d \\mu ( \\zeta ) , \\quad \\eta _ { \\mu } ( z ) = \\frac { \\psi _ { \\mu } ( z ) } { 1 + \\psi _ { \\mu } ( z ) } , \\end{align*}"}
{"id": "8183.png", "formula": "\\begin{align*} & \\eta _ { x , p ; \\delta } ( y ) = e ^ { i p ( y - x ) } \\eta _ \\delta ( y - x ) \\end{align*}"}
{"id": "528.png", "formula": "\\begin{align*} F ( \\dot { x } , x , t ) = - \\frac { \\partial } { \\partial x } \\left [ \\frac { \\partial \\Phi _ { g n } ( x , t ) } { \\partial t } \\right ] = \\frac { \\partial E _ { g n } ( \\dot { x } , x , t ) } { \\partial x } \\ . \\end{align*}"}
{"id": "5200.png", "formula": "\\begin{align*} \\iota _ \\Gamma ^ * \\tilde { \\mathfrak { o } } ^ \\pi = o _ N \\otimes ( \\tilde { \\mathfrak { o } } ^ \\pi _ { v ^ o } \\boxtimes \\tilde { \\mathfrak { o } } _ { v ^ c } ) , \\end{align*}"}
{"id": "1554.png", "formula": "\\begin{align*} \\tilde { \\alpha } = \\frac { \\sqrt { ( d \\tilde { x } ^ 1 ) ^ 2 + ( d \\tilde { x } ^ 2 ) ^ 2 + ( d \\tilde { x } ^ 3 ) ^ 2 } } { \\tilde { x } ^ 3 } . \\end{align*}"}
{"id": "3144.png", "formula": "\\begin{align*} \\rho \\left ( A \\right ) = \\int _ { E _ { 1 } } \\hat { \\rho } \\left ( A \\right ) \\mu _ { \\rho } \\left ( \\mathrm { d } \\hat { \\rho } \\right ) , A \\in \\mathcal { U } . \\end{align*}"}
{"id": "2119.png", "formula": "\\begin{align*} ( A x , y ) _ V = \\sum \\limits _ { \\begin{subarray} { 1 } { u , v \\in V ; } \\\\ { u \\neq v } \\end{subarray} } x ( u ) y ( v ) \\sum \\limits _ { e \\in E _ u \\cap E _ v } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } . \\end{align*}"}
{"id": "997.png", "formula": "\\begin{align*} S _ { \\alpha , \\beta } = \\left \\{ x = \\left ( \\begin{matrix} x _ { 1 } \\\\ x _ { 2 } \\end{matrix} \\right ) \\in \\mathbb { Z } ^ { 2 } \\mid 0 \\leq x _ { 1 } , x _ { 2 } < p q \\right \\} . \\end{align*}"}
{"id": "8558.png", "formula": "\\begin{align*} f \\mapsto \\sup _ k \\left ( ( 2 k ) ! ^ { - 1 } \\int f ^ { 2 k } \\ d \\mu \\right ) ^ { 1 / 2 k } = \\left \\bracevert f \\right \\bracevert _ { \\cosh _ 2 } \\ . \\end{align*}"}
{"id": "5708.png", "formula": "\\begin{align*} & \\pi _ { \\{ 3 , 4 , 5 \\} } = e _ 3 ( y _ 1 , y _ 2 , y _ 3 , y _ 4 , y _ 5 ) , \\\\ & \\pi _ { \\{ 2 , 4 , 5 \\} } = e _ 1 ( y _ 1 , y _ 2 ) e _ 2 ( y _ 1 , y _ 2 , y _ 3 , y _ 4 , y _ 5 ) \\end{align*}"}
{"id": "2410.png", "formula": "\\begin{align*} \\| L _ \\mu ( s ) \\| ^ 2 _ { \\mathrm F } & = n \\sum _ { l = 1 } ^ { n - 1 } ( - \\lambda _ { l , n } ) ^ { 2 \\mu } e ^ { - 2 \\lambda ^ 2 _ { l , n } ( t - s ) } \\| \\{ \\Sigma _ n ( \\mathbb U ( s ) ) - \\Sigma _ n ( U ( s ) ) \\} e _ l \\| ^ 2 . \\end{align*}"}
{"id": "8614.png", "formula": "\\begin{align*} G ( s ) = \\sum _ { n = 1 } ^ { N } \\frac { 1 } { s ^ { 2 } + 2 \\zeta _ n \\omega _ { n } s + \\omega _ { n } ^ { 2 } } , \\end{align*}"}
{"id": "6086.png", "formula": "\\begin{align*} f ( x ) & = x ^ 4 - 4 x ^ 3 + 3 x \\\\ & = ( x - 1 ) ^ 4 - 6 ( x - 1 ) ^ 2 - 5 ( x - 1 ) \\\\ & = x ( x - 1 ) ( x ^ 2 - 3 x - 3 ) . \\end{align*}"}
{"id": "6369.png", "formula": "\\begin{align*} f _ { \\ , \\kappa } \\ , ( x ) = \\sqrt { \\frac { 2 \\beta \\kappa } { \\pi } } \\left ( 1 + \\frac { 1 } { 2 } \\kappa \\right ) \\frac { \\Gamma \\ ! \\left ( \\frac { 1 } { 2 \\kappa } + \\frac { 1 } { 4 } \\right ) } { \\Gamma \\ ! \\left ( \\frac { 1 } { 2 \\kappa } - \\frac { 1 } { 4 } \\right ) } \\ , \\ , \\ , \\exp _ { \\kappa } ( - \\beta \\ , x ^ 2 ) \\ \\ . \\end{align*}"}
{"id": "7398.png", "formula": "\\begin{align*} \\psi _ n ( y ) = \\psi _ n ( T _ i y ) \\quad \\textrm { f o r a l l } \\ \\ i \\in \\{ 1 , \\cdots , 1 2 \\} \\ \\ \\textrm { a n d } \\ y \\in \\mathbb { R } ^ N . \\end{align*}"}
{"id": "5922.png", "formula": "\\begin{align*} { \\lambda _ k } = \\frac { \\partial } { { \\partial { \\eta _ k } } } w \\left ( 0 \\right ) \\end{align*}"}
{"id": "6291.png", "formula": "\\begin{align*} & \\int x \\cos ( x ; q ) d _ q x = - q ^ { - \\frac { 1 } { 2 } } x \\sin ( q ^ { - \\frac { 1 } { 2 } } x ; q ) - \\cos ( \\frac { x } { q } ; q ) , \\\\ & \\int x \\sin ( x ; q ) d _ q x = \\frac { x } { q } \\cos ( q ^ { - \\frac { 1 } { 2 } } x ; q ) - \\sin ( \\frac { x } { q } ; q ) , \\end{align*}"}
{"id": "1586.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } = 2 b ^ 2 f ( x ^ 1 ) \\Big [ \\delta _ { \\epsilon 1 } f ' \\left ( x ^ 1 \\right ) \\left \\lbrace f ( x ^ 1 ) f '' \\left ( x ^ 1 \\right ) + f '^ 2 ( x ^ 1 ) + \\sin ^ 2 \\left ( x ^ 2 \\right ) \\right \\rbrace + \\delta _ { \\epsilon 2 } f ( x ^ 1 ) \\sin \\left ( x ^ 2 \\right ) \\cos \\left ( x ^ 2 \\right ) \\Big ] . \\end{align*}"}
{"id": "7176.png", "formula": "\\begin{align*} Y _ t ( u , z ) = \\sum _ { n \\in \\frac { r } { T } + \\mathbb { Z } } u ( n ) z ^ { - n - 1 } \\in V [ g ] [ [ z ^ { \\frac { 1 } { T } } , z ^ { - \\frac { 1 } { T } } ] ] . \\end{align*}"}
{"id": "4658.png", "formula": "\\begin{align*} 0 < \\| \\Psi _ t \\| _ { q , h } = t ^ { - \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| \\Psi _ 1 \\| _ { q , h } < \\infty , \\end{align*}"}
{"id": "7340.png", "formula": "\\begin{align*} f ( w , x , y , z ) = f ( w , x , z , y ) . \\end{align*}"}
{"id": "1364.png", "formula": "\\begin{align*} i \\widehat { u } _ t = \\left \\{ \\mp \\frac { p + 1 } { 4 \\pi } \\int _ \\mathbb { T } | u | ^ { p - 1 } ( x , t ) \\ , d x \\right \\} \\frac { \\widehat { v } } { L _ t [ u ] } + i \\frac { \\widehat { v } _ t } { L _ t [ u ] } = \\mp \\widehat { \\mathcal { R } ^ 1 } [ v , \\cdots , v ] \\frac { 1 } { L _ t [ u ] } + i \\frac { \\widehat { v } _ t } { L _ t [ u ] } , \\end{align*}"}
{"id": "4645.png", "formula": "\\begin{align*} \\hat { A } : = J ^ { - 1 } A J , \\end{align*}"}
{"id": "8193.png", "formula": "\\begin{align*} C ( u , v ) = u ^ { \\dagger } \\cdot A \\cdot v , \\end{align*}"}
{"id": "8846.png", "formula": "\\begin{align*} y = \\frac { t ( t ' - y ) + t ' f _ t ( y ) } { t ' - t } \\leq \\frac { t \\max ( - f ( y ) , 0 ) } { - f ( t ) } + \\frac { t ' f _ t ( y ) } { t ' - t } . \\end{align*}"}
{"id": "6950.png", "formula": "\\begin{align*} E ( s ) : = \\ker ( | \\Gamma | - s I ) = \\ker ( H _ u ^ 2 - s ^ 2 I ) , E _ 1 ( s ) : = \\ker ( | \\Gamma _ 1 | - s I ) = \\ker ( K _ u ^ 2 - s ^ 2 I ) ; \\end{align*}"}
{"id": "7230.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } a _ { n + 1 } = p a _ n + 1 , \\ a _ 1 = 0 , \\\\ b _ { n + 1 } = p b _ n + 1 , \\ b _ 1 = 1 \\end{array} \\right . \\end{align*}"}
{"id": "814.png", "formula": "\\begin{align*} \\Theta \\equiv \\Theta ( p , s _ 1 , q , s _ 2 ) = \\begin{cases} \\min \\{ 1 , p s _ 1 / ( p - 1 ) \\} \\mbox { i f } q s _ 2 < p s _ 1 + 2 ( 1 - s _ 1 ) , \\\\ \\min \\{ 1 , q s _ 2 / ( q - 1 ) \\} \\mbox { i f } p s _ 1 < q s _ 2 . \\end{cases} \\end{align*}"}
{"id": "3297.png", "formula": "\\begin{align*} u ^ s _ { A , q } ( x , y ) = \\sum _ { \\substack { ( \\ell _ 1 , m _ 1 ) \\in \\Gamma \\\\ ( \\ell _ 2 , m _ 2 ) \\in \\Gamma } } \\gamma _ { \\ell _ 1 m _ 1 \\ell _ 2 m _ 2 } h ^ { ( 1 ) } _ { \\ell _ 1 } ( k \\vert x \\vert ) Y ^ { m _ 1 } _ { \\ell _ 1 } \\left ( \\widehat { x } \\right ) h ^ { ( 1 ) } _ { \\ell _ 2 } ( k \\vert y \\vert ) Y ^ { m _ 2 } _ { \\ell _ 2 } \\left ( \\widehat { y } \\right ) . \\end{align*}"}
{"id": "1113.png", "formula": "\\begin{align*} k = \\frac { 2 } { \\sqrt { 4 b - 1 } } , l = \\frac { 1 } { \\sqrt { 4 b - 1 } } . \\end{align*}"}
{"id": "5865.png", "formula": "\\begin{align*} C _ 6 ^ { \\frac { p q } { p - q } } & \\lesssim \\sum _ { k = - \\infty } ^ { M - 1 } \\bigg ( \\int _ { x _ k } ^ { x _ { k + 1 } } u ( t ) \\bigg ( \\int _ t ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } d t \\bigg ) \\Phi ( 0 , x _ k ) \\\\ & + \\sum _ { k = - \\infty } ^ { M - 1 } \\sup _ { y \\in ( x _ k , x _ { k + 1 } ) } \\bigg ( \\int _ y ^ { x _ { k + 1 } } u ( t ) \\bigg ( \\int _ t ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } d t \\bigg ) \\Phi ( x _ k , y ) \\\\ & = C _ { 6 , 1 } + C _ { 6 , 2 } . \\end{align*}"}
{"id": "3541.png", "formula": "\\begin{align*} G _ p = p \\delta _ 1 + ( 1 - p ) \\delta _ \\infty , p > p _ c \\end{align*}"}
{"id": "1704.png", "formula": "\\begin{align*} \\begin{aligned} G ( \\xi - \\eta , \\eta ) = & t _ 0 e ^ { - t _ 0 | \\xi | ^ 2 } \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ { k } t ^ { k } _ 0 } { ( k + 1 ) ! } \\sum _ { m = 0 } ^ { k - \\ell } \\sum _ { \\ell = 0 } ^ { k } C ^ \\ell _ k C ^ { m } _ { k - \\ell } | \\xi - \\eta | ^ { 2 \\ell } | \\eta | ^ { 2 m } ( - | \\xi | ^ 2 ) ^ { k - \\ell - m } . \\end{aligned} \\end{align*}"}
{"id": "1624.png", "formula": "\\begin{align*} u ( z , l ) = { \\underset { \\bar { z } \\in \\{ 1 , . . . , Z \\} } { \\textrm { m a x } } } \\biggl ( \\textrm { m i n } \\biggl \\{ g [ \\bar { z } , z , l ] , u ( \\bar { z } , l - 1 ) \\biggr \\} \\biggr ) , \\end{align*}"}
{"id": "1959.png", "formula": "\\begin{align*} \\frac { ( x _ { 1 } , \\ldots , x _ { p } ) } { ( y _ { 1 } , \\ldots , y _ { p } ) } = ( x _ { 1 } , \\ldots , x _ { p } ) \\cdot \\frac { \\mathbf { 1 } } { ( y _ { 1 } , \\ldots , y _ { p } ) } , \\end{align*}"}
{"id": "7517.png", "formula": "\\begin{align*} P ( E _ { x \\rightarrow y } | S _ { i } ) = { \\left ( \\frac { i } { m _ x } \\right ) } ^ { k _ x } , \\end{align*}"}
{"id": "2839.png", "formula": "\\begin{align*} \\beta = \\frac { h - 1 } { 1 - \\kappa h } . \\end{align*}"}
{"id": "6838.png", "formula": "\\begin{align*} d ( t ) : = a _ 0 ( t - 2 \\frac { e ^ t - 1 } { e ^ t + 1 } ) > 0 , t > 0 . \\end{align*}"}
{"id": "303.png", "formula": "\\begin{align*} { \\rm d } ( { \\rm L } _ n ) & \\ge { \\rm d } \\bigg ( \\bigcup _ { a \\in A } \\bigcup _ { c \\in C _ a ^ v } { \\rm L } _ { a c } \\bigg ) = \\sum _ { a \\in A } \\sum _ { c \\in C _ a ^ v } { \\rm d } ( { \\rm L } _ { a c } ) = \\sum _ { a \\in A } { \\rm d } ( { \\rm L } _ a ) \\ , \\sum _ { c \\in C _ a } \\frac { 1 } { c } , \\end{align*}"}
{"id": "2055.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : T N h a t } \\widehat { T } _ { N , t } \\Phi ( \\mu _ { \\mathbf { x } ^ N } ) = T _ { N , t } [ \\Phi \\circ \\boldsymbol { \\mu } _ N ] ( \\mathbf { x } ^ N ) , \\end{align*}"}
{"id": "5326.png", "formula": "\\begin{align*} \\sigma = \\sigma ( \\Lambda ) = \\Lambda ( [ 0 , 1 ] ) . \\end{align*}"}
{"id": "2548.png", "formula": "\\begin{align*} \\mathsf { T } _ { n } ( R ) = \\begin{pmatrix} R & 0 & \\cdots & 0 \\\\ R & R & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ R & R & \\cdots & R \\\\ \\end{pmatrix} \\end{align*}"}
{"id": "3887.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ p u _ 1 + \\Delta _ p u _ 2 = \\hat f & \\hbox { i n } \\Omega \\\\ u _ 1 - u _ 2 = g _ 1 - g _ 2 & \\hbox { o n } \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
{"id": "1965.png", "formula": "\\begin{align*} \\mathbf { d } _ { p } + ( - a _ { 0 } ^ { ( p ) } A _ { 0 } ^ { ( p ) } , - \\sum _ { j = p - 1 } ^ { p } a _ { 1 } ^ { ( j ) } A _ { j - p + 1 } ^ { ( p ) } , - \\sum _ { j = p - 2 } ^ { p } a _ { 2 } ^ { ( j ) } A _ { j - p + 2 } ^ { ( p ) } , \\ldots , - \\sum _ { j = 1 } ^ { p } a _ { p - 1 } ^ { ( j ) } A _ { j - 1 } ^ { ( p ) } ) . \\end{align*}"}
{"id": "5406.png", "formula": "\\begin{align*} \\frac { 1 } { \\gamma ^ { 1 / 2 } } = 1 - \\frac { m } { m + 1 } \\end{align*}"}
{"id": "829.png", "formula": "\\begin{align*} \\mathbf { H } ^ { r } = \\sum _ { k = 1 } ^ { K } x _ { k } ^ { r } \\boldsymbol { a } \\left ( \\theta _ { k } ^ { r } \\right ) \\boldsymbol { a } ^ { H } \\left ( \\theta _ { k } ^ { r } \\right ) , \\end{align*}"}
{"id": "4448.png", "formula": "\\begin{align*} c _ { \\lfloor \\frac { r - 1 } { 2 } \\rfloor , r - 1 } & = \\prod \\limits _ { w = 0 } ^ { r - k - 1 } ( n - 1 + 2 w ) \\frac { ( - 1 ) ^ { k - 1 } \\ , ( r - 1 ) ! } { 2 ^ { k - 1 } \\ , ( k - 1 ) ! \\ , 1 ! } \\\\ & = - \\prod \\limits _ { w = 0 } ^ { r - k - 1 } ( n - 1 + 2 w ) \\frac { ( - 1 ) ^ { k } \\ , r ! } { 2 ^ { k } \\ , k ! } \\mbox { u s i n g } r = 2 k , \\\\ & = - c _ { k , r } = - c _ { \\lfloor \\frac { r } { 2 } \\rfloor , r } . \\end{align*}"}
{"id": "4913.png", "formula": "\\begin{align*} \\varphi ( t ) = \\ ( ( \\alpha + \\delta + \\beta - 1 ) \\eta ( T ^ * - t ) \\ ) ^ { \\frac { 1 } { 1 - \\delta - \\beta - \\alpha } } , \\end{align*}"}
{"id": "8989.png", "formula": "\\begin{align*} { n + 1 \\brack k } = n { n \\brack k } + { n \\brack k - 1 } , { n \\brack 0 } = \\begin{cases} 1 , & n = 0 , \\\\ 0 , & n \\neq 0 . \\end{cases} \\end{align*}"}
{"id": "4208.png", "formula": "\\begin{align*} H \\Phi _ { \\nu , \\nu ' } = ( | \\nu | + | \\nu ' | + n ) \\Phi _ { \\nu , \\nu ' } . \\end{align*}"}
{"id": "8726.png", "formula": "\\begin{align*} \\omega ( a ) = \\omega ( c ) = 0 \\end{align*}"}
{"id": "6061.png", "formula": "\\begin{align*} f ^ { ( \\ell + j ) } ( x ) = \\left ( f ^ { ( \\ell ) } \\right ) ^ { ( j ) } ( x ) = g ( x ) ( x - \\lambda _ i ) ^ { \\mu ^ { ( \\ell ) } _ { i , j } } \\end{align*}"}
{"id": "8249.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } G . a _ { + } = \\lambda a _ { + } ; \\\\ G . a _ { - } = - \\lambda a _ { - } . \\end{array} \\right . \\end{align*}"}
{"id": "5960.png", "formula": "\\begin{align*} \\Pi _ 1 u ( t _ n ) = u ( t _ n ) , n = 0 , \\dots , N , \\Pi _ 1 u | _ { I _ n } \\in \\mathbb { P } _ 1 ( I _ n ) . \\end{align*}"}
{"id": "3020.png", "formula": "\\begin{align*} \\frac { b + n ( b _ 0 + b _ 1 ) + \\mu } { 4 n + 1 } = \\theta < \\frac { 4 n + 3 } { 4 ( 4 n + 1 ) } + \\epsilon . \\end{align*}"}
{"id": "982.png", "formula": "\\begin{align*} \\tau \\left ( \\theta ^ { k } \\beta \\right ) = H ^ { k } \\tau ( \\beta ) = H ^ { k } \\overline { \\beta } , 0 \\leq k \\leq n - 1 . \\end{align*}"}
{"id": "3591.png", "formula": "\\begin{align*} \\alpha _ { k , j } & = \\begin{cases} \\sum \\limits _ { i = 1 } ^ { n } a _ i \\cdot \\mathtt { w } ^ + ( \\phi _ i ) \\geqslant 1 - c & t _ { k , j } = f ^ \\sharp _ { k , j } \\\\ \\sum \\limits _ { i = 1 } ^ { n } a _ i \\cdot \\mathtt { w } ^ + ( \\phi _ i ) < - c & \\end{cases} \\end{align*}"}
{"id": "102.png", "formula": "\\begin{align*} \\dot C _ t ( V _ t ) ( \\varphi ) \\dot C _ t - \\frac { 1 } { 2 } \\ddot { C } _ t & \\geq \\Big [ \\frac { C _ t } { t } \\big ( C _ t ^ { - 1 } - \\chi _ t C _ t ^ { - 2 } \\big ) \\frac { C _ t } { t } + \\frac { 1 } { t } A C _ t \\Big ] \\dot C _ t \\\\ & = \\Big [ \\frac { 1 } { t } \\Big ( \\frac { 1 } { t } + A \\Big ) C _ t - \\frac { \\chi _ t } { t ^ 2 } \\Big ] \\dot C _ t \\\\ & = \\Big [ \\frac { 1 } { t } - \\frac { \\chi _ t } { t ^ 2 } \\Big ] \\dot C _ t = \\dot \\kappa _ t \\dot C _ t . \\end{align*}"}
{"id": "7597.png", "formula": "\\begin{align*} x ( t ) = \\frac { 2 t ^ 3 - 2 t ^ 2 } { 2 t - 1 } \\end{align*}"}
{"id": "6842.png", "formula": "\\begin{align*} p ( t , v , g ) = \\frac { 1 } { V _ F } p _ 0 ( t , g ) . \\end{align*}"}
{"id": "2730.png", "formula": "\\begin{align*} _ { \\Gamma } \\circ \\gamma = ( \\gamma _ t \\circ \\nabla ) ' & & & & \\mathbf { c u r l } _ { \\Gamma } \\circ \\gamma _ t = ( \\gamma _ n \\circ \\mathbf { c u r l } ) ' . \\end{align*}"}
{"id": "8921.png", "formula": "\\begin{align*} A ^ { k l } _ { i j } \\xi _ i \\xi _ j \\eta _ k \\eta _ l : = ( D _ { p _ k p _ l } A _ { i j } ) \\xi _ i \\xi _ j \\eta _ k \\eta _ l \\ge 0 \\end{align*}"}
{"id": "6698.png", "formula": "\\begin{align*} \\prescript { } { \\gamma } { \\mathcal { F } } _ \\tau f ( \\lambda ) : = \\int _ G e ^ \\tau _ { \\lambda , 1 } ( g ) f ( g ) \\ ; d g , \\ ; \\ ; \\ ; \\ ; \\lambda \\in \\mathfrak { a } ^ * _ \\mathbb { C } . \\end{align*}"}
{"id": "4963.png", "formula": "\\begin{align*} \\kappa _ 4 = \\sum _ { k = 0 } ^ { \\infty } \\int _ 0 ^ 1 \\int _ { 0 } ^ { 1 \\wedge ( y + k ) } [ ( y + k ) ^ \\alpha - ( y + k - z ) ^ \\alpha ] ^ 2 d z d y . \\end{align*}"}
{"id": "1344.png", "formula": "\\begin{align*} S ( x ) = S _ \\infty ( x ) ( 1 + \\eta N * \\rho ) , \\end{align*}"}
{"id": "7304.png", "formula": "\\begin{align*} \\int _ 0 ^ s e ^ { \\mu _ 3 s _ 1 } ( T - t ) ^ { 2 { \\sf d } _ 1 } \\sigma ^ 2 d s _ 1 < C \\left [ \\tfrac { 1 } { \\mu _ 3 } e ^ { \\mu _ 3 s _ 1 } ( T - t ) ^ { 2 { \\sf d } _ 1 } \\sigma ^ 2 \\right ] _ { s _ 1 = 0 } ^ { s _ 1 = s } \\lesssim \\tfrac { 1 } { \\mu _ 3 } e ^ { \\mu _ 3 s } ( T - t ) ^ { 2 { \\sf d } _ 1 } \\sigma ^ 2 . \\end{align*}"}
{"id": "2030.png", "formula": "\\begin{align*} \\iint _ { E \\times E } \\varphi _ 2 ( z _ 1 ' , z _ 2 ' ) \\Gamma ^ { ( 2 ) } ( z _ 1 , z _ 2 , \\dd z _ 1 ' , \\dd z _ 2 ' ) & = \\int _ { \\Theta } \\varphi _ 2 ( \\psi _ 1 ( z _ 1 , z _ 2 , \\theta ) , \\psi _ 2 ( z _ 1 , z _ 2 , \\theta ) ) \\nu ( \\dd \\theta ) \\\\ & = \\int _ { \\Theta } \\varphi _ 2 ( \\psi _ 2 ( z _ 2 , z _ 1 , \\theta ) , \\psi _ 1 ( z _ 2 , z _ 1 , \\theta ) ) \\nu ( \\dd \\theta ) , \\end{align*}"}
{"id": "3350.png", "formula": "\\begin{align*} \\varphi ( u _ j ) = ( w _ { 1 , j } , w _ { 2 , j } , \\hdots , w _ { L , j } ) \\in B \\end{align*}"}
{"id": "3454.png", "formula": "\\begin{align*} \\mathrm { T V } _ { q } ( X ) = \\mathrm { E } [ ( X - \\mathrm { T C E } _ { X } ( x _ { q } ) ) ^ { 2 } | X > x _ { q } ] \\end{align*}"}
{"id": "141.png", "formula": "\\begin{align*} \\| T _ { \\lambda ^ 3 } \\| _ { L ^ 1 \\cap L ^ \\infty } & \\leq c \\lambda ^ 3 \\Big [ \\Big ( \\frac { 1 } { m ^ 6 _ t } + \\frac { 1 } { m ^ 8 _ t } \\Big ) { \\bf 1 } _ { d = 2 } + \\Big ( \\frac { 1 } { m ^ 2 _ t } + \\frac { 1 } { m ^ 5 _ t } \\Big ) { \\bf 1 } _ { d = 3 } + P _ { 3 } \\big ( \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } \\big ) \\Big ] . \\end{align*}"}
{"id": "5182.png", "formula": "\\begin{align*} \\{ x y = 0 \\} / \\mu _ d , \\end{align*}"}
{"id": "376.png", "formula": "\\begin{align*} \\frac { \\| \\alpha / \\pi \\| _ 2 ^ 2 - 1 } { \\| \\alpha / \\pi \\| _ 2 ^ 2 } - \\pi ( A ) = 2 \\sum _ x \\pi ( x ) \\left ( \\mathbb { P } _ x [ T _ A \\le t _ { \\mathrm { m e d } } ] \\right ) ^ 2 - 2 \\frac { 1 } { \\| \\alpha / \\pi \\| _ 2 ^ 2 } ( 1 - e ^ { - \\lambda _ 1 t _ { \\mathrm { m e d } } } ) ^ { 2 } . \\end{align*}"}
{"id": "8275.png", "formula": "\\begin{align*} k = \\frac { 1 } { 2 \\hslash \\sqrt { \\beta / 3 } } \\sqrt { 1 - \\sqrt { 1 - \\frac { 1 6 } { 3 } m \\beta | E | } } \\end{align*}"}
{"id": "3368.png", "formula": "\\begin{align*} F _ m ( \\psi _ n ) ( x , y ) = ( ( r _ { n + 1 } / r _ n ) x , ( r _ { n + 1 } / r _ n - p _ n ) y ) ) . \\end{align*}"}
{"id": "6757.png", "formula": "\\begin{gather*} H ( \\Q ^ E | \\R ^ E ) = \\log 2 > 0 , \\\\ H ( \\Q ^ { D \\cup E \\cup F } | \\R ^ { D \\cup E \\cup F } ) = H ( \\Q ^ { D \\cup E } | \\R ^ { D \\cup E } ) = H ( \\Q ^ { E \\cup F } | \\R ^ { E \\cup F } ) = 0 . \\end{gather*}"}
{"id": "8367.png", "formula": "\\begin{align*} H ( e _ i ) ( b ) & = H ( e _ { t _ 0 } ) ( b ) = H ( e _ { t _ 0 } ) ( H ( u _ { p _ 1 } ) ( a ) ) = H ( e _ { t _ 0 } \\cdot u _ { p _ 1 } ) ( a ) = \\mathstrut \\\\ & = H ( e _ { t _ 1 } \\cdot v _ { q _ 1 } ) ( a ) = H ( e _ { t _ 1 } ) ( H ( v _ { q _ 1 } ) ( a ) ) = H ( e _ { t _ 1 } ) ( b ) , \\end{align*}"}
{"id": "1348.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + \\bigtriangleup u \\pm | u | ^ { p - 1 } u = 0 \\\\ u ( x , 0 ) = u _ 0 \\in H ^ s _ x ( \\mathbb { T } ) , \\end{cases} \\end{align*}"}
{"id": "4729.png", "formula": "\\begin{align*} H _ { w } e _ { ( k ) } H _ { \\omega _ { ( d ) } } H _ { y } = \\sum \\limits _ { \\substack { ( \\omega ' , \\chi , \\varpi ' ) \\in B _ { k , n } ^ { * } \\times \\mathfrak { S } _ { 2 k + 1 , n } \\times B _ { k , n } \\\\ \\ell ( \\omega ' ) + \\ell ( \\chi ) + \\ell ( \\varpi ' ) \\leq \\ell ( w ) + \\ell ( \\omega _ { ( d ) } ) + \\ell ( y ) } } r _ { \\omega ' } s _ { \\varpi ' } t ^ { \\chi } H _ { \\omega ' } e _ { ( k ) } H _ { \\chi } H _ { \\varpi ' } \\end{align*}"}
{"id": "1494.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\big ( \\log _ { \\lambda } ( 1 + t ) \\big ) ^ { k } = \\sum _ { n = k } ^ { \\infty } S _ { 1 , \\lambda } ( n , k ) \\frac { t ^ { n } } { n ! } , \\end{align*}"}
{"id": "2955.png", "formula": "\\begin{align*} K ^ { G ( K / E ) } = F _ \\infty ^ { G ( F _ \\infty / E ) } = E . \\end{align*}"}
{"id": "683.png", "formula": "\\begin{align*} \\partial _ s \\left ( e ^ { f _ o } v _ s \\right ) = \\Delta _ { g _ o } ^ { f _ o } \\left ( e ^ { f _ o } v _ s \\right ) . \\end{align*}"}
{"id": "7223.png", "formula": "\\begin{align*} a _ n = \\frac { p ^ { n - 1 } - 1 } { p - 1 } . \\end{align*}"}
{"id": "742.png", "formula": "\\begin{align*} \\mathcal A _ k : = \\big \\{ f \\in ( \\mathcal H , K _ 1 ) \\otimes ( \\mathcal H , K _ 2 ) : \\big ( \\big ( \\tfrac { \\partial } { \\partial \\zeta } \\big ) ^ { \\boldsymbol i } f ( z , \\zeta ) \\big ) _ { | \\Delta } = 0 , \\ ; | \\boldsymbol i | \\leq k \\big \\} , \\end{align*}"}
{"id": "1675.png", "formula": "\\begin{align*} & c _ 2 ( F ( \\mathbf { x } _ 0 ) , F ( \\mathbf { y } _ 0 ) , \\delta ) = c _ 1 ( g _ \\delta ( F ( \\mathbf { x } _ 0 ) ) , g _ \\delta ( F ( \\mathbf { y } _ 0 ) ) ) \\\\ \\neq \\ , & c _ 2 ( F ( \\mathbf { x } _ 1 ) , F ( \\mathbf { y } _ 1 ) , \\delta ) = c _ 1 ( g _ \\delta ( F ( \\mathbf { x } _ 1 ) ) , g _ \\delta ( F ( \\mathbf { y } _ 1 ) ) ) \\end{align*}"}
{"id": "3294.png", "formula": "\\begin{align*} u ^ \\infty _ { A , q } ( d , \\theta ) = \\underset { ( \\ell _ 1 , m _ 1 ) \\in \\Gamma } { \\sum } \\ ; \\underset { ( \\ell _ 2 , m _ 2 ) \\in \\Gamma } { \\sum } \\mu _ { \\ell _ 1 m _ 1 \\ell _ 2 m _ 2 } Y ^ { m _ 1 } _ { \\ell _ 1 } ( d ) Y ^ { m _ 2 } _ { \\ell _ 1 } ( \\theta ) , \\end{align*}"}
{"id": "2807.png", "formula": "\\begin{align*} \\tilde { w } ( \\kappa , h ) : = \\ \\frac { 2 ( 1 - h ) + 2 ( 1 - \\kappa h ) } { h ( 2 - h ) ( 2 - \\kappa h ) } \\end{align*}"}
{"id": "1565.png", "formula": "\\begin{align*} { R } ^ { i } _ { k } = 2 \\frac { \\partial { G } ^ i } { \\partial x ^ k } - y ^ j \\frac { \\partial ^ 2 { G } ^ i } { \\partial x ^ j \\partial y ^ k } + 2 { G } ^ j \\frac { \\partial ^ 2 { G } ^ i } { \\partial y ^ j \\partial y ^ k } - \\frac { \\partial { G } ^ i } { \\partial y ^ j } \\frac { \\partial { G } ^ j } { \\partial y ^ k } . \\end{align*}"}
{"id": "462.png", "formula": "\\begin{align*} \\Omega _ { \\mu } = \\{ r \\zeta : \\zeta \\in \\mathbb { T } , 0 \\le r < R _ { \\mu } ( \\zeta ) \\} . \\end{align*}"}
{"id": "1345.png", "formula": "\\begin{align*} C _ 1 - \\alpha \\langle x \\rangle \\leq M _ \\infty ( x ) : = \\log ( S _ \\infty ( x ) ) \\leq C _ 2 - \\alpha \\langle x \\rangle . \\end{align*}"}
{"id": "663.png", "formula": "\\begin{gather*} \\Box _ { g _ { i , t } } u _ { i , t } = 0 M \\times [ s , s ' ] , \\\\ u _ { i , s } = u ^ 0 _ i , \\end{gather*}"}
{"id": "7003.png", "formula": "\\begin{align*} \\widetilde U _ \\tau = \\bigsqcup _ { \\sigma } H _ \\sigma , \\end{align*}"}
{"id": "2279.png", "formula": "\\begin{align*} & \\sup _ { x \\geq 1 } \\Vert ( w , \\Omega ) x ^ { - \\sigma _ 0 } \\Vert _ { L _ \\eta ^ 2 } ^ 2 + \\Vert ( w , \\Omega ) x ^ { - \\sigma _ 0 - \\frac { 1 } { 2 } } \\Vert _ { L _ x ^ 2 L _ \\eta ^ 2 } ^ 2 + \\Vert ( w _ \\eta , \\Omega _ \\eta ) x ^ { - \\sigma _ 0 } \\Vert _ { L _ x ^ 2 L _ \\eta ^ 2 } ^ 2 \\\\ & \\lesssim R ( \\delta , \\sigma ) ^ 3 + R ( \\delta , \\sigma ) ^ 4 + R ( \\delta , \\sigma ) ^ 6 . \\end{align*}"}
{"id": "4741.png", "formula": "\\begin{align*} k _ { v , n } = \\min \\left \\{ v \\left ( \\psi _ n ^ 2 ( x ( P ) ) \\right ) , v \\left ( \\phi _ n ( x ( P ) ) \\right ) \\right \\} . \\end{align*}"}
{"id": "2842.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { h \\big [ \\kappa h ^ 2 - 2 h ( 1 + \\kappa ) + 3 \\big ] } { 2 L \\big [ 2 - h ( 1 + \\kappa ) \\big ] } \\big \\| g _ { 0 } \\big \\| ^ 2 + \\frac { h } { 2 L \\big [ 2 - h ( 1 + \\kappa ) \\big ] } \\| g _ 1 \\| ^ 2 { } \\leq { } f _ { 0 } - f _ { 1 } \\end{aligned} \\end{align*}"}
{"id": "4798.png", "formula": "\\begin{align*} \\sigma _ 0 = \\frac 1 2 ( \\sigma + \\sigma ' ) , \\end{align*}"}
{"id": "7754.png", "formula": "\\begin{align*} U = \\Xi \\Lambda K ^ T , \\end{align*}"}
{"id": "8711.png", "formula": "\\begin{align*} g ( x ^ { 1 } , x ^ { 2 } , \\dot { x } ^ { 1 } , \\dot { x } ^ { 2 } ) = \\sqrt { ( \\dot { x } ^ { 1 } ) ^ 2 + ( \\dot { x } ^ { 2 } ) ^ 2 } + b ( \\cos ( t + c ) \\dot { x } ^ 1 + \\sin ( t + c ) \\dot { x } ^ 2 ) . \\end{align*}"}
{"id": "6805.png", "formula": "\\begin{align*} \\left \\langle u , v \\right \\rangle _ { H ^ 2 _ { a } ( 0 , 1 ) } = \\int _ { 0 } ^ { 1 } f v \\ , d x \\Longleftrightarrow \\int _ { 0 } ^ { 1 } \\biggl ( u v + a u '' v '' \\biggr ) d x = \\int _ { 0 } ^ { 1 } f v \\ , d x , \\end{align*}"}
{"id": "2276.png", "formula": "\\begin{align*} R ( \\delta , \\sigma ) : = { \\rm { m a x } } \\{ \\Vert ( e _ \\sigma , z ^ l ( e _ \\delta - \\delta ) ) \\Vert _ { L ^ \\infty } , \\Vert z ^ l ( e ' _ \\delta , e ' _ \\sigma ) \\Vert _ { L ^ \\infty } , \\Vert z ^ l ( e ' _ \\delta , e ' _ \\sigma , e '' _ \\delta , e '' _ \\sigma ) \\Vert _ { L ^ 2 } \\} . \\end{align*}"}
{"id": "1531.png", "formula": "\\begin{align*} \\int _ M ^ x e ^ { \\lambda y } d F _ Z ( y ) & \\leq \\ , e ^ { \\lambda M } ( 1 - F _ Z ( M ) ) + \\lambda \\int _ M ^ x e ^ { \\lambda y } ( 1 - F _ Z ( y ) ) d y \\\\ [ 3 p t ] & \\leq \\ , V e ^ { \\lambda M } M ^ { - \\gamma } + \\lambda V \\int _ M ^ x e ^ { \\lambda y } y ^ { - \\gamma } d y \\\\ [ 3 p t ] & = \\ , V e ^ { 2 \\gamma } \\left ( \\frac { \\lambda } { 2 \\gamma } \\right ) ^ { \\gamma } + V e ^ { \\lambda x } x ^ { - \\gamma } \\int _ 0 ^ { \\lambda ( x - M ) } e ^ { - w } \\left ( 1 - \\frac { w } { \\lambda x } \\right ) ^ { - \\gamma } d w , \\end{align*}"}
{"id": "58.png", "formula": "\\begin{align*} ( \\alpha , 1 - \\zeta { \\pi ' _ n } ^ j ) _ { \\rho , L , n } & = ( \\dfrac { \\zeta { \\pi ' _ n } ^ j } { 1 - \\zeta { \\pi ' _ n } ^ j } \\alpha , ( \\zeta { \\pi ' _ n } ^ j ) ^ { - 1 } ) _ { \\rho , L , n } \\\\ & = - j ( \\dfrac { \\zeta { \\pi ' _ n } ^ j } { 1 - \\zeta { \\pi ' _ n } ^ j } \\alpha , \\pi ' _ n ) _ { \\rho , L , n } . \\end{align*}"}
{"id": "8419.png", "formula": "\\begin{gather*} t _ 1 ( \\tau , z ) = \\overline { \\theta _ { 1 , 0 } ^ 0 ( \\tau ) } \\theta _ { 1 , 0 } ( \\tau , z ) + \\overline { \\theta _ { 1 , 1 } ^ 0 ( \\tau ) } \\theta _ { 1 , 1 } ( \\tau , z ) \\end{gather*}"}
{"id": "5593.png", "formula": "\\begin{align*} u _ i ^ 1 = \\sum _ { k = 1 } ^ { n } ( \\tau ^ { M , 0 } ) _ { i , k } ^ { - 1 } d ^ 0 _ k . \\end{align*}"}
{"id": "6844.png", "formula": "\\begin{align*} p ( t , g ) : = p _ 0 ( t , g ) . \\end{align*}"}
{"id": "6608.png", "formula": "\\begin{align*} \\norm { \\rho _ t ( \\beta ) f _ t } ^ 2 & = \\Lambda _ \\Omega ( t ) \\int _ { \\chi \\in \\mathcal { X } _ p ^ { t e m p } / W } \\chi ( \\phi _ { \\beta ^ \\ast \\ast \\beta } ) \\ , d \\nu _ p ( \\chi ) + O _ \\Omega \\Big ( \\| \\beta ^ \\ast \\ast \\beta \\| _ { L ^ 1 } t ^ { d - \\delta } \\Big ) \\\\ & = \\Lambda _ \\Omega ( t ) \\int _ { \\chi \\in \\mathcal { X } _ p ^ { t e m p } / W } | \\beta ( \\omega _ { \\chi } ) | ^ 2 \\ , d \\nu _ p ( \\chi ) + O _ \\Omega \\Big ( \\| \\beta \\| ^ 2 _ { L ^ 1 } t ^ { d - \\delta } \\Big ) . \\end{align*}"}
{"id": "7533.png", "formula": "\\begin{align*} I : = & \\bigg \\| 1 _ { \\{ | T _ { n } - 1 | > 1 \\} } \\sup _ { t \\in [ 0 , 1 ] } | X _ { n } ( t ) - B ( t ) | \\bigg \\| _ { L ^ { \\frac { p } { 2 } } } \\\\ \\le & \\big ( \\mu ( | T _ { n } - 1 | > 1 ) \\big ) ^ { 1 / p } \\bigg \\| \\sup _ { t \\in [ 0 , 1 ] } | X _ { n } ( t ) - B ( t ) | \\bigg \\| _ { L ^ { p } } \\\\ \\le & \\big ( \\mu ( | T _ { n } - 1 | > 1 ) \\big ) ^ { 1 / p } \\bigg ( \\bigg \\| \\sup _ { t \\in [ 0 , 1 ] } | X _ { n } ( t ) | \\bigg \\| _ { L ^ { p } } + \\bigg \\| \\sup _ { t \\in [ 0 , 1 ] } | B ( t ) | \\bigg \\| _ { L ^ { p } } \\bigg ) \\\\ \\le & C n ^ { - \\frac { 1 } { 4 } } . \\end{align*}"}
{"id": "4566.png", "formula": "\\begin{align*} h ^ * _ P = ( h _ 1 ^ * , \\ldots , h _ s ^ * , 0 , \\ldots , 0 ) \\in \\Z ^ d \\ , , \\end{align*}"}
{"id": "608.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } u ( x _ i = j ) = I ^ { n } _ { x _ i } u + A ^ { n } _ { x _ i } u + D _ { x _ i , M } ^ { n } u + E ^ { n } _ { x _ i , M } u . \\end{align*}"}
{"id": "5551.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\rightarrow \\infty } \\left . \\sum \\limits _ { i = 0 } ^ { b r } \\dbinom { a r } { i } \\middle / r \\dbinom { a r } { b r } \\right . = 0 . \\end{align*}"}
{"id": "2151.png", "formula": "\\begin{align*} d \\nu ( s ) = 3 ^ { - 1 } s ^ { 2 } I _ { [ - 1 , 2 ] } ( s ) \\ , d \\lambda ( s ) . \\end{align*}"}
{"id": "4277.png", "formula": "\\begin{align*} b _ 2 = ( g _ 2 g _ 1 ^ { - 1 } ) b _ 1 \\sigma ( g _ 2 g _ 1 ^ { - 1 } ) ^ { - 1 } \\end{align*}"}
{"id": "4421.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { j = 1 } ^ e T _ { e , j } \\right ) \\cup \\left ( \\bigcup _ { \\ell = 1 } ^ e U _ { e , \\ell } \\right ) = [ 4 k d + 1 , 4 k d + 8 k e ] \\cup [ 4 k N + 1 , 4 k N + 4 e ] = [ 4 k d + 1 , 4 k N + 4 e ] , \\end{align*}"}
{"id": "4472.png", "formula": "\\begin{align*} x y ^ { ( n + 2 ) } ( x ) = - ( n + 1 ) y ^ { ( n + 1 ) } ( x ) + x y ^ { ( n ) } ( x ) + n y ^ { ( n - 1 ) } ( x ) . \\end{align*}"}
{"id": "1623.png", "formula": "\\begin{align*} P _ { o u t } & = 1 - \\textrm { P r } \\bigg [ \\bar { \\gamma _ 1 } \\ge 1 , \\bar { \\gamma _ 2 } \\ge 1 , . . . , \\bar { \\gamma _ i } \\ge 1 , . . . , \\bar { \\gamma _ N } \\ge 1 \\bigg ] \\\\ & = 1 - \\textrm { P r } \\bigg [ { \\underset { i \\in \\{ 1 , . . . , N \\} } { \\textrm { m i n } } } \\{ \\bar { \\gamma _ i } \\} \\ge 1 \\bigg ] . \\end{align*}"}
{"id": "7731.png", "formula": "\\begin{align*} \\Phi ( t , \\tau ) = I + \\int _ \\tau ^ { t } A ( \\tau ' ) \\Phi ( \\tau ' , \\tau ) d \\tau ' \\quad t \\geq \\tau \\geq 0 . \\end{align*}"}
{"id": "8859.png", "formula": "\\begin{align*} \\Bigl ( { \\rm t w o \\ s u m m a t i o n s \\ i n \\ \\eqref { b 1 } } \\Bigr ) = \\sum _ { k = 1 } ^ N & \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ( \\lambda _ j ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { k | k } \\right ) \\\\ - 2 \\sum _ { \\l - k > 1 } & \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { k | \\l } \\right ) \\det \\left ( ( \\lambda _ j ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { \\l | k } \\right ) . \\end{align*}"}
{"id": "2856.png", "formula": "\\begin{align*} \\begin{aligned} c ' ( h ) = - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\left [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\right ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 \\end{aligned} \\end{align*}"}
{"id": "5989.png", "formula": "\\begin{align*} u ( t ) = 1 + t ^ \\beta + H ( t - r ) t ^ \\beta , \\end{align*}"}
{"id": "4252.png", "formula": "\\begin{align*} - \\frac { \\mathrm { d } } { \\mathrm { d } x } \\log \\widetilde { \\Phi } ( x ) \\geq - \\frac { \\mathrm { d } } { \\mathrm { d } x } \\log f ( x ) = \\frac { 1 } { 2 ( x + 2 ) \\sqrt { \\log ( x + 2 ) } } . \\end{align*}"}
{"id": "4168.png", "formula": "\\begin{align*} \\Lambda _ k ^ \\lambda g = g \\times _ \\lambda \\varphi _ k ^ \\lambda . \\end{align*}"}
{"id": "7524.png", "formula": "\\begin{align*} b _ 1 \\sqrt { n _ 1 } + b _ 2 \\sqrt { n _ 2 } + \\cdots + b _ k \\sqrt { n _ k } = 0 \\end{align*}"}
{"id": "120.png", "formula": "\\begin{align*} \\eta _ t = C _ \\infty ( 0 ) - C ( 0 ) \\geq 0 , \\gamma _ t = \\| C _ \\infty ^ 3 \\| _ { L ^ 1 } - \\| C ^ 3 \\| _ { L ^ 1 } \\geq 0 . \\end{align*}"}
{"id": "4667.png", "formula": "\\begin{align*} R ( r ) = \\begin{cases} \\frac { 1 } { 2 } ( 1 + \\cos \\pi r ) , & 0 \\leq r \\leq 1 , \\\\ 0 , & r > 1 . \\end{cases} \\end{align*}"}
{"id": "1805.png", "formula": "\\begin{align*} \\begin{bmatrix} - A ^ T \\hat z \\\\ A \\hat x \\end{bmatrix} \\in \\begin{bmatrix} \\partial f ( \\hat x ) + \\nabla h ( \\hat x ) \\\\ \\partial g ^ \\ast ( \\hat z ) \\end{bmatrix} \\end{align*}"}
{"id": "4766.png", "formula": "\\begin{align*} b _ { i _ 1 , j _ 1 } = b _ { i _ 1 , j _ 2 } = b _ { i _ 2 , j _ 2 } = b _ { i _ 2 , j _ 3 } = b _ { i _ 3 , j _ 3 } = b _ { i _ 3 , j _ 1 } = 1 . \\end{align*}"}
{"id": "6092.png", "formula": "\\begin{align*} \\theta _ { N + 2 \\ , B } = \\sum _ { C = 1 } ^ { N + 1 } h _ { B C } \\theta _ C \\end{align*}"}
{"id": "3751.png", "formula": "\\begin{align*} = s ^ 2 \\sin \\theta \\big [ 1 - \\cos ( \\theta - \\tilde { \\theta } ) + \\sin \\theta \\sin \\tilde { \\theta } ( 1 - \\cos ( \\phi - \\tilde { \\phi } ) ) \\big ] \\gtrsim s ^ 2 \\sin \\theta | 1 - \\cos ( \\theta - \\tilde { \\theta } ) | . \\end{align*}"}
{"id": "6463.png", "formula": "\\begin{align*} \\langle \\Phi _ 0 ( x ) ^ { ( 0 ) } ( \\alpha ) , e \\rangle & = \\Phi _ 0 ( x ) ^ { ( 0 ) } [ \\langle \\alpha , e \\rangle ] - [ \\Phi _ 0 ( x ) ^ { ( 0 ) } , \\iota _ e ] ^ { ( - 1 ) } ( \\alpha ) \\\\ & = \\varrho ( x ) [ \\langle \\alpha , e \\rangle ] - \\langle \\alpha , \\nabla _ x ( e ) \\rangle . \\end{align*}"}
{"id": "8747.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t ^ n \\psi ( t ) = A _ { n } ( t ) \\psi ( t ) \\end{array} \\end{align*}"}
{"id": "6358.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\left \\langle J ' _ \\lambda ( v _ n ) , v _ n - v \\right \\rangle = 0 . \\end{align*}"}
{"id": "1299.png", "formula": "\\begin{align*} \\mathcal { G } ( r ( T ) , x ) & = x - \\sum _ { j = 1 } ^ { k } \\dfrac { \\alpha _ j } { \\mathcal { G } ( r ( T _ j ) , x ) } . \\end{align*}"}
{"id": "5727.png", "formula": "\\begin{align*} \\pi _ { [ 1 , 1 ] } \\cdot \\pi _ { [ 2 , 2 ] } = \\pi _ { [ 1 , 1 ] } \\cdot \\pi _ { \\{ 2 \\} } = \\pi _ { [ 1 , 1 ] } \\cdot ( y _ 1 + y _ 2 ) = \\pi _ { [ 1 , 1 ] } \\cdot 2 y _ 2 = 2 \\pi _ { [ 1 , 2 ] } , \\end{align*}"}
{"id": "5492.png", "formula": "\\begin{align*} V _ { \\theta } ^ { \\eta } = \\sum _ { s } \\eta ( s ) J _ { \\theta } ^ * ( s ) , \\end{align*}"}
{"id": "1077.png", "formula": "\\begin{gather*} \\lim _ { | y | \\to \\infty } \\left \\| ( 1 + ( w - x ) R ( \\lambda , A _ 2 ) ) ^ { - 1 } \\right \\| _ { \\infty } = 1 \\\\ \\lim _ { | y | \\to \\infty } \\left \\| ( 1 + ( x - w ) R ( \\mu , A _ 1 ) ) \\right \\| _ { \\infty } = 1 . \\end{gather*}"}
{"id": "7854.png", "formula": "\\begin{align*} \\varphi ( T ) = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 1 } ^ N \\varphi ( u _ n ^ * T u _ n ) = 0 . \\end{align*}"}
{"id": "57.png", "formula": "\\begin{align*} [ \\alpha , v _ n ] _ { L , n } = [ \\alpha _ m , v _ m ] _ { E _ { \\rho } ^ m , m } & = \\dfrac { 1 } { \\eta ^ m } \\operatorname { T } _ m ( \\lambda _ { \\rho } ( \\alpha _ m ) v _ m ^ { - 1 } ) \\cdot _ { \\rho } v _ m \\\\ & = \\dfrac { 1 } { \\eta ^ m } \\operatorname { T } _ m ( \\lambda _ { \\rho } ( \\alpha _ m ) v _ m ^ { - 1 } ) \\cdot _ { \\rho } ( \\eta ^ m \\cdot _ { \\rho } v _ { 2 m } ) \\\\ & = \\operatorname { T } _ m ( \\lambda _ { \\rho } ( \\alpha _ m ) v _ m ^ { - 1 } ) \\cdot _ { \\rho } v _ { 2 m } . \\end{align*}"}
{"id": "2638.png", "formula": "\\begin{align*} \\begin{vmatrix} x _ 0 & x _ 1 \\\\ a _ 0 & a _ 1 \\end{vmatrix} \\in J _ { ( 1 , 0 , 0 ; \\ , r _ 1 ) } \\ , \\ \\begin{vmatrix} y _ 0 & y _ 1 \\\\ b _ 0 & b _ 1 \\end{vmatrix} \\in J _ { ( 0 , 1 , 0 ; \\ , r _ 2 ) } \\ , \\ \\begin{vmatrix} z _ 0 & z _ 1 \\\\ c _ 0 & c _ 1 \\end{vmatrix} \\in J _ { ( 0 , 0 , 1 ; \\ , r _ 3 ) } \\ , \\end{align*}"}
{"id": "3630.png", "formula": "\\begin{align*} u \\wedge u ^ t ( \\cdot ) = \\min \\{ u ( \\cdot ) , u ^ t ( \\cdot ) \\} , u \\vee u ^ t ( \\cdot ) = \\max \\{ u ( \\cdot ) , u ^ t ( \\cdot ) \\} . \\end{align*}"}
{"id": "2009.png", "formula": "\\begin{align*} H _ { s , c , 2 } = \\{ n \\in H _ { s , c } \\ \\ P ( n ) > [ s c ^ { - 1 } ] \\} . \\end{align*}"}
{"id": "5307.png", "formula": "\\begin{align*} \\lambda _ { n , k } = \\int _ 0 ^ 1 x ^ { k - 2 } ( 1 - x ) ^ { n - k } \\Lambda ( d x ) . \\end{align*}"}
{"id": "7794.png", "formula": "\\begin{align*} \\widetilde { H } ( S _ { [ n ] } ( f _ 2 ) \\circ \\phi \\circ S _ { [ n ] } ( f _ 1 ) ) = \\left \\{ \\begin{array} { c c c } f _ 2 \\circ \\widetilde { H } ( \\phi ) \\circ f _ 1 & \\mbox { i f } & n \\ \\mbox { i s o d d } , \\\\ \\det ( f _ 1 ) \\det ( f _ 2 ) f _ 2 \\circ \\widetilde { H } ( \\phi ) \\circ f _ 1 & \\mbox { i f } & n \\ \\mbox { i s e v e n } . \\end{array} \\right . \\end{align*}"}
{"id": "907.png", "formula": "\\begin{align*} \\norm { I _ t - \\Tilde { \\Psi } _ t } _ { C ^ { s , \\alpha } ( M ) } = O ( t ^ { l + \\frac { 1 } { 2 } - \\frac { s + \\alpha } { 2 } } ) , \\end{align*}"}
{"id": "6518.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\exp \\Big \\{ - \\sum _ { i = 1 } ^ m \\mu ( ( A ( m ) ^ { { \\rho } i } , \\infty ) ) \\Big \\} < \\infty . \\end{align*}"}
{"id": "4204.png", "formula": "\\begin{align*} L f = \\mathcal G f \\end{align*}"}
{"id": "1585.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial z ^ j _ { \\eta } } \\frac { \\partial ^ 2 \\varphi ^ j } { \\partial x ^ { \\epsilon } \\partial x ^ { \\eta } } = \\frac { f ' ( x ^ 1 ) } { \\sqrt { 1 + f '^ 2 ( x ^ 1 ) } } \\delta _ { \\epsilon 1 } \\left [ f ( x ^ 1 ) f '' ( x ^ 1 ) + 1 + f '^ 2 ( x ^ 1 ) \\right ] , \\end{align*}"}
{"id": "252.png", "formula": "\\begin{align*} \\delta ^ { ( s ) } v _ k + \\frac { \\sum _ { i j } ( \\sum _ l \\Gamma ^ { k ( s ) } _ { i j , l } c _ l ) v _ i ^ { \\phi ^ s } v _ j ^ { \\phi ^ s } } { ( \\sum _ l r _ l c _ l ) ( \\sum _ l v _ l ^ { \\phi ^ s } ) } = 0 , \\ \\ k \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "1096.png", "formula": "\\begin{align*} 2 g ( \\nabla _ X Y , Z ) = X g ( Y , Z ) + Y g ( X , Z ) - Z g ( X , Y ) + g ( [ X , Y ] , Z ) - g ( Y , [ X , Z ] - 2 P _ X Z ) - g ( X , [ Y , Z ] ) . \\end{align*}"}
{"id": "3679.png", "formula": "\\begin{align*} K _ { r s } = - \\alpha ( \\beta _ { r r } + ( \\beta _ r ) ^ 2 ) \\end{align*}"}
{"id": "1057.png", "formula": "\\begin{align*} \\widetilde { B _ 0 } S _ { n } ( t ) x = \\int _ 0 ^ t \\widetilde { B _ 0 } T ( t - s , A ) \\tilde { B } S _ { n - 1 } ( t ) x \\ , \\mathrm { d } s , \\end{align*}"}
{"id": "1333.png", "formula": "\\begin{align*} \\tilde B ( | v - v _ * | , | \\tilde r | ) = | v - v _ * | ^ \\gamma \\tilde b ( | \\tilde r | ) , \\tilde r = u / | u | \\cdot \\omega . \\end{align*}"}
{"id": "4239.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log \\sup _ { u \\in V } p _ n ( u , v ) } { \\log n } = 0 \\end{align*}"}
{"id": "3914.png", "formula": "\\begin{align*} \\hat { \\pi } _ { h , n } ( x ) & : = \\frac { 1 } { n \\Delta _ n } \\frac { 1 } { \\prod _ { l = 1 } ^ d h _ l } \\sum _ { i = 0 } ^ { n - 1 } \\prod _ { l = 1 } ^ d K ( \\frac { x _ l - X _ { t _ i } ^ l } { h _ l } ) ( t _ { i + 1 } - t _ i ) \\\\ & = \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } \\mathbb { K } _ h ( x - X _ { t _ i } ) , \\end{align*}"}
{"id": "3312.png", "formula": "\\begin{align*} \\Re \\left \\langle \\begin{pmatrix} \\varphi \\\\ \\psi \\end{pmatrix} , B ( s ) \\begin{pmatrix} \\varphi \\\\ \\psi \\end{pmatrix} \\right \\rangle _ \\Gamma \\geq \\beta \\ , \\min ( 1 , | s | ^ 2 ) \\frac { \\Re s } { | s | ^ 2 } \\left ( \\| \\varphi \\| ^ 2 _ { - 1 / 2 , \\Gamma } + \\| \\psi \\| ^ 2 _ { 1 / 2 , \\Gamma } \\right ) \\end{align*}"}
{"id": "3659.png", "formula": "\\begin{align*} \\left ( \\int _ { \\R ^ N } u ^ q \\right ) ^ { \\frac { 1 } { q } } = \\norm { u } _ { L ^ { q } } \\leq C \\norm { \\nabla u } _ { L ^ p } ^ { \\beta } \\norm { u } _ { L ^ 1 } ^ { 1 - \\beta } , \\end{align*}"}
{"id": "7081.png", "formula": "\\begin{align*} \\begin{aligned} \\mathsf { T } _ { 1 } ( t _ { k } ) & = \\frac { 2 ( \\lambda - \\sum _ { i = 1 } ^ { n } \\beta _ { i } u _ { i } ( t _ { k } ) ) } { \\sum _ { i = 1 } ^ { n } \\alpha _ { i } } , \\\\ u _ i ( t _ { k + 1 } ) & = \\beta _ { i } u _ i ( t _ { k } ) + \\alpha _ { i } \\mathsf { T } _ { 1 } ( t _ { k } ) . \\end{aligned} \\end{align*}"}
{"id": "8078.png", "formula": "\\begin{align*} \\phi ( N \\times { \\mathbb R } ^ { \\dim M - 1 } ) \\subset U ^ { + } : = \\{ x \\in M \\mid \\operatorname { s g n } ( Z ) > 0 \\} . \\end{align*}"}
{"id": "568.png", "formula": "\\begin{align*} \\mathbb { E } ( \\overline { T } _ n ) = \\frac { 1 } { n } \\sum _ { 1 \\leq i \\leq n } \\mathbb { E } ( X _ i X _ { i + 1 } ) . \\end{align*}"}
{"id": "6361.png", "formula": "\\begin{align*} P _ { \\kappa } ( x ) = N _ { \\kappa } \\int _ 0 ^ { x } t ^ { n - 1 } \\exp _ { \\kappa } ( - x ) \\ \\ , \\end{align*}"}
{"id": "5767.png", "formula": "\\begin{align*} E ( G ^ 0 / A ) = E ( G ^ 0 / R ) E ( R / A ) \\neq 0 \\end{align*}"}
{"id": "7906.png", "formula": "\\begin{align*} ( L ( t , - z ) \\alpha , L ( t , z ) \\beta ) = ( \\alpha , \\beta ) , \\end{align*}"}
{"id": "5438.png", "formula": "\\begin{align*} \\begin{cases} f ^ k \\to f F ^ s _ { p _ 1 , q } , \\ , & \\| f ^ k \\| _ { L ^ { \\infty } ( \\R ^ n ) } \\leq C \\\\ g ^ k \\to g F ^ s _ { p , q } , \\ , & \\end{cases} \\end{align*}"}
{"id": "8715.png", "formula": "\\begin{align*} \\frac { \\partial h } { \\partial \\dot { x } ^ { 1 } } = - \\frac { ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } } { 2 } x ^ { 2 } + \\lambda \\left ( \\frac { \\dot { x } ^ { 1 } } { \\sqrt { ( \\dot { x } ^ { 1 } ) ^ { 2 } + ( \\dot { x } ^ { 2 } ) ^ { 2 } } } + b \\cos ( t + c ) \\right ) \\end{align*}"}
{"id": "5297.png", "formula": "\\begin{align*} \\sum _ { i \\in I } a _ i = \\sum _ { j = 1 } ^ h \\left [ r ( I _ j ) + ( | I _ j | - 1 ) r \\right ] = r ( I ) + ( | I | - 1 ) r \\end{align*}"}
{"id": "4085.png", "formula": "\\begin{align*} \\boldsymbol { F } ( n ) = \\beta \\mathbf { a } ( n ) . \\end{align*}"}
{"id": "6787.png", "formula": "\\begin{align*} & \\sum _ { i \\in M _ P - i ^ * } \\sum _ { j \\in N _ i } a _ { i j } x ^ * _ { i j } + \\sum _ { i j \\in \\bar P , i \\neq i ^ * } ( b - s ) x ^ * _ { i j } + \\sum _ { j \\in N _ { i ^ * } } a _ { i ^ * j } x ^ * _ { i ^ * j } \\\\ > & b + ( | M _ P - M _ 0 | - 2 ) ( b - s ) \\\\ = & s + ( | \\bar P | - 1 ) ( b - s ) . \\end{align*}"}
{"id": "2676.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } ( n - \\ell - 1 ) ^ 2 ( p - 1 ) ^ 2 p ^ { n - 4 } \\\\ & = ( p - 1 ) ^ 2 p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } ( n - \\ell - 1 ) ^ 2 \\\\ & = ( p - 1 ) ^ 2 p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { i = 1 } ^ { n - k - 2 } i ^ 2 \\\\ & = ( p - 1 ) ^ 2 p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\frac { 1 } { 6 } ( n - k - 2 ) ( n - k - 1 ) ( 2 ( n - k - 2 ) + 1 ) \\\\ & = \\frac { 1 } { 6 } ( p - 1 ) ^ 2 p ^ { n - 4 } \\cdot \\frac { 1 } { 2 } ( n - 2 ) ^ 2 ( n - 3 ) ( n - 1 ) . \\end{align*}"}
{"id": "7396.png", "formula": "\\begin{align*} \\begin{aligned} u \\mapsto \\int _ { \\mathbb { R } ^ { N } } \\frac { 1 } { 2 } \\left ( | \\nabla u | ^ { 2 } + { V _ { \\varepsilon } } ( y ) u ^ { 2 } \\right ) - \\frac { 1 } { p + 1 } | u | ^ { p + 1 } d y , \\end{aligned} \\end{align*}"}
{"id": "2718.png", "formula": "\\begin{align*} \\langle \\mathcal { N } _ \\perp \\rangle _ { \\psi _ \\mathrm { g s } } \\le \\ ; & C \\\\ \\langle \\mathrm { d } \\Gamma _ \\perp ( h _ \\mathrm { M F } - \\mu _ + ) \\rangle _ { \\psi _ \\mathrm { g s } } \\le \\ ; & C \\\\ \\langle \\mathcal { N } _ - \\rangle _ { \\psi _ \\mathrm { g s } } \\le \\ ; & C _ \\varepsilon \\min \\{ N , T ^ { - 1 - \\varepsilon } \\} . \\end{align*}"}
{"id": "8370.png", "formula": "\\begin{align*} \\alpha _ { i , v } = \\zeta ^ { a _ i v ^ T } . \\end{align*}"}
{"id": "2175.png", "formula": "\\begin{align*} \\frac { \\nu ( \\left \\{ \\alpha \\right \\} ) } { \\omega ^ { \\prime } ( h ( \\alpha ) ) } = \\nu ( \\left \\{ \\alpha \\right \\} ) + \\mu ( \\{ s _ { \\mu } \\} ) = \\mu \\boxplus \\nu ( \\{ \\alpha + s _ { \\mu } \\} ) . \\end{align*}"}
{"id": "8861.png", "formula": "\\begin{align*} \\begin{vmatrix} \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { k | k } \\right ) & \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { k | \\l } \\right ) \\\\ \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { \\l | k } \\right ) & \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { \\l | \\l } \\right ) \\end{vmatrix} & = f ( \\lambda _ i ( t ) ) \\det \\left ( ( \\lambda _ i ( t ) I _ N - H _ { \\alpha } ( t ) ) _ { k \\l | k \\l } \\right ) \\\\ & = 0 , \\end{align*}"}
{"id": "7341.png", "formula": "\\begin{align*} ( w x , y , z ) - ( w , x y , z ) + ( w , x , y z ) = w ( x , y , z ) + ( w , x , y ) z . \\end{align*}"}
{"id": "953.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { g } e _ { i } f _ { i } = n . \\end{align*}"}
{"id": "95.png", "formula": "\\begin{align*} C _ t = ( A + 1 / t ) ^ { - 1 } ( t > 0 ) , C _ 0 = 0 , \\end{align*}"}
{"id": "1507.png", "formula": "\\begin{align*} \\frac { d } { d x } \\Big ( e ^ { - x } y _ { p , \\lambda } ( x ) \\Big ) = \\phi _ { p , \\lambda } ( x ) . \\end{align*}"}
{"id": "5256.png", "formula": "\\begin{align*} \\# Z ( H ^ { \\Lambda _ Q } ) = - \\# Z ( H ^ { \\Lambda _ { Q ^ - } } ) . \\end{align*}"}
{"id": "1728.png", "formula": "\\begin{align*} F ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) = \\exp \\Bigg ( \\int _ C \\frac { e ^ { z s } } { ( e ^ { \\overline \\omega _ 1 s } - 1 ) ( e ^ { \\omega _ 2 s } - 1 ) } \\frac { d s } { s } \\Bigg ) , \\end{align*}"}
{"id": "6475.png", "formula": "\\begin{align*} U _ t = \\mathcal { A } U , U ( 0 ) = U _ 0 , \\end{align*}"}
{"id": "5669.png", "formula": "\\begin{align*} X _ { w _ J } \\cong \\prod _ { k = 1 } ^ { m } F l _ { n _ k } . \\end{align*}"}
{"id": "1556.png", "formula": "\\begin{align*} x ^ 1 = c _ 1 t + c _ 2 , ~ x ^ 2 = k _ 1 \\textnormal { a n d } x ^ 1 = k _ 2 , ~ x ^ 2 = c _ 3 t + c _ 4 , \\end{align*}"}
{"id": "3078.png", "formula": "\\begin{align*} f _ { \\mathbf { M ^ { \\ast } } _ { - \\xi _ { \\boldsymbol { s } k } } } ( \\boldsymbol { w } ) = c _ { n - 1 , \\xi _ { \\boldsymbol { s } k } } ^ { \\ast } \\left ( 1 + \\sqrt { \\boldsymbol { w } ^ { T } \\boldsymbol { w } + \\xi _ { \\boldsymbol { s } , k } ^ { 2 } } \\right ) \\exp \\left \\{ - \\sqrt { \\boldsymbol { w } ^ { T } \\boldsymbol { w } + \\xi _ { \\boldsymbol { s } , k } ^ { 2 } } \\right \\} , ~ \\boldsymbol { w } \\in \\mathbb { R } ^ { n - 1 } , \\end{align*}"}
{"id": "6980.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sum \\limits _ { k \\ge 1 } \\frac { a _ k } { \\lambda _ k ^ 2 ( \\lambda _ N ^ 2 + \\lambda _ { k } ^ { 2 } ) } = \\sum \\limits _ { k \\ge 1 } \\frac { a _ k } { \\lambda _ k ^ 4 } , \\end{align*}"}
{"id": "4925.png", "formula": "\\begin{align*} h _ { i j ; k } - h _ { i k ; j } = - \\bar R _ { \\alpha \\beta \\gamma \\delta } \\nu ^ \\alpha X _ { ; i } ^ \\beta X _ { ; j } ^ \\gamma X _ { ; k } ^ \\delta = 0 , \\end{align*}"}
{"id": "1234.png", "formula": "\\begin{align*} \\det ( x I - B _ 1 ( T ) ) & = \\prod _ { j = 1 } ^ { l ( T ) } \\det ( R _ j ) . \\end{align*}"}
{"id": "7728.png", "formula": "\\begin{align*} B _ { i i } ( t _ 1 : t _ 0 ) & \\geq \\eta _ i 0 \\leq t _ 0 \\leq t _ 1 \\leq t _ L , \\cr B _ { j j } ( t _ 1 : t _ 0 ) & \\geq \\eta _ j 0 \\leq t _ 0 \\leq t _ 1 \\leq \\sigma , \\cr \\sum _ { t = 0 } ^ { \\sigma - 1 } B _ { j i } ( t ) & \\geq \\delta \\delta \\in ( 0 , \\eta _ j ) . \\end{align*}"}
{"id": "3134.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\ell - 1 } \\eta _ k ^ { - 1 / s } \\le C _ d \\eta _ \\ell ^ { - 1 / s } \\quad C _ d : = \\frac { q _ c ^ { 1 / ( 2 s ) } } { \\big ( 1 - q _ c ^ { 1 / ( 2 s ) } \\big ) ( 1 - q _ c ) ^ { 1 / ( 2 s ) } } . \\end{align*}"}
{"id": "1748.png", "formula": "\\begin{align*} \\big ( \\frac { v } { w - t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { v } { w + t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { v } { - t } \\big ) > 0 , \\\\ \\big ( \\frac { - t \\tau / 2 } { w - t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { - t \\tau / 2 } { w + t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { - t \\tau / 2 } { - t } \\big ) = ( \\tau / 2 ) > 0 . \\end{align*}"}
{"id": "1072.png", "formula": "\\begin{align*} R ( w + i y , A ) \\left ( 1 - ( x - w ) R ( x + i y , A ) \\right ) = R ( x + i y , A ) \\end{align*}"}
{"id": "8110.png", "formula": "\\begin{align*} \\P ( Y ( t , x ) > R ) & \\geq \\P ( A _ N ) \\P _ N \\Biggl ( Y _ < ( \\tau _ N , \\eta _ N ) \\prod _ { i = 1 } ^ { N } u _ < ( \\tau _ { i } , \\eta _ { i } ; \\tau _ { i - 1 } , \\eta _ { i - 1 } ) \\zeta _ i > R \\Biggr ) \\\\ & \\geq C ^ N N ^ { - N } R ^ { - 1 - \\frac 2 d } \\log ^ { N - 1 } ( R ) \\\\ & = \\frac { R ^ { - 1 - \\frac 2 d } ( C \\log R ) ^ { K \\exp ( W ( \\frac 1 { d K } \\log R ) ) } } { \\log R ( K \\exp ( W ( \\frac 1 { d K } \\log R ) ) ) ^ { K \\exp ( W ( \\frac 1 { d K } \\log R ) ) } } . \\end{align*}"}
{"id": "3935.png", "formula": "\\begin{align*} | \\tilde { I } _ 2 | & \\le c \\frac { \\Delta _ n } { T _ n } \\sum _ { j = j _ { \\delta } + 1 } ^ { j _ D } ( \\frac { 1 } { t _ j } + 1 ) \\\\ & \\le c \\frac { \\Delta _ n } { T _ n } ( \\sum _ { t _ j \\le 1 , \\ , j = j _ { \\delta } + 1 } ^ { j _ D } \\frac { 1 } { t _ j } + \\sum _ { t _ j > 1 , \\ , j = j _ { \\delta } + 1 } ^ { j _ D } 1 ) \\\\ & \\le \\frac { c } { T _ n } ( | \\log D | + | \\log \\delta | + D ) . \\end{align*}"}
{"id": "482.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ N B ( x _ i , \\frac { 1 } { 2 } c ' ) \\subseteq B ( 0 , c + \\frac { 1 } { 2 } c ' ) . \\end{align*}"}
{"id": "3244.png", "formula": "\\begin{align*} \\kappa ^ p \\nu _ p ( \\{ ( a , r ) \\in \\omega \\times \\R _ + : \\ c _ d ' r | \\nabla f ( a ) | - C A r ^ 2 > \\kappa \\} ) = I _ 1 ( \\kappa ) - I _ 2 ( \\kappa ) \\ , , \\end{align*}"}
{"id": "6076.png", "formula": "\\begin{align*} N _ 2 = N _ { f _ 2 } ( \\Lambda _ 2 ) . \\end{align*}"}
{"id": "1485.png", "formula": "\\begin{align*} \\mathbb { A } _ { v _ 0 } ( \\boldsymbol { \\eta } , \\boldsymbol { \\zeta } , \\boldsymbol { \\alpha } , \\beta ) - \\lim _ { n \\to \\infty } \\dfrac { 1 } { n } \\sum _ { v \\neq v _ 0 } { { F _ v } } ( \\boldsymbol { \\alpha } , \\beta ) ( n ) = V ( \\boldsymbol { \\eta } , \\boldsymbol { \\zeta } , \\boldsymbol { \\alpha } , \\beta ) \\enspace . \\end{align*}"}
{"id": "3610.png", "formula": "\\begin{align*} & { \\rm ( a ) } \\ \\rho ( I ) = 1 ; \\quad { \\rm ( b ) } \\ \\rho _ { i c } ( I ) = 1 ; { \\rm ( c ) } \\ G \\mbox { i s b i p a r t i t e } ; \\quad { \\rm ( d ) } \\ \\rho ( I ^ \\vee ) = 1 . \\end{align*}"}
{"id": "3798.png", "formula": "\\begin{align*} E r r ^ 2 _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\sum _ { \\begin{subarray} { c } k _ 3 , j _ 3 \\in \\Z _ + , i _ 3 \\in \\{ 0 , 1 , 2 , 3 , 4 \\} , a _ 3 \\in \\{ 0 , 1 , 2 , 3 \\} \\\\ n _ 3 \\in [ - M _ t , 2 ] \\cap \\Z , \\mu _ 3 \\in \\{ + , - \\} , l _ 3 \\in [ - j _ 3 , 2 ] \\cap \\Z \\\\ \\end{subarray} } \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta + \\sigma ) + i \\mu _ 2 s | \\sigma | + i \\mu s | \\xi | + i \\mu _ 1 s | \\eta | } \\end{align*}"}
{"id": "2440.png", "formula": "\\begin{align*} T _ { f w ^ \\zeta _ 0 } = M _ { f w ^ \\zeta _ 0 } = M _ { f w _ 0 ^ \\zeta } + \\sum _ { g ' \\prec f w _ 0 ^ \\zeta } t _ { g ' f w _ 0 ^ \\zeta } ( q ) M _ { g ' } , \\end{align*}"}
{"id": "981.png", "formula": "\\begin{align*} \\tau ( \\alpha \\beta ) = \\tau ( \\alpha ) \\otimes \\tau ( \\beta ) = \\overline { \\alpha } \\otimes \\overline { \\beta } . \\end{align*}"}
{"id": "2258.png", "formula": "\\begin{align*} \\begin{cases} e '' _ \\sigma + \\frac { z } { 2 } e ' _ \\sigma = 0 , \\\\ e _ \\sigma ( 0 ) = 0 , e _ \\sigma ( \\infty ) = \\sigma . \\end{cases} \\end{align*}"}
{"id": "7284.png", "formula": "\\begin{align*} ( \\Delta - f _ 2 ' ( { \\sf U } _ \\infty ) ) \\theta & = - f ( { \\sf U } _ \\infty ) - \\sum _ { i = 0 } ^ N \\tfrac { f ^ { ( i ) } ( { \\sf U } _ \\infty ) } { i ! } ( \\theta _ 0 + \\cdots + \\theta _ { L - 1 } ) ^ i + \\sum _ { i = 2 } ^ N \\tfrac { f _ 2 ^ { ( i ) } ( { \\sf U } _ \\infty ) } { i ! } ( \\theta _ 0 + \\cdots + \\theta _ { L - 1 } ) ^ i . \\end{align*}"}
{"id": "8640.png", "formula": "\\begin{align*} R _ { g ^ { - 1 } * g } \\frac { d g } { d t } = \\mathcal { P } ( { \\rm A d } _ g a ( t ) ) , \\end{align*}"}
{"id": "2392.png", "formula": "\\begin{align*} \\left \\langle V _ { \\psi _ 1 } \\xi _ 1 , V _ { \\psi _ 2 } \\xi _ 2 \\right \\rangle _ { _ { L ^ 2 ( G _ 2 ) } } = \\langle \\xi _ 1 , \\xi _ 2 \\rangle _ { _ { L ^ 2 ( \\widehat { \\R ^ 2 } \\times \\widehat { \\R } ) } } \\langle T \\psi _ 2 , T \\psi _ 1 \\rangle _ { _ { L ^ 2 ( \\widehat { \\R ^ 2 } \\times \\widehat { \\R } ) } } . \\end{align*}"}
{"id": "1261.png", "formula": "\\begin{align*} W _ j = E _ j ( x , d - 1 ) ( \\forall j = \\overline { 0 , l ( T ) } ) , \\end{align*}"}
{"id": "7038.png", "formula": "\\begin{align*} E \\ni f \\mapsto P f : = \\lim _ { r \\to \\infty } C ( r ) f \\end{align*}"}
{"id": "7614.png", "formula": "\\begin{align*} \\dfrac { \\pi } { 2 } \\Bigl ( \\dfrac { 1 } { 2 } \\Bigr ) _ { r - 1 } \\dfrac { 1 } { ( n + r ) ^ { r - \\frac { 1 } { 2 } } } - \\dfrac { ( r - 1 ) ! } { n ^ r } < H _ r \\leq \\dfrac { \\pi } { 2 } \\Bigl ( \\dfrac { 1 } { 2 } \\Bigr ) _ { r - 1 } \\dfrac { 1 } { n ^ { r - \\frac { 1 } { 2 } } } - \\dfrac { ( r - 1 ) ! } { ( n + r ) ^ r } + \\sum _ { k = 1 } ^ { \\infty } \\dfrac { 1 } { k \\pi ^ k } \\Bigl ( \\dfrac { k } { 2 } \\Bigr ) _ r \\dfrac { 1 } { n ^ { r + \\frac { k } { 2 } } } . \\end{align*}"}
{"id": "7211.png", "formula": "\\begin{align*} \\alpha = s + y , \\ \\beta = s - y \\end{align*}"}
{"id": "5484.png", "formula": "\\begin{align*} G ( x , y ) = \\int _ 0 ^ \\infty H ( x , y , t ) d t . \\end{align*}"}
{"id": "1915.png", "formula": "\\begin{align*} S ^ { ( q ) } _ { [ n , j ] } & : = \\sum _ { \\gamma \\in \\mathcal { S } _ { [ n , j ] } } w ( \\gamma + q ) , \\\\ T ^ { ( q ) } _ { [ n , j ] } & : = \\sum _ { \\gamma \\in \\widehat { \\mathcal { S } } _ { [ n , j ] } } w ( \\gamma - q ) . \\end{align*}"}
{"id": "1948.png", "formula": "\\begin{align*} \\langle \\mathcal { H } _ { q } ^ { n + 1 } e _ { r } , e _ { 0 } \\rangle = \\langle \\mathcal { H } _ { q } ^ { n } ( \\mathcal { H } _ { q } \\ , e _ { r } ) , e _ { 0 } \\rangle = \\begin{cases} \\sum _ { k = 0 } ^ { p } a _ { q } ^ { ( k ) } \\langle \\mathcal { H } _ { q } ^ { n } e _ { k } , e _ { 0 } \\rangle , & \\mbox { i f } \\ , \\ , r = 0 , \\\\ [ 0 . 5 e m ] \\langle \\mathcal { H } _ { q } ^ { n } e _ { r - 1 } , e _ { 0 } \\rangle + \\sum _ { k = 0 } ^ { p } a _ { r + q } ^ { ( k ) } \\langle \\mathcal { H } _ { q } ^ { n } e _ { r + k } , e _ { 0 } \\rangle , & \\mbox { i f } \\ , \\ , r \\geq 1 . \\end{cases} \\end{align*}"}
{"id": "4779.png", "formula": "\\begin{align*} F ( \\mathbf { B } _ B ) = & ( \\kappa - \\nu ) ( \\kappa - \\nu - 1 ) ( \\kappa - 2 a - ( b > 1 ) - 2 ) \\\\ + & ( \\kappa - \\nu ) ( a + ( b > 1 ) ) ( \\kappa - 2 a - ( b > 1 ) - 1 ) \\\\ + & a ( \\kappa - \\nu ) ( \\kappa - 2 a - ( b > 1 ) - 1 ) a ( \\kappa - \\nu ) ( \\kappa - 2 a - ( b > 1 ) ) . \\end{align*}"}
{"id": "2087.png", "formula": "\\begin{align*} \\frac { \\log _ 2 ( \\widetilde { N _ H } ( q \\alpha ) ) } { ( R + q ) \\alpha } = \\frac { q } { q + R } \\frac { \\log _ 2 ( \\widetilde { N _ H } ( q \\alpha ) ) } { q \\alpha } > \\frac { t } { t + 1 } ( 1 + \\frac { 1 } { t } ) h = h . \\end{align*}"}
{"id": "2387.png", "formula": "\\begin{align*} H ^ D = \\left \\{ \\begin{pmatrix} a & 0 \\\\ 0 & b \\end{pmatrix} : a b \\neq 0 \\right \\} , H ^ S = \\left \\{ \\begin{pmatrix} a & b \\\\ - b & a \\end{pmatrix} : a ^ 2 + b ^ 2 > 0 \\right \\} , H ^ \\alpha = \\left \\{ \\begin{pmatrix} a & b \\\\ 0 & | a | ^ \\alpha \\end{pmatrix} : a \\neq 0 \\right \\} , \\end{align*}"}
{"id": "266.png", "formula": "\\begin{align*} ( B ( \\Lambda _ i ^ { \\nu } - 1 ) ) _ { k j } - ( B ( \\Lambda _ j ^ { \\nu } - 1 ) ) _ { k i } = \\pi L _ { i j k } ( \\Lambda ^ { \\nu } ) - \\pi ^ { \\nu } L _ { i j k } ^ { \\nu } , \\end{align*}"}
{"id": "8157.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } \\| \\chi _ { S _ n } e ^ { - i t H _ 0 } \\varphi \\| = 0 \\end{align*}"}
{"id": "1238.png", "formula": "\\begin{align*} \\mathcal { E } ( T ) & = 2 \\sqrt { \\dfrac { 7 + \\sqrt { 5 } } { 2 } } + 2 \\sqrt { \\dfrac { 7 - \\sqrt { 5 } } { 2 } } \\\\ & = \\sqrt { 1 4 + 2 \\sqrt { 5 } } + \\sqrt { 1 4 - 2 \\sqrt { 5 } } \\ , . \\end{align*}"}
{"id": "7520.png", "formula": "\\begin{align*} \\sum _ { i } \\mbox { P r } ( C _ { i } ) \\sum _ { j = 0 } ^ { k } \\binom { m - i } { j } \\left ( \\frac { \\binom { k } { j } ( ( j ! ) ( i ^ { ( k - j ) } ) ) } { m ^ { k } } \\right ) ^ { 2 } , \\end{align*}"}
{"id": "7394.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 U _ { h , i } ( y ) & \\le M \\sum _ { i = 1 } ^ 4 e ^ { - | y - h t _ i | } \\min \\{ | y - h t _ i | ^ { - \\left ( \\frac { N - 1 } { 2 } \\right ) } , 1 \\} \\\\ & \\le 4 M e ^ { - | y - h t _ 1 | } \\min \\{ | y - h t _ 1 | ^ { - \\left ( \\frac { N - 1 } { 2 } \\right ) } , 1 \\} . \\end{align*}"}
{"id": "272.png", "formula": "\\begin{align*} ( F ^ { \\nu } _ i ) _ { j k } - ( F ^ { \\nu } _ j ) _ { i k } = L ^ { \\nu } _ { i j k } , \\end{align*}"}
{"id": "3556.png", "formula": "\\begin{align*} w _ 2 ( \\pi ) & = \\sum \\limits _ { j = 1 } ^ n ( c _ j / 2 ) \\left ( \\sum \\limits _ { 1 \\leq i _ 1 < i _ 2 < \\cdots < i _ j \\leq n } ( t _ { i _ 1 } + t _ { i _ 2 } + \\cdots + t _ { i _ j } ) \\right ) \\\\ & = \\frac { 1 } { 2 } \\sum \\limits _ { j = 1 } ^ n c _ j \\binom { n - 1 } { j - 1 } \\left ( \\sum \\limits _ { i = 1 } ^ n t _ i \\right ) \\\\ & = \\frac { 1 } { 2 } \\sum \\limits _ { j = 1 } ^ n c _ j \\binom { n - 1 } { j - 1 } a _ 2 . \\end{align*}"}
{"id": "7770.png", "formula": "\\begin{align*} Z ^ k _ { j , , r s } = \\exp \\Big [ - \\frac { 1 } { 2 } \\big ( Q _ j ^ k ( \\alpha _ r ) - Q _ j ^ k ( \\alpha _ s ) \\big ) ^ 2 \\Big ] r , s = 1 , \\ldots , p . \\end{align*}"}
{"id": "7841.png", "formula": "\\begin{align*} \\abs { \\varphi ( q p x p q ) } & \\leq \\varphi ( p v \\abs { x } ^ { 1 / 2 } q \\abs { x } ^ { 1 / 2 } v ^ * p ) ^ { 1 / 2 } \\varphi ( p \\abs { x } ^ { 1 / 2 } q \\abs { x } ^ { 1 / 2 } p ) ^ { 1 / 2 } \\\\ & \\leq K \\| v | x | v ^ * \\| _ 1 ^ { 1 / 2 } \\| | x | \\| _ 1 ^ { 1 / 2 } = K \\| x \\| _ 1 . \\end{align*}"}
{"id": "7027.png", "formula": "\\begin{align*} S = \\begin{bmatrix} - v _ { i } \\sin ( \\psi _ i ) \\sec ^ 2 ( \\beta _ i ) & \\cos ( \\psi _ i ) - \\sin ( \\psi _ i ) \\tan ( \\beta _ i ) \\\\ v _ { i } \\cos ( \\psi _ i ) \\sec ^ 2 ( \\beta _ i ) & \\sin ( \\psi _ i ) + \\cos ( \\psi _ i ) \\tan ( \\beta _ i ) \\end{bmatrix} , \\end{align*}"}
{"id": "5783.png", "formula": "\\begin{align*} E ( G / H ) = E ( ( N H ) / H ) = E ( N / ( N \\cap H ) ) = \\pm 1 \\end{align*}"}
{"id": "1189.png", "formula": "\\begin{align*} J ' ( v ; w ) : = \\lim _ { t \\to 0 } \\frac { J ( v + t w ) - J ( v ) } { t } = \\int _ { \\R ^ { d } } \\langle w , j _ { q } ( v ) \\rangle d \\rho - L ( w ) . \\end{align*}"}
{"id": "3118.png", "formula": "\\begin{align*} a ( u , J I u - J z _ { \\mathrm { n c } } ) + a _ { \\mathrm { p w } } ( z _ { \\mathrm { n c } } , z _ { \\mathrm { n c } } - I u ) & = b ( \\lambda u , J I u - J z _ { \\mathrm { n c } } + ( 1 + \\delta ) ( z _ { \\mathrm { n c } } - I u ) ) \\\\ & = \\lambda b ( u , ( J - 1 ) ( I u - z _ { \\mathrm { n c } } ) ) + \\lambda b ( \\delta u , z _ { \\mathrm { n c } } - I u ) . \\end{align*}"}
{"id": "3932.png", "formula": "\\begin{align*} \\sup _ { ( a , b ) \\in \\Sigma } \\mathbb { E } [ | \\hat { \\pi } _ { h , n } ( x ) - \\pi ( x ) | ^ 2 ] & \\le c \\sum _ { j = 1 } ^ d h _ j ^ { * \\ , 2 \\beta _ j } + \\frac { c } { T _ n } \\frac { \\sum _ { l = 1 } ^ d | \\log ( h ^ * _ l ) | } { \\prod _ { l \\ge 3 } h ^ * _ l } \\\\ & = ( \\frac { \\log T _ n } { T _ n } ) ^ { \\frac { \\bar { \\beta } _ 3 } { ( 2 \\bar { \\beta } _ 3 + d - 2 ) } } \\end{align*}"}
{"id": "5924.png", "formula": "\\begin{align*} w \\left ( { { \\eta _ 1 } , . . . , { \\eta _ d } } \\right ) = \\mathop { \\lim } \\limits _ { T \\to \\infty } \\frac 1 T \\ln { \\left \\langle D _ { 1 } ^ { \\eta _ 1 } . . . D _ { d } ^ { \\eta _ d } \\right \\rangle } \\end{align*}"}
{"id": "2610.png", "formula": "\\begin{align*} \\begin{cases} r _ + + r _ - = 2 a + 4 k \\\\ 0 \\le r _ - \\le a + 3 k \\le r _ + \\le 2 a + 3 k \\end{cases} \\ . \\end{align*}"}
{"id": "2270.png", "formula": "\\begin{align*} & \\int _ 0 ^ r \\left [ ( \\phi '' + \\frac { z } { 2 } \\phi ' - \\frac { z } { 2 } e _ \\sigma \\psi ' ) \\cdot z \\phi ' + ( \\psi '' + \\frac { z } { 2 } \\psi ' - \\frac { z } { 2 } e _ \\sigma \\phi ' ) \\cdot z \\psi ' \\right ] { \\rm { d } } z \\\\ & = \\frac { 1 } { 2 } \\int _ 0 ^ r | z ( \\phi ' , \\psi ' ) | ^ 2 + \\int _ 0 ^ r z ( \\phi ' \\phi '' + \\psi ' \\psi '' ) - \\int _ 0 ^ r z ^ 2 e _ \\sigma \\phi ' \\psi ' . \\end{align*}"}
{"id": "4408.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { k = [ m _ n ] + 1 } ^ { [ m _ n ] + L + 1 } \\beta _ { k - [ m _ n ] - 1 } \\frac { 1 } { s _ n ^ k } z ^ k . \\end{align*}"}
{"id": "871.png", "formula": "\\begin{align*} f ( s , t ) f _ { m _ { s t } } ( s , t ) & = ( t s ) _ { m _ { s t } - 1 } f _ { m _ { s t } } ( s , t ) \\\\ & = ( t s ) _ { m _ { t s } - 1 } f _ { m _ { t s } - 1 } ( t , s ) \\\\ & = t f _ 1 ( t , s ) f _ 2 ( t , s ) \\cdots f _ { m _ { t s } - 1 } ( t , s ) \\\\ & = t f ( t , s ) \\\\ & = t ( s t ) _ { m _ { t s } - 1 } \\\\ & = ( t s ) _ { m _ { t s } } . \\end{align*}"}
{"id": "5535.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\abs { w _ i - \\lambda \\frac { z _ i - a } { 1 - \\bar { a } z _ i } } ^ 2 \\end{align*}"}
{"id": "4546.png", "formula": "\\begin{align*} J _ { 4 2 } & = - \\frac { 4 } { m n \\pi ^ { 2 } } \\int _ { 0 } ^ { \\pi - h _ { 1 } } \\int _ { h _ { 2 } } ^ { \\pi } H _ { x , z _ { 1 } , y , z _ { 2 } } ( t _ { 1 } + h _ { 1 } , t _ { 2 } ) S ( t _ 1 , t _ 2 ) \\\\ & \\times \\left ( \\frac { 1 } { \\left ( 2 \\sin \\frac { t _ { 2 } } { 2 } \\right ) ^ { 2 } } - \\frac { 1 } { t _ { 2 } { } ^ { 2 } } \\right ) \\frac { 1 } { \\left ( t _ { 1 } + h _ { 1 } \\right ) { } ^ { 2 } } d t _ { 1 } d t _ { 2 } . \\end{align*}"}
{"id": "1673.png", "formula": "\\begin{align*} d ^ n ( \\Phi ) ( \\vec { p } \\ , ) : = \\sum _ { i = 0 } ^ { n + 1 } ( - 1 ) ^ i \\pi _ { \\bigwedge \\vec { p } , \\bigwedge \\vec { p } ^ { \\ , i } } ( \\Phi ( \\vec { p } ^ { \\ : i } ) ) \\end{align*}"}
{"id": "7942.png", "formula": "\\begin{align*} y ^ d \\mapsto \\prod _ { i = 1 } ^ { l _ + } y _ i ^ { p _ i ^ + \\cdot d } \\prod _ { j \\in S _ + } x _ j ^ { D _ j \\cdot d } . \\end{align*}"}
{"id": "6553.png", "formula": "\\begin{align*} \\sum _ { m = m _ 0 } ^ \\infty \\prod _ { i = i _ 0 } ^ m \\sum _ { k = 0 } ^ \\infty \\mu ( k ) ( 1 - E _ 2 ( i , m ) ) ^ i \\precsim \\sum _ { m = m _ 0 } ^ \\infty \\prod _ { i = i _ 0 } ^ m \\mu ( [ 0 , 2 A ( m ) ^ { { { \\rho } } i } ] ) . \\end{align*}"}
{"id": "7965.png", "formula": "\\begin{align*} H _ { ( X , D ) } ( y ) = \\sum _ { d \\in \\mathbb K } \\sum _ { \\vec d = ( D _ i \\cdot d ) _ { i \\in I } \\in ( \\mathbb Z ) ^ { | I | } , f = [ d ] \\in \\mathbb K / \\mathbb L } H _ { ( X , D ) , f , \\vec d } \\textbf { 1 } _ { f } [ \\textbf { 1 } ] _ { \\vec d } . \\end{align*}"}
{"id": "6826.png", "formula": "\\begin{align*} p ( t , V _ F , g ) - p ( t , 0 , g ) = 0 , \\ \\forall t > 0 , g \\in \\mathbb { R } . \\end{align*}"}
{"id": "7773.png", "formula": "\\begin{align*} z ^ k _ { j , } ( \\alpha ^ * ) = z ^ k _ { j , } ( \\alpha ^ * , \\alpha ^ * ) - [ Z ( \\boldsymbol { \\alpha } , \\alpha ^ * ) ] ^ T [ Z _ { j , } ^ k ] ^ { - 1 } \\ , Z ( \\boldsymbol { \\alpha } , \\alpha ^ * ) , \\end{align*}"}
{"id": "6857.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\frac { d ( b ^ 2 ( t ) ) } { d t } & \\leq C | b | - ( 1 - g _ 1 ) b ^ 2 + g _ 1 \\delta _ 1 | b c | , \\\\ \\frac { 1 } { 2 } \\frac { d ( c ^ 2 ( t ) ) } { d t } & \\leq C | c | + 2 a _ 1 | b c | - c ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "339.png", "formula": "\\begin{align*} r ( g ) = \\min \\{ j \\in \\N \\ : \\ S y z ( g ) _ j \\ne 0 \\} \\leq d - 1 . \\end{align*}"}
{"id": "2003.png", "formula": "\\begin{align*} \\nu _ p \\left ( \\binom { m } { t } \\right ) = \\frac { \\sigma _ p ( t ) + \\sigma _ p ( m - t ) - \\sigma _ p ( m ) } { p - 1 } . \\end{align*}"}
{"id": "2881.png", "formula": "\\begin{align*} S ( x _ 1 , \\dotsc , x _ { n } ) = ( x _ { n } , x _ 1 , \\dotsc , x _ { n - 1 } ) . \\end{align*}"}
{"id": "4287.png", "formula": "\\begin{align*} [ K _ j J , e ^ { M J } ] = 0 \\end{align*}"}
{"id": "419.png", "formula": "\\begin{align*} a = H _ { \\mu } ( A + i f _ { \\mu } ( A ) ) , b = H _ { \\mu } ( B + i f _ { \\mu } ( B ) ) , \\end{align*}"}
{"id": "1068.png", "formula": "\\begin{align*} R ( z , A _ 2 ) - R ( z , A _ 1 ) = R ( z , A _ 2 ) B R ( z , A _ 1 ) , \\end{align*}"}
{"id": "4188.png", "formula": "\\begin{align*} \\Vert g _ { > \\iota } \\Vert _ p & \\lesssim R ^ { Q / q } \\Vert g _ { > \\iota } \\Vert _ 2 \\\\ & \\le R ^ { Q / q } \\bigg \\Vert \\sum _ { \\ell = \\iota + 1 } ^ \\infty F _ \\ell ^ { ( \\iota ) } ( L , U ) f \\bigg \\Vert _ 2 \\\\ & \\lesssim 2 ^ { \\iota ( Q - d _ 2 ) / q } \\norm { F ^ { ( \\iota ) } \\psi } _ 2 \\norm { f } _ p \\\\ & \\lesssim 2 ^ { - \\varepsilon \\iota } \\norm { F ^ { ( \\iota ) } } _ { L ^ 2 _ s } \\norm { f } _ p \\end{align*}"}
{"id": "7972.png", "formula": "\\begin{align*} H _ { ( F _ 1 , D _ 2 + D _ 3 + D _ 4 ) } ( y _ 1 , y _ 2 ) = e ^ { \\frac { P \\log y _ 1 + H \\log y _ 2 } { 2 \\pi i } } \\sum _ { d _ 1 , d _ 2 \\geq 0 } y _ 1 ^ { d _ 1 } y _ 2 ^ { d _ 2 } \\frac { 1 } { \\Gamma ( 1 + \\frac { H } { 2 \\pi i } + d _ 2 ) } [ \\textbf { 1 } ] _ { d _ 2 - d _ 1 , d _ 1 , d _ 1 } . \\end{align*}"}
{"id": "8821.png", "formula": "\\begin{align*} f _ t ( y ) : = \\left \\{ \\begin{aligned} & y - t & & \\ ; y \\geq t \\\\ & 0 & & \\ ; y \\leq t \\end{aligned} \\right . \\end{align*}"}
{"id": "7420.png", "formula": "\\begin{align*} \\begin{array} { c } \\rho v _ { t t } - \\alpha v _ { x x } + \\gamma \\beta p _ { x x } = 0 , \\\\ \\mu p _ { t t } - \\beta p _ { x x } + \\gamma \\beta v _ { x x } = 0 , \\end{array} \\end{align*}"}
{"id": "631.png", "formula": "\\begin{align*} I ( s , n ) = n ^ { 2 - 2 s } \\int _ 0 ^ 1 z ^ { 2 \\alpha - 2 s - 1 } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\Big ( \\frac { \\sin ^ 2 ( \\pi x ) } { \\pi ^ 2 } + \\frac { \\sin ^ 2 ( \\pi y ) } { \\pi ^ 2 } + z ^ 2 \\Big ) ^ { - \\alpha } d x d y d z . \\end{align*}"}
{"id": "4394.png", "formula": "\\begin{align*} & \\int _ { D _ j } | u _ { m , \\epsilon , j } | ^ { 2 } e ^ { v _ { \\epsilon } ( \\Psi _ m ) - \\varphi _ { m } - \\Psi _ m } \\leq \\int _ { D _ j } v '' _ { \\epsilon } ( \\Psi _ m ) | F | ^ 2 e ^ { - \\phi - \\varphi _ { m } - \\Psi _ m } . \\end{align*}"}
{"id": "2222.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } i _ { \\frac { \\partial } { \\partial r } } \\overline { \\omega } ^ { n + 1 } = ( \\Pi ^ { \\ast } \\omega _ { h } ) ^ { n } \\wedge \\eta = \\omega ^ { n } \\wedge \\eta . \\end{array} \\end{align*}"}
{"id": "5228.png", "formula": "\\begin{align*} & d \\left ( x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } ( ( a _ i + 1 ) x _ 1 d x _ 2 \\wedge \\cdots \\wedge d x _ n - ( - 1 ) ^ { i - 1 } ( a _ 1 + 1 ) x _ i d x _ 1 \\wedge \\cdots \\wedge \\widehat { d x _ i } \\wedge \\cdots \\wedge d x _ n \\right ) \\\\ = { } & x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } ( ( a _ 1 + 1 ) ( a _ i + 1 ) - ( a _ i + 1 ) ( a _ 1 + 1 ) ) d x _ 1 \\wedge \\cdots \\wedge d x _ n \\\\ = { } & 0 . \\end{align*}"}
{"id": "3812.png", "formula": "\\begin{align*} E l l ^ { \\mu , 4 ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) = \\sum _ { \\kappa \\in [ \\bar { \\kappa } , 2 ] \\cap \\Z } { } ^ \\kappa E l l ^ { \\mu , 4 ; l } _ { k , j , n } ( t _ 1 , t _ 2 ) , { } ^ \\kappa E l l ^ { \\mu , 4 ; l } _ { k , j , n } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot \\xi - i s \\hat { v } \\cdot \\xi } \\hat { g } ( s , \\xi , v ) \\varphi _ { \\kappa ; \\bar { \\kappa } } ( \\frac { \\xi \\cdot ( \\hat { v } - \\hat { V } ( s ) ) } { | \\hat { v } - \\hat { V } ( s ) | | \\xi | } ) \\end{align*}"}
{"id": "6284.png", "formula": "\\begin{align*} D _ { q ^ { - 1 } } v ( x ) - q A ( x ) v ( x ) = 1 . \\end{align*}"}
{"id": "4767.png", "formula": "\\begin{align*} c _ { i _ 1 , j _ 1 } - c _ { i _ 1 , j _ 2 } + c _ { i _ 2 , j _ 2 } - c _ { i _ 2 , j _ 3 } + c _ { i _ 3 , j _ 3 } - c _ { i _ 3 , j _ 1 } = 0 \\quad ( \\mathrm { m o d } \\ ; z ) . \\end{align*}"}
{"id": "403.png", "formula": "\\begin{align*} \\cal T ( p , t ) = \\log \\frac { | \\hat { \\sf g } _ t v | } { | v | } . \\end{align*}"}
{"id": "4653.png", "formula": "\\begin{align*} \\tilde p = \\tilde p _ \\delta = \\sum _ { n = 0 } ^ { \\infty } p _ n . \\end{align*}"}
{"id": "6890.png", "formula": "\\begin{align*} \\rho ( t , 0 ) = \\rho ( t , V _ F ) , t > 0 . \\end{align*}"}
{"id": "2447.png", "formula": "\\begin{align*} \\phi _ \\zeta ( M _ f ) : = M _ f S _ \\zeta , f \\in \\Z ^ { m | n } . \\end{align*}"}
{"id": "716.png", "formula": "\\begin{align*} \\| u \\| _ { k , \\alpha } : = \\displaystyle \\sum _ { l _ { 1 } + 2 l _ { 2 } \\leq k } \\left \\| | \\widetilde { \\nabla } ^ { \\ell _ 1 } \\partial ^ { \\ell _ { 2 } } _ { s } u | _ { \\widetilde { g } } \\right \\| _ { \\alpha } . \\end{align*}"}
{"id": "5878.png", "formula": "\\begin{align*} \\bar \\psi _ j ( x ) = \\bar \\psi _ j ( x _ 1 , \\ldots , x _ n ) : = \\sum _ { i = 1 } ^ n x _ i \\varphi ^ j _ i . \\end{align*}"}
{"id": "4869.png", "formula": "\\begin{align*} R ( N _ { 1 } , N _ { 2 } ) = \\sum \\log p _ { 1 } \\cdots \\log p _ { 1 0 } \\end{align*}"}
{"id": "2662.png", "formula": "\\begin{align*} b _ j = \\frac { ( b _ i - 1 ) a _ j } { a _ i - 1 } . \\end{align*}"}
{"id": "2444.png", "formula": "\\begin{align*} \\texttt { l } _ { g f } = \\sum _ { x \\in W ^ g } \\ell _ { g \\cdot x , f } q ^ { - \\ell ( x ) } , f , g \\in \\Z ^ { m | n } _ { \\zeta - } . \\end{align*}"}
{"id": "6852.png", "formula": "\\begin{align*} \\tilde { p } ( t , U _ F , g ) - \\tilde { p } ( t , 0 , g ) = 0 , g \\in \\mathbb { R } , t > 0 , N ( t ) = \\int _ { 0 } ^ { \\infty } g \\tilde { p } ( t , U _ F , g ) d g . \\end{align*}"}
{"id": "5096.png", "formula": "\\begin{align*} & E \\left [ ( \\sigma ' \\sigma ) ( X _ { \\eta _ n ( s _ 1 ) } ) ( \\sigma ' \\sigma ) ( X _ { \\eta _ n ( s _ 2 ) } ) \\Xi ^ { n , 2 } _ { s _ 1 } \\Xi ^ { n , 2 } _ { s _ 2 } \\right ] \\\\ & = E \\left [ ( \\sigma ' \\sigma ) ( X _ { \\eta _ n ( s _ 1 ) } ) ( \\sigma ' \\sigma ) ( X _ { \\eta _ n ( s _ 2 ) } ) \\Xi ^ { n , 2 } _ { s _ 1 } E [ \\Xi ^ { n , 2 } _ { s _ 2 } | \\mathcal { F } _ { \\eta _ n ( s _ 2 ) } \\right ] ] = 0 . \\end{align*}"}
{"id": "4745.png", "formula": "\\begin{align*} x ( n P ) - x ( P ) = \\frac { \\phi _ n ( x ( P ) ) } { \\psi _ n ^ 2 ( x ( P ) ) } - x ( P ) = x ( P ) + \\frac { \\psi _ { n + 1 } ( P ) \\psi _ { n - 1 } ( P ) } { \\psi _ n ^ 2 ( x ( P ) ) } - x ( P ) = \\frac { \\psi _ { n + 1 } ( P ) \\psi _ { n - 1 } ( P ) } { \\psi _ n ^ 2 ( x ( P ) ) } . \\end{align*}"}
{"id": "2564.png", "formula": "\\begin{align*} A _ { 1 2 } = T _ + \\ , \\ \\ A _ { 2 3 } = U _ + \\ , \\ \\ T _ 3 = \\dfrac { 1 } { 2 } ( A _ { 1 1 } - A _ { 2 2 } ) \\ , \\ \\ U _ 3 = \\dfrac { 1 } { 2 } ( A _ { 2 2 } - A _ { 3 3 } ) \\ , \\end{align*}"}
{"id": "3052.png", "formula": "\\begin{align*} c _ { n } ^ { \\ast } = \\frac { \\Gamma ( n / 2 ) } { ( 2 \\pi ) ^ { n / 2 } } \\left [ \\int _ { 0 } ^ { \\infty } s ^ { n / 2 - 1 } \\overline { G } _ { n } ( s ) \\mathrm { d } s \\right ] ^ { - 1 } \\end{align*}"}
{"id": "2373.png", "formula": "\\begin{align*} \\partial _ i L ( h _ \\theta ) - a _ { i 0 } = \\sum _ { k , j > 0 } a _ { i k j } Q _ \\theta ^ j . \\end{align*}"}
{"id": "3964.png", "formula": "\\begin{align*} j ( a * b ) ( \\gamma ) = \\sum _ { \\eta \\in G ^ { u } } j ( a ) ( \\eta ) \\ , j ( b ) ( \\eta ^ { - 1 } \\gamma ) . \\end{align*}"}
{"id": "3637.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\sup _ { x \\in \\R ^ N } \\int _ { x + [ 0 , 1 ) ^ N } u _ n = \\lim _ { k \\to \\infty } \\int _ { x _ k + [ 0 , 1 ) ^ N } u _ { n _ k } . \\end{align*}"}
{"id": "2526.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\{ g ( \\nabla _ X Z , Y ) + g ( \\nabla _ Y Z , X ) \\} + \\frac { 2 \\mu } { \\lambda ^ 2 } h ( \\tilde { X } , \\tilde { Y } ) = 0 . \\end{align*}"}
{"id": "6885.png", "formula": "\\begin{align*} | N _ 0 ( t ) - N ( B ( t ) , C ( t ) ) | & \\leq \\frac { 1 } { V _ F } \\int _ { \\mathbb { R } } | y | p _ { t , 0 } ( y ) d y = \\frac { 1 } { V _ F } e ^ { - t } \\int _ { \\mathbb { R } } | y | p _ { 0 , 0 } ( y ) d y \\\\ & = \\frac { 1 } { V _ F } e ^ { - t } \\int _ { \\mathbb { R } } \\int _ { 0 } ^ { V _ F } | y | p _ { } ( v , y ) d v d y \\leq C e ^ { - t } \\end{align*}"}
{"id": "7477.png", "formula": "\\begin{align*} z \\in \\mathrm { F i x } ( S _ { N } ) = \\mathrm { F i x } \\left ( \\mathcal { T } _ { \\left \\{ C _ { m , r } \\left ( N \\right ) \\left ( j \\right ) \\right \\} _ { j = 1 } ^ { r } } \\right ) = \\cap _ { j = 1 } ^ { r } C _ { f \\left ( \\left ( r - 1 \\right ) N + j - 1 \\right ) } \\subset C _ { f \\left ( \\left ( r - 1 \\right ) N + k \\right ) } = C _ { p } . \\end{align*}"}
{"id": "8722.png", "formula": "\\begin{align*} h _ { \\dot { x } ^ { 1 } \\dot { x } ^ { 1 } } = f _ { \\dot { x } ^ 1 \\dot { x } ^ { 1 } } + \\lambda g _ { \\dot { x } ^ { 1 } \\dot { x } ^ { 1 } } . \\end{align*}"}
{"id": "7381.png", "formula": "\\begin{align*} 0 & = \\int _ { { \\C } _ 1 } \\left ( \\sum _ { i = 1 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \\cdot t _ { i } \\phi _ n ( y ) \\right ) d y \\\\ & = \\int _ { \\{ y | y + h _ n t _ 1 \\in { \\C } _ 1 \\} } f ( | y | ) \\left ( y \\cdot t _ { 1 } \\right ) \\bar { \\phi } _ n ( y ) d y \\\\ & + \\int _ { { \\C } _ 1 } \\left ( \\sum _ { i = 2 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \\cdot t _ { i } \\phi _ n ( y ) \\right ) d y . \\end{align*}"}
{"id": "3459.png", "formula": "\\begin{align*} \\mathrm { D T K } _ { ( p , q ) } ( X ) = \\frac { \\mathrm { E } \\left [ ( X - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { 4 } | x _ { p } < X < x _ { q } \\right ] } { \\mathrm { D T V } _ { ( p , q ) } ^ { 2 } ( X ) } - 3 , \\end{align*}"}
{"id": "2140.png", "formula": "\\begin{align*} \\mu _ { n } = D _ { b _ { n } } \\mu ^ { \\boxplus n } \\boxplus \\delta _ { a _ { n } } \\end{align*}"}
{"id": "8891.png", "formula": "\\begin{align*} \\det ( \\sigma ^ 2 I _ n - A ^ H A ) = 0 . \\end{align*}"}
{"id": "238.png", "formula": "\\begin{align*} ( \\delta ^ { ( s ) } ) ^ 2 u = f ^ { ( 1 2 ) } ( u , \\delta ^ { ( s ) } u , ( \\delta ^ { ( s ) } u ) ^ { \\phi ^ s } ) , \\end{align*}"}
{"id": "3200.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } f ( x ) d x = \\int _ { S ^ { n - 1 } } \\int _ 0 ^ \\infty f ( \\mathcal O + r \\cdot \\xi ) r ^ { n - 1 } d r d \\mathcal H ^ { n - 1 } . \\end{align*}"}
{"id": "4657.png", "formula": "\\begin{align*} \\lambda R _ \\lambda f + \\tilde R _ \\lambda \\lambda q R _ \\lambda f - R _ \\lambda \\psi - R _ \\lambda q R _ \\lambda \\psi - \\tilde R _ \\lambda q f = f + \\tilde R _ \\lambda q f - \\tilde R _ \\lambda q f = f . \\end{align*}"}
{"id": "1501.png", "formula": "\\begin{align*} \\phi _ { n + 1 , \\lambda } ( x ) = x \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\phi _ { k , \\lambda } ( x ) ( 1 - \\lambda ) _ { n - k , \\lambda } , ( n \\ge 0 ) . \\end{align*}"}
{"id": "206.png", "formula": "\\begin{align*} \\alpha _ { n , i } ( f ^ { n } , g ^ { n } _ { i } ) \\coloneqq \\Pr \\left ( \\hat { H } _ { i } = 1 | H = 0 \\right ) i \\in \\{ 1 , 2 \\} , \\end{align*}"}
{"id": "1206.png", "formula": "\\begin{align*} { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\theta } ) & > \\frac { C } { 2 } \\frac { \\xi _ n } { \\sqrt n } - { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\mu ) \\\\ & = { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\mu ) + o _ { \\mathbb { P } } ( n ^ { - 1 / 2 } ) , \\end{align*}"}
{"id": "6617.png", "formula": "\\begin{align*} S _ { k } ( c ; \\xi ) = & { \\sum _ { a \\ , \\hbox { \\tiny m o d } \\ , k } } ^ { \\ ! \\ ! \\ ! * } \\sum _ { { \\tiny \\begin{tabular} { c } $ z \\ , \\hbox { \\tiny m o d } \\ , k \\ell $ \\\\ $ z = \\xi ( \\ell ) $ \\end{tabular} } } e _ { k \\ell } \\big ( a \\ell F _ h ( z ) + c \\cdot z \\big ) \\\\ = & { \\sum _ { a \\ , \\hbox { \\tiny m o d } \\ , k } } ^ { \\ ! \\ ! \\ ! * } \\ ; \\sum _ { z ' \\ , \\hbox { \\tiny m o d } \\ , k } e _ { k \\ell } \\big ( a \\ell F _ h ( \\xi + \\ell z ' ) + c \\cdot ( \\xi + \\ell z ' ) \\big ) . \\end{align*}"}
{"id": "6772.png", "formula": "\\begin{align*} I _ \\alpha ( n ) : = \\int _ 1 ^ \\infty ( \\log z ) ^ n e ^ { - \\alpha z } \\ , d z \\qquad ( \\alpha > 0 ) , \\end{align*}"}
{"id": "5627.png", "formula": "\\begin{align*} \\partial _ t v _ k = \\frac { 1 } { m } \\nabla \\cdot ( m \\nabla v _ k ^ k ) + \\nabla v _ k \\cdot \\vec { b } + F v _ k Q _ T . \\end{align*}"}
{"id": "6025.png", "formula": "\\begin{align*} x ^ 3 + a x ^ 2 = x ^ 3 + ( b - 3 ) x ^ 2 + ( 3 - 2 b ) x + ( b + c - 1 ) \\end{align*}"}
{"id": "1393.png", "formula": "\\begin{align*} \\| v ( t ) \\| _ { H ^ { 1 } _ x ( \\mathbb { T } ) } = \\| u ( t ) \\| _ { H ^ { 1 } _ x ( \\mathbb { T } ) } \\leq C ( \\| u _ 0 \\| _ { H ^ 1 _ x ( \\mathbb { T } ) } , \\| f \\| _ { H ^ { 1 } _ x ( \\mathbb { T } ) } , \\gamma ) \\end{align*}"}
{"id": "4422.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { i = 1 } ^ d S _ { d , i } \\right ) \\cup \\left ( \\bigcup _ { j = 1 } ^ e T _ { e , j } \\right ) \\cup \\left ( \\bigcup _ { \\ell = 1 } ^ e U _ { e , \\ell } \\right ) = [ 1 , ( 4 k + 2 ) N ] , \\end{align*}"}
{"id": "8686.png", "formula": "\\begin{align*} d V _ { \\min } = ( 1 - \\| \\beta \\| _ { \\alpha } ) ^ { n + 1 } d V _ { \\alpha } . \\end{align*}"}
{"id": "3614.png", "formula": "\\begin{align*} & V ( Q ( I ) ) = \\left \\{ \\left . \\frac { e _ { i _ 1 } + \\cdots + e _ { i _ k } } { k - s + d } \\right | s - d + 1 \\leq k \\leq s , \\ 1 \\leq i _ 1 < \\cdots < i _ k \\leq s \\right \\} , \\\\ & V ( Q ( I ^ \\vee ) ) = \\left \\{ \\left . \\frac { e _ { j _ 1 } + \\cdots + e _ { j _ { \\ell } } } { \\ell - d + 1 } \\right | d \\leq \\ell \\leq s , \\ 1 \\leq j _ 1 < \\cdots < j _ \\ell \\leq s \\right \\} . \\end{align*}"}
{"id": "3920.png", "formula": "\\begin{align*} \\Delta _ n ' : = \\sup _ { t \\in [ 0 , T _ n ] } \\sup _ { i , j = 1 , . . . , d } | \\varphi _ { n , i } ( t ) - \\varphi _ { n , j } ( t ) | . \\end{align*}"}
{"id": "1412.png", "formula": "\\begin{align*} q ( z ) = z ^ r - \\sigma _ 1 z ^ { r - 1 } + \\sigma _ 2 z ^ { r - 2 } - \\cdots + ( - 1 ) ^ { r - 1 } \\sigma _ { r - 1 } z + ( - 1 ) ^ { r } \\sigma _ { r } , \\end{align*}"}
{"id": "2679.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } \\binom { n - \\ell - 2 } { 2 } ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\\\ & = ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } \\binom { n - \\ell - 2 } { 2 } \\\\ & = ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\binom { n - k - 2 } { 3 } \\\\ & = ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } \\binom { n - 2 } { 4 } . \\end{align*}"}
{"id": "7636.png", "formula": "\\begin{align*} B _ { \\rm S T } \\ = \\ \\{ t ^ { n } , \\ n \\in \\mathbb { N } \\} . \\end{align*}"}
{"id": "4081.png", "formula": "\\begin{align*} E ( X _ 1 ^ 2 ) = \\varepsilon ^ 2 \\big [ P ( \\boldsymbol { X } = \\mathbf { s } ( 1 ) ) + P ( \\boldsymbol { X } = \\mathbf { s } ( 4 ) ) \\big ] . \\end{align*}"}
{"id": "2126.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow _ { \\sphericalangle } \\alpha } \\left | \\frac { f ( z ) - f ^ * ( \\alpha ) } { z - \\alpha } \\right | = + \\infty , \\end{align*}"}
{"id": "8602.png", "formula": "\\begin{align*} A Y + Y A ^ { * } \\leq 0 \\ \\ B = - A Y C ^ { * } . \\end{align*}"}
{"id": "1371.png", "formula": "\\begin{align*} \\left | \\underset { \\Gamma _ p } { \\int \\sum } \\overline { \\widehat { \\eta } } _ k \\prod _ { \\substack { j = 1 \\\\ o d d } } ^ { p } \\widehat { v } _ { k _ j } \\prod _ { \\substack { j = 2 \\\\ e v e n } } ^ p \\overline { \\widehat { v } } _ { k _ j } \\ , d \\Gamma \\right | . \\end{align*}"}
{"id": "7595.png", "formula": "\\begin{align*} \\frac { d Y } { d t } = A Y , Y ( t ) = \\begin{pmatrix} y _ { 1 1 } ( t ) & y _ { 1 2 } ( t ) \\\\ y _ { 2 1 } ( t ) & y _ { 2 2 } ( t ) \\end{pmatrix} \\in \\mathrm { S L } ( 2 , \\mathbb { R } ) , Y ( 0 ) = I _ 2 , \\end{align*}"}
{"id": "2105.png", "formula": "\\begin{align*} ( L y _ i ) ( v ) & = - \\sum \\limits _ { e \\in E _ v } \\frac { \\delta _ E ( e ) } { \\delta _ V ( v ) } \\frac { 1 } { | e | } y _ i ( v ) . \\end{align*}"}
{"id": "1752.png", "formula": "\\begin{align*} \\big ( \\frac { v + n w - n t \\tau / 2 } { w - t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { v + n w - n t \\tau / 2 } { w + t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { v + n w - n t \\tau / 2 } { - t } \\big ) > 0 , \\\\ \\big ( \\frac { - t \\tau / 2 } { w - t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { - t \\tau / 2 } { w + t \\tau / 2 } \\big ) > 0 , \\big ( \\frac { - t \\tau / 2 } { - t } \\big ) = ( \\tau / 2 ) > 0 . \\end{align*}"}
{"id": "3641.png", "formula": "\\begin{align*} \\frac 1 2 \\sum _ { i \\in \\N } m ( ( \\mu ^ i _ n ) _ { n \\in \\sigma _ i ( \\N ) } ) \\le \\sum _ { i \\in \\N } \\norm { \\mu ^ i } = \\lim _ { \\ell \\to \\infty } \\norm { \\mu _ { \\sigma ( \\ell ) } ^ b } \\leq \\liminf _ { \\ell \\to \\infty } \\norm { \\mu _ { \\sigma ( \\ell ) } } < \\infty . \\end{align*}"}
{"id": "5179.png", "formula": "\\begin{align*} \\int _ { \\Xi _ { \\mu } } e ^ { W _ { \\mathbf { y } } / \\hbar } f d x _ 1 \\wedge \\cdots \\wedge d x _ a = \\delta _ { \\mu 0 } + t _ { \\mu } ( \\mathbf { y } ) \\hbar ^ { - 1 } + O ( \\hbar ^ { - 2 } ) \\end{align*}"}
{"id": "7455.png", "formula": "\\begin{align*} i \\delta \\lambda ^ n u ^ n _ x = - i \\lambda ^ n w ^ n + c \\left ( ( 1 - m ) w ^ n + m \\int _ 0 ^ { \\infty } \\sigma ( s ) \\eta ^ n ( s ) d s \\right ) _ { x x } + f ^ { 5 , n } + \\delta f ^ { 1 , n } _ x \\end{align*}"}
{"id": "7133.png", "formula": "\\begin{align*} x _ { e } = - \\sum _ { u \\in S ( v _ e ) } x _ { \\phi ( u ) } + \\sum _ { u \\in S ( r _ e ) } x _ { \\phi ( u ) } , \\end{align*}"}
{"id": "7078.png", "formula": "\\begin{align*} \\theta _ { i } : = \\phi \\min \\nolimits _ { 0 < \\gamma \\leq \\gamma ^ { \\max } _ { i } } \\{ \\gamma ^ { - 1 } + K \\varphi _ { i } ( \\gamma ) \\} , \\forall i \\in \\mathcal { N } , \\end{align*}"}
{"id": "4464.png", "formula": "\\begin{align*} \\psi _ k ( \\xi ) & = \\psi ( \\xi / 2 ^ k ) - \\psi ( \\xi / 2 ^ { k - 1 } ) , \\psi _ { \\leq k } ( \\xi ) = \\psi ( \\xi / 2 ^ k ) , \\psi _ { \\geq k } ( \\xi ) = 1 - \\psi ( \\xi / 2 ^ { k - 1 } ) , \\\\ \\tilde \\psi _ k ( \\xi ) & = \\psi _ { k - 1 } ( \\xi ) + \\psi _ k ( \\xi ) + \\psi _ { k + 1 } ( \\xi ) . \\end{align*}"}
{"id": "5503.png", "formula": "\\begin{align*} \\theta _ { t + 1 } = \\theta _ { t + 1 / 2 } - \\beta _ t w _ { \\rm a } ( t ) . \\end{align*}"}
{"id": "7978.png", "formula": "\\begin{align*} H _ { ( X _ + , D _ + ) , ( d _ 2 , \\cdots , d _ 2 ) } = H _ { ( X _ - , D _ - ) , ( d _ 1 - d _ 2 , \\cdots , d _ 1 - d _ 2 ) } \\end{align*}"}
{"id": "41.png", "formula": "\\begin{align*} \\Tilde { \\mathcal { N } } ( f ) ( v _ n ) = \\operatorname { N } _ { n + 1 , n } ( f ( v _ { n + 1 } ) ) = \\operatorname { N } _ { n + 1 , n } ( \\beta ' ) = \\beta . \\end{align*}"}
{"id": "3823.png", "formula": "\\begin{align*} { } _ { 1 ; 0 } ^ { \\kappa } E l l H ^ { \\mu , 4 ; l } _ { k , j , n } ( t _ 1 , t _ 2 ) : = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot \\xi } \\big [ \\big ( \\sum _ { \\begin{subarray} { c } k _ 1 \\in \\Z _ + , \\mu _ 1 \\in \\{ + , - \\} , n _ 1 \\in [ - 2 M _ t / 1 5 , 2 ] \\cap \\Z , \\\\ \\end{subarray} } \\mathcal { F } \\big ( { } _ { } ^ z T _ { k _ 1 , n _ 1 } ^ { \\mu _ 1 } ( B ) ( s , \\cdot , V ( s ) ) f ( s , \\cdot , v ) ) ( \\xi ) \\end{align*}"}
{"id": "7825.png", "formula": "\\begin{align*} f ( \\gamma \\cup \\delta ) = f ( \\gamma ) \\star f ( \\delta ) , \\end{align*}"}
{"id": "6167.png", "formula": "\\begin{align*} d i s t ^ { 2 } ( x , \\mathbf { S } _ { p } ^ { 1 } ) = | z _ { 1 } | ^ { 2 k } + | z _ { 2 } | ^ { 2 k } : = r _ { \\mathbf { S } _ { \\mathbf { p } } ^ { 1 } } ^ { 2 k } . \\end{align*}"}
{"id": "7684.png", "formula": "\\begin{align*} \\tilde { y } _ k : = y _ { k - 1 } - \\frac { 1 } { \\sigma _ { k - 1 } } g ( x _ k ) , \\tilde { Z } _ k : = \\left [ Z _ { k - 1 } - \\frac { 1 } { \\sigma _ { k - 1 } } X ( x _ k ) \\right ] _ { + } , \\end{align*}"}
{"id": "2942.png", "formula": "\\begin{align*} & ( n = 3 ) \\mathcal { N } ( \\psi ) = \\pm [ \\frac { 1 } { | x | ^ { 3 / 2 - \\delta } } * | \\psi | ^ 2 ] ( x ) \\psi ( x ) , \\delta \\in ( 0 , \\frac { 3 } { 2 } ) . \\\\ & ( n = 3 ) \\mathcal { N } ( \\psi ) = \\pm | \\psi | \\psi . \\end{align*}"}
{"id": "1101.png", "formula": "\\begin{align*} ( \\mbox { H o r } P _ U V ) _ { ( x , t ) } = 0 . \\end{align*}"}
{"id": "7626.png", "formula": "\\begin{align*} n / 2 - 5 \\gamma n \\leq | X ' _ { R } | , | Y ' _ { R } | \\leq n / 2 + 5 \\gamma n \\ , . \\end{align*}"}
{"id": "2518.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { 1 } { 2 } \\{ g ( \\nabla _ U \\xi , V ) + g ( \\nabla _ V \\xi , U ) \\} + R i c ^ \\nu ( U , V ) - ( m - n ) \\| H \\| ^ 2 g ( U , V ) \\\\ + d i v ( H ) g ( U , V ) + \\mu g ( U , V ) = 0 . \\end{array} \\end{align*}"}
{"id": "2978.png", "formula": "\\begin{align*} T _ 0 ( d ) = \\frac { \\Gamma ( \\frac s 2 ) ^ 2 d ^ s L ( s , \\chi _ d ) ^ 2 } { \\Gamma ( s ) \\zeta ( 2 s ) } . \\end{align*}"}
{"id": "2243.png", "formula": "\\begin{align*} ( \\mathbf { U } ^ 0 , \\mathbf { B } ^ 0 , P ^ 0 ) = ( 1 , 0 , \\sigma , 0 , 0 ) , \\end{align*}"}
{"id": "4701.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty M _ k ( n ) q ^ n = \\sum _ { n = k } ^ \\infty \\frac { q ^ { { k \\choose 2 } + ( k + 1 ) n } } { ( q ; q ) _ n } \\begin{bmatrix} n - 1 \\\\ k - 1 \\end{bmatrix} . \\end{align*}"}
{"id": "326.png", "formula": "\\begin{align*} f ( B ( v ) _ q ) = \\sum _ { b \\in B ( v ) _ q } \\frac { 1 } { b \\log b } \\ & \\ge \\ \\frac { 1 } { 1 + v } \\sum _ { b \\in B ( v ) _ q } \\frac { 1 } { b \\log P ( b ) } . \\end{align*}"}
{"id": "6212.png", "formula": "\\begin{align*} ( D _ q f ) ( x ) = \\frac { f ( x ) - f ( q x ) } { ( 1 - q ) x } , x \\neq 0 , \\end{align*}"}
{"id": "7689.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma _ { k - 1 } } | P ( x _ k ) | = \\frac { 1 } { 2 \\sigma _ { k - 1 } } \\left ( \\norm { g ( x _ k ) } ^ 2 + \\norm { \\left [ - X ( x _ k ) \\right ] _ { + } } _ { \\mathrm { F } } ^ 2 \\right ) \\to 0 . \\end{align*}"}
{"id": "8667.png", "formula": "\\begin{align*} \\int _ R ^ \\infty Q ( r \\omega ) | u ( r ) | ^ 2 \\ , d r & = \\int _ 0 ^ \\infty W ( r ) | \\tilde u ( r ) | ^ 2 \\ , d r \\leq 4 \\left ( \\sup _ { s > 0 } s \\int _ s ^ \\infty W ( t ) \\ , d t \\right ) \\int _ 0 ^ \\infty | \\tilde u ' ( r ) | ^ 2 \\ , d r \\\\ & = 4 \\left ( \\sup _ { s > R } s \\int _ s ^ \\infty Q ( t \\omega ) \\ , d t \\right ) \\left ( \\int _ R ^ \\infty | \\tilde u ' ( r ) | ^ 2 \\ , d r + c _ R | u ( R ) | ^ 2 \\right ) . \\end{align*}"}
{"id": "4384.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi _ 1 < 0 \\} } | \\tilde F - f _ 0 F ^ { 1 + \\delta } | ^ 2 e ^ { - \\varphi } = \\int _ { \\{ \\Psi _ 1 < 0 \\} } | \\frac { \\tilde F } { F ^ { 1 + \\delta } } - f _ 0 | ^ 2 = \\int _ { \\{ \\Psi _ 1 < 0 \\} } | \\frac { \\tilde F } { F ^ { 1 + \\delta } } | ^ 2 - \\int _ { \\{ \\Psi _ 1 < 0 \\} } | f _ 0 | ^ 2 . \\end{align*}"}
{"id": "7959.png", "formula": "\\begin{align*} I _ { ( X _ - , D _ - ) } ( y , z ) = z e ^ { t _ - / z } \\sum _ { d \\in \\mathbb K _ { - } } \\tilde y ^ { d } \\left ( \\prod _ { i \\in M _ 0 } \\frac { \\prod _ { a \\leq 0 , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } { \\prod _ { a \\leq D _ i \\cdot d , \\langle a \\rangle = \\langle D _ i \\cdot d \\rangle } ( \\bar { D } _ i + a z ) } \\right ) \\textbf { 1 } _ { [ - d ] } I _ { D _ - , d } , \\end{align*}"}
{"id": "8083.png", "formula": "\\begin{align*} e ^ { i t } \\cdot ( r , \\theta ) : = ( r , \\theta + t ) , \\end{align*}"}
{"id": "2097.png", "formula": "\\begin{align*} \\mathcal Q ( x ) ( v ) & = \\sum \\limits _ { e \\in E _ v } \\frac { ( a v g ( x ) ) ( e ) } { | e | } \\frac { \\delta _ E ( e ) } { \\delta _ V ( v ) } \\\\ & = \\sum \\limits _ { e \\in E _ v } \\sum \\limits _ { u \\in e } x ( u ) \\\\ & = \\sum \\limits _ { e \\in E } B _ { v e } \\sum \\limits _ { u \\in V } B _ { u e } x ( u ) = ( ( B B ^ T ) x ) ( v ) , \\end{align*}"}
{"id": "3934.png", "formula": "\\begin{align*} { } & ( \\frac { 1 } { n } ) ^ { \\frac { 1 } { 1 + \\alpha } - \\frac { \\bar { \\beta } } { 2 \\bar { \\beta } + d } \\frac { d - 2 } { \\bar { \\beta _ 3 } } } \\\\ & = ( \\frac { 1 } { n } ) ^ { \\frac { \\bar { \\beta } ( 2 \\bar { \\beta _ 3 } + d - 2 ) } { \\bar { \\beta _ 3 } ( 2 \\bar { \\beta } + d ) } - \\frac { \\bar { \\beta } ( d - 2 ) } { \\bar { \\beta _ 3 } ( 2 \\bar { \\beta } + d ) } } \\\\ & = ( \\frac { 1 } { n } ) ^ { \\frac { 2 \\bar { \\beta } } { 2 \\bar { \\beta } + d } } , \\end{align*}"}
{"id": "5040.png", "formula": "\\begin{align*} Q ^ { n , 6 } _ \\tau = Q ^ { n , 7 } _ \\tau + Q ^ { n , 8 } _ \\tau , \\end{align*}"}
{"id": "3639.png", "formula": "\\begin{align*} \\tau _ { - x ^ i _ n } \\mu ^ { i - 1 } _ n = \\tau _ { - x ^ i _ n } \\mu _ n - \\sum _ { 0 \\le j < i } \\tau _ { - x ^ i _ n + x ^ j _ n } \\mu ^ j , \\end{align*}"}
{"id": "9000.png", "formula": "\\begin{align*} P _ { k , n } ( z ) = \\sum _ { j = k + 1 } ^ n ( - z ) ^ { j - k - 1 } \\binom { j - 1 } { k } { n \\brack j } . \\end{align*}"}
{"id": "4128.png", "formula": "\\begin{align*} D ^ { i } : = P _ { \\ker ( S ^ { i + 1 } ) } d _ { V } ^ { i } A ^ { i } = P _ { \\Upsilon ^ { i + 1 } } d _ { V } ^ { i } A ^ { i } , \\end{align*}"}
{"id": "3522.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast } = \\frac { \\xi _ { p } ^ { 2 } ( 1 + | \\xi _ { p } | ) \\exp ( - | \\xi _ { p } | ) - \\xi _ { q } ^ { 2 } ( 1 + | \\xi _ { q } | ) \\exp ( - | \\xi _ { q } | ) } { 2 F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "2070.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n s d e c a n o n i c a l } \\dd \\mathsf { X } ^ i _ t = b ( \\mathsf { X } ^ i _ t , f _ t ) \\dd t + \\dd \\overline { B } ^ i _ t , \\end{align*}"}
{"id": "2229.png", "formula": "\\begin{align*} A ( t ) = ( I - \\kappa e ^ { - a ( t , s ) } ) ^ { - 1 } \\partial _ t a ( t , s ) \\end{align*}"}
{"id": "7412.png", "formula": "\\begin{align*} 1 - z ^ m = m ( 1 - z ) + O ( m | 1 - z | ) ( m \\in \\mathbb { N } , \\ , \\ , \\ , z \\in \\mathbb { C } , \\ , \\ , \\ , | 1 - z | \\le 1 / m ) , \\end{align*}"}
{"id": "7015.png", "formula": "\\begin{align*} b _ { 2 k } ( M ) = h _ k ( P ) . \\end{align*}"}
{"id": "3633.png", "formula": "\\begin{align*} F ( y ) = \\int _ y ^ 1 \\frac { \\dd t } { \\rho ( t ) } \\in [ 0 , + \\infty ] . \\end{align*}"}
{"id": "1697.png", "formula": "\\begin{align*} \\partial _ t u - \\Delta u = f , u ( 0 , x ) = u _ 0 ( x ) . \\end{align*}"}
{"id": "2601.png", "formula": "\\begin{align*} \\sum _ { i _ 1 , . . . , i _ p = 1 } ^ 3 c _ { i _ 1 , . . . , i _ p } \\ , e _ { i _ 1 } \\otimes . . . \\otimes e _ { i _ p } \\in \\mathcal H _ { p , 0 } \\iff c _ { i _ { f ( 1 ) } , . . . , i _ { f ( p ) } } = c _ { i _ 1 , . . . , i _ p } \\end{align*}"}
{"id": "23.png", "formula": "\\begin{align*} \\Tilde { \\sigma _ i } \\Tilde { \\Phi } _ L ( \\beta ) = h _ i \\Tilde { \\sigma _ j } . \\end{align*}"}
{"id": "3460.png", "formula": "\\begin{align*} f _ { X } ( x ) : = \\frac { c _ { 1 } } { \\sigma } g _ { 1 } \\left \\{ \\frac { 1 } { 2 } \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right \\} , ~ x \\in \\mathbb { R } , \\end{align*}"}
{"id": "353.png", "formula": "\\begin{align*} q _ A \\geq \\frac { 2 } { 2 - \\beta } + ( \\ell _ { \\delta } ( A ) - 1 ) + o ( 1 ) = \\ell _ { \\delta } ( A ) + \\beta ' + o ( 1 ) . \\end{align*}"}
{"id": "6165.png", "formula": "\\begin{align*} \\mathbb { Z } _ { r } & \\rightarrow \\mathbb { C } ^ { 2 } \\\\ \\gamma & \\mapsto \\begin{bmatrix} e ^ { 2 \\pi i \\frac { 1 } { r } } & 0 \\\\ 0 & e ^ { 2 \\pi i \\frac { a } { r } } , \\end{bmatrix} \\end{align*}"}
{"id": "5797.png", "formula": "\\begin{align*} ( a , x ) \\ast ( b , y ) = ( a b f ( x , y ) , x y ) . \\end{align*}"}
{"id": "224.png", "formula": "\\begin{align*} F = I n v \\circ ( 0 , 1 , \\gamma ) , \\ c , \\gamma \\not \\in \\{ 0 , 1 \\} , \\ n \\ge 4 \\end{align*}"}
{"id": "5188.png", "formula": "\\begin{align*} J : = \\omega _ C \\otimes S ^ \\vee \\end{align*}"}
{"id": "8151.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\sup \\limits _ { t \\geq 0 } \\| P _ { \\delta } ( W _ { n ; \\textrm { f a r } } ) e ^ { - i t H } \\varphi \\| = 0 \\end{align*}"}
{"id": "5301.png", "formula": "\\begin{align*} \\{ I _ 1 , \\ldots , I _ h \\} = { \\bf a } ( \\{ Q _ 1 , \\ldots , Q _ h \\} ) . \\end{align*}"}
{"id": "7735.png", "formula": "\\begin{align*} A ( t _ 1 : t _ 0 ) = \\prod _ { k = t _ 0 } ^ { t _ 1 - 1 } ( 1 - \\bar a _ k ) \\stackrel { ( a ) } { \\geq } \\prod _ { k = t _ 0 } ^ { t _ 1 - 1 } e ^ { - M ( \\gamma ) \\bar a _ k } = e ^ { - M ( \\gamma ) \\sum _ { k = t _ 0 } ^ { t _ 1 - 1 } \\bar a _ k } \\geq e ^ { - M ( \\gamma ) \\sum _ { k = 0 } ^ { \\infty } \\bar a _ k } \\geq e ^ { - M ( \\gamma ) \\Delta } > 0 , \\end{align*}"}
{"id": "8763.png", "formula": "\\begin{align*} q ^ { r - 2 a } [ 2 \\ell - r ] & \\frac { q ^ { 2 - 2 r } K ^ { - 2 } - 1 } { q ^ { 4 c - 2 r + 2 } K ^ { - 2 } - 1 } + q ^ { 2 \\ell - a } [ a ] + q ^ { 2 c + r - 2 \\ell - a } [ r - 2 c - a ] \\frac { q ^ { 2 - 2 r } K ^ { - 2 } - 1 } { q ^ { 4 c - 2 r + 2 } K ^ { - 2 } - 1 } \\\\ & + q ^ { 2 c - 2 \\ell } [ 2 c ] \\frac { q ^ { - 2 a } K ^ { - 2 } - 1 } { q ^ { 4 c - 2 r + 2 } K ^ { - 2 } - 1 } + q ^ { 1 - r - 2 a } [ 2 \\ell - r + 1 ] \\frac { ( q ^ { 4 c } - 1 ) K ^ { - 2 } } { q ^ { 4 c - 2 r + 2 } K ^ { - 2 } - 1 } = [ 2 \\ell ] . \\end{align*}"}
{"id": "222.png", "formula": "\\begin{align*} b & = F ( x + a ) + c F ( x ) = ( A _ 1 \\circ F '' ) ( x + a ) + c ( A _ 1 \\circ F '' ) ( x ) = A _ 1 ( F '' ( x + a ) ) + c A _ 1 ( F '' ( x ) ) \\\\ & = u _ 1 F '' ( x + a ) + v _ 1 + c ( u _ 1 F '' ( x ) + v ) = u _ 1 ( F '' ( x + a ) + c F '' ( x ) ) + ( c + 1 ) v _ 1 \\end{align*}"}
{"id": "1573.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 C ^ 2 } { \\partial z ^ i _ { \\epsilon } \\partial z ^ j _ { \\eta } } = \\frac { \\partial } { \\partial z ^ j _ { \\eta } } \\left ( 2 C \\frac { \\partial C } { \\partial z ^ i _ { \\epsilon } } \\right ) = 2 \\frac { \\partial C } { \\partial z ^ i _ { \\epsilon } } \\frac { \\partial C } { \\partial z ^ j _ { \\eta } } + 2 C \\frac { \\partial ^ 2 C } { \\partial z ^ i _ { \\epsilon } \\partial z ^ j _ { \\eta } } . \\end{align*}"}
{"id": "1263.png", "formula": "\\begin{align*} n ( T , k + 1 - j ) - n ( T , k - j ) & = ( d - 2 ) ( d - 1 ) ^ { k - 1 - j } , \\end{align*}"}
{"id": "1361.png", "formula": "\\begin{align*} L _ t [ u ] = \\exp \\left ( \\mp i \\frac { p + 1 } { 4 \\pi } \\int _ 0 ^ t \\int _ \\mathbb { T } | u | ^ { p - 1 } ( x , s ) \\ , d x d s \\right ) , \\end{align*}"}
{"id": "5996.png", "formula": "\\begin{align*} ( A _ h u , v ) _ { L ^ 2 ( \\Omega ) } = ( \\nabla u , \\nabla v ) _ { L ^ 2 ( \\Omega ) } u , v \\in V _ h . \\end{align*}"}
{"id": "8864.png", "formula": "\\begin{align*} \\dfrac { \\partial \\det A } { \\partial a _ { k k } } = \\det \\left ( A _ { k | k } \\right ) , \\ \\ \\dfrac { \\partial \\det A } { \\partial a _ { k \\l } } = ( - 1 ) ^ { k + \\l } 2 \\det \\left ( A _ { k | \\l } \\right ) . \\end{align*}"}
{"id": "5214.png", "formula": "\\begin{align*} \\int _ { \\Xi _ { \\mu } } x ^ { \\mu ' } e ^ { ( x _ 1 ^ { r _ 1 } + \\cdots + x _ n ^ { r _ n } ) / \\hbar } \\Omega = \\delta _ { \\mu \\mu ' } . \\end{align*}"}
{"id": "8710.png", "formula": "\\begin{align*} f ( x ^ { 1 } , x ^ { 2 } , \\dot { x } ^ 1 , \\dot { x } ^ { 2 } ) = \\frac { ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } } { 2 } ( x ^ { 1 } \\dot { x } ^ { 2 } - x ^ { 2 } \\dot { x } ^ { 1 } ) \\end{align*}"}
{"id": "1506.png", "formula": "\\begin{align*} \\begin{aligned} y _ { p , \\lambda } ( x ) & = Y _ { \\lambda } ( x , p ) - e ^ { x } \\phi _ { p , \\lambda } ( x ) \\\\ & = \\sum _ { n = 2 } ^ { \\infty } \\Big ( ( 1 ) _ { p , \\lambda } + ( 2 ) _ { p , \\lambda } + \\cdots + ( n - 1 ) _ { p , \\lambda } \\Big ) \\frac { x ^ { n } } { n ! } , ( p \\in \\mathbb { N } ) . \\end{aligned} \\end{align*}"}
{"id": "2936.png", "formula": "\\begin{align*} w \\lim \\limits _ { t \\to \\infty } e ^ { - i H _ a t } ( I _ d - J _ a ) P _ c ( H _ a ) U ( t , 0 ) \\psi ( 0 ) = 0 , \\end{align*}"}
{"id": "6634.png", "formula": "\\begin{align*} \\rho _ { p , \\ell } ( \\beta _ h ) \\phi _ w ( g ^ { - 1 } , u ) = & \\int _ { X _ { p , \\ell } } \\phi _ w \\ , d \\mu _ { p , \\ell } \\\\ & \\ ; + O _ { p , w , \\ell , \\delta , \\theta , \\epsilon } \\Big ( \\| g \\| _ E ^ \\theta h ^ { 1 - ( \\theta - 4 ) \\delta } + \\| g \\| _ E h ^ { - 1 / 2 + 3 \\delta + \\epsilon } \\Big ) . \\end{align*}"}
{"id": "1217.png", "formula": "\\begin{align*} \\mathcal { G } ( v , x ) & = x - \\sum _ { w \\in c ( v ) } \\dfrac { 1 } { \\mathcal { G } ( w , x ) } ( \\forall v \\in V ( T ) ) , \\end{align*}"}
{"id": "3197.png", "formula": "\\begin{align*} R _ { p } ( h , \\mathcal O ) : = \\int _ { X \\setminus B ( \\mathcal O , 1 ) } h ^ { \\frac { p } { 1 - p } } d \\mu R _ { 1 } ( h , \\mathcal O ) : = \\| h ^ { - 1 } \\| _ { L ^ \\infty ( X \\setminus B ( \\mathcal O , 1 ) ) } . \\end{align*}"}
{"id": "1372.png", "formula": "\\begin{align*} N ^ { - s - \\varepsilon - } \\| P _ { N } \\eta \\| _ { X ^ { 0 , 1 / 2 - } } N ^ { s - } _ { 1 } \\| P _ { N _ { 1 } } v \\| _ { X ^ { 0 , 1 / 2 - } _ T } \\prod _ { j = 2 } ^ 3 N _ { j } ^ { \\frac { p - 5 } { 2 ( p - 1 ) } + \\varepsilon } \\| P _ { N _ { j } } v \\| _ { X ^ { 0 , 1 / 2 + } _ T } \\prod _ { j \\geq 4 } ^ p N ^ { s - } _ { j } \\| P _ { N _ { j } } v \\| _ { X ^ { 0 , 1 / 2 + } _ T } . \\end{align*}"}
{"id": "3315.png", "formula": "\\begin{align*} \\delta ( e ^ { - z } ) = z + \\O ( z ^ { p + 1 } ) \\end{align*}"}
{"id": "4613.png", "formula": "\\begin{align*} g [ \\alpha ] ^ { - 1 } g ^ { - 1 } = [ ( c ^ { - 1 } e ^ { - 1 } ) ( f ^ { - 1 } a ^ { - 1 } c ^ { - 1 } e ^ { - 1 } ) ( e c ) ] \\end{align*}"}
{"id": "5877.png", "formula": "\\begin{align*} | t | d _ n ^ { \\gamma | t | } ( t \\mathcal { K } ) _ X = d _ n ^ { \\gamma } ( \\mathcal { K } ) _ X , \\mbox { w h e r e } { t \\mathcal K } : = \\{ t f : \\ , f \\in \\mathcal K \\} . \\end{align*}"}
{"id": "7298.png", "formula": "\\begin{align*} { \\cal A } & = \\{ { \\sf a } \\in C ( [ 0 , T ] ) , \\ | { \\sf a } ( t ) - { \\sf a } _ 0 ( t ) | \\leq ( T - t ) ^ { \\frac { { \\sf d } _ 1 } { 2 } } { \\sf a } _ 0 ( t ) \\} , \\\\ & { \\sf a } _ 0 ( t ) = \\frac { 6 - n } { 2 { \\sf A } _ 1 } \\int _ t ^ T \\eta ( t _ 1 ) ^ \\frac { 2 } { 1 - q } d t _ 1 = \\kappa _ 1 ( T - t ) \\eta ^ \\frac { 2 } { 1 - q } . \\end{align*}"}
{"id": "5805.png", "formula": "\\begin{align*} b _ { i | v } = 2 / v . \\end{align*}"}
{"id": "1960.png", "formula": "\\begin{align*} ( x _ { 1 } , x _ { 2 } , \\ldots , x _ { p } ) = \\frac { \\mathbf { 1 } } { ( \\frac { x _ 2 } { x _ 1 } , \\frac { x _ 3 } { x _ 1 } , \\ldots , \\frac { x _ { p } } { x _ { 1 } } , \\frac { 1 } { x _ 1 } ) } . \\end{align*}"}
{"id": "760.png", "formula": "\\begin{align*} & \\mu _ i ( x ) - \\mu _ i ( y ) = r _ i x _ i \\left ( 1 - \\sum _ { j = 1 } ^ { d } a _ { i j } x _ j ^ + \\right ) - r _ i y _ i \\left ( 1 - \\sum _ { j = 1 } ^ { d } a _ { i j } y _ j ^ + \\right ) \\\\ & = r _ i ( x _ i - y _ i ) - r _ i \\left [ \\left ( x _ i \\sum _ { j = 1 } ^ { d } a _ { i j } x _ j ^ + \\right ) - \\left ( y _ i \\sum _ { j = 1 } ^ { d } a _ { i j } y _ j ^ + \\right ) \\right ] . \\end{align*}"}
{"id": "6093.png", "formula": "\\begin{align*} { \\bar \\Omega } _ { A B } = d \\theta _ { A B } - \\sum _ { C = 1 } ^ { N + 1 } \\theta _ { A C } \\wedge \\theta _ { C B } - \\theta _ { A \\ , N + 2 } \\wedge \\theta _ { N + 2 \\ , B } \\end{align*}"}
{"id": "5844.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ M a _ k \\bigg ( \\sum _ { i = k } ^ M a _ i \\bigg ) ^ s b _ k & = \\sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \\sum _ { i = k } ^ M a _ i \\big ( \\sum _ { j = i } ^ M a _ j \\big ) ^ s + \\bigg ( \\sum _ { k = N } ^ M a _ k \\big ( \\sum _ { i = k } ^ M a _ i \\big ) ^ s \\bigg ) b _ N \\\\ & \\approx \\sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \\bigg ( \\sum _ { i = k } ^ M a _ i \\bigg ) ^ { s + 1 } + \\bigg ( \\sum _ { k = N } ^ M a _ k \\bigg ) ^ { s + 1 } b _ N , \\end{align*}"}
{"id": "5866.png", "formula": "\\begin{align*} c _ k = \\int _ { x _ k } ^ { x _ { k + 1 } } u ( t ) \\bigg ( \\int _ t ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } d t , \\end{align*}"}
{"id": "182.png", "formula": "\\begin{align*} \\sup _ { 0 < h < 1 } \\bigg \\{ R ^ { \\frac { N } { p } - 2 } \\sum _ { 1 < | z _ \\gamma | \\leq R } | z _ \\gamma | ^ 2 \\omega _ { \\gamma , h } + R ^ { \\frac { N } { p } } \\sum _ { | z _ \\gamma | > R } \\omega _ { \\gamma , h } \\bigg \\} = o _ R ( 1 ) , \\end{align*}"}
{"id": "1743.png", "formula": "\\begin{align*} \\log G ( z \\ , | \\ , \\omega _ 1 , \\widetilde { \\omega } _ 1 , \\omega _ 2 ) = \\int _ C \\frac { - e ^ { ( z + \\overline \\omega _ 1 ) s } } { ( e ^ { \\omega _ 1 s } - 1 ) ( e ^ { \\widetilde \\omega _ 1 s } - 1 ) ( e ^ { \\omega _ 2 s } - 1 ) } \\cdot \\frac { d s } { s } , \\end{align*}"}
{"id": "8875.png", "formula": "\\begin{align*} \\sum _ { j ( \\neq i ) } \\frac { 1 } { \\lambda _ i - \\lambda _ j } = \\frac { \\sum _ { j ( \\neq i ) } \\prod _ { k ( \\neq i , j ) } ( \\lambda _ i - \\lambda _ k ) } { \\prod _ { j ( \\neq i ) } ( \\lambda _ i - \\lambda _ j ) } , \\end{align*}"}
{"id": "2521.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } g ( A _ X Y + A _ Y X , U ) + R i c ( X , Y ) + \\mu \\frac { 1 } { \\lambda ^ 2 } h ( \\tilde { X } , \\tilde { Y } ) = 0 , \\end{align*}"}
{"id": "3477.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast } & = \\frac { c _ { 1 } \\left [ \\overline { G } _ { ( 2 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { p } ^ { 2 } \\right ) - \\overline { G } _ { ( 2 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { q } ^ { 2 } \\right ) \\right ] } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "1419.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } Q _ { N } \\Big [ \\varepsilon _ 0 \\frac { \\gamma } { \\beta + \\gamma } N ^ { 4 / 3 } ( \\log N ) & ^ { - 2 / 3 } \\leq R _ { N , d , D } \\\\ & \\leq K _ { 0 } \\Big ( \\frac { \\beta + \\gamma } { \\beta } \\Big ) ^ { 1 / 2 } N ^ { 4 / 3 } ( \\log N ) ^ { 4 / 3 } \\Big ] = 1 . \\end{align*}"}
{"id": "7401.png", "formula": "\\begin{align*} \\begin{aligned} c _ n & : = - \\frac { \\int _ { { \\C } _ 1 } \\left ( \\sum _ { i = 2 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \\cdot t _ { i } \\psi _ n ( y ) \\right ) d y } { \\int _ { { \\C } _ 1 } \\left ( \\sum _ { i = 1 } ^ 4 f ( | y - h _ n t _ { i } | ) ( y - h _ n t _ { i } ) \\cdot t _ { i } \\right ) ^ 2 d y } . \\end{aligned} \\end{align*}"}
{"id": "2585.png", "formula": "\\begin{align*} \\mathcal E = \\mathcal E ( \\mathbb C P ^ 2 , \\mathbb C P ^ 1 , \\pi ) \\ : = \\ \\mathbb C P ^ 1 \\hookrightarrow \\mathcal E \\xrightarrow { \\pi } \\mathbb C P ^ 2 \\ . \\end{align*}"}
{"id": "598.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { j = 1 } ^ { N ' } \\sum _ { k = 0 } ^ { n _ j } b _ { j k } x ^ { \\alpha _ j } \\log ^ k ( x ) + \\sum _ { k = 0 } ^ { n _ 0 } b _ { 0 k } \\log ^ k ( x ) + o ( x ^ { \\alpha _ { N ' } } \\log ^ { n _ { N ' } } ( x ) ) \\end{align*}"}
{"id": "1983.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / 2 } } | V _ { k , r } | ^ 2 \\ , d x & = \\int _ { B _ { 1 / 2 } } \\big | V + ( 1 - \\eta _ r ) ( U _ k - U _ 0 ) \\big | ^ 2 \\ , d x \\\\ & \\le 2 \\int _ { B _ { 1 / 2 } } \\big ( | V | ^ 2 + | U _ k | ^ 2 + | U _ 0 | ^ 2 \\big ) \\ , d x \\\\ & \\le \\frac { 1 } { \\mu } \\Big ( \\| v \\| ^ 2 _ { H ^ 1 ( B _ { 1 / 2 } ) } + C ( N , M ) \\Big ) = C ( N , M , \\mu ) \\Big ( \\| v \\| ^ 2 _ { H ^ 1 ( B _ { 1 / 2 } ) } + 1 \\Big ) . \\end{align*}"}
{"id": "2401.png", "formula": "\\begin{align*} \\| e ^ { - \\frac { 1 } { 2 } A _ n ^ 2 ( t - s ) } ( - A _ n ) ^ { \\gamma } \\| _ { 2 } = \\max _ { 1 \\le j \\le n - 1 } e ^ { - \\frac { 1 } { 2 } \\lambda _ { j , n } ^ 2 ( t - s ) } ( - \\lambda _ { j , n } ) ^ \\gamma \\le C ( t - s ) ^ { - \\frac { \\gamma } { 2 } } , \\quad \\gamma > 0 , \\end{align*}"}
{"id": "7638.png", "formula": "\\begin{align*} \\prod _ { k \\in F } f ^ { - 1 } \\circ ( \\chi ^ { - 1 } ) ^ k \\circ f \\circ \\chi ^ k & = \\prod _ { k \\in F } ( ( \\chi ^ { - 1 } ) ^ k ) ^ f \\circ \\chi ^ k \\\\ & = \\prod _ { k \\in F } ( ( \\chi ^ { - 1 } ) ^ k ) ^ f \\circ \\prod _ { k \\in F } \\chi ^ k \\\\ & = [ f , \\chi ^ F ] , \\end{align*}"}
{"id": "5795.png", "formula": "\\begin{align*} [ G , G ] = [ N , G ] . \\end{align*}"}
{"id": "5346.png", "formula": "\\begin{align*} ( \\Lambda _ { B _ 1 } - \\Lambda _ { B _ 2 } ) [ f ] [ g ] = ( B _ 1 - B _ 2 ) ( u _ f , u _ g ^ * ) \\end{align*}"}
{"id": "7941.png", "formula": "\\begin{align*} l _ \\pm = \\dim H ^ 2 ( X _ \\pm ; \\mathbb R ) = r - \\# ( S _ \\pm ) l = r - 1 - \\# ( S _ 0 ) . \\end{align*}"}
{"id": "7530.png", "formula": "\\begin{align*} X _ { n } ( t ) = B ( T _ { k } ) + \\bigg ( \\frac { t V _ { n } - V _ { k } } { V _ { k + 1 } - V _ { k } } \\bigg ) \\big ( B ( T _ { k + 1 } ) - B ( T _ { k } ) \\big ) , \\hbox { i f } ~ V _ { k } \\leq t V _ { n } < V _ { k + 1 } . \\end{align*}"}
{"id": "6987.png", "formula": "\\begin{align*} G _ { N } ( z ) : = \\prod \\limits _ { k = 1 } ^ { N - 1 } \\Bigg ( \\frac { \\mu _ { k } ^ { 2 } - z } { \\lambda _ { k + 1 } ^ { 2 } - z } \\Bigg ) \\end{align*}"}
{"id": "5853.png", "formula": "\\begin{align*} \\sup _ { k \\leq M - 1 } 2 ^ { - \\frac { k } { p } } \\bigg ( \\int _ { x _ k } ^ { \\infty } u \\bigg ) ^ { \\frac { 1 } { q } } V _ r ( 0 , x _ k ) \\approx \\sup _ { k \\leq M - 1 } 2 ^ { - \\frac { k } { p } } \\bigg ( \\int _ { x _ k } ^ { \\infty } u \\bigg ) ^ { \\frac { 1 } { q } } V _ r ( x _ { k - 1 } , x _ k ) = A _ 1 . \\end{align*}"}
{"id": "7985.png", "formula": "\\begin{align*} \\hat { \\Gamma } _ { ( Y _ + , D _ { Y , + } ) } : = \\bigoplus _ { \\vec d \\in ( \\mathbb Z ) ^ { | I _ + | } , f \\in \\mathbb K _ { + } / \\mathbb L } \\frac { \\prod _ { i \\in \\bar { I } _ + } \\Gamma ( 1 + \\bar D _ i - \\langle D _ i \\cdot f \\rangle ) } { \\prod _ { i = 1 } ^ k \\Gamma ( 1 + v _ i ) } \\prod _ { i \\in I _ + , d _ i < 0 } \\frac { 1 } { \\bar D _ i - d _ i } \\textbf { 1 } _ { f } [ \\textbf 1 ] _ { ( d _ i ) _ { i \\in I _ + } } . \\end{align*}"}
{"id": "8915.png", "formula": "\\begin{align*} u _ n '' = ( \\lambda _ n - u _ n ^ { p - 2 } ) u _ n \\implies - u _ n '' + \\frac { \\lambda _ n } { 2 } u _ n \\le 0 \\end{align*}"}
{"id": "3496.png", "formula": "\\begin{align*} L _ { 2 } & = \\frac { F _ { Y _ { ( 1 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "4639.png", "formula": "\\begin{align*} \\| B ^ n x \\| _ \\infty & = \\sup _ { k \\in \\N } \\left | w ^ { - n k + \\frac { n ( n + 1 ) } { 2 } } x _ { k - n } \\right | = \\sup _ { k \\ge n + 1 } \\left | w ^ { - n k + \\frac { n ( n + 1 ) } { 2 } } x _ { k - n } \\right | \\\\ & \\le \\sup _ { k \\ge n + 1 } { \\left | w \\right | } ^ { - n k + \\frac { n ( n + 1 ) } { 2 } } \\sup _ { k \\ge n + 1 } \\left | x _ { k - n } \\right | = { \\left | w \\right | } ^ { - n ( n + 1 ) + \\frac { n ( n + 1 ) } { 2 } } \\| x \\| _ \\infty \\\\ & = { \\left | w \\right | } ^ { - \\frac { n ( n + 1 ) } { 2 } } \\| x \\| _ \\infty , \\end{align*}"}
{"id": "2028.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : s y m g a m m a 2 } \\Gamma ^ { ( 2 ) } ( z _ 1 , z _ 2 , \\dd z _ 1 ' , \\dd z _ 2 ' ) = \\Gamma ^ { ( 2 ) } ( z _ 2 , z _ 1 , \\dd z _ 2 ' , \\dd z _ 1 ' ) . \\end{align*}"}
{"id": "1210.png", "formula": "\\begin{align*} \\big \\{ h \\in \\dot { H } ^ { - 1 , p } ( \\rho ) : \\| h - \\ell \\| _ { \\dot { H } ^ { - 1 , p } ( \\rho ) } \\le r \\big \\} = \\bigcap _ { \\substack { \\varphi \\in C _ 0 ^ { \\infty } : \\\\ \\| \\varphi \\| _ { \\dot { H } ^ { 1 , q } ( \\rho ) } \\le 1 } } \\big \\{ h \\in \\dot { H } ^ { - 1 , p } ( \\rho ) : | ( h - \\ell ) ( \\varphi ) | \\le r \\big \\} . \\end{align*}"}
{"id": "1987.png", "formula": "\\begin{align*} \\mathbf { T ^ 2 _ { k } } = \\int _ { B _ { 1 / 2 } } \\nabla ( \\eta _ r ( U _ k - U _ 0 ) ) \\cdot \\nabla ( U _ 0 + V ) \\ , d x = \\int _ { B _ { 1 / 2 } } & \\eta _ r \\nabla ( U _ 0 + V ) \\cdot \\nabla ( U _ k - U _ 0 ) \\ , d x \\\\ & + \\int _ { B _ { 1 / 2 } } ( U _ k - U _ 0 ) \\nabla \\eta _ r \\cdot \\nabla ( U _ 0 + V ) \\ , d x . \\end{align*}"}
{"id": "8379.png", "formula": "\\begin{align*} F ( 4 \\lambda - 1 , 2 , 2 ) = 4 \\lambda \\end{align*}"}
{"id": "5898.png", "formula": "\\begin{align*} \\tilde { \\mu } _ { V X ^ { ( n ) } } ( A ) : = \\mathbb { P } \\Big ( \\Big ( \\frac { 1 } { 1 + \\alpha } \\Big ) ^ { 1 / p } \\sum _ { j = 1 } ^ n Z _ j V _ { \\bullet , j } \\in A \\Big ) , \\end{align*}"}
{"id": "1311.png", "formula": "\\begin{align*} B ( | v - v _ * | , \\cos \\theta ) = | v - v _ * | ^ \\gamma b \\left ( \\cos \\theta \\right ) , \\gamma = \\frac { s - ( 2 d - 1 ) } { s - 1 } , \\end{align*}"}
{"id": "460.png", "formula": "\\begin{align*} \\Sigma _ { \\mu \\boxtimes \\nu } ( z ) = \\Sigma _ { \\mu } ( z ) \\Sigma _ { \\nu } ( z ) , \\quad | z | < \\min \\{ \\rho _ { \\mu } , \\rho _ { \\nu } \\} . \\end{align*}"}
{"id": "6733.png", "formula": "\\begin{align*} \\mu \\int _ { \\Omega } \\varepsilon y _ t : \\varepsilon v + \\nu \\int _ { \\Omega } \\left ( 0 , \\abs { \\varepsilon y _ t } - g \\right ) \\ , \\dfrac { \\varepsilon y _ t } { \\abs { \\varepsilon y _ t } } : \\varepsilon v = \\int _ { \\Omega } ( u + t h ) \\cdot v , \\forall v \\in Y . \\end{align*}"}
{"id": "7624.png", "formula": "\\begin{align*} | Z | & \\geq \\left | \\bigcup _ { p \\in [ r + 1 ] } N _ { i } ( w _ p , Z ) \\right | \\\\ & \\geq ( r + 1 ) \\frac { n } { r } - \\sum _ { 1 \\leq q < p \\leq r + 1 } | N _ { i } ( w _ q , Z ) \\cap N _ { i } ( w _ p , Z ) | \\\\ & \\geq | Z | \\Big ( \\frac { r + 1 } { r } - \\frac { \\binom { r + 1 } { 2 } } { r ^ 3 } \\Big ) > | Z | . \\end{align*}"}
{"id": "2501.png", "formula": "\\begin{align*} H e s s f ( X , Y ) = g ( h _ f ( X ) , Y ) , ~ \\forall X , Y \\in \\Gamma ( T M ) . \\end{align*}"}
{"id": "6499.png", "formula": "\\begin{align*} \\int _ 0 ^ { c _ 1 } h ' \\left ( \\abs { v } ^ 2 + \\abs { u _ x + b v _ x } ^ 2 \\right ) d x = o ( 1 ) . \\end{align*}"}
{"id": "327.png", "formula": "\\begin{align*} { \\rm d } ( { \\rm L } _ b ) = \\frac { 1 } { b } \\prod _ { p < P ( b ) } \\Big ( 1 - \\frac { 1 } { p } \\Big ) = \\frac { e ^ { - \\gamma } + o _ y ( 1 ) } { b \\log P ( b ) } . \\end{align*}"}
{"id": "4196.png", "formula": "\\begin{align*} \\kappa _ \\gamma ( y , v ) : = \\sum _ { \\ell = - 1 } ^ \\iota \\int _ { A _ \\ell ( y ) } | K _ \\ell ^ { ( \\iota ) } ( ( x , u ) ^ { - 1 } ( y , v ) ) | \\ , d ( x , u ) . \\end{align*}"}
{"id": "8156.png", "formula": "\\begin{align*} & ( | \\eta _ \\delta ^ \\perp | ^ 2 * \\chi _ { B _ { n } } ) ( x ) = \\int \\limits _ { B _ { n } } | \\eta _ \\delta ^ \\perp ( y - x ) | ^ 2 \\ , d y \\leq C \\int \\limits _ { B _ { n } } \\| x - y \\| ^ { - \\ell } \\ , d y \\leq C \\int \\limits _ { B _ { n } } ( 2 n - \\| y \\| ) ^ { - \\ell } \\ , d y \\\\ & \\leq C \\int \\limits _ { 0 } ^ n ( 2 n - r ) ^ { - \\ell } r ^ { d - k - 1 } \\ , d r \\leq C n ^ { - \\ell + d - k } \\end{align*}"}
{"id": "6232.png", "formula": "\\begin{align*} & \\int \\frac { 1 } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( q x [ n - 1 ] _ q + c [ n ] _ q \\right ) \\widetilde { h } _ n ( x ; q ) d _ q x = \\\\ & \\frac { 1 - q } { ( - x ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\widetilde { h } _ n ( x ; q ) - q ^ { 1 - n } [ n ] _ q ( c + x ) \\widetilde { h } _ { n - 1 } ( x ; q ) \\right ) , \\end{align*}"}
{"id": "6874.png", "formula": "\\begin{align*} G _ t ( x ) : = \\frac { 1 } { \\sqrt { 2 \\pi C ( t ) } } \\exp \\left ( - \\frac { ( x - B ( t ) ) ^ 2 } { 2 C ( t ) } \\right ) \\end{align*}"}
{"id": "7655.png", "formula": "\\begin{align*} [ v ] & = [ \\lambda _ { \\delta } u _ 1 + \\mu _ { \\delta } u _ 2 ] \\in S \\cap [ \\rho _ 2 ] , \\\\ [ w ] & = [ \\lambda _ { \\theta } u _ 1 + \\mu _ { \\theta } u _ 2 ] \\in S \\cap [ \\rho _ 2 ] . \\end{align*}"}
{"id": "1058.png", "formula": "\\begin{align*} \\max \\left \\{ \\frac { p q } { n p + q } , 1 \\right \\} = 1 . \\end{align*}"}
{"id": "1222.png", "formula": "\\begin{align*} \\mathcal { F } ( v , x ) & = x - \\beta ( v ) - \\sum _ { w \\in c ( v ) } \\dfrac { 1 } { \\mathcal { F } ( w , x ) } ( \\forall v \\in V ( T ) ) , \\end{align*}"}
{"id": "2711.png", "formula": "\\begin{align*} U _ N : \\mathcal { H } _ N \\to \\ell ^ 2 ( \\mathcal { F } _ \\perp ) , U _ N \\psi _ N = \\bigoplus _ { s = 0 } ^ N \\ ; \\bigoplus _ { d = - N + s , - N + s + 2 , \\dots } ^ { N - s - 2 , N - s } \\Phi _ { s , d } . \\end{align*}"}
{"id": "2118.png", "formula": "\\begin{align*} \\det ( A _ G ) & = ( - 1 ) ^ { | V | - l - 1 } \\left [ ( t - 1 ) ^ 2 - | W | l t - ( t - 1 ) ( t - 1 - | W | l + l ) \\right ] \\alpha ^ { | V | } ( t - 1 ) ^ { ( l - 1 ) } | E | ^ { | W | - 1 } \\\\ & = ( - 1 ) ^ { | V | - l - 1 } l [ 1 - ( t + w ) ] \\alpha ^ { | V | } ( t - 1 ) ^ { ( l - 1 ) } | E | ^ { | W | - 1 } . \\end{align*}"}
{"id": "7713.png", "formula": "\\begin{align*} = \\sum _ { p ^ { \\nu } \\le x ^ { 1 / k } } ( \\log p ) \\left ( \\frac { x } { p ^ { \\nu k } } P _ { k - 1 } \\Big ( \\log \\frac { x } { p ^ { \\nu k } } \\Big ) + O \\Big ( \\Big ( \\frac { x } { p ^ { \\nu k } } \\Big ) ^ { \\theta _ k + \\varepsilon } \\Big ) \\right ) . \\end{align*}"}
{"id": "1619.png", "formula": "\\begin{align*} \\gamma _ i = \\textrm { m i n } \\{ \\gamma _ { i , 1 } , \\gamma _ { i , 2 } , . . . , \\gamma _ { i , l } , . . . , \\gamma _ { i , L } \\} , \\end{align*}"}
{"id": "8402.png", "formula": "\\begin{gather*} ( \\gamma , ( \\lambda , \\mu ) ) ( \\gamma ' , ( \\lambda ' , \\mu ' ) ) = \\left ( \\gamma \\gamma ' , ( \\lambda , \\mu ) \\gamma ' + ( \\lambda ' , \\mu ' ) \\right ) . \\end{gather*}"}
{"id": "3009.png", "formula": "\\begin{align*} A _ { C , \\Delta } ( p ) = \\frac { 1 } { n } - \\left ( \\Omega \\cdot C \\right ) _ p . \\end{align*}"}
{"id": "5913.png", "formula": "\\begin{align*} T ( r , f _ j ) & = O ( \\log ^ + T _ 1 ( r ) ) + O ( \\log r ) + O ( 1 ) \\\\ & = O ( \\log ^ + T _ { \\exp _ A f } ( r ) ) + O ( \\log r ) + O ( 1 ) | | . \\end{align*}"}
{"id": "7918.png", "formula": "\\begin{align*} \\mathcal A _ \\omega : = \\{ I \\subset \\{ 1 , \\ldots , m \\} | \\omega \\in \\angle _ I \\} . \\end{align*}"}
{"id": "1557.png", "formula": "\\begin{align*} \\sigma _ { B H } ( x ) = \\frac { v o l ( B ^ n ( 1 ) ) } { v o l \\left \\lbrace ( y ^ i ) \\in T _ x M : F ( x , y ) < 1 \\right \\rbrace } , \\end{align*}"}
{"id": "2977.png", "formula": "\\begin{align*} T _ 0 ( d ) = \\frac { \\Gamma ( \\frac s 2 ) ^ 2 } { 4 \\Gamma ( s ) } d ^ s L ( s , \\chi _ d ) \\sum _ { c = 1 } ^ \\infty \\frac { K ^ + ( d , 0 ; 4 c ) } { c ^ { s + 1 / 2 } } . \\end{align*}"}
{"id": "1530.png", "formula": "\\begin{align*} \\cosh R = \\cosh r \\cosh r ' - \\sinh r \\sinh r ' \\cos \\theta _ R ( r , r ' ) . \\end{align*}"}
{"id": "2961.png", "formula": "\\begin{align*} 1 2 \\sqrt \\pi | b ( 1 ) | ^ 2 = \\langle \\varphi , \\varphi \\rangle ^ { - 1 } \\int _ 0 ^ \\infty \\varphi ( i y ) y ^ { - 1 } \\ , d y . \\end{align*}"}
{"id": "7469.png", "formula": "\\begin{align*} Q : = S _ { j _ { f } } \\dots S _ { 0 } \\end{align*}"}
{"id": "2598.png", "formula": "\\begin{align*} \\psi _ 1 = P B _ { 2 1 } x ^ 1 _ 1 { x ^ 2 _ 2 } ^ \\ast \\ , \\ \\ \\psi _ 2 = P B _ { 1 2 } x ^ 2 _ 1 { x ^ 1 _ 2 } ^ \\ast \\ . \\end{align*}"}
{"id": "5884.png", "formula": "\\begin{align*} d _ n ^ { \\gamma _ n } ( \\mathcal K ) _ X \\geq C ( \\log _ 2 n ) ^ { - \\alpha } , n = 1 , 2 , \\dots . \\end{align*}"}
{"id": "4322.png", "formula": "\\begin{align*} & - \\int _ { \\partial \\Sigma \\cap L } e ^ { - i \\hat { \\theta } } d ( f _ L F ) - \\int _ { \\partial \\Sigma \\cap L _ 0 } e ^ { - i \\hat { \\theta } } d ( f _ { L _ 0 } F ) \\\\ & = \\sum _ { } e ^ { - i \\hat { \\theta } } F f | ^ + _ - , \\end{align*}"}
{"id": "4829.png", "formula": "\\begin{align*} R ( G ) = \\bigoplus _ { \\rho \\in \\operatorname { I r r } ( G ) } \\Q \\ , \\rho \\ ; , \\end{align*}"}
{"id": "417.png", "formula": "\\begin{align*} \\Omega _ { \\mu } = \\{ z \\in \\mathbb { H } : z + \\varphi _ { \\mu } ( z ) \\in \\mathbb { H } \\} . \\end{align*}"}
{"id": "6248.png", "formula": "\\begin{align*} & \\int f ( t ) h ( t / q ) \\left ( - \\frac { q ( 1 + q ) + t } { q ^ 2 ( 1 - q ) t } u ( t / q ) + \\frac { 1 } { q ( 1 - q ) ^ 2 t } \\right ) y ( t ) d _ q t \\\\ & = f ( x / q ) h ( x / q ) \\left ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \\right ) . \\end{align*}"}
{"id": "444.png", "formula": "\\begin{align*} \\Sigma _ { \\mu } ( z ) = \\gamma \\exp \\left [ \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) \\right ] , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
{"id": "8107.png", "formula": "\\begin{align*} \\limsup _ { x \\to \\infty } \\frac { \\sup _ { y \\in \\Z ^ d , \\lvert y \\rvert \\leq x } Y ( t , y ) } { f ( x ) } = \\infty \\limsup _ { x \\to \\infty } \\frac { \\sup _ { y \\in \\Z ^ d , \\lvert y \\rvert \\leq x } Y ( t , y ) } { f ( x ) } = 0 , \\end{align*}"}
{"id": "4297.png", "formula": "\\begin{align*} Z ( L ) = \\int _ L \\Omega . \\end{align*}"}
{"id": "320.png", "formula": "\\begin{align*} \\sup _ { A \\textnormal { p r i m i t i v e } } \\overline { { \\rm d } } ( A ) \\ ; = \\ ; \\frac { 1 } { 2 } . \\end{align*}"}
{"id": "4181.png", "formula": "\\begin{align*} f = \\sum _ { j = 0 } ^ \\infty f _ j f _ j : = f \\chi _ { B _ j } . \\end{align*}"}
{"id": "5066.png", "formula": "\\begin{align*} L ^ { n , 2 } _ \\tau + L ^ { n , 3 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) ( s - \\eta _ n ( s ) ) ^ \\alpha \\Lambda ^ n _ s d s , \\end{align*}"}
{"id": "4676.png", "formula": "\\begin{align*} \\dot { u } ^ { - 1 } h \\pi _ J ( g ) \\i g & = { } ^ J \\dot { u } ^ { - 1 } h _ 1 b _ 1 t _ 1 \\pi _ J ( g ) \\i g \\in { } ^ J \\dot { u } ^ { - 1 } \\pi _ J ( g _ 1 ) \\i g _ 1 B ^ + \\subset U ^ - B ^ + . \\end{align*}"}
{"id": "6619.png", "formula": "\\begin{align*} I _ { k , \\ell } ( c ) = ( h \\ell ) ^ 4 \\int _ { \\mathbb { R } ^ 4 } w ( \\ell x ) H \\big ( h ^ { - 1 } k , \\ell ^ 2 N ( x ) - 1 \\big ) e _ { k } ( - h c \\cdot x ) \\ , d x . \\end{align*}"}
{"id": "1309.png", "formula": "\\begin{align*} v ' = \\frac { v + v _ * } { 2 } + \\frac { | v - v _ * | } { 2 } \\sigma v _ * ' = \\frac { v + v _ * } { 2 } - \\frac { | v - v _ * | } { 2 } \\sigma . \\end{align*}"}
{"id": "6280.png", "formula": "\\begin{align*} f ( q ^ k x ) = q ^ { \\frac { k ( k - 1 ) } { 2 } } x ^ k f ( x ) ( k \\in \\mathbb { N } _ 0 ) . \\end{align*}"}
{"id": "436.png", "formula": "\\begin{align*} K _ { - } = \\{ s + i t \\in K : t \\le f _ { \\nu } ( s _ { 0 } ) - \\varepsilon \\} \\end{align*}"}
{"id": "3304.png", "formula": "\\begin{align*} s ^ 2 U - \\Delta U = 0 \\Omega ^ + , \\end{align*}"}
{"id": "5030.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } E \\left [ | Q ^ { n , 3 } _ \\tau | ^ 2 \\right ] = 0 . \\end{align*}"}
{"id": "3786.png", "formula": "\\begin{align*} { } _ { } ^ z T _ { k , j ; n , l } ^ { \\mu ; m , 2 } ( B ) ( t , x , \\zeta ) = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\big ( ( t - s ) \\mathbf { P } _ 3 \\big ( \\mathfrak { K } ^ { \\mu , B } _ { k ; n } ( y , \\omega , v , \\zeta ) \\big ) + \\mathbf { P } _ 3 \\big ( \\mathfrak { K } ^ { e r r ; \\mu , B } _ { k ; n } ( y , v , \\zeta ) \\big ) \\big ) f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "1147.png", "formula": "\\begin{align*} M ( X _ { n - 5 } ' ) = & M ( \\Delta _ n ) \\setminus \\{ \\sigma \\} \\cup \\{ \\sigma \\setminus \\{ v ^ { i _ 1 } , \\ldots , v ^ { i _ { n - 5 } } , v ^ { p } \\} , \\sigma \\setminus \\{ v ^ { i _ 1 } , \\ldots , v ^ { i _ { n - 5 } } , v ^ { n } \\} \\} \\cup \\\\ & \\{ \\sigma \\setminus \\{ v ^ { l _ 1 } , v ^ { l _ 2 } , \\ldots , v ^ { l _ { n - 5 } } \\} : \\{ l _ 1 , l _ 2 , \\ldots , l _ { n - 5 } \\} \\subseteq \\{ 4 , 5 , \\ldots , n - 1 \\} \\setminus \\{ i _ 1 , \\ldots , i _ { n - 5 } \\} \\} . \\end{align*}"}
{"id": "3488.png", "formula": "\\begin{align*} \\overline { G } _ { ( 1 ) } ( u ) = \\frac { m } { m - 1 } \\left ( 1 + \\frac { 2 u } { m } \\right ) ^ { - ( m - 1 ) / 2 } \\end{align*}"}
{"id": "1191.png", "formula": "\\begin{align*} h ( \\varphi ) = \\int _ { \\R ^ d } \\varphi d h = \\int _ { \\R ^ d } \\varphi d \\left ( \\frac { \\mu _ t - \\nu _ t } { t } \\right ) = \\frac { 1 } { t } \\int _ { \\R ^ d } \\varphi d ( \\mu _ t - \\nu _ t ) . \\end{align*}"}
{"id": "6929.png", "formula": "\\begin{align*} \\Gamma S = S ^ * \\Gamma ; \\end{align*}"}
{"id": "6542.png", "formula": "\\begin{align*} S _ 1 \\leq \\sum \\limits _ { i = 0 } ^ { \\infty } \\mu \\left ( \\left [ a _ i , \\infty \\right ) \\right ) < \\infty . \\end{align*}"}
{"id": "7406.png", "formula": "\\begin{align*} \\frac { \\sum _ { k = 1 } ^ N f ( S _ k p / q ) - E ( f ) N } { \\sqrt { N } } \\overset { d } { \\to } \\mathcal { N } ( 0 , \\sigma ^ 2 ) \\end{align*}"}
{"id": "4419.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ d S _ { d , i } = [ 1 , 4 k d ] \\cup [ 4 k N + 4 e + 1 , ( 4 k + 2 ) N ] . \\end{align*}"}
{"id": "8280.png", "formula": "\\begin{align*} k \\coth ( k a / 2 ) = k ' \\coth ( k ' a / 2 ) \\end{align*}"}
{"id": "8198.png", "formula": "\\begin{align*} \\lambda _ { \\pm } = \\frac { \\sqrt { 1 + \\left ( \\frac { 4 \\beta \\hslash ^ { 2 } } { 3 a ^ { 2 } } \\right ) ^ { 2 } } \\pm 1 } { 4 \\beta \\hslash ^ { 2 } / 3 a ^ { 2 } } , \\end{align*}"}
{"id": "8853.png", "formula": "\\begin{align*} T _ 0 ^ { x , \\alpha } = \\infty , \\ \\ a . s . , \\end{align*}"}
{"id": "7613.png", "formula": "\\begin{align*} H _ r = ( - 1 ) ^ { r - 1 } \\Delta ^ r ( \\widehat { \\mu } ( n ) - 2 \\log \\widehat { \\mu } ( n ) ) - \\sum _ { k = 1 } ^ { \\infty } ( - 1 ) ^ { r - 1 } \\Delta ^ r \\Bigl ( \\dfrac { 1 } { k \\widehat { \\mu } ( n ) ^ k } \\Bigr ) \\end{align*}"}
{"id": "8115.png", "formula": "\\begin{align*} \\int _ 1 ^ { R ^ { N + 1 } } \\frac { 1 } { u } ( \\log u ) ^ i \\ , \\dd u = \\frac { ( N + 1 ) ^ { i + 1 } } { i + 1 } ( \\log R ) ^ { i + 1 } \\end{align*}"}
{"id": "8847.png", "formula": "\\begin{align*} \\| f _ t \\circ \\psi \\| = \\int _ { - x _ 0 } ^ { x _ 0 } ( \\psi ( x ) - t ) d x \\leq \\| f _ { t ' } \\circ \\psi \\| + ( t ' - t ) \\int _ { - x _ 0 } ^ { x _ 0 } d x < \\infty . \\end{align*}"}
{"id": "1517.png", "formula": "\\begin{align*} D _ { r } p ( x ) = \\frac { 1 } { r ! } \\Big ( \\frac { d } { d x } \\Big ) ^ { r } \\bigg [ x ^ { r } p ( x ) \\bigg ] . \\end{align*}"}
{"id": "1153.png", "formula": "\\begin{align*} ( f ^ \\dagger f , f ^ \\dagger f ) = | f ^ \\dagger f | ^ 2 = \\sum _ { \\mathcal { I } } \\sum _ { \\mathcal { J } } \\left ( f ^ \\mathcal { I } f ^ \\mathcal { J } \\right ) ^ 2 \\end{align*}"}
{"id": "5538.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } S _ n ^ { ( p ) } S _ n ^ { ( p ) } = \\sum _ { \\gamma \\in \\hat { \\Gamma } _ 0 ^ n } \\left | \\gamma ' ( 0 ) \\right | ^ p \\left | \\hat { \\rho } ( \\gamma ) ' ( 0 ) \\right | ^ { 2 - p } . \\end{align*}"}
{"id": "6622.png", "formula": "\\begin{align*} I _ { g , k , \\ell } ( c ) : = h ^ 4 \\int _ { \\mathbb { R } ^ 4 } w ( g ^ { - 1 } x ) H ( h ^ { - 1 } k , N ( x ) - 1 ) e _ { k \\ell } ( - h c \\cdot x ) \\ , d x . \\end{align*}"}
{"id": "3440.png", "formula": "\\begin{align*} H ^ k ( V , T _ V \\otimes \\mathcal { O } ( - Y _ { \\vert { V } } ) ) & = H ^ k ( V , \\Omega _ V ^ { n - 1 } \\otimes \\mathcal { O } ( - Y _ { \\vert { V } } - K _ { \\mathbb { P } \\vert { V } } ) ) \\\\ & \\subset H ^ k ( V , \\Omega _ V ^ { n - 1 } \\otimes \\mathcal { O } ( - m Y _ { \\vert { V } } - m K _ { \\mathbb { P } \\vert { V } } ) ) \\end{align*}"}
{"id": "5841.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ { M } a _ k \\bigg ( \\sum _ { j = k } ^ { M } a _ j \\bigg ) ^ { \\beta - 1 } \\approx \\bigg ( \\sum _ { k = N } ^ { M } a _ k \\bigg ) ^ { \\beta } \\end{align*}"}
{"id": "1118.png", "formula": "\\begin{align*} f _ { W | S } ^ * ( w | s ) = \\delta ( w - g ^ { - 1 } ( s | \\rho , \\lambda ) ) , \\end{align*}"}
{"id": "3370.png", "formula": "\\begin{align*} \\eta \\circ \\lambda \\circ ( \\alpha _ i ) _ { \\ast } = ( \\beta ^ { ( 2 ) } _ i ) _ { \\ast } \\circ ( \\iota ^ { A _ i } _ 2 ) _ { \\ast } = \\mu \\circ \\rho \\circ ( \\alpha _ i ) _ { \\ast } . \\end{align*}"}
{"id": "6881.png", "formula": "\\begin{align*} N ( t ) & = \\int _ { 0 } ^ { \\infty } g p ( t , V _ F , g ) d g = \\frac { 1 } { V _ F } \\int _ { 0 } ^ { \\infty } g \\left ( \\sum _ { k = - \\infty } ^ { + \\infty } p _ k ( t , g ) \\right ) d g . \\\\ & = \\sum _ { k = - \\infty } ^ { + \\infty } \\frac { 1 } { V _ F } \\int _ { 0 } ^ { \\infty } g p _ k ( t , g ) d g = \\sum _ { k = - \\infty } ^ { + \\infty } N _ k ( t ) , \\end{align*}"}
{"id": "3498.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast } & = \\frac { \\Gamma \\left ( ( m + 1 ) / 2 \\right ) m ^ { 3 / 2 } \\left [ \\left ( 1 + \\frac { \\xi _ { p } ^ { 2 } } { m } \\right ) ^ { - ( m - 3 ) / 2 } - \\left ( 1 + \\frac { \\xi _ { q } ^ { 2 } } { m } \\right ) ^ { - ( m - 3 ) / 2 } \\right ] } { \\Gamma ( m / 2 ) ( m - 1 ) ( m - 3 ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "3456.png", "formula": "\\begin{align*} \\mathrm { D T V } _ { ( p , q ) } ( X ) = \\mathrm { E } [ ( X - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { 2 } | x _ { p } < X < x _ { q } ] , \\end{align*}"}
{"id": "3885.png", "formula": "\\begin{align*} T _ { k , l } ( s ) = \\min \\{ \\max \\{ s - k , 0 \\} , l - k \\} \\hbox { i n } [ 0 , + \\infty ) \\end{align*}"}
{"id": "4662.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } t ^ { \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| \\tilde P _ t f \\| _ { q , h } = 0 . \\end{align*}"}
{"id": "6837.png", "formula": "\\begin{align*} D ( t ) & \\geq \\left [ \\int _ { 0 } ^ { t } a _ 0 d s - \\frac { ( \\int _ { 0 } ^ { t } e ^ { s - t } a _ 0 d s ) ^ 2 } { \\int _ { 0 } ^ { t } e ^ { 2 ( s - t ) } a _ 0 d s } \\right ] + \\left [ \\int _ { 0 } ^ { t } a _ 1 N ( s ) d s - \\frac { ( \\int _ { 0 } ^ { t } e ^ { s - t } a _ 1 N ( s ) d s ) ^ 2 } { \\int _ { 0 } ^ { t } e ^ { 2 ( s - t ) } a _ 1 N ( s ) d s } \\right ] \\\\ & \\geq \\left [ \\int _ { 0 } ^ { t } a _ 0 d s - \\frac { ( \\int _ { 0 } ^ { t } e ^ { s - t } a _ 0 d s ) ^ 2 } { \\int _ { 0 } ^ { t } e ^ { 2 ( s - t ) } a _ 0 d s } \\right ] = a _ 0 ( t - 2 \\frac { e ^ t - 1 } { e ^ t + 1 } ) . \\end{align*}"}
{"id": "6590.png", "formula": "\\begin{align*} h _ { \\mu , \\nu } : = ( g _ { j , \\mu , 1 } \\circ \\pi , \\ , \\ldots , \\ , g _ { j , \\mu , \\nu - 1 } \\circ \\pi , \\ , d g _ { j , \\mu , \\nu } \\circ \\iota , \\ , g _ { j , \\mu , \\nu + 1 } \\circ \\pi , \\ , \\ldots , \\ , g _ { j , \\mu , k } \\circ \\pi ) . \\end{align*}"}
{"id": "6142.png", "formula": "\\begin{align*} \\begin{aligned} & v _ h \\to v \\quad \\textrm { s t r o n g l y i n } H ^ { 1 } _ { \\# } \\quad \\textrm { a s } h \\to 0 , \\\\ & q _ h \\to q \\quad \\textrm { s t r o n g l y i n } L ^ 2 _ { \\# } \\quad \\textrm { a s } h \\to 0 ; \\end{aligned} \\end{align*}"}
{"id": "912.png", "formula": "\\begin{align*} \\begin{aligned} h _ 1 - \\dfrac { \\mathrm { t r } _ g h _ 1 } { n } \\cdot g & = - A _ 1 ( g ) + \\dfrac { \\mathrm { t r } _ g A _ 1 ( g ) } { n } \\cdot g , \\\\ h _ 2 - \\dfrac { \\mathrm { t r } _ g h _ 2 } { n } \\cdot g & = - A _ 2 ( g ) + \\dfrac { \\mathrm { t r } _ g A _ 2 ( g ) } { n } \\cdot g - A _ { 1 , 1 } ( h _ 1 ) + \\dfrac { \\mathrm { t r } _ g A _ { 1 , 1 } ( h _ 1 ) } { n } \\cdot g , \\\\ \\cdots \\cdots & = \\cdots \\cdots . \\end{aligned} \\end{align*}"}
{"id": "4262.png", "formula": "\\begin{align*} \\gamma _ \\rho : = \\frac { \\int \\limits _ { \\mathbb R ^ n } W _ \\rho ( x ) \\sum _ { i = 2 } ^ d W ^ 2 _ \\rho ( \\hat \\Theta _ i ( x ) ) \\partial _ \\rho W _ \\rho ( x ) d x } { \\int \\limits _ { \\mathbb R ^ n } \\ ( \\partial _ \\rho W _ \\rho ( x ) \\ ) ^ 2 d x } . \\end{align*}"}
{"id": "2468.png", "formula": "\\begin{align*} \\operatorname { c o d i m } \\Phi \\left ( { A } \\right ) = \\frac { | \\Delta _ { 1 } | - | \\Delta _ { 1 } \\backslash { A } ^ { \\perp } | } { 2 } - | { A } | = \\frac { | \\Delta _ { 1 } \\cap { A } ^ { \\perp } | } { 2 } - | { A } | . \\end{align*}"}
{"id": "4570.png", "formula": "\\begin{align*} C _ R ( J / I ) = \\left ( \\bigcap _ { e > 0 } J ^ { [ p ^ e ] } : _ S ( I ^ { [ p ^ e ] } : _ S I ) \\right ) / I . \\end{align*}"}
{"id": "3524.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast \\ast } = \\frac { \\xi _ { p } ^ { 3 } ( 1 + | \\xi _ { p } | ) \\exp ( - | \\xi _ { p } | ) - \\xi _ { q } ^ { 3 } ( 1 + | \\xi _ { q } | ) \\exp ( - | \\xi _ { q } | ) } { 2 F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "5165.png", "formula": "\\begin{align*} m ^ { - 1 } \\sum _ { i = 0 } ^ { m - 1 } \\log ( \\rho _ i ) < 0 \\iff \\prod _ { i = 0 } ^ { m - 1 } \\rho _ i < 1 . \\end{align*}"}
{"id": "6875.png", "formula": "\\begin{align*} \\begin{aligned} B ( t ) & = \\int _ { 0 } ^ { t } e ^ { - ( t - s ) } g _ { i n } ( s ) d s = \\int _ { 0 } ^ { t } e ^ { - ( t - s ) } ( g _ 0 + g _ 1 N ( s ) ) d s . \\\\ C ( t ) & = 2 \\int _ { 0 } ^ { t } e ^ { - 2 ( t - s ) } a ( s ) d s = 2 \\int _ { 0 } ^ { t } e ^ { - 2 ( t - s ) } ( a _ 0 + a _ 1 N ( s ) ) d s . \\end{aligned} \\end{align*}"}
{"id": "8417.png", "formula": "\\begin{gather*} \\theta _ { m , r } ^ 0 ( \\tau ) : = \\theta _ { m , r } ( \\tau , 0 ) \\end{gather*}"}
{"id": "2191.png", "formula": "\\begin{align*} \\tau ^ { \\alpha } ( T _ 1 , \\dots , T _ a ) = \\left ( \\tau ^ { \\alpha _ 1 } ( T _ 1 ) , \\dots , \\tau ^ { \\alpha _ a } ( T _ a ) \\right ) , \\end{align*}"}
{"id": "1017.png", "formula": "\\begin{align*} x | _ S = ( x _ { s _ 1 } , x _ { s _ 2 } , \\dots ) \\end{align*}"}
{"id": "6957.png", "formula": "\\begin{align*} F ^ { \\{ \\alpha \\} } = \\frac { F } { 1 + \\alpha F } , \\end{align*}"}
{"id": "2174.png", "formula": "\\begin{align*} ( ( \\mu \\boxplus \\nu ) _ { } ) = ( p _ { \\mu \\boxplus \\nu } ) = \\overline { h ( V ) } . \\end{align*}"}
{"id": "3475.png", "formula": "\\begin{align*} L _ { 1 } = \\frac { c _ { 1 } \\left [ \\xi _ { p } \\overline { G } _ { ( 1 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { p } ^ { 2 } \\right ) - \\xi _ { q } \\overline { G } _ { ( 1 ) } \\left ( \\frac { 1 } { 2 } \\xi _ { q } ^ { 2 } \\right ) \\right ] } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , ~ L _ { 2 } = \\frac { F _ { Y _ { ( 1 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "1157.png", "formula": "\\begin{align*} T [ f \\alpha + g ] = T [ f ] \\alpha + T [ g ] \\end{align*}"}
{"id": "4781.png", "formula": "\\begin{align*} F ( \\mathbf { B } _ B ) = & 4 ( \\kappa - 3 \\nu / 4 ) ( \\kappa - 3 \\nu / 4 - 1 ) ( \\kappa - \\nu / 2 - 2 ) + 4 ( \\kappa - 3 \\nu / 4 ) ( \\nu / 4 ) ( \\kappa - \\nu / 2 - 1 ) \\\\ + & 4 ( \\nu / 4 ) ( \\kappa - 3 \\nu / 4 ) ( \\kappa - \\nu / 2 - 1 ) + 4 ( \\nu / 4 ) ( \\nu / 4 ) ( \\kappa - \\nu / 2 ) . \\end{align*}"}
{"id": "6055.png", "formula": "\\begin{align*} \\Lambda = ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ m ) . \\end{align*}"}
{"id": "6936.png", "formula": "\\begin{align*} \\| x \\| ^ 2 = \\| ( \\Sigma ^ * ) ^ n x \\| ^ 2 = \\sum _ { k = 0 } ^ { n - 1 } \\| ( \\Sigma ^ * ) ^ k x \\| ^ 2 . \\end{align*}"}
{"id": "4873.png", "formula": "\\begin{align*} \\begin{aligned} B ( n , q ) = B ( n , q ; \\chi _ { 0 } , \\ldots , \\chi _ { 0 } ) , A ( n , q ) = \\frac { B ( n , q ) } { \\varphi ^ { 5 } ( q ) } , \\mathfrak { S } ( n ) = \\sum \\limits _ { q = 1 } ^ { \\infty } A ( n , q ) . \\end{aligned} \\end{align*}"}
{"id": "6850.png", "formula": "\\begin{align*} p ^ * ( v , g ) : = \\frac { 1 } { V _ F } \\frac { 1 } { \\sqrt { 2 \\pi c ^ * } } \\exp \\left ( - \\frac { ( g - b ^ * ) ^ 2 } { 2 c ^ * } \\right ) , \\end{align*}"}
{"id": "5646.png", "formula": "\\begin{align*} & \\inf _ { \\mathbb { R } ^ n } f > 0 \\ \\ f \\in C ^ 3 ( \\mathbb { R } ^ n \\setminus D ) , \\\\ & \\exists \\ \\beta > 2 \\ \\ \\limsup _ { | x | \\rightarrow \\infty } | x | ^ { \\beta + m } | D ^ m ( f ( x ) - 1 ) | < \\infty , \\ m = 0 , 1 , 2 , 3 , \\end{align*}"}
{"id": "4855.png", "formula": "\\begin{align*} N = p _ { 1 } + p _ { 2 } + 2 ^ { v _ { 1 } } + 2 ^ { v _ { 2 } } + \\cdots + 2 ^ { v _ { k } } . \\end{align*}"}
{"id": "8576.png", "formula": "\\begin{align*} m = n _ { k _ 1 } + n _ { k _ 2 } + \\ldots + n _ { k _ l } , \\end{align*}"}
{"id": "3535.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast } = \\frac { \\Gamma ( t - 1 ) \\left [ \\xi _ { p } ^ { 2 } \\left ( 1 + \\xi _ { p } ^ { 2 } \\right ) ^ { - ( t - 1 ) } - \\xi _ { q } ^ { 2 } \\left ( 1 + \\xi _ { q } ^ { 2 } \\right ) ^ { - ( t - 1 ) } \\right ] } { 2 \\Gamma ( t - \\frac { 1 } { 2 } ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "5122.png", "formula": "\\begin{align*} C = P _ n \\Lambda \\otimes _ { P _ { n - 1 } \\Lambda } \\Q \\end{align*}"}
{"id": "2280.png", "formula": "\\begin{align*} g _ e ^ 1 = \\sigma v _ e ^ 1 . \\end{align*}"}
{"id": "104.png", "formula": "\\begin{align*} \\chi ^ { \\epsilon , L } _ t ( \\lambda , \\mu , m ^ 2 ) : = \\max _ { x \\in \\Lambda _ { \\epsilon , L } } \\epsilon ^ d \\sum _ { y \\in \\Lambda _ { \\epsilon , L } } \\big < \\varphi _ x \\varphi _ y \\big > ^ { \\epsilon , L } _ { \\lambda , \\mu + 1 / t , m ^ 2 } = \\epsilon ^ d \\sum _ { x \\in \\Lambda _ { \\epsilon , L } } \\big < \\varphi _ 0 \\varphi _ x \\big > ^ { \\epsilon , L } _ { \\lambda , \\mu + 1 / t , m ^ 2 } , \\end{align*}"}
{"id": "8036.png", "formula": "\\begin{align*} \\lim _ { y _ { \\mathrm { r } + 1 } \\rightarrow 0 } \\bar I _ { \\mathcal O _ { \\mathbb P } ( - 1 ) \\oplus \\mathcal O _ { \\mathbb P } ( 1 - \\sum _ { i = 1 } ^ k w _ i ) } ( y ) = 0 , \\end{align*}"}
{"id": "4316.png", "formula": "\\begin{align*} \\zeta = \\omega ( J _ X \\frac { \\partial u } { \\partial s } + \\sqrt { - 1 } \\frac { \\partial u } { \\partial s } , Y ) d s = \\omega ( J _ X \\frac { \\partial u } { \\partial s } , Y ) d s . \\end{align*}"}
{"id": "3992.png", "formula": "\\begin{align*} a ' b ' = a ' g ' ~ ~ ~ \\mbox { a n d } ~ ~ ~ a '' b '' = a '' g '' . \\end{align*}"}
{"id": "2223.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } g _ { i } ^ { T } = \\frac { 1 } { 2 } d \\eta _ { i } = \\pi ^ { \\ast } ( h _ { i } ) = \\pi ^ { \\ast } ( \\omega _ { h _ { i } } ) \\end{array} \\end{align*}"}
{"id": "6013.png", "formula": "\\begin{align*} f ( x ) & = 3 x ^ 5 - 7 x ^ 4 + 4 x ^ 3 = x ^ 3 ( x - 1 ) ( 3 x - 4 ) \\\\ f ' ( x ) & = 1 5 x ^ 4 - 2 8 x ^ 3 + 1 2 x ^ 2 = x ^ 2 \\left ( 3 x - 2 \\right ) \\left ( 5 x - 6 \\right ) \\\\ f '' ( x ) & = 6 0 x ^ 3 - 8 4 x ^ 2 + 2 4 x = 1 2 x ( x - 1 ) ( 5 x - 2 ) \\\\ f ^ { ( 3 ) } ( x ) & = 1 8 0 x ^ 2 - 1 6 8 x + 2 4 = 1 2 \\left ( 1 5 x ^ 2 - 1 4 x + 2 \\right ) \\\\ f ^ { ( 4 ) } ( x ) & = 3 6 0 x - 1 6 8 = 2 4 ( 1 5 x - 7 ) \\\\ f ^ { ( 5 ) } ( x ) & = 3 6 0 . & \\end{align*}"}
{"id": "8844.png", "formula": "\\begin{align*} y \\leq \\frac { t f ( y ) } { f ( t ) } = \\frac { t \\max ( - f ( y ) , 0 ) } { - f ( t ) } \\leq \\frac { t \\max ( - f ( y ) , 0 ) } { - f ( t ) } + \\frac { t ' f _ t ( y ) } { t ' - t } \\end{align*}"}
{"id": "4891.png", "formula": "\\begin{gather*} \\bigwedge _ { i = 1 } ^ k \\bigwedge _ { j = 1 } ^ { n - k - 1 } \\neg a _ { i , j } \\lor a _ { i , j + 1 } \\qquad \\bigwedge _ { i = 0 } ^ k \\bigwedge _ { j = 1 } ^ { n - k } \\neg a _ { i , j } \\lor a _ { i + 1 , j } \\lor \\neg b _ { i + j } \\\\ \\bigwedge _ { i = 1 } ^ { k - 1 } \\bigwedge _ { j = 1 } ^ { n - k } a _ { i , j } \\lor \\neg a _ { i + 1 , j } \\qquad \\bigwedge _ { i = 1 } ^ k \\bigwedge _ { j = 0 } ^ { n - k } a _ { i , j } \\lor \\neg a _ { i , j + 1 } \\lor b _ { i + j } \\end{gather*}"}
{"id": "4206.png", "formula": "\\begin{align*} N = \\sum _ { j = 1 } ^ n ( a _ j \\partial _ { b _ j } - b _ j \\partial _ { a _ j } ) . \\end{align*}"}
{"id": "2808.png", "formula": "\\begin{align*} \\begin{aligned} \\min _ { 0 { } \\leq { } i { } \\leq { } N } & \\| \\nabla f ( x _ i ) \\| ^ 2 { } \\leq { } \\ , 2 L \\big ( f ( x _ 0 ) - f _ * \\big ) \\max \\left \\{ ( 1 - h ) ^ { 2 N } \\ , \\ \\frac { 1 } { q + N \\frac { h ( 2 - h ) ( 2 - \\kappa h ) } { 2 - h \\left ( 1 + \\kappa \\right ) } } \\right \\} \\end{aligned} \\end{align*}"}
{"id": "4225.png", "formula": "\\begin{align*} ( a ( 1 ) , a ( 2 ) , \\ldots ) = ( b _ 1 + 2 , b _ 1 + 2 \\cdot 2 , \\ldots , b _ 1 + 2 \\cdot d _ 1 , b _ 2 + 2 ^ 2 , b _ 2 + 2 ^ 2 \\cdot 2 , \\ldots , b _ 2 + 2 ^ 2 \\cdot d _ 2 , b _ 3 + 2 ^ 3 , b _ 3 + 2 ^ 3 \\cdot 2 , \\ldots ) \\end{align*}"}
{"id": "7585.png", "formula": "\\begin{align*} Y _ { k + 1 } = e ^ { \\Omega _ k } \\ , Y _ k . \\end{align*}"}
{"id": "380.png", "formula": "\\begin{align*} \\frac 2 5 \\sum _ x \\pi ( x ) ( \\mathbb { P } _ x [ T _ A \\le t ] ) ^ { 2 } & \\le \\sum _ x \\pi ( x ) ( \\mathbb { P } _ x [ T _ A \\le \\tau ] ) ^ { 2 } \\le \\sum _ x \\pi ( x ) \\left ( \\sum _ { a \\in A } \\mathbb { P } _ x [ T _ a \\le \\tau ] \\right ) ^ { 2 } \\\\ & \\le k ^ { 2 } \\sum _ x \\pi ( x ) \\mathbb { ( P } _ x [ T _ o \\le \\tau ] ) ^ { 2 } , \\end{align*}"}
{"id": "4338.png", "formula": "\\begin{align*} \\partial _ s u + J ( t , u ) ( \\partial _ t u - X _ H ) = 0 , \\end{align*}"}
{"id": "5018.png", "formula": "\\begin{align*} | \\psi _ { n , 1 } ( u , s _ 2 ) | = | ( s _ 2 - \\eta _ n ( u ) ) ^ { \\alpha } - ( s _ 2 - u ) ^ \\alpha | \\le \\frac { \\alpha } n \\delta ^ { \\alpha - 1 } . \\end{align*}"}
{"id": "3177.png", "formula": "\\begin{align*} e _ { \\Phi ^ { \\mathfrak { b } , \\gamma } } \\left ( \\rho \\right ) = \\int _ { \\mathbb { S } \\times \\mathbb { D } _ { \\varepsilon } } e _ { \\mathcal { K } _ { \\gamma } \\left ( \\Psi , f \\right ) } \\left ( \\rho \\right ) \\mathfrak { b } \\left ( \\mathrm { d } \\left ( \\Psi , f \\right ) \\right ) \\ , \\rho \\in E _ { 1 } \\ . \\end{align*}"}
{"id": "5717.png", "formula": "\\begin{align*} e _ { k } ( y _ 1 , \\ldots , y _ { b } ) = e _ { k } ( y _ 1 , \\ldots , y _ { b - 1 } ) + e _ { k - 1 } ( y _ 1 , \\ldots , y _ { b - 1 } ) y _ { b } ( 1 \\le k \\le b \\le n - 1 ) , \\end{align*}"}
{"id": "2425.png", "formula": "\\begin{align*} \\| b \\| = \\| \\delta ( a ) \\| & \\leq \\| a \\| = \\| q _ A ( a ) \\| \\\\ & = \\| \\tilde \\delta ( q _ A ( a ) ) \\| = \\| q _ B ( \\delta ( a ) ) \\| \\\\ & = \\| q _ B ( b ) \\| , \\end{align*}"}
{"id": "5852.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } \\int _ { x _ k } ^ { x _ { k + 1 } } \\bigg ( \\int _ { x _ k } ^ t f ^ r v \\bigg ) ^ { \\frac { q } { r } } u ( t ) d t \\bigg ) ^ { \\frac { 1 } { q } } \\leq \\mathfrak { C '' } \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ k \\bigg ( \\int _ { x _ k } ^ { x _ { k + 1 } } f \\bigg ) ^ p \\bigg ) ^ { \\frac { 1 } { p } } \\end{align*}"}
{"id": "8903.png", "formula": "\\begin{align*} Q ( \\varphi ; U , \\mathcal { F } ) : = \\int _ { \\mathcal { F } } \\left ( | \\varphi ' | ^ 2 + ( \\lambda - ( p - 1 ) \\rho | U | ^ { p - 2 } ) \\varphi ^ 2 \\right ) \\ , d x , \\forall \\varphi \\in H ^ 1 ( \\mathcal { F } ) \\cap C _ c ( \\mathcal { F } ) . \\end{align*}"}
{"id": "3105.png", "formula": "\\begin{align*} a _ { \\mathrm { p w } } ( u _ { \\mathrm { n c } } , v _ { \\mathrm { n c } } ) = \\lambda _ h \\Big ( \\frac { u _ { \\mathrm { n c } } } { 1 - \\lambda _ h \\kappa _ { m } ^ 2 h _ { \\mathcal { T } } ^ { 2 { m } } } , v _ { \\mathrm { n c } } \\Big ) _ { L ^ 2 ( \\Omega ) } \\qquad v _ { \\mathrm { n c } } \\in V ( \\mathcal { T } ) \\end{align*}"}
{"id": "8767.png", "formula": "\\begin{align*} \\frac { 1 } { q ( p _ 1 , p _ 2 ) } = \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } - 1 . \\end{align*}"}
{"id": "5476.png", "formula": "\\begin{align*} \\Delta ( f \\circ \\delta ) = f ' ( \\delta ) \\Delta \\delta + f '' ( \\delta ) \\norm { \\nabla \\delta } ^ 2 . \\end{align*}"}
{"id": "8148.png", "formula": "\\begin{align*} & \\| \\varphi _ { n ; \\textrm { i n } } \\| = \\| P _ { \\delta } ( W _ { n ; \\textrm { i n } } ) e ^ { - i t _ n H } \\varphi \\| \\\\ & \\leq \\| P _ { \\delta } ( W _ { n ; \\textrm { i n } } ) ( e ^ { - i t _ n H } - e ^ { - i t _ n H _ 0 } ) \\varphi \\| + \\| P _ { \\delta } ( W _ { n ; \\textrm { i n } } ) e ^ { - i t _ n H _ 0 } \\varphi \\| \\end{align*}"}
{"id": "8003.png", "formula": "\\begin{align*} H _ { ( X , K 3 ) , ( d , 0 ) } ( y , 0 ) = H _ { ( Q _ 4 , K 3 ) , d } ( y ) . \\end{align*}"}
{"id": "4274.png", "formula": "\\begin{align*} V _ L = \\bigoplus _ i V _ { \\lambda _ i , 0 } \\qquad V _ L = \\bigoplus _ i V _ { \\lambda _ i , 1 } . \\end{align*}"}
{"id": "930.png", "formula": "\\begin{align*} E ( \\Psi _ t ) ( 0 , - \\frac { 1 } { 2 } h ) + Q ( v , v ) = v \\end{align*}"}
{"id": "5983.png", "formula": "\\begin{align*} \\hat G ( 0 ) = A U _ 0 - f ( 0 ) , \\hat G ( t _ j ) = 0 , \\ ; j = 1 , \\dots , N . \\end{align*}"}
{"id": "4904.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & p & q & \\cdots & s \\\\ p & q & r & \\cdots & t \\end{pmatrix} . \\end{align*}"}
{"id": "7696.png", "formula": "\\begin{align*} E V _ { \\mathcal { S } } ( B , \\nu ) = ( A _ { \\mathcal { R } _ { B , \\nu } } , \\mu _ { \\mathcal { R } _ { B , \\nu } } ) , \\end{align*}"}
{"id": "3458.png", "formula": "\\begin{align*} \\mathrm { D T S } _ { ( p , q ) } ( X ) = \\frac { \\mathrm { E } \\left [ ( X - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { 3 } | x _ { p } < X < x _ { q } \\right ] } { \\mathrm { D T V } _ { ( p , q ) } ^ { 3 / 2 } ( X ) } , \\end{align*}"}
{"id": "8773.png", "formula": "\\begin{align*} \\tilde { Y } _ O ( P , G ) & : = \\sup \\{ \\| \\phi _ 1 * \\phi _ 2 * \\cdots * \\phi _ N \\| _ { q ( P ) } \\mid ( \\phi _ 1 , \\phi _ 2 , \\cdots , \\phi _ N ) \\in \\tilde { \\mathcal { B } } \\} , \\\\ \\tilde { Y } _ R ( P , G ) & : = \\inf \\{ \\| \\phi _ 1 * \\phi _ 2 * \\cdots * \\phi _ N \\| _ { q ( P ) } \\mid ( \\phi _ 1 , \\phi _ 2 , \\cdots , \\phi _ N ) \\in \\tilde { \\mathcal { B } } \\} . \\end{align*}"}
{"id": "3108.png", "formula": "\\begin{align*} & \\Vert \\kappa _ { m } ^ { - 1 } h _ { \\mathcal { T } } ^ { - { m } } ( \\psi - \\phi ) \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 = \\Vert \\kappa _ { m } ^ { - 1 } h _ { \\mathcal { T } } ^ { - { m } } \\delta \\phi \\Vert ^ 2 _ { L ^ 2 ( \\Omega ) } = \\lambda _ h ^ 2 \\kappa _ { m } ^ 2 \\Vert h _ { \\mathcal { T } } ^ { m } ( 1 + \\delta ) \\phi \\Vert ^ 2 _ { L ^ 2 ( \\Omega ) } . \\end{align*}"}
{"id": "775.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } | a _ n | \\phi _ { n } ( r ) \\leq \\sum _ { n = 1 } ^ { \\infty } M ( n ) \\phi _ { n } ( r ) . \\end{align*}"}
{"id": "5536.png", "formula": "\\begin{align*} \\hat { A } ( z ) = \\lambda ^ A \\frac { z - a ^ A } { 1 - \\overline { a ^ A } z } , \\end{align*}"}
{"id": "7778.png", "formula": "\\begin{align*} \\tilde U = \\tilde \\Xi \\tilde \\Lambda \\tilde K \\end{align*}"}
{"id": "6726.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 } \\dfrac { \\mathsf { m } ( E + t H ) - \\mathsf { m } ( E ) } { t } & = \\lim _ { t \\rightarrow 0 } \\dfrac { E + t H } { \\abs { E + t H } } \\left ( \\dfrac { \\abs { E + t H } - \\abs { E } } { t } \\right ) = \\dfrac { E } { \\abs { E } ^ 2 } E : H . \\end{align*}"}
{"id": "4730.png", "formula": "\\begin{align*} ( 1 ) ~ ~ \\overline { C _ { d } } & = C _ { d } , \\\\ ( 2 ) ~ ~ C _ { d } & = H _ { d } + \\sum \\limits _ { \\substack { d ' \\in I _ { k , n } \\\\ \\ell ( d ' ) < \\ell ( d ) } } p _ { d ' , d } H _ { d ' } , ~ \\mbox { w h e r e } p _ { d ' , d } \\in q ^ { - 1 } \\mathbb { Z } [ q ^ { - 1 } ] . \\end{align*}"}
{"id": "5969.png", "formula": "\\begin{align*} C ^ 1 _ { T , \\beta } = 2 \\max ( 2 ^ { 1 - \\beta } \\Gamma ( 1 - \\beta ) T ^ { \\beta } , 1 ) \\end{align*}"}
{"id": "2170.png", "formula": "\\begin{align*} \\omega ^ { \\prime } ( h ( \\alpha ) ) = \\frac { 1 } { H ^ { \\prime } ( \\alpha ) } = \\frac { 1 } { 1 - g ( \\alpha ) } = \\begin{cases} 1 / ( 1 - I _ { 2 } I _ { 3 } ) & \\alpha \\in A , \\\\ 1 / ( 1 - I _ { 1 } I _ { 4 } ) & \\alpha \\in B \\cup C . \\end{cases} \\end{align*}"}
{"id": "3643.png", "formula": "\\begin{align*} \\limsup _ n \\sum _ { i < k _ n } \\mu _ n ( \\tilde B _ n ^ i ) & \\leq \\limsup _ n \\sum _ { i < k _ n } \\mu _ n ( B _ n ^ i ) \\\\ & = \\sum _ { i < k } \\norm { \\mu ^ i } \\leq \\sum _ { i < k } \\liminf _ n \\mu _ n ( \\tilde B _ n ^ i ) \\leq \\liminf _ n \\sum _ { i < k _ n } \\mu _ n ( \\tilde B _ n ^ i ) \\end{align*}"}
{"id": "2938.png", "formula": "\\begin{align*} & D _ H B \\equiv i [ H , B ] + \\frac { \\partial B } { \\partial t } . \\\\ & B = B ^ * , \\end{align*}"}
{"id": "3626.png", "formula": "\\begin{align*} v _ \\mathbf { x } ( q _ i , w ) = \\dfrac { | \\{ [ w ' \\ ! : \\ ! \\mathbf { x } ' \\ ! : \\ ! q ' ] \\mid [ w ' \\ ! : \\ ! \\mathbf { x } ' \\ ! : \\ ! q ' ] \\prec [ w \\ ! : \\ ! \\mathbf { x } \\ ! : \\ ! q _ i ] \\} | } { 2 \\cdot n \\cdot | W | } \\end{align*}"}
{"id": "5466.png", "formula": "\\begin{align*} b \\star ( a _ 1 a _ 2 ) = l ( b a _ 1 a _ 2 ) = & l ( a ' b a _ 2 ) \\stackrel { ( L 2 ) } = a ' \\cdot l ( b a _ 2 ) \\\\ = & l ( b a _ 1 ) \\cdot l ( b a _ 2 ) = ( b \\star a _ 1 ) ( b \\star a _ 2 ) . \\end{align*}"}
{"id": "7811.png", "formula": "\\begin{align*} \\psi = \\psi _ k \\circ \\psi _ { k - 1 } \\circ \\cdots \\circ \\psi _ 1 , \\end{align*}"}
{"id": "7852.png", "formula": "\\begin{align*} \\lVert e _ Q a _ 1 ^ * u _ n a _ 2 e _ Q \\rVert _ { \\infty , 1 } & = \\lVert E _ Q ( a _ 1 ^ * u _ n c _ a ) \\rVert _ 1 \\to 0 . \\end{align*}"}
{"id": "3195.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } y _ { 2 3 } ^ 2 + y _ { 2 4 } ^ 2 = - 2 a \\\\ y _ { 1 3 } y _ { 2 3 } + y _ { 1 4 } y _ { 2 4 } = b \\\\ y _ { 1 3 } ^ 2 + y _ { 1 4 } ^ 2 = - 2 c \\end{array} \\right . . \\end{align*}"}
{"id": "5272.png", "formula": "\\begin{align*} X _ { Q , \\bullet } = - X _ { Q ^ { + r } , \\bullet } = - X _ { Q ^ { + s } , \\bullet } . \\end{align*}"}
{"id": "7790.png", "formula": "\\begin{align*} \\rho _ { u + \\delta } ( x ) = x - ( u + \\delta , x ) ( u + \\delta ) . \\end{align*}"}
{"id": "1937.png", "formula": "\\begin{align*} R _ { 0 } ( z ) & = \\frac { 1 } { z - \\sum _ { \\ell = 0 } ^ { p } a _ { - \\ell } \\ , S _ { p - \\ell - 1 } ^ { ( 1 ) } ( z ) \\ , T _ { \\ell - 1 } ^ { ( 1 ) } ( z ) } \\\\ R _ { j } ( z ) & = R _ { 0 } ( z ) \\ , S _ { j - 1 } ^ { ( 1 ) } ( z ) 1 \\leq j \\leq p . \\end{align*}"}
{"id": "2847.png", "formula": "\\begin{align*} \\bar { x } _ i : = x _ i - \\frac { - \\kappa } { 1 - \\kappa } \\frac { h _ i } { L } U \\in \\big [ x _ { i + 1 } , x _ { i } \\big ] , i \\in \\big \\{ 0 , \\dots , N - 1 \\big \\} . \\end{align*}"}
{"id": "7255.png", "formula": "\\begin{align*} \\mathbb { E } ( \\tau ) L ^ k ( f , \\mu ) & \\geq \\int _ { \\tilde \\Omega ^ k } d ( f ( e ) , f ( \\gamma _ { \\tau ( \\tilde \\omega ) } ^ k ( \\tilde \\omega ) ) ) d \\mathbb { P } ( \\tilde \\omega ) \\\\ & = \\sum _ { i = 1 } ^ { \\infty } b _ i \\left ( \\int _ { \\Gamma } d ( f ( e ) , f ( \\gamma ) ) d \\mu ^ { i k } ( \\gamma ) \\right ) \\end{align*}"}
{"id": "9016.png", "formula": "\\begin{align*} \\Delta _ 1 ( a , b ) = \\frac { \\pi } { 2 b } \\left ( \\ln \\left ( \\frac { a } { b } \\right ) - \\gamma \\right ) + \\frac { 2 } { b } \\beta ^ { \\prime } ( 1 ) . \\end{align*}"}
{"id": "4314.png", "formula": "\\begin{align*} \\dim ( \\ker \\bar { \\partial } \\subset W ^ { 1 , 2 ; \\mu } ) = \\dim \\Gamma ( \\mathbb { P } ^ 1 ( \\R ) , \\mathcal { E } \\otimes \\mathcal { O } ( \\frac { \\mu } { \\pi } ) ) = K ( \\frac { \\mu } { \\pi } + 1 ) + \\sum _ 1 ^ K n _ i . \\end{align*}"}
{"id": "3616.png", "formula": "\\begin{align*} & \\big \\langle \\sum _ { i \\in A _ 1 \\cup A _ 2 } e _ i , \\ , e _ { i _ 1 } + \\cdots + e _ { i _ k } \\big \\rangle = | ( A _ 1 \\cup A _ 2 ) \\cap \\{ i _ 1 , \\ldots , i _ k \\} | = d - s + k , \\mbox { a n d } \\\\ & e _ i \\in H \\mbox { f o r } i \\in \\{ 1 , \\ldots , s \\} \\setminus \\{ i _ 1 , \\ldots , i _ k \\} . \\end{align*}"}
{"id": "1906.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { C } _ { [ n , r , s ] } } w ( \\gamma ) = a _ { - s } ^ { ( r + s ) } \\sum _ { k \\in \\mathbb { Z } } \\sum _ { \\ell \\in \\mathbb { Z } } A _ { [ k - 1 , r - 1 ] } ^ { ( 1 ) } B _ { [ \\ell - k - 1 , s - 1 ] } ^ { ( 1 ) } W _ { [ n - \\ell - 1 , 0 ] } \\end{align*}"}
{"id": "8718.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\omega } { d t ^ { 2 } } + \\omega - \\frac { \\mu a ^ { 2 } } { \\lambda } = 0 , \\end{align*}"}
{"id": "809.png", "formula": "\\begin{align*} ( z f ' ( z ) ) ' = G ' ( z ) \\psi ( \\omega ( z ) ) . \\end{align*}"}
{"id": "3452.png", "formula": "\\begin{align*} \\mathrm { T C S } _ { q } ( X ) = \\frac { \\mathrm { E } \\left [ ( X - \\mathrm { T C E } _ { q } ( X ) ) ^ { 3 } | X > x _ { q } \\right ] } { \\mathrm { T V } _ { q } ^ { 3 / 2 } ( X ) } \\end{align*}"}
{"id": "6008.png", "formula": "\\begin{align*} M _ { f _ 2 } ( 1 ) = M _ { f _ 1 } ( \\lambda _ 2 ) = ( \\mu _ { 2 , 0 } , \\mu _ { 2 , 1 } , \\mu _ { 2 , 2 } , \\mu _ { 2 , 3 } , \\mu _ { 2 , 4 } ) = ( 0 , 1 , 0 , 1 , 0 ) . \\end{align*}"}
{"id": "5509.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\ \\d u ^ { n } ( t ) = & \\left [ ( 1 + i \\alpha ) \\Delta u ^ { n } ( t ) - ( \\gamma ( t ) + i \\beta ) P _ { n } | u ^ { n } ( t ) | ^ { 2 } u ^ { n } ( t ) + P _ n f ( t , u ^ { n } ( t ) ) \\right ] \\d t \\\\ & + P _ n g ( t , u ^ { n } ( t ) ) \\d W ^ { n } ( t ) \\\\ \\ u ^ { n } ( s ) = & P _ { n } \\zeta _ { s } . \\end{aligned} \\right . \\end{align*}"}
{"id": "1111.png", "formula": "\\begin{align*} \\mbox { H o r } P _ { \\frac { \\partial } { \\partial m _ i } } \\frac { \\partial } { \\partial m _ i } = \\frac { 1 } { 2 n \\sigma } \\frac { \\partial } { \\partial \\sigma } \\mbox { o r } - \\frac { 1 } { 2 n \\sigma } \\frac { \\partial } { \\partial \\sigma } , \\end{align*}"}
{"id": "5798.png", "formula": "\\begin{align*} ( A , X ) \\ast ( B , Y ) = ( A B + h _ s ( X , Y ) E _ { 1 3 } , X Y ) \\end{align*}"}
{"id": "2499.png", "formula": "\\begin{align*} g ( X , Y ) = \\sum _ { i = 1 } ^ { m } g ( X , e _ i ) g ( Y , e _ i ) . \\end{align*}"}
{"id": "3835.png", "formula": "\\begin{align*} E r r U ^ { 1 ; 2 } _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i \\widetilde { \\Phi } ^ 1 _ { \\mu _ 1 , \\mu _ 2 } ( \\xi , \\eta , \\sigma ; s , X ( s ) , v ) } \\clubsuit K _ { k _ 2 , j _ 2 , n _ 2 } ^ { \\mu , i _ 2 } ( s , \\sigma , V ( s ) ) \\cdot { } _ { } ^ 1 \\clubsuit E l l U ^ 1 ( s , \\xi , \\eta , v , V ( s ) ) \\end{align*}"}
{"id": "5408.png", "formula": "\\begin{align*} \\Delta ^ 1 _ h f ( x ) = f ( x + h ) - f ( x ) , \\Delta ^ 2 _ h f ( x ) = f ( x + 2 h ) - 2 f ( x + h ) + f ( x ) , \\end{align*}"}
{"id": "4505.png", "formula": "\\begin{align*} 0 = & \\nabla _ 1 k _ { 2 3 } - \\nabla _ 2 k _ { 1 3 } , \\\\ 0 = & \\nabla _ \\alpha k _ { \\beta \\beta } - \\nabla _ \\beta k _ { \\alpha \\beta } , \\\\ 0 = & \\nabla _ \\alpha k _ { 3 3 } - \\nabla _ 3 k _ { \\alpha 3 } . \\end{align*}"}
{"id": "6653.png", "formula": "\\begin{align*} G _ X ( \\alpha ) = e ^ { \\lambda _ Z ( e ^ { \\alpha } - 1 - \\alpha ) } \\ , . \\ , \\end{align*}"}
{"id": "1904.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { V } _ { [ n , \\ell ] } } w ( \\gamma ) = \\sum _ { k \\in \\mathbb { Z } } a _ { 0 } ^ { ( \\ell ) } A ^ { ( 1 ) } _ { [ k - 1 , \\ell - 1 ] } W _ { [ n - k - 1 , 0 ] } , 1 \\leq \\ell \\leq p . \\end{align*}"}
{"id": "8448.png", "formula": "\\begin{align*} & \\left \\{ ( x , y ) \\in I _ { n , \\beta } ( w ) \\times J _ { n , \\beta } ( k ) : | T _ { \\beta } ^ n x - f ( x , y ) | < \\varphi ( n ) \\right \\} \\\\ & \\subset \\left \\{ x \\in I _ { n , \\beta } ( w ) : | T _ { \\beta } ^ n x - f ( x , k \\varphi ( n ) / \\beta ^ n ) | < 2 \\varphi ( n ) \\right \\} \\times J _ { n , \\beta } ( k ) \\\\ & : = \\tilde { I } _ { n , \\beta } ( w , k ) \\times J _ { n , \\beta } ( k ) , \\end{align*}"}
{"id": "8045.png", "formula": "\\begin{align*} I _ { K _ X } ( q ) = ( d H ) q ^ { H / z } \\sum _ { \\substack { n \\in \\mathbb Q _ { \\geq 0 } \\\\ \\exists j : n w _ j \\in \\mathbb Z } } q ^ n \\frac { \\prod _ { k = 1 } ^ { n d } ( d H + k z ) ( - 1 ) ^ { n d ^ \\prime } \\prod _ { k = 1 } ^ { n d ^ \\prime } ( d ^ \\prime H + k z ) } { \\prod _ { i = 1 } ^ N \\prod _ { 0 < k \\leq n w _ i , \\langle k \\rangle = \\langle n w _ i \\rangle } ( w _ i H + k z ) } \\textbf { 1 } _ { \\langle - n \\rangle } . \\end{align*}"}
{"id": "7265.png", "formula": "\\begin{align*} \\int _ 0 ^ R \\psi _ 3 ^ { ( R ) } Z _ 1 r ^ { n - 1 } d r = ( \\psi _ 3 ^ { ( R ) } , Z _ 1 ) _ { L _ { } ^ 2 } = ( \\psi _ 3 ^ { ( R ) } , Z _ 1 - \\alpha \\psi _ 2 ^ { ( R ) } ) _ { L _ { } ^ 2 } \\end{align*}"}
{"id": "2438.png", "formula": "\\begin{align*} M _ f S _ \\zeta = M _ { f \\cdot \\tau } H _ { \\tau ^ { - 1 } } S _ \\zeta = q ^ { - \\ell ( \\tau ) } M _ { f \\cdot \\tau } S _ \\zeta = q ^ { - \\ell ( \\tau ) } \\widetilde { N } _ { f \\cdot \\tau } . \\end{align*}"}
{"id": "995.png", "formula": "\\begin{align*} \\varphi ( \\alpha , \\beta ) = \\left ( \\left | \\operatorname { d e t } \\left ( H ^ { * } ( \\overline { \\alpha } ) \\right ) \\right | - 1 \\right ) \\left ( \\left | \\operatorname { d e t } \\left ( H ^ { * } ( \\overline { \\beta } ) \\right ) \\right | - 1 \\right ) . \\end{align*}"}
{"id": "1884.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { L } _ { [ n , j ] } } w ( \\gamma ) = a _ { 0 } ^ { ( j ) } A _ { [ n - 1 , j ] } , 0 \\leq j \\leq p . \\end{align*}"}
{"id": "7425.png", "formula": "\\begin{align*} g ( t ) = e ^ { - k t } , k > 0 , \\end{align*}"}
{"id": "8982.png", "formula": "\\begin{align*} \\| u _ { q , r } + 1 \\| ^ 2 _ { W ^ { 1 , 2 } ( \\tilde K ) } = \\| \\nabla u _ { q , r } \\| ^ 2 _ { L ^ 2 ( \\tilde K ) } + \\| u _ { q , r } + 1 \\| ^ 2 _ { L ^ 2 ( \\tilde K ) } \\leq \\| \\nabla u _ { q , r } \\| ^ 2 _ { L ^ 2 ( \\tilde K ) } + 2 \\| u _ { q , r } \\| ^ 2 _ { L ^ 2 ( \\tilde K ) } + 2 V _ g ( \\tilde K ) , \\end{align*}"}
{"id": "1830.png", "formula": "\\begin{align*} m ( z ) = \\cfrac { 1 } { z - b _ { 0 } - \\cfrac { a _ 0 } { z - b _ { 1 } - \\cfrac { a _ { 1 } } { \\ddots } } } . \\end{align*}"}
{"id": "878.png", "formula": "\\begin{align*} \\C _ { B _ n } ( \\sigma _ 1 ) = \\langle \\sigma _ 1 , \\sigma _ 3 , \\sigma _ 4 , \\ldots , \\sigma _ { n - 1 } , \\sigma _ 2 \\sigma _ 1 \\sigma _ 1 \\sigma _ 2 \\rangle . \\end{align*}"}
{"id": "1149.png", "formula": "\\begin{align*} ( A + B ) ^ \\dagger = A ^ \\dagger + B ^ \\dagger , ( \\lambda A ) ^ \\dagger = \\lambda ^ \\dagger A ^ \\dagger = \\lambda A ^ \\dagger , A ^ { \\dagger \\dagger } = A , ( A B ) ^ \\dagger = B ^ \\dagger A ^ \\dagger , \\end{align*}"}
{"id": "5560.png", "formula": "\\begin{align*} 0 = \\nabla ^ * \\nabla \\omega - 2 W _ + ( \\omega ) + \\frac { s } { 3 } \\omega \\end{align*}"}
{"id": "3000.png", "formula": "\\begin{align*} Q ( f ) = \\sum _ { \\substack { a , b \\geq 0 \\\\ a + b \\geq s } } c _ { a , b } \\phi ^ a ( f - r ) ^ b \\ , \\end{align*}"}
{"id": "5569.png", "formula": "\\begin{align*} K ^ { \\phi } _ t f ( x ) = Q ^ \\phi _ { 1 - e ^ { - t } } d _ { e ^ { - t } } f ( x ) , \\end{align*}"}
{"id": "1812.png", "formula": "\\begin{align*} \\begin{array} { l c c } & \\mbox { m a t r i x n o r m } & \\mbox { L i p s c h i t z c o n s t a n t } \\\\ \\mbox { B r e g m a n C o n d a t - - V \\ ~ u } & \\| A \\| _ { 1 , 2 } = \\sqrt 2 & L _ 1 = \\max _ { i , j } | ( C ^ T C ) _ { i j } | \\\\ \\mbox { B r e g m a n P D 3 O } & \\| A \\| _ 2 \\leq 2 & L _ 2 = \\| C \\| _ 2 ^ 2 . \\end{array} \\end{align*}"}
{"id": "8995.png", "formula": "\\begin{align*} \\lambda _ n = \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n - 1 } \\delta _ { n - k } + \\frac { 1 } { n } \\lambda _ 1 , n \\geq 2 . \\end{align*}"}
{"id": "1328.png", "formula": "\\begin{align*} u _ \\infty ( x ) = \\frac { 1 } { \\rho _ f ( x ) } \\int _ \\R v f _ \\infty \\d v = 0 . \\end{align*}"}
{"id": "4966.png", "formula": "\\begin{align*} A ^ n _ t = n ^ { \\alpha + \\frac 1 2 } \\int ^ t _ 0 [ ( t - \\eta _ n ( s ) ) ^ { \\alpha } - ( t - s ) ^ { \\alpha } ] \\sigma ( X ^ n _ { \\eta _ { n } ( s ) } ) \\ , d W _ s , \\end{align*}"}
{"id": "3135.png", "formula": "\\begin{align*} \\varepsilon : = \\varepsilon _ 7 : = \\min \\big \\{ \\varepsilon _ 6 , ( 2 \\Lambda _ 1 ^ 2 M _ 3 ) ^ { - 1 / ( 2 \\sigma ) } , ( ( c _ 3 - \\beta ) / ( 1 + 1 / \\beta ) ) ^ { 1 / ( 2 \\sigma ) } \\big \\} . \\end{align*}"}
{"id": "7678.png", "formula": "\\begin{align*} \\phi _ { k + 1 } : = \\frac { 1 } { 2 } \\phi _ { k } , \\ \\psi _ { k + 1 } : = \\psi _ { k } , \\ \\gamma _ { k + 1 } : = \\gamma _ { k } , \\ y _ { k + 1 } : = \\overline { y } _ { k + 1 } , \\ Z _ { k + 1 } : = \\overline { Z } _ { k + 1 } , \\end{align*}"}
{"id": "3402.png", "formula": "\\begin{align*} \\sum _ { G \\in I ^ { C _ \\ell } } \\mu ( U _ { C _ \\ell , G } ) = \\lim _ { g \\to \\infty } \\lim _ { L \\to \\infty } \\sum _ { G \\in I ^ { C _ \\ell } } \\mu _ { g , L } ( U _ { C _ \\ell , G } ) = 1 . \\end{align*}"}
{"id": "8966.png", "formula": "\\begin{align*} C _ 1 \\sum _ { i = 1 } ^ p \\int _ { f _ i ^ { - 1 } ( E _ i ) } f _ i ^ \\ast \\rho ^ { - \\delta - \\frac { n } { p } + j } | \\partial _ j f _ i ^ \\ast u | d \\mathbf { x } \\leq \\int _ M \\rho ^ { - \\delta - \\frac { n } { p } + j } | \\hat \\nabla ^ j u | d V _ { \\hat g } \\leq C _ 2 \\sum _ { i = 1 } ^ p \\int _ { f _ i ^ { - 1 } ( E _ i ) } f _ i ^ \\ast \\rho ^ { - \\delta - \\frac { n } { p } + j } | \\partial _ j f _ i ^ \\ast u | d \\mathbf { x } \\end{align*}"}
{"id": "2948.png", "formula": "\\begin{align*} & N ( \\psi ) = V ( x , t ) + c | \\psi | ^ m , \\ , \\ , c > 0 \\\\ & | x | ^ { 3 + 0 } | V ( x , t ) | \\leq c < \\infty , \\ , \\ , | x | > 1 \\\\ & | x | ^ { 3 + 0 } | \\nabla V ( x , t ) | \\leq c < \\infty , \\ , \\ , | x | > 1 \\\\ & | \\nabla V ( x , t ) | + | V ( x , t ) | \\lesssim 1 . \\end{align*}"}
{"id": "2524.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( L _ Z g ) ( X , Y ) + \\frac { 1 } { \\lambda ^ 2 } \\{ R i c ^ N ( \\tilde { X } , \\tilde { Y } ) + f _ 3 h ( \\tilde { X } , \\tilde { Y } ) \\} = 0 , \\end{align*}"}
{"id": "377.png", "formula": "\\begin{align*} \\P _ \\pi [ T _ { A } > t ] = \\langle P _ { t , B } 1 _ { B } , 1 _ { B } \\rangle _ { \\pi } = \\sum _ { i = 1 } ^ m c _ { i } ^ { 2 } e ^ { - t ( 1 - \\gamma _ i ) } = \\frac { 1 } { \\| \\alpha / \\pi \\| _ 2 ^ 2 } e ^ { - t / \\mathbb { E } _ { \\alpha } [ T _ A ] } + \\sum _ { i = 2 } ^ m c _ { i } ^ { 2 } e ^ { - t ( 1 - \\gamma _ i ) } \\end{align*}"}
{"id": "8290.png", "formula": "\\begin{align*} k ' = \\frac { 1 } { 2 \\hslash \\sqrt { \\beta / 3 } } \\sqrt { \\frac { \\sqrt { \\frac { 1 6 } { 3 } m \\beta | E | } + 1 } { 2 } } . \\end{align*}"}
{"id": "4470.png", "formula": "\\begin{align*} K _ 0 ( x ) & = - \\left [ \\ln \\left ( \\frac { | x | } 2 \\right ) + \\gamma \\right ] I _ 0 ( x ) + \\sum \\limits _ { k = 1 } ^ \\infty b _ k x ^ { 2 k } , \\end{align*}"}
{"id": "7158.png", "formula": "\\begin{align*} g _ { 2 1 } ( g _ { 1 1 } + g _ { 2 1 } ) = 0 g _ { 2 2 } ( g _ { 1 2 } + g _ { 2 2 } ) = 0 . \\end{align*}"}
{"id": "210.png", "formula": "\\begin{align*} \\mathcal { K } ^ { n } _ { 0 , j } = \\left \\{ \\pi _ { \\bar { X } , \\bar { Y } _ { [ 0 : 2 ] } , \\bar { Z } _ j } \\in \\mathcal { P } _ { n } \\left ( \\mathcal { X } \\times \\mathcal { Y } _ { [ 0 : 2 ] } \\times \\mathcal { Z } _ j \\right ) \\ : \\ \\pi _ { \\bar { X } , \\bar { Y } _ { [ 0 : 2 ] } } \\stackrel { \\delta ^ { \\prime } _ { n } } { \\approx } p _ { X , Y _ { [ 0 : 2 ] } } \\wedge \\pi _ { \\bar { Y } _ { 0 } , \\bar { Y } _ { j } , \\bar { Z } _ j } \\stackrel { \\delta ^ { \\prime } _ { n } } { \\approx } p _ { Y _ { 0 } , Y _ { j } , Z _ j } \\right \\} , \\end{align*}"}
{"id": "7970.png", "formula": "\\begin{align*} I _ { F _ 1 } ( y _ 1 , y _ 2 , z ) = z e ^ { ( P \\log y _ 1 + H \\log y _ 2 ) / z } \\sum _ { d _ 1 , d _ 2 \\geq 0 } y _ 1 ^ { d _ 1 } y _ 2 ^ { d _ 2 } \\frac { \\prod _ { a = - \\infty } ^ 0 ( H - P + a z ) } { \\prod _ { a = 1 } ^ { d _ 2 } ( H + a z ) \\prod _ { a = - \\infty } ^ { d _ 2 - d _ 1 } ( H - P + a z ) \\prod _ { a = 1 } ^ { d _ 1 } ( P + a z ) ^ 2 } . \\end{align*}"}
{"id": "573.png", "formula": "\\begin{align*} M _ 1 = f _ k ( i ) f _ k ( i + 1 ) + O _ \\alpha \\Big ( \\frac { \\tau _ 3 ( i ) \\tau _ 3 ( i + 1 ) } { \\sqrt { i } } \\Big ) . \\end{align*}"}
{"id": "7166.png", "formula": "\\begin{align*} \\mathcal { L } ( V ) = \\mathbb { C } [ t ^ { \\frac { 1 } { T } } , t ^ { - \\frac { 1 } { T } } ] \\otimes V . \\end{align*}"}
{"id": "8111.png", "formula": "\\begin{align*} \\P ( Y ( t , x ) > R ) & \\leq e ^ { - t \\lvert D \\rvert } R ^ { - 1 - \\frac 2 d } \\sum _ { N = 1 } ^ \\infty \\frac { ( C \\log R ) ^ N } { N ! N ^ N } \\sum _ { L = N } ^ \\infty \\frac { C ^ { L - N } } { ( L - N ) ! } \\leq C R ^ { - 1 - \\frac 2 d } e ^ { C ( \\log R ) ^ { 1 / 2 } } , \\end{align*}"}
{"id": "5912.png", "formula": "\\begin{align*} T ( r , f _ j ) & = \\frac 1 { 2 \\pi } \\int _ { | z | = r } \\log ^ + | f _ j | \\ , d \\theta \\leq \\frac 1 { 4 \\pi } \\int _ { | z | = r } \\log ( 1 + | f _ j | ^ 2 ) \\ , d \\theta \\\\ & = \\frac 1 { 2 } \\log \\left ( 1 + \\frac 1 { 2 \\pi } \\int _ { | z | = r } | f _ j | ^ 2 \\ , d \\theta \\right ) \\leq \\frac 1 2 \\log ^ + T _ 2 ( r ) + O ( 1 ) \\\\ & = O ( \\log ^ + T _ { \\exp _ A f } ( r ) ) + O ( 1 ) . \\end{align*}"}
{"id": "4397.png", "formula": "\\begin{align*} & \\lim _ { \\epsilon \\to 0 } \\int _ { D _ j } ( v '' _ { \\epsilon } ( \\Psi ) ) | f | ^ 2 e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) - \\Psi } \\\\ = & \\int _ { D _ j } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ { 0 } - B < \\Psi < - t _ { 0 } \\} } | f | ^ 2 e ^ { - u ( - v _ { t _ 0 , B } ( \\Psi ) ) - \\Psi } \\\\ \\leq & \\left ( \\sup _ { D _ j } e ^ { - u ( - v _ { t _ 0 , B } ( \\Psi ) ) } \\right ) \\int _ { D _ j } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ { 0 } - B < \\Psi < - t _ { 0 } \\} } | f | ^ 2 e ^ { - \\Psi } \\\\ < & + \\infty . \\end{align*}"}
{"id": "3128.png", "formula": "\\begin{align*} \\sum _ { k = \\ell } ^ { \\ell + m } \\eta _ k ^ 2 & \\le \\Lambda _ c \\eta _ \\ell ^ 2 \\quad \\quad \\eta _ { \\ell + m } ^ 2 \\le \\frac { q _ c ^ m } { 1 - q _ c } \\eta _ { \\ell } ^ 2 \\quad \\ell , m \\in \\mathbb { N } _ 0 . \\end{align*}"}
{"id": "1590.png", "formula": "\\begin{align*} E = b ^ 2 \\sum \\limits _ { k = 1 } ^ { 3 } ( - 1 ) ^ { \\gamma + \\tau } z ^ { k } _ { \\tilde { \\gamma } } z ^ { k } _ { \\tilde { \\tau } } \\left [ \\cos ^ 2 \\theta z ^ { 1 } _ { \\gamma } z ^ { 1 } _ { \\tau } + \\sin ^ 2 \\theta z ^ { 2 } _ { \\gamma } z ^ { 2 } _ { \\tau } + 2 \\sin \\theta \\cos \\theta z ^ { 1 } _ { \\gamma } z ^ { 2 } _ { \\tau } \\right ] . \\end{align*}"}
{"id": "6705.png", "formula": "\\begin{align*} \\phi ( \\pi ) : = \\phi ( \\sigma _ \\pi , \\lambda _ \\pi ) \\vert _ { W _ \\pi } \\in ( W _ \\pi ) . \\end{align*}"}
{"id": "8992.png", "formula": "\\begin{align*} B _ { n } ( x _ 1 , x _ 2 , \\ldots , x _ n ) = \\sum _ { \\pi ( n ) } \\frac { n ! } { \\prod _ { r = 1 } ^ n k _ r ! } \\prod _ { r = 1 } ^ { n } \\left ( \\frac { x _ r } { r ! } \\right ) ^ { k _ r } , \\end{align*}"}
{"id": "5527.png", "formula": "\\begin{align*} f ' ( \\gamma ( 0 ) ) = f ' ( 0 ) \\frac { \\rho ( \\gamma ) ' ( 0 ) } { \\gamma ' ( 0 ) } . \\end{align*}"}
{"id": "4291.png", "formula": "\\begin{align*} \\langle \\otimes _ { l = 1 } ^ { m } \\mathfrak { c } \\rangle _ { t } = \\exp \\biggl ( - \\frac { t } { 2 } \\sum \\limits _ { j = 1 } ^ N \\sum \\limits _ { i , p = 1 } ^ m \\otimes _ { l = 1 } ^ m ( E K _ { j } ) ^ { \\delta _ { i l } + \\delta _ { p l } } \\biggr ) \\langle \\otimes _ { l = 1 } ^ { m } \\mathfrak { c } \\rangle _ { 0 } , \\end{align*}"}
{"id": "452.png", "formula": "\\begin{align*} \\frac { 1 } { x _ { 0 } } = \\Phi ( r _ { 0 } e ^ { i h _ { \\nu } ( r _ { 0 } ) } ) \\end{align*}"}
{"id": "4510.png", "formula": "\\begin{align*} R _ { 1 2 1 2 } = 2 K + h _ { 1 1 } h _ { 2 2 } - h _ { 1 2 } ^ 2 . \\end{align*}"}
{"id": "3383.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { s _ 0 L _ t } { X _ t } = 1 , \\end{align*}"}
{"id": "1035.png", "formula": "\\begin{align*} \\int _ { n - 1 } ^ { n } \\| \\tilde { B } T ( s , A ) \\| _ { q } \\ , \\mathrm { d } s & = \\int _ { 0 } ^ { 1 } \\| \\tilde { B } T ( s , A ) T ( n - 1 , A ) \\| _ { q } \\ , \\mathrm { d } s \\\\ & \\leq \\| T ( n - 1 , A ) \\| _ { \\infty } \\int _ { 0 } ^ { 1 } \\| \\tilde { B } T ( s , A ) \\| _ { q } \\ , \\mathrm { d } s , \\end{align*}"}
{"id": "3983.png", "formula": "\\begin{align*} I _ { \\bar { b } } ^ t + b _ k { a } _ \\delta , \\qquad { \\rm w i t h ~ c a r d i n a l i t y } ~ ~ | I _ { \\bar { b } } ^ t + b _ k { a } _ \\delta | = | I _ { \\bar { b } } ^ t | = q ^ { d _ { \\bar { b } } t } ; \\end{align*}"}
{"id": "8493.png", "formula": "\\begin{align*} \\frac { \\log ( 3 M ^ 2 / \\delta ) } { 2 \\varepsilon ^ 2 } = \\frac { \\log ( M ) } { \\varepsilon ^ 2 } + \\frac { \\log \\left ( 3 / \\delta \\right ) } { 2 \\varepsilon ^ 2 } . \\end{align*}"}
{"id": "1982.png", "formula": "\\begin{align*} A _ + ( 0 ) = : a ^ 2 \\\\ A _ - ( 0 ) = : b ^ 2 . \\end{align*}"}
{"id": "6849.png", "formula": "\\begin{align*} p ^ * ( g ) : = \\frac { 1 } { \\sqrt { 2 \\pi c ^ * } } \\exp \\left ( - \\frac { ( g - b ^ * ) ^ 2 } { 2 c ^ * } \\right ) , \\end{align*}"}
{"id": "612.png", "formula": "\\begin{align*} \\int \\frac { d x } { a + b \\sin ^ 2 ( x ) } = \\frac { \\mathrm { s i g n } ( a ) } { \\sqrt { a ( a + b ) } } \\arctan \\Big ( \\sqrt { \\frac { a + b } { a } } \\tan ( x ) \\Big ) . \\end{align*}"}
{"id": "2742.png", "formula": "\\begin{align*} & \\limsup _ { R \\rightarrow \\infty } \\int _ { B _ { R } ^ { c } ( 0 ) } \\frac { | u ( x ) - u ( y ) | ^ { p - 1 } } { | x - y | ^ { n + \\sigma p } } d y \\\\ \\leq & \\limsup _ { R \\rightarrow \\infty } \\left ( \\frac { R } { R - | x | } \\right ) ^ { n + \\sigma p } \\int _ { B _ { R } ^ { c } ( 0 ) } \\frac { C ( p ) ( u ^ { p - 1 } ( x ) + u ^ { p - 1 } ( y ) ) } { | y | ^ { n + \\sigma p } } d y = 0 , \\end{align*}"}
{"id": "2216.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial } { \\partial t } \\omega ( t ) = - \\mathrm { R i c } _ { \\omega ( t ) } ^ { T } - \\omega ( t ) , \\omega ( 0 ) = \\omega _ { 0 } . \\end{array} \\end{align*}"}
{"id": "8552.png", "formula": "\\begin{align*} \\xi = \\int ( \\delta _ x - \\delta _ y ) \\ a ( d x , d y ) \\ . \\end{align*}"}
{"id": "7484.png", "formula": "\\begin{align*} a _ k \\psi ^ { ( n + 1 ) } ( k _ 1 , \\ldots , k _ n ) = \\sqrt { n + 1 } \\psi ^ { ( n + 1 ) } ( k , k _ 1 , \\ldots , k _ n ) \\end{align*}"}
{"id": "1559.png", "formula": "\\begin{align*} F ( x , y ) = \\left ( \\varphi ^ * \\tilde { F } \\right ) ( x , y ) = \\tilde { F } \\left ( \\varphi ( x ) , \\varphi _ * ( y ) \\right ) , \\forall ( x , y ) \\in T M . \\end{align*}"}
{"id": "2687.png", "formula": "\\begin{align*} u _ { r , \\alpha } : = \\frac { u _ { 2 \\alpha + 1 } + u _ { 2 \\alpha + 2 } } { \\sqrt { 2 } } , u _ { \\ell , \\alpha } : = \\frac { u _ { 2 \\alpha + 1 } - u _ { 2 \\alpha + 2 } } { \\sqrt { 2 } } \\end{align*}"}
{"id": "8229.png", "formula": "\\begin{align*} k _ { \\pm } ^ { 2 } = \\frac { 1 + \\sqrt { 1 \\pm 1 6 i m \\beta \\lambda / 3 } } { 4 \\beta \\hslash ^ { 2 } / 3 } \\end{align*}"}
{"id": "6682.png", "formula": "\\begin{align*} \\begin{aligned} & \\| h ' _ 1 - h ' _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } + \\| h _ 1 - h _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } \\leq ( T _ 0 + 1 ) \\| h ' _ 1 - h ' _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } + | h _ { 0 1 } - h _ { 0 2 } | , \\\\ & \\| g ' _ 1 - g ' _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } + \\| g _ 1 - g _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } \\leq ( T _ 0 + 1 ) \\| g ' _ 1 - g ' _ 2 \\| _ { C ^ 0 ( [ 0 , T _ 0 ] ) } + | g _ { 0 1 } - g _ { 0 2 } | , \\end{aligned} \\end{align*}"}
{"id": "1537.png", "formula": "\\begin{align*} n e ^ { - \\alpha R } \\kappa \\sqrt { \\ss } \\int _ { \\widehat { r } } ^ R e ^ { - \\beta r _ 0 } \\sinh ( \\alpha r _ 0 ) d r _ 0 = \\begin{cases} O \\Big ( n \\Big ( \\frac { \\kappa \\sqrt { \\ss } } { e ^ { \\beta R } } \\Big ) ^ { 1 \\wedge \\frac { \\alpha } { \\beta } } \\Big ) , & \\\\ [ 8 p t ] \\Theta \\Big ( n \\frac { \\kappa \\sqrt { \\ss } } { e ^ { \\beta R } } \\log \\big ( \\frac { e ^ { \\beta R } } { \\kappa \\sqrt { \\ss } } \\big ) \\Big ) , & \\end{cases} \\end{align*}"}
{"id": "1797.png", "formula": "\\begin{align*} \\eta ( x , z ) = f ( x ) + h ( x ) + \\gamma \\| A x - b \\| + b ^ T z + ( f + h ) ^ * ( - A ^ T z ) \\end{align*}"}
{"id": "7732.png", "formula": "\\begin{align*} \\Phi ( t _ 2 , t _ 1 ) = \\Phi ( t _ 2 , \\tau ) \\Phi ( \\tau , t _ 1 ) \\quad t _ 2 \\geq \\tau \\geq t _ 1 \\geq 0 . \\end{align*}"}
{"id": "1453.png", "formula": "\\begin{align*} \\Delta ( z ) = ( - 1 ) ^ { r m } \\times \\ z ^ { r m ( n + 1 ) } \\ \\ P _ { r m } ( z ) \\times \\ 1 / z ^ { r m ( n + 1 ) } \\ \\ \\Delta _ { 1 , r m + 1 } ( z ) \\enspace . \\end{align*}"}
{"id": "6041.png", "formula": "\\begin{align*} \\Lambda = \\left ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ m \\right ) \\end{align*}"}
{"id": "7043.png", "formula": "\\begin{align*} t _ 5 ( 5 ) = & | \\{ ( 5 ) , ( 4 , 1 ) , ( 3 , 1 , 1 ) , ( 2 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 1 , 1 ) \\} | = 5 , \\\\ t _ 6 ( 5 ) = & | \\{ ( 4 , 2 ) , ( 3 , 2 , 1 ) , ( 2 , 2 , 1 , 1 ) \\} | = 3 , \\\\ t _ 7 ( 5 ) = & | \\{ ( 4 , 3 ) , ( 3 , 3 , 1 ) , ( 3 , 2 , 2 ) , ( 2 , 2 , 2 , 1 ) \\} | = 4 , \\\\ t _ 8 ( 5 ) = & | \\{ ( 4 , 4 ) , ( 3 , 3 , 2 ) , ( 2 , 2 , 2 , 2 ) \\} | = 3 , \\\\ t _ 9 ( 5 ) = & | \\{ ( 3 , 3 , 3 ) \\} | = 1 \\end{align*}"}
{"id": "7155.png", "formula": "\\begin{align*} ( g _ { 2 1 } y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 ) \\{ ( g _ { 1 1 } + g _ { 2 1 } ) y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 \\} ^ { a - 1 } \\\\ = g _ { 2 3 } ^ { a } y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } . \\end{align*}"}
{"id": "1595.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial z ^ i _ { \\epsilon } } v ^ i = 2 b ^ 2 f ( x ^ 1 ) \\cos \\left ( x ^ 2 - \\theta \\right ) \\Big [ - \\delta _ { \\epsilon 1 } f ( x ^ 1 ) f ' ( x ^ 1 ) \\cos \\left ( x ^ 2 - \\theta \\right ) + \\delta _ { \\epsilon 2 } \\left \\lbrace 1 + f '^ 2 \\left ( x ^ 1 \\right ) \\right \\rbrace \\sin \\left ( x ^ 2 - \\theta \\right ) \\Big ] . \\end{align*}"}
{"id": "4054.png", "formula": "\\begin{align*} & A ( X \\otimes X ) = 0 , & & ( X ^ * \\otimes X ^ * ) A = 0 , & & ( \\Psi \\otimes \\Psi ) ( 1 - A ) = 0 , & & ( 1 - A ) ( \\Psi ^ * \\otimes \\Psi ^ * ) = 0 , \\end{align*}"}
{"id": "4870.png", "formula": "\\begin{align*} \\begin{aligned} & \\omega N _ { 1 } < p _ { 1 } \\leq N _ { 1 } , \\ p _ { 2 } , p _ { 3 } \\sim U _ { 1 } , \\ p _ { 4 } , p _ { 5 } \\sim V _ { 1 } , \\\\ & \\omega N _ { 2 } < p _ { 6 } \\leq N _ { 2 } , \\ p _ { 7 } , p _ { 8 } \\sim U _ { 2 } , \\ p _ { 9 } , p _ { 1 0 } \\sim V _ { 2 } , \\ v _ { j } \\leq L \\end{aligned} \\end{align*}"}
{"id": "7206.png", "formula": "\\begin{align*} \\int _ { \\R } g ( x ) d x = 0 . \\end{align*}"}
{"id": "2851.png", "formula": "\\begin{align*} \\begin{aligned} h _ * = \\arg \\max _ { 0 < h { } \\leq { } \\bar { h } ( \\kappa ) } p ( h , \\kappa ) . \\end{aligned} \\end{align*}"}
{"id": "6383.png", "formula": "\\begin{align*} \\beta ( \\xi _ 1 , \\xi _ 2 ) : = \\lim _ { n \\to \\infty } \\langle \\ , [ U ( \\gamma _ n ) ] ( \\xi _ 1 ) , \\xi _ 2 \\ , \\rangle \\end{align*}"}
{"id": "6680.png", "formula": "\\begin{align*} \\begin{aligned} & \\| z _ 1 - z _ 2 \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } + \\| w _ 1 - w _ 2 \\| _ { C ^ { \\frac { 1 + \\beta _ 0 } { 2 } , 1 + \\beta _ 0 } ( D ) } \\\\ & \\leq C _ 3 \\big ( \\| h _ 1 - h _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + \\| g _ 1 - g _ 2 \\| _ { C ^ 1 ( [ 0 , T _ 0 ] ) } + \\| z _ { 0 1 } - z _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } + \\| w _ { 0 1 } - w _ { 0 2 } \\| _ { C ^ { 2 } ( [ 0 , 1 ] ) } \\big ) \\\\ \\end{aligned} \\end{align*}"}
{"id": "5636.png", "formula": "\\begin{align*} \\mathcal { J } _ { 2 , 1 } = - 2 \\mathcal { I } _ { 2 , 1 } + 2 \\iint _ { Q _ T } [ \\nabla m \\cdot \\nabla p ] | \\nabla p | ^ 2 - p \\Delta p [ \\nabla \\cdot ( m \\nabla p ) ] . \\end{align*}"}
{"id": "5332.png", "formula": "\\begin{align*} A \\phi ( x ) = \\nabla \\cdot ( Q \\nabla \\phi ) ( x ) + a \\cdot \\nabla \\phi ( x ) + \\int _ { \\R ^ d } \\left ( \\phi ( x + y ) - \\phi ( x ) - \\nabla \\phi ( x ) y 1 _ { \\{ | y | < 1 \\} } \\right ) \\nu ( d y ) , \\end{align*}"}
{"id": "5210.png", "formula": "\\begin{align*} \\left ( X = \\C ^ n , W = \\sum _ { i = 1 } ^ n x _ i ^ { r _ i } \\right ) . \\end{align*}"}
{"id": "1295.png", "formula": "\\begin{align*} P ( T , x ) & = \\prod _ { v \\in V ( T ) } \\mathcal { G } ( v , x ) \\\\ & = \\mathcal { G } ( r ( T ) , x ) \\prod _ { j = 1 } ^ { d } \\left ( \\prod _ { v \\in V ( T _ j ) } \\mathcal { G } ( v , x ) \\right ) \\\\ & = \\mathcal { G } ( r ( T ) , x ) \\prod _ { j = 1 } ^ { d } P ( T _ j , x ) . \\end{align*}"}
{"id": "2460.png", "formula": "\\begin{align*} \\gg ^ { \\mu } = \\left \\{ g \\in \\gg \\mid \\left [ h , g \\right ] = \\mu ( h ) g \\right \\} . \\end{align*}"}
{"id": "5743.png", "formula": "\\begin{align*} \\pi _ b \\cdot \\pi _ { [ a , b ] } = \\pi _ { [ a - 1 , b ] } + ( b - a + 1 ) \\pi _ { [ a , b + 1 ] } . \\end{align*}"}
{"id": "8720.png", "formula": "\\begin{align*} g _ { \\dot { x } ^ { 1 } \\dot { x } ^ { 1 } } = \\frac { \\cos ^ { 2 } t } { a } , g _ { \\dot { x } ^ { 2 } \\dot { x } ^ { 2 } } = \\frac { \\sin ^ { 2 } t } { a } , g _ { x ^ { 1 } \\dot { x } ^ { 2 } } = g _ { \\dot { x } ^ { 1 } x ^ { 2 } } = 0 . \\end{align*}"}
{"id": "1199.png", "formula": "\\begin{align*} ( \\mu * \\gamma _ { \\sigma } ) ( B _ y ) = \\int _ { B _ y } f _ { \\mu } ( z ) d z \\ge \\inf _ { z \\in B _ y } f _ \\mu ( z ) \\times \\int _ { B _ y } d z \\ge c _ 3 e ^ { - c _ 2 ^ 2 } f _ { \\mu } ( 0 ) e ^ { - 2 ( 1 + c _ 1 ) ( | x | ^ 2 + 9 ) } , \\end{align*}"}
{"id": "7686.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ i ^ k : = \\lambda _ i \\big ( \\sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) \\big ) . \\end{align*}"}
{"id": "51.png", "formula": "\\begin{align*} \\dfrac { ( f ^ { \\tilde { \\phi } ^ { - n } } \\circ \\rho _ { \\eta } ( X ) ) ' } { f ^ { \\tilde { \\phi } ^ { - n } } \\circ \\rho _ { \\eta } ( X ) } = \\eta \\dfrac { ( f ^ { \\tilde { \\phi } ^ { - n } } ) ' } { f ^ { \\tilde { \\phi } ^ { - n } } } \\circ \\rho _ { \\eta } ( X ) . \\end{align*}"}
{"id": "5711.png", "formula": "\\begin{align*} ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \\ldots , y _ i ) = 0 ( 1 \\le k \\le i ) \\end{align*}"}
{"id": "2636.png", "formula": "\\begin{align*} | I | \\leqslant \\sum _ { i = 1 } ^ m r _ i ( Y _ i ) \\end{align*}"}
{"id": "3777.png", "formula": "\\begin{align*} G _ { k , j ; n , l , r } ^ { \\star ; m , i ; p , q } ( t , x , \\zeta ) : = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ 0 ^ \\pi \\int _ { 0 } ^ { 2 \\pi } \\big [ i \\mu \\big ( \\omega ^ { m ; E } _ { j , l , r } ( t - s , v , \\omega ) + \\hat { \\zeta } \\times \\omega ^ { m ; B } _ { j , l , r } ( t - s , v , \\omega ) \\big ) K _ { k ; n } ^ { } ( \\mathfrak { m } ) ( y , \\zeta ) + \\big ( c ^ { a ; m , E } _ { j , l , r } ( t - s , v , \\omega ) \\end{align*}"}
{"id": "3288.png", "formula": "\\begin{align*} \\int _ { D } e ^ { - i \\varphi } q ( x ) u _ { 1 } u _ 2 d x = \\int _ { D } q ( x ) \\ ; e ^ { i x \\cdot \\xi } \\ , d x + \\mathcal { R } ' _ { 1 } ( \\xi , s ) + \\mathcal { R } ' _ { 2 } ( \\xi , s ) , \\end{align*}"}
{"id": "954.png", "formula": "\\begin{align*} S _ { 1 } = \\left \\{ \\sum _ { i = 1 } ^ { n } a _ { i } \\alpha _ { i } \\mid 0 \\leq a _ { i } < p q , \\ a _ { i } \\in \\mathbb { Z } , \\ 1 \\leq i \\leq n \\right \\} . \\end{align*}"}
{"id": "629.png", "formula": "\\begin{align*} & N : = M + 1 , \\ \\textup { f o r t h e e x p a n s i o n o f } I ^ { \\frac { n } { 2 } } _ { y } \\circ E ^ { \\frac { n } { 2 } } _ { x , M } f ^ { - \\alpha } , \\\\ & N ' : = 2 ( M + 1 ) , \\ \\textup { f o r t h e e x p a n s i o n o f } E ^ { \\frac { n } { 2 } } _ { y , M } \\circ E ^ { \\frac { n } { 2 } } _ { x , M } f ^ { - \\alpha } . \\end{align*}"}
{"id": "4764.png", "formula": "\\begin{align*} \\left | \\mathcal K _ { \\kappa , 2 } \\right | = \\frac 1 2 \\left ( \\sum _ { i = 0 } ^ { \\lfloor \\kappa / 2 \\rfloor } ( \\kappa - 2 i + 1 ) + \\binom { \\kappa + 3 } { 3 } \\right ) . \\end{align*}"}
{"id": "4099.png", "formula": "\\begin{align*} { B } w ^ \\prime ( t ) - A ^ \\ast w ( t ) - Q ^ * B w ( t ) = g ( t ) , \\ t \\in \\ , [ 0 , T ] , w ( T ) = 0 . \\end{align*}"}
{"id": "450.png", "formula": "\\begin{align*} \\Omega _ { \\rho , \\theta } = \\left \\{ r e ^ { i t } : \\rho < r < \\frac { 1 } { \\rho } , t \\in ( \\theta , 2 \\pi - \\theta ) \\right \\} , \\end{align*}"}
{"id": "7009.png", "formula": "\\begin{align*} \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } S ( h ) = n ! \\cdot \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } V o l ( h ) = 0 . \\end{align*}"}
{"id": "216.png", "formula": "\\begin{align*} L _ { B } ( x ^ { n } , b _ { \\mathcal { T } } ) = \\frac { 1 } { p ^ { U } _ { \\mathcal { T } } } \\sum _ { \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } \\in \\mathcal { P } _ { n } ( \\mathcal { Y } _ { \\mathcal { T } } | \\pi _ { \\bar { X } } ) } N _ { \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } } ( x ^ n , b _ { \\mathcal { T } } ) l _ { \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } } ( x ^ n ) , \\end{align*}"}
{"id": "1880.png", "formula": "\\begin{align*} A _ { [ n , j ] } = \\sum _ { k = i } ^ { n - j + i } \\sum _ { \\gamma \\in \\widetilde { \\mathcal { D } } _ { [ n , j , k ] } } w ( \\gamma ) = \\sum _ { k = i } ^ { n - j + i } A _ { [ k , i ] } A _ { [ n - k - 1 , j - i - 1 ] } ^ { ( i + 1 ) } , n \\geq j , \\end{align*}"}
{"id": "8232.png", "formula": "\\begin{align*} | z | ^ { 2 } = a ^ { 2 } + b ^ { 2 } = z ^ { * } z = \\sqrt { 1 ^ { 2 } + ( 1 6 m \\beta \\lambda / 3 ) ^ { 2 } } \\end{align*}"}
{"id": "3905.png", "formula": "\\begin{align*} q _ { i } ^ \\star = \\left \\{ \\begin{array} { l l } 1 & i \\leq \\lfloor t I \\rfloor \\\\ t I - \\lfloor t I \\rfloor & i = \\lfloor t I \\rfloor + 1 \\\\ 0 & i > \\lfloor t I \\rfloor + 1 . \\end{array} \\right . \\end{align*}"}
{"id": "7012.png", "formula": "\\begin{align*} \\partial _ { i _ 1 } ^ { k _ 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } S ( h ) = - \\sum _ { l \\ne i _ j } \\langle \\chi , \\Lambda ( \\rho _ { l } ) \\rangle \\partial _ l \\partial _ { i _ 1 } ^ { k _ 1 - 1 } \\ldots \\partial _ { i _ s } ^ { k _ s } S ( h ) . \\end{align*}"}
{"id": "8739.png", "formula": "\\begin{align*} f ( k + 1 ) - f ( k ) = \\frac { 1 } { 2 ^ { \\tau ^ * } } \\left ( 2 ^ { 2 ( \\rho - \\tau ^ * ) } - 1 \\right ) + \\frac { \\epsilon ^ 2 \\cdot 2 ^ { 2 ( \\rho - 1 ) } } { 8 } \\left ( \\frac { 2 k + 1 } { k ^ 2 ( k + 1 ) ^ 2 } \\right ) \\ge 0 . \\end{align*}"}
{"id": "5629.png", "formula": "\\begin{align*} w _ t = \\nabla \\lambda _ t \\cdot \\nabla p + ( k - 1 ) \\left [ w ( w + F ) + ( \\nabla w + \\nabla F ) \\cdot ( p \\nabla \\lambda + 2 \\nabla p ) + p ( \\Delta w + \\Delta F ) \\right ] + \\end{align*}"}
{"id": "7247.png", "formula": "\\begin{align*} L _ { \\Gamma } : \\begin{cases} \\ell ^ { 2 } ( V ) \\rightarrow \\ell ^ 2 ( V ) \\\\ L _ \\Gamma \\varphi ( v ) = \\varphi ( v ) - \\frac { 1 } { | \\mathcal { E } ( \\{ v \\} ) | } \\sum _ { \\{ v , w \\} \\in E } \\varphi ( w ) \\end{cases} \\end{align*}"}
{"id": "6477.png", "formula": "\\begin{align*} u _ x = v _ x = 0 \\ \\ \\ \\ ( b _ 1 , b _ 2 ) \\ \\ \\ \\ y _ x = z _ x = 0 \\ \\ \\ \\ ( d _ 1 , d _ 2 ) . \\end{align*}"}
{"id": "6023.png", "formula": "\\begin{align*} f ( x ) = x ^ 3 + a x ^ 2 . \\end{align*}"}
{"id": "3712.png", "formula": "\\begin{align*} t ( v \\Uparrow u ) = \\cup \\{ t ( u ' ) \\mid u ' \\in { \\downarrow _ 1 } ( v ) : q ( u ) \\prec q ( u ' ) \\} \\cup \\{ i \\} . \\end{align*}"}
{"id": "8103.png", "formula": "\\begin{align*} { \\mathfrak X } ( G / H \\times { \\mathbb R } ^ { k } ) ^ { G } = { \\mathfrak X } ( G / H \\times { \\mathbb R } ^ { k } ) ^ { \\mathfrak g } , \\end{align*}"}
{"id": "67.png", "formula": "\\begin{align*} | \\langle A \\cdot u , v \\rangle | & = | \\langle u , A ^ T \\cdot v \\rangle | \\leq \\| u \\| _ { - r } \\| A ^ T \\cdot v \\| _ r \\leq c \\| u \\| _ { - r } \\| A ^ T \\| _ s \\| v \\| _ r = c \\| u \\| _ { - r } \\| A \\| _ s , \\end{align*}"}
{"id": "2013.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n m a r t n i g a l e p r o b l e m } M ^ \\varphi _ t : = \\varphi ( \\mathsf { X } _ t ) - \\varphi ( \\mathsf { X } _ 0 ) - \\int _ 0 ^ t L _ { f _ s } \\varphi ( \\mathsf { X } _ s ) \\dd s , \\end{align*}"}
{"id": "3382.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { B ( X _ t ) } { t } = \\lim _ { t \\to \\infty } \\frac { B ( \\| Z _ t \\| _ { d + 1 } ) } { t } = \\frac { s _ 0 \\sigma ^ 2 } { 2 c _ 0 } , , \\end{align*}"}
{"id": "8755.png", "formula": "\\begin{align*} F \\cdot F ^ { ( n ) } = [ n + 1 ] F ^ { ( n + 1 ) } , \\end{align*}"}
{"id": "7270.png", "formula": "\\begin{align*} u _ t = \\Delta _ x u - | u | ^ { q - 1 } u . \\end{align*}"}
{"id": "3900.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to + \\infty } \\Phi _ \\rho ( \\mu , 1 ) = \\| u \\| _ { \\infty , B _ 1 ( 0 ) } = \\sup _ { B _ 1 ( 0 ) } u . \\end{align*}"}
{"id": "7364.png", "formula": "\\begin{align*} { \\lim _ { | y | \\to \\infty } \\left ( e ^ { | y | } | y | ^ { \\left ( \\frac { N - 1 } { 2 } \\right ) } { U } _ 0 ( | y | ) \\right ) = \\alpha , } \\end{align*}"}
{"id": "2681.png", "formula": "\\begin{align*} H _ N : = \\sum _ { j = 1 } ^ N \\big ( - \\Delta _ { x _ j } + V _ \\mathrm { D W } ( x _ j ) \\big ) + \\frac { \\lambda } { { N - 1 } } \\sum _ { 1 \\le i < j \\le N } w ( x _ i - x _ j ) \\end{align*}"}
{"id": "1313.png", "formula": "\\begin{align*} M ( v ) : = \\frac { 1 } { ( 2 \\pi ) ^ { - d / 2 } } e ^ { - \\frac { | v | ^ 2 } { 2 } } , \\end{align*}"}
{"id": "2570.png", "formula": "\\begin{align*} A _ { j k } ^ { ( 1 ) } + A _ { j k } ^ { ( 2 ) } + A _ { j k } ^ { ( 3 ) } = 0 \\mbox { o n } \\ \\ \\mathcal H ^ 0 \\ . \\end{align*}"}
{"id": "1808.png", "formula": "\\begin{align*} \\theta _ k = \\frac { \\tilde \\theta _ k } { 2 } \\geq \\frac { - L + \\sqrt { L ^ 2 + 4 \\delta ^ 2 \\beta \\| A \\| ^ 2 } } { 4 \\tau _ { k - 1 } \\beta \\| A \\| ^ 2 } . \\end{align*}"}
{"id": "6450.png", "formula": "\\begin{align*} C _ 1 ( [ x , y ] _ \\mathfrak { g } , z ) + \\nabla _ { \\varrho ( x ) } ^ 2 ( C _ 1 ( y , z ) ) + \\circlearrowleft ( x , y , z ) = d t _ 2 ( C _ 2 ( x , y , z ) ) \\end{align*}"}
{"id": "916.png", "formula": "\\begin{align*} P ( u ) \\cdot v _ 0 = \\begin{bmatrix} \\nabla u \\\\ \\nabla \\nabla u \\end{bmatrix} v _ 0 = \\begin{bmatrix} f \\\\ h \\end{bmatrix} , v _ 0 ( x ) \\perp \\mathrm { K e r } P ( u ) ( x ) \\end{align*}"}
{"id": "1802.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\mbox { B r e g m a n p r i m a l C o n d a t - - V \\ ~ { u } a l g o r i t h m } & \\tilde x = 2 x ^ { ( k + 1 ) } - x ^ { ( k ) } & \\tilde z = z ^ { ( k ) } \\\\ \\mbox { B r e g m a n d u a l C o n d a t - - V \\ ~ { u } a l g o r i t h m } & \\tilde x = x ^ { ( k ) } & \\tilde z = 2 z ^ { ( k + 1 ) } - z ^ { ( k ) } . \\end{array} \\end{align*}"}
{"id": "1454.png", "formula": "\\begin{align*} \\vec { q } _ { \\ell } = { } ^ t \\Biggl ( \\psi _ { { 1 , r - 1 } } ( t ^ n P _ { \\ell } ( t ) ) , \\ldots , \\psi _ { { 1 , 0 } } ( t ^ n P _ { \\ell } ( t ) ) , \\ldots , \\psi _ { { m , r - 1 } } ( t ^ n P _ { \\ell } ( t ) ) , \\ldots , \\psi _ { { m , 0 } } ( t ^ n P _ { \\ell } ( t ) ) \\Biggr ) \\enspace . \\end{align*}"}
{"id": "1796.png", "formula": "\\begin{align*} \\begin{array} [ t ] { l l } \\mbox { m i n i m i z e } & f ( x ) + h ( x ) \\\\ \\mbox { s u b j e c t t o } & A x = b \\end{array} \\begin{array} [ t ] { l l } \\mbox { m a x i m i z e } & - b ^ T z - ( f + h ) ^ * ( - A ^ T z ) . \\end{array} \\end{align*}"}
{"id": "5874.png", "formula": "\\begin{align*} d _ n ( { t \\mathcal K } , N ) _ X = | t | d _ n ( { \\mathcal K } , N ) _ X \\ \\mbox { a n d } \\ d _ n ( t \\mathcal { K } ) _ X = | t | d _ n ( \\mathcal { K } ) _ X , \\end{align*}"}
{"id": "6432.png", "formula": "\\begin{align*} x \\in \\ , & \\mathfrak { g } [ 1 ] \\longmapsto \\Phi _ 0 ( x ) = { \\mathcal { L } } _ { \\varrho ( x ) } \\in \\mathfrak { X } _ 0 ( E ) [ 1 ] \\\\ x \\wedge y \\in \\wedge ^ 2 & \\mathfrak g [ 1 ] \\longmapsto \\Phi _ 1 ( x , y ) = \\iota _ { \\chi ( x , y ) } \\in \\mathfrak { X } _ { - 1 } ( E ) [ 1 ] \\end{align*}"}
{"id": "8835.png", "formula": "\\begin{align*} f ( y ) = f _ + ' ( 0 ) y + \\int _ { 0 } ^ { \\infty } f _ t ( y ) d \\nu ( t ) \\end{align*}"}
{"id": "2858.png", "formula": "\\begin{align*} f ( x ) & = H _ { \\delta } ( \\| A x - b \\| _ 2 ) + \\frac { 1 } { 2 } \\mu \\| x \\| ^ 2 \\\\ \\nabla f ( x ) & = \\nabla H _ { \\delta } ( \\| A x - b \\| _ 2 ) + \\mu x \\\\ f & \\in \\mathcal { F } _ { \\mu , \\ , \\delta ^ { - 1 } \\ , \\| A ^ T A \\| _ 2 + \\mu } \\end{align*}"}
{"id": "463.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\min _ { 1 \\le i \\le k _ { n } } \\mu _ { n , i } ( \\{ \\zeta \\in \\mathbb { T } : | \\zeta - 1 | < \\varepsilon \\} ) = 1 \\end{align*}"}
{"id": "6670.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\overline { \\chi } _ t - A _ 2 \\overline { \\chi } _ { i i } - ( B _ 2 + g ' C _ 2 + D _ 2 ) \\overline { \\chi } _ i = a _ { 0 2 } \\varphi ^ { q } - \\lambda _ { 0 2 } \\sqrt { A _ 2 } ^ \\alpha | \\chi _ i | ^ { \\alpha } , & t > 0 , \\ 0 < i < g _ 0 , \\\\ \\overline { \\chi } _ i ( t , 0 ) = \\overline { \\chi } ( t , g _ 0 ) = 0 , & t > 0 , \\\\ \\overline { \\chi } ( 0 , i ) = \\chi _ 0 ( i ) , & 0 \\leq i \\leq g _ 0 , \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "5232.png", "formula": "\\begin{align*} \\begin{aligned} x & \\longmapsto x \\exp ( ( a + 1 ) g x ^ a y ^ b ) \\\\ y & \\longmapsto y \\exp ( - ( a + 1 ) g x ^ a y ^ b ) . \\end{aligned} \\end{align*}"}
{"id": "2165.png", "formula": "\\begin{align*} I _ { 1 } = \\int _ { \\mathbb { R } } \\frac { d \\nu ( s ) } { s ^ { 2 } } = \\lim _ { y \\rightarrow 0 ^ { + } } \\frac { - \\Im G _ { \\nu } ( i y ) } { y } \\in ( 0 , + \\infty ] \\end{align*}"}
{"id": "416.png", "formula": "\\begin{align*} \\varphi _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } = \\varphi _ { \\mu _ { 1 } } + \\varphi _ { \\mu _ { 2 } } \\end{align*}"}
{"id": "356.png", "formula": "\\begin{align*} \\delta _ { } : = D ^ { 1 / 2 } . \\end{align*}"}
{"id": "1195.png", "formula": "\\begin{align*} \\| \\nabla \\varphi \\| _ { L ^ q ( \\nu _ t ; \\R ^ d ) } ^ q = \\int _ { \\R ^ d } | \\nabla \\varphi | ^ q d \\mu + t \\int _ { \\R ^ d } | \\nabla \\varphi | ^ q d h _ 2 = \\| \\nabla \\varphi \\| _ { L ^ q ( \\mu ; \\R ^ d ) } ^ q + O ( t ) , \\ t \\downarrow 0 . \\end{align*}"}
{"id": "8894.png", "formula": "\\begin{align*} H _ v : = \\left [ \\partial _ 1 F ( v ) , \\dots , \\partial _ n F ( v ) \\right ] . \\end{align*}"}
{"id": "5401.png", "formula": "\\begin{align*} ( ( - \\Delta ) ^ s + q _ { \\gamma } ) v & = 0 \\quad \\quad \\Omega \\\\ v & = f \\quad \\quad \\Omega _ e \\end{align*}"}
{"id": "2394.png", "formula": "\\begin{align*} \\partial _ t u _ R + \\Delta ^ 2 u _ R = \\Delta f _ R ( u _ R ) + \\sigma ( u _ R ) \\dot { W } , R \\ge 1 , \\end{align*}"}
{"id": "2240.png", "formula": "\\begin{align*} ( H , G ) | _ { Y = 0 } = ( 0 , 0 ) . \\end{align*}"}
{"id": "5801.png", "formula": "\\begin{align*} b _ { i | v } = \\frac { F _ { v - i , i } + F _ { i , v - i } } { a ( v ) } = \\frac { 2 F _ { v - i , i } } { a ( v ) } . \\end{align*}"}
{"id": "3499.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast \\ast } = \\frac { \\Gamma \\left ( ( m + 1 ) / 2 \\right ) \\sqrt { m } \\left [ \\xi _ { p } ^ { 3 } \\left ( 1 + \\frac { \\xi _ { p } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } - \\xi _ { q } ^ { 3 } \\left ( 1 + \\frac { \\xi _ { q } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } \\right ] } { \\Gamma ( m / 2 ) ( m - 1 ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "918.png", "formula": "\\begin{align*} v ' = v _ 0 - \\frac { \\langle v _ 0 , w \\rangle _ { \\mathbb { R } ^ N } } { \\langle w , w \\rangle _ { \\mathbb { R } ^ N } } w . \\end{align*}"}
{"id": "1949.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { S } _ { 1 } ( n + 1 , 0 ) } w ( \\gamma ) = \\sum _ { k = 0 } ^ { p } a _ { q } ^ { ( k ) } \\Big ( \\sum _ { \\gamma \\in \\mathcal { S } _ { 1 } ( n , k ) } w ( \\gamma ) \\Big ) = \\sum _ { k = 0 } ^ { p } a _ { q } ^ { ( k ) } \\langle \\mathcal { H } _ { q } ^ { n } e _ { k } , e _ { 0 } \\rangle = \\langle \\mathcal { H } _ { q } ^ { n + 1 } e _ { 0 } , e _ { 0 } \\rangle \\end{align*}"}
{"id": "2251.png", "formula": "\\begin{align*} \\hat { u _ p ^ 0 } = u _ p ^ 0 + \\delta , \\hat { h _ p ^ 0 } = h _ p ^ 0 + \\sigma \\end{align*}"}
{"id": "4644.png", "formula": "\\begin{align*} \\hat { A } : = J ^ { - 1 } A J , \\end{align*}"}
{"id": "2866.png", "formula": "\\begin{align*} ( \\phi ^ t _ { X _ H } ) ^ * \\omega _ Q = \\omega _ Q \\ ; . \\end{align*}"}
{"id": "448.png", "formula": "\\begin{align*} \\Phi ( \\eta _ { \\mu } ( z ) ) = z \\end{align*}"}
{"id": "3089.png", "formula": "\\begin{align*} \\mathrm { M D T E } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { P } ) = \\left ( \\begin{array} { c c c c c c c c c c c } 3 . 4 4 5 9 \\\\ 4 . 6 0 4 7 \\end{array} \\right ) , \\mathrm { M D T C o v } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { P } ) = \\left ( \\begin{array} { c c c c c c c c c c c } 0 . 6 0 0 8 & 0 . 2 4 3 6 \\\\ 0 . 2 4 3 6 & 0 . 2 7 6 0 \\end{array} \\right ) . \\end{align*}"}
{"id": "7310.png", "formula": "\\begin{align*} w _ 0 = \\sum _ { j = 0 } ^ J \\sum _ { i = 0 } ^ J \\alpha _ j { \\sf A } _ { j i } e _ i ( z ) + \\sum _ { j = 0 } ^ J \\alpha _ j e _ { j , J } ^ \\bot . \\end{align*}"}
{"id": "753.png", "formula": "\\begin{align*} u _ d = u _ d ^ { ( 0 ) } + \\sum _ { n = 1 } ^ { \\infty } ( u _ d ^ { ( n + 1 ) } - u _ d ^ { ( n ) } ) = u _ d ^ { ( 0 ) } + \\sum _ { n = 1 } ^ { \\infty } \\big ( \\Phi _ d ( u _ d ^ { ( n ) } ) - \\Phi _ d ( u _ d ^ { ( n - 1 ) } ) \\big ) , \\end{align*}"}
{"id": "2797.png", "formula": "\\begin{align*} U = \\sqrt { \\frac { 2 L \\ \\Delta } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\Big ( 2 h _ i - h _ i ^ 2 \\frac { - \\kappa } { 1 - \\kappa } \\Big ) } } \\end{align*}"}
{"id": "7056.png", "formula": "\\begin{align*} \\sum _ { M \\geq 1 } g _ r ( M ) \\ , z ^ M & = \\sum _ { n \\geq 0 } \\sum _ { d = 1 } ^ { r - 1 } \\frac { z ^ { n r + d } } { ( 1 - z ) ^ { n ( r - 1 ) + d } } = \\sum _ { d = 1 } ^ { r - 1 } \\left ( \\frac { z } { 1 - z } \\right ) ^ d \\sum _ { n \\geq 0 } \\left ( \\frac { z ^ r } { ( 1 - z ) ^ { r - 1 } } \\right ) ^ n \\\\ & = \\frac { z } { 1 - z } \\frac { 1 - \\left ( \\frac { z } { 1 - z } \\right ) ^ { r - 1 } } { 1 - \\frac { z } { 1 - z } } \\frac 1 { 1 - \\frac { z ^ r } { ( 1 - z ) ^ { r - 1 } } } . \\end{align*}"}
{"id": "3081.png", "formula": "\\begin{align*} \\frac { c _ { n } } { c _ { n - 2 , \\xi _ { \\boldsymbol { s } k } , \\xi _ { \\boldsymbol { t } l } } ^ { \\ast \\ast } } = & \\frac { 2 ^ { ( n - 6 ) / 2 } } { \\Gamma ( n - 1 ) \\pi } \\bigg \\{ \\int _ { 0 } ^ { \\infty } t ^ { \\frac { n - 4 } { 2 } } \\left [ ( 2 + 3 \\sqrt { 2 } ) \\left ( t + \\frac { \\xi _ { \\boldsymbol { s } , k } ^ { 2 } } { 2 } + \\frac { \\xi _ { \\boldsymbol { t } , l } ^ { 2 } } { 2 } \\right ) + 3 \\right ] \\exp \\left ( - \\sqrt { 2 t + \\xi _ { \\boldsymbol { s } , k } ^ { 2 } + \\xi _ { \\boldsymbol { t } , l } ^ { 2 } } \\right ) \\mathrm { d } t \\bigg \\} . \\end{align*}"}
{"id": "896.png", "formula": "\\begin{align*} \\norm { h } ^ 2 = u ( a ) . \\end{align*}"}
{"id": "3743.png", "formula": "\\begin{align*} J ^ { e s s ; 0 } _ { k , m ' } ( t , x ) : = \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\mathcal { K } _ { k , j } ^ { e s s } ( y , \\omega , v ) f ( s , x - y + ( t - s ) \\omega , v ) \\varphi _ { m ' ; - k } ( t - s ) \\varphi _ { m ; - 1 0 M _ t } ( t - s ) d \\omega d y d v \\big | _ { s = 0 } ^ t \\end{align*}"}
{"id": "1449.png", "formula": "\\begin{align*} F _ { s } ( \\alpha _ i / z ) = \\sum _ { k = 0 } ^ { \\infty } \\dfrac { 1 } { ( k + x + 1 ) ^ { r - s } } \\cdot \\dfrac { \\alpha ^ { k + 1 } _ i } { z ^ { k + 1 } } = \\Phi _ { r - s } ( x , \\alpha _ i / z ) \\ \\ ( 1 \\le i \\le m , 0 \\le s \\le r - 1 ) \\enspace , \\end{align*}"}
{"id": "7470.png", "formula": "\\begin{gather*} y _ { n } : = Q \\left ( y _ { n - 1 } \\right ) \\end{gather*}"}
{"id": "1710.png", "formula": "\\begin{align*} \\{ \\sigma : \\Gamma \\to \\{ \\pm 1 \\} \\ | \\ \\sigma ( \\gamma _ 1 + \\gamma _ 2 ) = ( - 1 ) ^ { \\langle \\gamma _ 1 , \\gamma _ 2 \\rangle } \\ \\sigma ( \\gamma _ 1 ) \\sigma ( \\gamma _ 2 ) \\} \\ . \\end{align*}"}
{"id": "1601.png", "formula": "\\begin{align*} v o l ( D ^ 2 _ x ( 1 ) ) : = \\frac { v o l ( B ^ 2 ( 1 ) ) b ^ 2 A ^ { \\epsilon \\eta } z ^ 3 _ { \\epsilon } z ^ 3 _ { \\eta } } { 2 \\sqrt { \\det A } } . \\end{align*}"}
{"id": "6611.png", "formula": "\\begin{align*} & \\sum _ { x \\in \\xi + \\ell \\mathbb { Z } ^ 4 } w ( h ^ { - 1 } x ) e _ k ( a F _ h ( x ) ) H \\big ( h ^ { - 1 } k , h ^ { - 2 } F _ h ( x ) \\big ) \\\\ = & \\sum _ { { \\tiny b \\ , \\hbox { \\tiny m o d } \\ , k } } \\sum _ { { \\tiny \\begin{tabular} { c } $ z \\ , \\hbox { \\tiny m o d } \\ , k \\ell $ \\\\ $ z = \\xi ( \\ell ) , z = b ( k ) $ \\end{tabular} } } e _ k ( a F _ h ( b ) ) \\left ( \\sum _ { y \\in \\mathbb { Z } ^ 4 } f ( y ) \\right ) , \\end{align*}"}
{"id": "792.png", "formula": "\\begin{align*} \\frac { \\beta ( 1 + r ^ m ) } { ( 1 - r ^ m ) ^ { 2 + 1 } } + \\frac { ( 1 - \\beta ) r ^ m } { ( 1 - r ^ m ) ^ { 2 } } + \\frac { r ( 1 + r ^ 2 ) } { ( 1 - r ^ 2 ) ^ 2 } = \\frac { 1 } { 4 } . \\end{align*}"}
{"id": "3977.png", "formula": "\\begin{align*} c = ( c _ 0 , c _ 1 , \\cdots , c _ { n - 1 } ) \\in C ~ \\implies ~ \\rho ( c ) = ( c _ { n - 1 } , c _ 0 , \\cdots , c _ { n - 2 } ) \\in C . \\end{align*}"}
{"id": "4286.png", "formula": "\\begin{align*} \\rho ^ { \\rm ( s t ) } = \\exp \\biggl ( \\frac 1 2 \\mathfrak { a } ^ { T } M \\mathfrak { a } + s \\biggr ) \\end{align*}"}
{"id": "1249.png", "formula": "\\begin{align*} Y _ 0 & = 1 , \\\\ Y _ 1 & = x - 1 , \\\\ Y _ j & = ( x - c _ { l ( T ) + 1 - j } - 1 ) Y _ { j - 1 } - c _ { l ( T ) + 1 - j } Y _ { j - 2 } ( \\forall j \\in \\overline { 2 , l ( T ) - 1 } ) , \\\\ Y _ { l ( T ) } & = ( x - c _ 1 ) Y _ { l ( T ) - 1 } - c _ 1 Y _ { l ( T ) - 2 } , \\end{align*}"}
{"id": "6506.png", "formula": "\\begin{align*} b \\nabla v = 0 \\ \\ \\ \\ \\Omega \\nabla v = \\nabla u = 0 \\omega _ b . \\end{align*}"}
{"id": "2188.png", "formula": "\\begin{gather*} I _ 1 = \\{ \\ , 1 , 2 , 3 , 4 , 5 \\ , \\} , I _ 2 = \\{ \\ , 6 \\ , \\} , I _ 3 = \\{ \\ , 7 , 8 \\ , \\} , \\\\ J _ 1 = \\{ \\ , 1 \\ , \\} , J _ 2 = \\{ \\ , 2 \\ , \\} , J _ 3 = \\{ \\ , 3 \\ , \\} , J _ 4 = \\{ \\ , 4 , 5 , 6 , 7 , 8 \\ , \\} , \\end{gather*}"}
{"id": "4530.png", "formula": "\\begin{align*} F ( u , x _ 1 , x _ 2 ) : = \\frac { 1 } { 2 } | \\nabla u | ^ { - 2 } + \\frac { 1 } { 2 } l _ { x _ 1 } ^ 2 + \\frac { 1 } { 2 } l _ { x _ 2 } ^ 2 + l _ { x _ 1 } x _ 2 \\psi ( u ) - l _ { x _ 2 } x _ 1 \\psi ( u ) - l _ u . \\end{align*}"}
{"id": "2419.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb P \\big ( R _ 2 ^ { ( n ) } \\ge 3 ^ { - 2 p } \\big ) = 0 . \\end{align*}"}
{"id": "1366.png", "formula": "\\begin{align*} \\widehat { T _ \\ell } [ v _ 1 , \\cdots , v _ p ] _ k = \\sum _ { \\substack { k = k _ 1 - k _ 2 + \\cdots - k _ { p - 1 } + k _ p \\\\ | k _ \\ell | \\gg k _ 2 ^ * \\\\ C a s e \\ , B } } \\frac { \\prod _ { \\substack { j = 1 \\\\ o d d } } ^ { p } \\widehat { v } _ { k _ i } \\prod _ { \\substack { j = 2 \\\\ e v e n } } ^ p \\overline { \\widehat { v } _ { k _ i } } } { \\Phi ( k , k _ 1 \\cdots , k _ p ) } , \\end{align*}"}
{"id": "75.png", "formula": "\\begin{align*} D _ A w ( t ) = v ( t ) \\end{align*}"}
{"id": "2374.png", "formula": "\\begin{align*} I = \\{ l \\mid 1 \\leq l \\leq d \\mbox { s u c h t h a t } l = p ^ i \\mbox { f o r s o m e } i \\in \\N \\} \\end{align*}"}
{"id": "3269.png", "formula": "\\begin{align*} A ( x ) : = ( A _ 2 - A _ 1 ) ( x ) , q ( x ) : = ( q _ 2 - q _ 1 ) ( x ) , x \\in \\R ^ 3 . \\end{align*}"}
{"id": "1867.png", "formula": "\\begin{align*} \\mbox { c a r d } \\ , \\{ \\mbox { u p s t e p s i n } \\ , \\gamma \\} = p \\times \\mbox { c a r d } \\ , \\{ \\mbox { d o w n s t e p s i n } \\ , \\gamma \\} + j . \\end{align*}"}
{"id": "5906.png", "formula": "\\begin{align*} \\tilde { \\mu } _ { V X ^ { ( n ) } } ( A ) : = \\mathbb { P } \\Big ( \\Big ( \\frac { 1 } { \\beta } \\Big ) ^ { 1 / p } \\sum _ { j = 1 } ^ n Z _ j V _ { \\bullet , j } \\in A \\Big ) , \\end{align*}"}
{"id": "7926.png", "formula": "\\begin{align*} \\mathbb K : = \\left \\{ f \\in \\mathbb L \\otimes \\mathbb Q | \\left \\{ i \\in \\{ 1 , \\ldots , m \\} | D _ i \\cdot f \\in \\mathbb Z \\right \\} \\in \\mathcal A \\right \\} . \\end{align*}"}
{"id": "7390.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 c _ i = 0 . \\end{align*}"}
{"id": "4886.png", "formula": "\\begin{align*} J & = \\iint \\limits _ { [ 0 , 1 ] ^ { 2 } } \\left | f _ { 1 } ^ { 2 } ( \\alpha _ { 1 } ) f _ { 2 } ^ { 2 } ( \\alpha _ { 2 } ) G ^ { 4 } ( \\alpha _ { 1 } + \\alpha _ { 2 } ) \\right | d \\alpha _ { 1 } d \\alpha _ { 2 } \\\\ & = \\sum \\limits _ { 1 \\leq m _ { 1 } , \\ldots m _ { 4 } \\leq L } \\prod \\limits _ { i = 1 } ^ { 2 } t _ { i } ( 2 ^ { m _ { 1 } } + 2 ^ { m _ { 2 } } - 2 ^ { m _ { 3 } } - 2 ^ { m _ { 4 } } ) , \\end{align*}"}
{"id": "6078.png", "formula": "\\begin{align*} f _ 2 ( x ) & = x ( x - 2 ) ( x - 3 ) ( x + 6 ) ^ 5 \\\\ & = x ^ 8 + 2 5 x ^ 7 + 2 1 6 x ^ 6 + 5 4 0 x ^ 5 - 2 1 6 0 x ^ 4 - 1 1 6 6 4 x ^ 3 + 4 6 6 5 6 x . \\end{align*}"}
{"id": "289.png", "formula": "\\begin{align*} f ( A ) = \\sum _ { a \\in A } \\frac { 1 } { a \\log { a } } < \\sum _ { a \\in A } \\frac { 1 } { a \\log P ( a ) } \\approx e ^ \\gamma \\sum _ { a \\in A } \\frac { 1 } { a } \\prod _ { p < P ( a ) } \\Big ( 1 - \\frac { 1 } { p } \\Big ) . \\end{align*}"}
{"id": "6382.png", "formula": "\\begin{align*} \\{ 1 , \\ldots , k + 1 \\} = R _ 1 \\cup \\cdots \\cup R _ p \\end{align*}"}
{"id": "7135.png", "formula": "\\begin{align*} x _ { e } = - \\sum _ { \\{ i , c \\} \\in E ( \\mathsf { P } _ { n + 2 } ) \\atop b < i < c } x _ { i c } + \\sum _ { \\{ j , d \\} \\in E ( \\mathsf { P } _ { n + 2 } ) \\atop c \\leq j < d } x _ { j d } . \\end{align*}"}
{"id": "2061.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : P P l a w e m p i r i c a l p r o c e s s } F ^ N _ t = ( \\boldsymbol { \\mu } _ N ) _ \\# S ^ N _ t ( f ^ N _ 0 ) , \\overline { F } { } ^ N _ t : = ( \\overline { S } _ t \\circ \\boldsymbol { \\mu } _ N ) _ \\# f ^ N _ 0 , \\end{align*}"}
{"id": "5756.png", "formula": "\\begin{align*} \\pi _ { i _ 1 } \\pi _ J & = \\pi _ { J _ 1 } \\cdots \\pi _ { J _ { k - 1 } } ( \\pi _ { i _ 1 } \\pi _ { J _ k } ) \\pi _ { J _ { k + 1 } } \\cdots \\pi _ { J _ m } \\\\ & = \\pi _ { J _ 1 } \\cdots \\pi _ { J _ { k - 1 } } \\left ( \\binom { | J _ k | + 1 } { 1 } \\pi _ { J ' ( i _ 1 ) } \\right ) \\pi _ { J _ { k + 1 } } \\cdots \\pi _ { J _ m } \\end{align*}"}
{"id": "8213.png", "formula": "\\begin{align*} k = \\frac { \\sqrt { \\sqrt { 1 + \\frac { 1 6 } { 3 } \\beta m E } - 1 } } { 2 \\hslash \\sqrt { \\beta / 3 } } , k ' = \\frac { \\sqrt { \\sqrt { 1 + \\frac { 1 6 } { 3 } \\beta m E } + 1 } } { 2 \\hslash \\sqrt { \\beta / 3 } } . \\end{align*}"}
{"id": "5861.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\bigg ( \\int _ t ^ { \\infty } u \\bigg ) ^ { \\frac { 1 } { 1 - q } } V _ r ( 0 , t ) ^ { \\frac { q } { 1 - q } } = 0 . \\end{align*}"}
{"id": "1177.png", "formula": "\\begin{align*} \\| \\varphi ( \\cdot + a ) \\| _ { \\dot { H } ^ { 1 , q } ( \\mu ^ { - a } * \\gamma _ { \\sigma } ) } ^ q & = \\int _ { \\R ^ d } | \\nabla \\varphi ( \\cdot + a ) | ^ q d ( \\mu ^ { - a } * \\gamma _ { \\sigma } ) \\\\ & = \\int _ { \\R ^ d } | \\nabla \\varphi | ^ q d ( \\mu * \\gamma _ { \\sigma } ) = \\| \\varphi \\| _ { \\dot { H } ^ { 1 , q } ( \\mu * \\gamma _ { \\sigma } ) } ^ q . \\end{align*}"}
{"id": "8311.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } j | c _ j | = O \\left ( m v \\left ( \\frac { 1 } { m } \\right ) \\right ) , \\end{align*}"}
{"id": "2935.png", "formula": "\\begin{align*} \\max \\left \\{ \\sum _ { j = 1 } ^ m F ( x ^ j ) : \\pi \\in \\Pi \\land x ^ j \\in P ( \\chi _ { H ^ j ( \\pi ) } ) \\right \\} \\end{align*}"}
{"id": "4988.png", "formula": "\\begin{align*} \\kappa _ 2 ^ 2 \\int _ 0 ^ \\tau \\left | \\sum _ { i = 1 } ^ Q \\rho _ i \\mathbf { 1 } _ { [ 0 , t _ i ] } ( \\tau ) ( t _ i - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ( X _ s ) \\right | ^ 2 d s . \\end{align*}"}
{"id": "1524.png", "formula": "\\begin{align*} \\Delta _ { h } : = \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + \\frac { \\alpha } { 2 } \\frac { 1 } { \\tanh ( \\alpha r ) } \\frac { \\partial } { \\partial r } + \\frac { 1 } { 2 \\sinh ^ 2 ( \\beta r ) } \\frac { \\partial ^ 2 } { \\partial \\theta ^ 2 } \\end{align*}"}
{"id": "3201.png", "formula": "\\begin{align*} \\int _ { G } f ( g ) d g = \\int _ S \\int _ 0 ^ \\infty f ( \\gamma _ \\xi ( t ) ) t ^ { Q - 1 } d t d \\sigma ( \\xi ) \\end{align*}"}
{"id": "5703.png", "formula": "\\begin{align*} \\alpha _ { i } \\cdot \\varpi _ J = 0 , \\end{align*}"}
{"id": "3486.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast } = \\frac { \\phi ( \\xi _ { p } ) - \\phi ( \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , ~ L _ { 1 } ^ { \\ast \\ast } = \\frac { \\xi _ { p } ^ { 3 } \\phi ( \\xi _ { p } ) - \\xi _ { q } ^ { 3 } \\phi ( \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "4757.png", "formula": "\\begin{align*} \\Delta ( z ) = \\sum _ { n = 1 } ^ \\infty \\tau ( n ) q ^ n , \\end{align*}"}
{"id": "3206.png", "formula": "\\begin{align*} E _ { k , t } ^ { } = \\sum _ { i = 1 } ^ n \\tau \\xi ^ { } _ k \\left ( f ^ { } _ { k , t , i } \\right ) ^ 3 , \\end{align*}"}
{"id": "7876.png", "formula": "\\begin{align*} f _ + ^ * ( K _ { X _ + } + D _ + ) = f _ - ^ * ( K _ { X _ - } + D _ - ) . \\end{align*}"}
{"id": "2469.png", "formula": "\\begin{align*} \\left [ u , \\left [ x , z \\right ] \\right ] = \\left ( - 1 \\right ) ^ { p \\left ( u \\right ) } \\left [ x , \\left [ u , z \\right ] \\right ] \\in \\left [ x , \\gg \\right ] . \\end{align*}"}
{"id": "5203.png", "formula": "\\begin{align*} \\iota _ { \\Lambda } ^ * o _ { 0 , k _ 1 , k _ 2 , 1 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I } } = o _ N \\otimes \\left ( o _ { v _ 1 } \\boxtimes o _ { v _ 2 } \\right ) , \\end{align*}"}
{"id": "3004.png", "formula": "\\begin{align*} \\vec { b } \\vec { A } _ k = c _ k \\equiv 0 \\pmod { u _ k } \\ , \\end{align*}"}
{"id": "4917.png", "formula": "\\begin{align*} R ( U , Y ) Z = \\nabla _ U \\nabla _ Y Z - \\nabla _ Y \\nabla _ U Z - \\nabla _ { [ U , Y ] } Z , \\end{align*}"}
{"id": "556.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i ) = \\mathbb { P } ( P _ i \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } ) , \\end{align*}"}
{"id": "1317.png", "formula": "\\begin{align*} \\int _ { \\{ v \\cdot n _ x < 0 \\} } R ( u \\to v ; x ) | v \\cdot n _ x | \\d v = 1 . \\end{align*}"}
{"id": "8757.png", "formula": "\\begin{align*} \\Delta ( E ) = E \\otimes 1 + K \\otimes E , \\Delta ( F ) = 1 \\otimes F + F \\otimes K ^ { - 1 } , \\Delta ( K ) = K \\otimes K . \\end{align*}"}
{"id": "7995.png", "formula": "\\begin{align*} f _ { ( \\mathbb P ^ 3 , K 3 ) } ( t ) = \\sum _ { d \\geq 0 } \\frac { ( 4 d ) ! } { ( d ! ) ^ 4 } t ^ { 4 d } , f _ { ( Q _ 4 , K 3 ) } ( t ) = \\sum _ { d \\geq 0 } \\frac { ( 4 d ) ! } { ( d ! ) ^ 4 } t ^ { d } . \\end{align*}"}
{"id": "7618.png", "formula": "\\begin{align*} H _ r < \\dfrac { C ( r ) } { n ^ { r - \\frac { 1 } { 2 } } } - \\dfrac { ( r - 1 ) ! } { 2 ^ r \\ n ^ r } + \\dfrac { \\widehat { C _ 2 } ( r ) } { n ^ { r + \\frac { 1 } { 2 } } } + \\dfrac { r } { 1 0 ^ r \\ n ^ { r + \\frac { 1 } { 2 } } } = \\dfrac { C ( r ) } { n ^ { r - \\frac { 1 } { 2 } } } - \\dfrac { ( r - 1 ) ! } { 2 ^ r \\ n ^ r } + \\underset { = C _ 2 ( r ) } { \\underbrace { \\Bigl ( \\widehat { C _ 2 } ( r ) + \\dfrac { r } { 1 0 ^ r } \\Bigr ) } } \\dfrac { 1 } { n ^ { r + \\frac { 1 } { 2 } } } . \\end{align*}"}
{"id": "7258.png", "formula": "\\begin{align*} \\limsup _ { t \\to T } ( T - t ) ^ \\frac { 1 } { p - 1 } \\| u ( t ) \\| _ \\infty & < \\infty ( ) , \\\\ \\limsup _ { t \\to T } ( T - t ) ^ \\frac { 1 } { p - 1 } \\| u ( t ) \\| _ \\infty & = \\infty ( ) . \\end{align*}"}
{"id": "2517.png", "formula": "\\begin{align*} \\begin{array} { l l } \\frac { 1 } { 2 } \\{ g ( \\nabla _ U \\xi , V ) + g ( \\nabla _ V \\xi , U ) \\} + R i c ^ \\nu ( U , V ) - ( m - n ) g ( U , V ) \\| H \\| ^ 2 \\\\ + \\sum \\limits _ { j = 1 } ^ { n } g ( \\nabla _ { X _ j } H , X _ j ) g ( U , V ) + \\mu g ( U , V ) = 0 . \\end{array} \\end{align*}"}
{"id": "90.png", "formula": "\\begin{align*} \\chi ^ { \\epsilon , L } ( \\lambda , \\mu ) : = \\epsilon ^ d \\sum _ { x \\in \\Lambda _ { \\epsilon , L } } \\langle \\varphi _ 0 \\varphi _ x \\rangle ^ { \\epsilon , L } _ { \\lambda , \\mu } \\leq \\bar \\chi . \\end{align*}"}
{"id": "5396.png", "formula": "\\begin{align*} B _ { \\gamma } \\colon H ^ s ( \\R ^ n ) \\times H ^ s ( \\R ^ n ) \\to \\R , B _ { \\gamma } ( u , v ) \\vcentcolon = \\int _ { \\R ^ { 2 n } } \\Theta _ { \\gamma } \\nabla ^ s u \\cdot \\nabla ^ s v \\ , d x d y \\end{align*}"}
{"id": "5386.png", "formula": "\\begin{align*} \\langle q , v ^ 2 \\rangle = \\langle q _ s , v ^ 2 \\rangle + \\langle q _ 0 , v ^ 2 \\rangle \\geq \\langle q _ s , v ^ 2 \\rangle \\end{align*}"}
{"id": "4174.png", "formula": "\\begin{align*} ( \\cos ( t \\sqrt L ) f , g ) = 0 \\quad | t | < d _ { \\mathrm { C C } } ( U , V ) . \\end{align*}"}
{"id": "7861.png", "formula": "\\begin{align*} \\| e _ B ( T - T _ n ) e _ B \\| _ { \\infty , 1 } = \\sup _ { a , b \\in ( B ) _ 1 } \\langle e _ B ( T - T _ n ) e _ B \\hat a , b \\rangle \\leq \\| T - T _ n \\| _ { \\infty , 1 } , \\end{align*}"}
{"id": "7215.png", "formula": "\\begin{align*} w ( | x | , t ) : = \\left \\{ \\begin{array} { l l l l } 1 & \\mbox { i f } & \\ a > 0 , \\\\ \\log ^ { - 1 } ( t + | x | + 3 R ) & \\mbox { i f } & \\ a = 0 , \\\\ ( t + | x | + 3 R ) ^ { a } & \\mbox { i f } & \\ a < 0 . \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "8073.png", "formula": "\\begin{align*} \\phi ( N \\times \\{ 0 \\} ) = S ^ { 1 } \\cdot x . \\end{align*}"}
{"id": "2849.png", "formula": "\\begin{align*} \\begin{aligned} x _ i { } & = { } \\frac { U _ * } { L } + \\frac { U _ * } { L } \\sum \\limits _ { j = \\textbf { \\textit { i } } } ^ { N - 1 } h _ j \\\\ g _ i { } & = { } U _ * \\\\ f _ i { } & = { } \\Delta - \\frac { U _ * ^ 2 } { 2 L } \\sum \\limits _ { j = 0 } ^ { i - 1 } h _ j \\ , \\Big ( 2 - h _ j \\frac { - \\kappa } { 1 - \\kappa } \\Big ) \\end{aligned} \\end{align*}"}
{"id": "796.png", "formula": "\\begin{align*} M _ { r } ^ { N } ( G ) \\leq \\sum _ { n = N } ^ { \\infty } r ^ { 2 n - 1 } = \\frac { 1 } { r } \\left ( \\frac { r ^ { 2 N } } { 1 - r ^ 2 } \\right ) . \\end{align*}"}
{"id": "678.png", "formula": "\\begin{align*} \\partial _ s V _ s & = - \\partial _ s X _ s - [ V _ s , \\Phi _ { s * } X ] \\\\ & = - \\partial _ s X _ s + [ X _ s , \\widehat \\psi _ { s * } X ] \\end{align*}"}
{"id": "6029.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 + a x ^ 3 . \\end{align*}"}
{"id": "9014.png", "formula": "\\begin{align*} \\beta ^ { \\prime } ( 0 ) = \\frac { \\gamma } { 2 } + \\frac { 1 } { 2 } \\ln \\left ( \\frac { \\pi } { 2 } \\right ) - \\frac { 2 } { \\pi } \\beta ^ { \\prime } ( 1 ) . \\end{align*}"}
{"id": "8136.png", "formula": "\\begin{align*} \\eta = \\eta ^ \\parallel \\otimes \\eta ^ \\perp \\end{align*}"}
{"id": "7110.png", "formula": "\\begin{align*} \\dot { x } = f ( t , x ) + g ( t , x ) u \\end{align*}"}
{"id": "1430.png", "formula": "\\begin{align*} t ' ( w ) = \\begin{pmatrix} e ^ x & & 0 & & 0 \\\\ 0 & & ( e - 1 ) y + 1 & & 0 \\\\ 0 & & 0 & & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "739.png", "formula": "\\begin{align*} K _ 1 ^ 2 \\partial _ i \\bar { \\partial } _ j \\log K _ 1 & = K _ 2 ^ 2 \\partial _ i \\bar { \\partial } _ j \\log K _ 2 + K _ 3 ^ 2 \\partial _ i \\bar { \\partial } _ j \\log K _ 3 \\\\ & \\quad \\quad + K _ 2 \\partial _ i \\bar { \\partial } _ j K _ 3 + K _ 3 \\partial _ i \\bar { \\partial } _ j K _ 2 - \\partial _ i K _ 2 \\bar { \\partial } _ j K _ 3 - \\partial _ i K _ 3 \\bar { \\partial } _ j K _ 2 . \\end{align*}"}
{"id": "2538.png", "formula": "\\begin{align*} R i c ( e _ 2 , e _ 2 ) = - 2 e ^ { - 2 x _ 2 } - 1 , \\end{align*}"}
{"id": "564.png", "formula": "\\begin{align*} T _ k ( l , i , j , \\alpha ) = & f _ k ( j ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) } { \\sqrt { j - i } } \\Big ) , \\end{align*}"}
{"id": "7026.png", "formula": "\\begin{align*} \\begin{bmatrix} \\omega _ i ^ 0 \\\\ a _ { i } ^ 0 \\end{bmatrix} = S ^ { - 1 } \\begin{bmatrix} a _ { x , i } + \\dot { y } _ i \\dot { \\psi } _ i \\\\ a _ { y , i } - \\dot { x } _ i \\dot { \\psi } _ i \\end{bmatrix} , \\end{align*}"}
{"id": "872.png", "formula": "\\begin{align*} & f _ { m _ { s t } } ( s , t ) * f _ { m _ { s t } - 1 } ( s , t ) * \\cdots * f _ 1 ( s , t ) = f _ 0 ( s , t ) \\\\ & u * f _ { m _ { s t } - 1 } ( s , t ) * f _ { m _ { s t } - 2 } ( s , t ) * \\cdots * f _ 1 ( s , t ) * f _ 0 ( s , t ) \\\\ & = u * f _ { m _ { t s } - 1 } ( t , s ) * f _ { m _ { t s } - 2 } ( t , s ) * \\cdots * f _ 1 ( t , s ) * f _ 0 ( t , s ) \\end{align*}"}
{"id": "4380.png", "formula": "\\begin{align*} & \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ \\Psi _ 1 \\le 2 \\log r _ 1 \\} \\cap D _ j } | f | ^ 2 \\\\ = & \\lim _ { p \\rightarrow 1 + 0 } \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ p \\Psi _ 1 < 2 \\log r _ 1 \\} \\cap D _ j } | f | ^ 2 \\\\ \\ge & \\limsup _ { p \\rightarrow 1 + 0 } G _ { j , p } ( 0 ) \\\\ \\ge & C _ j \\end{align*}"}
{"id": "3838.png", "formula": "\\begin{align*} L a s t U ^ b _ { i , i _ 1 , i _ 2 , i _ 3 } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { ( \\R ^ 3 ) ^ 4 } e ^ { i \\widetilde { \\Phi } ^ b _ { \\mu _ 1 , \\mu _ 2 , \\mu _ 3 } ( \\xi , \\eta , \\sigma , \\kappa ; s , X ( s ) , v ) } K ( s , X ( s ) , V ( s ) ) \\cdot { } _ { } ^ 3 \\clubsuit E l l U ^ b ( s , \\xi , \\eta , \\sigma , v , V ( s ) ) \\end{align*}"}
{"id": "180.png", "formula": "\\begin{align*} \\widetilde { U } ( t ) = U ^ { j - 1 } + \\frac { t - t _ { j - 1 } } { t _ j - t _ { j - 1 } } \\Big ( U ^ { j } - U ^ { j - 1 } \\Big ) = \\frac { t - t _ { j - 1 } } { \\Delta t _ j } U ^ j + \\frac { t _ j - t } { \\Delta t _ j } U ^ { j - 1 } , \\end{align*}"}
{"id": "9002.png", "formula": "\\begin{align*} { n + 1 \\brack r + 1 } = \\frac { n ! } { r ! } B _ r \\left ( H _ n ^ { ( 1 ) } , - H _ n ^ { ( 2 ) } , \\ldots , ( - 1 ) ^ { r - 1 } ( r - 1 ) ! H _ n ^ { ( r ) } \\right ) . \\end{align*}"}
{"id": "5015.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } E [ | R ^ { n , 2 } _ \\tau | ^ 2 ] = 0 . \\end{align*}"}
{"id": "8687.png", "formula": "\\begin{align*} J ( x ) = A + \\lambda L = \\int _ { t _ 0 } ^ { t _ 1 } h ( x ^ 1 , x ^ 2 , \\dot { x } ^ 1 , \\dot { x } ^ 2 , \\lambda ) d t , \\end{align*}"}
{"id": "6715.png", "formula": "\\begin{align*} L _ { \\sigma , \\lambda } \\circ \\phi ( - \\lambda , - \\sigma ) = \\phi ( \\sigma , \\lambda ) \\circ L _ { \\sigma , \\lambda } . \\end{align*}"}
{"id": "4490.png", "formula": "\\begin{align*} \\mathcal { B } = ~ [ ( 1 + ( \\eta _ 1 + \\eta _ 2 + \\eta _ 3 ) ^ 2 ) ^ { 3 / 2 } - ( 1 + \\eta _ 3 ^ 2 ) ^ { 3 / 2 } ] ( 1 + \\eta _ 1 ^ 2 ) ^ { 3 / 2 } [ ( 1 + \\eta _ 3 ^ 2 ) ^ { 3 / 2 } - ( 1 + \\eta _ 1 ^ 2 ) ^ { 3 / 2 } ] ( 1 + ( \\eta _ 1 + \\eta _ 2 + \\eta _ 3 ) ^ 2 ) ^ { 3 / 2 } . \\end{align*}"}
{"id": "1674.png", "formula": "\\begin{align*} C _ s ( \\tau ) & = e ( \\tau ) - \\sum _ { i < s + 1 } ( - 1 ) ^ i A _ s ( \\tau ^ i ) , \\\\ S _ s ( \\tau ) & = d \\left ( C _ s ( \\tau ) * F ( \\tau ) \\right ) , \\textnormal { a n d } \\\\ A _ { s + 1 } ( \\tau ) & = ( - 1 ) ^ { s + 1 } C _ s ( \\tau ) * F ( \\tau ) . \\end{align*}"}
{"id": "5941.png", "formula": "\\begin{align*} { \\cal J } \\left [ X \\right ] = { \\rm D e t } \\left ( { \\frac { { \\delta { A _ { i j } } \\left ( t \\right ) } } { { \\delta { X _ { k p } } \\left ( { t ' } \\right ) } } } \\right ) \\end{align*}"}
{"id": "370.png", "formula": "\\begin{align*} B : = \\bigcup _ { a \\in A } \\phi _ a ^ { - 1 } ( A \\setminus \\{ a \\} ) . \\end{align*}"}
{"id": "3939.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d r } } \\prod _ { i = 1 } ^ d | K _ { h _ i } ( x _ i - u _ i ^ { q _ i } ) | \\prod _ { j = 1 } ^ { r - 1 } p _ { w _ { j + 1 } - w _ { j } } ( u ^ { j } , u ^ { j + 1 } ) d u ^ 1 \\dots d u ^ r \\le C , \\end{align*}"}
{"id": "1399.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} B & { \\bf 0 } \\\\ { \\bf 0 } & 1 / 2 \\end{matrix} \\right ] , B = \\left [ \\begin{matrix} 1 & 1 \\\\ \\epsilon & 1 \\end{matrix} \\right ] . \\end{align*}"}
{"id": "8395.png", "formula": "\\begin{align*} g _ 1 < g _ 2 \\Leftrightarrow \\begin{cases} g _ 1 ^ { - 1 } g _ 2 \\notin H \\ & p ( g _ 1 ) \\prec p ( g _ 2 ) , \\\\ g _ 1 ^ { - 1 } g _ 2 \\in H & s ^ { - 1 } g _ 1 < _ H s ^ { - 1 } g _ 2 \\textit { w h e r e $ s \\in S $ i s s u c h t h a t $ s H = g _ 1 H = g _ 2 H $ } . \\end{cases} \\end{align*}"}
{"id": "174.png", "formula": "\\begin{align*} \\| C ^ 3 _ \\infty \\| _ { L ^ 1 } - \\| C ^ 3 _ t \\| _ { L ^ 1 } & \\leq c \\int _ 0 ^ 1 \\frac { 1 } { t \\big ( m ^ 2 + \\frac { \\alpha } { t } \\big ) ^ { 4 - d } } \\ , d \\alpha \\\\ & \\leq \\frac { c } { m ^ 2 ( m ^ 2 t + 1 ) } { \\bf 1 } _ { d = 2 } + c \\log \\Big ( 1 + \\frac { 1 } { m ^ 2 t } \\Big ) { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "4218.png", "formula": "\\begin{align*} Z _ n = Z _ n ^ * \\leq \\alpha Z ^ * _ { n - 1 } \\end{align*}"}
{"id": "7668.png", "formula": "\\begin{align*} \\mathsf Z _ { S _ { 5 , 5 , 1 } } ( q ) = \\prod _ { n > 0 } \\ , \\left ( 1 - q ^ n \\right ) ^ { - \\Omega _ n [ S _ { 5 , 5 , 1 } ] } , \\end{align*}"}
{"id": "3548.png", "formula": "\\begin{align*} m = M ^ { - 1 } \\cdot a . \\end{align*}"}
{"id": "2987.png", "formula": "\\begin{align*} c ( s ) = \\frac { 2 | m | ^ { 1 / 2 - s } \\sigma _ { 2 s - 1 } ( | m | ) } { ( 2 s - 1 ) \\Lambda ( 2 s - 1 ) } , \\end{align*}"}
{"id": "8015.png", "formula": "\\begin{align*} K _ { \\tilde X } = f _ - ^ * K _ { X _ - } + \\left ( 1 + \\sum _ { i \\in M _ - } D _ i \\cdot e \\right ) E = f _ + ^ * K _ { X _ + } + \\left ( 1 - \\sum _ { i \\in M _ + } D _ i \\cdot e \\right ) E . \\end{align*}"}
{"id": "8964.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { W ^ { 1 , 2 } _ { \\delta ^ \\ast } ( M ) } = \\| \\nabla u \\| ^ 2 _ { L ^ 2 ( M ) } + \\| u \\| ^ 2 _ { L ^ 2 _ { \\delta ^ \\ast } ( M ) } , \\end{align*}"}
{"id": "173.png", "formula": "\\begin{align*} & 0 \\leq C _ \\infty ( 0 ) - C _ t ( 0 ) = : \\eta _ t \\leq c \\log \\Big ( 1 + \\frac { 1 } { m ^ 2 t } \\Big ) { \\bf 1 } _ { d = 2 } + c m \\Big ( \\sqrt { 1 + \\frac { 1 } { t m ^ 2 } } - 1 \\Big ) { \\bf 1 } _ { d = 3 } , \\\\ & 0 \\leq \\| C ^ 3 _ \\infty \\| _ { L ^ 1 } - \\| C ^ 3 _ t \\| _ { L ^ 1 } : = \\gamma _ t \\leq \\frac { c } { m ^ 2 ( m ^ 2 t + 1 ) } { \\bf 1 } _ { d = 2 } + c \\log \\Big ( 1 + \\frac { 1 } { m ^ 2 t } \\Big ) { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "5212.png", "formula": "\\begin{align*} \\{ \\Xi _ { \\mu } : = \\Xi ^ { r _ 1 } _ { \\mu _ 1 } \\times \\cdots \\times \\Xi ^ { r _ n } _ { \\mu _ n } \\ , | \\ , \\mu \\in D \\} . \\end{align*}"}
{"id": "2336.png", "formula": "\\begin{align*} \\overline F _ p = \\sum _ { i \\in I \\cup \\{ 0 \\} } a _ i Q _ \\theta ^ i \\end{align*}"}
{"id": "6610.png", "formula": "\\begin{align*} \\delta _ n = c _ Q Q ^ { - 2 } \\sum _ { k = 1 } ^ \\infty { \\sum _ { a \\ , \\hbox { \\tiny m o d } \\ , k } } ^ { \\ ! \\ ! \\ ! * } e _ k ( a n ) H ( Q ^ { - 1 } k , Q ^ { - 2 } n ) , \\end{align*}"}
{"id": "3461.png", "formula": "\\begin{align*} c _ { 1 } & = \\frac { \\Gamma ( 1 / 2 ) } { ( 2 \\pi ) ^ { 1 / 2 } } \\left [ \\int _ { 0 } ^ { \\infty } s ^ { - 1 / 2 } g _ { 1 } ( s ) \\mathrm { d } s \\right ] ^ { - 1 } \\\\ & = \\frac { 1 } { \\sqrt { 2 } } \\left [ \\int _ { 0 } ^ { \\infty } s ^ { - 1 / 2 } g _ { 1 } ( s ) \\mathrm { d } s \\right ] ^ { - 1 } . \\end{align*}"}
{"id": "782.png", "formula": "\\begin{align*} e ^ { D r ^ m } ( \\beta + ( 1 - \\beta ( 1 - D ) ) r ^ m ) + \\sum _ { n = 2 } ^ { \\infty } \\prod _ { k = 0 } ^ { n - 2 } \\frac { D } { k + 1 } \\phi _ { n } ( r ) = e ^ { - D } - \\phi _ { 1 } ( r ) , \\end{align*}"}
{"id": "583.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) = & \\sum _ { \\substack { 0 \\leq l \\leq i \\\\ \\gcd ( l , i ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } C _ { \\alpha } ( i , l ) \\sum _ { \\substack { 0 \\leq m _ 1 \\leq 1 \\\\ \\gcd ( l + m _ 1 , i + 1 ) \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } C _ { \\alpha } ( 1 , m _ 1 ) \\\\ & \\Big ( f _ k ( j ) f _ k ( j + 1 ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) \\tau _ 3 ( j + 1 ) } { \\sqrt { j - i - 1 } } \\Big ) \\Big ) . \\end{align*}"}
{"id": "7603.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\frac { d x } { d t } & = 5 \\sin 1 0 t - x + y , \\\\ \\frac { d y } { d t } & = 5 \\cos 1 0 t + x + y , \\end{aligned} \\right . \\end{align*}"}
{"id": "8360.png", "formula": "\\begin{align*} u \\times v \\mapsto \\langle u | v \\rangle _ t = e ^ { t ( ( \\tilde { f } ^ { | u \\rangle } , \\tilde { f } ^ { | v \\rangle } ) _ { { } _ { \\mathfrak { J } } } + 4 \\pi ) } = e ^ { - t 4 \\pi ( \\lambda \\textrm { c o t h } \\lambda - 1 ) } \\end{align*}"}
{"id": "7481.png", "formula": "\\begin{align*} \\cap _ { i = 1 } ^ { m } C _ { i } \\subset \\cap _ { n = 0 } ^ { \\infty } \\mathrm { F i x } ( S _ { n } ) . \\end{align*}"}
{"id": "6263.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { x } f ( t ) h ( t / q ) \\left ( - \\frac { q ( 1 + q ) + t } { q ^ 2 ( 1 - q ) t } u ( t / q ) + \\frac { 1 } { q ( 1 - q ) ^ 2 t } \\right ) y ( t ) \\ , d _ q t \\\\ & = \\frac { f ( x ) } { c q ( 1 - q ) } \\sum _ { n = 0 } ^ { \\infty } q ^ { \\frac { n ( n - 3 ) } { 2 } } x ^ { n - 1 } \\left ( 1 - q + q c + q ^ { n + 2 } x \\right ) y ( q ^ n x ) = G ( x ) . \\end{align*}"}
{"id": "7664.png", "formula": "\\begin{align*} s _ 0 = 1 , s _ 1 = n H , s _ 2 = \\left [ n ^ 2 - \\binom { n } { 2 } \\right ] H ^ 2 , s _ 3 = \\left [ n ^ 3 + \\binom { n } { 3 } - 2 n \\binom { n } { 2 } \\right ] H ^ 3 . \\end{align*}"}
{"id": "6183.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\widehat { \\omega } _ { t } = \\frac { 1 } { T _ { 0 } } ( ( T _ { 0 } - t ) \\omega _ { 0 } - t \\psi ^ { \\ast } \\omega _ { N } ) \\in \\lbrack \\omega ( t ) ] _ { B } = [ \\omega _ { 0 } ] _ { B } - t c _ { 1 } ^ { B } ( M ) . \\end{array} \\end{align*}"}
{"id": "8942.png", "formula": "\\begin{align*} G [ u ] : = G ( \\cdot , u , D u ) = 0 , { \\rm o n } \\ \\partial \\Omega , \\end{align*}"}
{"id": "8721.png", "formula": "\\begin{align*} U : = g _ { x ^ { 1 } \\dot { x } ^ { 2 } } - g _ { \\dot { x } ^ { 1 } x ^ { 2 } } - g _ { \\dot { x } ^ { 1 } \\dot { x } ^ { 1 } } \\frac { d } { d t } \\left ( \\frac { \\dot { x } ^ { 1 } } { \\dot { x } ^ { 2 } } \\right ) . \\end{align*}"}
{"id": "7550.png", "formula": "\\begin{align*} \\rho ( x _ 1 , x _ 2 ) \\circ \\rho ( x _ 3 , x _ 4 ) & = \\rho ( [ x _ 1 , x _ 2 , x _ 3 ] _ \\mathfrak { g } , x _ 4 ) + \\rho ( x _ 3 , [ x _ 1 , x _ 2 , x _ 4 ] _ \\mathfrak { g } ) + \\rho ( x _ 3 , x _ 4 ) \\circ \\rho ( x _ 1 , x _ 2 ) , \\\\ \\rho ( x _ 1 , [ x _ 2 , x _ 3 , x _ 4 ] _ \\mathfrak { g } ) & = \\rho ( x _ 3 , x _ 4 ) \\circ \\rho ( x _ 1 , x _ 2 ) - \\rho ( x _ 2 , x _ 4 ) \\circ \\rho ( x _ 1 , x _ 3 ) + \\rho ( x _ 2 , x _ 3 ) \\circ \\rho ( x _ 1 , x _ 4 ) . \\end{align*}"}
{"id": "1861.png", "formula": "\\begin{align*} z A _ { 0 } ( z ) - 1 & = \\sum _ { j = 0 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { j } ( z ) , \\\\ A _ { j } ( z ) & = A _ { i } ( z ) \\ , A _ { j - i - 1 } ^ { ( i + 1 ) } ( z ) , 0 \\leq i < j \\leq p , \\end{align*}"}
{"id": "945.png", "formula": "\\begin{align*} p _ { \\varepsilon } ( \\rho ) = \\begin{cases} p ( \\rho ) & \\mbox { f o r } \\rho \\in [ 0 , \\overline { \\rho } - \\varepsilon ] \\\\ p ( \\overline { \\rho } - \\varepsilon ) + | ( \\rho - \\overline { \\rho } + \\varepsilon ) ^ { + } | ^ { \\gamma } & \\mbox { f o r } \\rho \\in ( \\overline { \\rho } - \\varepsilon , \\infty ) , \\end{cases} \\end{align*}"}
{"id": "6146.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } ( \\| u _ h ^ m \\| _ { 2 } ^ 2 - \\| u _ h ^ { m - 1 } \\| _ { 2 } ^ 2 ) + \\nu \\Delta t \\| \\nabla u _ { h } ^ { m , \\frac { 1 } { 2 } } \\| _ { 2 } ^ 2 = 0 . \\end{align*}"}
{"id": "7929.png", "formula": "\\begin{align*} M _ { \\pm } : = \\{ i \\in \\{ 1 , \\ldots , m \\} : \\pm D _ i \\cdot e > 0 \\} ; \\end{align*}"}
{"id": "820.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty f ^ p ( x ) w ( x ) d x & \\leq \\frac { \\gamma A } { ( \\beta + 1 ) ^ { p / p _ 0 } } \\int _ 0 ^ \\infty g ^ p ( x ) w ( x ) d x \\\\ & = K \\int _ 0 ^ \\infty g ^ p ( x ) w ( x ) d x , \\end{align*}"}
{"id": "6848.png", "formula": "\\begin{align*} N ( b , c ) : = \\frac { 1 } { V _ F } \\int _ { 0 } ^ { + \\infty } g \\frac { 1 } { \\sqrt { 2 \\pi c } } \\exp \\left ( - \\frac { ( g - b ) ^ 2 } { 2 c } \\right ) d g \\geq 0 , b \\in \\mathbb { R } , \\ , c > 0 . \\end{align*}"}
{"id": "3886.png", "formula": "\\begin{align*} - \\Delta _ p u _ i - \\lambda ^ i | u _ i | ^ { p - 2 } u _ i = f \\Omega , \\end{align*}"}
{"id": "8869.png", "formula": "\\begin{align*} \\nabla f \\cdot \\nabla f ( \\lambda ) = 2 \\sum _ { k = 1 } ^ N \\det \\left ( ( \\lambda I _ N - H ) _ { k | k } \\right ) ^ 2 + 4 \\sum _ { k = 1 } ^ { N - 1 } \\det \\left ( ( \\lambda I _ N - H ) _ { k | k + 1 } \\right ) ^ 2 . \\end{align*}"}
{"id": "2305.png", "formula": "\\begin{align*} ( u , v , h , g ) = ( - \\phi _ y , \\phi _ x , - \\psi _ y , \\psi _ x ) . \\end{align*}"}
{"id": "6098.png", "formula": "\\begin{align*} { \\bar \\Omega } _ { n + 1 \\ , n + 2 } = c ( p ) \\ , \\omega _ { n + 1 } \\wedge \\omega _ { n + 2 } \\end{align*}"}
{"id": "8760.png", "formula": "\\begin{align*} & [ 2 k + 2 a - 2 \\ell + 2 ] [ 2 a - 2 \\ell + 2 ] [ 2 k - 2 \\ell + 2 ] + [ 2 k + 2 a - 2 \\ell + 3 ] ^ 2 [ 2 \\ell ] - [ 2 k + 1 ] ^ 2 [ 2 \\ell ] \\\\ = & [ 2 a - 2 \\ell + 2 ] [ 2 k + 2 ] [ 2 k + 2 a + 2 ] . \\end{align*}"}
{"id": "8314.png", "formula": "\\begin{align*} \\frac { \\sigma _ { r - 1 } ( n ) } { n ^ { r - 1 } \\zeta ( r ) } = \\sum _ { m = 1 } ^ { \\infty } \\frac { c _ m ( n ) } { m ^ r } \\end{align*}"}
{"id": "8120.png", "formula": "\\begin{align*} Y ^ { ( 0 ) } ( t , x ) & = Y ^ { ( 0 , m , \\beta ) } ( t , x ) = Y _ < ( t , x ) , \\\\ Y ^ { ( 0 , \\beta ) } _ < ( t , x ) & = Y ^ { ( 0 , \\beta ) } ( t , x ) = 1 , u ^ { ( 0 , \\beta ) } _ < ( s , y ; t , x ) = g ( t - s , x - y ) \\end{align*}"}
{"id": "8777.png", "formula": "\\begin{align*} \\phi _ 1 * \\phi _ 2 ( g ' ) \\geq \\| \\phi _ 1 \\| _ { p _ 1 } ^ { p _ 1 / p _ 2 ' } \\| \\phi _ 2 \\| _ { p _ 2 } ^ { p _ 2 / p _ 1 ' } \\left ( \\int _ { G } ^ { } \\phi _ 1 ( g ) ^ { p _ 1 } \\phi _ 2 ( g ^ { - 1 } g ' ) ^ { p _ 2 } d g \\right ) ^ { 1 / p } = \\left ( \\int _ { G } ^ { } \\phi _ 1 ( g ) ^ { p _ 1 } \\phi _ 2 ( g ^ { - 1 } g ' ) ^ { p _ 2 } d g \\right ) ^ { 1 / p } \\end{align*}"}
{"id": "4194.png", "formula": "\\begin{align*} F _ \\ell ^ { ( \\iota ) } ( L , U ) f _ { m } ^ { ( \\ell ) } ( x , u ) = f _ { m } ^ { ( \\ell ) } * K _ \\ell ^ { ( \\iota ) } ( x , u ) , \\end{align*}"}
{"id": "4714.png", "formula": "\\begin{align*} H _ { l , r } ^ { + } = \\begin{cases} H _ l H _ { l + 1 } \\cdots H _ { r } & \\hbox { i f } l \\leq r , \\\\ H _ l H _ { l - 1 } \\cdots H _ { r } & \\hbox { i f } l > r , \\end{cases} \\end{align*}"}
{"id": "3929.png", "formula": "\\begin{align*} V a r ( \\hat { \\pi } _ { h , n } ( x ) ) & \\le \\frac { c } { T _ n } \\big [ \\frac { 1 } { \\prod _ { l = 1 } ^ d h _ l } ( \\delta _ 1 + \\Delta _ n ) + \\frac { 1 } { \\prod _ { l \\ge k _ 0 + 1 } h _ l } \\delta _ 1 ^ { 1 - \\frac { k _ 0 } { 2 } } + \\delta _ 2 ^ { 1 - \\frac { d } { 2 } } + D + \\frac { 1 } { ( \\prod _ { l = 1 } ^ d h _ l ) ^ 2 } e ^ { - \\rho D } \\big ] . \\end{align*}"}
{"id": "3678.png", "formula": "\\begin{align*} \\mathcal { R } = 2 K _ { r s } + 2 ( n - 2 ) K _ { r i } + 2 ( n - 2 ) K _ { i s } + ( n - 2 ) ( n - 3 ) K _ { i j } \\end{align*}"}
{"id": "2462.png", "formula": "\\begin{align*} \\dim \\left [ \\gg ^ { \\mu } , x \\right ] = \\frac { \\dim \\gg ^ { \\mu } } { 2 } . \\end{align*}"}
{"id": "7121.png", "formula": "\\begin{align*} & \\gcd ( F _ n , F _ m ) = F _ { \\gcd ( n , m ) } \\mbox { f o r a l l $ m , n \\ge 0 $ , $ ( m , n ) \\ne ( 0 , 0 ) $ } ; \\\\ & F _ { n + m } = F _ { n - 1 } F _ m + F _ { n } F _ { m + 1 } \\mbox { f o r a l l $ m \\ge 0 $ a n d $ n \\ge 1 $ } ; \\\\ & F _ n ^ 2 - F _ { n - r } F _ { n + r } = ( - 1 ) ^ { n - r } F _ r ^ 2 \\mbox { f o r a l l $ n \\ge r \\ge 0 $ } \\mbox { ( C a t a l a n i d e n t i t y ) } ; \\\\ & F _ n ^ 2 - F _ { n - 1 } F _ { n + 1 } = ( - 1 ) ^ { n - 1 } \\mbox { f o r a l l $ n \\ge 1 $ } \\mbox { ( C a s s i n i i d e n t i t y ) } . \\end{align*}"}
{"id": "1246.png", "formula": "\\begin{align*} Q ( T , x ) & = \\prod _ { j = 1 } ^ { l ( T ) } \\mathcal { H } _ j ^ { n ( T , j ) } . \\end{align*}"}
{"id": "4192.png", "formula": "\\begin{align*} g _ { m } ^ { ( \\ell ) } : = \\chi _ { B _ { 3 R } ^ { d _ { \\mathrm { C C } } } ( 0 ) } F _ \\ell ^ { ( \\iota ) } ( L , U ) f _ { m } ^ { ( \\ell ) } \\end{align*}"}
{"id": "8269.png", "formula": "\\begin{align*} A _ { 0 } = \\frac { \\cosh ( k ' a / 2 ) } { \\sqrt { \\frac { a } { 2 } + a \\cosh ^ { 2 } ( k ' a / 2 ) - \\frac { 3 \\sinh ( k ' a / 2 ) \\cosh ( k ' a / 2 ) } { k ' } } } \\end{align*}"}
{"id": "5903.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\sup _ { V \\in \\mathbb { V } _ { n , k } } \\mathbb { E } \\Big \\lVert \\sum _ { j = 1 } ^ n n ^ { 1 / p } \\frac { Z _ j } { ( \\lVert Z ^ { ( n ) } \\rVert _ p ^ p + W _ n ) ^ { 1 / p } } V _ { \\bullet , j } \\Big \\rVert _ 2 < \\infty , \\end{align*}"}
{"id": "458.png", "formula": "\\begin{align*} \\frac { d \\mu } { d m } ( \\xi ) = \\lim _ { r \\uparrow 1 } \\Re \\frac { 1 + \\eta _ { \\mu } ( r \\ , \\overline { \\xi } ) } { 1 - \\eta _ { \\mu } ( r \\ , \\overline { \\xi } ) } , \\quad \\xi \\in \\mathbb { T } . \\end{align*}"}
{"id": "8679.png", "formula": "\\begin{align*} \\sigma _ { H T } ( x ) = \\frac { 1 } { v o l ( B ^ n ( 1 ) ) } \\int _ { F ( x , y ) < 1 } \\det ( g _ { i j } ( x , y ) ) d y . \\end{align*}"}
{"id": "713.png", "formula": "\\begin{align*} [ u ] _ { \\alpha } : = \\sup _ { M ^ 2 _ T } \\left \\{ \\dfrac { | u ( p , t ) - u ( p ' , t ' ) | } { d ( p , p ' ) ^ { \\alpha } + | t - t ' | ^ { \\alpha / 2 } } \\right \\} < \\infty , \\end{align*}"}
{"id": "5334.png", "formula": "\\begin{align*} A ^ * \\varphi ( x ) = \\int _ { \\R ^ d } \\varphi ( x - y ) \\nu ( d y ) \\end{align*}"}
{"id": "8302.png", "formula": "\\begin{align*} \\hat { H } \\phi _ { n } ^ { \\pm } ( x ) = E _ { n } ^ { \\pm } \\phi _ { n } ^ { \\pm } ( x ) = E _ { n \\pm 1 } ^ { \\mp } \\phi _ { n } ^ { \\pm } ( x ) , \\end{align*}"}
{"id": "4790.png", "formula": "\\begin{align*} \\left ( \\varrho , u \\right ) | _ { t = 0 } = \\left ( \\varrho _ { 0 } ( x ) , u _ { 0 } ( x ) \\right ) , \\ x \\in \\R ^ { d } . \\end{align*}"}
{"id": "1245.png", "formula": "\\begin{align*} \\mathcal { H } _ { l ( T ) } & = x - 1 , \\\\ \\mathcal { H } _ j & = x - ( c _ j + 1 ) - \\dfrac { c _ j } { \\mathcal { H } _ { j + 1 } } ( \\forall j \\in \\overline { 2 , l ( T ) - 1 } ) , \\\\ \\mathcal { H } _ 1 & = x - c _ 1 - \\dfrac { c _ 1 } { \\mathcal { H } _ 2 } , \\end{align*}"}
{"id": "8415.png", "formula": "\\begin{gather*} \\check h ( \\tau ) : = h _ 0 ( 4 \\tau ) + h _ 1 ( 4 \\tau ) = \\sum _ D C _ \\phi ( D ) q ^ { - D } , \\end{gather*}"}
{"id": "3945.png", "formula": "\\begin{align*} | \\tilde { k } ( t , s ) | & \\le \\frac { c } { t - s } \\frac { 1 } { \\prod _ { l \\ge 3 } h _ l ^ * } \\int _ { \\R ^ d } \\prod _ { m = 1 } ^ 2 K _ { h _ m ^ * } ( x _ m - z _ m ) \\prod _ { l = 3 } ^ d K _ { h _ { \\sigma ( l ) } ^ * } ( x _ { \\sigma ( l ) } - y ^ l _ { \\sigma ( l ) } ) \\\\ & \\times \\int _ { \\R ^ { d ( d - 2 ) } } \\prod _ { l = 4 } ^ d p _ { w _ l - w _ { l - 1 } } ( y ^ { l - 1 } , y ^ l ) p _ { s - w _ d } ( y ^ d , z ) \\pi ( y ^ 3 ) d z d y ^ 3 . . . d y ^ d . \\end{align*}"}
{"id": "934.png", "formula": "\\begin{align*} \\Xi _ n ( \\sigma ) : = [ \\theta _ { i j } ] _ { 1 \\leq i , j \\leq n } \\end{align*}"}
{"id": "8265.png", "formula": "\\begin{align*} \\psi _ { B _ { k } } = B _ { k } \\left [ \\frac { \\sin ( k x ) } { \\sin ( k a / 2 ) } - \\frac { \\sinh ( k ' x ) } { \\sinh ( k ' a / 2 ) } \\right ] , \\end{align*}"}
{"id": "1055.png", "formula": "\\begin{align*} W ( t ) = \\sum _ { n = \\ell + 1 } ^ \\infty \\widetilde { B _ 0 } S _ n ( t ) \\in \\mathcal { C } _ 1 , \\end{align*}"}
{"id": "8091.png", "formula": "\\begin{align*} g ^ { - 1 } = ( g ^ { i j } ) = \\begin{pmatrix} 1 / a ^ { 2 } \\sin ^ { 2 } ( \\chi ) & 0 & 0 \\\\ 0 & 1 / \\cos ^ { 2 } ( \\chi ) & 0 \\\\ 0 & 0 & 1 / ( a ^ { 2 } \\cos ^ { 2 } ( \\chi ) + \\sin ^ { 2 } ( \\chi ) ) \\end{pmatrix} . \\end{align*}"}
{"id": "6151.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\| u ^ { \\Delta t } _ { h } - v ^ { \\Delta t } _ { h } \\| _ { 2 } ^ 2 \\ , d t & = \\sum _ { m = 1 } ^ N \\| u _ h ^ { m } - u _ { h } ^ { m - 1 } \\| _ { 2 } ^ 2 \\int _ { t _ { m - 1 } } ^ { t _ m } \\left ( \\frac { 1 } { 2 } - \\frac { t - t _ { m - 1 } } { \\Delta t } \\right ) ^ 2 d t \\\\ & \\leq \\frac { \\Delta t } { 1 2 } \\sum _ { m = 1 } ^ N \\| u _ { h } ^ m - u _ { h } ^ { m - 1 } \\| _ { 2 } ^ 2 . \\end{align*}"}
{"id": "6189.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\omega _ { M } : = \\psi ^ { \\ast } \\omega _ { N } + \\frac { \\sqrt { - 1 } } { 2 \\pi } \\frac { \\varepsilon _ { 0 } } { r _ { S ^ { 1 } } ^ { 2 } } \\sum _ { i , j = 1 } ^ { 2 } ( \\delta _ { i j } - \\frac { \\overline { z } _ { i } z _ { j } } { r _ { S ^ { 1 } } ^ { 2 } } ) d z _ { i } \\wedge d \\overline { z } _ { j } \\end{array} \\end{align*}"}
{"id": "1278.png", "formula": "\\begin{align*} \\Psi ( E _ { j + 1 } ( x , d - 1 ) ) - \\Psi ( E _ j ( x , d - 1 ) ) & = f _ j \\ , \\sqrt { d - 1 } \\ , , \\end{align*}"}
{"id": "8681.png", "formula": "\\begin{align*} d V _ { \\min } = \\sigma _ { \\min } ( x ) d x , \\end{align*}"}
{"id": "6767.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\tau ( n ) q ^ n = q \\prod _ { j = 1 } ^ \\infty ( 1 - q ^ j ) ^ { 2 4 } \\end{align*}"}
{"id": "4876.png", "formula": "\\begin{align*} \\mathfrak { J } _ { i } ( n ) = \\sum \\limits _ { m _ { 1 } + \\cdots + m _ { 5 } = n \\atop U _ { i } ^ { 3 } < m _ { 2 } , m _ { 3 } \\leq 8 U _ { i } ^ { 3 } , V _ { i } ^ { 3 } < m _ { 4 } , m _ { 5 } \\leq 8 V _ { i } ^ { 3 } } ( m _ { 2 } \\ldots m _ { 5 } ) ^ { - 2 / 3 } \\end{align*}"}
{"id": "4535.png", "formula": "\\begin{align*} \\sigma _ { m , n } ( f ; x , y ) = \\frac { 1 } { \\pi ^ { 2 } } \\int _ { 0 } ^ { \\pi } \\int _ { 0 } ^ { \\pi } h _ { x , y } ( t _ { 1 } , t _ { 2 } ) F _ { m , n } ( t _ { 1 } , t _ { 2 } ) \\ , d t _ { 1 } d t _ { 2 } , \\end{align*}"}
{"id": "8406.png", "formula": "\\begin{gather*} \\overline { \\varrho _ m ( \\gamma , \\upsilon ) } \\theta _ m | _ { \\frac 1 2 , m } ( \\gamma , \\upsilon ) = \\theta _ m \\end{gather*}"}
{"id": "4606.png", "formula": "\\begin{align*} & ( e _ 1 , f _ 1 ) = - 1 + \\frac { 1 } { \\alpha } , ( e _ 2 , f _ 2 ) = - 1 + \\alpha , ( e _ 3 , f _ 3 ) = 1 , \\\\ & ( \\psi _ { \\alpha _ 1 , \\beta _ 1 , \\gamma _ 1 } , \\psi _ { \\alpha _ 2 , \\beta _ 2 , \\gamma _ 2 } ) = - 2 \\epsilon _ { \\alpha _ 1 , \\alpha _ 2 } \\epsilon _ { \\beta _ 1 , \\beta _ 2 } \\epsilon _ { \\gamma _ 1 , \\gamma _ 2 } . \\end{align*}"}
{"id": "8598.png", "formula": "\\begin{align*} E _ 0 = E , E _ { j + 1 } = E _ j \\cap ( 2 ^ { - k _ j } \\oplus E _ j ) , j = 0 , 1 , \\ldots , l - 1 . \\end{align*}"}
{"id": "5840.png", "formula": "\\begin{align*} \\sup _ { N \\leq k \\leq M } a _ k \\bigg ( \\sum _ { i = k } ^ M b _ i \\bigg ) ^ \\alpha \\approx \\sup _ { N \\leq k \\leq M } a _ k b _ k ^ \\alpha . \\end{align*}"}
{"id": "3882.png", "formula": "\\begin{align*} \\| \\hat { u } _ n \\| _ { p - 1 } = 1 , \\ \\sup _ { n \\in \\mathbb N } \\| a _ n \\| _ \\infty < \\infty , \\ \\| \\hat { f } _ n \\| _ { L ^ 1 ( \\Omega ) } + \\| \\hat { g } _ n \\| _ \\infty \\to 0 \\hbox { a s } n \\to + \\infty \\end{align*}"}
{"id": "4576.png", "formula": "\\begin{align*} \\Psi \\left ( F _ * ^ e \\left ( c ( I ^ { [ p ^ e ] } : I ) \\right ) \\right ) = ( F _ * ^ e c \\cdot \\Psi ) ( I ^ { [ p ^ e ] } : I ) \\not \\subseteq J . \\end{align*}"}
{"id": "4920.png", "formula": "\\begin{align*} R ( W , U , Y , Z ) = g ( R ( W , U ) Y , Z ) . \\end{align*}"}
{"id": "4691.png", "formula": "\\begin{align*} & ( a ; q ) _ n = \\begin{cases} 1 , & \\\\ ( 1 - a ) ( 1 - a q ) \\cdots ( 1 - a q ^ { n - 1 } ) , & \\end{cases} \\\\ & ( a ; q ) _ \\infty = \\lim _ { n \\to \\infty } ( a ; q ) _ n . \\end{align*}"}
{"id": "7692.png", "formula": "\\begin{align*} e v _ { \\mathcal { S } } ( B ) = A _ { \\mathcal { R } _ { B } } \\in \\mathcal { B } ^ * ( X ) , \\end{align*}"}
{"id": "7170.png", "formula": "\\begin{align*} [ u + D \\mathcal { L } ( V , g ) , v + D \\mathcal { L } ( V , g ) ] = u _ 0 v + D \\mathcal { L } ( V , g ) \\end{align*}"}
{"id": "7561.png", "formula": "\\begin{align*} S ^ m ( f _ 1 , \\dots , f _ m ) ( x ) & \\ge C _ m | x | ^ { 1 - m } \\int _ { B ^ { m - 1 } ( 0 , \\frac 1 { 3 0 0 m } ) } | \\tilde { u } | ^ { - \\frac 1 p } \\left ( - \\log ( | \\tilde { u } | ) \\right ) ^ { - \\frac { 2 } { p } } d \\tilde { u } \\\\ & \\gtrsim \\begin{cases} | x | ^ { 1 - m } & \\frac 1 p = m - 1 , \\\\ \\infty & \\frac 1 p > m - 1 . \\end{cases} \\end{align*}"}
{"id": "7314.png", "formula": "\\begin{align*} e ^ { 2 ( \\frac { \\gamma } { 2 } + J + \\frac { 1 } { 2 } { \\sf c } _ 1 ) \\tau } \\| w _ 2 ^ \\bot \\| _ \\rho ^ 2 - \\underbrace { e ^ { 2 ( \\frac { \\gamma } { 2 } + J + \\frac { 1 } { 2 } { \\sf c } _ 1 ) \\tau } \\| w _ 2 ^ \\bot \\| _ \\rho ^ 2 | _ { \\tau = - \\log T } } _ { = T ^ { - 2 ( \\frac { \\gamma } { 2 } + J + \\frac { 1 } { 2 } { \\sf c } _ 1 ) } \\| w _ 0 ^ \\bot \\| _ \\rho ^ 2 } \\lesssim \\int _ 0 ^ t ( T - t ) ^ { - 1 + { \\sf c } _ 1 } d t \\lesssim T ^ { { \\sf c } _ 1 } . \\end{align*}"}
{"id": "5872.png", "formula": "\\begin{align*} \\sum _ { i = - \\infty } ^ k 2 ^ { - i \\frac { q } { p - q } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \\frac { p q } { p - q } } \\leq \\bigg ( \\sum _ { i = - \\infty } ^ { k } 2 ^ { - i \\frac { r } { p - r } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \\frac { p r } { p - r } } \\bigg ) ^ { \\frac { q ( p - r ) } { r ( p - q ) } } . \\end{align*}"}
{"id": "8451.png", "formula": "\\begin{align*} \\mathcal { H } ^ s _ { \\infty } ( F ) & \\ll \\liminf _ { N \\to \\infty } \\sum _ { n = N } ^ { \\infty } \\beta _ 1 ^ n \\beta _ 2 ^ n \\beta _ 2 ^ { - n ( 1 - ( 1 + \\theta _ 1 ) \\log _ { \\beta _ 2 } \\beta _ 1 ) } \\beta _ 1 ^ { - n ( 1 + \\theta _ { 1 } ) s } \\\\ & = \\liminf _ { N \\to \\infty } \\sum _ { n = N } ^ { \\infty } \\beta _ 2 ^ { n \\left ( 2 + \\theta _ 1 - ( 1 + \\theta _ { 1 } ) s \\right ) \\log _ { \\beta _ 2 } \\beta _ 1 } \\\\ & = 0 . \\end{align*}"}
{"id": "7483.png", "formula": "\\begin{align*} x _ { * } \\in \\cap _ { j = 1 } ^ { k } \\mathrm { F i x } ( T _ { j } ) \\subset \\mathrm { F i x } ( Q ) = \\cap _ { n = 1 } ^ { j _ { f } } \\mathrm { F i x } ( S _ { n } ) = \\cap _ { i = 1 } ^ { m } C _ { i } . \\end{align*}"}
{"id": "1741.png", "formula": "\\begin{align*} g ^ c _ { k - 2 } ( z , \\omega _ 1 , \\widetilde \\omega _ 1 ) = \\int _ { c \\cdot C } \\frac { - e ^ { ( z + \\overline \\omega _ 1 ) s } \\ , s ^ { k - 2 } } { ( e ^ { \\omega _ 1 s } - 1 ) ( e ^ { \\widetilde \\omega _ 1 s } - 1 ) } \\ d s , \\end{align*}"}
{"id": "2550.png", "formula": "\\begin{align*} U _ 3 = \\dfrac { 3 } { 4 } Y - \\dfrac { 1 } { 2 } T _ 3 \\ \\ \\ , \\ \\ \\ V _ 3 = \\dfrac { 3 } { 4 } Y + \\dfrac { 1 } { 2 } T _ 3 \\ , \\end{align*}"}
{"id": "8177.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\| \\chi _ { S _ { t ^ \\beta } ^ c } J U _ 0 ( t ) f \\| = 0 \\end{align*}"}
{"id": "5469.png", "formula": "\\begin{align*} g _ { r , x } ( z ) & = g ( x ) \\mathrm { d i s t } ( x , z ) \\leq \\frac { r } { 2 } , \\\\ g _ { r , x } & = g X \\setminus B ( x , r ) . \\end{align*}"}
{"id": "2048.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : O m e g a i n f t y } \\Omega _ \\infty \\left ( f ^ N , f \\right ) : = \\mathcal { W } _ p { \\left ( \\mathrm { L a w } ( \\mu _ { \\mathcal { X } ^ N } ) , \\delta _ f \\right ) } \\underset { N \\to + \\infty } { \\longrightarrow } 0 , \\end{align*}"}
{"id": "4413.png", "formula": "\\begin{align*} S _ { d , i } = \\{ i , d + i , 2 d + i , \\ldots , ( 4 k - 1 ) d + i \\} . \\end{align*}"}
{"id": "2540.png", "formula": "\\begin{align*} R i c ( X _ 1 , Y _ 1 ) = - a _ 1 a _ 3 e ^ { - 6 x _ 2 } - a _ 1 a _ 3 e ^ { - 2 x _ 2 } - a _ 1 a _ 3 e ^ { - 4 x _ 2 } + a _ 1 a _ 3 - 2 a _ 2 a _ 4 e ^ { - 2 x _ 2 } - a _ 2 a _ 4 . \\end{align*}"}
{"id": "3552.png", "formula": "\\begin{align*} r _ l = \\sum _ { \\tau \\in Y _ l } 2 m _ { \\tau } . \\end{align*}"}
{"id": "8425.png", "formula": "\\begin{gather*} y ^ 2 = x ^ 3 - 1 3 3 9 2 D ^ 2 x - 1 0 8 0 4 3 2 D ^ 3 \\end{gather*}"}
{"id": "4455.png", "formula": "\\begin{align*} ( R f ^ { i _ 1 \\dots i _ k } ) _ { p _ 1 q _ 1 \\dots p _ { m - k } q _ { m - k } } = 0 \\mbox { i n } U . \\end{align*}"}
{"id": "7216.png", "formula": "\\begin{align*} \\| U \\| _ 2 : = \\sup _ { ( x , t ) \\in \\R \\times [ 0 , T ] } w ( | x | , t ) | U ( x , t ) | . \\end{align*}"}
{"id": "1339.png", "formula": "\\begin{align*} f _ \\infty ( x , v ) = Z ^ { - 1 } e ^ { - \\Phi ( x ) - \\frac { | v | ^ 2 } { 2 } } , Z = \\int e ^ { - \\Phi ( x ) - \\frac { | v | ^ 2 } { 2 } } \\d z . \\end{align*}"}
{"id": "8346.png", "formula": "\\begin{align*} \\widetilde { \\varphi } = { w _ 1 } ^ + \\tilde { f } _ + + { w _ 1 } ^ - \\tilde { f } _ - + w _ { r ^ { - 2 } } \\tilde { f } _ { 0 + } + w _ { r ^ { 2 } } \\tilde { f } _ { 0 - } . \\end{align*}"}
{"id": "3788.png", "formula": "\\begin{align*} { } _ { } ^ z T _ { k , j ; n , l } ^ { S , i ; \\mu ; m , 2 } ( B ) ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\phi _ { m ; - 1 0 M _ t , c _ t } ( t - s ) \\big [ ( t - s ) K _ { k ; n } ^ { \\mu } ( 1 ) ( y , \\zeta , \\omega ) + \\widetilde { K } _ { k ; n } ^ { \\mu } ( 1 ) ( y , \\zeta ) \\big ] f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "5109.png", "formula": "\\begin{align*} \\psi ^ 2 _ { n , 1 } ( s , \\eta _ n ( t ) ) & = [ ( \\eta _ n ( t ) - \\eta _ n ( s ) ) ^ \\alpha - ( \\eta _ n ( t ) - s ) ^ \\alpha ] ^ 2 \\\\ & \\le ( \\eta _ n ( t ) - s ) ^ { 2 \\alpha - 2 } ( s - \\eta _ n ( s ) ) ^ 2 \\le ( \\delta / 2 ) ^ { 2 \\alpha - 2 } n ^ { - 2 } , \\end{align*}"}
{"id": "8044.png", "formula": "\\begin{align*} J _ W : = \\left ( \\exp \\left ( 2 \\pi i q _ 1 \\right ) , \\ldots , \\exp \\left ( 2 \\pi i q _ N \\right ) \\right ) \\in ( \\mathbb C ^ \\times ) ^ N . \\end{align*}"}
{"id": "4411.png", "formula": "\\begin{align*} a _ 1 \\left ( \\sum _ { k = [ m _ n ] + 1 } ^ { [ m _ { n + 1 } ] } | \\hat { g } ( k ) | ^ 2 \\right ) ^ { 1 / 2 } \\leq \\frac { 1 } { 2 ^ { [ m _ { n + 1 } ] - [ m _ n ] } } \\sum _ { \\Theta _ n \\in \\Delta _ n } M _ 1 ( g _ { \\Theta _ n } , s _ n ) \\leq M _ 1 ( g _ { { \\tilde { \\Theta } } _ n } , s _ n ) . \\end{align*}"}
{"id": "5367.png", "formula": "\\begin{align*} M _ { \\geq \\kappa } ( H ^ s \\to H ^ { - s } ) \\vcentcolon = \\{ \\ , a \\in M ( H ^ s \\to H ^ { - s } ) \\ , ; \\ , a ( f ^ 2 ) \\geq \\kappa \\norm { f } _ { L ^ 2 ( \\R ^ n ) } ^ 2 \\forall f \\in C _ c ^ \\infty ( \\R ^ n ) \\ , \\} . \\end{align*}"}
{"id": "6935.png", "formula": "\\begin{align*} I - \\Sigma \\Sigma ^ * = q q ^ * , \\end{align*}"}
{"id": "8274.png", "formula": "\\begin{align*} \\tanh ( k ' a / 2 ) = k ' a / 2 \\end{align*}"}
{"id": "5495.png", "formula": "\\begin{align*} \\omega _ { t + 1 } = \\omega _ t + \\alpha _ t \\left ( c _ t ' + \\gamma \\phi ( s _ t '' , a _ t '' ) ^ T \\omega _ t - \\phi ( s _ t ' , a _ t ' ) ^ T \\omega _ t \\right ) \\phi ( s _ t ' , a _ t ' ) , \\end{align*}"}
{"id": "3397.png", "formula": "\\begin{align*} E _ { 3 , H _ { a b } } ^ { 0 , 2 } = E _ { 3 , H _ a } ^ { 0 , 2 } \\times E _ { 3 , H _ b } ^ { 0 , 2 } \\end{align*}"}
{"id": "7920.png", "formula": "\\begin{align*} X _ \\omega : = [ \\mathcal U _ \\omega / K ] , \\end{align*}"}
{"id": "8268.png", "formula": "\\begin{align*} k = 0 , k ' = \\frac { \\sqrt { 2 } } { 2 \\hslash \\sqrt { \\beta / 3 } } . \\end{align*}"}
{"id": "4386.png", "formula": "\\begin{align*} \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ \\Psi = - \\infty \\} } | f | ^ 2 = \\lim _ { p \\rightarrow 0 + 0 } \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ p \\Psi < 2 \\log r _ 1 \\} } | f | ^ 2 \\ge G ( 0 ; \\Psi , I _ + ( 2 a _ o ^ f ( \\Psi ) \\Psi ) _ o , f ) . \\end{align*}"}
{"id": "3565.png", "formula": "\\begin{align*} w ( \\pi ) = \\begin{cases} ( 1 + \\delta s _ 1 ) ( 1 + t _ 1 ) ^ { c _ 1 / 2 } , & q \\equiv 1 \\pmod 4 , \\\\ ( 1 + v _ 1 ) ^ { c _ 1 } , & q \\equiv 3 \\pmod 4 , \\end{cases} \\end{align*}"}
{"id": "1692.png", "formula": "\\begin{align*} ( x y ) z = x ( y z ) + ( - 1 ) ^ { \\vert y \\vert \\vert z \\vert } x ( z y ) , \\end{align*}"}
{"id": "3102.png", "formula": "\\begin{align*} \\eta ^ 2 ( T ) = | T | ^ { 2 { m } / 3 } \\Vert \\lambda _ h u _ { \\mathrm { n c } } \\Vert ^ 2 _ { L ^ 2 ( T ) } + | T | ^ { 1 / 3 } \\sum _ { F \\in \\mathcal { F } ( T ) } \\Vert [ { D } ^ { m } _ { \\mathrm { p w } } u _ { \\mathrm { n c } } ] _ F \\times \\nu _ F \\Vert ^ 2 _ { L ^ 2 ( F ) } \\end{align*}"}
{"id": "3833.png", "formula": "\\begin{align*} E r r U ^ { 1 ; 0 } _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\sum _ { a = 1 , 2 } ( - 1 ) ^ a \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i \\widetilde { \\Phi } ^ 1 _ { \\mu _ 1 , \\mu _ 2 } ( \\xi , \\eta , \\sigma ; t _ a , X ( t _ a ) , v ) } ( \\Phi ^ 1 _ { \\mu _ 1 , \\mu _ 2 } ( \\xi , \\eta , \\sigma ; V ( t _ a ) , v ) ) ^ { - 1 } \\end{align*}"}
{"id": "671.png", "formula": "\\begin{align*} \\left | \\left . \\int u _ { 1 , t } d \\nu ^ 2 _ { t } \\ , \\right | ^ { t = \\bar s } _ { t = s } \\right | \\le C ( C _ 0 , D , Y ) \\varepsilon ( 1 + { s ' - s } ) . \\end{align*}"}
{"id": "5331.png", "formula": "\\begin{align*} E ^ 2 _ j = \\{ Z _ { i _ j , t } ( t , B ( x _ { i _ j } , \\epsilon ) ) \\geq p ( t , \\epsilon ) | \\pi _ { i _ j } | / 2 \\} , \\end{align*}"}
{"id": "4581.png", "formula": "\\begin{align*} C _ R ( J ) R [ x ] = C _ { R [ x ] } ( J [ x ] ) = C _ { R [ x ] } ( J ' ) \\end{align*}"}
{"id": "3657.png", "formula": "\\begin{align*} \\lambda = m ^ { \\frac { s / p - 1 } { 1 + N - s N / p } } . \\end{align*}"}
{"id": "5845.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { k = N } ^ M a _ k ^ q v _ k ^ q \\bigg ) ^ { \\frac { 1 } { q } } \\leq C \\bigg ( \\sum _ { k = N } ^ M a _ k ^ p w _ k ^ p \\bigg ) ^ { \\frac { 1 } { p } } \\end{align*}"}
{"id": "495.png", "formula": "\\begin{gather*} f _ i ( v ) = \\begin{cases} s _ i v & s _ i v > v s _ i v \\le _ L w \\\\ 0 & \\end{cases} e _ i ( v ) = \\begin{cases} s _ i v & s _ i v < v \\\\ 0 & \\end{cases} \\end{gather*}"}
{"id": "3007.png", "formula": "\\begin{align*} T = Q ( z + f ) = \\sum _ { t = 0 } ^ { \\ell } ( z + f ) ^ t Q ^ { ( t ) } = \\sum _ { t = 0 } ^ { \\ell } \\sum _ { u = 0 } ^ { t } \\binom { t } { u } z ^ u f ^ { t - u } Q ^ { ( t ) } = \\sum _ { u = 0 } ^ { \\ell } z ^ u T _ u \\ , \\end{align*}"}
{"id": "5541.png", "formula": "\\begin{align*} \\mathbb { E } [ \\ell ] = \\dfrac { \\ell } { 2 ^ \\ell } \\left ( 2 ^ { \\ell - 1 } + \\dbinom { \\ell - 1 } { ( \\ell - 1 ) / 2 } \\right ) . \\end{align*}"}
{"id": "3062.png", "formula": "\\begin{align*} \\mathrm { ( i ) } ~ ~ \\mathrm { E } [ \\mathbf { Y } | \\boldsymbol { a } < \\mathbf { Y } \\leq \\boldsymbol { b } ] = \\boldsymbol { \\mu } + \\frac { \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } \\boldsymbol { \\delta } } { F _ { \\mathbf { Z } } ( \\boldsymbol { \\xi _ { a } } , \\boldsymbol { \\xi _ { b } } ) } , \\end{align*}"}
{"id": "3854.png", "formula": "\\begin{align*} & \\tilde { N } = N ^ 1 \\cup N ^ 2 , \\tilde { E } = E ^ 1 \\cup E ^ 2 , \\tilde { h } _ 0 = h _ 0 ^ 1 \\cup h _ 0 ^ 2 \\tilde { H } = H ^ 1 \\cup H ^ 2 , \\\\ & \\tilde { L } : \\tilde { N } \\to \\{ 1 , . . . , d \\} \\mbox { s u c h t h a t } \\tilde { L } | _ { N ^ i } = L ^ i . \\end{align*}"}
{"id": "6865.png", "formula": "\\begin{align*} b ^ * = \\beta ( c ^ * ) \\geq g _ 0 > 0 . \\end{align*}"}
{"id": "8969.png", "formula": "\\begin{align*} \\left \\vert \\int _ \\Omega R \\xi _ E ^ 2 u ^ 2 d V _ g \\right \\vert = \\left \\vert \\int _ { \\Omega \\cap E } R \\xi _ E ^ 2 u ^ 2 d V _ g \\right \\vert \\sim \\left \\vert \\int _ { \\tilde M } \\left ( R \\xi _ E ^ 2 u ^ 2 \\right ) ^ { \\sim } d V _ { \\tilde g } \\right \\vert . \\end{align*}"}
{"id": "1135.png", "formula": "\\begin{align*} u _ t = w _ t + 2 \\alpha s _ t , ~ ~ ~ ~ ~ t = 1 , 2 , \\ldots , n . \\end{align*}"}
{"id": "7649.png", "formula": "\\begin{align*} \\alpha ^ 1 _ { [ v ] } F ^ 1 _ { [ v ] } + \\cdots + \\alpha ^ n _ { [ v ] } F ^ n _ { [ v ] } = 0 , [ v ] \\in S , \\end{align*}"}
{"id": "3644.png", "formula": "\\begin{align*} r \\alpha = \\frac { r - 1 } { 1 + \\frac 1 N - \\frac 1 p } \\geq p . \\end{align*}"}
{"id": "2040.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : D 0 e x p r e s s i o n } D _ p ( \\mu , \\nu ) : = { \\left ( \\sum _ { k = 1 } ^ { + \\infty } \\frac { 1 } { 2 ^ k } | \\langle \\mu - \\nu , \\varphi ^ k \\rangle | ^ p \\right ) } ^ { 1 / p } . \\end{align*}"}
{"id": "3846.png", "formula": "\\begin{align*} \\partial _ { ( 1 ) } = & \\partial _ \\mu , \\\\ \\partial _ { ( a _ 1 , . . . , a _ { k - 1 } , a _ k ) } = & \\begin{cases} \\nabla _ { v _ { a _ k } } \\cdot \\partial _ { ( a _ 1 , . . . , a _ { k - 1 } ) } & a _ k \\leq \\max \\{ a _ 1 , . . . , a _ { k - 1 } \\} , \\\\ \\partial _ \\mu \\cdot \\partial _ { ( a _ 1 , . . . , a _ { k - 1 } ) } & a _ k > \\max \\{ a _ 1 , . . . , a _ { k - 1 } \\} . \\end{cases} \\end{align*}"}
{"id": "2545.png", "formula": "\\begin{align*} \\begin{array} { l l } \\nabla _ { e _ i } e _ j = 0 ~ . \\end{array} \\end{align*}"}
{"id": "7616.png", "formula": "\\begin{align*} S ( r ) < \\sum _ { k = 2 r } ^ { \\infty } \\dfrac { 1 } { k \\pi ^ k } \\Bigl ( \\dfrac { k } { 2 } \\Bigr ) ^ r e ^ { \\frac { r ( r - 1 ) } { k } } \\dfrac { 1 } { r ^ k } \\leq \\dfrac { e ^ { \\frac { r - 1 } { 2 } } } { 2 ^ r } \\sum _ { k = 2 r } ^ { \\infty } \\dfrac { k ^ { r - 1 } } { ( \\pi r ) ^ k } \\ \\ \\Bigl ( , e ^ { \\frac { r ( r - 1 ) } { k } } \\leq e ^ { \\frac { r - 1 } { 2 } } \\forall \\ k \\geq 2 r \\Bigr ) . \\end{align*}"}
{"id": "1329.png", "formula": "\\begin{align*} f _ t ( x , v ) = e ^ { - t } f _ 0 ( x - v t , v ) + \\int _ 0 ^ t e ^ { - ( t - s ) } \\rho ( x - v ( t - s ) ) M _ { T ( x - v ( t - s ) ) } ( v ) \\d s , \\end{align*}"}
{"id": "1398.png", "formula": "\\begin{align*} A [ X _ 1 , X _ 2 ] = [ X _ 1 , X _ 2 ] \\begin{pmatrix} \\Lambda _ 1 & \\ \\\\ \\ & \\Lambda _ 2 \\end{pmatrix} , \\end{align*}"}
{"id": "7599.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { d x } { d t } & = g _ 1 ( t ) + \\left ( g _ 3 ( t ) + g _ 7 ( t ) \\right ) x + g _ 4 ( t ) y + g _ 8 ( t ) x ^ 2 + g _ 9 ( t ) x y , \\\\ \\frac { d y } { d t } & = g _ 2 ( t ) + g _ 5 ( t ) x + \\left ( g _ 6 ( t ) + g _ 7 ( t ) \\right ) y + g _ 8 ( t ) x y + g _ 9 ( t ) y ^ 2 . \\end{aligned} \\right . \\end{align*}"}
{"id": "4980.png", "formula": "\\begin{align*} n ^ { \\alpha + \\frac 1 2 } Y ^ n _ t = R ^ n _ t + \\widetilde { A } ^ n _ t + \\widetilde { C } ^ n _ t + \\widetilde { Z } ^ n _ t , \\end{align*}"}
{"id": "1844.png", "formula": "\\begin{align*} \\mathbf { c } _ { k } : = \\begin{cases} ( 1 , 1 , \\ldots , 1 ) , & 1 \\leq k \\leq p , \\\\ [ 0 . 5 e m ] ( - a _ { k - p - 1 } ^ { ( p ) } , 1 , \\ldots , 1 ) , & k \\geq p + 1 , \\end{cases} \\end{align*}"}
{"id": "6891.png", "formula": "\\begin{align*} \\lim _ { \\bar { \\rho } \\rightarrow ( { 1 } / { g _ 1 } ) ^ { - } } N ^ * ( \\bar { \\rho } ) = + \\infty . \\end{align*}"}
{"id": "5281.png", "formula": "\\begin{align*} O _ i ( P ^ { C o n t } ) = \\left \\langle \\prod _ { j \\in I _ i } \\tau _ { d _ j } ^ { ( a _ j , b _ j ) } \\sigma _ 1 ^ { k _ 1 ( i ) } \\sigma _ 2 ^ { k _ 2 ( i ) } \\sigma _ { 1 2 } \\right \\rangle ^ { \\mathbf { s } ^ { \\Gamma _ { 0 , k _ 1 ( i ) , k _ 2 ( i ) , 1 , I _ i } , o } } . \\end{align*}"}
{"id": "6106.png", "formula": "\\begin{align*} \\beta \\frac { 1 - \\varepsilon \\| b \\| } { 1 + \\varepsilon \\beta } \\le \\beta _ h : = \\inf _ { x _ h \\in X _ h \\setminus \\{ 0 \\} } \\sup _ { y _ h \\in Y _ h \\setminus \\{ 0 \\} } \\frac { b ( x _ h , y _ h ) } { \\| x _ h \\| _ { H } \\| y _ h \\| _ H } . \\end{align*}"}
{"id": "5031.png", "formula": "\\begin{align*} Q ^ { n , 4 } _ \\tau = n ^ { 2 \\alpha + 1 } \\sum _ { j = 0 } ^ { \\lfloor n \\tau \\rfloor } \\xi _ { j , n } , \\end{align*}"}
{"id": "4712.png", "formula": "\\begin{align*} \\mathfrak { S } _ n = \\mathfrak { S } _ { n - 1 } \\bigsqcup ( \\bigsqcup _ { r = 1 } ^ { n - 1 } s _ { r , n - 1 } \\mathfrak { S } _ { n - 1 } ) \\hbox { ~ ~ ~ ~ ( a d i s j o i n t u n i o n ) , } \\end{align*}"}
{"id": "5268.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ h \\# Z ( { \\bf s } ^ { \\Lambda _ { Q , j } } ) = \\prod _ { i = 1 } ^ h \\left \\langle \\prod _ { \\ell \\in I _ i } \\tau ^ { ( a _ \\ell , b _ \\ell ) } _ { d _ \\ell } \\sigma _ 1 ^ { k _ 1 ( i ) } \\sigma _ 2 ^ { k _ 2 ( i ) } \\sigma _ { 1 2 } \\right \\rangle ^ { \\mathbf { s } ^ { \\Gamma _ { 0 , k _ 1 ( i ) , k _ 2 ( i ) , 1 , I _ i } } , o } \\end{align*}"}
{"id": "2722.png", "formula": "\\begin{align*} W _ { r , \\le M } : = \\ ; & Q _ { r , \\le M } \\big ( Q _ { r , \\le M } ( h _ \\mathrm { M F } - \\mu _ + + 2 \\lambda K _ { 1 1 } ) Q _ { r , \\le M } \\big ) ^ { - 1 } Q _ { r , \\le M } \\\\ W _ { \\ell , \\le M } : = \\ ; & Q _ { \\ell , \\le M } \\big ( Q _ { \\ell , \\le M } ( h _ \\mathrm { M F } - \\mu _ + + 2 \\lambda K _ { 2 2 } ) Q _ { \\ell , \\le M } \\big ) ^ { - 1 } Q _ { \\ell , \\le M } . \\end{align*}"}
{"id": "7837.png", "formula": "\\begin{align*} \\langle \\mu ^ * ( J a P _ { \\hat { 1 } } b J ) \\hat { 1 } , \\hat { 1 } \\rangle & = { \\rm T r } ( \\mu ^ * ( J a J P _ { \\hat { 1 } } J b J ) P _ { \\hat { 1 } } ) = { \\rm T r } ( \\mu ^ * ( P _ { \\hat { 1 } } ) J b J P _ { \\hat { 1 } } J a J ) \\\\ & = \\overline { { \\rm T r } ( \\mu ( a P _ { \\hat { 1 } } b ) P _ { \\hat { 1 } } ) } = \\langle \\hat { 1 } , \\mu ( a P _ { \\hat 1 } b ) \\hat { 1 } \\rangle . \\end{align*}"}
{"id": "8131.png", "formula": "\\begin{align*} \\psi _ n = \\psi _ { n , \\textrm { i n } } + \\psi _ { n , \\textrm { o u t } } + o ( 1 ) \\end{align*}"}
{"id": "1408.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} 1 & 0 & 0 \\\\ 1 & 1 - \\delta & 0 \\\\ 0 & 0 & 1 - 2 \\delta \\end{matrix} \\right ] , \\Delta A = \\left [ \\begin{matrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & \\epsilon & 0 \\end{matrix} \\right ] . \\end{align*}"}
{"id": "1516.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - x } \\mathrm { L i } _ { p , \\lambda } ( x ) & = \\sum _ { l = 0 } ^ { \\infty } x ^ { l } \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - \\lambda ) ^ { k - 1 } ( 1 ) _ { k , 1 / \\lambda } } { ( k - 1 ) ! k ^ { p } } x ^ { k } \\\\ & = \\sum _ { n = 1 } ^ { \\infty } \\sum _ { k = 1 } ^ { n } \\frac { ( - \\lambda ) ^ { k - 1 } ( 1 ) _ { k , 1 / \\lambda } } { k ^ { p } ( k - 1 ) ! } x ^ { n } . \\end{align*}"}
{"id": "1924.png", "formula": "\\begin{align*} \\begin{aligned} a _ { n } ^ { ( k ) } & = 0 , 0 \\leq k \\leq p - 1 , n \\in \\mathbb { Z } , \\\\ a _ { n } ^ { ( p ) } & = a _ { n } , n \\in \\mathbb { Z } , \\end{aligned} \\end{align*}"}
{"id": "6385.png", "formula": "\\begin{align*} T _ h ^ { ( n ) } = \\left \\{ \\begin{array} { l l } T _ h , & \\mbox { i f $ h \\neq h _ o $ , } \\\\ U ( \\gamma _ n ) , & \\mbox { i f $ h = h _ o $ . } \\end{array} \\right . \\end{align*}"}
{"id": "1659.png", "formula": "\\begin{align*} D _ { B } : = B \\setminus \\bigcup _ { \\{ \\gamma \\in I : B \\in S _ { \\gamma } \\} } D _ { \\gamma } . \\end{align*}"}
{"id": "1629.png", "formula": "\\begin{align*} [ A , M ] = \\{ N \\in \\mathcal { D } : \\exists k \\ , ( r _ k ( N ) = A ) \\mathrm { \\ a n d \\ } N \\le M \\} , \\end{align*}"}
{"id": "7076.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { \\delta } ( t ) = \\lambda - \\mathbf { u } ( t ) , \\dot { w } _ { i } ( t ) = u _ { i } ( t ) - \\gamma _ { i } ( t ) , \\end{aligned} \\end{align*}"}
{"id": "6993.png", "formula": "\\begin{align*} \\alpha = P _ 1 \\widehat { d x _ 1 } \\wedge \\dots \\wedge d x _ n + \\dots + P _ n d x _ 1 \\wedge \\dots \\wedge \\ \\widehat { d x _ n } . \\end{align*}"}
{"id": "7536.png", "formula": "\\begin{align*} \\bigg \\| \\max _ { 1 \\le k \\le n } \\big | V _ { k } - \\frac { V _ { k } } { V _ { n } } \\big | \\bigg \\| _ { L ^ { \\gamma p } } ^ { \\gamma } = \\| V _ { n } - 1 \\| _ { L ^ { \\gamma p } } ^ { \\gamma } \\le C n ^ { - \\frac { \\gamma } { 2 } } . \\end{align*}"}
{"id": "8510.png", "formula": "\\begin{align*} \\log _ Q ( | C | ) = \\sum \\limits _ { j = 1 } ^ { u } \\left ( \\frac { \\ln ( q _ j ) } { \\ln ( Q ) } \\right ) \\dim ( C _ j ) . \\end{align*}"}
{"id": "420.png", "formula": "\\begin{align*} - \\frac { 1 } { \\pi } \\Im G _ { \\mu } ( t ) = - \\frac { 1 } { \\pi } \\Im \\frac { 1 } { s + i f _ { \\mu } ( s ) } = \\frac { 1 } { \\pi } \\frac { f _ { \\mu } ( s ) } { s ^ { 2 } + f _ { \\mu } ( s ) ^ { 2 } } . \\end{align*}"}
{"id": "8774.png", "formula": "\\begin{align*} Y _ O ( P , G ) & : = \\sup \\{ \\| \\phi _ 1 * \\phi _ 2 \\| _ { q ( P ) } \\mid ( \\phi _ 1 , \\phi _ 2 ) \\in \\mathcal { B } \\} , \\\\ Y _ R ( P , G ) & : = \\inf \\{ \\| \\phi _ 1 * \\phi _ 2 \\| _ { q ( P ) } \\mid ( \\phi _ 1 , \\phi _ 2 ) \\in \\mathcal { B } \\} . \\end{align*}"}
{"id": "1045.png", "formula": "\\begin{align*} \\tilde { B } S _ 1 ( t ) x = \\int _ 0 ^ t \\tilde { B } T ( t - s , A ) \\tilde { B } T ( s , A ) x \\ , \\mathrm { d } s , \\end{align*}"}
{"id": "251.png", "formula": "\\begin{align*} \\delta ^ { ( s ) } w _ k + \\frac { \\sum _ { i j } ( \\sum _ l \\Gamma ^ { k ( s ) } _ { i j , l } c _ l ) v _ i ^ { \\phi ^ s } w _ j ^ { \\phi ^ s } } { ( \\sum _ l r _ l c _ l ) ( \\sum _ l v _ l ^ { \\phi ^ s } ) } = 0 , \\ \\ k \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "8025.png", "formula": "\\begin{align*} Y _ + = X _ { \\tilde { \\omega } _ + } , Y _ - = X _ { \\tilde { \\omega } _ - } . \\end{align*}"}
{"id": "8495.png", "formula": "\\begin{align*} \\rho ( 1 ) = \\frac { 2 b d k \\log ( \\frac { 4 b d k } { 1 } ) } { 1 } + \\frac { \\log ( \\frac { 3 } { \\delta } ) } { 2 } \\leq n . \\end{align*}"}
{"id": "3519.png", "formula": "\\begin{align*} & \\mathrm { D T S } _ { ( p , q ) } ( X ) \\\\ & = \\mathrm { D T V } _ { ( p , q ) } ^ { - 3 / 2 } ( X ) \\bigg \\{ \\sum _ { k = 2 } ^ { 3 } \\binom { 3 } { k } ( - \\mathrm { D T E } _ { ( p , q ) } ( X ) ) ^ { 3 - k } [ \\mu ^ { k } + k \\mu ^ { k - 1 } \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) ] \\\\ & + 2 \\mathrm { D T E } _ { ( p , q ) } ^ { 3 } ( X ) + 3 [ \\mu - \\mathrm { D T E } _ { ( p , q ) } ( X ) ] \\sigma ^ { 2 } \\left ( L _ { 1 } + 2 L _ { 2 } \\right ) + \\sigma ^ { 3 } \\left ( L _ { 1 } ^ { \\ast } + 2 L _ { 2 } ^ { \\ast } \\right ) \\bigg \\} , \\end{align*}"}
{"id": "5948.png", "formula": "\\begin{align*} { \\cal J } = \\prod \\limits _ t \\left | \\frac { \\d A _ { i j \\ , t } } { \\d X _ { k m \\ , t } } \\right | \\end{align*}"}
{"id": "1716.png", "formula": "\\begin{align*} & \\mathbb { S } _ q ( \\ell _ n ) ( x _ { \\beta ^ \\vee } ) = ( 1 - x _ { \\beta + n \\delta } ) ^ { - 1 } x _ { \\beta ^ \\vee } , \\\\ & \\mathbb { S } _ q ( \\ell _ n ) ( x _ { \\delta ^ \\vee } ) = \\prod _ { k = 0 } ^ { n - 1 } ( 1 - ( q ^ { \\frac { 1 } { 2 } } ) ^ { 1 - n + 2 k } x _ { \\beta + n \\delta } ) ^ { - 1 } x _ { \\delta ^ \\vee } , \\\\ & \\mathbb { S } _ q ( \\ell _ { \\infty } ) ( x _ { \\beta ^ \\vee } ) = x _ { \\beta ^ \\vee } . \\end{align*}"}
{"id": "1714.png", "formula": "\\begin{align*} \\mathbb { E } _ q ( x ) \\ \\mathbb { E } _ q ( q x ) ^ { - 1 } = 1 - x . \\end{align*}"}
{"id": "268.png", "formula": "\\begin{align*} ( D ^ { \\nu } _ i ) _ { j k } : = \\frac { 1 } { 2 } ( ( C ^ { \\nu } _ i ) _ { j k } + ( C ^ { \\nu } _ j ) _ { i k } - ( C ^ { \\nu } _ k ) _ { i j } ) . \\end{align*}"}
{"id": "338.png", "formula": "\\begin{align*} S y z ( g ) = \\{ a = ( a _ 1 , \\ldots , a _ n ) \\in R ^ n \\ : \\ a _ 1 g _ 1 + \\ldots + a _ n g _ n = 0 \\} . \\end{align*}"}
{"id": "122.png", "formula": "\\begin{align*} \\| C \\| _ { L ^ 1 } = \\frac { 1 } { m ^ 2 _ t } , \\| C ^ 2 \\| _ { L ^ 1 } \\leq \\frac { c } { m ^ 2 _ t } { \\bf 1 } _ { d = 2 } + \\frac { c } { m _ t } { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "578.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i < j \\leq n } \\mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) = \\sum _ { 1 \\leq i \\leq n } \\sum _ { i + 1 < j \\leq n } \\mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) + O ( n ) , \\end{align*}"}
{"id": "5959.png", "formula": "\\begin{align*} \\hat U ( t _ n ) = U _ n , n = 0 , \\dots , N , \\hat U | _ { I _ n } \\in \\mathbb { P } _ 1 ( I _ n ) , \\end{align*}"}
{"id": "3132.png", "formula": "\\begin{align*} | \\mathcal { T } _ \\ell | - | \\mathcal { T } _ 0 | \\le \\Lambda _ { \\mathrm { B D d V } } \\sum _ { k = 0 } ^ { \\ell - 1 } | \\mathcal { M } _ k | \\end{align*}"}
{"id": "8659.png", "formula": "\\begin{align*} \\sup \\left \\{ \\int _ 0 ^ \\infty f g \\ , d r : \\ f \\geq 0 \\ , , \\ \\overline \\mu _ \\beta ( f ) \\leq 1 \\right \\} = \\beta \\ , \\underline \\nu _ \\beta ( g ) \\ , . \\end{align*}"}
{"id": "3042.png", "formula": "\\begin{align*} R ^ { G } _ { T _ { w } } \\theta _ { w } ( s ) = \\frac { 1 } { | C _ G ( s ) ^ { F } | } \\sum _ { \\{ h \\in G ^ { F } \\mid h s h ^ { - 1 } \\in T _ { w } \\} } Q ^ { C _ G ( s ) } _ { C _ { h ^ { - 1 } T _ { w } h } ( s ) } ( 1 ) \\theta _ { w } ( h s h ^ { - 1 } ) . \\end{align*}"}
{"id": "4634.png", "formula": "\\begin{align*} A B x = x , \\ x \\in Y . \\end{align*}"}
{"id": "8516.png", "formula": "\\begin{align*} r _ n \\sim \\sum _ { i = 1 } ^ { \\infty } \\frac { B _ { 2 i } } { ( 2 i - 1 ) 2 i } n ^ { - 2 i + 1 } , \\end{align*}"}
{"id": "2167.png", "formula": "\\begin{align*} \\lim _ { y \\rightarrow 0 ^ { + } } \\int _ { \\mathbb { R } } \\frac { 1 + s ^ { 2 } } { | 1 - s G _ { \\nu } ( i y ) | ^ { 2 } } \\ , d \\sigma _ { \\mu } ( s ) = 0 . \\end{align*}"}
{"id": "4154.png", "formula": "\\begin{align*} J : = \\begin{pmatrix} 0 & I _ { n } \\\\ - I _ { n } & 0 \\end{pmatrix} . \\end{align*}"}
{"id": "1172.png", "formula": "\\begin{align*} \\hat \\Theta _ n : = \\left \\{ \\theta \\in \\Theta : { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\theta } ) \\leq \\inf _ { \\theta ' \\in \\Theta } { \\sf W } _ p ^ { ( \\sigma ) } ( \\hat \\mu _ n , \\nu _ { \\theta ' } ) + n ^ { - 1 / 2 } \\lambda _ n \\right \\} , \\end{align*}"}
{"id": "5174.png", "formula": "\\begin{align*} \\hat \\sigma ^ { - 1 } ( i ) = \\begin{cases} \\sigma ^ { - 1 } ( i ) , & i < j _ \\sigma \\\\ \\sigma ^ { - 1 } ( i + 1 ) , & i \\in [ j _ \\sigma , m - 1 ] , \\end{cases} \\end{align*}"}
{"id": "935.png", "formula": "\\begin{align*} \\langle \\nabla _ i \\Psi _ t , \\nabla _ j \\Psi _ t \\rangle = \\delta _ i ^ j + O ( t ) . \\end{align*}"}
{"id": "5472.png", "formula": "\\begin{align*} f _ i ( z ) & = \\left \\lbrace \\begin{matrix} ( f _ { i - 1 } ) _ { r , x } ( z ) & z \\in B ( x , r ) x \\in X _ i \\\\ f _ { i - 1 } ( z ) & \\end{matrix} \\right . \\end{align*}"}
{"id": "5908.png", "formula": "\\begin{align*} \\tilde { \\mu } _ { V X ^ { ( n ) } } ( A ) & \\le \\mu _ { V X ^ { ( n ) } } ( A _ \\varepsilon ) + \\mathbb { P } \\Big ( \\Big \\lVert \\Big ( \\frac { 1 } { \\beta } \\Big ) ^ { 1 / p } \\sum _ { j = 1 } ^ n Z _ j V _ { \\bullet , j } - \\sum _ { j = 1 } ^ n n ^ { \\kappa / p } \\frac { Z _ j } { ( \\lVert Z ^ { ( n ) } \\rVert _ p ^ p + W _ n ) ^ { 1 / p } } V _ { \\bullet , j } \\Big \\rVert _ 2 \\ge \\varepsilon \\Big ) . \\end{align*}"}
{"id": "8026.png", "formula": "\\begin{align*} [ X _ \\infty ] = \\bar { \\tilde D } _ { m + 2 } . \\end{align*}"}
{"id": "7838.png", "formula": "\\begin{align*} \\langle \\mu ^ * ( J T J ) \\hat { 1 } , \\hat { 1 } \\rangle = \\langle \\hat { 1 } , \\mu ( T ) \\hat { 1 } \\rangle , \\end{align*}"}
{"id": "304.png", "formula": "\\begin{align*} { \\rm d } ( { \\rm L } _ n ) & \\ge \\frac { 1 } { \\sqrt { v } \\ , r _ q } \\sum _ { a \\in A } { \\rm d } ( { \\rm L } _ a ) = \\frac { 1 } { \\sqrt { v } \\ , r _ q } \\ , { \\rm d } ( { \\rm L } _ A ) \\end{align*}"}
{"id": "4558.png", "formula": "\\begin{align*} 0 = \\sum _ { ( v , w ) \\ , \\in \\ , E ( G ) } ( - 1 + 1 ) = \\sum _ { ( v , w ) \\ , : \\ , v } ( - 1 ) + \\sum _ { ( v , w ) \\ , : \\ , w } 1 \\ , . \\end{align*}"}
{"id": "6917.png", "formula": "\\begin{align*} \\frac { \\mathbb { P } \\Big \\{ W _ j > 0 \\ \\Big | \\ | W _ j | , \\boldsymbol { W } _ { - j } \\Big \\} } { \\mathbb { P } \\Big \\{ W _ j < 0 \\ \\Big | \\ | W _ j | , \\boldsymbol { W } _ { - j } \\Big \\} } = & \\frac { \\mathbb { P } \\Big \\{ W _ j > 0 \\ \\Big | \\ | W _ j | \\Big \\} } { \\mathbb { P } \\Big \\{ W _ j < 0 \\ \\Big | \\ | W _ j | \\Big \\} } a . s . \\end{align*}"}
{"id": "1098.png", "formula": "\\begin{align*} \\mbox { H o r } D _ V W = - \\frac { G ( V , W ) } { f } \\mbox { g r a d } f + \\mbox { H o r } P _ V W , \\end{align*}"}
{"id": "8826.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha ( t _ 1 , t _ 2 ) d t _ 1 d t _ 2 = \\phi _ 1 * \\phi _ 2 ( g ) \\end{align*}"}
{"id": "7196.png", "formula": "\\begin{align*} \\langle f , I ^ { \\circ } ( w _ 1 , z ) w _ 2 \\rangle = 0 , \\mbox { f o r a n y } \\ f \\in ( M ^ 3 ) ' , \\ w _ 1 \\in M ^ 1 , \\ w _ 2 \\in M ^ 2 ( 0 ) . \\end{align*}"}
{"id": "2284.png", "formula": "\\begin{align*} v _ { e x } ^ 1 - u _ { e Y } ^ 1 = 0 , u _ { e x } ^ 1 + v _ { e Y } ^ 1 = 0 . \\end{align*}"}
{"id": "963.png", "formula": "\\begin{align*} \\mathbb { Z } [ \\theta ] = \\{ \\sum _ { i = 0 } ^ { n - 1 } a _ { i } \\theta ^ { i } \\mid a _ { i } \\in \\mathbb { Z } , \\ 1 \\leq i \\leq n \\} . \\end{align*}"}
{"id": "5216.png", "formula": "\\begin{align*} A _ I : = \\Q [ \\{ u _ { i , j } \\ , | \\ , i \\in I , j \\in \\Z _ { \\ge 0 } \\} ] / \\langle u _ { i , j } u _ { i , j ' } \\ , | \\ , i \\in I , j , j ' \\in \\Z _ { \\ge 0 } \\rangle . \\end{align*}"}
{"id": "2547.png", "formula": "\\begin{align*} R : = \\begin{pmatrix} A & { _ A M _ B } \\\\ 0 & B \\end{pmatrix} \\end{align*}"}
{"id": "8270.png", "formula": "\\begin{align*} B _ { 0 } = \\frac { \\sinh ( k ' a / 2 ) } { \\sqrt { - \\frac { a } { 2 } + \\left ( \\frac { a } { 3 } + \\frac { 8 } { a k '^ { 2 } } \\right ) \\sinh ^ { 2 } ( k ' a / 2 ) - \\frac { 3 \\sinh ( k ' a / 2 ) \\cosh ( k ' a / 2 ) } { k ' } } } . \\end{align*}"}
{"id": "3311.png", "formula": "\\begin{align*} u ( 0 ) = u _ 0 \\ ; ( H ^ 1 ( \\Omega ^ - ) ) \\partial _ t u ( 0 ) = v _ 0 \\ ; ( L ^ 2 ( \\Omega ^ - ) ) , \\end{align*}"}
{"id": "3638.png", "formula": "\\begin{align*} \\mu ( B _ 1 ( x ) ) \\le \\liminf _ { \\ell \\to \\infty } \\tau _ { - x _ { \\sigma ( \\ell ) } } \\mu _ { \\sigma ( \\ell ) } ( B _ 1 ( x ) ) = \\liminf _ { \\ell \\to \\infty } \\mu _ { \\sigma ( \\ell ) } ( B _ 1 ( x + x _ { \\sigma ( \\ell ) } ) ) = 0 , \\end{align*}"}
{"id": "5773.png", "formula": "\\begin{align*} E ( G / A ) = E ( G / N _ G ( A ) ) E ( N _ G ( A ) / A ) = E ( N _ G ( A ) / A ) \\end{align*}"}
{"id": "402.png", "formula": "\\begin{align*} \\sf K ( x ) = \\int _ 0 ^ { r ( x ) } K ( x , s ) d s , \\end{align*}"}
{"id": "4705.png", "formula": "\\begin{align*} C _ 2 ( n ) & = \\sum _ { j = - \\infty } ^ \\infty ( - 1 ) ^ j | \\{ \\pi = ( G _ j , \\pi _ 2 , \\pi _ 3 ) \\mid \\ell ( \\pi _ 2 ) = \\ell ( \\pi _ 3 ) \\} | \\\\ & = \\sum _ { j = - \\infty } ^ \\infty ( - 1 ) ^ j u _ 2 \\big ( n - j ( 3 j - 1 ) / 2 \\big ) . \\end{align*}"}
{"id": "6571.png", "formula": "\\begin{align*} \\frac { f ( x _ 0 + t y _ 0 ) - f ( x _ 0 ) } { t } \\ ; = \\ ; \\ell ( x _ 0 , y _ 0 , t ) \\ ; \\to \\ ; \\ell ( x _ 0 , y _ 0 , 0 ) \\ ; = \\ ; g ( x _ 0 , y _ 0 ) \\end{align*}"}
{"id": "2268.png", "formula": "\\begin{align*} I _ 5 & = \\int _ 0 ^ r ( \\bar \\phi + e _ \\delta - \\delta ) ( 1 - \\delta + \\bar \\phi + e _ \\delta ) \\cdot ( e '' _ \\delta \\phi + e '' _ \\sigma \\psi ) { \\rm { d } } z \\\\ & \\leq \\Vert ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert ( e '' _ \\delta , e '' _ \\sigma ) \\Vert _ { L ^ 2 } \\cdot \\Vert ( \\bar \\phi + e _ \\delta - \\delta ) ( 1 - \\delta + \\bar \\phi + e _ \\delta ) \\Vert _ { L ^ \\infty } , \\end{align*}"}
{"id": "2824.png", "formula": "\\begin{align*} \\begin{aligned} - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\left [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\right ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 = 0 \\end{aligned} \\end{align*}"}
{"id": "7415.png", "formula": "\\begin{align*} \\Pr \\left ( \\max _ { 2 \\le R < r } \\left | \\sum _ { i = 2 } ^ R E _ i \\right | \\ge t n 2 ^ { ( 1 / 2 - c ( 2 - \\gamma ) / 8 ) n } \\right ) \\ll \\left ( \\frac { \\log H } { t H } \\right ) ^ { 1 / \\gamma } 2 ^ { ( 1 - 1 / ( 2 \\gamma ) + c ( 2 - \\gamma ) / ( 8 \\gamma ) ) n } . \\end{align*}"}
{"id": "5093.png", "formula": "\\begin{align*} P ^ { n , 2 } _ \\tau = n ^ { \\alpha + \\frac 1 2 } \\int ^ { \\tau } _ 0 ( t - s ) ^ { \\alpha } [ ( \\sigma ' ( X _ s ) - \\sigma ' ( X _ { \\eta _ n ( s ) } ) ] \\sigma ( X _ { \\eta _ n ( s ) } ) \\Xi ^ { n , 2 } _ s d s . \\end{align*}"}
{"id": "1463.png", "formula": "\\begin{align*} { { C _ { n , u , m } = } } C _ { u , m } : = \\Psi ( { \\hat { P } } _ { u } ) \\enspace . \\end{align*}"}
{"id": "1545.png", "formula": "\\begin{align*} \\pi _ { m } = \\begin{cases} p _ { m } \\pi _ { m } + p _ { m + 1 } \\pi _ { m + 1 } , & \\\\ ( 1 - p _ { m - 1 } ) \\pi _ { m - 1 } + p _ { m + 1 } \\pi _ { m + 1 } , & \\\\ ( 1 - p _ { m - 1 } ) \\pi _ { m - 1 } + ( 1 - p _ { m } ) \\pi _ { m } , & \\end{cases} \\end{align*}"}
{"id": "6855.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d b ( t ) } { d t } & = g _ 0 + g _ 1 N ( b ( t ) , c ( t ) ) - b ( t ) = g _ 0 - ( 1 - g _ 1 ) b + g _ 1 ( N - b ) , \\\\ \\frac { d c ( t ) } { d t } & = 2 a _ 0 + 2 a _ 1 N ( b ( t ) , c ( t ) ) - 2 c ( t ) = 2 a _ 0 + 2 a _ 1 N - 2 c . \\end{aligned} \\end{align*}"}
{"id": "2515.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\{ g ( \\nabla _ U \\xi , V ) + g ( \\nabla _ V \\xi , U ) \\} + R i c ^ \\nu ( U , V ) + \\mu g ( U , V ) = 0 . \\end{align*}"}
{"id": "8879.png", "formula": "\\begin{align*} f ( \\lambda ) = \\lambda ^ N + \\sum _ { k = 1 } ^ { N } ( - 1 ) ^ k \\lambda ^ { N - k } \\sum _ { 1 \\le j _ 1 < \\cdots < j _ k \\le N } \\underset { 1 \\le \\ell , m \\le k } { { \\rm d e t } } ( A _ { j _ \\ell j _ m } ) . \\end{align*}"}
{"id": "1350.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + \\bigtriangleup u - | u | ^ { p - 1 } u + i \\gamma u = f ( x ) \\\\ u ( x , 0 ) = u _ 0 \\in H ^ s _ x ( \\mathbb { T } ) , \\end{cases} \\end{align*}"}
{"id": "5860.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 + } \\bigg ( \\int _ t ^ { \\infty } u \\bigg ) ^ { \\frac { 1 } { 1 - q } } V _ r ( 0 , t ) ^ { \\frac { q } { 1 - q } } = 0 . \\end{align*}"}
{"id": "7058.png", "formula": "\\begin{align*} Z ^ { \\pm } : ( \\dot { x } , \\dot { y } ) = ( X ^ { \\pm } ( x , y ) , Y ^ { \\pm } ( x , y ) ) , ( x , y ) \\in \\Sigma ^ { \\pm } , \\end{align*}"}
{"id": "3783.png", "formula": "\\begin{align*} H ^ { m ; p , i ; f } _ { k , j ; l , n } ( t , x , \\zeta ) : = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } 2 ^ { 3 k + 2 n } ( 1 + 2 ^ k | y \\cdot \\tilde { \\zeta } | ) ^ { - N _ 0 ^ 3 } ( 1 + 2 ^ { k + n } ( | y \\cdot \\tilde { \\zeta } _ 1 | + | y \\cdot \\tilde { \\zeta } _ 2 | ) ) ^ { - N _ 0 ^ 3 } f ( s , x - y + ( t - s ) \\omega , v ) \\end{align*}"}
{"id": "8567.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty a _ n ^ 2 < \\infty , \\end{align*}"}
{"id": "6504.png", "formula": "\\begin{align*} E ' ( t ) = - \\int _ { \\Omega } b | \\nabla u _ { t } | ^ 2 d x , \\end{align*}"}
{"id": "5835.png", "formula": "\\begin{align*} \\sup _ { N \\leq k \\leq M } a _ k \\sup _ { N \\leq i \\leq k } b _ i = \\sup _ { N \\leq k \\leq M } a _ k b _ k . \\end{align*}"}
{"id": "6244.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n q ^ n \\left ( c - 1 + q ^ 2 + q ^ n x \\right ) A i _ q ( q ^ n x ) \\\\ & = \\frac { q c + x } { q ( 1 + q ) } \\ , _ 1 \\phi _ 1 ( 0 ; - q ^ 2 ; q , - x ) - ( 1 - q ) A i _ q ( \\frac { x } { q } ) , \\end{align*}"}
{"id": "3687.png", "formula": "\\begin{align*} \\beta ( s , r ) = ( 1 - \\varphi ( r ) ) f ( s , 0 ) , \\end{align*}"}
{"id": "2104.png", "formula": "\\begin{align*} ( L y _ i ) ( v ) & = \\begin{cases} - \\sum \\limits _ { e \\in E _ k } \\frac { \\delta _ E ( e ) } { c } \\frac { 1 } { | e | } y _ i ( v ) & v \\in W , \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "297.png", "formula": "\\begin{align*} \\frac { 1 } { a \\log a } \\ \\le \\ \\frac { 1 } { 1 + v } \\frac { 1 } { a \\log P ( a ) } = \\frac { e ^ \\gamma } { \\mu _ { P ( a ) } } \\frac { 1 } { ( 1 + v ) a } \\prod _ { p < P ( a ) } \\Big ( 1 - \\frac { 1 } { p } \\Big ) = \\frac { e ^ \\gamma } { \\mu _ { P ( a ) } } \\frac { { \\rm d } ( { \\rm L } _ a ) } { 1 + v } . \\end{align*}"}
{"id": "5688.png", "formula": "\\begin{align*} m _ { 5 , 1 , 1 , 1 } ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) & = x _ 1 ^ 5 x _ 2 x _ 3 x _ 4 + x _ 2 ^ 5 x _ 1 x _ 3 x _ 4 + x _ 3 ^ 5 x _ 1 x _ 2 x _ 4 + x _ 4 ^ 5 x _ 1 x _ 2 x _ 3 . \\end{align*}"}
{"id": "5734.png", "formula": "\\begin{align*} \\pi _ { [ a , i ] } \\cdot \\pi _ { [ i + 1 , b ] } & = \\binom { b - a + 1 } { i - a + 1 } \\pi _ { [ a - 1 , b - 1 ] } + \\binom { b - a } { i - a } \\pi _ { [ a , b - 1 ] } y _ b + \\binom { b - a } { i - a + 1 } \\pi _ { [ a , b - 1 ] } y _ { b } \\\\ & = \\binom { b - a + 1 } { i - a + 1 } \\pi _ { [ a - 1 , b - 1 ] } + \\binom { b - a + 1 } { i - a + 1 } \\pi _ { [ a , b - 1 ] } y _ { b } \\\\ & = \\binom { b - a + 1 } { i - a + 1 } \\pi _ { [ a , b ] } , \\end{align*}"}
{"id": "3955.png", "formula": "\\begin{align*} C \\prod _ { i = 1 } ^ d \\left ( \\int _ { \\R ^ { r } } | K _ { h _ i } ( x _ i - u _ i ^ { q _ i } ) | \\prod _ { j = 1 } ^ { r - 1 } \\frac { 1 } { ( w _ { j + 1 } - w _ j ) ^ { 1 / 2 } } e ^ { - \\lambda _ 0 \\frac { ( u ^ { j + 1 } _ i - u ^ j _ i ) ^ 2 } { w _ { j + 1 } - w _ j } } d u ^ 1 _ i \\dots d u ^ r _ i \\right ) . \\end{align*}"}
{"id": "3652.png", "formula": "\\begin{align*} e ( \\{ x _ 0 \\} ) \\geq H ( x _ 0 , m _ v ) + \\sum _ { 0 \\leq i < k } H ( x _ 0 , m _ i ) \\geq H \\Bigl ( x _ 0 , m _ v + \\sum _ { 0 \\leq i < k } m _ i \\Bigr ) = H ( x _ 0 , u ( \\{ x _ 0 \\} ) ) . \\end{align*}"}
{"id": "8330.png", "formula": "\\begin{align*} \\delta ( P _ { k , l } ) = \\sum _ { s - r = l - k + n } b _ { r , s } ( k , l ) P _ { r , s } \\ , , \\end{align*}"}
{"id": "8543.png", "formula": "\\begin{align*} \\epsilon _ p = \\frac { 2 p + 1 } { 2 } \\log \\left ( \\frac { p + 1 } { p } \\right ) - 1 = \\frac { 1 } { 2 x } \\log \\left ( \\frac { 1 + x } { 1 - x } \\right ) - 1 , \\end{align*}"}
{"id": "1044.png", "formula": "\\begin{align*} \\begin{aligned} S _ { 1 } ( t ) x & = \\int _ { 0 } ^ { t } T ( t - s , A ) \\tilde { B } T ( s , A ) x \\ , \\textnormal { d } s \\\\ S _ { n } ( t ) x & = \\int _ { 0 } ^ { t } T ( t - s , A ) \\tilde { B } S _ { n - 1 } ( s ) x \\ , \\textnormal { d } s n > 1 . \\end{aligned} \\end{align*}"}
{"id": "1643.png", "formula": "\\begin{align*} p ( x , y , t ) \\ , = \\ , 2 \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\log \\tau ( x , y , t ) \\end{align*}"}
{"id": "6422.png", "formula": "\\begin{align*} \\Phi _ 0 ( [ x , y ] _ \\mathfrak { g } ) - [ \\Phi _ 0 ( x ) , \\Phi _ 0 ( y ) ] = [ Q , \\Phi _ 1 ( x , y ) ] , \\ ; \\forall x , y \\in \\mathfrak { g } [ 1 ] . \\end{align*}"}
{"id": "4116.png", "formula": "\\begin{align*} S ^ { i + 1 , j } d ^ { i , j } = - d ^ { i + 1 , j - 1 } S ^ { i , j } , \\forall i , j \\geq 0 . \\end{align*}"}
{"id": "1666.png", "formula": "\\begin{align*} f _ { j } = { f } _ { j } ^ 0 - \\sum _ { \\{ p : j \\in S _ p \\} } ( { f } _ { j } ^ 0 ) | _ { C _ { p , S _ p } } . \\end{align*}"}
{"id": "7099.png", "formula": "\\begin{align*} ( k , v , t ) \\mapsto \\begin{cases} ( k , v , 0 ) , & t = 0 , \\\\ & \\\\ \\left ( k \\exp ( t v ) , t \\right ) , & t > 0 , \\end{cases} \\end{align*}"}
{"id": "8180.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\sup _ { t \\geq 0 } \\| \\chi _ { S _ { R } ^ c } e ^ { - i t H } \\phi \\| \\geq \\lim _ { R \\rightarrow \\infty } \\| \\chi _ { S _ { R } ^ c } e ^ { - i R ^ \\frac { 1 } { \\beta } H } \\phi \\| = \\| \\hat { f } \\| \\end{align*}"}
{"id": "3537.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast \\ast } = \\frac { \\Gamma ( t - 1 ) \\left [ \\xi _ { p } ^ { 3 } \\left ( 1 + \\xi _ { p } ^ { 2 } \\right ) ^ { - ( t - 1 ) } - \\xi _ { q } ^ { 3 } \\left ( 1 + \\xi _ { q } ^ { 2 } \\right ) ^ { - ( t - 1 ) } \\right ] } { 2 \\Gamma ( t - \\frac { 1 } { 2 } ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "3075.png", "formula": "\\begin{align*} c _ { n } ^ { \\ast \\ast } = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } \\Psi _ { 1 } ^ { \\ast } ( - 1 , \\frac { n } { 2 } + 1 , 1 ) } . \\end{align*}"}
{"id": "4468.png", "formula": "\\begin{align*} \\| f \\| _ { B ^ { a , b } } = \\sum \\limits _ { j \\in \\Z } ( 2 ^ { a j } + 2 ^ { b j } ) \\| P _ j f \\| _ { L ^ \\infty } . \\end{align*}"}
{"id": "6753.png", "formula": "\\begin{align*} H ^ { 0 } \\left ( X , K _ X \\right ) = H ^ { 0 } \\left ( Y , K _ { Y } \\right ) \\oplus \\bigoplus _ { \\chi \\ne \\chi _ { 0 0 0 } } { H ^ { 0 } \\left ( Y , K _ { Y } + L _ { \\chi } \\right ) } . \\end{align*}"}
{"id": "2212.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { d } { d t } g ^ { T } ( x , t ) = - \\mathrm { R i c } ^ { T } ( x , t ) + \\varkappa g ^ { T } ( x , t ) \\end{array} \\end{align*}"}
{"id": "6064.png", "formula": "\\begin{align*} f ( x ) = ( x - \\lambda ) ^ p g ( x ) . \\end{align*}"}
{"id": "7423.png", "formula": "\\begin{align*} \\begin{array} { c } v ( 0 , t ) = \\alpha v _ x ( L , t ) - \\gamma \\beta p _ x ( L , t ) = 0 , \\\\ p ( 0 , t ) = \\beta p _ x ( L , t ) - \\gamma \\beta v _ x ( L , t ) = 0 . \\end{array} \\end{align*}"}
{"id": "3673.png", "formula": "\\begin{align*} H ^ { j ; 2 } _ { k , \\tilde { k } ; m , l } ( t _ 1 , t _ 2 ) = \\sum _ { a = 1 , 2 } ( - 1 ) ^ { a - 1 } V _ 3 ( t _ a ) \\int _ 0 ^ { t _ a } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( t _ a ) \\cdot \\xi + i ( t _ a - \\tau ) | \\xi | } \\frac { ( \\hat { v } _ 3 \\hat { v } \\cdot \\xi - \\xi _ 3 ) } { | \\xi | ( | \\xi | + \\hat { V } ( t _ a ) \\cdot \\xi ) } \\varphi _ { l ; \\bar { l } } ( \\frac { \\xi \\times v } { | \\xi | | v | } ) \\varphi _ { m ; - 1 0 M _ t } ( t _ a - \\tau ) \\end{align*}"}
{"id": "7828.png", "formula": "\\begin{align*} \\mathbb S ( M ) = \\{ T \\in \\mathbb B ( L ^ 2 M ) \\mid [ T , J x J ] \\in \\mathbb K ^ { \\infty , 1 } ( M ) , \\ { \\rm f o r \\ a l l \\ } x \\in M \\} . \\end{align*}"}
{"id": "3150.png", "formula": "\\begin{align*} \\left \\Vert U _ { L } ^ { \\Phi } - U _ { L } ^ { \\Psi } \\right \\Vert _ { \\mathcal { U } } = \\left \\Vert U _ { L } ^ { \\Phi - \\Psi } \\right \\Vert _ { \\mathcal { U } } \\leq \\left \\vert \\Lambda _ { L } \\right \\vert \\left \\Vert \\Phi - \\Psi \\right \\Vert _ { \\mathcal { W } _ { 1 } } \\ , L \\in \\mathbb { N } _ { 0 } , \\end{align*}"}
{"id": "3894.png", "formula": "\\begin{align*} I ' ( w ) [ \\phi ] = \\int _ \\Omega | \\nabla w ^ \\frac { 1 } { p } | ^ { p - 2 } \\langle \\nabla w ^ \\frac { 1 } { p } , \\nabla ( w ^ \\frac { 1 - p } { p } \\phi ) \\rangle , I '' ( w ) [ \\phi , \\phi ] = \\int _ \\Omega \\rho ( w , \\phi ) \\end{align*}"}
{"id": "4158.png", "formula": "\\begin{align*} \\pi _ \\lambda ( a , b , t ) \\varphi ( \\xi ) = e ^ { i \\lambda t } e ^ { i \\lambda ( a \\xi + \\frac { 1 } { 2 } a b ) } \\varphi ( \\xi + b ) , \\end{align*}"}
{"id": "3792.png", "formula": "\\begin{align*} H y p _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta ) + i \\mu s | \\xi | + i \\mu _ 1 s | \\eta | } \\clubsuit K _ { k _ 1 , j _ 1 ; n _ 1 } ^ { \\mu _ 1 , i _ 1 } ( s , \\eta , V ( s ) ) \\cdot { } ^ 1 \\clubsuit K ( s , \\xi , X ( s ) , V ( s ) ) d \\xi d \\eta d s . \\end{align*}"}
{"id": "8867.png", "formula": "\\begin{align*} \\det \\left ( ( \\lambda I _ N - H ) _ { k | k + 1 } \\right ) = - y _ { k k + 1 } \\det \\left ( ( \\lambda I _ N - H ) _ { k k + 1 | k k + 1 } \\right ) . \\end{align*}"}
{"id": "3213.png", "formula": "\\begin{align*} Q ^ { \\pi } ( s , a ) = \\lim _ { T \\rightarrow \\infty } \\mathbb { E } ^ { \\pi } \\left \\{ \\sum _ { t = 0 } ^ T \\gamma ^ t c _ { a _ { \\pi } } ( s _ t ) \\Big | s _ 0 = s , a _ 0 = a \\right \\} . \\end{align*}"}
{"id": "1661.png", "formula": "\\begin{align*} r _ { A ( U ) } ( f ) : = \\sup \\{ | \\lambda | : \\lambda \\in \\sigma _ { A ( U ) } ( f ) \\} . \\end{align*}"}
{"id": "5819.png", "formula": "\\begin{align*} \\pi ( x , q , { \\bf { a } } ) = \\# \\{ p \\leq x : p _ n \\equiv a _ n \\bmod q \\} . \\end{align*}"}
{"id": "3088.png", "formula": "\\begin{align*} \\boldsymbol { \\delta } = \\left ( \\begin{array} { c c c c c c c c c c c } 0 . 4 1 7 2 \\\\ 0 . 4 8 7 5 \\end{array} \\right ) , \\boldsymbol { \\Omega } = \\left ( \\begin{array} { c c c c c c c c c c c } 0 . 7 8 8 6 & 0 . 0 9 9 3 \\\\ 0 . 0 9 9 3 & 0 . 6 6 8 8 \\end{array} \\right ) . \\end{align*}"}
{"id": "565.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ j ) = & f _ k ( j ) \\bigg ( f _ k ( i ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( i ) } { \\sqrt { i } } \\Big ) \\bigg ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) } { \\sqrt { j - i } } \\Big ) \\\\ = & f _ k ( i ) f _ k ( j ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) \\tau _ 3 ( i ) } { \\sqrt { i } } \\Big ) + O _ { \\alpha } \\Big ( \\frac { \\tau _ 3 ( j ) } { \\sqrt { j - i } } \\Big ) , \\end{align*}"}
{"id": "1280.png", "formula": "\\begin{align*} \\Psi ( & E _ { j + 1 } ( x , d - 1 ) ) - \\Psi ( E _ j ( x , d - 1 ) ) = \\\\ & = 2 \\sqrt { d - 1 } \\left ( \\cot \\left ( \\dfrac { \\pi } { 2 j + 4 } \\right ) - 1 \\right ) - 2 \\sqrt { d - 1 } \\left ( \\csc \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) - 1 \\right ) \\\\ & = 2 \\sqrt { d - 1 } \\left ( \\cot \\left ( \\dfrac { \\pi } { 2 j + 4 } \\right ) - \\csc \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) \\right ) \\\\ & = f _ j \\ , \\sqrt { d - 1 } \\ , . \\end{align*}"}
{"id": "6127.png", "formula": "\\begin{align*} r _ i & \\le r - 1 \\ \\tag * { ( H 3 ) } \\\\ | H _ i | & \\ge r _ i + 1 \\ \\tag * { ( H 4 ) } \\end{align*}"}
{"id": "3795.png", "formula": "\\begin{align*} \\Phi _ 2 ( \\xi , \\eta , \\zeta ) : = \\hat { \\zeta } \\cdot ( \\xi + \\eta ) + + \\mu | \\xi | + \\mu _ 1 | \\eta | . \\end{align*}"}
{"id": "7540.png", "formula": "\\begin{align*} \\mathrm { d } X = \\left \\{ \\bar { g } ( X ) - \\frac { 1 } { 2 } h ( X ) h ' ( X ) \\int _ { M } v ^ { 2 } \\mathrm { d } \\mu \\right \\} \\mathrm { d } t + h ( X ) \\circ \\mathrm { d } W , ~ X ( 0 ) = \\xi . \\end{align*}"}
{"id": "5154.png", "formula": "\\begin{align*} [ D \\vec { x } ] _ { w _ i } = & \\sum _ { v _ j \\in V } d _ G ( v _ i , v _ j ) \\vec { x } _ { v _ j } \\\\ = & \\sum _ { \\ell = 1 , \\ell \\neq i } ^ k \\sum _ { v _ j \\in W _ \\ell } d _ G ( v _ i , v _ j ) \\vec { x } _ { v _ j } + \\sum _ { v _ j \\in W _ i , v _ j \\neq w _ i } d _ G ( v _ i , v _ j ) \\vec { x } _ { v _ j } \\\\ = & \\sum _ { \\ell = 1 , \\ell \\neq i } ^ k | W _ \\ell | d _ G ( v _ i , w _ \\ell ) \\vec { x } _ { w _ \\ell } + c _ i ( | W _ i | - 1 ) \\vec { x } _ { w _ i } \\\\ = & [ D / \\mathcal { W } \\vec { y } ] _ { v _ i } . \\end{align*}"}
{"id": "4410.png", "formula": "\\begin{align*} c _ 1 \\Vert f \\Vert _ 1 \\leq \\sum _ { k = 0 } ^ { \\infty } \\delta _ k | \\alpha _ k | \\leq c _ 2 \\Vert f \\Vert _ 1 . \\end{align*}"}
{"id": "9007.png", "formula": "\\begin{align*} \\delta ^ { \\star } _ { m , n } : = \\begin{cases} 1 , & \\textup { f o r } \\ , \\ , n \\neq 1 , \\\\ 0 , & \\textup { f o r } \\ , \\ , n = 1 . \\end{cases} \\end{align*}"}
{"id": "8713.png", "formula": "\\begin{align*} g _ { \\dot { x } ^ { 1 } } = - \\sin t + b \\cos ( t + c ) , ~ ~ g _ { \\dot { x } ^ { 2 } } = \\cos t + b \\sin ( t + c ) . \\end{align*}"}
{"id": "6920.png", "formula": "\\begin{align*} f - q | _ g = & \\ - L ^ 2 ( p | _ { L ^ 2 ( x y ) } ) z + L ^ 2 ( p | _ { L ^ 2 ( x ) y } z ) \\\\ \\equiv & \\ - L ^ 2 ( p | _ { L ^ 2 ( x ) y } ) z + L ^ 2 ( p | _ { L ^ 2 ( x ) y } ) z \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "500.png", "formula": "\\begin{align*} e _ 2 ( b ) = \\begin{psmallmatrix} & 0 & \\\\ 0 & & 0 \\\\ & 1 & \\end{psmallmatrix} \\otimes \\begin{psmallmatrix} & 0 & \\\\ 0 & & 1 \\\\ & 1 & \\end{psmallmatrix} \\otimes \\begin{psmallmatrix} & 0 & \\\\ 1 & & 1 \\\\ & 1 & \\end{psmallmatrix} \\otimes \\begin{psmallmatrix} & 0 & \\\\ 1 & & 1 \\\\ & 1 & \\end{psmallmatrix} \\ , . \\end{align*}"}
{"id": "6476.png", "formula": "\\begin{align*} u ( 0 ) = u ( L ) = y ( 0 ) = y ( L ) = 0 . \\end{align*}"}
{"id": "5377.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u _ j + \\sum _ { | \\alpha | \\leq m } a _ \\alpha ( D ^ \\alpha u _ j ) & = F \\mbox { i n } \\ ; \\ ; \\Omega , \\\\ u _ j & = f _ j \\mbox { i n } \\ ; \\ ; \\Omega _ e \\end{align*}"}
{"id": "6640.png", "formula": "\\begin{align*} 0 \\leq x _ { t } \\leq P _ { \\ , } \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } x _ { t } \\leq P _ { \\ , } \\ , , \\ , \\end{align*}"}
{"id": "8734.png", "formula": "\\begin{align*} h _ { \\dot { x } ^ 1 \\dot { x } ^ 1 } = \\frac { \\lambda ( \\dot { x } ^ 2 ) ^ 2 } { ( ( \\dot { x } ^ 1 ) ^ 2 + ( \\dot { x } ^ 2 ) ^ 2 ) ^ { \\frac { 3 } { 2 } } } , ~ h _ { \\dot { x } ^ 1 \\dot { x } ^ 2 } = - \\frac { \\dot { x } ^ 1 \\dot { x } ^ 2 } { ( ( \\dot { x } ^ 1 ) ^ 2 + ( \\dot { x } ^ 2 ) ^ 2 ) ^ { \\frac { 3 } { 2 } } } , ~ h _ { \\dot { x } ^ 2 \\dot { x } ^ 2 } = \\frac { \\lambda ( \\dot { x } ^ 1 ) ^ 2 } { ( ( \\dot { x } ^ 1 ) ^ 2 + ( \\dot { x } ^ 2 ) ^ 2 ) ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
{"id": "5430.png", "formula": "\\begin{align*} \\| u \\| _ { F ^ 0 _ { \\infty , \\infty } } = \\sup _ { j \\in \\N _ 0 , x \\in \\R ^ n } | u _ j ( x ) | \\leq C \\| u \\| _ { L ^ { \\infty } ( \\R ^ n ) } . \\end{align*}"}
{"id": "4627.png", "formula": "\\begin{align*} \\gamma = \\alpha _ 0 \\bar \\beta _ 1 \\alpha _ 1 \\cdots \\bar \\beta _ { n } \\alpha _ n \\end{align*}"}
{"id": "5521.png", "formula": "\\begin{align*} \\abs { f ' ( z ) } = \\abs { \\frac { a } { 2 \\lambda } } \\abs { g ' ( \\textbf { z } ) } \\abs { \\textbf { z } + \\lambda i } ^ 2 . \\end{align*}"}
{"id": "3309.png", "formula": "\\begin{align*} - \\frac 1 2 \\varphi - K ^ t ( \\partial _ t ) \\varphi + \\partial _ t ^ { - 1 } W ( \\partial _ t ) \\psi = 0 . \\end{align*}"}
{"id": "5995.png", "formula": "\\begin{align*} f ( t , x ) = u ( t , x ) + \\Gamma ( \\beta + 1 ) ( 1 + H ( t - r ) ) \\sin x . \\end{align*}"}
{"id": "515.png", "formula": "\\begin{align*} L _ { n s } ( \\dot { x } , x , t ) = { \\frac { 1 } { g _ 1 ( t ) \\dot { x } + g _ 2 ( t ) x + g _ 3 ( t ) } } \\ , \\end{align*}"}
{"id": "5991.png", "formula": "\\begin{align*} E ^ 2 _ { } ( t _ { n + 1 } ) \\geq \\theta \\max _ { j = 1 , \\dots , N } E ^ 2 _ { } ( t _ { j } ) . \\end{align*}"}
{"id": "5604.png", "formula": "\\begin{align*} x + y \\equiv 0 \\pmod 4 . \\end{align*}"}
{"id": "7795.png", "formula": "\\begin{align*} \\widetilde { H } ( \\phi ) _ 0 = \\left \\{ \\begin{array} { c c c } t ^ { n - 1 } \\phi _ 2 , & \\mbox { i f } & n \\ \\mbox { i s o d d } , \\\\ \\det ( \\widetilde { H } ( \\phi ) ) t ^ { n - 1 } \\phi _ 2 , & \\mbox { i f } & n \\ \\mbox { i s e v e n } . \\end{array} \\right . \\end{align*}"}
{"id": "4293.png", "formula": "\\begin{align*} [ K _ j E , e ^ { M E } ] = 0 \\end{align*}"}
{"id": "2053.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : l i o u v i l l e m c k e a n } \\partial _ t f ^ N _ t = - \\nabla \\cdot ( \\mathbf { b } ^ N f ^ N _ t ) + \\frac { \\sigma ^ 2 } { 2 } \\Delta f ^ N _ t , \\end{align*}"}
{"id": "2974.png", "formula": "\\begin{align*} F _ { m , Q } ( z , s ) = \\sum _ { \\substack { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma \\\\ \\gamma \\neq \\gamma _ 1 , \\gamma _ 2 } } f _ m ( \\gamma z , s ) . \\end{align*}"}
{"id": "5315.png", "formula": "\\begin{align*} \\int _ { v ( t ) } ^ \\infty \\psi _ \\Lambda ( u ) ^ { - 1 } d u = t . \\end{align*}"}
{"id": "7559.png", "formula": "\\begin{align*} S ^ 2 ( f , g ) ( x ) \\geq & \\int _ { 0 } ^ { 1 } | x - x y | ^ { - 1 / 2 } \\left ( - \\log ( | x - x y | ) \\right ) ^ { - 1 } \\frac { d y } { \\sqrt { 1 - y ^ 2 } } \\\\ & \\geq \\frac { \\sqrt { x } } { \\sqrt { 2 } } \\int _ { 0 } ^ { 1 } ( x - x y ) ^ { - 1 } \\left ( - \\log ( x - x y ) \\right ) ^ { - 1 } d y \\\\ & \\geq \\frac { 1 } { \\sqrt { 2 x } } \\int _ { 0 } ^ { x } u ^ { - 1 } \\left ( - \\log ( u ) \\right ) ^ { - 1 } d u = + \\infty , \\end{align*}"}
{"id": "6671.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\widehat { \\varphi } _ t - A _ 1 ( i , h _ 2 ) \\widehat { \\varphi } _ { i i } - ( B ( i , h _ 2 ) + h _ 2 ' C ( i , h _ 2 ) + D ( i , h _ 2 ) ) \\widehat { \\varphi } _ i = K ( i , t ) , & t > 0 , \\ 0 < i < h _ 0 , \\\\ \\widehat { \\varphi } _ i ( t , 0 ) = \\widehat { \\varphi } ( t , h _ 0 ) = 0 , & t > 0 , \\\\ \\widehat { \\varphi } ( 0 , i ) = 0 , & 0 \\leq i \\leq h _ 0 , \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "554.png", "formula": "\\begin{align*} \\sum \\limits _ { 1 \\leq i \\leq n } f _ k ( n ) f _ k ( n + 1 ) & = N \\sum _ { w \\leq N } \\frac { g _ k ( w ) \\tau ^ * ( w ) } { w ^ 2 } + N \\sum _ { w > N } \\frac { g _ k ( w ) \\tau ^ * ( w ; N ) } { w ^ 2 } + O _ { \\varepsilon } ( N ^ { \\varepsilon } ) \\\\ & = N \\sum _ { w \\leq N } \\frac { g _ k ( w ) \\tau ^ * ( w ) } { w ^ 2 } + O _ { \\varepsilon } ( N ^ { \\varepsilon } ) , \\end{align*}"}
{"id": "4018.png", "formula": "\\begin{align*} \\epsilon ^ * ( n , R ) = e ^ { - n E ( R ) + o ( n ) } . \\end{align*}"}
{"id": "4852.png", "formula": "\\begin{align*} \\delta _ X ( f \\wedge g ) & = 2 [ a b , f _ 1 ] _ X \\otimes g _ 1 - 2 [ a b , g _ 1 ] _ X \\otimes f _ 1 - 2 a b \\otimes [ f _ 1 , g _ 1 ] _ X . \\end{align*}"}
{"id": "5183.png", "formula": "\\begin{align*} \\zeta \\cdot ( x , y ) = ( \\zeta x , \\zeta ^ { - 1 } y ) . \\end{align*}"}
{"id": "8534.png", "formula": "\\begin{align*} \\log ( p + 1 ) & = A _ p + b _ p - \\epsilon _ p \\\\ A _ p & = \\int _ { p } ^ { p + 1 } \\log x \\ , d x \\\\ b _ p & = \\frac { 1 } { 2 } \\big ( \\log ( p + 1 ) - \\log ( p ) \\big ) \\\\ \\epsilon _ p & = \\int _ { p } ^ { p + 1 } \\log x \\ , d x - \\log p - \\frac { 1 } { 2 } \\big ( \\log ( p + 1 ) - \\log ( p ) \\big ) \\\\ & = \\cdots \\\\ & = \\frac { 2 p + 1 } 2 \\log \\frac { p + 1 } p - 1 \\end{align*}"}
{"id": "7522.png", "formula": "\\begin{align*} E _ i = \\frac { 1 } { | C | } H _ i H _ i ^ \\top \\end{align*}"}
{"id": "6893.png", "formula": "\\begin{align*} \\begin{cases} g _ 0 + ( g _ 1 \\bar { \\rho } V _ F ) N ( b ^ * , c ^ * ) - b ^ * = 0 , \\\\ a _ 0 + ( a _ 1 \\bar { \\rho } V _ F ) N ( b ^ * , c ^ * ) - c ^ * = 0 , \\end{cases} \\end{align*}"}
{"id": "2365.png", "formula": "\\begin{align*} \\nu ( l ( h _ { \\sigma ' } ) ) \\geq \\beta _ b + b \\gamma _ { \\sigma ' } > \\beta _ b + b \\gamma _ { \\sigma } = \\nu ( f ) \\end{align*}"}
{"id": "7777.png", "formula": "\\begin{align*} \\tilde { U } = U + E , \\end{align*}"}
{"id": "5920.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { T \\to \\infty } \\left ( \\frac 1 T \\ln E _ k \\right ) = \\lambda _ 1 + \\dots + \\lambda _ k , \\ \\ k = 1 \\dots d \\end{align*}"}
{"id": "2481.png", "formula": "\\begin{align*} K _ r ( x , y , z ) & = \\left ( \\sum _ { i = 1 } ^ { n - 2 } \\frac { 3 } { \\varepsilon } ( z _ i - \\Phi _ i ( x , y ) ) ^ 2 \\right ) ^ r , \\\\ W _ r ( x , y , z ) & = F _ { p , q } ( x , y ) + K _ r ( x , y , z ) . \\end{align*}"}
{"id": "811.png", "formula": "\\begin{align*} A _ \\ell ( u , v , S _ 1 \\times S _ 2 ) = \\int _ { S _ 1 \\times S _ 2 } \\frac { [ u ( x ) - u ( y ) ] ^ { \\ell - 1 } ( v ( x ) - v ( y ) ) } { | x - y | ^ { N + \\ell s } } ~ d x d y . \\end{align*}"}
{"id": "7380.png", "formula": "\\begin{align*} f ( s ) = - \\frac { U _ 0 ( s ) ^ { p - 1 } U _ 0 ' ( s ) } { s } . \\end{align*}"}
{"id": "7628.png", "formula": "\\begin{align*} \\log \\Delta ^ { \\alpha \\vect s + \\alpha ' \\vect s ' } _ \\Omega = \\alpha \\log \\Delta ^ { \\vect s } _ \\Omega + \\alpha ' \\log \\Delta ^ { \\vect s ' } _ \\Omega \\end{align*}"}
{"id": "3949.png", "formula": "\\begin{align*} { } I _ 4 & \\le \\frac { c } { ( \\prod _ { l = 1 } ^ d h _ l ^ * ) ^ 2 } \\frac { 1 } { T _ n ^ 2 } \\int _ 0 ^ { T _ n } \\int _ { D } ^ { T _ n } e ^ { - \\rho s ' } e ^ { \\rho \\Delta _ n } d t d s ' \\le \\frac { c } { T _ n ( \\prod _ { l = 1 } ^ d h _ l ^ * ) ^ 2 } e ^ { - \\rho D } \\end{align*}"}
{"id": "8388.png", "formula": "\\begin{align*} \\lim _ { | z | \\to \\infty } \\frac { | X ^ \\epsilon ( t , z ) - z | } { | z | } = 0 . \\end{align*}"}
{"id": "6719.png", "formula": "\\begin{align*} p _ m ( \\lambda , k ) = ( k \\lambda ) ^ { m - 1 } + \\sum _ { \\substack { l = 0 \\\\ l \\equiv m - 1 } } ^ { m - 3 } p ^ m _ l ( \\lambda ^ 2 + k ^ 2 ) ( k \\lambda ) ^ l , \\end{align*}"}
{"id": "5070.png", "formula": "\\begin{align*} \\Lambda ^ n _ s = ( W _ s - W _ { \\eta _ n ( s ) } ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) + I _ 2 \\left ( \\mathbf { 1 } _ { [ \\eta _ n ( s ) , s ] } \\otimes \\psi _ { n , 1 } ( \\cdot , s ) \\right ) , \\end{align*}"}
{"id": "1818.png", "formula": "\\begin{align*} w ( ( n , m ) \\rightarrow ( n + 1 , m + 1 ) ) & = 1 , \\\\ w ( ( n , m ) \\rightarrow ( n + 1 , m ) ) & = b _ { m } , \\\\ w ( ( n , m ) \\rightarrow ( n + 1 , m - 1 ) ) & = a _ { m - 1 } , \\ , \\ , \\ , \\ , m \\geq 1 . \\end{align*}"}
{"id": "5583.png", "formula": "\\begin{align*} \\widetilde { G } ( x , x _ 0 ) = \\frac { G ( x , x _ 0 ) } { g ( x ) \\ , g ( x _ 0 ) } \\ge 1 , \\forall x \\in \\Omega . \\end{align*}"}
{"id": "83.png", "formula": "\\begin{align*} W ^ s ( ( 0 , x ) ; F ) & = \\{ 0 \\} \\times W ^ s ( x ; P ) , \\\\ W ^ u ( ( 0 , x ) ; F ) & = \\bigcup _ { - \\infty < s < 1 } \\{ r \\} \\times \\Phi ^ { \\tilde P } _ { t _ + ( r ) } \\big ( W ^ u ( x ; P ) \\big ) , \\\\ W ^ s ( ( 1 , y ) ; F ) & = \\bigcup _ { 0 < s < + \\infty } \\{ r \\} \\times \\Phi ^ { P } _ { t _ - ( r ) } \\big ( W ^ s ( y ; \\tilde P ) \\big ) , \\\\ W ^ u ( ( 1 , y ) ; F ) & = \\{ 1 \\} \\times W ^ u ( y ; \\tilde P ) , \\end{align*}"}
{"id": "3956.png", "formula": "\\begin{align*} j ( a ^ { * } ) ( \\gamma ) = \\overline { j ( a ) ( \\gamma ^ { - 1 } ) } j ( a * b ) ( \\gamma ) = \\sum _ { \\eta \\in G ^ { r ( \\gamma ) } } j ( a ) ( \\eta ) \\ , j ( b ) ( \\eta ^ { - 1 } \\gamma ) . \\end{align*}"}
{"id": "1480.png", "formula": "\\begin{align*} v _ p ( k ! ) = \\sum _ { \\ell = 1 } ^ { C _ { p , k } } \\left \\lfloor \\dfrac { k } { p ^ { \\ell } } \\right \\rfloor \\le v _ p ( N _ k ) \\le \\sum _ { \\ell = 1 } ^ { C _ { p , k } } \\left ( 1 + \\left \\lfloor \\dfrac { k } { p ^ { \\ell } } \\right \\rfloor \\right ) = v _ p ( k ! ) + C _ { p , k } , \\end{align*}"}
{"id": "970.png", "formula": "\\begin{align*} \\overline { e } _ { 1 } = \\left ( \\begin{array} { c } 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array} \\right ) , \\overline { e } _ { 2 } = \\left ( \\begin{array} { l } 0 \\\\ 1 \\\\ \\vdots \\\\ 0 \\end{array} \\right ) , \\cdots , \\overline { e } _ { n } = \\left ( \\begin{array} { l } 0 \\\\ 0 \\\\ \\vdots \\\\ 1 \\end{array} \\right ) . \\end{align*}"}
{"id": "6392.png", "formula": "\\begin{align*} a _ i : = \\min ( V _ i ) , \\ \\ b _ i : = \\max ( V _ i ) . \\end{align*}"}
{"id": "4365.png", "formula": "\\begin{align*} s _ 0 - \\ell _ r b = \\ell e _ r + a - \\ell _ r ( a h _ r - \\nu e _ r ) = ( \\ell + \\ell ' _ r a + \\ell _ r \\nu ) e _ r = ( \\ell + c ) e _ r , \\end{align*}"}
{"id": "5915.png", "formula": "\\begin{align*} P _ j ( f _ j , \\hat \\phi ) = P _ j ( f _ j , \\hat \\phi _ 1 , \\ldots , \\hat \\phi _ n ) = 0 , 1 \\leq j \\leq n . \\end{align*}"}
{"id": "188.png", "formula": "\\begin{align*} ( \\hat { M } ^ 1 , \\hat { M } ^ 2 ) = g ( Y _ { 1 : n } ) . \\end{align*}"}
{"id": "6871.png", "formula": "\\begin{align*} 2 \\sqrt { 2 } e ^ { x ^ 2 / 2 } ( x ^ 2 - 3 ) \\int _ 0 ^ { x / \\sqrt { 2 } } e ^ { - t ^ 2 } d t + 2 x & \\leq 2 \\sqrt { 2 } e ^ { x ^ 2 / 2 } ( x ^ 2 - 3 ) \\frac { x } { \\sqrt { 2 } } e ^ { - x ^ 2 / 2 } + 2 x \\\\ & = 2 x ( x ^ 2 - 3 ) + 2 x \\leq - 4 x + 2 x \\leq 0 , \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
{"id": "805.png", "formula": "\\begin{align*} | g ( | z | \\leq r ) | = | f ( \\omega ( | z | \\leq r ) ) | \\leq \\max _ { | z | = r } | f ( | z | \\leq r ) | = h _ { \\psi } ( r ) , \\end{align*}"}
{"id": "197.png", "formula": "\\begin{align*} & \\P ^ { \\theta } ( \\tilde { \\pi } _ { 0 : n - 1 } , e _ { 1 : n } ) = \\prod _ { t = 0 } ^ { n - 1 } \\P ^ { \\theta } ( \\tilde { \\pi } _ t , e _ { t + 1 } | \\tilde { \\pi } _ { 0 : t - 1 } , e _ { 1 : t } ) \\\\ & = \\prod _ { t = 0 } ^ { n - 1 } 1 _ { \\theta _ { t + 1 } [ \\tilde { \\pi } _ t ] } ( e _ { t + 1 } ) \\times \\\\ & \\sum _ { y _ t , m ^ 1 , m ^ 2 } \\pi _ t ( m ^ 1 , m ^ 2 ) Q ( y _ t | e ^ 1 _ t ( m ^ 1 ) , e ^ 2 _ t ( m ^ 2 ) ) 1 _ { \\tilde { F } ( \\tilde { \\pi } _ { t - 1 } , e _ t , y _ t ) } ( \\tilde { \\pi } _ t ) \\end{align*}"}
{"id": "868.png", "formula": "\\begin{align*} x ^ { - 1 } s x & \\simeq _ R ^ f z w w ^ { - 1 } y ^ { - 1 } s y w w ^ { - 1 } z ^ { - 1 } & ( \\ ; x = _ G y w w ^ { - 1 } z ^ { - 1 } ) \\\\ & \\simeq _ R ^ f z w w ^ { - 1 } z ^ { - 1 } t z w w ^ { - 1 } z ^ { - 1 } & ( \\ ; w ^ { - 1 } y ^ { - 1 } s y w \\simeq _ R ^ f w ^ { - 1 } z ^ { - 1 } t z w ) \\\\ & \\simeq _ R ^ f t & ( \\ ; z w w ^ { - 1 } z ^ { - 1 } = _ G \\epsilon , \\ ; \\ ; \\epsilon \\ ; ) . \\end{align*}"}
{"id": "1362.png", "formula": "\\begin{align*} \\int _ \\mathbb { T } | u | ^ { p - 1 } ( x , t ) \\ , d x = 2 \\pi \\sum _ { 0 = k _ 1 - k _ 2 + \\cdots - k _ { p - 1 } } \\widehat { u } _ { k _ 1 } \\overline { \\widehat { u } } _ { k _ 2 } \\cdots \\widehat { u } _ { k _ { p - 2 } } \\overline { \\widehat { u } } _ { k _ { p - 1 } } = \\int _ \\mathbb { T } | v | ^ { p - 1 } ( x , t ) \\ , d x , \\end{align*}"}
{"id": "3141.png", "formula": "\\begin{align*} \\alpha _ { x } ( a _ { y , \\mathrm { s } } ) = a _ { y + x , \\mathrm { s } } \\ , y \\in \\mathfrak { L } , \\ ; \\mathrm { s } \\in \\mathrm { S } . \\end{align*}"}
{"id": "5413.png", "formula": "\\begin{align*} \\langle \\Lambda _ { \\gamma } f , g \\rangle \\vcentcolon = B _ { \\gamma } ( u _ f , g ) , \\langle \\Lambda _ q f , g \\rangle \\vcentcolon = B _ q ( v _ f , g ) , \\end{align*}"}
{"id": "5568.png", "formula": "\\begin{align*} M _ { n , k } = Y ^ { \\alpha _ 1 } _ n ( t _ 1 ) \\cdots Y ^ { \\alpha _ k } _ n ( t _ k ) , \\ M _ k = X ^ { \\alpha _ 1 } _ \\phi ( t _ 1 , x ) \\cdots X ^ { \\alpha _ k } _ \\phi ( t _ k , x ) , \\end{align*}"}
{"id": "5921.png", "formula": "\\begin{align*} w \\left ( { { \\eta _ 1 } , . . . , { \\eta _ d } } \\right ) = \\mathop { \\lim } \\limits _ { T \\to \\infty } \\frac 1 T \\ln { \\left \\langle E _ { 1 } ^ { \\eta _ 1 - \\eta _ 2 } E _ { 2 } ^ { \\eta _ 2 - \\eta _ 3 } . . . E _ { d - 1 } ^ { \\eta _ { d - 1 } - \\eta _ d } E _ { d } ^ { \\eta _ d } \\right \\rangle } \\end{align*}"}
{"id": "949.png", "formula": "\\begin{align*} E ( \\rho _ R | \\rho _ { \\infty } ) = H '' ( \\xi ) | \\rho _ R - \\rho _ { \\infty } | ^ 2 = \\frac { p ' ( \\xi ) } { \\xi } | \\rho _ R - \\rho _ { \\infty } | ^ 2 , \\mbox { f o r s o m e } \\xi \\in ( 0 , \\overline { \\rho } ) . \\end{align*}"}
{"id": "8770.png", "formula": "\\begin{align*} \\frac { 1 } { q ( P ) } = 1 - N + \\sum _ { k = 1 } ^ N \\frac { 1 } { p _ k } . \\end{align*}"}
{"id": "2683.png", "formula": "\\begin{align*} \\mathcal { E } ^ \\mathrm { H } [ u _ + ] = \\min \\Big \\{ \\mathcal { E } ^ \\mathrm { H } [ u ] \\ ; | \\ ; u \\in H ^ 1 ( \\mathbb { R } ^ d ) \\cap L ^ 2 ( \\mathbb { R } ^ d , V _ \\mathrm { D W } ( x ) d x ) , \\ ; \\| u \\| _ { L ^ 2 } = 1 \\Big \\} . \\end{align*}"}
{"id": "5957.png", "formula": "\\begin{align*} a ( \\hat \\eta ) \\Omega _ \\phi x & = \\check \\eta x \\\\ a ( \\check \\eta ) \\eta & = \\hat \\eta \\otimes _ \\phi \\eta \\end{align*}"}
{"id": "4246.png", "formula": "\\begin{align*} \\mathbf { P } _ x ( \\tau _ \\dagger = \\infty \\mid X ) = \\prod _ { i = 0 } ^ \\infty ( 1 - K ( X _ i ) ) , \\sum _ { i = 0 } ^ \\infty K ( X _ i ) < \\infty . \\end{align*}"}
{"id": "5909.png", "formula": "\\begin{align*} \\xi _ n : = \\Big ( \\Big ( \\frac { 1 } { \\beta } \\Big ) ^ { 1 / p } - \\frac { n ^ { \\kappa / p } } { ( \\lVert Z ^ { ( n ) } \\rVert _ p ^ p + W _ n ) ^ { 1 / p } } \\Big ) ^ 2 \\longrightarrow 0 \\end{align*}"}
{"id": "862.png", "formula": "\\begin{align*} \\textrm { r a n k } \\left ( \\mathbf { V } \\right ) = \\sum _ { m } u \\left ( \\lambda _ { m } \\right ) \\end{align*}"}
{"id": "4456.png", "formula": "\\begin{align*} I ^ 0 ( \\phi _ { \\eta } ) = 0 \\mbox { a n d } \\phi _ { \\eta } = 0 \\mbox { i n } V _ H . \\end{align*}"}
{"id": "8278.png", "formula": "\\begin{align*} \\psi _ { B _ { k } } = B _ { k } \\left [ \\frac { \\sinh ( k x ) } { \\sinh ( k a / 2 ) } - \\frac { \\sinh ( k ' x ) } { \\sinh ( k ' a / 2 ) } \\right ] , \\end{align*}"}
{"id": "7372.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ 4 U _ { h , i } ^ { p - 1 } ( T _ k y ) \\frac { \\partial U _ { h , i } ( T _ k y ) } { \\partial h } \\\\ & = \\sum _ { i = 1 } ^ 4 f ( | T _ k y - h t _ i | ) ( T _ k y - h t _ i ) \\cdot t _ i \\\\ & = \\sum _ { i = 1 } ^ 4 f ( | T _ k ( y - h t _ { k _ i } ) | ) ( T _ k ( y - h t _ { k _ i } ) ) \\cdot T _ k ( t _ { k _ i } ) \\\\ & = \\sum _ { i = 1 } ^ 4 f ( | y - h t _ { k _ i } | ) ( y - h _ n t _ { k _ i } ) \\cdot t _ { k _ i } = \\sum _ { i = 1 } ^ 4 U _ { h , i } ^ { p - 1 } ( y ) \\frac { \\partial U _ { h , i } ( y ) } { \\partial h } . \\end{align*}"}
{"id": "1179.png", "formula": "\\begin{align*} \\mathfrak { F } = \\{ f * \\phi _ \\sigma : f \\in \\dot { H } ^ { 1 , q } ( \\gamma _ { \\sigma } ) , \\| f \\| _ { L ^ { q } ( \\gamma _ { \\sigma } ) } \\le 1 , \\| f \\| _ { \\dot { H } ^ { 1 , q } ( \\gamma _ { \\sigma } ) } \\le C \\} \\end{align*}"}
{"id": "6123.png", "formula": "\\begin{align*} ( - 1 ) ^ { n - k } \\frac { k } { n } a _ k = \\sum _ { 1 \\leq i _ 1 < i _ 2 < \\dots < i _ { n - k } \\leq n - 1 } \\omega _ { i _ 1 } \\omega _ { i _ 2 } \\cdots \\omega _ { i _ { n - k } } , \\ , \\ , \\ , k = 1 , \\dots , n - 1 . \\end{align*}"}
{"id": "7606.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\dot { \\boldsymbol { x } } ( t ) & = A ( t ) \\boldsymbol { x } ( t ) + B ( t ) \\boldsymbol { u } ( t ) , \\\\ \\boldsymbol { y } ( t ) & = C ( t ) \\boldsymbol { x } ( t ) + D ( t ) \\boldsymbol { u } ( t ) , \\end{aligned} \\right . \\end{align*}"}
{"id": "6464.png", "formula": "\\begin{align*} \\frac { \\dd \\Xi _ t } { \\dd t } = \\Bar { Q } \\circ H _ t + H _ t \\circ Q _ { \\mathfrak { g } } . \\end{align*}"}
{"id": "8518.png", "formula": "\\begin{align*} y _ { k + 1 } ( x ) = \\frac { 2 k + 1 } { k + 1 } x y _ k ( x ) - \\frac { k } { k + 1 } y _ { k - 1 } ( x ) , k > 0 , \\end{align*}"}
{"id": "7783.png", "formula": "\\begin{align*} \\sin \\theta ( { \\mathcal C } _ 1 , { \\mathcal C } _ 2 ) = \\lVert ( I - P _ { { \\mathcal C } _ 2 } ) P _ { { \\mathcal C } _ 1 } \\rVert _ S . \\end{align*}"}
{"id": "1266.png", "formula": "\\begin{align*} \\mathcal { E } ( \\mathcal { B } _ { d , k } ) & = \\sum _ { j = 1 } ^ { k - 1 } f _ j ( d - 1 ) ^ { k - \\frac { 1 } { 2 } - j } , \\end{align*}"}
{"id": "8630.png", "formula": "\\begin{align*} \\frac { \\partial \\psi } { \\partial t } = - { \\rm i } \\widehat { H } ( t ) \\psi = - \\sum _ { \\alpha = 1 } ^ r b _ { \\alpha } ( t ) { \\rm i } \\widehat { H } _ \\alpha \\psi , \\psi \\in \\mathcal { H } , t \\in \\mathbb { R } . \\end{align*}"}
{"id": "7566.png", "formula": "\\begin{align*} \\frac { d x } { d t } = b _ 1 ( t ) + b _ 2 ( t ) x + b _ 3 ( t ) x ^ 2 , x \\in \\mathbb { R } , \\end{align*}"}
{"id": "7766.png", "formula": "\\begin{align*} \\mathcal { Q } _ { j , } ^ k \\sim \\mathcal { G P } ( q ^ k _ { j , } , z ^ k _ { j , } + \\eta ) k = 1 , \\ldots , m , \\ j = 1 , \\ldots , L , \\end{align*}"}
{"id": "2592.png", "formula": "\\begin{align*} J = \\dfrac { r _ + - r _ - } { 2 } \\ . \\end{align*}"}
{"id": "2124.png", "formula": "\\begin{align*} c = \\liminf _ { z \\rightarrow \\alpha } \\frac { \\Im f ( z ) } { \\Im z } < + \\infty \\end{align*}"}
{"id": "6724.png", "formula": "\\begin{align*} \\underset { t \\rightarrow 0 } { } \\dfrac { \\mathsf { m } ( E + t H ) - \\mathsf { m } ( E ) } { t } = \\underset { t \\rightarrow 0 } { } \\dfrac { 1 } { t } \\left \\{ t H - g \\dfrac { E + t H } { \\abs { E + t H } } + g \\dfrac { E } { \\abs { E } } \\right \\} = H - g \\mathsf { F } ' ( E ; H ) . \\end{align*}"}
{"id": "2083.png", "formula": "\\begin{align*} h ( Y ) = m n h ( X ) . \\end{align*}"}
{"id": "4050.png", "formula": "\\begin{align*} \\hat { \\beta } ( y ) + \\hat { \\beta } ( z ) = \\hat { \\beta } ( y + z ) y z \\mathbb { R } ^ { n } . \\end{align*}"}
{"id": "3745.png", "formula": "\\begin{align*} \\lesssim 2 ^ { 2 \\epsilon M _ t } \\min \\big \\{ 2 ^ { m - l } 2 ^ { \\epsilon M _ t + 2 l + j + 2 \\tilde { \\alpha } _ t M _ t } , 2 ^ { - 2 m - j - 3 l } \\big \\} \\mathbf { 1 } _ { m \\in ( - 1 0 M _ t , \\epsilon M _ t ] \\cap \\Z } + 2 ^ { m + 4 j } \\mathbf { 1 } _ { m = - 1 0 M _ t } \\end{align*}"}
{"id": "3189.png", "formula": "\\begin{align*} \\sum _ { z \\in \\mathbb { Z } ^ { d } } \\gamma ^ { d } f \\left ( \\gamma z + a \\right ) = \\sum _ { z \\in \\mathbb { Z } ^ { d } } \\gamma ^ { d } f \\left ( \\gamma ( z + b ^ { ( a , \\gamma ) } ) \\right ) \\ . \\end{align*}"}
{"id": "4385.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi _ 1 < - t \\} \\cap U } | \\tilde F - F ^ { 1 + \\delta } | ^ 2 e ^ { - \\varphi - \\Psi _ 1 } = \\int _ { \\{ \\Psi _ 1 < - t \\} \\cap U } | \\frac { \\tilde F } { F ^ { 1 + \\delta } } - 1 | ^ 2 e ^ { - \\Psi _ 1 } < + \\infty . \\end{align*}"}
{"id": "234.png", "formula": "\\begin{align*} \\delta ^ { ( s ) } u = f \\left ( u , u ^ { \\phi ^ s } \\right ) . \\end{align*}"}
{"id": "3724.png", "formula": "\\begin{align*} \\forall s \\in [ 0 , t ] , x , v \\in \\R ^ 3 , X ( X ( x , v , 0 , s ) , V ( x , v , 0 , s ) , s , 0 ) = x , V ( X ( x , v , 0 , s ) , V ( x , v , 0 , s ) , s , 0 ) = v . \\end{align*}"}
{"id": "2225.png", "formula": "\\begin{align*} C _ { M } ^ { B } = \\{ [ \\alpha ] _ { B } \\in H _ { B } ^ { 1 , 1 } ( M , \\mathbb { R } ) | \\exists \\omega > 0 [ \\omega ] _ { B } = [ \\alpha ] _ { B } \\} . \\end{align*}"}
{"id": "1074.png", "formula": "\\begin{align*} \\left \\| \\sum _ { k = 0 } ^ \\infty R ( x + i y , A ) ^ k ( x - w ) ^ k \\right \\| _ { \\infty } \\leq 2 \\end{align*}"}
{"id": "4666.png", "formula": "\\begin{align*} g _ { i j } ( \\theta ) = \\sum \\limits _ { k = 1 } ^ d \\frac { \\partial \\phi ^ k } { \\partial \\theta _ i } ( \\theta ) \\frac { \\partial \\phi ^ k } { \\partial \\theta _ j } ( \\theta ) , i , j = 1 , . . . , m . \\end{align*}"}
{"id": "2064.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : L i n f t y a g a i n s t m o n o m i a l } \\mathcal { L } _ \\infty \\Phi _ k ( \\nu ) = \\sum _ { i = 1 } ^ k \\langle Q ( \\nu ) , \\varphi ^ i \\rangle \\prod _ { j \\ne i } \\langle \\nu , \\varphi ^ j \\rangle , \\end{align*}"}
{"id": "5771.png", "formula": "\\begin{align*} H / H ^ 0 = K _ 1 / K _ 1 ^ 0 = N _ K ( T ) / T = W ( K ) . \\end{align*}"}
{"id": "112.png", "formula": "\\begin{align*} m ^ 2 _ t : = m ^ 2 + \\frac { 1 } { t } , \\end{align*}"}
{"id": "3555.png", "formula": "\\begin{align*} \\beta _ { n } ^ * ( v _ i ) = \\begin{cases} v _ i ' , & , \\\\ 0 , & . \\end{cases} \\end{align*}"}
{"id": "4946.png", "formula": "\\begin{align*} h ( x , t ) = c ( t ) \\end{align*}"}
{"id": "3866.png", "formula": "\\begin{align*} E ^ { T , \\Upsilon , Y } \\subseteq \\bigg ( \\bigcup _ { i = 1 } ^ n E ^ { T _ i , \\Upsilon _ i , Y _ i } \\bigg ) \\cup \\Big \\{ \\tilde { \\phi } ^ { T , \\Upsilon , Y } \\big [ \\tilde { h } _ j ^ { Y } \\big ] : \\tilde { h } _ j ^ { Y } \\in Z ^ a \\big [ Y _ 1 , . . . , Y _ n \\big ] , \\tilde { h } _ j ^ { \\Upsilon } \\in Z ^ a \\big [ \\Upsilon _ 1 , . . . , \\Upsilon _ n \\big ] \\Big \\} \\end{align*}"}
{"id": "2633.png", "formula": "\\begin{align*} \\max \\{ | S | : S \\in \\bigcap _ { i = 1 } ^ n \\mathcal { F } _ i \\} & = \\min \\{ r _ 1 ( X ) + r _ 2 ( X _ 1 \\backslash X ) + \\dots + r _ { n - 1 } ( X _ { n - 2 } \\backslash X _ { n - 3 } ) + r _ n ( E \\backslash X _ { n - 2 } ) : \\\\ & X \\subseteq X _ 1 \\subseteq X _ 2 \\subseteq \\dots \\subseteq X _ { n - 2 } \\subseteq E \\} \\end{align*}"}
{"id": "6287.png", "formula": "\\begin{align*} & \\int \\dfrac { x \\cos ( x ; q ) } { ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\ , _ 2 \\phi _ 1 ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 , q ^ 2 ; 0 ; q ^ 2 , q ) d _ q x \\\\ & = \\frac { - \\sqrt { q } } { ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\dfrac { \\sqrt { q } \\cos ( \\frac { x } { q } ; q ) } { ( 1 - q ) } + x \\sin ( q ^ { \\frac { - 1 } { 2 } } x ; q ) \\ , _ 2 \\phi _ 1 ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 , q ^ 2 ; 0 ; q ^ 2 , q ) \\right ) , \\end{align*}"}
{"id": "2218.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\bigtriangledown ^ { T } H = \\frac { 1 } { u + A } \\bigtriangledown ^ { T } | \\bigtriangledown ^ { T } u | ^ { 2 } - \\frac { | \\bigtriangledown ^ { T } u | ^ { 2 } } { ( u + A ) ^ { 2 } } \\bigtriangledown ^ { T } u + \\bigtriangledown ^ { T } t r _ { \\omega } \\widehat { \\omega } _ { \\infty } . \\end{array} \\end{align*}"}
{"id": "7818.png", "formula": "\\begin{align*} \\phi ^ { - 1 } e _ { \\phi ( \\lambda ) } \\phi = \\frac { t ( \\lambda , \\lambda ) } { 2 } e _ \\lambda ^ \\vee , \\end{align*}"}
{"id": "2306.png", "formula": "\\begin{align*} \\begin{cases} ( \\phi , \\psi ) | _ { y = 0 } = ( \\phi , \\psi ) | _ { y \\rightarrow \\infty } = ( \\phi , \\psi ) | _ { x = 1 } = ( \\phi , \\psi ) | _ { x \\rightarrow \\infty } = ( 0 , 0 ) , \\\\ ( \\phi _ x , \\psi _ x ) | _ { x = 1 } = ( \\phi _ x , \\psi _ x ) | _ { x \\rightarrow \\infty } = ( 0 , 0 ) . \\end{cases} \\end{align*}"}
{"id": "1744.png", "formula": "\\begin{align*} F ^ * ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) \\cdot F ^ * ( z \\ , | \\ , \\overline \\omega _ 1 , - \\omega _ 2 ) = \\prod _ { k \\geq 0 } \\big ( 1 - x _ 2 ( q _ 2 \\widetilde { q } _ 2 ) ^ { k / 2 } \\big ) \\cdot \\prod _ { k \\geq 1 } \\big ( 1 - x _ 2 ^ { - 1 } ( q _ 2 \\widetilde { q } _ 2 ) ^ { k / 2 } \\big ) ^ { - 1 } , \\end{align*}"}
{"id": "3215.png", "formula": "\\begin{align*} & \\sum _ { t = 1 } ^ \\infty \\alpha _ k ^ { t } = \\infty , ~ ~ \\sum _ { t = 1 } ^ \\infty ( \\alpha _ k ^ { t } ) ^ 2 < \\infty , \\forall k \\in { \\cal K } \\\\ & \\sum _ { t = 1 } ^ \\infty \\alpha ^ { t } = \\infty , ~ ~ \\sum _ { t = 1 } ^ \\infty ( \\alpha ^ { t } ) ^ 2 < \\infty . \\end{align*}"}
{"id": "8480.png", "formula": "\\begin{align*} y _ k = \\frac { U } { m } - \\frac { L ( m - 1 ) } { 2 } , \\end{align*}"}
{"id": "2073.png", "formula": "\\begin{align*} \\label [ I ] { d e f : m o d i f i c a t i o n } \\forall t \\geq 0 , P ( X _ t = Y _ t ) = 1 , \\end{align*}"}
{"id": "8017.png", "formula": "\\begin{align*} f _ + ^ * ( K _ { X _ + } + D _ + ) = f _ - ^ * ( K _ { X _ - } + D _ - ) . \\end{align*}"}
{"id": "6201.png", "formula": "\\begin{align*} & \\int F ( x ) \\Big ( \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ q k ( x ) + p ( x ) D _ q k ( x ) + r ( x ) k ( x ) \\Big ) y ( x ) d _ q x \\\\ & = F ( x ) \\Big ( y ( x ) D _ { q ^ { - 1 } } k ( x ) - k ( x ) D _ { q ^ { - 1 } } y ( x ) \\Big ) , \\end{align*}"}
{"id": "4393.png", "formula": "\\begin{align*} \\int _ { D _ j } ( \\eta + g ^ { - 1 } ) | \\bar \\partial _ { \\Phi } ^ * \\alpha | ^ 2 e ^ { - \\Phi } \\ge \\int _ { D _ j } v '' _ { \\epsilon } ( \\Psi _ m ) | \\alpha \\llcorner ( \\bar \\partial \\Psi _ m ) ^ { \\sharp } | ^ 2 e ^ { - \\Phi } . \\end{align*}"}
{"id": "5879.png", "formula": "\\begin{align*} \\| \\bar \\psi ( z ) \\| _ Y = \\| \\bar \\psi ( z ) - \\bar \\psi ( 0 ) \\| _ Y \\leq \\sqrt { n } \\| z \\| _ H \\leq \\sqrt { n } , \\end{align*}"}
{"id": "4910.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { \\partial X } { \\partial t } ( x , t ) = \\psi u ^ \\alpha \\rho ^ \\delta f ^ { - \\beta } ( x , t ) \\nu ( x , t ) , \\\\ & X ( \\cdot , 0 ) = X _ 0 , \\end{cases} \\end{align*}"}
{"id": "2901.png", "formula": "\\begin{align*} N _ F \\leq \\min \\{ N _ { h _ 1 } , N _ { h _ 2 } , \\ldots , N _ { h _ n } \\} = N _ h = 2 ^ { n - 2 } \\end{align*}"}
{"id": "8228.png", "formula": "\\begin{align*} - \\frac { \\hslash ^ { 2 } } { 2 m } k _ { \\pm } ^ { 2 } + \\frac { \\beta \\hslash ^ { 4 } } { 3 m } k _ { \\pm } ^ { 4 } = \\pm i \\lambda \\Leftrightarrow \\frac { 2 \\beta \\hslash ^ { 2 } } { 3 } k _ { \\pm } ^ { 4 } - k _ { \\pm } ^ { 2 } \\mp \\frac { 2 m i \\lambda } { \\hslash ^ { 2 } } = 0 , \\end{align*}"}
{"id": "7467.png", "formula": "\\begin{align*} S _ { n } : = \\mathcal { T } _ { \\left \\{ C _ { m , r } \\left ( n \\right ) \\left ( j \\right ) \\right \\} _ { j = 1 } ^ { r } } \\end{align*}"}
{"id": "6404.png", "formula": "\\begin{align*} \\{ 1 , \\ldots , \\ell + 1 \\} \\setminus \\{ \\tau _ { \\pi } ( 2 ) , \\ldots , \\tau _ { \\pi } ( \\ell + 1 ) \\} = \\{ 1 , \\ldots , \\ell + 1 \\} \\setminus \\{ j _ 1 + 1 , \\ldots , j _ { \\ell } + 1 \\} \\end{align*}"}
{"id": "6601.png", "formula": "\\begin{align*} \\left < \\tau ( \\beta ^ S _ h ) \\phi , \\phi \\right > = \\frac { 1 } { m _ { G _ S } ( B ^ S _ h ) } \\int _ { G _ S / L _ S } \\int _ { B ^ S _ h } \\phi ( g ^ { - 1 } x L _ S ) \\phi ( x L _ S ) \\ , d m _ { G _ S } ( g ) d \\nu ( x L _ S ) . \\end{align*}"}
{"id": "8203.png", "formula": "\\begin{align*} \\hat { H } \\psi = E \\psi , \\end{align*}"}
{"id": "1224.png", "formula": "\\begin{align*} A ( T ) & = \\begin{bmatrix} O & D _ { l ( T ) } & O & \\cdots & O & O \\\\ D _ { l ( T ) } ^ T & O & D _ { l ( T ) - 1 } & \\cdots & O & O \\\\ O & D _ { l ( T ) - 1 } ^ T & O & \\cdots & O & O \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ O & O & O & \\cdots & O & D _ 2 \\\\ O & O & O & \\cdots & D _ 2 ^ T & O \\end{bmatrix} , \\end{align*}"}
{"id": "5344.png", "formula": "\\begin{align*} | \\langle \\Lambda _ B [ f ] , [ g ] \\rangle | & = | B ( u _ { f - \\phi } , g - \\psi ) | \\leq C \\| u _ { f - \\phi } \\| _ { H ^ s ( \\R ^ n ) } \\| g - \\psi \\| _ { H ^ s ( \\R ^ n ) } \\\\ & \\leq C \\| f - \\phi \\| _ { H ^ s ( \\R ^ n ) } \\| g - \\psi \\| _ { H ^ s ( \\R ^ n ) } \\end{align*}"}
{"id": "8189.png", "formula": "\\begin{align*} P _ \\delta ( E ) = P ^ \\parallel _ \\delta ( E ^ \\parallel ) \\otimes P ^ \\perp _ \\delta ( E ^ \\perp ) \\end{align*}"}
{"id": "7813.png", "formula": "\\begin{align*} \\gamma \\star \\delta : = \\rho _ \\tau ( \\rho _ \\tau ( \\gamma ) \\cup \\rho _ \\tau ( \\delta ) ) . \\end{align*}"}
{"id": "1325.png", "formula": "\\begin{align*} \\beta ( t , d ) & = \\beta ( t ) = e ^ { - t } \\mbox { w h e n } \\phi _ d = e ^ { \\tilde \\tau ( z ) } , \\\\ \\beta ( t , d ) & = \\beta ( t ) = ( 1 + t ) ^ { - \\xi } , \\mbox { w h e n } \\phi _ d = ( 1 + \\tilde \\tau ( z ) ) ^ \\xi , \\ , \\xi > 1 . \\end{align*}"}
{"id": "6539.png", "formula": "\\begin{align*} \\inf \\limits _ { m \\in \\N } \\P \\{ m \\ \\mathrm { i s \\ d r y } \\} = \\inf \\limits _ { m \\in \\N } \\prod _ { i = 0 } ^ { m - 1 } \\left ( 1 - \\P \\{ \\ell ^ { ( A ) } _ i > m - i \\} \\right ) > 0 . \\end{align*}"}
{"id": "4465.png", "formula": "\\begin{align*} 1 = \\sum \\limits _ { k \\in \\mathbb Z } \\psi _ k ( \\xi ) , \\end{align*}"}
{"id": "8913.png", "formula": "\\begin{align*} \\begin{cases} - V '' + V = V ^ { p - 1 } , V \\ge 0 & , \\\\ \\sum _ { i = 1 } ^ m V _ i ' ( 0 ^ + ) = 0 ; \\end{cases} \\end{align*}"}
{"id": "3072.png", "formula": "\\begin{align*} c _ { n } ^ { \\ast } & = \\frac { ( m + n - 2 ) \\Gamma ( n / 2 ) } { ( 2 \\pi ) ^ { n / 2 } m } \\left [ \\int _ { 0 } ^ { \\infty } t ^ { n / 2 - 1 } \\left ( 1 + \\frac { 2 t } { m } \\right ) ^ { - ( m + n - 2 ) / 2 } \\mathrm { d } t \\right ] ^ { - 1 } \\\\ & = \\frac { ( m + n - 2 ) \\Gamma ( n / 2 ) } { ( m \\pi ) ^ { n / 2 } m B ( \\frac { n } { 2 } , ~ \\frac { m - 2 } { 2 } ) } , ~ i f ~ m > 2 \\end{align*}"}
{"id": "4854.png", "formula": "\\begin{align*} & \\mathcal { L } _ { V _ a } b - \\mathcal { L } _ { V _ b } a = - 2 d ^ \\Lambda a \\wedge d ^ \\Lambda b \\\\ + & \\iota _ X ( d ^ \\Lambda a \\wedge d b - d ^ \\Lambda b \\wedge d a ) + d \\iota _ X ( d ^ \\Lambda a \\wedge b - d ^ \\Lambda b \\wedge a ) + d d ^ \\Lambda b \\wedge \\iota _ X a - d d ^ \\Lambda a \\wedge \\iota _ X b \\end{align*}"}
{"id": "3862.png", "formula": "\\begin{align*} N : = & \\Big \\{ 0 , 1 , . . . , i \\Big \\} , E : = \\Big \\{ ( 1 , 0 ) , . . . ( i , 0 ) \\Big \\} , L : N \\to \\{ 1 , . . . , d \\} , \\\\ h _ 0 : = & \\{ 0 \\} \\cup \\Big \\{ j \\in \\{ 1 , . . . , i \\} : a _ j = 0 \\Big \\} , H : = \\Big \\{ \\big \\{ j \\in \\{ 1 , . . . , i \\} : a _ j = k \\big \\} : k = 1 , . . . , m [ a ] \\Big \\} . \\end{align*}"}
{"id": "1807.png", "formula": "\\begin{align*} \\tau _ k = \\bar \\theta _ k \\tau _ { k - 1 } \\geq \\tau _ { k - 1 } \\geq \\tau _ \\mathrm { m i n } , \\sigma _ k = \\bar \\theta _ k \\sigma _ { k - 1 } \\geq \\sigma _ { k - 1 } \\geq \\sigma _ \\mathrm { m i n } . \\end{align*}"}
{"id": "4592.png", "formula": "\\begin{align*} \\partial _ t \\mathcal { U } ( s , t ) = \\mathcal { L } ( t ) \\mathcal { U } ( s , t ) , \\mathcal { U } ( s , s ) = I . \\end{align*}"}
{"id": "3604.png", "formula": "\\begin{align*} & \\langle ( x _ 1 , \\ldots , x _ s ) , u _ i \\rangle \\geq x _ { s + 1 } = 0 , \\ i = 1 , \\ldots , m , \\ x _ { s + 1 } \\geq x _ { s + 3 } \\\\ & x _ i \\geq 0 , \\ i = 1 , \\ldots , s , \\ x _ { s + 3 } \\geq 0 \\\\ & d _ j x _ { s + 2 } - \\langle ( x _ 1 , \\ldots , x _ s ) , \\gamma _ j \\rangle = d _ j - x _ 1 \\gamma _ { j , 1 } = d _ j \\geq x _ { s + 3 } = 0 , \\ , x _ { s + 2 } = 1 , \\end{align*}"}
{"id": "656.png", "formula": "\\begin{align*} D P _ g ( w ) = - \\left ( D L _ { ( g , P ( g ) ) } ( 0 , \\cdot ) \\right ) ^ { - 1 } \\circ D L _ { ( g , P ( g ) ) } ( w , 0 ) . \\end{align*}"}
{"id": "2051.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : o p t i m a l c o u p l i n g j u m p T } X ^ i _ { T ^ i _ n } = \\mathsf { T } { \\Big ( \\overline { X } { } ^ i _ { T ^ i _ n } \\Big ) } , \\end{align*}"}
{"id": "8878.png", "formula": "\\begin{align*} f ( \\lambda ) = \\lambda ^ N + \\sum _ { k = 1 } ^ N ( - 1 ) ^ k \\lambda ^ { N - k } \\sum _ { 1 \\le j _ 1 < \\cdots < j _ k \\le N } \\ \\prod _ { \\ell = 1 } ^ k \\lambda _ { j _ \\ell } . \\end{align*}"}
{"id": "6988.png", "formula": "\\begin{align*} \\prod _ { k \\ge 1 } \\frac { \\mu _ k ^ 2 } { \\lambda _ { k + 1 } ^ 2 } = \\infty , \\end{align*}"}
{"id": "3067.png", "formula": "\\begin{align*} \\mathrm { ( I ) } ~ \\mathrm { M D T E } _ { ( \\boldsymbol { a } , \\boldsymbol { b } ) } ( \\mathbf { Y } ) & = \\boldsymbol { \\mu } + \\frac { \\mathbf { \\Sigma } ^ { \\frac { 1 } { 2 } } \\boldsymbol { \\delta } } { F _ { \\mathbf { Z } } ( \\boldsymbol { \\xi _ { a } } , \\boldsymbol { \\xi _ { b } } ) } , \\end{align*}"}
{"id": "4166.png", "formula": "\\begin{align*} \\Phi _ { \\nu , \\nu ' } ^ \\lambda ( z ) = \\lambda ^ { n / 2 } \\Phi ^ 1 _ { \\nu , \\nu ' } ( \\lambda ^ { 1 / 2 } z ) \\end{align*}"}
{"id": "8166.png", "formula": "\\begin{align*} \\varphi ( x , y ) : = \\psi _ 0 ( x ) \\psi _ 1 ( y ) \\end{align*}"}
{"id": "3281.png", "formula": "\\begin{align*} \\int _ D e ^ { - i \\varphi } q ( x ) u _ 1 u _ 2 \\ , d x = \\int _ D q ( x ) e ^ { i x \\cdot \\xi } \\ , d x + \\mathcal { R } ' ( \\xi , s ) , \\end{align*}"}
{"id": "4574.png", "formula": "\\begin{align*} F _ * ^ e \\left ( \\frac { s u ^ { p ^ e - 1 } } { u ^ { p ^ e } } \\right ) = \\frac { 1 } { u } \\cdot \\frac { F _ * ^ e ( s u ^ { p ^ e - 1 } ) } { 1 } = \\frac { F _ * ^ e ( s u ^ { p ^ e - 1 } ) } { u } . \\end{align*}"}
{"id": "5881.png", "formula": "\\begin{align*} e _ n ( { \\mathcal K } ) _ X { \\geq \\frac { C } { [ \\log _ 2 n ] ^ \\alpha } } , n = 1 , 2 , \\ldots , \\hbox { t h e n } d _ n ( { \\mathcal K } ) _ X \\geq C ' \\frac { 1 } { [ \\log _ 2 n ] ^ \\alpha } , \\end{align*}"}
{"id": "2783.png", "formula": "\\begin{align*} \\begin{aligned} \\tfrac { \\sigma _ 0 L } { h _ 0 B } = 1 - \\tfrac { 1 - \\kappa h _ 0 } { 2 \\left ( 1 - \\kappa \\right ) } \\ , \\ , , \\tfrac { \\alpha _ 0 } { B } = 1 \\end{aligned} \\end{align*}"}
{"id": "8819.png", "formula": "\\begin{align*} \\psi ( 0 ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\alpha _ { t _ 1 , t _ 2 } ( 0 ) d t _ 1 d t _ 2 \\end{align*}"}
{"id": "8761.png", "formula": "\\begin{align*} \\lfloor x \\rfloor & = \\max \\{ m \\in \\mathbb { Z } \\mid m \\leq x \\} , \\lceil x \\rceil = \\min \\{ n \\in \\mathbb { Z } \\mid n \\geq x \\} . \\end{align*}"}
{"id": "6277.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } ( - q ) ^ k A i _ q ( q ^ k x ) = \\frac { 1 } { 1 + q } \\ , _ 1 \\phi _ 1 ( 0 ; - q ^ 2 ; q , - x ) . \\end{align*}"}
{"id": "550.png", "formula": "\\begin{align*} \\sum \\limits _ { 1 \\leq n \\leq N } f _ k ( n ) = \\sum _ { w \\leq N } \\frac { g _ k ( w ) } { w } \\Big ( \\frac { N } { w } + O ( 1 ) \\Big ) , \\end{align*}"}
{"id": "1385.png", "formula": "\\begin{align*} \\begin{cases} i v _ t + \\bigtriangleup v + i \\gamma v - \\mathcal { R } ^ 2 [ v , \\cdots , v ] - \\mathcal { N R } [ v , \\cdots , v ] = F \\\\ v ( x , 0 ) = u _ 0 \\in H ^ 1 _ x ( \\mathbb { T } ) , \\end{cases} \\end{align*}"}
{"id": "1456.png", "formula": "\\begin{align*} \\Delta ( z ) = ( - 1 ) ^ { r m } \\dfrac { 1 } { ( r m n + r m ) ! } \\left ( \\dfrac { d } { d z } \\right ) ^ { r m n + r m } P _ { m , r m } ( z ) \\cdot { \\rm { d e t } } { \\begin{pmatrix} \\vec { q } _ { 0 } \\ \\cdots \\ \\vec { q } _ { r m - 1 } \\end{pmatrix} } \\enspace . \\end{align*}"}
{"id": "7940.png", "formula": "\\begin{align*} \\tilde { \\mathbb L } ^ \\vee _ \\pm = \\left ( H ^ 2 ( X _ \\pm ; \\mathbb R ) \\cap \\tilde { \\mathbb L } ^ \\vee _ \\pm \\right ) \\oplus \\bigoplus _ { j \\in S _ \\pm } \\mathbb Z D _ j . \\end{align*}"}
{"id": "4923.png", "formula": "\\begin{align*} \\nu _ { ; i } ^ \\alpha = h _ i ^ k X _ { ; k } ^ \\alpha . \\end{align*}"}
{"id": "3149.png", "formula": "\\begin{align*} e _ { \\Phi } \\left ( \\rho \\right ) = \\lim \\limits _ { L \\rightarrow \\infty } \\frac { \\rho \\left ( U _ { L } ^ { \\Phi } \\right ) } { \\left \\vert \\Lambda _ { L } \\right \\vert } = \\rho \\left ( \\mathfrak { e } _ { \\Phi } \\right ) \\ , \\end{align*}"}
{"id": "2294.png", "formula": "\\begin{align*} I _ 2 = & \\left [ \\overline u _ e ^ i ( x ) \\varphi ( x , 0 ) + \\overline h _ e ^ i ( x ) \\psi ( x , 0 ) \\right ] x ^ { - \\frac { 1 } { 2 } - 2 \\sigma _ i } \\\\ \\leq & | ( \\varphi , \\psi ) ( x , 0 ) x ^ { - \\frac { 3 } { 4 } + \\sigma _ { i - 1 } - \\frac { 1 } { 2 } - 2 \\sigma _ i } | \\leq \\delta _ 0 | ( \\varphi , \\psi ) ( x , 0 ) x ^ { - \\frac { 3 } { 4 } - \\sigma _ i } | ^ 2 + C | x ^ { - \\frac { 1 } { 2 } - \\sigma ' } | ^ 2 , \\end{align*}"}
{"id": "1881.png", "formula": "\\begin{align*} A _ { [ n , j ] } = \\sum _ { k \\in \\mathbb { Z } } A _ { [ k , i ] } A _ { [ n - k - 1 , j - i - 1 ] } ^ { ( i + 1 ) } , n \\in \\mathbb { Z } . \\end{align*}"}
{"id": "1788.png", "formula": "\\begin{align*} y _ n ^ * ( u _ i ) = ( P _ n ^ * \\ : y ^ * ) ( u _ i ) = y ^ * ( P _ n \\ : u _ i ) = 0 \\end{align*}"}
{"id": "7128.png", "formula": "\\begin{align*} s _ 1 ( y ) : = 1 s _ n ( y ) : = \\sum _ { k \\geq 1 } s ( n , k ) y ^ k ( n \\geq 2 ) , \\end{align*}"}
{"id": "3642.png", "formula": "\\begin{align*} \\tau _ { - x _ n } \\mu ^ i _ n = \\tau _ { - x _ n } \\mu _ n - \\sum _ { 0 \\le j \\leq i } \\tau _ { x ^ j _ n - x _ n } \\mu ^ j , \\end{align*}"}
{"id": "4326.png", "formula": "\\begin{align*} \\int _ \\Sigma \\omega + \\sum _ { } = 0 . \\end{align*}"}
{"id": "212.png", "formula": "\\begin{align*} \\mathcal { K } ^ { n } _ { 2 , j } = \\left \\{ \\pi _ { X , Y _ { [ 0 : 2 ] } , Z _ j } \\in \\mathcal { P } \\left ( \\mathcal { X } \\times \\mathcal { Y } _ { [ 0 : 2 ] } \\times \\mathcal { Z } _ j \\right ) \\ : \\ \\pi _ { Z _ j } \\stackrel { \\delta ^ { \\prime } _ { n } } { \\approx } p _ { Z _ j } \\right \\} . \\end{align*}"}
{"id": "1887.png", "formula": "\\begin{align*} \\begin{aligned} A _ { 0 } ^ { ( k ) } ( z ) & = \\frac { 1 } { z - a _ { k } ^ { ( 0 ) } - \\sum _ { j = 1 } ^ { p } a _ { k } ^ { ( j ) } \\ , A _ { j - 1 } ^ { ( k + 1 ) } ( z ) } \\\\ A _ { j } ^ { ( k ) } ( z ) & = A _ { 0 } ^ { ( k ) } ( z ) \\ , A ^ { ( k + 1 ) } _ { j - 1 } ( z ) 1 \\leq j \\leq p . \\end{aligned} \\end{align*}"}
{"id": "4668.png", "formula": "\\begin{align*} h _ { i j } ( \\omega ) = \\sum \\limits _ { k = 1 } ^ d \\frac { \\partial \\psi ^ k } { \\partial \\omega _ i } ( \\omega ) \\frac { \\partial \\psi ^ k } { \\partial \\omega _ j } ( \\omega ) , i , j = 1 , . . . , m - 1 ; \\end{align*}"}
{"id": "543.png", "formula": "\\begin{align*} g _ k ( n ) : = \\sum _ { \\substack { r d = n \\\\ d \\ { \\rm { i s } } \\ k - { \\rm { f r e e } } } } \\mu ( r ) , \\end{align*}"}
{"id": "2405.png", "formula": "\\begin{align*} J _ 2 ( t ) & = n \\sum _ { l = 1 } ^ { n - 1 } ( - \\lambda _ { l , n } ) ^ { - 1 } \\| \\{ \\Sigma _ n ( \\mathbb U ( t ) ) - \\Sigma _ n ( U ( t ) ) \\} e _ l \\| ^ 2 \\\\ & \\le C \\left \\| \\mathbb U ( t ) - U ( t ) \\right \\| ^ 2 \\le C \\| \\mathbb U ( t ) - \\tilde U ( t ) \\| ^ 2 + C \\| E ( t ) \\| ^ 2 , \\end{align*}"}
{"id": "3506.png", "formula": "\\begin{align*} c _ { ( 2 ) } ^ { \\ast } = \\frac { 1 } { ( 2 \\pi ) ^ { 1 / 2 } \\Psi _ { 1 } ^ { \\ast } ( - 1 , \\frac { 3 } { 2 } , 1 ) } . \\end{align*}"}
{"id": "1216.png", "formula": "\\begin{align*} \\Delta ( E ) = E \\otimes 1 + 1 \\otimes E \\end{align*}"}
{"id": "2675.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } ( \\ell - k - 1 ) ( n - \\ell - 1 ) ( p - 1 ) ^ 2 p ^ { n - 4 } \\\\ & = ( p - 1 ) ^ 2 p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { \\ell = k + 1 } ^ { n - 2 } ( \\ell - k - 1 ) ( n - \\ell - 1 ) \\\\ & = ( p - 1 ) ^ 2 p ^ { n - 4 } \\sum _ { k = 1 } ^ { n - 3 } \\binom { n - k - 1 } { 3 } \\\\ & = ( p - 1 ) ^ 2 p ^ { n - 4 } \\binom { n - 1 } { 4 } . \\end{align*}"}
{"id": "3324.png", "formula": "\\begin{align*} c ( x ) = \\left \\{ \\begin{array} { c c } 2 & x _ 1 \\in ( - 2 . 5 , 1 ) x _ 2 \\in ( 1 . 5 , 2 . 5 ) \\\\ 1 & . \\end{array} \\right . \\end{align*}"}
{"id": "2617.png", "formula": "\\begin{align*} n + m = 2 p - a - 2 k \\end{align*}"}
{"id": "121.png", "formula": "\\begin{align*} \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } : = \\| E \\| _ { L ^ 1 } + \\| E \\| _ { L ^ \\infty } . \\end{align*}"}
{"id": "7805.png", "formula": "\\begin{align*} \\gamma : = \\kappa ( E _ 0 ) \\sqrt { t d _ { X _ 0 \\times Y _ 0 } } \\end{align*}"}
{"id": "1757.png", "formula": "\\begin{align*} \\begin{array} { c c c c c c } ( n _ m + 0 ) _ { m = 1 } ^ \\infty & ( n _ m + 1 ) _ { m = 1 } ^ \\infty & ( n _ m + 2 ) _ { m = 2 } ^ \\infty & ( n _ m + 3 ) _ { m = 3 } ^ \\infty & ( n _ m + 4 ) _ { m = 4 } ^ \\infty & \\dots \\\\ 1 & 2 & & & & \\\\ 3 & 4 & 5 & & & \\\\ 6 & 7 & 8 & 9 & & \\\\ 1 0 & 1 1 & 1 2 & 1 3 & 1 4 & \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{array} \\end{align*}"}
{"id": "5270.png", "formula": "\\begin{align*} \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { k _ 1 ( 0 ) + r } a ^ { \\widehat { Q } ^ { + r } } + \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { k _ 2 ( 0 ) + s } b ^ { \\widehat { Q } ^ { + s } } = 1 + \\sum _ { \\substack { j = 1 \\\\ 2 \\notin I _ j } } ^ h \\left ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { Q } _ j } + \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { Q } _ j } \\right ) . \\end{align*}"}
{"id": "4034.png", "formula": "\\begin{align*} \\Lambda ( a _ n ) & = \\Lambda ( D ) + ( a _ n - D ) \\Lambda ' ( D ) + \\frac { ( a _ n - D ) ^ 2 } { 2 } \\Lambda '' ( D ) \\\\ & + \\frac { ( a _ n - D ) ^ 3 } { 6 } \\Lambda ''' ( D ) + O ( | a _ n - D | ^ 4 ) . \\end{align*}"}
{"id": "2089.png", "formula": "\\begin{align*} \\pi ( y ) _ { i , j } = \\begin{cases} ( y _ 2 ) _ { i , j } & \\quad ( y _ 1 ) _ { i , j } = 1 \\\\ ( y _ 3 ) _ { i , j } & \\quad ( y _ 1 ) _ { i , j } \\neq 1 \\\\ \\end{cases} \\end{align*}"}
{"id": "8185.png", "formula": "\\begin{align*} P _ \\delta ( E ) \\psi = 0 \\end{align*}"}
{"id": "1387.png", "formula": "\\begin{align*} T = T ( \\| u _ 0 \\| _ { H ^ 1 _ x ( \\mathbb { T } ) } , \\| f \\| _ { H ^ { 1 } _ x ( \\mathbb { T } ) } , \\gamma ) \\end{align*}"}
{"id": "2533.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( L _ { Z _ 1 } g ) ( X _ 1 , Y _ 1 ) = \\frac { 1 } { 2 } \\Big \\{ g ( \\nabla _ { X _ 1 } Z _ 1 , Y _ 1 ) + g ( \\nabla _ { Y _ 1 } Z _ 1 , X _ 1 ) \\Big \\} . \\end{align*}"}
{"id": "8043.png", "formula": "\\begin{align*} W ( \\lambda ^ { w _ 1 } x _ 1 , \\ldots , \\lambda ^ { w _ N } x _ N ) = \\lambda ^ d W ( x _ 1 , \\ldots , x _ N ) , \\end{align*}"}
{"id": "7213.png", "formula": "\\begin{align*} I _ { 1 3 } ( x , t ) , I _ { 1 4 } ( x , t ) \\le C \\left \\{ \\begin{array} { l l } 1 & \\mbox { i f } \\ a > 0 , \\\\ \\log ( T + 3 R ) & \\mbox { i f } \\ a = 0 , \\\\ ( T + R ) ^ { - a } & \\mbox { i f } \\ a < 0 . \\\\ \\end{array} \\right . \\end{align*}"}
{"id": "3936.png", "formula": "\\begin{align*} \\mathbb { E } [ | \\hat { \\pi } _ { h , n } ( x ) - \\pi ( x ) | ^ 2 ] & \\le c ( h _ 1 ^ { 2 \\beta _ 1 } + h _ 2 ^ { 2 \\beta _ 2 } ) + \\frac { c } { T _ n } \\sum _ { j = 1 } ^ 2 | \\log h _ j | + \\frac { c } { T _ n } \\frac { \\Delta _ n } { h _ 1 h _ 2 } \\\\ & \\le c ( h _ 1 ^ { 2 \\beta _ 1 } + h _ 2 ^ { 2 \\beta _ 2 } ) + c ( \\frac { 1 } { n } ) ^ { 1 - \\frac { 1 } { \\bar { \\beta } + 1 } } \\frac { \\sum _ { j = 1 } ^ 2 | \\log h _ j | } { \\log ( n \\Delta _ n ) } + \\frac { c } { n h _ 1 h _ 2 } . \\end{align*}"}
{"id": "8775.png", "formula": "\\begin{align*} Y _ O ( p _ 1 , G ) = \\tilde { Y } _ O ( p _ 1 , G ) = Y _ R ( p _ 1 , G ) = \\tilde { Y } _ R ( p _ 1 , G ) = C ( p _ 1 ) = 1 . \\end{align*}"}
{"id": "6817.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } u '''' v \\ , d x = \\int _ { 0 } ^ { 1 } u '' v '' d x . \\end{align*}"}
{"id": "5123.png", "formula": "\\begin{align*} 0 = V _ 0 \\subseteq V _ 1 \\subseteq \\ldots \\subseteq V _ n = S H _ * ( X ; \\Q ) \\end{align*}"}
{"id": "1032.png", "formula": "\\begin{align*} R ( \\lambda , A ) = ( \\lambda - A ) ^ { - 1 } \\end{align*}"}
{"id": "5150.png", "formula": "\\begin{align*} [ D \\vec { x } ] _ { v _ i } = \\sum _ { j = 1 } ^ n d _ G ( v _ i , v _ j ) \\vec { x } _ { v _ j } , \\end{align*}"}
{"id": "6117.png", "formula": "\\begin{align*} P ( z ) = a _ n z ^ n + a _ { n - 1 } z ^ { n - 1 } + \\cdots + a _ 1 z + a _ 0 , \\ , \\ , a _ j \\in \\mathbb C _ p , \\ , \\ , j = 0 , 1 , \\dots , n . \\end{align*}"}
{"id": "4538.png", "formula": "\\begin{align*} D _ { m , n } ( x , y ) & = \\sigma _ { m , 2 m ; n , 2 n } ( f ; x , y ) - f ( x , y ) \\\\ & = \\frac { 4 } { m n \\pi ^ { 2 } } \\int _ { 0 } ^ { \\pi } \\int _ { 0 } ^ { \\pi } \\phi _ { x , y } ( t _ { 1 } , t _ { 2 } ) \\frac { S ( t _ { 1 } , t _ { 2 } ) } { \\left ( 4 \\sin \\frac { t _ { 1 } } { 2 } \\sin \\frac { t _ { 2 } } { 2 } \\right ) ^ { 2 } } d t _ { 1 } d t _ { 2 } , \\end{align*}"}
{"id": "7107.png", "formula": "\\begin{align*} \\tau _ x ( f \\star g ) = \\tau _ x ( g \\star f ) , \\end{align*}"}
{"id": "8108.png", "formula": "\\begin{align*} \\P _ N \\Biggl ( \\prod _ { i = 1 } ^ { N } ( \\Delta \\tau _ i ) ^ { - \\frac d 2 } > C ^ { - N } R \\Biggr ) \\geq C ^ N N ^ { ( 1 + \\frac d 2 ) N } R ^ { - 1 - \\frac 2 d } \\biggl ( \\log \\frac { t R ^ { \\frac 2 { d N } } } { C N } \\biggr ) ^ { N - 1 } . \\end{align*}"}
{"id": "6192.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\omega ^ { \\prime } = \\omega _ { N } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\phi _ { T _ { 0 } } \\geq 0 . \\end{array} \\end{align*}"}
{"id": "4571.png", "formula": "\\begin{align*} ( F _ * ^ e \\psi ) ( F _ * ^ { e + d } r ) = ( F _ * ^ e \\psi ) ( F _ * ^ e ( F _ * ^ d r ) ) = F _ * ^ e ( \\psi ( F _ * ^ d r ) ) . \\end{align*}"}
{"id": "140.png", "formula": "\\begin{align*} & \\| C ( C \\star C ^ 2 ) \\| _ { L ^ 1 } \\leq \\| C ^ 2 \\| ^ 2 _ { L ^ 1 } \\leq \\frac { c } { m _ t ^ 4 } { \\bf 1 } _ { d = 2 } + \\frac { c } { m ^ 2 _ t } { \\bf 1 } _ { d = 3 } , \\\\ & \\| C ( C \\star C ) \\| _ { L ^ 1 } \\leq \\| C ^ 2 \\| _ { L ^ 1 } \\| C \\| _ { L ^ 1 } \\leq \\frac { c } { m _ t ^ 4 } { \\bf 1 } _ { d = 2 } + \\frac { c } { m _ t ^ { 3 } } { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "8385.png", "formula": "\\begin{align*} F ^ * ( 1 0 , 2 , 4 ) = 1 2 8 . \\end{align*}"}
{"id": "4983.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E [ N ^ n _ { \\eta _ n ( t ) } ( W _ b - W _ a ) ] = 0 . \\end{align*}"}
{"id": "6575.png", "formula": "\\begin{align*} f _ i ( y , w _ i ) - f ( x , v _ i ) = f _ i ( y , t _ i p _ i ) = t _ i f _ i ( y , p _ i ) \\in t _ i Q _ i \\ , . \\end{align*}"}
{"id": "8935.png", "formula": "\\begin{align*} \\mathcal F [ u ] : = \\det [ D ^ 2 u - g _ { x x } ( \\cdot , Y ( \\cdot , u , D u ) , Z ( \\cdot , u , D u ) ) ] = \\det E ( \\cdot , Y , Z ) \\psi ( \\cdot , u , D u ) , \\end{align*}"}
{"id": "4744.png", "formula": "\\begin{align*} \\widehat { \\lambda } ( ( n + 1 ) P ) = - \\widehat { \\lambda } ( ( n - 1 ) P ) + 2 \\widehat { \\lambda } ( n P ) + 2 \\widehat { \\lambda } ( P ) + v ( x ( n P ) - x ( P ) ) - v ( \\Delta ) / 6 . \\end{align*}"}
{"id": "3406.png", "formula": "\\begin{align*} \\sigma ( g _ 1 , g _ 2 ) + \\sigma ( g _ 1 g _ 2 , g _ 3 ) = \\sigma ( g _ 1 , g _ 2 g _ 3 ) + \\sigma ( g _ 2 , g _ 3 ) . \\end{align*}"}
{"id": "3720.png", "formula": "\\begin{align*} \\phi ^ { [ \\lambda _ k \\lambda '' v ] } : = ( \\lambda _ k \\lambda '' ) \\cdot \\phi . \\end{align*}"}
{"id": "2535.png", "formula": "\\begin{align*} g ( X _ 1 , Y _ 1 ) = g ( a _ 1 e _ 1 + a _ 2 e _ 2 , a _ 3 e _ 1 + a _ 4 e _ 2 ) = ( a _ 1 a _ 3 e ^ { - 2 x _ 2 } + a _ 2 a _ 4 ) , \\end{align*}"}
{"id": "8632.png", "formula": "\\begin{align*} \\varphi ( v ) \\psi = \\frac { d } { d t } \\bigg | _ { t = 0 } \\Phi ( { \\exp ( - t v ) } , \\psi ) , \\forall \\psi \\in D _ \\mathfrak { V } , \\forall v \\in \\mathfrak { g } . \\end{align*}"}
{"id": "5140.png", "formula": "\\begin{align*} \\mathcal { G } ( 2 c ) \\oplus \\mathcal { G } ( 1 2 c ) = \\mathcal { G } ( 4 c ) \\oplus \\mathcal { G } ( 1 0 c ) = \\mathcal { G } ( 6 c ) \\oplus \\mathcal { G } ( 8 c ) = 6 \\end{align*}"}
{"id": "4406.png", "formula": "\\begin{align*} \\int _ 0 ^ { s _ n } r ^ { m _ n } d \\mu = b \\int _ { s _ n } ^ R r ^ { m _ n } d \\mu \\ \\ \\ \\mbox { a n d } \\ \\ \\ \\int _ 0 ^ { s _ n } r ^ { m _ { n + 1 } } d \\mu = \\int _ { s _ n } ^ R r ^ { m _ { n + 1 } } d \\mu . \\end{align*}"}
{"id": "2182.png", "formula": "\\begin{align*} z = H ( \\omega ( z ) ) = \\omega ( z ) + ( \\varphi _ { \\mu } \\circ F _ { \\mu \\boxplus \\nu } ) ( z ) , z \\in \\mathbb { C } ^ { + } . \\end{align*}"}
{"id": "6604.png", "formula": "\\begin{align*} \\Lambda _ { \\Omega } ( t ) = C ( \\Omega ) t ^ d + O ( t ^ { d - 1 } ) \\end{align*}"}
{"id": "6116.png", "formula": "\\begin{align*} O _ j ^ { h _ 1 ^ { - 1 } } \\subseteq ( H \\cap P _ i ^ { g _ j } ) ^ { h _ 1 ^ { - 1 } } = H ^ { h _ 1 ^ { - 1 } } \\cap P _ i ^ { p '' g } = H \\cap P _ i ^ g \\end{align*}"}
{"id": "4834.png", "formula": "\\begin{align*} \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } g . \\varphi _ k \\in R _ \\mathrm { r a t } ( H _ k ) = \\mathcal { S } ( H _ k ) \\ ; . \\end{align*}"}
{"id": "7143.png", "formula": "\\begin{align*} \\mathcal { I } _ D & = \\langle x _ 1 ^ { a } , x _ 2 ( x _ 1 + x _ 2 ) ^ { b - 1 } , x _ 3 ( x _ 1 + x _ 2 + x _ 3 ) ^ { a - 1 } \\rangle , \\\\ \\mathcal { I } _ { \\widetilde { D } } & = \\langle y _ 1 ^ { a } , y _ 2 ( y _ 1 + y _ 2 ) ^ { a - 1 } , y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { b - 1 } \\rangle . \\end{align*}"}
{"id": "7734.png", "formula": "\\begin{align*} B _ { j i } ( \\sigma : 0 ) & = \\tilde B _ { j i } ( \\sigma : 0 ) + B _ { j j } ( \\sigma : k _ L + 1 ) B _ { j i } ( k _ L ) B _ { i i } ( k _ L : 0 ) \\cr & \\geq \\frac { 1 } { 2 } \\eta _ i ( \\eta _ j - \\varepsilon ) ( \\delta - \\varepsilon ) + \\eta _ j \\varepsilon \\eta _ i \\cr & = \\frac { 1 } { 2 } \\eta _ i \\varepsilon ^ 2 + \\frac { 1 } { 2 } \\eta _ i ( \\eta _ j - \\delta ) \\varepsilon + \\frac { 1 } { 2 } \\eta _ i \\eta _ j \\delta \\geq \\frac { 1 } { 2 } \\eta _ i \\eta _ j \\delta , \\end{align*}"}
{"id": "3950.png", "formula": "\\begin{align*} \\xi _ { ( w _ i ) _ i , \\phi } ^ { \\mathbf { G } } ( y _ { \\sigma ( 1 ) } ^ 1 , \\dots , y _ { \\sigma ( d ) } ^ d ) = \\int _ { \\R ^ { d ( d - 1 ) } } \\phi ( y ^ 1 ) \\prod _ { i = 1 } ^ { d - 1 } g _ { ( w _ { i + 1 } - w _ i ) \\tilde { a } ( y _ i ) } ( y ^ { i + 1 } - y ^ i ) d \\widehat { y } ^ 1 \\dots d \\widehat { y } ^ d . \\end{align*}"}
{"id": "5429.png", "formula": "\\begin{align*} \\psi _ j ( \\xi ) = \\psi _ 0 ( \\xi / 2 ^ j ) - \\psi _ 0 ( \\xi / 2 ^ { j - 1 } ) . \\end{align*}"}
{"id": "4323.png", "formula": "\\begin{align*} \\mathcal { S } ( L ) = \\int _ { \\mathcal { M } } \\mathcal { I } , \\mathcal { I } = \\int _ { \\Sigma } e ^ { - i \\hat { \\theta } } F \\omega + \\sum _ { } e ^ { - i \\hat { \\theta } } F f | ^ + _ - . \\end{align*}"}
{"id": "1693.png", "formula": "\\begin{align*} ( { \\rm r e s p . } ~ ~ x ( y z ) = ( x y ) z + ( - 1 ) ^ { \\vert x \\vert \\vert y \\vert } ( y x ) z ) , \\end{align*}"}
{"id": "7784.png", "formula": "\\begin{align*} \\sin \\theta \\big ( R ( \\tilde \\Xi _ 1 ) , R ( \\Xi _ 1 ) \\big ) \\le \\frac { \\lVert E \\rVert _ S } { \\delta } = { \\mathcal P } _ { B 1 } , \\end{align*}"}
{"id": "198.png", "formula": "\\begin{align*} x ( \\{ v _ 1 , v _ 3 \\} ) = x ( \\{ v _ 2 , v _ 3 \\} ) = 0 \\end{align*}"}
{"id": "4260.png", "formula": "\\begin{align*} \\lim _ { r \\to + \\infty } r ^ { \\frac { n - 1 } { 2 } } e ^ { r } U ( r ) = \\mathfrak u > 0 \\ \\hbox { a n d } \\ \\lim _ { r \\to + \\infty } \\frac { U ' ( r ) } { U ( r ) } = - 1 , \\end{align*}"}
{"id": "3442.png", "formula": "\\begin{align*} 0 = H ^ { n - 1 } ( \\tilde { \\mathbb { P } } , \\Omega _ { \\tilde { \\mathbb { P } } } ^ 1 \\otimes \\mathcal { O } ( \\tilde { Y } + K _ { \\tilde { \\mathbb { P } } } ) ) = H ^ { n } ( \\tilde { \\mathbb { P } } , \\Omega _ { \\tilde { \\mathbb { P } } } ^ 1 \\otimes \\mathcal { O } ( \\tilde { Y } + K _ { \\tilde { \\mathbb { P } } } ) ) \\end{align*}"}
{"id": "3441.png", "formula": "\\begin{align*} N _ { Y _ f / \\mathcal { X } } \\cong \\bigoplus \\limits _ { i = 1 } ^ { l ( \\Delta ) - 1 } \\mathcal { O } _ { Y _ f } \\cong T _ { B , f } \\otimes _ { \\mathbb { C } } \\mathcal { O } _ { Y _ f } . \\end{align*}"}
{"id": "606.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } u ( i ) & = \\int _ 0 ^ n u ( x ) d x + \\frac { 1 } { 2 } \\big ( u ( 0 ) + u ( n ) \\big ) + \\sum _ { j = 1 } ^ { M } \\frac { B _ { 2 j } } { ( 2 j ) ! } \\big ( u ^ { ( 2 j - 1 ) } ( n ) - u ^ { ( 2 j - 1 ) } ( 0 ) \\big ) \\\\ & + \\frac { 1 } { ( 2 M + 1 ) ! } \\int _ 0 ^ n B _ { 2 M + 1 } ( x - [ x ] ) u ^ { ( 2 M + 1 ) } ( x ) d x , \\end{align*}"}
{"id": "5187.png", "formula": "\\begin{align*} \\widehat { I } _ 0 = I _ 0 \\cup \\mathcal { R } . \\end{align*}"}
{"id": "1069.png", "formula": "\\begin{align*} \\lim _ { | y | \\to \\infty } \\| R ( x + i y , A _ 2 ) - R ( x + i y , A _ 1 ) \\| _ q = 0 \\forall x < - \\omega _ 0 ( A _ j ) . \\end{align*}"}
{"id": "425.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\frac { 1 + x ^ { 2 } } { | s + i t - x | ^ { 2 } } d \\sigma ( x ) = 1 , t = f _ { \\mu } ( s ) , \\end{align*}"}
{"id": "2615.png", "formula": "\\begin{align*} \\begin{cases} n = p - k - a \\ , \\ \\ m = q - k \\\\ n = p - k - a + 1 \\ , \\ \\ m = q - k - 1 \\\\ \\vdots \\\\ n = p - k \\ , \\ \\ m = q - a - k \\\\ \\end{cases} \\ . \\end{align*}"}
{"id": "846.png", "formula": "\\begin{align*} x _ { B , m } ^ { r , p o s t } = \\int _ { x _ { m } ^ { r } } x _ { m } ^ { r } \\hat { p } ( x _ { m } ^ { r } | \\boldsymbol { y } ) , \\end{align*}"}
{"id": "3414.png", "formula": "\\begin{align*} G = \\langle g _ 1 , \\ldots , g _ r | r _ 1 , \\ldots , r _ t \\rangle , \\end{align*}"}
{"id": "267.png", "formula": "\\begin{align*} L _ { i j k } ^ { 1 } = L _ { i j k } ( 1 ) . \\end{align*}"}
{"id": "7680.png", "formula": "\\begin{align*} C : = \\left \\{ y \\in \\Re ^ { m } | \\ , - y _ { \\max } e \\leq y \\leq y _ { \\max } e \\right \\} \\mbox { a n d } D : = \\left \\{ Z \\in \\S ^ { d } | \\ , O \\preceq Z \\preceq z _ { \\max } I \\right \\} , \\end{align*}"}
{"id": "3354.png", "formula": "\\begin{align*} \\lambda _ { 1 , 2 } ^ s ( t ) = \\left \\{ \\begin{aligned} & \\lambda _ 1 ( 2 s t + 1 - s ) & : 0 \\leq t \\leq \\frac { 1 } { 2 } \\\\ & \\lambda _ 2 ( 2 t - 1 ) & : \\frac { 1 } { 2 } \\leq t \\leq 1 \\\\ \\end{aligned} \\right \\} \\lambda _ i ^ s ( t ) = \\lambda _ i ( t ( 1 - s ) ) . \\end{align*}"}
{"id": "7976.png", "formula": "\\begin{align*} & H _ { ( X _ + , D _ + ) } ( y _ 1 , y _ 2 ) \\\\ = & e ^ { \\frac { \\xi _ + \\log y _ 1 + h _ + \\log y _ 2 } { 2 \\pi i } } \\sum _ { d _ 1 , d _ 2 \\geq 0 } \\frac { y _ 1 ^ { d _ 1 } y _ 2 ^ { d _ 2 } [ \\textbf { 1 } ] _ { ( d _ 2 , \\cdots , d _ 2 ) } } { \\Gamma ( 1 + \\frac { h _ + } { 2 \\pi i } + d _ 2 ) ^ { r ^ \\prime + 1 } \\Gamma ( 1 + \\frac { \\xi _ + - h _ + } { 2 \\pi i } + d _ 1 - d _ 2 ) ^ { r ^ \\prime + 1 } \\Gamma ( 1 + \\frac { \\xi _ + } { 2 \\pi i } + d _ 1 ) } . \\end{align*}"}
{"id": "4273.png", "formula": "\\begin{align*} V _ L = \\bigoplus _ i V _ { \\lambda _ i } . \\end{align*}"}
{"id": "4150.png", "formula": "\\begin{align*} ( X _ j f ) ^ \\mu = X _ j ^ \\mu f ^ \\mu , \\end{align*}"}
{"id": "7059.png", "formula": "\\begin{align*} Z ^ \\pm = \\begin{cases} \\dot { x } = - y + \\sum _ { k = 2 } ^ { n } P ^ \\pm _ k ( x , y ) , \\\\ \\dot { y } = x + \\sum _ { k = 2 } ^ { n } Q ^ \\pm _ k ( x , y ) , \\end{cases} \\ \\ ( x , y ) \\in \\Sigma ^ { \\pm } , \\end{align*}"}
{"id": "5409.png", "formula": "\\begin{align*} \\Delta ^ 2 _ h ( f \\ast \\rho _ { \\epsilon } ) = ( \\Delta ^ 1 _ h f ) \\ast ( \\Delta ^ 1 _ h \\rho _ { \\epsilon } ) . \\end{align*}"}
{"id": "2304.png", "formula": "\\begin{align*} \\begin{cases} N _ 1 \\ { \\rm i s ~ c h o s e n ~ w i t h ~ } N _ 1 - M _ 1 = 1 0 0 , \\\\ N _ 2 \\ { \\rm i s ~ c h o s e n ~ w i t h ~ } N _ 2 - M _ 2 = 2 N _ 1 , \\\\ N _ k = N _ 2 , \\ k = 3 , \\cdots , 6 , \\end{cases} \\end{align*}"}
{"id": "3293.png", "formula": "\\begin{align*} u _ { A , q } ^ s ( x , y ) & = \\frac { 1 } { 4 \\pi } \\frac { e ^ { i k \\vert x \\vert } } { \\vert x \\vert } \\frac { e ^ { i k \\vert y \\vert } } { \\vert y \\vert } u ^ \\infty _ { - A , q } \\left ( \\widehat { y } , - \\widehat { x } \\right ) + \\frac { O ( 1 ) } { | x | ^ 2 } \\| u _ { A , q } ( \\cdot , y ) \\| _ { L ^ 2 ( D ) } \\cr & + \\frac { O ( 1 ) } { | x | | y | ^ 2 } \\| u _ { - A , q } ( \\cdot , - \\hat x ) \\| _ { L ^ 2 ( D ) } . \\end{align*}"}
{"id": "1958.png", "formula": "\\begin{align*} \\mathbf { 1 } = ( 1 , \\ldots , 1 ) . \\end{align*}"}
{"id": "6577.png", "formula": "\\begin{align*} E ' : = \\bigcup _ { x \\in M } ( E _ x ) ' \\ , ; \\end{align*}"}
{"id": "6285.png", "formula": "\\begin{align*} \\int \\frac { f ( x ) } { g ( x / q ) } v ( x / q ) r ( x ) y ( x ) d _ q x = \\frac { f ( x / q ) } { g ( x / q ) } \\Big ( y ( x / q ) - v ( x / q ) D _ { q ^ { - 1 } } y ( x ) \\Big ) , \\end{align*}"}
{"id": "7695.png", "formula": "\\begin{align*} \\int _ { X } f ( x ) d M _ { \\mathcal { R } } ( \\nu ) = \\int _ { X } L _ { \\mathcal { R } } ( f ) ( x ) d \\nu , \\end{align*}"}
{"id": "7802.png", "formula": "\\begin{align*} \\psi _ E : = \\widetilde { H } _ 0 ( [ \\kappa ( E ) \\sqrt { t d _ { X \\times Y } } ] _ * ) . \\end{align*}"}
{"id": "8329.png", "formula": "\\begin{align*} \\delta ( P _ { k , l } ) = ( \\beta _ k - \\beta _ l ) P _ { k , l } \\ , . \\end{align*}"}
{"id": "1605.png", "formula": "\\begin{align*} \\frac { \\partial C } { \\partial z ^ i _ { \\epsilon } } v ^ i = 0 , \\end{align*}"}
{"id": "7782.png", "formula": "\\begin{align*} \\min _ { \\mathbf { y } \\in { \\mathcal C } _ 1 } \\lVert \\mathbf { x } - \\mathbf { y } \\rVert _ 2 = \\lVert ( I - P _ { { \\mathcal C } _ 1 } \\mathbf { x } ) \\rVert _ 2 . \\end{align*}"}
{"id": "6209.png", "formula": "\\begin{align*} \\int f ( x ) h ( x ) \\left ( u ' ( x ) + u ^ 2 ( x ) + p ( x ) u ( x ) + r ( x ) \\right ) y ( x ) d x = f ( x ) h ( x ) \\left ( u ( x ) y ( x ) - y ' ( x ) \\right ) , \\end{align*}"}
{"id": "7886.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ n r _ i ^ { s _ { i , - } } \\right ) \\left \\langle \\gamma _ 1 \\bar { \\psi } ^ { a _ 1 } , \\ldots , \\gamma _ m \\bar { \\psi } ^ { a _ m } \\right \\rangle _ { g , \\{ \\vec s ^ j \\} _ { j = 1 } ^ m , \\beta } ^ { X _ { D , \\vec r } } \\end{align*}"}
{"id": "1151.png", "formula": "\\begin{align*} ( f g , f g ) = ( g g ^ \\dagger , f ^ \\dagger f ) , \\end{align*}"}
{"id": "145.png", "formula": "\\begin{align*} \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } \\leq \\frac { c \\lambda \\eta _ t } { m _ t } { \\bf 1 } _ { d = 3 } + f _ { \\lambda , \\mu , m ^ 2 , t } \\big ( \\| E \\| _ { L ^ 1 \\cap L ^ \\infty } ) . \\end{align*}"}
{"id": "840.png", "formula": "\\begin{align*} p ( \\boldsymbol { s } , \\boldsymbol { s } ^ { r } , \\boldsymbol { s } ^ { c } , \\boldsymbol { x } ^ { r } , \\boldsymbol { x } ^ { c } ) = p ( \\boldsymbol { s } ) \\prod _ { m } \\mathop { p ( s _ { m } ^ { r } | s _ { m } ) } \\prod _ { m } \\mathop { p ( s _ { m } ^ { c } | s _ { m } ) } \\prod _ { m } p ( x _ { m } ^ { r } | s _ { m } ^ { r } ) p ( x _ { m } ^ { c } | s _ { m } ^ { c } ) . \\end{align*}"}
{"id": "8822.png", "formula": "\\begin{align*} f _ t \\left ( \\int _ { \\Omega } ^ { } \\alpha ( \\omega ) d \\omega \\right ) \\leq \\int _ { \\Omega } ^ { } f _ { T ( \\omega ) } \\circ \\alpha ( \\omega ) d \\omega , t : = \\int _ { \\Omega } ^ { } T ( \\omega ) d \\omega \\end{align*}"}
{"id": "1647.png", "formula": "\\begin{align*} \\theta _ { \\mathcal { C } } ( \\mathbf { z } ) = a _ { 0 0 0 } & + a _ { 1 0 0 } \\exp [ z _ 1 ] + a _ { 0 1 0 } \\exp [ z _ 2 ] + a _ { 0 0 1 } \\exp [ z _ 3 ] + a _ { 1 1 0 } \\exp [ z _ 1 + z _ 2 ] \\\\ & + a _ { 1 0 1 } \\exp [ z _ 1 + z _ 3 ] + a _ { 0 1 1 } \\exp [ z _ 2 + z _ 3 ] + a _ { 1 1 1 } \\exp [ z _ 1 + z _ 2 + z _ 3 ] \\end{align*}"}
{"id": "6417.png", "formula": "\\begin{align*} 0 = \\left ( \\Bar { Q } \\circ ( \\Bar \\Phi \\circ Q _ \\mathfrak { g } - \\Bar { Q } \\circ \\Bar \\Phi ) \\right ) ^ { ( n + 1 ) } & = \\Bar { Q } ^ { ( 0 ) } \\circ ( \\Bar \\Phi \\circ Q _ \\mathfrak { g } - \\Bar { Q } \\circ \\Bar \\Phi ) ^ { ( n + 1 ) } \\\\ & = [ Q , ( \\Bar \\Phi \\circ Q _ \\mathfrak { g } - \\Bar { Q } \\circ \\Bar \\Phi ) ^ { ( n + 1 ) } ] , \\end{align*}"}
{"id": "2910.png", "formula": "\\begin{align*} F ( A x + e ) = B G ( x ) + d . \\end{align*}"}
{"id": "1934.png", "formula": "\\begin{align*} [ w ^ { n } ] \\ , h ( w ) ^ { k } = \\frac { k } { n } \\ , [ t ^ { n - k } ] \\ , \\phi ( t ) ^ { n } , n , k \\geq 1 . \\end{align*}"}
{"id": "7088.png", "formula": "\\begin{align*} \\delta ( t _ { k + 1 } ) = \\left \\{ \\begin{aligned} & \\delta ( t _ { k } ) & & \\beta ^ { \\top } e ( t _ { k } ) \\geq 0 \\\\ & \\delta ( t _ { k } ) - 0 . 5 \\beta ^ { \\top } e ( t _ { k } ) \\mathsf { T } _ { k } & & \\beta ^ { \\top } e ( t _ { k } ) < 0 . \\end{aligned} \\right . \\end{align*}"}
{"id": "5107.png", "formula": "\\begin{align*} \\Lambda ^ { ( 2 ) } _ { n , \\delta } = \\exp \\left ( i \\mu Y ^ { n , 1 } _ { t } + i \\lambda n ^ { \\alpha + \\frac 1 2 } \\int _ { t - \\delta } ^ { \\eta _ n ( t ) } \\psi _ { n , 1 } ( s , \\eta _ n ( t ) ) d W _ s \\right ) . \\end{align*}"}
{"id": "5001.png", "formula": "\\begin{align*} \\Psi ^ { n , 3 } _ s = \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\ , \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) ^ 2 , \\end{align*}"}
{"id": "7846.png", "formula": "\\begin{align*} \\langle \\mu ( T _ \\xi ^ * ) \\widehat { a b } , \\hat { c } \\rangle = \\langle \\mu ( L _ \\xi ) \\hat { a } , c \\xi b ^ * \\rangle . \\end{align*}"}
{"id": "5811.png", "formula": "\\begin{align*} n _ 0 = x _ 0 ^ 2 + 4 y ^ 2 + 1 2 z ^ 2 . \\end{align*}"}
{"id": "858.png", "formula": "\\begin{align*} \\gamma _ { m } ^ { b } = \\frac { \\rho _ { 1 , 0 } ( ( \\pi _ { R , m } ^ { i n } ) ^ { - 1 } - 1 ) ( ( \\gamma _ { m + 1 } ^ { b } ) ^ { - 1 } - 1 ) + ( 1 - \\rho _ { 1 , 0 } ) } { ( \\rho _ { 0 , 0 } + \\rho _ { 1 , 0 } ) ( ( \\pi _ { m + 1 } ^ { i n } ) ^ { - 1 } - 1 ) ( ( \\gamma _ { m + 1 } ^ { b } ) ^ { - 1 } - 1 ) + \\rho _ { 1 , 1 } + \\rho _ { 0 1 } } , \\end{align*}"}
{"id": "7392.png", "formula": "\\begin{align*} & { L _ { \\varepsilon , h } ( \\phi ) } = \\Gamma _ { \\varepsilon , h } ( \\phi ) \\end{align*}"}
{"id": "2530.png", "formula": "\\begin{align*} s ^ { K e r F _ \\ast } + ( m - n ) \\mu - ( m - n ) ^ 2 \\| H \\| ^ 2 + ( m - n ) d i v ( H ) = 0 , \\end{align*}"}
{"id": "2618.png", "formula": "\\begin{align*} W _ A \\star W _ R = W _ { A R } \\end{align*}"}
{"id": "1768.png", "formula": "\\begin{align*} | \\sum _ { j = 1 } ^ { m _ 1 } y _ n ^ * ( x _ j ^ k ) - \\sum _ { j = 1 } ^ { m _ 1 } y ^ * ( x _ j ^ k ) | < \\epsilon / 4 . \\end{align*}"}
{"id": "5248.png", "formula": "\\begin{align*} \\Lambda _ { Q , 0 } : = \\Gamma _ { 0 , k _ 1 ( 0 ) + 1 , k _ 2 ( 0 ) + 1 , 0 , \\{ ( a _ i , b _ i ) \\} _ { i \\in I _ 0 } } ; \\end{align*}"}
{"id": "526.png", "formula": "\\begin{align*} \\frac { 2 \\ddot { x } } { C _ 1 [ f ( t ) \\dot { x } - a _ o x + C _ 2 ] ^ 3 } = F ( \\dot { x } , x , t ) \\ , \\end{align*}"}
{"id": "4439.png", "formula": "\\begin{align*} \\L I _ m ^ k f , g \\R = \\L f , ( I _ m ^ k ) ^ * g \\R \\end{align*}"}
{"id": "6017.png", "formula": "\\begin{align*} g _ { j + 1 } ( x ) = { \\mu _ j } g _ j ( x ) + ( x - \\lambda ) g ' _ j ( x ) \\end{align*}"}
{"id": "8557.png", "formula": "\\begin{align*} \\Psi ( y ) = x y - \\Phi ( \\phi ^ { - 1 } ( y ) ) = \\inf _ x \\left ( x y - \\Phi ( x ) \\right ) \\ , \\end{align*}"}
{"id": "5894.png", "formula": "\\begin{align*} I ( \\nu ) = - \\frac { 1 } { 2 } \\log \\mathrm { d e t } ( I _ k - A A ^ T ) \\end{align*}"}
{"id": "4396.png", "formula": "\\begin{align*} & \\lim _ { m \\rightarrow + \\infty } \\int _ { D _ j } ( v '' _ { \\epsilon } ( \\Psi _ m ) ) | f F ^ { 1 + \\delta } | ^ 2 e ^ { - \\phi - \\varphi _ { m } - \\Psi _ m } \\\\ = & \\int _ { D _ j } ( v '' _ { \\epsilon } ( \\Psi ) ) | f F ^ { 1 + \\delta } | ^ 2 e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) - \\varphi - \\Psi } \\\\ < & + \\infty . \\end{align*}"}
{"id": "226.png", "formula": "\\begin{align*} ( b \\gamma + c ) a + \\gamma ( b \\gamma + c + 1 ) & = a ( a ^ { - 1 } \\gamma + c ( \\gamma + 1 ) ) + a ^ { - 1 } \\gamma ^ 2 + c \\gamma ( \\gamma + 1 ) + \\gamma \\\\ & = a ^ { - 1 } \\left ( c ( \\gamma + 1 ) a ^ 2 + c \\gamma ( \\gamma + 1 ) a + \\gamma ^ 2 \\right ) \\\\ ( b \\gamma + 1 ) a + \\gamma ( b \\gamma + c + 1 ) & = a ( a ^ { - 1 } \\gamma + c \\gamma + 1 ) + a ^ { - 1 } \\gamma ^ 2 + c \\gamma ( \\gamma + 1 ) + \\gamma \\\\ & = a ^ { - 1 } \\left ( ( c \\gamma + 1 ) a ^ 2 + c \\gamma ( \\gamma + 1 ) a + \\gamma ^ 2 \\right ) \\end{align*}"}
{"id": "3278.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\tilde { A } _ j , q _ j } = \\mathcal { N } _ { A _ j , q _ j } , j = 1 , 2 . \\end{align*}"}
{"id": "4256.png", "formula": "\\begin{align*} V ( | x | ) = 1 + \\frac { \\mathfrak v _ \\infty } { | x | ^ { \\nu } } + \\mathcal O \\ ( \\frac { 1 } { | x | ^ { \\nu + \\epsilon } } \\ ) \\hbox { $ C ^ 1 $ - u n i f o r m l y a s } \\ | x | \\to + \\infty , \\end{align*}"}
{"id": "8429.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { s \\to - \\infty } P _ { s , \\tau } \\varphi \\left ( { \\xi , j } \\right ) = \\left ( { \\varphi , \\mu _ \\tau } \\right ) . \\end{align*}"}
{"id": "4153.png", "formula": "\\begin{align*} \\omega _ \\mu ( T _ { \\bar \\mu } z , T _ { \\bar \\mu } w ) = | \\mu | \\omega ( z , w ) z , w \\in \\R ^ { 2 n } , \\end{align*}"}
{"id": "3214.png", "formula": "\\begin{align*} Q ^ * ( s , a ) & = \\mathbb { E } \\Big \\{ c ( s , a ) + \\gamma \\min _ { a ^ \\prime } Q ^ { * } ( s _ { t + 1 } , a ^ \\prime ) \\Big | s _ t = s , a _ t = a \\Big \\} , \\end{align*}"}
{"id": "7277.png", "formula": "\\begin{align*} u ( x , t ) & = \\lambda ^ { - \\frac { n - 2 } { 2 } } { \\sf Q } ( y ) + \\lambda ( t ) ^ { - \\frac { n - 2 } { 2 } } \\sigma { \\sf A } _ 1 ( 1 + o ) = \\lambda ^ { - \\frac { n - 2 } { 2 } } \\sigma { \\sf A } _ 1 ( 1 + o ) | y | \\to \\infty . \\end{align*}"}
{"id": "1448.png", "formula": "\\begin{align*} R _ { \\ell , i , s } ( z ) & = \\displaystyle P _ { \\ell } ( z ) \\psi _ { i , s } \\left ( \\dfrac { 1 } { z - t } \\right ) - P _ { \\ell , i , s } ( z ) = \\displaystyle \\psi _ { i , s } \\left ( \\dfrac { P _ { \\ell } ( t ) } { z - t } \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\dfrac { \\psi _ { i , s } ( t ^ k P _ { \\ell } ( t ) ) } { z ^ { k + 1 } } \\enspace . \\end{align*}"}
{"id": "2358.png", "formula": "\\begin{align*} l ( b ) = l ( a ) + \\sum _ { i = 1 } ^ { \\deg _ X l } \\partial _ i l ( a ) ( b - a ) ^ i . \\end{align*}"}
{"id": "1297.png", "formula": "\\begin{align*} \\mathcal { G } _ 1 ( r ( T ) , x ) & = P ( T , x ) , \\\\ \\mathcal { G } _ 2 ( r ( T ) , x ) & = \\prod _ { j = 1 } ^ { d } P ( T _ j , x ) , \\end{align*}"}
{"id": "7917.png", "formula": "\\begin{align*} ( \\mathbb C ^ \\times ) ^ { I } \\times ( \\mathbb C ) ^ { \\bar { I } } : = \\left \\{ ( z _ 1 , \\ldots , z _ m ) | z _ i \\neq 0 i \\in I \\right \\} . \\end{align*}"}
{"id": "1186.png", "formula": "\\begin{align*} \\begin{cases} & \\int _ { \\R ^ { d } } \\langle \\nabla \\varphi , E \\rangle d \\rho = h ( \\varphi ) , \\forall \\varphi \\in C _ { 0 } ^ { \\infty } , \\\\ & j _ p ( E ) \\in \\overline { \\{ \\nabla \\varphi : \\varphi \\in C _ { 0 } ^ { \\infty } \\} } ^ { L ^ q ( \\rho ; \\R ^ d ) } . \\end{cases} \\end{align*}"}
{"id": "4247.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\log p ^ \\dagger _ n ( X _ n ^ \\dagger , X _ 0 ^ \\dagger ) } { \\log \\mathbb { G } _ \\dagger ( X _ n ^ \\dagger , X _ 0 ^ \\dagger ) } = 1 \\end{align*}"}
{"id": "4383.png", "formula": "\\begin{align*} & \\lim _ { B \\rightarrow 0 + 0 } \\left ( \\left ( \\frac { 1 } { \\delta } + 1 \\right ) e ^ { B } - 1 \\right ) \\frac { G ( 0 ) - G ( B ) } { B } \\\\ = & \\lim _ { B \\rightarrow 0 + 0 } \\left ( \\left ( \\frac { 1 } { \\delta } + 1 \\right ) e ^ { B } - 1 \\right ) \\frac { G ( - \\log 1 ) - G ( - \\log e ^ { - B } ) } { 1 - e ^ { - B } } \\cdot \\frac { 1 - e ^ { - B } } { B } \\\\ \\le & \\frac { 1 } { \\delta } G ( 0 ) . \\end{align*}"}
{"id": "5190.png", "formula": "\\begin{align*} \\mathcal { R } : = & \\{ \\hbox { $ q $ i s a c o n t r a c t e d b o u n d a r y n o d e o f m u l t i p l i c i t y $ 0 $ } \\} \\\\ & \\cup \\{ \\hbox { $ q $ i s a h a l f - n o d e o f $ \\widehat { C } $ w h i c h i s o f m u l t i p l i c i t y $ 0 $ b u t n o t a n a n c h o r } \\} . \\end{align*}"}
{"id": "7346.png", "formula": "\\begin{align*} [ x y , z ] + [ y z , x ] + [ z x , y ] & = ( x y ) z - z ( x y ) + ( y z ) x - x ( y z ) + ( z x ) y - y ( z x ) & \\\\ & = ( x , y , z ) + ( y , z , x ) + ( z , x , y ) = 3 ( x , y , z ) = 0 . & \\end{align*}"}
{"id": "4951.png", "formula": "\\begin{align*} \\| K ( x - y ) - K ( x ) \\| _ { \\ell ^ s ( \\mathbb { Z } ^ + ) } & = \\| \\Phi _ k ( x , y ) - \\Phi _ { k - 1 } ( x , y ) \\| _ { \\ell ^ s ( \\mathbb { Z } ^ + ) } \\\\ & \\leq \\| \\Phi _ k ( x , y ) \\| _ { \\ell ^ s ( \\mathbb { Z } ^ + ) } + \\| \\Phi _ { k - 1 } ( x , y ) \\| _ { \\ell ^ s ( \\mathbb { Z } ^ + ) } \\\\ & \\leq 2 \\| \\Phi _ { k - 1 } ( x , y ) \\| _ { \\ell ^ s ( \\mathbb { Z } ^ + ) } , \\end{align*}"}
{"id": "1264.png", "formula": "\\begin{align*} \\sigma ^ { * } ( \\mathcal { B } _ { d , k } ) & = \\left \\{ 2 \\sqrt { d - 1 } \\cos \\left ( \\frac { h } { j + 1 } \\pi \\right ) \\colon 1 \\le h \\le j \\le k \\right \\} . \\end{align*}"}
{"id": "2116.png", "formula": "\\begin{align*} y _ i ( v ) = \\begin{cases} 1 & v = v _ 0 , \\\\ - 1 & v = v _ i , \\\\ 0 & \\end{cases} \\end{align*}"}
{"id": "4665.png", "formula": "\\begin{align*} d i v ( F ) = \\frac { 1 } { \\sqrt { d e t \\ G } } \\sum \\limits _ { k = 1 } ^ d \\sum \\limits _ { i , j = 1 } ^ m \\frac { \\partial } { \\partial \\theta _ i } ( \\sqrt { d e t \\ G } g ^ { i j } F ^ k ( \\Phi ( \\theta ) ) \\frac { \\partial \\phi ^ k } { \\partial \\theta _ j } ) \\end{align*}"}
{"id": "5052.png", "formula": "\\begin{align*} K ^ { n , 3 } _ \\tau & = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ { ( \\delta + \\frac 1 n ) \\wedge \\tau } \\gamma _ s [ ( \\sigma ' ( X _ s ) \\sigma ( X _ { \\eta _ n ( s ) - \\delta } ) ) ^ 2 - ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) ] \\left ( \\int _ { \\eta _ n ( s ) - \\delta } ^ { \\eta _ n ( s ) } \\psi _ { n , 2 } ( u , s ) d W _ u \\right ) ^ 2 d s \\end{align*}"}
{"id": "6794.png", "formula": "\\begin{align*} u '''' ( x ) = g ( x ) f ( u ( x ) ) , \\quad \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , 0 \\le x \\le 1 , \\end{align*}"}
{"id": "6491.png", "formula": "\\begin{align*} u = y = 0 ( c _ 1 , c _ 2 ) . \\end{align*}"}
{"id": "7575.png", "formula": "\\begin{align*} \\frac { d Y } { d t } = A ( t ) Y ( t ) , Y ( 0 ) = I , A ( t ) = \\sum _ { \\alpha = 1 } ^ { r } b _ \\alpha ( t ) M _ \\alpha , \\end{align*}"}
{"id": "6529.png", "formula": "\\begin{align*} \\P \\{ N _ t \\geq m + i \\} \\leq C _ t e ^ { - t } \\frac { t ^ { m + i } } { ( m + i ) ! } < C _ t e ^ { - t } \\frac { t ^ { m + i } e ^ { m + i } } { ( m + i ) ^ { m + i } } = C _ t e ^ { - t } \\left ( \\frac { t e } { m + i } \\right ) ^ { m + i } . \\end{align*}"}
{"id": "1120.png", "formula": "\\begin{align*} \\Delta = \\sum _ { i = 0 } ^ { k - 1 } \\int _ { a _ i } ^ { a _ { i + 1 } } \\mbox { d } s \\cdot f _ S ( s ) \\left [ \\lambda \\omega _ i s - C ( \\lambda \\omega _ i ) - \\frac { 1 } { 2 } \\lambda ^ 2 \\sigma _ N ^ 2 \\omega _ i ^ 2 + \\frac { \\rho } { 2 } ( P _ w - \\omega _ i ^ 2 ) \\right ] \\end{align*}"}
{"id": "4423.png", "formula": "\\begin{align*} \\mathcal { F } ( d ) = \\{ C ' ( S _ { d , i } ) \\mid i \\in [ 1 , d ] \\} \\cup \\{ C ( T _ { e , j } ) \\mid j \\in [ 1 , e ] \\} \\cup \\{ C ( U _ { e , \\ell } ) \\mid \\ell \\in [ 1 , e ] \\} \\end{align*}"}
{"id": "8384.png", "formula": "\\begin{align*} F ^ * ( k , 2 , 4 ) = F ( k , 2 , 4 ) . \\end{align*}"}
{"id": "4149.png", "formula": "\\begin{align*} L = - ( X _ 1 ^ 2 + \\dots + X _ { d _ 1 } ^ 2 ) . \\end{align*}"}
{"id": "2393.png", "formula": "\\begin{align*} ( f _ r ) & _ { \\{ J , J ^ 2 + I _ 1 , \\ldots , J ^ { 2 ^ { a - 1 } } + I _ 1 , I _ 1 , \\ldots , I _ k , I _ k ^ 2 , \\ldots , I _ k ^ { 2 ^ { b - 1 } } , I _ k ^ { 2 ^ b } \\} } \\\\ = ( f _ r ) & _ { \\{ J , J ^ 2 + I _ 1 \\} } \\vee \\ldots \\vee ( f _ r ) _ { \\{ J ^ { 2 ^ { a - 1 } } + I _ 1 , I _ 1 \\} } \\vee ( f _ r ) _ { \\{ I _ 1 , \\ldots , I _ k \\} } \\vee ( f _ r ) _ { \\{ I _ k , I _ k ^ 2 \\} } \\vee \\ldots \\vee ( f _ r ) _ { \\{ I _ k ^ { 2 ^ { b - 1 } } , 0 \\} } \\\\ = ( f _ r ) & _ { \\mathbf { I } } . \\end{align*}"}
{"id": "6658.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u _ 0 \\in C ^ { 2 } ( [ 0 , h _ 0 ] ) , v _ 0 \\in C ^ { 2 } ( [ 0 , g _ 0 ] ) , u ' _ 0 ( 0 ) = v ' _ 0 ( 0 ) = 0 , \\\\ u _ 0 ( h _ 0 ) = 0 , u _ 0 ( r ) > 0 \\ \\ u ' _ 0 ( r ) < 0 , \\ \\ r \\in ( 0 , h _ 0 ] , \\\\ v _ 0 ( g _ 0 ) = 0 , v _ 0 ( r ) > 0 \\ \\ v ' _ 0 ( r ) < 0 , \\ \\ r \\in ( 0 , g _ 0 ] . \\end{array} \\right . \\end{align*}"}
{"id": "2058.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m o d i f i e d p h i } \\left \\{ \\begin{array} { r c l } \\partial _ s \\tilde { \\varphi } + a \\cdot \\nabla _ x \\tilde { \\varphi } & = & 0 \\\\ \\tilde { \\varphi } ( s = t , x ) & = & \\varphi ( x ) \\end{array} \\right . \\end{align*}"}
{"id": "4692.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty u _ 0 ( n ) \\ , q ^ n = \\sum _ { k = 1 } ^ \\infty \\frac { q ^ k } { ( q ; q ) _ { k - 1 } ( q ; q ) _ k } . \\end{align*}"}
{"id": "405.png", "formula": "\\begin{align*} \\cal L _ { \\varphi + i \\eta } = \\lambda _ \\eta Q _ \\eta + N _ \\eta \\end{align*}"}
{"id": "8812.png", "formula": "\\begin{align*} \\phi _ 1 * \\phi _ 2 ( g ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\int _ { G } ^ { } 1 _ { L _ 1 ( t _ 1 ) } ( g ' ) 1 _ { L _ 2 ( t _ 2 ) } ( g '^ { - 1 } g ) d g ' d t _ 1 d t _ 2 = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } 1 _ { L _ 1 ( t _ 1 ) } * 1 _ { L _ 2 ( t _ 2 ) } ( g ) d t _ 1 d t _ 2 . \\end{align*}"}
{"id": "8467.png", "formula": "\\begin{align*} \\left \\| f \\right \\| _ 1 = \\left \\| f \\cdot \\ 1 \\right \\| _ 1 \\le \\left \\| f \\right \\| _ 2 \\left \\| \\ 1 \\right \\| _ 2 = \\left \\| f \\right \\| _ 2 . \\end{align*}"}
{"id": "8260.png", "formula": "\\begin{align*} V _ { U } = \\left \\lbrace v \\in \\mathcal { H } | \\left ( \\begin{array} { c c } U & - 1 \\\\ 0 & 0 \\end{array} \\right ) ( D P ) v = 0 \\right \\rbrace , \\end{align*}"}
{"id": "4838.png", "formula": "\\begin{align*} \\frac { 1 } { [ K : \\Q ] } \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } g . \\varphi _ k & = \\frac { 1 } { p ^ { r - 1 } ( p - 1 ) } \\Big ( \\sum _ { k = 0 } ^ { p ^ r - 1 } \\varphi _ k - \\sum _ { k = 0 } ^ { p ^ { r - 1 } - 1 } \\varphi _ { k p } \\Big ) \\\\ & = \\frac { 1 } { p ^ { r - 1 } ( p - 1 ) } \\Big ( \\mathrm { I n d } ^ G _ { \\{ e \\} } \\mathbb { 1 } - \\mathrm { I n d } ^ G _ { \\langle a ^ { p ^ { r - 1 } } \\rangle } \\mathbb { 1 } \\Big ) \\in \\mathcal { S } ( G ) \\ ; , \\end{align*}"}
{"id": "2813.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\big \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\big \\} { } \\leq { } \\frac { 2 L \\ , \\big [ f ( x _ 0 ) - f _ * \\big ] } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left [ 2 h _ i - \\frac { h _ i ^ 2 } { 2 } \\max ( 1 , h _ i ) \\right ] } \\end{align*}"}
{"id": "1634.png", "formula": "\\begin{align*} r [ B , M ] ^ * = \\bigcup _ { m \\le n } r _ n [ B , M ] ^ * . \\end{align*}"}
{"id": "3156.png", "formula": "\\begin{align*} \\mathrm { P } _ { \\mathfrak { m } } ^ { \\sharp } = - \\inf f _ { \\mathfrak { m } } ^ { \\sharp } \\left ( E _ { 1 } \\right ) \\in \\mathbb { R } , \\end{align*}"}
{"id": "808.png", "formula": "\\begin{align*} k _ { \\psi } ( r ^ m ) + R ^ N ( r ) + k _ { \\psi } ( - 1 ) = 0 , \\end{align*}"}
{"id": "5117.png", "formula": "\\begin{align*} \\left [ ( t - \\eta _ n ( s ) ) ^ { \\alpha } - ( t - s ) ^ { \\alpha } \\right ] ^ 2 & \\leq C \\left [ ( t - \\eta _ n ( s ) ) ^ { 2 \\alpha } + ( t - s ) ^ { 2 \\alpha } \\right ] \\\\ & \\leq \\begin{cases} C n ^ { - 2 \\alpha } & \\alpha \\geq 0 \\\\ C ( t - s ) ^ { 2 \\alpha } & \\alpha < 0 \\end{cases} . \\end{align*}"}
{"id": "8471.png", "formula": "\\begin{align*} y _ i & \\geq 0 ( \\forall i ) \\\\ \\left | y _ { i + 1 } - y _ i \\right | & \\leq L ( \\forall 1 \\leq i \\leq n - 1 ) \\\\ \\sum _ { i = 1 } ^ n y _ i & = U . \\end{align*}"}
{"id": "2154.png", "formula": "\\begin{align*} d \\nu ( s ) = ( 3 / 8 ) \\left [ s ^ { 2 } I _ { [ - 1 , 1 ] } ( s ) + s ^ { - 2 } I _ { \\mathbb { R } \\setminus [ - 1 , 1 ] } \\right ] \\ , d \\lambda ( s ) . \\end{align*}"}
{"id": "4479.png", "formula": "\\begin{align*} X _ T = \\{ u \\in L ^ \\infty ( 0 , T ; H ^ s ( \\R ) ) \\mid \\| u \\| _ { L ^ \\infty _ T H ^ s } \\leq 2 \\bar C , \\| u \\| _ { L ^ \\infty _ T H ^ 3 } < 1 \\} . \\end{align*}"}
{"id": "4275.png", "formula": "\\begin{align*} \\Phi ( g ) = ( b \\sigma ) ^ r g = b \\cdot \\sigma ( b ) \\cdots \\sigma ^ { r - 1 } ( b ) \\cdot \\sigma ^ r ( g ) . \\end{align*}"}
{"id": "1041.png", "formula": "\\begin{align*} \\omega _ 0 ( A ) = \\lim _ { t \\to \\infty } \\frac { \\log \\phi ( t ) } { t } . \\end{align*}"}
{"id": "5732.png", "formula": "\\begin{align*} \\pi _ { [ a , i ] } \\pi _ { [ i + 1 , b - 1 ] } y _ { b } = \\binom { b - a } { i - a + 1 } \\pi _ { [ a , b - 1 ] } y _ { b } \\end{align*}"}
{"id": "1143.png", "formula": "\\begin{align*} \\omega ( u ) : = \\sum _ { i = 1 } ^ m \\chi ( u _ i \\neq y ) \\cdot ( i + | \\{ j \\le i : u _ j = z \\} | ) , \\end{align*}"}
{"id": "4776.png", "formula": "\\begin{align*} \\mathcal T _ { \\kappa , 3 } ^ { 1 , 2 } = \\bigcup _ { i , j \\in \\mathbb N \\colon \\atop { 2 ( i + j ) \\leq \\kappa } } \\{ [ n _ 0 , n _ 1 , n _ 2 , n _ 3 , n _ 4 , n _ 5 , n _ 4 , n _ 7 ] \\in \\mathcal S _ { \\kappa , 3 } : n _ 1 = n _ 2 = i , n _ 5 = n _ 6 = j \\} , \\end{align*}"}
{"id": "4358.png", "formula": "\\begin{align*} y ^ { \\nu _ { r - 1 } ( a _ { s _ 0 } ) } R _ { r - 1 } ( a _ { s _ 0 } ) + y ^ { \\nu _ { r - 1 } ( g q ) } R _ { r - 1 } ( g q ) = y ^ { \\nu _ { r - 1 } ( h ) } R _ { r - 1 } ( h ) . \\end{align*}"}
{"id": "4299.png", "formula": "\\begin{align*} L _ { \\phi , A } = \\{ ( z _ 1 ( y ) x _ 1 , . . . , z _ n ( y ) x _ n ) : y \\in \\R , x _ k \\in \\R , x ^ 2 _ 1 + \\ldots + x _ n ^ 2 = 1 \\} . \\end{align*}"}
{"id": "7370.png", "formula": "\\begin{align*} \\begin{aligned} { E } _ h : = \\left \\{ \\phi \\in H _ s ~ \\Big | ~ \\int _ { \\mathbb { R } ^ N } \\sum _ { i = 1 } ^ 4 U _ { h , i } ^ { p - 1 } \\frac { \\partial U _ { h , i } } { \\partial h } \\ , \\phi ~ d y = 0 \\right \\} , \\end{aligned} \\end{align*}"}
{"id": "3289.png", "formula": "\\begin{align*} \\int _ B Q _ { A , q } v _ 1 ( I - T _ { - A , q } ) ^ { - 1 } v _ 2 d x = \\int _ B Q _ { - A , q } v _ 2 ( I - T _ { A , q } ) ^ { - 1 } v _ 1 d x , \\end{align*}"}
{"id": "8728.png", "formula": "\\begin{align*} \\omega ( t ) = c _ 1 \\cos t + c _ 2 \\sin t + \\frac { \\mu a ^ 2 } { \\lambda } . \\end{align*}"}
{"id": "719.png", "formula": "\\begin{align*} \\left ( \\partial _ s + L \\right ) \\omega = \\varphi . \\end{align*}"}
{"id": "2729.png", "formula": "\\begin{align*} \\mathsf { K } _ { k + 1 } \\circ \\S ^ t _ k = - ( \\S ^ n _ { k + 1 } ) ' \\circ \\mathsf { K } _ { k } , \\end{align*}"}
{"id": "3161.png", "formula": "\\begin{align*} \\mathit { \\Omega } _ { \\mathfrak { m } } ^ { \\sharp } \\doteq \\left \\{ \\omega \\in E _ { 1 } : \\exists \\{ \\rho _ { n } \\} _ { n = 1 } ^ { \\infty } \\subseteq E _ { 1 } \\mathrm { \\ } ^ { \\ast } \\ \\omega \\underset { n \\rightarrow \\infty } { \\lim } f _ { \\mathfrak { m } } ^ { \\sharp } ( \\rho _ { n } ) = \\inf \\ , f _ { \\mathfrak { m } } ^ { \\sharp } ( E _ { 1 } ) \\right \\} \\end{align*}"}
{"id": "1746.png", "formula": "\\begin{align*} & \\log F ( z + \\omega _ 2 \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) - \\log F ( z \\ , | \\ , \\overline \\omega _ 1 , \\omega _ 2 ) = f _ { - 1 } ^ c ( z , \\overline \\omega _ 1 ) , \\\\ & \\log G ( z + \\omega _ 2 \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) - \\log G ( z + \\omega _ 2 \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) = g _ { - 1 } ^ c ( z , \\omega _ 1 , \\omega _ 2 ) , \\end{align*}"}
{"id": "2551.png", "formula": "\\begin{align*} [ T _ 3 , U _ 3 ] = [ U _ 3 , V _ 3 ] = [ V _ 3 , T _ 3 ] = 0 \\ , \\end{align*}"}
{"id": "466.png", "formula": "\\begin{align*} P ( Q ^ \\gamma ( \\infty ) \\geq k ) = \\rho ^ k , k \\geq 0 . \\end{align*}"}
{"id": "830.png", "formula": "\\begin{align*} \\boldsymbol { y } _ { q } ^ { c } = \\mathbf { h } ^ { c } u _ { q } + \\boldsymbol { n } _ { q } ^ { c } , \\end{align*}"}
{"id": "7417.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N f ( S _ k \\alpha ) = \\sum _ { m = 0 } ^ { n - 1 } \\sum _ { i = 2 } ^ { r _ m - 1 } ( T _ { m , i } + D _ { m , i } ) + \\sum _ { i = 2 } ^ R ( T _ { n , i } + D _ { n , i } ) + O ( N ^ { 1 / 2 - c } ) . \\end{align*}"}
{"id": "3645.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\int _ { \\Omega _ n } f ( \\Phi _ n , u _ n , \\nabla u _ n ) & \\geq \\liminf _ { n \\to \\infty } \\int _ { \\{ u _ n > 0 \\} } \\frac { f ( \\Phi _ n , u _ n , \\nabla u _ n ) } { u _ n } u _ n \\\\ & = \\liminf _ { n \\to \\infty } \\int _ { \\R ^ N \\times \\R \\times \\R ^ N } g ( x , u , \\xi ) \\dd \\nu _ n ( x , u , \\xi ) \\\\ & \\geq \\int _ { \\R ^ N } g ( x , u , \\xi ) \\dd \\delta _ { ( x _ 0 , 0 , 0 ) } = f ' _ - ( x _ 0 , 0 ^ + , 0 ) , \\end{align*}"}
{"id": "6292.png", "formula": "\\begin{align*} & \\int \\frac { x p _ n ( x ; - 1 ; q ) } { ( q ^ 2 - x ^ 2 ) } \\left ( \\ , _ 2 \\phi _ 1 ( \\frac { x ^ 2 } { q ^ 2 } , q ^ { 2 } ; x ^ 2 ; q ^ 2 , q ) - 1 \\right ) d _ q x = \\\\ & ( q ^ { - n } - 1 ) [ n + 1 ] _ q \\left ( p _ n ( \\frac { x } { q } ; - 1 ; q ) + \\frac { x } { q } \\ , _ 2 \\phi _ 1 ( \\frac { x ^ 2 } { q ^ 2 } , q ^ { 2 } ; x ^ 2 ; q ^ 2 , q ) \\ , _ 3 \\phi _ 2 ( q ^ { - n } , q ^ { n + 1 } , x ; q ; - q ; q , q ) \\right ) . \\end{align*}"}
{"id": "2844.png", "formula": "\\begin{align*} P ( h ) = \\frac { 1 } { 2 L } \\frac { h \\big [ \\kappa h ^ 2 - 2 h ( 1 + \\kappa ) + 4 \\big ] } { 2 - h ( 1 + \\kappa ) } = \\frac { 1 } { 2 L } \\frac { h ( 2 - h ) ( 2 - \\kappa h ) } { 2 - h ( 1 + \\kappa ) } . \\end{align*}"}
{"id": "5828.png", "formula": "\\begin{align*} \\frac { y _ { n + 1 } - y _ { n } } { x _ { n + 1 } - x _ { n } } = \\frac { \\left ( p _ { n + 1 } ^ 2 - 1 \\right ) } { \\left ( ( p _ { n + 1 } ^ 2 - 1 ) - 1 \\right ) } \\frac { y _ { n } } { x _ { n } } = \\frac { \\left ( p _ { n + 1 } ^ 2 - 1 \\right ) } { \\left ( p _ { n + 1 } ^ 2 - 2 \\right ) } \\frac { y _ { n } } { x _ { n } } , \\end{align*}"}
{"id": "2031.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : N e w t o n p o t e n t i a l } \\left \\{ \\begin{array} { r c l } \\displaystyle { \\frac { \\dd X ^ i _ t } { \\dd t } } & = & V ^ i _ t \\\\ \\displaystyle { \\frac { \\dd V ^ i _ t } { \\dd t } } & = & \\displaystyle { - \\sum _ { j = 1 } ^ { N } \\nabla V ( | X ^ j _ t - X ^ i _ t | ) } \\end{array} \\right . , \\end{align*}"}
{"id": "759.png", "formula": "\\begin{align*} \\bar { r } = \\sup _ { r \\in \\N } r _ i , \\bar { a } = \\left ( \\sup _ { i \\in \\N } \\sum _ { j \\in \\N } \\lvert a _ { i j } \\rvert ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } , \\alpha = 2 \\bar { r } + 3 . \\end{align*}"}
{"id": "7228.png", "formula": "\\begin{align*} u ( x , t ) \\ge M _ n \\left ( \\frac { 1 + t - x } { 1 + l _ n R } \\right ) ^ { - b \\cdot a _ n } \\left \\{ \\log \\left ( \\frac { 1 + t - x } { 1 + l _ n R } \\right ) \\right \\} ^ { b _ n } \\quad \\mbox { i n } \\ D _ n \\end{align*}"}
{"id": "8191.png", "formula": "\\begin{align*} \\frac { \\partial \\psi } { \\partial t } = \\hat { H } \\psi ; \\psi | _ { \\ , t = 0 } = \\psi _ { 0 } , \\end{align*}"}
{"id": "3493.png", "formula": "\\begin{align*} & \\mathrm { D T V } _ { ( p , q ) } ( X ) \\\\ & = - \\mathrm { D T E } _ { ( p , q ) } ^ { 2 } ( X ) + \\mu ^ { 2 } + 2 \\mu \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sigma ^ { 2 } \\left ( L _ { 1 } + \\frac { m } { m - 2 } L _ { 2 } \\right ) , ~ m > 2 , \\end{align*}"}
{"id": "208.png", "formula": "\\begin{align*} P ( y ^ { n } _ { [ 0 : 2 ] } | x ^ { n } , f _ { [ 0 : 2 ] } ) = \\frac { P ( y ^ { n } _ { [ 0 : 2 ] } , x ^ { n } , f _ { [ 0 : 2 ] } ) } { \\sum _ { y ^ { n } _ { [ 0 : 2 ] } \\in \\mathcal { Y } ^ { n } _ { [ 0 : 2 ] } } P ( y ^ { n } _ { [ 0 : 2 ] } , x ^ { n } , f _ { [ 0 : 2 ] } ) } = \\frac { \\phi ( x ^ { n } ) p ( y ^ { n } _ { [ 0 : 2 ] } | x ^ { n } ) P ( f _ { [ 0 : 2 ] } | y ^ { n } _ { [ 0 : 2 ] } ) } { \\sum _ { y ^ { n } _ { [ 0 : 2 ] } \\in \\mathcal { Y } ^ { n } _ { [ 0 : 2 ] } } \\phi ( x ^ { n } ) p ( y ^ { n } _ { [ 0 : 2 ] } | x ^ { n } ) P ( f _ { [ 0 : 2 ] } | y ^ { n } _ { [ 0 : 2 ] } ) } , \\end{align*}"}
{"id": "8459.png", "formula": "\\begin{align*} p \\left ( x _ 1 , x _ 2 , \\ldots , x _ d \\right ) = \\sum _ { i = 1 } ^ k w _ i p _ { i , 1 } \\left ( x _ 1 \\right ) p _ { i , 2 } \\left ( x _ 2 \\right ) \\cdots p _ { i , d } \\left ( x _ d \\right ) . \\end{align*}"}
{"id": "78.png", "formula": "\\begin{align*} Y ( t ) = D _ A \\tilde w ( t ) - v ( t ) = : \\tilde v ( t ) , \\forall t \\in \\R , \\end{align*}"}
{"id": "2799.png", "formula": "\\begin{align*} \\bar { x } _ i : = x _ i - \\frac { - \\kappa } { 1 - \\kappa } \\frac { h _ i } { L } U \\in \\big [ x _ { i + 1 } , x _ { i } \\big ] , i \\in \\big \\{ 0 , \\dots , N - 1 \\big \\} . \\end{align*}"}
{"id": "3658.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } u = \\int _ { \\R ^ N } u ^ { \\alpha q } u ^ { ( 1 - \\alpha ) s } \\leq \\left ( \\int _ { \\R ^ N } u ^ q \\right ) ^ \\alpha \\left ( \\int _ { \\R ^ N } u ^ s \\right ) ^ { 1 - \\alpha } . \\end{align*}"}
{"id": "2162.png", "formula": "\\begin{align*} h ^ { - 1 } ( s ) = \\sum \\nolimits _ { n = 0 } ^ { \\infty } ( \\Re a _ { n } ) ( s - s _ { 0 } ) ^ { n } , s \\in J . \\end{align*}"}
{"id": "3898.png", "formula": "\\begin{align*} \\kappa = \\begin{cases} \\frac { N - 2 + p } { N - p } & \\hbox { i f } 1 < p < 2 \\hbox { a n d } \\mathcal G = 0 \\\\ \\frac { N ( p - 1 ) } { N ( p - 1 ) - p } & \\hbox { i f e i t h e r } 2 \\leq p < N \\hbox { o r } p = N \\geq 3 \\\\ 2 & \\hbox { i f } p = N = 2 . \\end{cases} \\end{align*}"}
{"id": "504.png", "formula": "\\begin{align*} ( t _ x ( \\Phi ) ) ( x ) = \\max _ { y \\leq x } ( \\Phi ( y ) ) + \\min _ { z \\geq x } ( \\Phi ( z ) ) - \\Phi ( x ) , \\end{align*}"}
{"id": "7718.png", "formula": "\\begin{align*} A ( k ) = \\begin{bmatrix} A _ S ( k ) & A _ { S \\bar { S } } ( k ) \\\\ A _ { \\bar { S } S } ( k ) & A _ { \\bar { S } } ( k ) \\end{bmatrix} ~ ~ ~ ~ ~ \\pi ( k ) = \\begin{bmatrix} \\pi _ { S } ( k ) \\\\ \\pi _ { \\bar { S } } ( k ) \\end{bmatrix} \\end{align*}"}
{"id": "6486.png", "formula": "\\begin{align*} u = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) y = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) . \\end{align*}"}
{"id": "7544.png", "formula": "\\begin{align*} \\mathcal M ^ - & = \\{ \\vec x : r _ j \\le x _ j \\le v _ j , \\ r _ i \\le x _ i \\le s _ i \\ ( i \\neq j ) \\} \\\\ \\mathcal M ^ + & = \\{ \\vec x : v _ j \\le x _ j \\le s _ j , \\ r _ i \\le x _ i \\le s _ i \\ ( i \\neq j ) \\} \\end{align*}"}
{"id": "2510.png", "formula": "\\begin{align*} g ( ( \\nabla _ U A ) _ { X _ j } X _ j , V ) = g ( \\nabla _ U A _ { X _ j } X _ j - A ( \\nabla _ U X _ j , X _ j ) - A ( X _ j , \\nabla _ U X _ j ) , V ) . \\end{align*}"}
{"id": "6040.png", "formula": "\\begin{align*} \\lambda _ i = r \\kappa _ i + s \\end{align*}"}
{"id": "7136.png", "formula": "\\begin{align*} H ^ * ( X _ { D } ) = \\mathbb { Z } [ x _ { 2 3 } , x _ { 3 7 } , x _ { 6 7 } , x _ { 8 9 } ] / \\langle x _ { 2 3 } ^ { 3 } , x _ { 3 7 } ( - x _ { 2 3 } + x _ { 3 7 } + x _ { 6 7 } ) , x _ { 6 7 } ^ 4 , x _ { 8 9 } ^ 2 ( - x _ { 3 7 } - x _ { 6 7 } + x _ { 8 9 } ) \\rangle . \\end{align*}"}
{"id": "1765.png", "formula": "\\begin{align*} | f _ k ( y ^ * ) - \\sum _ { j = 1 } ^ { m _ 1 } y ^ * ( x ^ k _ j ) | < \\epsilon / 4 . \\end{align*}"}
{"id": "3250.png", "formula": "\\begin{align*} Q _ { A , q } v : = i \\ ; \\textrm { d i v } ( A v ) + i A \\cdot \\nabla v - ( \\abs { A } ^ 2 + q ) v , v \\in H ^ 1 _ { \\mathrm { l o c } } ( \\R ^ 3 ) . \\end{align*}"}
{"id": "7840.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\sup _ { j } \\abs { \\varphi ( \\mu _ j ( p _ i ^ \\perp v ) ) } = 0 , \\end{align*}"}
{"id": "8580.png", "formula": "\\begin{align*} & 0 \\le s , t \\le l , \\\\ & \\{ j _ 1 , \\ldots , j _ s \\} \\cap \\{ k _ 1 , \\ldots , k _ t \\} = \\varnothing , \\\\ & j _ 1 > \\ldots > j _ s , k _ 1 > \\ldots > k _ t , \\end{align*}"}
{"id": "118.png", "formula": "\\begin{align*} Q _ t : = S _ t \\big ( S _ t ^ 2 \\star S _ t ^ 2 \\big ) . \\end{align*}"}
{"id": "6843.png", "formula": "\\begin{align*} N ( t ) = \\int _ { 0 } ^ { + \\infty } g p ( t , V _ F , g ) d g = \\frac { 1 } { V _ F } \\int _ { 0 } ^ { + \\infty } g p _ 0 ( t , g ) d g . \\end{align*}"}
{"id": "8463.png", "formula": "\\begin{align*} \\left \\| \\hat { p } - p \\right \\| ^ 2 _ 2 = \\left \\| \\hat { p } \\right \\| _ 2 ^ 2 - 2 \\left < p , \\hat { p } \\right > + \\left \\| p \\right \\| _ 2 ^ 2 . \\end{align*}"}
{"id": "5207.png", "formula": "\\begin{align*} k _ 1 + k _ { 1 2 } = r m _ 1 + 1 ; k _ 2 + k _ { 1 2 } = s m _ 2 + 1 \\end{align*}"}
{"id": "610.png", "formula": "\\begin{align*} \\textup { T r } ( \\Delta _ n + z ^ 2 ) ^ { - \\alpha } & = n ^ { - 2 \\alpha + 2 } \\int _ 0 ^ 1 \\int _ 0 ^ 1 f ( x , y , 1 , z / n ) ^ { - \\alpha } d x d y \\\\ & + 2 \\ , I ^ { n } _ { y } \\circ E ^ { n } _ { x , M } f ^ { - \\alpha } + E ^ { n } _ { y , M } \\circ E ^ { n } _ { x , M } f ^ { - \\alpha } , \\end{align*}"}
{"id": "1773.png", "formula": "\\begin{align*} K ( \\alpha , A , z + a f _ k ) = \\operatorname { K e r } ( z + a f _ k ) \\cap \\bigcap _ { n = 1 } ^ \\infty K ( \\alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \\end{align*}"}
{"id": "240.png", "formula": "\\begin{align*} ( \\delta ^ { ( s ) } ) ^ i u = f ^ { ( i ) } ( u , u ^ { \\phi ^ s } , \\ldots , u ^ { \\phi ^ { i s } } ) , \\end{align*}"}
{"id": "3336.png", "formula": "\\begin{align*} ( s _ i ^ { - 1 } - I d ) ( v _ i ) = \\beta \\circ ( I d - s _ j ) [ \\beta ^ { - 1 } ( v _ 2 ) ] \\end{align*}"}
{"id": "3049.png", "formula": "\\begin{align*} f _ { \\boldsymbol { Y } } ( \\boldsymbol { y } ) = \\frac { 2 c _ { n } } { \\sqrt { | \\boldsymbol { \\Sigma } | } } g _ { n } \\left \\{ \\frac { 1 } { 2 } ( \\boldsymbol { y } - \\boldsymbol { \\mu } ) ^ { T } \\mathbf { \\Sigma } ^ { - 1 } ( \\boldsymbol { y } - \\boldsymbol { \\mu } ) \\right \\} H \\left ( \\mathbf { \\Sigma } ^ { - \\frac { 1 } { 2 } } ( \\boldsymbol { y } - \\boldsymbol { \\mu } ) \\right ) , ~ \\boldsymbol { y } \\in \\mathbb { R } ^ { n } , \\end{align*}"}
{"id": "1394.png", "formula": "\\begin{align*} S _ t u _ 0 = L _ { t } ^ { - 1 } [ S _ t u _ 0 ] W ^ \\gamma _ t u _ 0 + N _ t u _ 0 . \\end{align*}"}
{"id": "5300.png", "formula": "\\begin{align*} \\mathrm { C o n t } ( \\{ Q _ 1 , \\ldots , Q _ h \\} ) : = ( - 1 ) ^ { h - 1 } \\left ( \\prod _ { j = 1 } ^ { h - 1 } ( j r + 1 - b _ I ) \\right ) \\left ( \\prod _ { j = 1 } ^ h \\prod _ { i = 1 } ^ { | I _ j | - 1 } r - b _ { I _ j } + i \\right ) , \\end{align*}"}
{"id": "39.png", "formula": "\\begin{align*} \\Tilde { \\mathcal { N } } ( f ) \\circ \\rho _ { \\eta } ( X ) = \\Pi _ { w \\in W _ { \\rho } ^ { m _ 0 } } f ( X + w ) , \\end{align*}"}
{"id": "4137.png", "formula": "\\begin{align*} \\Theta ( G ) & \\leq \\vartheta ( G ) , \\\\ \\vartheta ( G \\boxtimes H ) & = \\vartheta ( G ) \\vartheta ( H ) , \\end{align*}"}
{"id": "1968.png", "formula": "\\begin{align*} \\mathbf { v } _ { k } ( z ) = \\frac { \\mathbf { c } _ { k + 1 } } { \\mathbf { d } _ { k + 1 } ( z ) + \\mathbf { v } _ { k + 1 } ( z ) } . \\end{align*}"}
{"id": "3198.png", "formula": "\\begin{align*} \\int _ { X \\setminus B ( \\mathcal O , 1 ) } | f | d \\mu = \\int _ { \\mathbb S } \\int _ 1 ^ \\infty | f ( \\gamma _ \\xi ( t ) ) | t ^ { Q - 1 } \\ d t d \\sigma ( \\xi ) \\end{align*}"}
{"id": "2132.png", "formula": "\\begin{align*} V = \\{ x \\in \\mathbb { R } : g ( x ) > b \\} . \\end{align*}"}
{"id": "7964.png", "formula": "\\begin{align*} \\deg _ 0 ( \\phi ) = 2 p \\phi , \\phi \\in H ^ { 2 p } ( D _ { I _ { \\vec s } } ) ; \\end{align*}"}
{"id": "1969.png", "formula": "\\begin{align*} \\widehat { \\mathbf { c } } _ { n } & = \\begin{cases} \\mathbf { 1 } , & 1 \\leq n \\leq p , \\\\ ( - a _ { n - p - 1 } , 1 , \\ldots , 1 ) , & n \\geq p + 1 , \\end{cases} \\\\ \\widehat { \\mathbf { d } } _ { n } ( z ) & = ( 0 , \\ldots , 0 , z ) , n \\geq 1 . \\end{align*}"}
{"id": "172.png", "formula": "\\begin{align*} \\| C _ { ( m ) } ( C _ { ( m ) } ^ 2 \\star C _ { ( m ) } ^ 2 ) \\| _ { L ^ 1 } = \\frac { 1 } { L ^ { d } } \\sum _ { k \\in \\Lambda ^ * } \\frac { 1 } { m ^ 2 + \\theta ( k ) } \\Big ( \\frac { 1 } { L ^ { d } } \\sum _ { p \\in \\Lambda ^ * } \\frac { 1 } { m ^ 2 + \\theta ( p ) } \\frac { 1 } { m ^ 2 + \\theta ( k - p ) } \\Big ) ^ 2 . \\end{align*}"}
{"id": "4965.png", "formula": "\\begin{align*} Y ^ n _ t = A ^ n _ t + Y ^ { n , 1 } _ t = : A ^ n _ t + C ^ n _ t + Z ^ n _ t , \\end{align*}"}
{"id": "4300.png", "formula": "\\begin{align*} \\mathcal { M } _ { H _ 0 } ( H ) = \\mathcal { M } _ { H _ 0 ' } ( H ) + \\mathcal { M } _ { H _ 0 } ( H _ 0 ' ) . \\end{align*}"}
{"id": "1546.png", "formula": "\\begin{align*} \\P ( Z \\ge 1 2 \\rho L _ { } \\tau ) & \\le e ^ { - 1 2 h \\rho L _ { } \\tau } 4 ^ { \\tau } = \\big ( 4 / e ^ { 3 } \\big ) ^ { \\tau } \\leq e ^ { - \\tau } . \\end{align*}"}
{"id": "2228.png", "formula": "\\begin{align*} A ( t ) = ( I + \\kappa U ( s , t ) ) \\partial _ t { \\rm L o g } ( U ( t , s ) + \\kappa I ) \\end{align*}"}
{"id": "6969.png", "formula": "\\begin{align*} a _ n = ( \\lambda _ n ^ 2 - \\mu _ n ^ 2 ) \\prod _ { k \\ne n } \\left ( \\frac { \\lambda _ n ^ 2 - \\mu _ k ^ 2 } { \\lambda _ n ^ 2 - \\lambda _ k ^ 2 } \\right ) \\end{align*}"}
{"id": "150.png", "formula": "\\begin{align*} \\sup _ { t > 0 } \\frac { c \\eta _ t } { m _ t } \\leq c ( \\mu ) { \\bf 1 } _ { d = 2 } + c _ 0 \\quad \\lim _ { \\mu \\rightarrow \\infty } c ( \\mu ) = 0 , \\end{align*}"}
{"id": "974.png", "formula": "\\begin{align*} \\overline { f } = \\left ( \\begin{array} { c } f _ { 0 } \\\\ f _ { 1 } \\\\ \\vdots \\\\ f _ { n - 1 } \\end{array} \\right ) \\in \\mathbb { R } ^ { n } \\stackrel { t } { \\longrightarrow } f ( x ) = f _ { 0 } + f _ { 1 } x + \\cdots + f _ { n - 1 } x ^ { n - 1 } \\in \\mathbb { R } [ x ] / \\langle \\phi ( x ) \\rangle \\end{align*}"}
{"id": "7280.png", "formula": "\\begin{align*} \\| u ( t ) \\| _ \\infty = \\lambda ( t ) ^ { - \\frac { n - 2 } { 2 } } = ( T - t ) ^ { - \\frac { n - 2 } { 6 - n } \\Gamma _ J } ( J \\in \\N ) . \\end{align*}"}
{"id": "351.png", "formula": "\\begin{align*} S ( \\delta _ { } ) = o ( 1 ) . \\end{align*}"}
{"id": "2835.png", "formula": "\\begin{align*} \\min _ { 0 \\leq i \\leq N } \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\} { } \\leq { } \\frac { q } { S _ * } \\| \\nabla f ( x _ N ) \\| ^ 2 + \\sum _ { i = 0 } ^ { N - 1 } \\frac { p _ i } { S _ * } \\min \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\ , , \\ , \\| \\nabla f ( x _ { i + 1 } ) \\| ^ 2 \\} . \\end{align*}"}
{"id": "2613.png", "formula": "\\begin{align*} Z ^ { ( 0 , 0 ) } _ { ( 0 , 0 , 0 ) , 0 } \\equiv 1 \\end{align*}"}
{"id": "5321.png", "formula": "\\begin{align*} N ^ t _ s : = \\# \\Pi ^ t ( s ) \\end{align*}"}
{"id": "5494.png", "formula": "\\begin{align*} \\theta _ { t + 1 } = \\theta _ t - \\beta _ t Q _ { \\theta _ t } ^ * ( s _ t , a _ t ) \\frac { d } { d \\theta } \\log \\pi _ { \\theta _ t } ( a _ t | s _ t ) , \\end{align*}"}
{"id": "5944.png", "formula": "\\begin{align*} K _ { ( i j ) ( k p ) } = - a { \\delta _ { i j } } { \\delta _ { k p } } + b { \\delta _ { i k } } { \\delta _ { j p } } + c { \\delta _ { i p } } { \\delta _ { j k } } \\end{align*}"}
{"id": "1631.png", "formula": "\\begin{align*} r _ k [ A , M ] = \\{ r _ k ( N ) : N \\in [ A , M ] \\} . \\end{align*}"}
{"id": "603.png", "formula": "\\begin{align*} \\zeta ( \\mathcal { L } _ n , s ) & = \\pi ^ { 2 s - 1 / 2 } \\frac { \\Gamma ( 1 / 2 - s ) } { \\Gamma ( 1 - s ) } n ^ { 1 - 2 s } + 2 \\zeta _ R ( 2 s ) + \\frac { 2 s } { 3 } \\pi ^ 2 \\zeta _ R ( 2 s - 2 ) n ^ { - 2 } + o ( n ^ { - 2 } ) \\\\ & = \\pi ^ { 2 s - 1 / 2 } \\frac { \\Gamma ( 1 / 2 - s ) } { \\Gamma ( 1 - s ) } n ^ { 1 - 2 s } + \\zeta ( \\Delta _ { \\mathbb { S } ^ 1 } , s ) + \\frac { s } { 3 } \\pi ^ 2 \\zeta ( \\Delta _ { \\mathbb { S } ^ 1 } , s - 1 ) n ^ { - 2 } + o ( n ^ { - 2 } ) . \\end{align*}"}
{"id": "7410.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 0 } ^ { N - 1 } \\varphi ( 2 \\pi x ) ^ k \\varphi ( 2 \\pi y ) ^ { N - k - 1 } \\right | \\le \\frac { 2 } { \\left | \\varphi ( 2 \\pi x ) - \\varphi ( 2 \\pi y ) \\right | } \\ll \\frac { 1 } { \\| x - y \\| } . \\end{align*}"}
{"id": "8184.png", "formula": "\\begin{align*} ( P _ \\delta ( E ) \\psi ) ( y ) = [ ( | \\eta _ \\delta | ^ 2 * \\chi _ { A } ) \\psi ] ( y ) \\end{align*}"}
{"id": "1283.png", "formula": "\\begin{align*} W _ j = H e _ j ( x ) ( \\forall j = \\overline { 0 , l ( T ) } ) , \\end{align*}"}
{"id": "2347.png", "formula": "\\begin{align*} { i \\choose j } a _ i h ^ j = a _ { i j 0 } + a _ { i j 1 } Q ' + \\ldots + a _ { i j n } Q '^ n \\end{align*}"}
{"id": "1402.png", "formula": "\\begin{align*} X _ 1 = \\begin{pmatrix} 1 & 1 & 1 \\\\ \\epsilon ^ { \\frac { 1 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 1 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 2 } { 3 } } \\\\ \\epsilon ^ { \\frac { 2 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 2 } { 3 } } & \\epsilon ^ { \\frac { 1 } { 3 } } e ^ { i 2 \\pi \\frac { 4 } { 3 } } \\\\ 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 \\end{pmatrix} \\end{align*}"}
{"id": "4222.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } a ( k ) - a ( k - 1 ) = + \\infty \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { 1 \\vee \\log ( \\psi ^ { - 1 } ( a ( k ) ) ) } = \\infty , \\end{align*}"}
{"id": "3918.png", "formula": "\\begin{align*} { } \\hat { \\pi } _ { h , { T _ n } } ^ a ( x ) & = \\frac { 1 } { T _ n \\prod _ { l = 1 } ^ d h _ l } \\int _ 0 ^ { T _ n } \\prod _ { l = 1 } ^ d K ( \\frac { x _ l - X _ { \\varphi _ { n , l } ( u ) } ^ l } { h _ l } ) d u \\\\ & = : \\frac { 1 } { T _ n } \\int _ 0 ^ { T _ n } \\prod _ { l = 1 } ^ d K _ { h _ l } ( x _ l - X _ { \\varphi _ { n , l } ( u ) } ^ l ) d u . \\end{align*}"}
{"id": "847.png", "formula": "\\begin{align*} v _ { B , m } ^ { p o s t } = \\int _ { x _ { m } ^ { r } } | x _ { m } - x _ { B , m } ^ { r , p o s t } | ^ { 2 } \\hat { p } ( x _ { m } ^ { r } | \\boldsymbol { y } ) , \\end{align*}"}
{"id": "7839.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\sup _ { j } \\abs { \\varphi ( \\mu _ j ( ( x - x _ i ) v ) } = 0 . \\end{align*}"}
{"id": "5792.png", "formula": "\\begin{align*} [ a , b ] = a ^ { - 1 } b ^ { - 1 } a b = ( b b ^ { - 1 } ) a ^ { - 1 } ( b b ^ { - 1 } ) b ^ { - 1 } a b = b ( b ^ { - 1 } a ^ { - 1 } b ) b ^ { - 1 } ( b ^ { - 1 } a b ) = b \\bar { a } b ^ { - 1 } \\bar { a } ^ { - 1 } \\in [ B , A ] _ 1 \\end{align*}"}
{"id": "2840.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 2 - h ( 1 + \\kappa ) } { 1 - \\kappa h } \\big ( f _ { 0 } - f _ { 1 } \\big ) & { } \\geq { } \\frac { 1 } { 2 L ( 1 - \\kappa ) } \\frac { h ( 1 - \\kappa ) [ \\kappa h ^ 2 - 2 h ( 1 + \\kappa ) + 3 ] } { 1 - \\kappa h } \\big \\| g _ { 0 } \\big \\| ^ 2 \\\\ & + \\frac { h } { 2 L \\big ( 1 - \\kappa \\big ) } \\frac { 1 - \\kappa } { 1 - \\kappa h } \\| g _ 1 \\| ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "4903.png", "formula": "\\begin{align*} \\lvert v _ { n , 1 } \\rvert ^ 2 \\prod _ { k = 1 } ^ { n - 1 } { \\lambda _ k } ( A ) = \\prod _ { k = 1 } ^ { n - 1 } { \\lambda _ k } ( M _ 1 ) . \\end{align*}"}
{"id": "4950.png", "formula": "\\begin{align*} \\mathcal { V } _ s f ( x ) & = \\| K \\ast f ( x ) \\| _ { \\ell ^ s ( { \\mathbb { Z } ^ + } ) } \\\\ & = \\left ( \\sum _ { k = 1 } ^ \\infty | \\phi _ k \\ast f ( x ) - \\phi _ { k - 1 } \\ast f ( x ) | ^ s \\right ) ^ { 1 / s } \\\\ & = \\left ( \\sum _ { k = 1 } ^ \\infty | A _ { n _ k } f ( x ) - A _ { n _ { k - 1 } } f ( x ) | ^ s \\right ) ^ { 1 / s } \\end{align*}"}
{"id": "4412.png", "formula": "\\begin{align*} ( a , b ) = \\left \\{ \\begin{array} { l l } ( 4 , - 2 ) , & \\mbox { i f } \\ell \\equiv 0 \\pmod { 4 } , \\\\ ( 2 4 , - 1 8 ) , & \\mbox { i f } \\ell \\equiv 2 \\pmod { 4 } . \\end{array} \\right . \\end{align*}"}
{"id": "5757.png", "formula": "\\begin{align*} \\pi _ { i _ 1 } \\pi _ J = \\binom { | J _ k | + 1 } { 1 } \\pi _ { J ' } \\end{align*}"}
{"id": "4477.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } k \\\\ \\vec { i } \\end{array} \\right ) = \\frac { k ! } { i _ 1 ! i _ 2 ! \\cdots i _ l ! \\cdots } , \\end{align*}"}
{"id": "2897.png", "formula": "\\begin{align*} x _ 4 & = x _ 2 + y _ 4 \\\\ x _ 6 & = x _ 2 + y _ 4 + y _ 6 \\\\ & \\cdots \\cdots \\\\ x _ { k - 1 } & = x _ 2 + y _ 4 + y _ 6 + \\cdots + y _ { k - 1 } . \\end{align*}"}
{"id": "1874.png", "formula": "\\begin{align*} A _ { [ n , j ] } = \\sum _ { \\gamma \\in \\mathcal { D } _ { [ n , j ] } } w ( \\gamma ) = \\sum _ { k = 0 } ^ { n - j } \\sum _ { \\gamma \\in \\mathcal { D } _ { [ n , j , k ] } } w ( \\gamma ) = \\sum _ { k = 0 } ^ { n - j } A _ { [ k , 0 ] } \\ , A _ { [ n - k - 1 , j - 1 ] } ^ { ( 1 ) } , n \\geq j . \\end{align*}"}
{"id": "2954.png", "formula": "\\begin{align*} Y ^ { \\prime \\prime } + \\frac { 1 } { t } Y ^ \\prime = 0 \\end{align*}"}
{"id": "137.png", "formula": "\\begin{align*} \\big \\| C \\star \\psi \\star S \\big \\| _ { L ^ 1 } & \\leq c \\Big ( \\frac { 1 } { m ^ 4 _ t } { \\bf 1 } _ { d = 2 } + \\big ( \\frac { 1 } { m _ t ^ { 1 / 2 } } + \\frac { 1 } { m _ t ^ { 5 / 2 } } \\big ) { \\bf 1 } _ { d = 3 } \\Big ) \\Big ( \\| E \\| _ { L ^ 1 } + \\frac { 1 } { m ^ 2 _ t } \\Big ) . \\end{align*}"}
{"id": "6205.png", "formula": "\\begin{align*} D _ q F ( x ) = p ( x ) F ( x ) \\end{align*}"}
{"id": "3587.png", "formula": "\\begin{align*} \\mathtt { m } \\left ( \\bigwedge \\limits _ { t \\in J } \\mathsf { c l } ' _ t \\right ) = \\mathtt { w } ^ + \\left ( \\bigwedge \\limits _ { t \\in J } \\mathsf { c l } ' _ t \\right ) - \\sum \\limits _ { \\bigwedge \\limits _ { t ' \\in J ' } \\mathsf { c l } ' _ { t ' } \\not \\dashv ~ \\vdash _ { \\mathsf { F D E } } \\bigwedge \\limits _ { t \\in J } \\mathsf { c l } ' _ t } \\mathtt { m } \\left ( \\bigwedge \\limits _ { t ' \\in J ' } \\mathsf { c l } ' _ { t ' } \\right ) \\end{align*}"}
{"id": "6188.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\psi ^ { \\ast } r _ { S ^ { 1 } } ^ { 2 } = | s | _ { h } ^ { 2 } . \\end{array} \\end{align*}"}
{"id": "6375.png", "formula": "\\begin{align*} \\mathrm { S p e c t r u m } ( A _ 0 ) = \\{ t \\in ( 0 , 1 ) \\mid \\ , \\exists \\ , 1 \\leq i \\leq d \\mbox { s u c h t h a t } w _ i = t \\} . \\end{align*}"}
{"id": "8016.png", "formula": "\\begin{align*} f _ + ^ * K _ { X _ + } = f _ - ^ * K _ { X _ - } + \\left ( \\sum _ { i = 1 } ^ m D _ i \\cdot e \\right ) E . \\end{align*}"}
{"id": "1923.png", "formula": "\\begin{align*} z S _ { 0 } ( z ) - 1 & = a _ { 0 } \\ , S _ { p } ( z ) \\\\ S _ { j } ( z ) & = S _ { i } ( z ) \\ , S _ { j - i - 1 } ^ { ( i + 1 ) } ( z ) 0 \\leq i < j \\leq p . \\end{align*}"}
{"id": "3663.png", "formula": "\\begin{align*} \\forall s \\in [ 0 , t ] , x , v \\in \\R ^ 3 , X ( X ( x , v , 0 , s ) , V ( x , v , 0 , s ) , s , 0 ) = x , V ( X ( x , v , 0 , s ) , V ( x , v , 0 , s ) , s , 0 ) = v . \\end{align*}"}
{"id": "6223.png", "formula": "\\begin{align*} D _ q u ( x ) + u ( x ) u ( q x ) + \\tilde { A } ( x ) u ( q x ) + r ( x ) = 0 , \\end{align*}"}
{"id": "3148.png", "formula": "\\begin{align*} \\mathcal { W } _ { \\Lambda } \\doteq \\left \\{ \\Phi \\in \\mathcal { W } _ { 1 } : \\Phi _ { \\mathcal { Z } } = 0 \\mathcal { Z } \\nsubseteq \\Lambda \\mathcal { Z } \\ni 0 \\right \\} \\end{align*}"}
{"id": "8538.png", "formula": "\\begin{align*} P _ k ^ * ( - x ) & = ( - 1 ) ^ { k - 1 } P _ k ^ * ( x ) \\\\ P _ k ( - x ) & = ( - 1 ) ^ { k } P _ k ( x ) \\\\ p _ k ( x ) & = x \\frac { P _ k ^ * ( x ) } { P _ k ( x ) } - 1 = \\\\ ( 2 p + 1 ) ^ 2 & = 4 p ( p + 1 ) + 1 = 4 z + 1 \\end{align*}"}
{"id": "6478.png", "formula": "\\begin{align*} u _ x = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) y _ x = 0 \\ \\ \\ \\ ( c _ 2 , L ) \\end{align*}"}
{"id": "1011.png", "formula": "\\begin{align*} G ( [ 0 ; a _ 1 , a _ 2 , a _ 3 \\dots ] ) = [ 0 ; a _ 2 , a _ 3 , \\dots ] . \\end{align*}"}
{"id": "4547.png", "formula": "\\begin{align*} \\Vert J _ { 4 4 } ^ { \\left ( 1 \\right ) } \\Vert _ { p } = \\mathcal { O } ( 1 ) ( v ( | z _ { 1 } | ) + ( v ( | z _ { 2 } | ) ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) . \\end{align*}"}
{"id": "919.png", "formula": "\\begin{align*} ( \\Delta - e ) ( \\nabla u \\cdot \\nabla v ) + ( \\Delta - e ) ( \\nabla u \\cdot \\nabla v ) + ( \\Delta - e ) ( \\nabla v \\cdot \\nabla v ) = ( \\Delta - e ) f , \\end{align*}"}
{"id": "6457.png", "formula": "\\begin{align*} \\Delta ' \\circ \\mathcal H = ( \\mathcal H \\otimes \\Phi ) \\circ \\Delta + ( \\Phi \\otimes \\mathcal H ) \\circ \\Delta . \\end{align*}"}
{"id": "8372.png", "formula": "\\begin{align*} 0 = u u ^ * + u _ { ( r ) } u _ { ( s ) } ^ * . \\end{align*}"}
{"id": "7869.png", "formula": "\\begin{align*} \\Gamma ( x ^ * , y ) = \\Delta ^ { 1 / 2 } ( x ^ * ) y + x ^ * \\Delta ^ { 1 / 2 } ( y ) - \\Delta ^ { 1 / 2 } ( x ^ * y ) , \\end{align*}"}
{"id": "6174.png", "formula": "\\begin{align*} \\frac { \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } + \\varphi _ { \\alpha \\overline { \\beta } } ) } { \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } ) } = e ^ { - \\kappa \\varphi + F } . \\end{align*}"}
{"id": "3831.png", "formula": "\\begin{align*} E r r U ^ 3 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \\sum _ { \\begin{subarray} { c } k _ 2 \\in \\Z _ + , n _ 2 \\in [ - M _ t , 2 ] \\cap \\Z \\\\ \\mu _ 2 \\in \\{ + , - \\} , i _ 2 \\in \\{ 0 , 1 , 2 , 3 , 4 \\} \\end{subarray} } \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot \\xi - i s \\hat { v } \\cdot \\eta + i s \\mu _ 1 | \\xi - \\eta | } \\big ( \\sum _ { l _ 2 \\in [ - M _ t , 2 ] \\cap \\Z } I n i _ { k _ 2 , j _ 2 , n _ 2 } ^ { \\mu _ 2 , i _ 2 } ( s , X ( s ) , V ( s ) ) \\end{align*}"}
{"id": "18.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { \\rho } = ( \\alpha _ n , \\beta _ n ) _ { \\rho , E _ { \\rho } ^ n , n } = \\frac { 1 } { \\eta ^ n } T _ { L | K } ( \\lambda _ { \\rho } ( \\alpha _ n ) \\delta _ n ( \\beta _ n ) ) \\cdot _ { \\rho } v _ n \\end{align*}"}
{"id": "1614.png", "formula": "\\begin{align*} g = ( g _ { i j } ) = \\begin{pmatrix} \\frac { \\left \\lbrace a ^ 2 ( y ^ 1 ) ^ 4 + 3 \\tilde { b } ^ 2 ( y ^ 2 ) ^ 4 \\right \\rbrace } { ( y ^ 1 ) ^ 4 } & \\frac { - 4 \\tilde { b } ^ 2 ( y ^ 2 ) ^ 3 } { ( y ^ 1 ) ^ 3 } \\\\ \\frac { - 4 \\tilde { b } ^ 2 ( y ^ 2 ) ^ 3 } { ( y ^ 1 ) ^ 3 } & \\frac { 2 \\tilde { b } \\left \\lbrace a ( y ^ 1 ) ^ 2 + 3 \\tilde { b } ^ 2 ( y ^ 2 ) ^ 2 \\right \\rbrace } { ( y ^ 1 ) ^ 2 } \\end{pmatrix} , \\end{align*}"}
{"id": "1046.png", "formula": "\\begin{align*} V _ k ( t ) x = \\sum _ { n = k + 1 } ^ \\infty S _ n ( t ) x \\end{align*}"}
{"id": "334.png", "formula": "\\begin{align*} \\overline { \\delta } ( A ) = \\overline { \\delta } ( A \\cap { \\rm L } _ { B \\cap ( z , \\infty ) } ) \\ \\le \\ \\delta ( { \\rm L } _ { B \\cap ( z , \\infty ) } ) < \\overline { \\delta } ( A ) , \\end{align*}"}
{"id": "823.png", "formula": "\\begin{align*} Q B _ { \\beta , \\psi , \\infty } : = \\bigcup _ { p > 0 } Q B _ { \\beta , \\psi , p } \\end{align*}"}
{"id": "7266.png", "formula": "\\begin{align*} u _ t = \\Delta _ x u + { \\bf b } ( x , t ) \\cdot \\nabla _ x u + V ( x , t ) u + f ( x , t ) \\qquad Q , \\end{align*}"}
{"id": "7351.png", "formula": "\\begin{align*} \\psi ( z ) + \\frac { z \\psi ' ( z ) } { \\psi ( z ) } = \\phi ( z ) . \\end{align*}"}
{"id": "2345.png", "formula": "\\begin{align*} f = a _ 0 + a _ 1 Q + \\ldots + a _ r Q ^ r \\mbox { a n d } f = b _ 0 + b _ 1 Q ' + \\ldots + b _ r Q '^ r \\end{align*}"}
{"id": "3071.png", "formula": "\\begin{align*} f _ { \\mathbf { M ^ { \\ast \\ast } } _ { - \\xi _ { \\boldsymbol { s } k } , \\xi _ { \\boldsymbol { t } l } } } ( \\boldsymbol { u } ) = c _ { n - 2 , \\xi _ { \\boldsymbol { s } k } , \\xi _ { \\boldsymbol { t } l } } ^ { \\ast \\ast } \\exp \\left \\{ - \\frac { 1 } { 2 } \\boldsymbol { u } ^ { T } \\boldsymbol { u } - \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { s } k } ^ { 2 } - \\frac { 1 } { 2 } \\xi _ { \\boldsymbol { t } l } ^ { 2 } \\right \\} = \\phi _ { n - 2 } ( \\boldsymbol { u } ) , ~ \\boldsymbol { u } \\in \\mathbb { R } ^ { n - 2 } , \\end{align*}"}
{"id": "1484.png", "formula": "\\begin{align*} - \\log \\vert R _ { \\ell , i , s } ( \\beta ) \\vert _ { v _ 0 } & \\leq n \\log \\vert \\beta \\vert _ { v _ 0 } - ( r m + 1 ) n \\log \\| \\boldsymbol { \\alpha } \\| _ { v _ 0 } - n c ( x , v _ 0 ) + ( 1 - \\varepsilon _ v ) \\limsup _ { n \\to \\infty } \\dfrac { 1 } { n } \\log | D _ { \\boldsymbol { c } , r m n } \\cdot D ' _ { \\boldsymbol { c } , r m n } | ^ { - 1 } _ v + o ( n ) \\\\ & = \\mathbb { A } _ { v _ 0 } ( \\boldsymbol { \\eta } , \\boldsymbol { \\zeta } , \\boldsymbol { \\alpha } , \\beta ) n + o ( n ) \\enspace . \\end{align*}"}
{"id": "7661.png", "formula": "\\begin{align*} \\widetilde s ( \\mathcal E ) = \\sum _ { 0 \\leq i \\leq 4 } \\widetilde s _ i ( \\mathcal E ) = ( 1 + H ) ^ { - n } , \\end{align*}"}
{"id": "1321.png", "formula": "\\begin{align*} \\dot x & = v , \\\\ \\dot v & = - \\varepsilon ^ 2 \\nabla \\Phi ( x ) , \\end{align*}"}
{"id": "3130.png", "formula": "\\begin{align*} M : = \\sup _ { N \\in \\mathbb { N } _ 0 } ( N + 1 ) ^ s \\min _ { \\mathcal { T } \\in \\mathbb { T } ( N ) } \\eta ( \\mathcal { T } ) < \\infty . \\end{align*}"}
{"id": "4446.png", "formula": "\\begin{align*} m ! \\ , N _ 0 ( R f ) _ { i _ 1 j _ 1 \\dots i _ m j _ m } = \\sum _ { l = 0 } ^ { \\lfloor \\frac { m } { 2 } \\rfloor } c _ { l , m } ( R ( i ^ l j ^ l N _ m f ) ) _ { i _ 1 j _ 1 \\dots i _ m j _ m } , \\end{align*}"}
{"id": "3211.png", "formula": "\\begin{align*} c _ { a _ { \\pi } } ( s _ t ) \\triangleq w _ 1 q _ { t } ^ { } + w _ 2 E _ t ^ { } + \\sum _ { k = 1 } ^ K \\Big ( w _ 3 q _ { k , t } ^ { } + w _ 4 E _ { k , t } \\Big ) \\end{align*}"}
{"id": "7919.png", "formula": "\\begin{align*} \\mathcal U _ \\omega : = \\bigcup _ { I \\in \\mathcal A _ \\omega } ( \\mathbb C ^ \\times ) ^ { I } \\times ( \\mathbb C ) ^ { \\bar { I } } . \\end{align*}"}
{"id": "5000.png", "formula": "\\begin{align*} \\Psi ^ { n , 2 } _ s = \\left ( \\int ^ s _ { \\eta _ n ( s ) } \\left ( s - \\eta _ n ( u ) \\right ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) ^ 2 , \\end{align*}"}
{"id": "6147.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { N } \\| u _ h ^ m - u _ h ^ { m - 1 } \\| _ { 2 } ^ 2 \\leq C \\left ( \\Delta t + \\frac { 1 } { h ^ { 1 / 2 } } \\right ) . \\end{align*}"}
{"id": "2786.png", "formula": "\\begin{align*} x _ { i + 1 } = x _ i - \\tfrac { h _ i } { L } g _ i , \\forall i = 0 , \\dots , N - 1 \\end{align*}"}
{"id": "7730.png", "formula": "\\begin{align*} x ( t ) = \\Phi ( t , \\tau ) x ( \\tau ) \\quad t \\geq \\tau \\geq 0 , \\end{align*}"}
{"id": "1403.png", "formula": "\\begin{align*} \\textrm { s i n } \\Theta ( Q , \\widetilde Q ) = \\left \\{ \\sin \\left ( \\cos ^ { - 1 } ( \\zeta _ 1 ) \\right ) , \\sin \\left ( \\cos ^ { - 1 } ( \\zeta _ 2 ) \\right ) , \\cdots , \\sin \\left ( \\cos ^ { - 1 } ( \\zeta _ r ) \\right ) \\right \\} . \\end{align*}"}
{"id": "767.png", "formula": "\\begin{align*} P _ { \\rm F A } ^ { } ( \\lambda ) = { \\rm P r } \\left ( \\left . { \\sum _ { m = 1 } ^ { M } ( T _ { { m } } - { c } ) ^ 2 } > \\frac { M } { 4 } ( \\alpha - \\lambda ) ^ 2 \\right | \\mathcal { H } _ 0 \\right ) . \\end{align*}"}
{"id": "3700.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ { \\Omega _ \\Gamma } } \\chi _ \\alpha ( x _ n ) \\big ( 1 - \\chi _ \\Gamma \\big ) \\partial _ { x _ n } ( R + V ) \\ , d \\mu = 0 \\end{align*}"}
{"id": "539.png", "formula": "\\begin{align*} \\sum \\limits _ { r \\mid n } \\mu ( r ) = \\begin{cases} 1 , & n = 1 , \\\\ 0 , & { \\rm o t h e r w i s e } , \\end{cases} \\end{align*}"}
{"id": "1198.png", "formula": "\\begin{align*} \\begin{cases} & , \\\\ & | g ( x ) - g ( 0 ) | \\le C ( 1 + | x | ^ { \\frac { 2 p } { \\beta } } ) | x | , \\ \\forall x \\in \\R ^ d , \\\\ & | \\nabla g ( x ) | \\le C ( 1 + | x | ^ { \\frac { 2 p } { \\beta } } ) \\ \\ x \\in \\R ^ d . \\end{cases} \\end{align*}"}
{"id": "4488.png", "formula": "\\begin{align*} \\partial _ \\xi \\Phi ( \\eta _ 1 , \\eta _ 2 , \\xi ) = - \\frac { 1 } { [ 1 + ( \\xi - \\eta _ 1 - \\eta _ 2 ) ^ 2 ] ^ { 3 / 2 } } + \\frac { 1 } { ( 1 + \\xi ^ 2 ) ^ { 3 / 2 } } , \\\\ \\partial _ { \\eta _ 1 } \\Phi ( \\eta _ 1 , \\eta _ 2 , \\xi ) = - \\frac { 1 } { ( 1 + \\eta _ 1 ^ 2 ) ^ { 3 / 2 } } + \\frac { 1 } { [ 1 + ( \\xi - \\eta _ 1 - \\eta _ 2 ) ^ 2 ] ^ { 3 / 2 } } . \\end{align*}"}
{"id": "5163.png", "formula": "\\begin{align*} N ' _ { W ' W ' } & = \\left ( | W ' | + 1 - s _ { W ' } | W ' | + \\sum _ { h ( T _ V ) \\ge 1 } \\frac { m _ V | W ' | } { m _ V | V | + 1 } + \\sum _ { h ( T _ { V ' } ) = 0 } \\frac { | W ' | } { | V ' | + 1 } \\right ) \\\\ & \\le 1 + | W ' | \\left ( 1 - s _ { W ' } + \\sum _ { h ( T _ V ) \\ge 1 } \\frac { 1 } { | V | - 2 } + \\sum _ { h ( T _ { V ' } ) = 0 } \\frac { 1 } { | V ' | + 1 } \\right ) \\\\ & \\le 1 + | W ' | \\left ( 1 - \\frac 3 4 s _ { W ' } \\right ) . \\\\ \\end{align*}"}
{"id": "6109.png", "formula": "\\begin{align*} U _ 0 = A \\iota ( z _ r ) \\ldots \\iota ( z _ { r + s } ) B \\end{align*}"}
{"id": "2117.png", "formula": "\\begin{align*} | W | x ^ 2 + ( t + l - l | W | - 1 ) x - l t = 0 \\end{align*}"}
{"id": "1635.png", "formula": "\\begin{align*} & \\varepsilon _ * = \\varepsilon _ { \\vec \\gamma } , \\ \\ k _ * = k _ { \\vec \\gamma } , \\ \\ t _ { \\mathbf { i } } = p _ { \\vec \\gamma } ( \\mathbf { i } ) , \\cr t _ { i , j } & = p _ { \\vec { \\gamma } } ( i , \\delta _ { \\vec { \\gamma } } ( j ) ) \\mathrm { \\ f o r \\ } i < \\mathbf { i } , \\ j < k _ * . \\end{align*}"}
{"id": "1701.png", "formula": "\\begin{align*} | y - 2 ^ { | k | + 2 N } e _ 1 | & = | y - 2 ^ { | j | + 2 N } e _ 1 + 2 ^ { | j | + 2 N } e _ 1 - 2 ^ { | k | + 2 N } e _ 1 | \\\\ & \\ge | 2 ^ { | j | + 2 N } e _ 1 - 2 ^ { | k | + 2 N } e _ 1 | - | y - 2 ^ { | j | + 2 N } e _ 1 | \\ge 2 ^ { 2 N - 1 } . \\end{align*}"}
{"id": "7868.png", "formula": "\\begin{align*} \\Delta = \\delta ^ * \\overline { \\delta } , \\ \\ \\ \\ \\ \\zeta _ \\alpha = \\left ( \\frac { \\alpha } { \\alpha + \\Delta } \\right ) ^ { 1 / 2 } , \\ \\ \\ \\ \\ \\tilde \\delta _ \\alpha = \\frac { 1 } { \\sqrt { \\alpha } } \\overline \\delta \\circ \\zeta _ \\alpha , \\ \\ \\ \\ \\ \\tilde \\Delta _ \\alpha = \\frac { 1 } { \\sqrt { \\alpha } } \\Delta ^ { 1 / 2 } \\circ \\zeta _ \\alpha , \\ \\ \\ \\ \\ \\theta _ \\alpha = 1 - \\tilde \\Delta _ \\alpha . \\end{align*}"}
{"id": "8186.png", "formula": "\\begin{align*} P _ { \\frac { \\delta } { 2 } } ( E ) P _ { \\frac { \\delta } { 2 } } ( F ) = 0 \\end{align*}"}
{"id": "8629.png", "formula": "\\begin{align*} \\widehat H ( t ) : = \\sum _ { \\alpha = 1 } ^ r b _ { \\alpha } ( t ) \\widehat { H } _ { \\alpha } , \\end{align*}"}
{"id": "5201.png", "formula": "\\begin{align*} \\mathfrak { o } ^ { \\pi ' } = ( - 1 ) ^ { m | B _ 2 | } \\mathfrak { o } ^ { \\pi ' _ 1 } _ { 0 , B _ 1 , I _ 1 } \\boxtimes \\mathfrak { o } ^ { \\pi ' _ 2 } _ { 0 , \\{ h _ 2 \\} \\cup B _ 2 , I _ 2 } . \\end{align*}"}
{"id": "3836.png", "formula": "\\begin{align*} E r r U ^ { 2 ; 2 } _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i \\widetilde { \\Phi } ^ 2 _ { \\mu _ 1 , \\mu _ 2 } ( \\xi , \\eta , \\sigma ; s , X ( s ) , v ) } \\clubsuit K _ { k _ 2 , j _ 2 , n _ 2 } ^ { \\mu , i _ 2 } ( s , \\sigma , V ( s ) ) \\cdot { } _ { } ^ 1 \\clubsuit E l l U ^ 2 ( s , \\xi , \\eta + \\sigma , v , V ( s ) ) \\end{align*}"}
{"id": "9006.png", "formula": "\\begin{align*} \\zeta ( n , x ) - \\zeta \\left ( n , x + \\frac { 1 } { 2 } \\right ) = \\frac { ( - 1 ) ^ { n } } { ( n - 1 ) ! } \\left ( \\psi _ { n - 1 } \\left ( x \\right ) - \\psi _ { n - 1 } \\left ( x + \\frac { 1 } { 2 } \\right ) \\right ) , \\end{align*}"}
{"id": "6221.png", "formula": "\\begin{align*} & \\int F ( x ) k ( q x ) \\Big ( D _ q u ( x ) + u ( x ) u ( q x ) + \\tilde { A } ( x ) u ( q x ) + r ( x ) \\Big ) y ( x ) d _ q x = \\\\ & F ( x ) k ( x ) \\Big ( y ( x ) u ( x ) - D _ { q ^ { - 1 } } y ( x ) \\Big ) , \\end{align*}"}
{"id": "1698.png", "formula": "\\begin{align*} u = U _ 0 + U _ 1 + U _ 2 , \\end{align*}"}
{"id": "799.png", "formula": "\\begin{align*} | g ( z ^ m ) | \\leq f ( r ^ m ) \\leq \\hat { f } _ 0 ( r ^ m ) , f _ 0 ( z ) = \\int _ { 0 } ^ { z } \\frac { \\psi ( t ) } { 1 - t ^ 2 } d t . \\end{align*}"}
{"id": "7152.png", "formula": "\\begin{align*} ( g _ { 2 1 } y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 ) \\{ ( g _ { 1 1 } + g _ { 2 1 } ) y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 \\} ^ { b - 1 } = \\alpha y _ 1 ^ a + g _ { 2 3 } ^ b y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { b - 1 } , \\end{align*}"}
{"id": "1809.png", "formula": "\\begin{align*} \\tilde x = 2 x ^ { ( k + 1 ) } - x ^ { ( k ) } + \\tau ( \\nabla h ( x ^ { ( k ) } ) - \\nabla h ( x ^ { ( k + 1 ) } ) ) , \\tilde z = z ^ { ( k ) } . \\end{align*}"}
{"id": "4113.png", "formula": "\\begin{align*} \\left \\| F ^ \\ast ( z ) { c _ 1 \\choose c _ 2 } \\right \\| ^ 2 + \\left \\| \\Theta ^ \\ast ( z ) { c _ 1 \\choose c _ 2 } \\right \\| ^ 2 & = | c _ 1 + c _ 2 | ^ 2 + | b _ 1 ( z ) c _ 1 | ^ 2 + | b _ 2 ( z ) c _ 2 | ^ 2 \\\\ & \\ge ( | c _ 1 | - | c _ 2 | ) ^ 2 + \\delta ^ 2 \\min \\{ | c _ 1 | ^ 2 , | c _ 2 | ^ 2 \\} \\\\ & \\ge \\frac { \\delta ^ 2 } { 1 6 } \\ , ( | c _ 1 | ^ 2 + | c _ 2 | ^ 2 ) \\end{align*}"}
{"id": "7878.png", "formula": "\\begin{align*} D _ { Z , - } = ( D _ { 1 , - } + D _ - ) \\cap Z _ - . \\end{align*}"}
{"id": "6484.png", "formula": "\\begin{align*} u _ x = v _ x = 0 \\ \\ \\ \\ ( b _ 1 , b _ 2 ) \\ \\ \\ \\ y _ x = z _ x = 0 \\ \\ \\ \\ ( d _ 1 , d _ 2 ) . \\end{align*}"}
{"id": "3295.png", "formula": "\\begin{align*} u ^ s _ { A , q } ( x , y ) & = \\underset { ( \\ell _ 1 , m _ 1 ) \\in \\Gamma } \\sum \\alpha _ { \\ell _ 1 m _ 1 } ( y ) h ^ { ( 1 ) } _ { \\ell _ 1 } ( k \\vert x \\vert ) Y ^ { m _ 1 } _ { \\ell _ 1 } \\left ( \\widehat { x } \\right ) . \\end{align*}"}
{"id": "4008.png", "formula": "\\begin{align*} t _ { k } ^ { ( h ) } = h + 2 H k = t _ { k - 1 } ^ { ( h ) } + 2 H , \\end{align*}"}
{"id": "6272.png", "formula": "\\begin{align*} \\int \\dfrac { ( \\frac { x } { a } , \\frac { x } { b } ; q ) _ \\infty } { ( x ; q ) _ \\infty } p _ n ( x ; a , b ; q ) d _ q x = \\dfrac { a b q ^ 2 ( 1 - q ) } { ( 1 - a q ) ( 1 - b q ) } \\dfrac { ( \\frac { x } { a q } , \\frac { x } { b q } ; q ) _ \\infty } { ( x ; q ) _ \\infty } p _ { n - 1 } ( x ; a q , b q ; q ) . \\end{align*}"}
{"id": "2692.png", "formula": "\\begin{align*} H _ \\mathrm { B H } = \\frac { \\mu _ + - \\mu _ - } { 2 } \\big ( a ^ * _ 1 a _ 2 + a ^ * _ 2 a _ 1 \\big ) + \\frac { \\lambda \\ , w _ { 1 1 1 1 } } { 2 ( N - 1 ) } \\big ( a ^ * _ 1 a ^ * _ 1 a _ 1 a _ 1 + a ^ * _ 2 a ^ * _ 2 a _ 2 a _ 2 \\big ) . \\end{align*}"}
{"id": "6148.png", "formula": "\\begin{align*} ( u _ h ^ m - u _ h ^ { m - 1 } , u _ { h } ^ { m , 1 / 2 } ) = \\frac { 1 } { 2 } ( u _ h ^ m - u _ h ^ { m - 1 } , u _ h ^ m + u _ h ^ { m - 1 } ) = \\frac { 1 } { 2 } ( \\| u _ h ^ m \\| _ 2 ^ 2 - \\| u _ h ^ { m - 1 } \\| _ 2 ^ 2 ) . \\end{align*}"}
{"id": "7185.png", "formula": "\\begin{align*} & n = \\delta ( j _ 4 ) + \\delta ( j _ 2 ) + j - \\delta ( j _ 2 + j _ 4 ) + \\frac { j _ 2 + j _ 4 } { T } - \\frac { [ j _ 2 + j _ 4 ] } { T } \\end{align*}"}
{"id": "7086.png", "formula": "\\begin{align*} \\delta ( t _ { k + 1 } ) & = \\delta ( t _ { k } ) + \\int ^ { t _ { k + 1 } } _ { t _ { k } } ( \\lambda - u ( t ) ) d t \\\\ & = \\delta ( t _ { k } ) + \\lambda \\mathsf { T } _ { k } - 0 . 5 ( \\beta ^ { \\top } u ( t _ { k } ) + \\mathbf { 1 } ^ { \\top } u ( t _ { k + 1 } ) ) \\mathsf { T } _ { k } . \\end{align*}"}
{"id": "2171.png", "formula": "\\begin{align*} d ( \\mu \\boxplus \\nu ) _ { } ( h ( x ) ) = \\pi ^ { - 1 } \\left [ I _ { V } ( x ) + I _ { B \\cup C } ( x ) \\right ] \\left ( - \\Im G _ { \\mu \\boxplus \\nu } \\right ) ^ { * } ( h ( x ) ) \\ , d \\lambda ( h ( x ) ) . \\end{align*}"}
{"id": "3280.png", "formula": "\\begin{align*} \\mathcal { R } = - \\int _ B & e ^ { - i \\varphi } \\big [ i ( A + \\nabla \\vartheta ) \\cdot ( u _ 1 \\nabla u _ 2 - u _ 2 \\nabla u _ 1 ) \\\\ & + \\Big ( ( A + \\nabla \\vartheta ) \\cdot ( A _ 1 + A _ 2 ) - q + ( A + \\nabla \\vartheta ) \\cdot \\nabla \\varphi \\Big ) u _ 1 u _ 2 \\big ] \\ , d x . \\end{align*}"}
{"id": "2409.png", "formula": "\\begin{gather*} K _ { \\mu } ( s ) : = e ^ { - A _ n ^ 2 ( t - s ) } ( - A _ n ) ^ { 1 + \\mu } \\left \\{ F _ n ( \\mathbb U ( s ) ) - F _ n ( U ( s ) ) \\right \\} , \\\\ L _ { \\mu } ( s ) : = \\sqrt { n } ( - A _ n ) ^ { \\mu } e ^ { - A _ n ^ 2 ( t - s ) } \\left \\{ \\Sigma _ n ( \\mathbb U ( s ) ) - \\Sigma _ n ( U ( s ) ) \\right \\} . \\end{gather*}"}
{"id": "1284.png", "formula": "\\begin{align*} P ( \\mathcal { A } _ k , x ) & = H e _ k ( x ) \\prod _ { j = 2 } ^ { k - 1 } H e _ j ( x ) ^ { \\frac { ( j - 1 ) ( k - 1 ) ! } { j ! } } . \\end{align*}"}
{"id": "5506.png", "formula": "\\begin{align*} W _ { 2 } ( \\mu _ { 1 } ( t ) , \\mu _ { 2 } ( t ) ) & = W _ { 2 } ( \\mu ( t , - n , \\mu _ { 1 } ( - n ) ) , \\mu ( t , - n , \\mu _ { 2 } ( - n ) ) ) \\\\ & \\leq \\left ( \\mathbb E \\| u ( t , - n , \\zeta _ { n , 1 } ) - u ( t , - n , \\zeta _ { n , 2 } ) \\| ^ { 2 } \\right ) ^ { 1 / 2 } \\\\ & \\leq { \\rm { e } } ^ { - ( \\lambda _ { * } - \\lambda _ { f } - \\frac { L _ { g } ^ { 2 } } { 2 } ) ( t + n ) } \\left ( \\mathbb E \\| \\zeta _ { n , 1 } - \\zeta _ { n , 2 } \\| ^ { 2 } \\right ) ^ { 1 / 2 } \\rightarrow 0 { \\rm { a s } } ~ n \\rightarrow \\infty . \\end{align*}"}
{"id": "1221.png", "formula": "\\begin{align*} B _ 1 ( T ) & = A ( T ) + \\mathrm { d i a g } ( \\beta ( u _ 1 ) , \\beta ( u _ 2 ) , \\ldots , \\beta ( u _ { n ( T ) } ) ) , \\\\ B _ 2 ( T ) & = - A ( T ) + \\mathrm { d i a g } ( \\beta ( u _ 1 ) , \\beta ( u _ 2 ) , \\ldots , \\beta ( u _ { n ( T ) } ) ) . \\end{align*}"}
{"id": "6979.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ N ^ 2 } \\prod \\limits _ { k = 1 } ^ { \\infty } \\Bigg ( \\frac { \\lambda _ N ^ 2 + \\mu _ { k } ^ { 2 } } { \\lambda _ N ^ 2 + \\lambda _ { k } ^ { 2 } } \\Bigg ) = \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { a _ k } { \\lambda _ k ^ 2 ( \\lambda _ N ^ 2 + \\lambda _ { k } ^ { 2 } ) } . \\end{align*}"}
{"id": "2103.png", "formula": "\\begin{align*} 0 = ( L z , z ) _ V = - \\sum _ { e \\in E } \\frac { 1 } { 2 } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum _ { u , v \\in e } ( z ( u ) - z ( v ) ) ^ 2 . \\end{align*}"}
{"id": "3352.png", "formula": "\\begin{align*} \\eta ( e ^ { 2 \\pi i t } ) = [ \\lambda _ 1 ( t ) , \\lambda _ 2 ( t ) , \\ldots , \\lambda _ a ( t ) ] \\end{align*}"}
{"id": "3942.png", "formula": "\\begin{align*} V a r ( \\hat { \\pi } ^ a _ { \\tilde { h } ^ * , T _ n } ( x ) ) & = \\frac { 2 } { T _ n ^ 2 } \\int _ 0 ^ { T _ n } \\int _ 0 ^ { t } k ( t , s ) 1 _ { s < t } \\ , \\big ( 1 _ { | t - s | \\le 3 \\Delta _ n } \\\\ & + 1 _ { 3 \\Delta _ n \\le | t - s | \\le D } + 1 _ { D \\le | t - s | \\le T _ n } \\big ) d s d t \\\\ & = \\sum _ { j = 1 } ^ 3 \\hat { I } _ j . \\end{align*}"}
{"id": "2879.png", "formula": "\\begin{align*} f _ i ( x ) = f \\circ S ^ { ( 1 - i ) m } ( x ) , i \\in \\Z . \\end{align*}"}
{"id": "3694.png", "formula": "\\begin{align*} \\chi _ \\alpha ( x _ n ) = \\begin{cases} & 0 | x _ n | \\geq 2 \\alpha \\\\ & 1 | x _ n | \\leq \\alpha / 2 , \\end{cases} \\end{align*}"}
{"id": "6788.png", "formula": "\\begin{align*} m = \\sum _ { j \\ge 0 } m \\chi _ { U _ j ( B ) } = \\sum _ { j \\ge 0 } m _ j . \\end{align*}"}
{"id": "1935.png", "formula": "\\begin{align*} S _ { j } \\left ( \\frac { 1 } { w } \\right ) = \\sum _ { m = 0 } ^ { \\infty } C _ { [ m , j ] } w ^ { m ( p + 1 ) + j + 1 } = h ( w ) ^ { j + 1 } , \\end{align*}"}
{"id": "1161.png", "formula": "\\begin{align*} f _ * = \\sum _ { 2 k = 0 } ^ n ( - 1 ) ^ { 2 k } f _ { 2 k } = \\sum _ { 2 k = 0 } ^ n f _ { 2 k } = f _ * , \\end{align*}"}
{"id": "6359.png", "formula": "\\begin{align*} p _ { \\kappa } ( x ) = N _ { \\kappa } \\ , \\ , x ^ { n - 1 } \\exp _ { \\kappa } ( - x ) \\ \\ , \\end{align*}"}
{"id": "5918.png", "formula": "\\begin{align*} \\frac { d ^ k } { d z ^ k } F ( z ) = \\frac { d ^ k } { d z ^ k } F _ { \\mathrm { m } } ( z ) + O ( 1 ) , | z | \\geq r _ 0 , ~ k \\geq 0 . \\end{align*}"}
{"id": "17.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ { \\rho } = ( \\alpha _ n , \\beta _ n ) _ { \\rho , E _ { \\rho } ^ n , n } \\end{align*}"}
{"id": "7999.png", "formula": "\\begin{align*} f _ { ( X , K 3 ) } ( y _ 1 , y _ 2 ) = \\sum _ { d _ 1 , d _ 2 \\geq 0 } \\frac { ( 4 d _ 1 ) ! ( d _ 1 + d _ 2 ) ! } { ( d _ 1 ! ) ^ 5 ( d _ 2 ! ) } y _ 1 ^ { d _ 1 } y _ 2 ^ { d _ 2 } . \\end{align*}"}
{"id": "709.png", "formula": "\\begin{align*} & \\overline { \\Gamma } ^ \\alpha _ { 0 0 } = 0 , \\ \\overline { \\Gamma } ^ 0 _ { 0 i } = 0 , \\ \\overline { \\Gamma } ^ k _ { i j } = \\widetilde { \\Gamma } ^ k _ { i j } , \\\\ & \\overline { \\Gamma } ^ 0 _ { i j } = f ( t ) f ' ( t ) \\widetilde { g } _ { i j } , \\ \\overline { \\Gamma } ^ k _ { 0 i } = \\frac { f ( t ) f ' ( t ) } { f ( t ) ^ 2 } \\delta ^ k _ i . \\end{align*}"}
{"id": "7949.png", "formula": "\\begin{align*} D _ + = \\sum _ { i \\in ( M _ + \\cup M _ - ) } \\bar { D } _ i \\subset X _ + D _ - = \\sum _ { i \\in ( M _ + \\cup M _ - ) } \\bar { D } _ i \\subset X _ - . \\end{align*}"}
{"id": "5993.png", "formula": "\\begin{align*} e ^ { \\max } _ { } = \\max _ { j = 1 , \\dots , N } E ^ 2 _ { } ( t _ j ) . \\end{align*}"}
{"id": "6028.png", "formula": "\\begin{align*} f ( x ) = x ^ 3 - \\frac { 3 } { 2 } x ^ 2 = ( x - 1 ) ^ 3 + \\frac { 3 } { 2 } ( x - 1 ) ^ 2 - \\frac { 1 } { 2 } \\end{align*}"}
{"id": "7052.png", "formula": "\\begin{align*} h _ 1 ( M ) = & h _ 1 ( M - 1 ) + h _ 1 ( M - 2 ) + F _ { M - 3 } + F _ { M - 4 } + \\cdots + F _ 1 + 1 \\\\ = & h _ 1 ( M - 1 ) + h _ 1 ( M - 2 ) + F _ { M - 1 } . \\end{align*}"}
{"id": "1310.png", "formula": "\\begin{align*} \\cos \\theta : = \\frac { ( v _ * ' - v ' ) \\cdot ( v _ * - v ) } { | v _ * - v | ^ 2 } = \\frac { v - v _ * } { | v - v _ * | } \\cdot \\sigma . \\end{align*}"}
{"id": "5722.png", "formula": "\\begin{align*} ( y _ i - y _ { i + 1 } ) \\cdot \\pi _ { [ a , b ] } & = ( y _ i - y _ { i + 1 } ) \\pi _ { [ a - 1 , b - 1 ] } + ( y _ i - y _ { i + 1 } ) \\pi _ { [ a , b - 1 ] } y _ b . \\end{align*}"}
{"id": "6536.png", "formula": "\\begin{align*} \\frac { s _ k } { s _ { k + 1 } } = \\frac { k + 1 } { \\AA ( i , { i + k + 1 } ) } \\frac { k - j + 1 } { k - j + 2 } \\left [ \\frac { \\big ( \\AA ( i , { i + k + 1 } ) \\big ) } { \\big ( \\AA ( i , { i + k } ) \\big ) } \\right ] ^ k \\geq \\frac { k + 1 } { 2 e ( \\ln k + 2 ) } . \\end{align*}"}
{"id": "4031.png", "formula": "\\begin{align*} \\underline { \\gamma } & = n \\chi ( \\epsilon _ n ) + O ( 1 ) \\\\ \\overline { \\gamma } & = n \\chi ( \\epsilon _ n ) + O ( 1 ) . \\end{align*}"}
{"id": "3218.png", "formula": "\\begin{align*} E _ * = E _ * \\otimes \\mathcal { O } _ X \\xhookrightarrow { } E _ * \\otimes ( E _ * \\otimes E ^ \\vee _ * ) = ( E _ * \\otimes E _ * ) \\otimes E ^ \\vee _ * \\xrightarrow { \\psi \\otimes I d } L _ * \\otimes E ^ \\vee _ * . \\end{align*}"}
{"id": "2705.png", "formula": "\\begin{align*} c _ d : = \\frac { 1 } { Z _ N } e ^ { - d ^ 2 / 4 \\sigma _ N ^ 2 } , \\qquad | d | \\le \\sigma _ N ^ 2 . \\end{align*}"}
{"id": "1165.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\Phi ( \\theta + t _ n h _ n ) - \\Phi ( \\theta ) } { t _ n } = \\Phi ' _ { \\theta } ( h ) \\end{align*}"}
{"id": "5987.png", "formula": "\\begin{align*} t _ j = T ( j / N ) ^ k \\end{align*}"}
{"id": "2532.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( L _ { Z _ 1 } g ) ( X _ 1 , Y _ 1 ) + R i c ( X _ 1 , Y _ 1 ) + \\mu g ( X _ 1 , Y _ 1 ) = 0 , \\end{align*}"}
{"id": "2189.png", "formula": "\\begin{align*} d ^ { \\tt s t } = d ^ { C ^ { \\tt s t } } _ { A ^ { \\tt s t } } , \\end{align*}"}
{"id": "1670.png", "formula": "\\begin{align*} f _ { i } = { f } _ { i } ^ 0 + \\sum _ { j = 1 } ^ k \\ \\sum _ { \\{ p : j \\in S _ p \\} } \\rho _ p { f } _ { j } ^ 0 \\ . \\end{align*}"}
{"id": "3709.png", "formula": "\\begin{align*} C _ { \\gamma _ 1 } \\cap C _ { \\gamma _ 2 } = \\{ \\gamma _ 1 \\} \\times C _ J = \\{ \\gamma _ 2 \\} \\times C _ J , \\end{align*}"}
{"id": "5761.png", "formula": "\\begin{align*} ( a , x ) \\ast ( b , y ) = ( a + b + h _ s ( x , y ) , \\ x y ) . \\end{align*}"}
{"id": "7164.png", "formula": "\\begin{align*} & \\phi ( K f _ 1 ) = 0 \\ , , \\\\ & [ \\phi ( f _ 1 ) , \\phi ( f _ 2 ) ] = 0 \\ , f _ 1 \\perp \\ , f _ 2 \\ , . \\end{align*}"}
{"id": "2632.png", "formula": "\\begin{align*} \\Hat { r } ( X ) \\\\ & = \\min _ { A \\subseteq X } \\bigl ( | X \\backslash A | + \\sum _ { i = 1 } ^ n r _ i ^ * ( A ) \\bigr ) \\\\ & = \\min _ { A \\subseteq X } \\bigl ( | X \\backslash A | + \\sum _ { i = 1 } ^ n ( | A | + r _ i ( E \\backslash A ) - r _ i ( E ) \\bigr ) \\\\ & = \\min _ { A \\subseteq X } \\biggl ( ( n - 1 ) | A | + | X | + \\sum _ { i = 1 } ^ n \\bigl ( r _ i ( E \\backslash A ) - r _ i ( E ) \\bigr ) \\biggr ) \\end{align*}"}
{"id": "3355.png", "formula": "\\begin{align*} \\varphi ( f ) ( t ) = u ( t ) \\begin{pmatrix} f ( \\lambda _ 1 ( t ) ) & 0 & \\hdots & 0 \\\\ 0 & f ( \\lambda _ 2 ( t ) ) & \\hdots & 0 \\\\ \\hdots & \\hdots & \\hdots & \\hdots \\\\ 0 & 0 & \\hdots & f ( \\lambda _ a ( t ) ) \\end{pmatrix} u ( t ) ^ { \\ast } . \\end{align*}"}
{"id": "3399.png", "formula": "\\begin{align*} \\sum _ { e _ i = 0 } ^ { \\infty } \\sum _ { e _ { i ' } = 0 } ^ \\infty ( - 1 ) ^ { e _ i + e _ { i ' } } M _ { i , i ' } ( e _ i , e _ { i ' } ) Q _ i ( e _ i ) Q _ { i ' } ( e _ { i ' } ) = w _ { V _ i } w _ { V _ i ' } , \\end{align*}"}
{"id": "8200.png", "formula": "\\begin{align*} \\varphi _ { 0 } ^ { + } = A . \\varphi _ { 0 } ^ { - } , \\end{align*}"}
{"id": "6243.png", "formula": "\\begin{align*} & \\int \\frac { x k ( q x ) y ( x ) } { ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\Big ( D _ q u ( x ) + u ( x ) u ( q x ) + \\frac { 1 - q ^ { 1 - \\nu } ( 1 - q ) [ \\nu ] ^ 2 } { x } u ( q x ) + \\frac { q x ^ 2 - q ^ { 1 - \\nu } [ \\nu ] ^ 2 } { x ^ 2 } \\Big ) d _ q x \\\\ & = \\frac { x k ( x ) } { ( - x ^ 2 ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\Big ( y ( x ) u ( x ) - D _ { q ^ { - 1 } } y ( x ) \\Big ) . \\end{align*}"}
{"id": "2576.png", "formula": "\\begin{align*} \\mathbf S : = \\mathbf S _ { 1 2 } - \\dfrac { 1 } { 3 } C _ 3 ^ { ( 1 ) } + \\dfrac { 1 } { 3 } C _ 3 ^ { ( 2 ) } \\ , \\end{align*}"}
{"id": "5076.png", "formula": "\\begin{align*} M ^ { n , 2 } _ s = \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u , \\end{align*}"}
{"id": "2315.png", "formula": "\\begin{align*} G ( { \\bf s } _ { \\alpha , \\delta } ) = { \\bf s } _ { \\frac { \\alpha } { 1 + \\alpha } , \\frac { \\delta } { 1 + \\alpha } } , \\widetilde { G } ( { \\bf s } _ { \\alpha , \\delta } ) = { \\bf s } _ { \\frac { \\alpha } { 1 + \\alpha } , \\frac { \\alpha + \\delta } { 1 + \\alpha } } E ( { \\bf s } _ { \\alpha , \\delta } ) = { \\bf s ' } _ { 1 - \\alpha , 1 - \\delta } . \\end{align*}"}
{"id": "1081.png", "formula": "\\begin{align*} k ( t ) = C \\mathrm { e } ^ { a t } t ^ { - d / 2 } , \\end{align*}"}
{"id": "1763.png", "formula": "\\begin{align*} | y _ n ^ * ( x _ j ^ k ) | = a _ j | f _ { p ( j ) } ( y _ n ^ * ) | \\leq C \\ : a _ j . \\end{align*}"}
{"id": "5956.png", "formula": "\\begin{align*} x \\eta = x \\xi ' \\otimes \\Omega _ \\phi = \\delta ( x ) \\otimes \\Omega _ \\phi + \\xi ' x \\otimes \\Omega _ \\phi = \\delta ( x ) \\otimes \\Omega _ \\phi + \\eta \\sigma ^ \\phi _ { i / 2 } ( x ) \\end{align*}"}
{"id": "4235.png", "formula": "\\begin{align*} \\mathbf { P } ( F _ k ^ c \\cap C _ k \\cap G _ k \\cap B _ k \\setminus \\cup _ { i = k _ 1 } ^ k A _ i ) \\leq \\mathbf { P } ( F _ k ^ c \\cap C _ k ) \\leq \\sqrt { e ^ { - a ( k ) + a ( k - 1 ) + 1 } } \\end{align*}"}
{"id": "6579.png", "formula": "\\begin{align*} \\phi ( \\beta ( g ^ { - 1 } , \\psi ^ { - 1 } ( \\alpha ( g , x ) , v ) ) ) = ( x , A ( g , x , v ) ) \\mbox { f o r a l l $ g \\in U $ , $ x \\in V $ , a n d $ v \\in F $ , } \\end{align*}"}
{"id": "7238.png", "formula": "\\begin{align*} y ( t ) & = \\int _ 0 ^ 1 w ( s ) \\partial _ s x ( t , s ) d s \\end{align*}"}
{"id": "972.png", "formula": "\\begin{align*} H ^ { * } \\left ( \\overline { e } _ { k } \\right ) & = \\left [ H \\overline { e } _ { k - 1 } , H ^ { 2 } \\overline { e } _ { k - 1 } , \\cdots , H ^ { n } \\overline { e } _ { k - 1 } \\right ] \\\\ & = H \\left [ \\overline { e } _ { k - 1 } , H \\overline { e } _ { k - 1 } , \\cdots , H ^ { n - 1 } \\overline { e } _ { k - 1 } \\right ] \\\\ & = H H ^ { * } ( \\overline { e } _ { k - 1 } ) = H H ^ { k - 2 } = H ^ { k - 1 } . \\end{align*}"}
{"id": "1690.png", "formula": "\\begin{align*} \\psi ( p , t ) = \\psi _ p ( t ) = \\psi _ p ( 0 ) - \\int _ 0 ^ t c ( s ) \\ , d s . \\end{align*}"}
{"id": "5848.png", "formula": "\\begin{align*} B ( a , b ) = \\sup _ { h \\in \\mathcal { M } ^ + ( a , b ) } \\frac { \\bigg ( \\int _ a ^ b \\bigg ( \\int _ a ^ t h ^ r v \\bigg ) ^ { \\frac { q } { r } } u ( t ) d t \\bigg ) ^ { \\frac { 1 } { q } } } { \\int _ a ^ b h } . \\end{align*}"}
{"id": "7879.png", "formula": "\\begin{align*} I _ { D } ( y ) = \\lim _ { \\tilde y _ { r + 1 } \\rightarrow 0 } L \\circ \\bar I _ { \\tilde D } ( \\tilde y ) , \\end{align*}"}
{"id": "5376.png", "formula": "\\begin{align*} B ^ * _ P ( v , w ) \\vcentcolon = \\langle ( - \\Delta ) ^ { s / 2 } v , ( - \\Delta ) ^ { s / 2 } w \\rangle + \\sum _ { | \\alpha | \\leq m } \\langle a _ \\alpha , v ( D ^ \\alpha w ) \\rangle , \\end{align*}"}
{"id": "7753.png", "formula": "\\begin{align*} { \\bf U } ^ k ( \\alpha _ i ) = [ \\tilde u _ { k , 1 } ^ { \\alpha _ i } , \\tilde u _ { k , 2 } ^ { \\alpha _ i } , \\ldots , \\tilde u _ { k , N _ h } ^ { \\alpha _ i } ] ^ T \\in \\mathbb { R } ^ { N _ h } k = 1 , \\ldots , m , \\end{align*}"}
{"id": "2864.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & = - \\tfrac { 1 } { N } \\sum _ { i = 1 } ^ N y _ i \\log \\big ( \\sigma ( A _ i x ) \\big ) + ( 1 - y _ i ) \\log \\big ( 1 - \\sigma ( A _ i x ) \\big ) + \\beta \\sum _ { i = 1 } ^ N e _ { \\lambda , \\sigma } l _ 0 ( [ x ] _ i ) \\\\ \\nabla f ( x ) & = \\tfrac { 1 } { N } A ^ T \\big ( \\sigma ( A _ i x ) - y \\big ) + \\beta \\nabla e _ { \\lambda , \\sigma } l _ 0 ( x ) \\end{aligned} \\end{align*}"}
{"id": "7257.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u + | u | ^ { p - 1 } u - | u | ^ { q - 1 } u & \\R ^ n \\times ( 0 , T ) , \\\\ u | _ { t = 0 } = u _ 0 ( x ) & \\R ^ n . \\end{cases} \\end{align*}"}
{"id": "278.png", "formula": "\\begin{align*} ( B ( \\Lambda _ i - 1 ) ) _ { k j } - ( B ( \\Lambda _ j - 1 ) ) _ { k i } = \\pi L _ { i j k } ( \\Lambda ) , \\end{align*}"}
{"id": "7343.png", "formula": "\\begin{align*} f ( w , x , y , z ) = - f ( z , w , x , y ) = f ( y , z , w , x ) , \\end{align*}"}
{"id": "2181.png", "formula": "\\begin{align*} G _ { \\mu \\boxplus \\nu } ( s _ { 0 } ) = \\lim _ { y \\rightarrow 0 ^ { + } } G _ { \\nu } \\left ( \\omega ( s _ { 0 } + i y ) \\right ) = G _ { \\nu } ^ { * } ( \\alpha ) \\in \\mathbb { R } \\setminus \\left \\{ 0 \\right \\} . \\end{align*}"}
{"id": "8399.png", "formula": "\\begin{gather*} \\Gamma _ 0 ( N ) _ Q : = \\left \\{ \\gamma \\in \\Gamma _ 0 ( N ) \\mid Q | \\gamma = Q \\right \\} . \\end{gather*}"}
{"id": "5161.png", "formula": "\\begin{align*} h ( v ) = 1 + \\max _ { w | } { h ( w ) } . \\end{align*}"}
{"id": "7320.png", "formula": "\\begin{align*} \\lambda ^ { - 1 } \\eta ( T - t ) ^ { { \\sf q } _ 1 } = ( T - t ) ^ { - { \\sf q } _ 2 } { \\sf l } _ 1 . \\end{align*}"}
{"id": "5086.png", "formula": "\\begin{align*} S ^ { n , M , 1 } _ \\tau = n ^ { \\alpha + \\frac 1 2 } \\sum _ { i = 0 } ^ { M - 1 } \\sigma ' ( X _ { \\tau _ i } ) \\int _ { \\tau _ i } ^ { \\tau _ { i + 1 } } ( t - s ) ^ \\alpha \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) d s \\end{align*}"}
{"id": "8474.png", "formula": "\\begin{align*} U & = \\sum _ { i = 1 } ^ n y _ i \\\\ & = \\sum _ { i = k } ^ n y _ k + L \\sum _ { i = k + 1 } ^ n ( i - k ) \\\\ \\Rightarrow U & = ( n - k + 1 ) y _ k + \\frac { L ( n - k ) ( n - k + 1 ) } { 2 } . \\end{align*}"}
{"id": "7863.png", "formula": "\\begin{align*} \\psi ( u T ) & = \\varphi ( e _ 1 u T e _ 1 ) = \\varphi ( e _ 1 ( u J u J ) T ( J u ^ * J ) e _ 1 ) = \\varphi ( e _ 1 T ( J u ^ * J ) e _ 1 ) \\\\ & = \\varphi ( e _ 1 T ( J u ^ * J ) ( u J u J ) e _ 1 ) = \\psi ( T u ) . \\end{align*}"}
{"id": "5074.png", "formula": "\\begin{align*} G ^ n _ \\tau & : = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s \\left ( \\sigma ' ( X _ s ) \\right ) ^ 2 \\ , \\left ( \\int ^ s _ { \\eta _ n ( s ) } ( s - \\eta _ n ( u ) ) ^ { \\alpha } \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) \\ , \\\\ & \\times \\left ( \\int ^ { \\eta _ n ( s ) } _ 0 \\psi _ { n , 2 } ( u , s ) \\sigma ( X _ { \\eta _ n ( u ) } ) \\ , d W _ u \\right ) d s , \\end{align*}"}
{"id": "3808.png", "formula": "\\begin{align*} { } _ 1 ^ { 1 } H y p ^ { \\mu , i } _ { k , j ; n } ( t _ 1 , t _ 2 ) = \\sum _ { l \\in [ n + \\epsilon M _ t / 2 , 2 ] \\cap \\Z } \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } ( E ( s , X ( s ) - y ) + \\hat { v } \\times B ( s , X ( s ) - y ) ) \\cdot \\nabla _ v \\mathcal { K } ^ { \\mu , i } _ { k , j , n } ( y , v , X ( s ) , V ( s ) ) \\end{align*}"}
{"id": "2736.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\sigma p } ( \\mathbb { R } ^ { n } ) : = \\left \\{ u \\in L ^ { p - 1 } _ { l o c } ( \\mathbb { R } ^ { n } ) \\ , \\Big | \\ ; \\int _ { \\mathbb { R } ^ { n } } \\frac { | u ( x ) | ^ { p - 1 } } { 1 + | x | ^ { n + \\sigma p } } d x < \\infty \\right \\} . \\end{align*}"}
{"id": "8800.png", "formula": "\\begin{align*} ( \\phi ^ * ) ^ { - 1 } ( \\mathbb { R } _ { > t } ) = \\left ( - \\frac { \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) } { 2 } , \\frac { \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) } { 2 } \\right ) . \\end{align*}"}
{"id": "4672.png", "formula": "\\begin{align*} \\kappa _ { n } = ( m - 1 ) \\frac { \\sum \\limits _ { i = 1 } ^ { m - 1 } \\kappa _ i } { m - 1 } = \\sum \\limits _ { i = 1 } ^ { m - 1 } \\kappa _ i = \\sum \\limits _ { i = 1 } ^ { m - 1 } \\sum \\limits _ { j = 1 } ^ { m - 1 } \\sum \\limits _ { k = 1 } ^ { m - 1 } \\tau _ { i j } \\tau _ { i k } l _ { j k } = \\sum \\limits _ { i = 1 } ^ { m - 1 } \\sum \\limits _ { j = 1 } ^ { m - 1 } h ^ { i j } l _ { i j } , \\end{align*}"}
{"id": "4075.png", "formula": "\\begin{align*} \\sum _ { X \\in M '' } P _ 0 ( X ) = \\sum _ { ( i _ n , i _ { n + 1 } , \\ldots , i _ q ) \\in I } P _ 0 \\big ( D [ i _ 0 , i _ 1 , \\ldots , i _ n , i _ { n + 1 } , \\ldots , i _ q ] \\big ) . \\end{align*}"}
{"id": "8392.png", "formula": "\\begin{align*} P _ f = \\begin{cases} P ^ { - 1 } , & f \\in P ^ { - 1 } \\\\ P & f \\in P . \\end{cases} \\end{align*}"}
{"id": "7179.png", "formula": "\\begin{align*} \\theta ^ 2 = 1 . \\end{align*}"}
{"id": "1664.png", "formula": "\\begin{align*} { f } _ { i } ^ 0 ( \\alpha ) + \\sum _ { j \\in S _ { p } } { f } _ { j } ^ 0 ( \\alpha ) = f ( \\gamma _ p ) . \\end{align*}"}
{"id": "2894.png", "formula": "\\begin{align*} g _ 1 = g _ { k + 1 } ( 1 , x _ 2 , \\ldots , x _ { k + 1 } ) . \\end{align*}"}
{"id": "1821.png", "formula": "\\begin{align*} \\int \\frac { 1 } { z - x } \\ , d \\mu ( x ) = \\cfrac { 1 } { z - b _ { 0 } - \\cfrac { a _ 0 } { z - b _ { 1 } - \\cfrac { a _ { 1 } } { \\ddots } } } \\end{align*}"}
{"id": "3571.png", "formula": "\\begin{align*} [ n ] _ q ! & = \\prod _ { i = 1 } ^ { n } \\dfrac { q ^ i - 1 } { q - 1 } \\\\ & = ( q + 1 ) ( q ^ 2 + q + 1 ) \\cdots ( q ^ { n - 1 } + q ^ { n - 2 } + \\cdots + q + 1 ) . \\end{align*}"}
{"id": "1651.png", "formula": "\\begin{align*} \\kappa _ { 2 i - 1 } \\ , = \\ , \\frac { u _ i ^ 2 + v _ i } { 2 u _ i } \\kappa _ { 2 i } \\ , = \\ , \\frac { v _ i - u _ i ^ 2 } { 2 u _ i } , \\end{align*}"}
{"id": "7222.png", "formula": "\\begin{align*} a _ { n + 1 } = p a _ n + 1 , \\ a _ 1 = 0 , \\end{align*}"}
{"id": "2934.png", "formula": "\\begin{align*} \\max \\left \\{ \\sum _ { j = 1 } ^ m F ( x ^ j ) : y \\in P ( \\Pi ) \\land Z ( y ) = h ^ 1 , h ^ 2 , \\ldots , h ^ m \\land x ^ j \\in P ( h ^ j ) \\right \\} , \\end{align*}"}
{"id": "1140.png", "formula": "\\begin{align*} & ( a ) _ n = ( a ; q ) _ n = ( 1 - a ) ( 1 - a q ) \\cdots ( 1 - a q ^ { n - 1 } ) , \\\\ & ( a ) _ { \\infty } = ( a ; q ) _ { \\infty } = \\lim _ { n \\to \\infty } ( a ; q ) _ { n } , ( a ) _ 0 = 1 . \\end{align*}"}
{"id": "6483.png", "formula": "\\begin{align*} u = 0 \\ \\ \\ \\ ( c _ 2 , L ) \\ \\ \\ \\ y = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) . \\end{align*}"}
{"id": "7944.png", "formula": "\\begin{align*} y ^ d \\mapsto \\prod _ { i = 1 } ^ { \\mathrm { r } } y _ i ^ { p _ i ^ + \\cdot d } . \\end{align*}"}
{"id": "5191.png", "formula": "\\begin{align*} & \\frac { 2 \\sum _ { i \\in I _ 1 } a _ i + \\sum _ { j \\in B _ 1 } b _ j + t _ 1 + 2 } { r } + \\frac { 2 \\sum _ { i \\in I _ 2 } a _ i + \\sum _ { j \\in B _ 2 } b _ j + t _ 2 + 2 } { r } \\\\ = { } & \\frac { 2 \\sum _ { i \\in I } a _ i + \\sum _ { j \\in B } b _ j + 2 + t _ 1 + t _ 2 + 2 } { r } \\equiv | D | + 1 \\pmod 2 , \\end{align*}"}
{"id": "8483.png", "formula": "\\begin{align*} y ^ * _ i & = \\left ( ( i - n + m ) L + \\frac { U } { m } - \\frac { L ( m + 1 ) } { 2 } \\right ) _ + \\\\ & = \\left ( ( i - 1 ) L + \\frac { U } { n - 1 } - \\frac { L n } { 2 } \\right ) _ + \\\\ & = \\left ( ( i - 1 ) L + \\frac { L n } { 2 } - \\frac { L n } { 2 } \\right ) _ + \\\\ & = ( i - 1 ) L \\end{align*}"}
{"id": "5283.png", "formula": "\\begin{align*} \\sum _ { \\substack { j = 1 \\\\ 2 \\notin I _ j } } ^ h & \\left ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { P } ^ { C o n t } _ j } + \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { P } ^ { C o n t } _ j } \\right ) + \\sum _ { j = 1 } ^ h \\mathbf { 1 } _ { 2 \\in I _ j } \\\\ & - \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { r + k _ 1 ( 0 ) } a ^ { \\widehat { P } ^ { C o n t , + r } } - \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { s + k _ 2 ( 0 ) } b ^ { \\widehat { P } ^ { C o n t , + s } } . \\end{align*}"}
{"id": "6856.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\frac { d ( b ^ 2 ( t ) ) } { d t } & = b ( g _ 0 + g _ 1 N - b ) \\leq C | b | - ( 1 - g _ 1 ) b ^ 2 + g _ 1 | b | \\sqrt { c } , \\\\ \\frac { 1 } { 2 } \\frac { d ( c ^ 2 ( t ) ) } { d t } & = c ( 2 a _ 0 + 2 a _ 1 N - 2 c ) \\leq C | c | + 2 a _ 1 | b c | + 2 a _ 1 c \\sqrt { c } - 2 c ^ 2 . \\end{aligned} \\end{align*}"}
{"id": "5323.png", "formula": "\\begin{align*} | \\pi ^ t _ i ( s ) | : = \\lim _ { n \\to \\infty } n ^ { - 1 } \\sum _ { j = 1 } ^ n 1 ( j \\in \\pi ^ t _ i ( s ) ) . \\end{align*}"}
{"id": "6789.png", "formula": "\\begin{align*} \\varphi _ j ( D ) = 2 ^ { - 2 \\ell j } ( - \\Delta ) ^ \\ell \\widetilde \\varphi _ k ( D ) \\end{align*}"}
{"id": "1218.png", "formula": "\\begin{align*} P ( T , x ) & = \\prod _ { v \\in V ( T ) } \\mathcal { G } ( v , x ) . \\end{align*}"}
{"id": "7671.png", "formula": "\\begin{align*} [ L ^ 2 ( 0 , T ; H ^ 1 ) \\cap L ^ 4 ( 0 , T ; L ^ 4 ) ] ^ * = L ^ 2 ( 0 , T ; H ^ { - 1 } ) + L ^ { 4 / 3 } ( 0 , T ; L ^ { 4 / 3 } ) , \\end{align*}"}
{"id": "2696.png", "formula": "\\begin{align*} \\langle v , K _ { 1 1 } u \\rangle = \\ ; & \\frac { 1 } { 2 } \\langle v \\otimes u _ 1 , w \\ , u _ 1 \\otimes u \\rangle \\\\ \\langle v , K _ { 2 2 } u \\rangle = \\ ; & \\frac { 1 } { 2 } \\langle v \\otimes u _ 2 , w \\ , u _ 2 \\otimes u \\rangle . \\end{align*}"}
{"id": "7850.png", "formula": "\\begin{align*} \\varphi \\circ { \\rm A d } ( u ) ( T ) = \\varphi \\circ { \\rm A d } ( J u ^ * J ) ( T ) = \\varphi ( T ) . \\end{align*}"}
{"id": "2015.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : m c k e a n v l a s o v } \\forall i \\in \\{ 1 , \\ldots , N \\} , \\dd X ^ { i , N } _ t = b \\big ( X ^ { i , N } _ t , \\mu _ { \\mathcal { X } ^ N _ t } \\big ) \\dd t + \\sigma \\big ( X ^ { i , N } _ t , \\mu _ { \\mathcal { X } _ t } \\big ) \\dd B ^ { i } _ t \\end{align*}"}
{"id": "2912.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\ldots , \\alpha _ n ) \\cdot C = ( 1 , 0 , \\ldots , 0 ) , \\end{align*}"}
{"id": "2226.png", "formula": "\\begin{align*} \\begin{array} { l l } e ^ { A } e ^ { B } = \\exp \\left [ A + B + \\frac { 1 } { 2 } [ A , B ] + \\frac { 1 } { 1 2 } [ A , [ A , B ] ] - \\frac { 1 } { 1 2 } [ B , [ A , B ] ] + \\cdots \\right ] , \\end{array} \\end{align*}"}
{"id": "652.png", "formula": "\\begin{align*} \\partial _ a q ( a , b ) ^ 2 = - \\dfrac { \\pi ^ 4 \\big ( 4 a ^ 3 - 6 a ^ 2 + 2 a + 4 b ^ 2 a - 2 b ^ 2 \\big ) } { \\left ( a ^ 4 - 2 a ^ 3 + 2 b ^ 2 a ^ 2 + a ^ 2 - 2 b ^ 2 a + b ^ 4 + b ^ 2 \\right ) ^ 2 } . \\end{align*}"}
{"id": "6525.png", "formula": "\\begin{align*} \\sum \\limits _ { i = j } ^ \\infty \\frac { 1 } { a _ i } \\leq \\frac { C _ a } { a _ j } , \\ \\ \\ j \\in \\N . \\end{align*}"}
{"id": "5149.png", "formula": "\\begin{align*} \\mathcal { N } ( n , 2 , \\{ 1 , 2 c \\} ) & = \\mathcal { N } ( 2 c q + r + 1 , 2 , \\{ 1 , 2 c \\} ) \\\\ & = \\mathcal { N } ( 6 q + r ' + 1 , 2 , \\{ 1 , 6 \\} ) & \\ref { t h r e e } \\end{align*}"}
{"id": "1304.png", "formula": "\\begin{align*} \\| \\mu ( E ) - \\nu ( E ) \\| _ { T V } : = \\sup _ { E \\in \\Sigma } | \\mu ( E ) - \\nu ( E ) | , \\end{align*}"}
{"id": "3393.png", "formula": "\\begin{align*} \\partial _ x \\partial _ x g ( x , y ) & = \\gamma \\left ( \\frac { 2 b ' ( x ) ^ 2 } { b ( x ) ^ 3 } - \\frac { b '' ( x ) } { b ( x ) ^ 2 } \\right ) \\| y \\| ^ 2 _ d , \\\\ \\partial _ { y _ i } \\partial _ { y _ j } g ( x , y ) & = \\frac { 2 \\gamma } { b ( x ) } \\ 1 { i = j } , \\\\ \\partial _ x \\partial _ { y _ i } g ( x , y ) & = - 2 \\gamma \\frac { b ' ( x ) } { b ( x ) ^ 2 } y _ i . \\end{align*}"}
{"id": "8168.png", "formula": "\\begin{align*} H _ 1 & = - \\Delta + c V _ 0 ( x ) \\\\ H _ x & = - \\frac { d ^ 2 } { d x ^ 2 } + c V _ 0 ( x ) \\end{align*}"}
{"id": "1894.png", "formula": "\\begin{align*} \\mathcal { U } _ { [ n , j ] } : = \\{ \\gamma \\in \\mathcal { P } _ { [ n , 0 ] } : \\mbox { t h e f i r s t s t e p i n $ \\gamma $ i s } \\ , \\ , ( 0 , 0 ) \\rightarrow ( 1 , - j ) \\} , - 1 \\leq j \\leq p . \\end{align*}"}
{"id": "4813.png", "formula": "\\begin{align*} J _ n : & = \\int _ { ( n - 1 ) / n } ^ { 1 } ( Q ( u ) - Q ( u _ { n , n } ) ) ^ 2 d u \\le \\int _ { ( n - 1 ) / n } ^ { 1 } ( Q ( u ) - q ) ^ 2 d u \\\\ & = \\int _ { ( n - 1 ) / n } ^ { 1 } Q ( u ) ^ 2 d u - 2 q \\int _ { ( n - 1 ) / n } ^ { 1 } Q ( u ) d u + q ^ 2 n ^ { - 1 } . \\end{align*}"}
{"id": "7368.png", "formula": "\\begin{align*} { { W } _ { h } } ( y ) : = { \\sum ^ 4 _ { i = 1 } } U _ { h , i } ( y ) = { \\sum ^ 4 _ { i = 1 } } U _ 0 ( y - h t _ i ) . \\end{align*}"}
{"id": "7278.png", "formula": "\\begin{align*} \\lambda ^ { - \\frac { n - 2 } { 2 } } { \\sf Q } ( y ) + \\lambda ^ { - \\frac { n - 2 } { 2 } } \\sigma { \\sf A } _ 1 & = \\pm \\eta ^ \\frac { 2 } { 1 - q } { \\sf U } ( \\xi ) | y | \\to \\infty , \\ | \\xi | \\to 0 , \\\\ \\pm \\eta ^ \\frac { 2 } { 1 - q } { \\sf U } ( \\xi ) & = \\pm { \\sf U } _ \\infty ( x ) + \\Theta _ J ( x , t ) | \\xi | \\to \\infty , \\ | z | \\to 0 . \\end{align*}"}
{"id": "4242.png", "formula": "\\begin{align*} \\mathbb { P } _ x ( X _ n = o d ( o , X _ m ) > r 0 \\leq m \\leq n ) & \\leq \\sum _ { m = 0 } ^ n \\sum _ { z \\in V } \\mathbb { P } _ x ( \\kappa = m , X _ \\kappa = z ) \\mathbb { P } _ z ( X _ { n - m } = o ) \\\\ & \\leq \\sqrt { \\frac { c ( o ) + K ( o ) } { c ( z ) + K ( z ) } } \\exp \\left [ - \\frac { r ^ 2 } { 2 n } \\right ] \\leq \\sqrt { \\frac { 2 c ( o ) } { c _ { \\min } } } \\exp \\left [ - \\frac { r ^ 2 } { 2 n } \\right ] , \\end{align*}"}
{"id": "6210.png", "formula": "\\begin{align*} u ' ( x ) + u ^ 2 ( x ) + p ( x ) u ( x ) = 0 , \\end{align*}"}
{"id": "3171.png", "formula": "\\begin{align*} \\Vert \\mathfrak { R } _ { \\Phi } ^ { f , \\gamma } \\Vert _ { \\mathcal { U } } \\leq \\mathfrak { F } _ { \\gamma } \\left ( \\Phi , f \\right ) \\lim _ { \\gamma \\rightarrow 0 ^ { + } } \\Vert \\mathfrak { R } _ { \\Phi } ^ { f , \\gamma } \\Vert _ { \\mathcal { U } } = 0 \\ . \\end{align*}"}
{"id": "748.png", "formula": "\\begin{align*} ( ( 1 \\otimes x _ 1 ) \\cdot ( 1 \\otimes x _ 2 ) ) _ { \\varphi _ { \\ast } \\lambda } = 1 \\otimes x _ 1 x _ 2 + \\sum _ { i } \\varphi ( m _ i ) \\otimes y _ i . \\end{align*}"}
{"id": "7377.png", "formula": "\\begin{align*} \\bar { \\phi } _ n ( y ) = \\phi _ n ( y + h _ n t _ 1 ) . \\end{align*}"}
{"id": "5936.png", "formula": "\\begin{align*} J ( \\xi _ T ) = { { \\cal L } ( \\xi _ T , 0 , 0 , \\dots ) } + C \\end{align*}"}
{"id": "8547.png", "formula": "\\begin{align*} \\begin{aligned} \\dot x ( t ) & = A x ( t ) + B u ( t ) + w ( t ) , \\\\ y ( t ) & = C x ( t ) + v ( t ) , \\end{aligned} \\end{align*}"}
{"id": "3581.png", "formula": "\\begin{align*} m _ \\pi & = \\dfrac { q ^ { ( 1 / 2 ) ( n ( n - 1 ) ) } - q ^ { ( 1 / 2 ) ( n - 1 ) ( n - 2 ) } } { 2 } \\\\ & = ( 1 / 2 ) \\cdot q ^ { ( 1 / 2 ) ( n - 1 ) ( n - 2 ) } ( q ^ { n - 1 } - 1 ) \\\\ & = \\dfrac { q - 1 } { 2 } \\cdot q ^ { ( 1 / 2 ) ( n - 1 ) ( n - 2 ) } \\cdot ( 1 + q + q ^ 2 + \\cdots + q ^ { n - 2 } ) . \\end{align*}"}
{"id": "3489.png", "formula": "\\begin{align*} \\overline { G } _ { ( 2 ) } ( u ) = \\frac { m ^ { 2 } } { ( m - 1 ) ( m - 3 ) } \\left ( 1 + \\frac { 2 u } { m } \\right ) ^ { - ( m - 3 ) / 2 } . \\end{align*}"}
{"id": "983.png", "formula": "\\begin{align*} \\tau ( \\alpha \\beta ) = \\sum _ { k = 0 } ^ { n - 1 } \\alpha _ { k } \\tau \\left ( \\theta ^ { k } \\beta \\right ) = \\sum _ { k = 0 } ^ { n - 1 } \\alpha _ { k } H ^ { k } \\overline { \\beta } = H ^ { * } ( \\overline { \\alpha } ) \\overline { \\beta } = \\overline { \\alpha } \\otimes \\overline { \\beta } , \\end{align*}"}
{"id": "3366.png", "formula": "\\begin{align*} \\psi _ n ( f ) ( z ) = \\begin{pmatrix} f ( 1 ) & 0 \\\\ 0 & f ( z ) \\end{pmatrix} \\end{align*}"}
{"id": "3116.png", "formula": "\\begin{align*} \\Vert u - J z _ { \\mathrm { n c } } \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 & = a ( u , { w } - J I { w } ) + a _ { \\mathrm { p w } } ( { w } , z _ { \\mathrm { n c } } - J z _ { \\mathrm { n c } } ) + a ( u , J I { w } ) - a _ { \\mathrm { p w } } ( { w } , z _ { \\mathrm { n c } } ) . \\end{align*}"}
{"id": "3348.png", "formula": "\\begin{align*} a \\cdot b = a + b - a b \\end{align*}"}
{"id": "6888.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } B ( t ) = + \\infty . \\end{align*}"}
{"id": "1962.png", "formula": "\\begin{gather*} \\mathbf { d } _ { 1 } = ( 0 , \\ldots , 0 , z - a _ { 0 } ^ { ( 0 ) } ) \\\\ \\mathbf { d } _ { 2 } = ( 0 , \\ldots , 0 , - a _ { 0 } ^ { ( 1 ) } , z - a _ { 1 } ^ { ( 0 ) } ) \\\\ \\mathbf { d } _ { 3 } = ( 0 , \\ldots , 0 , - a _ { 0 } ^ { ( 2 ) } , - a _ { 1 } ^ { ( 1 ) } , z - a _ { 2 } ^ { ( 0 ) } ) \\end{gather*}"}
{"id": "3038.png", "formula": "\\begin{align*} R ^ G _ { \\varphi } \\theta : = | W _ M | ^ { - 1 } \\sum _ { w \\in W _ M } \\tilde { \\varphi } ( w F ) R ^ G _ { T _ w } \\theta _ { T _ w } , \\end{align*}"}
{"id": "6095.png", "formula": "\\begin{align*} { \\bar \\Omega } _ { A B } + \\Psi _ { A B } = \\Omega _ { A B } , \\mbox { f o r } 1 \\le A , B \\le N + 1 . \\end{align*}"}
{"id": "6121.png", "formula": "\\begin{align*} | \\omega _ k | \\leq r _ k = r \\max \\left \\{ \\left | \\frac { j } { n } \\right | ^ { \\frac { 1 } { n - j } } , \\ , \\ , j = 1 , 2 , \\dots , k \\right \\} . \\end{align*}"}
{"id": "6621.png", "formula": "\\begin{align*} \\hbox { N } _ h ( N , w ; \\xi ) = c _ h h ^ { - 2 } \\ell ^ { - 4 } \\sum _ { k = 1 } ^ \\infty k ^ { - 4 } S _ { k } ( 0 ; \\xi ) I _ { k , \\ell } ( 0 ) + O _ { w , \\ell , \\epsilon } \\big ( h ^ { 3 / 2 + \\epsilon } \\big ) . \\end{align*}"}
{"id": "3547.png", "formula": "\\begin{align*} a = M \\cdot m . \\end{align*}"}
{"id": "6644.png", "formula": "\\begin{align*} \\epsilon _ n = \\frac { a } { n ^ { \\frac { 1 } { 2 } ( 1 - b ) } } \\ ; , \\ , \\end{align*}"}
{"id": "8197.png", "formula": "\\begin{align*} V _ { U } = \\left \\lbrace v \\in \\mathcal { H } \\left ( \\begin{array} { c c } U & - I \\\\ 0 & 0 \\end{array} \\right ) ( D P ) v = 0 \\right \\rbrace . \\end{align*}"}
{"id": "5089.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\sup _ { n \\ge M } \\sup _ { \\tau \\in [ 0 , t ] } \\| S ^ { n , M , 2 } _ \\tau \\| _ 2 = 0 . \\end{align*}"}
{"id": "5112.png", "formula": "\\begin{align*} \\Lambda ^ { ( 4 ) } _ { n , \\delta } = \\exp \\left ( i \\mu n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { t - \\delta } ( t - s ) ^ \\alpha \\xi ^ n _ s d W _ s + i \\lambda n ^ { \\alpha + \\frac 1 2 } \\int _ { t - \\delta } ^ { \\eta _ n ( t ) } \\psi _ { n , 1 } ( s , \\eta _ n ( t ) ) d W _ s \\right ) . \\end{align*}"}
{"id": "4637.png", "formula": "\\begin{align*} \\forall \\ , m \\in \\N : \\ \\| A ^ n e _ { n + m } \\| & = \\left \\| \\left ( \\left [ \\prod _ { j = k } ^ { k + n - 1 } w ^ j \\right ] \\delta _ { ( n + m ) ( k + n ) } \\right ) _ { k \\in \\N } \\right \\| _ \\infty = \\prod _ { j = 1 } ^ { m + n - 1 } | w | ^ j \\\\ & = | w | ^ { \\sum _ { j = 1 } ^ { m + n - 1 } j } = | w | ^ { \\frac { ( m + n ) ( m + n - 1 ) } { 2 } } \\to \\infty , \\ m \\to \\infty , \\end{align*}"}
{"id": "8352.png", "formula": "\\begin{align*} \\sum \\limits _ { i } ^ { N } \\alpha _ i = 0 . \\end{align*}"}
{"id": "6178.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\partial } { \\partial t } \\omega ( t ) = - \\mathrm { R i c } ^ { T } ( \\omega ( t ) ) , \\omega ( 0 ) = \\omega _ { 0 } . \\end{array} \\end{align*}"}
{"id": "201.png", "formula": "\\begin{align*} p _ { \\bar { X } , \\bar { Y } } ( a , b ) = p _ { \\bar { Y } | \\bar { X } } ( b | a ) p _ { \\bar { X } } ( a ) , \\end{align*}"}
{"id": "1940.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { H } _ { q } e _ { 0 } = \\sum _ { k = 0 } ^ { p } a _ { q } ^ { ( k ) } e _ { k } , \\\\ \\mathcal { H } _ { q } e _ { n } = e _ { n - 1 } + \\sum _ { k = 0 } ^ { p } a _ { n + q } ^ { ( k ) } \\ , e _ { n + k } , n \\geq 1 , \\end{cases} \\end{align*}"}
{"id": "4521.png", "formula": "\\begin{align*} \\tilde { g } = 2 d \\tau d u + g \\end{align*}"}
{"id": "5299.png", "formula": "\\begin{align*} { \\bf a } ( \\{ \\{ 1 , 3 \\} , \\{ 2 , 4 \\} \\} ) = { \\bf a } ( \\{ \\{ 1 , 4 \\} , \\{ 2 , 3 \\} \\} ) = \\{ \\{ 1 , 2 \\} , \\{ 1 , 2 \\} \\} . \\end{align*}"}
{"id": "2459.png", "formula": "\\begin{align*} \\gg = \\oplus _ { \\mu } \\gg ^ { \\mu } , \\end{align*}"}
{"id": "3752.png", "formula": "\\begin{align*} \\| B _ { k ; j , l } ^ { e s s ; m } ( t , x ) \\| _ { L ^ 2 } \\lesssim \\sup _ { g \\in L ^ 2 , \\| g \\| _ { L ^ 2 } = 1 } \\big | \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } ( t - s ) \\big | { ( \\hat { v } \\times \\xi ) } \\big | \\big | \\overline { \\hat { g } ( \\xi ) } \\big | \\big | \\hat { f } ( s , \\xi , v ) \\big | \\varphi _ { j , l } ( v , { \\omega } ) \\varphi _ { m , k , l } ^ { e s s } ( \\omega , \\xi ) \\varphi _ { m ; - 1 0 M _ t } ( t - s ) \\end{align*}"}
{"id": "1109.png", "formula": "\\begin{align*} l ( \\sigma ) \\left ( - \\frac { 2 } { \\sigma ^ 3 } \\right ) = \\frac { 5 } { \\sigma ^ 4 } + G ( P _ X V , P _ X V ) . \\end{align*}"}
{"id": "1735.png", "formula": "\\begin{align*} f _ { k - 2 } ( z , \\overline \\omega _ 1 ) = \\int _ C \\frac { e ^ { z s } \\ , s ^ { k - 2 } } { e ^ { \\overline \\omega _ 1 s } - 1 } \\ d s , \\end{align*}"}
{"id": "3523.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast } & = \\frac { ( 3 + \\xi _ { p } ^ { 2 } + 3 | \\xi _ { p } | ) \\exp ( - | \\xi _ { p } | ) - ( 3 + \\xi _ { q } ^ { 2 } + 3 | \\xi _ { q } | ) \\exp ( - | \\xi _ { q } | ) } { 2 F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "771.png", "formula": "\\begin{align*} \\left ( \\frac { \\sum _ { m = 1 } ^ { M } ( T _ { { m } } - \\alpha ) } { M } \\right ) ^ 2 \\le \\frac { \\sum _ { m = 1 } ^ { M } ( T _ { { m } } - \\alpha ) ^ 2 } { M } \\end{align*}"}
{"id": "914.png", "formula": "\\begin{align*} ( \\Psi _ { t , \\eta _ i } ^ { q ( t ) } ) ^ * g _ { \\mathrm { c a n } } = \\frac { \\mathrm { t r } _ g ( \\Psi _ { t , \\eta _ i } ^ { q ( t ) } ) ^ * g _ \\mathrm { c a n } } { n } \\cdot g + O ( t ^ l ) \\end{align*}"}
{"id": "5545.png", "formula": "\\begin{align*} M _ { n , k } ( n ) = \\dbinom { 2 n } { k } - \\dbinom { 2 n } { k - 1 } . \\end{align*}"}
{"id": "1091.png", "formula": "\\begin{align*} \\tilde { g } _ { i j } ( \\xi ) : = \\int \\partial _ i \\tilde { l } ^ { ( \\alpha ) } \\partial _ j \\tilde { l } ^ { ( - \\alpha ) } \\ , d x , \\quad \\tilde { \\Gamma } ^ { ( \\alpha ) } _ { i j , k } = \\int \\partial _ i \\partial _ j \\tilde { l } ^ { ( \\alpha ) } \\partial _ k \\tilde { l } ^ { ( - \\alpha ) } \\ , d x . \\end{align*}"}
{"id": "6090.png", "formula": "\\begin{align*} d \\theta _ A = \\sum \\theta _ { A B } \\wedge \\theta _ B , 1 \\le A , B \\le N + 2 . \\end{align*}"}
{"id": "264.png", "formula": "\\begin{align*} \\Lambda _ i ^ { \\nu + 1 } = \\Lambda _ i ^ { \\nu } + \\pi ^ { \\nu } Z ^ { \\nu } _ i \\end{align*}"}
{"id": "5193.png", "formula": "\\begin{align*} e _ 1 = \\frac { \\sum a _ i - ( r - 2 ) } { r } \\in \\Z , ~ ~ e _ 2 = \\frac { \\sum b _ i - ( s - 2 ) } { s } \\in \\Z . \\end{align*}"}
{"id": "5596.png", "formula": "\\begin{align*} \\theta ( s ) = \\left \\{ \\begin{aligned} 1 & 0 \\leq s < t \\\\ \\frac { t - s } { \\delta } + 1 & t \\leq s < t + \\delta \\\\ 0 & s > t + \\delta . \\end{aligned} \\right . \\end{align*}"}
{"id": "3650.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } e _ n ( B _ { R _ \\ell } ( x _ 0 ) ) = e ( B _ { R _ \\ell } ( x _ 0 ) ) \\quad \\lim _ { n \\to \\infty } \\int _ { B _ { r _ \\ell } ( x _ 0 ) } u _ n = u ( B _ { r _ \\ell } ( x _ 0 ) ) . \\end{align*}"}
{"id": "3097.png", "formula": "\\begin{align*} ( h \\rightharpoonup f ) ( m ) = \\sum _ { ( h ) } h _ 1 \\rightharpoonup f ( S h _ 2 \\rightharpoonup m ) \\end{align*}"}
{"id": "3850.png", "formula": "\\begin{align*} R _ { n } ^ { ( x _ 0 , y _ 0 ) , \\Pi ^ { \\mu , \\nu } } ( f ) \\lesssim & \\sum _ { j = 0 } ^ n O \\bigg ( | x - y | ^ { j + 1 } \\cdot \\int | v - u | ^ { n - j } d \\Pi ^ { \\mu , \\nu } ( u , v ) \\bigg ) , \\\\ & + \\sum _ { j = 0 } ^ n O \\bigg ( | x - y | ^ { j } \\cdot \\int | v - u | ^ { n - j + 1 } d \\Pi ^ { \\mu , \\nu } ( u , v ) \\bigg ) . \\end{align*}"}
{"id": "7922.png", "formula": "\\begin{align*} 0 \\longrightarrow \\mathbb { L } : = ( \\beta ) \\longrightarrow \\mathbb { Z } ^ { m } \\stackrel { \\beta } { \\longrightarrow } N \\longrightarrow 0 , \\end{align*}"}
{"id": "6674.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } w _ { t } - \\dfrac { \\Delta w } { h _ 1 ^ 2 } + A _ 1 ( t , s ) w _ { s } = A _ 2 ( t , s ) z + F _ 1 ( t , s ) , \\ \\ 0 < t < T _ 0 , \\ 0 < s < 1 , \\\\ w _ { s } ( t , 0 ) = w ( t , 1 ) = 0 , \\ \\ 0 < t < T _ 0 , \\\\ w ( 0 , s ) = u _ { 0 1 } - u _ { 0 2 } = w _ { 0 } ( s ) , \\ \\ 0 \\leq s \\leq 1 , \\end{array} \\right . \\end{align*}"}
{"id": "3119.png", "formula": "\\begin{align*} c _ b ^ { - 2 } \\Vert u _ { \\mathrm { n c } } \\Vert _ { L ^ 2 ( T ) } ^ 2 \\le \\Vert u _ { \\mathrm { n c } } \\Vert _ { b _ T } ^ 2 = ( u _ { \\mathrm { n c } } , u ) _ { b _ T } - ( u _ { \\mathrm { n c } } , u - u _ { \\mathrm { n c } } ) _ { b _ T } . \\end{align*}"}
{"id": "749.png", "formula": "\\begin{align*} \\mathrm { d } \\alpha _ { \\lambda , \\xi } ( x _ 1 , x _ 2 , x _ 3 ) = \\sum _ { \\circlearrowleft x _ 1 , x _ 2 , x _ 3 } \\alpha _ { \\lambda , \\xi } ( x _ 1 , x _ 2 x _ 3 ) + x _ 1 \\alpha _ { \\lambda , \\xi } ( x _ 2 , x _ 3 ) . \\end{align*}"}
{"id": "4061.png", "formula": "\\begin{align*} & \\psi _ i \\psi _ j = - q _ { i j } \\psi _ j \\psi _ i , & & \\sum _ { i = 1 } ^ r q _ i \\psi _ i \\psi _ { - i } = 0 . \\end{align*}"}
{"id": "3677.png", "formula": "\\begin{align*} g = \\frac { d r ^ 2 } { \\alpha ( r ) } + \\sin ^ 2 r \\cdot h _ { \\mathbb { S } ^ { n - 2 } } + e ^ { 2 \\beta ( s , r ) } \\cdot d s ^ 2 . \\end{align*}"}
{"id": "8872.png", "formula": "\\begin{align*} & 2 f _ { \\lambda } ( \\lambda ) f _ { \\lambda \\lambda } ( \\lambda ) - \\nabla f _ { \\lambda } \\cdot \\nabla f ( \\lambda ) \\\\ & = - f _ { \\lambda } ( \\lambda ) \\Delta f ( \\lambda ) - f ( \\lambda ) \\Delta f _ { \\lambda } ( \\lambda ) + 2 \\sum _ { \\l - k > 1 } \\dfrac { \\partial } { \\partial \\lambda } \\left ( \\det \\left ( ( \\lambda I _ N - H ) _ { k | k } \\right ) \\det \\left ( ( \\lambda I _ N - H ) _ { \\l | \\l } \\right ) \\right ) . \\end{align*}"}
{"id": "363.png", "formula": "\\begin{align*} \\delta _ j : = D \\exp \\left ( - \\frac { \\log n } { b ( n ) ^ { j / 4 } } \\right ) = D n ^ { - 1 / b ( n ) ^ { j / 4 } } . \\end{align*}"}
{"id": "2145.png", "formula": "\\begin{align*} \\int _ { ( - \\infty , - n ^ m ) } \\frac { n s ^ { 2 } \\ , d \\sigma _ { \\mu } ^ { - } ( s ) } { ( \\widetilde { x } _ n - s ) ^ { 2 } } & = n \\sum _ { k = n + 1 } ^ \\infty \\left ( \\frac { k ^ m } { \\widetilde { x } _ n + k ^ m } \\right ) ^ 2 \\sigma _ \\mu ^ - ( \\{ - k ^ m \\} ) \\\\ & \\leq 1 6 n ^ 3 \\sigma _ \\mu ^ - ( ( - \\infty , - n ^ m ) ) = \\frac { 1 6 \\widetilde { \\ell } ( n ^ m ) } { n ^ { a m - 3 } } \\end{align*}"}
{"id": "7122.png", "formula": "\\begin{align*} \\pi ( F _ 4 , 1 ) = 8 , \\pi ( F _ 5 , 1 ) = 2 0 , \\pi ( F _ 6 , 1 ) = 1 2 , \\pi ( F _ 7 , 1 ) = 2 8 . \\end{align*}"}
{"id": "4452.png", "formula": "\\begin{align*} \\sum \\limits _ { l = r } ^ k ( - 1 ) ^ { l - r } \\binom { k } { l } \\binom { l } { r } = 0 . \\end{align*}"}
{"id": "4831.png", "formula": "\\begin{align*} \\rho = \\frac { 1 } { [ K : \\Q ] } \\sum _ { g \\in \\mathrm { G a l } ( K / \\Q ) } g . \\rho \\ ; . \\end{align*}"}
{"id": "3666.png", "formula": "\\begin{align*} \\beta _ t = \\sup _ { s \\in [ 0 , t ] } \\beta _ t ( s ) \\leq 1 - 2 \\epsilon , \\alpha _ t = \\sup _ { s \\in [ 0 , t ] } \\alpha _ t ( s ) \\leq \\alpha ^ { \\star } - \\epsilon . \\end{align*}"}
{"id": "6479.png", "formula": "\\begin{align*} u = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) y = 0 \\ \\ \\ \\ ( c _ 2 , L ) . \\end{align*}"}
{"id": "7643.png", "formula": "\\begin{align*} Y _ { n , s , m } = V \\left ( \\sum _ { i = 1 } ^ n z _ i ( F _ i \\cdot \\underline { y } ) \\right ) . \\end{align*}"}
{"id": "2664.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 2 } ( n - 2 ) p ^ { n - 3 } ( p - 1 ) ( n - k - 1 ) & = ( n - 2 ) p ^ { n - 3 } ( p - 1 ) \\left ( ( n - 2 ) + ( n - 3 ) + \\ldots + 1 \\right ) \\\\ & = ( n - 2 ) p ^ { n - 3 } ( p - 1 ) \\binom { n - 1 } { 2 } . \\end{align*}"}
{"id": "4482.png", "formula": "\\begin{align*} \\partial _ \\xi e ^ { i x \\xi + i p ( \\xi ) t } = i [ x + t p ' ( \\xi ) ] e ^ { i x \\xi + i p ( \\xi ) t } . \\end{align*}"}
{"id": "4132.png", "formula": "\\begin{align*} A ^ { i } B ^ { i } ( \\phi ) = \\psi = \\phi - d _ V ^ { i - 1 } { G } ^ { i } \\phi - { G } ^ { i + 1 } d _ V ^ { i } \\phi . \\end{align*}"}
{"id": "6325.png", "formula": "\\begin{align*} \\frac { 1 } { q } D _ { q ^ { - 1 } } D _ { q } y ( x ) + \\dfrac { 1 - q x } { q x ^ 2 ( 1 - q ) } D _ { q ^ { - 1 } } y ( x ) + \\frac { 1 } { x ^ 2 ( 1 - q ) ^ 2 } y ( x ) = 0 . \\end{align*}"}
{"id": "834.png", "formula": "\\begin{align*} p ( x _ { m } ^ { r } | s _ { m } ^ { r } ) = ( 1 - s _ { m } ^ { r } ) \\delta ( x _ { m } ^ { r } ) + s _ { m } ^ { r } \\mathcal { C N } \\left ( x _ { m } ^ { r } ; 0 , ( \\sigma _ { m } ^ { r } ) ^ { 2 } \\right ) \\end{align*}"}
{"id": "6298.png", "formula": "\\begin{align*} & \\int \\frac { x ^ 2 } { ( x ^ 2 q ^ { - 1 } ( 1 - q ) ; q ^ 2 ) _ \\infty } \\cos ( x ; q ) d _ q x = \\frac { q } { ( \\frac { x ^ 2 } { q } ( 1 - q ) ; q ^ 2 ) _ \\infty } \\Big ( x \\cos ( \\frac { x } { q } ; q ) + \\sqrt { q } \\sin ( q ^ { \\frac { - 1 } { 2 } } x ; q ) \\Big ) , \\end{align*}"}
{"id": "1470.png", "formula": "\\begin{align*} \\bigcirc _ { w = 0 } ^ i ( \\theta _ t + \\gamma _ { r - i + w } ) ^ { - 1 } \\circ ( \\theta _ t + \\xi _ { i , 1 } ) \\circ \\cdots \\circ ( \\theta _ t + \\xi _ { i , \\ell _ i } ) = \\bigcirc _ { w ' = 0 } ^ j ( \\theta _ t + \\gamma _ { r - j + w ' } ) ^ { - 1 } \\circ ( \\theta _ t + \\xi _ { j , 1 } ) \\circ \\cdots \\circ ( \\theta _ t + \\xi _ { j , \\ell _ j } ) \\enspace , \\end{align*}"}
{"id": "6413.png", "formula": "\\begin{align*} L ( f ) = \\sum _ { i = 1 } ^ k X _ i [ f ] \\Theta _ i , \\forall f \\in \\mathcal { O } . \\end{align*}"}
{"id": "6785.png", "formula": "\\begin{align*} & s - a _ { i ^ * n _ { i ^ * } } + ( | M _ P - M _ 0 | - 1 ) ( b - s ) + a _ { i ^ * n _ { i ^ * } } \\cdot \\frac { a _ { i ^ * j ' } } { a _ { i ^ * n _ { i ^ * } } + b - s } \\cdot \\frac { a _ { i ^ * n _ { i ^ * } } + b - s } { a _ { i ^ * j ' } } \\\\ & = b + ( | M _ P - M _ 0 | - 2 ) ( b - s ) . \\end{align*}"}
{"id": "1576.png", "formula": "\\begin{align*} v o l ( D ^ 2 _ x ( 1 ) ) : = \\frac { v o l ( B ^ 2 ( 1 ) ) b ^ 2 A ^ { \\epsilon \\eta } z ^ 1 _ { \\epsilon } z ^ 1 _ { \\eta } } { 2 \\sqrt { \\det A } } . \\end{align*}"}
{"id": "7062.png", "formula": "\\begin{align*} Z _ j = \\begin{cases} \\dot { x } = - y + d _ j x + l _ j \\ , x ^ 2 + m _ j \\ , x \\ , y + n _ j \\ , y ^ 2 , \\\\ \\dot { y } = x ( 1 - y ) , \\end{cases} \\ \\ ( x , y ) \\in \\Sigma ^ { \\mathcal { V } } _ j , \\end{align*}"}
{"id": "247.png", "formula": "\\begin{align*} \\mathcal A \\rightarrow \\mathcal A / \\mathcal P \\simeq R _ { \\pi } \\stackrel { c ^ * } { \\longrightarrow } R , \\ \\ \\ F \\mapsto F _ { c } : = c ^ * ( F ( 1 ) ) \\end{align*}"}
{"id": "881.png", "formula": "\\begin{align*} \\chi _ k = \\min _ { S _ k \\subset V \\atop \\dim ( S _ k ) = k } \\max _ { 0 \\not = u \\in S _ k } \\frac { \\| u \\| _ V } { \\| u \\| _ { L _ 2 ( D ) } } . \\end{align*}"}
{"id": "2364.png", "formula": "\\begin{align*} \\nu ( f ) = \\beta _ b + b \\gamma _ \\sigma . \\end{align*}"}
{"id": "7462.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\left \\{ x _ { n } - y _ { n } \\right \\} _ { n = 0 } ^ { \\infty } \\ , \\ , \\mathrm { i s } \\ , \\mathrm { \\ , b o u n d e d } \\\\ \\left \\Vert x _ { n } - y _ { n } \\right \\Vert - \\left \\Vert T x _ { n } - T y _ { n } \\right \\Vert \\rightarrow 0 \\end{array} \\right \\} \\Longrightarrow x _ { n } - y _ { n } - \\left ( T x _ { n } - T y _ { n } \\right ) \\rightarrow 0 . \\end{align*}"}
{"id": "7339.png", "formula": "\\begin{align*} ( x , y , z ) & = - ( x , y , z ) + ( y , x , z ) + ( x , z , y ) & \\\\ & = - ( x y ) z + x ( y z ) + ( y x ) z - y ( x z ) + ( x z ) y - x ( z y ) & \\\\ & = - ( x y ) z + ( y x ) z + x ( y z ) - x ( z y ) + ( x z ) y - y ( x z ) & \\\\ & = - [ x , y ] z + x [ y , z ] + [ x z , y ] . \\end{align*}"}
{"id": "3930.png", "formula": "\\begin{align*} V a r ( \\hat { \\pi } _ { h , n } ( x ) ) & \\le \\frac { c } { T _ n } \\big [ \\frac { 1 } { \\prod _ { l = 1 } ^ d h _ l } ( \\delta _ 1 + \\Delta _ n ) + \\frac { \\delta _ 2 ^ { \\frac { 1 } { 2 } } } { \\prod _ { l \\ge 2 } h _ l } + \\frac { 1 } { \\prod _ { l \\ge 4 } h _ l \\ , \\delta _ 2 ^ { \\frac { 1 } { 2 } } } + D + \\frac { 1 } { ( \\prod _ { l = 1 } ^ d h _ l ) ^ 2 } e ^ { - \\rho D } \\big ] . \\end{align*}"}
{"id": "5713.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\le p \\le i , \\ 0 \\le q \\le n - i \\\\ p + q = k } } ( y _ i - y _ { i + 1 } ) e _ { p } ( y _ 1 , \\ldots , y _ i ) e _ { q } ( y _ { i + 1 } , \\ldots , y _ n ) = 0 . \\end{align*}"}
{"id": "657.png", "formula": "\\begin{align*} \\| A \\| _ { \\mathcal B _ 1 } = \\sup _ { u \\neq 0 } \\frac { \\| A ( u ) \\| _ { C ^ { k - 4 , \\gamma } _ { g _ o } } } { \\| u \\| _ { C ^ { k , \\gamma } _ { g _ o } } } . \\end{align*}"}
{"id": "3287.png", "formula": "\\begin{align*} u _ 1 u _ 2 = e ^ { i x \\cdot \\xi } e ^ { i ( \\varphi _ 1 + \\varphi _ 2 ) } + e ^ { i x \\cdot \\xi } \\big ( e ^ { i \\varphi _ { 2 } } r _ { 1 } + e ^ { i \\varphi _ { 1 } } r _ { 2 } + r _ 1 r _ 2 \\big ) . \\end{align*}"}
{"id": "1745.png", "formula": "\\begin{align*} G ^ * ( z \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) \\cdot G ^ * ( z \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , - \\omega _ 2 ) = \\prod _ { k _ 1 \\geq 0 , k _ 2 \\geq 0 } \\big ( 1 - x _ 2 q _ 2 ^ { k _ 1 + \\frac { 1 } { 2 } } \\widetilde { q } _ 2 ^ { k _ 2 + \\frac { 1 } { 2 } } \\big ) \\cdot \\prod _ { k _ 1 \\geq 0 , k _ 2 \\geq 0 } \\big ( 1 - x _ 2 ^ { - 1 } q _ 2 ^ { k _ 1 + \\frac { 1 } { 2 } } \\widetilde { q } _ 2 ^ { k _ 2 + \\frac { 1 } { 2 } } \\big ) . \\end{align*}"}
{"id": "8697.png", "formula": "\\begin{align*} A _ { m i n } = \\frac { ( 1 - b ) ^ 3 } { 2 } \\int \\limits _ { t _ { 0 } } ^ { t _ { 1 } } ( x ^ { 1 } \\dot { x } ^ { 2 } - x ^ { 2 } \\dot { x } ^ { 1 } ) d t . \\end{align*}"}
{"id": "5468.png", "formula": "\\begin{align*} ( \\chi ' \\ast \\kappa ) ( b ) & = & \\ , \\ , \\ast \\\\ & = \\chi ' ( b ) \\kappa ( b ) & ( \\ref { a c t i o n . e q . } ) \\\\ & = a ^ { - 1 } \\chi ( b ) \\kappa ( b ) a \\kappa ( b ) ^ { - 1 } \\kappa ( b ) \\\\ & = a ^ { - 1 } \\chi ( b ) \\kappa ( b ) a & \\ , \\ , \\ast \\\\ & = a ^ { - 1 } ( \\chi \\ast \\kappa ) ( b ) \\ , a . \\end{align*}"}
{"id": "3305.png", "formula": "\\begin{align*} U = - S ( s ) \\partial _ \\nu ^ + U + D ( s ) \\gamma ^ + U . \\end{align*}"}
{"id": "1667.png", "formula": "\\begin{align*} | f _ { i } ( \\alpha ) | = | f _ { i } ^ 0 ( \\alpha ) | < \\| f \\| _ { \\infty } + \\epsilon . \\end{align*}"}
{"id": "6268.png", "formula": "\\begin{align*} D _ { q } u ( x ) + u ( x ) u ( q x ) + D ( x ) u ( q x ) = 0 , \\end{align*}"}
{"id": "2616.png", "formula": "\\begin{align*} Q ( p , 0 ) \\otimes Q ( 0 , q ) = \\bigoplus _ { n = 0 } ^ { \\min \\{ p , q \\} } Q ( p - n , q - n ) \\ , \\end{align*}"}
{"id": "85.png", "formula": "\\begin{align*} \\Big | \\sum _ { k \\neq 0 } k | k | ^ { 2 s - 2 } A _ { - k } u _ k \\Big | ^ 2 & \\leq \\Big | \\sum _ { k \\neq 0 } | k | ^ { 2 s - 1 } \\| A _ { - k } \\| | u _ k | \\Big | ^ 2 \\\\ & \\leq \\Big ( \\sum _ { k \\neq 0 } | k | ^ { 2 s } \\| A _ { - k } \\| ^ 2 \\Big ) \\cdot \\Big ( \\sum _ { k \\neq 0 } | k | ^ { 2 s - 2 } | u _ k | ^ 2 \\Big ) \\\\ & = \\| A \\| _ s ^ 2 \\cdot \\| u \\| _ { s - 1 } ^ 2 . \\end{align*}"}
{"id": "721.png", "formula": "\\begin{align*} | A | ^ 2 _ { g _ X } = g _ X ^ { i l } g _ X ^ { j q } A _ { i j } A _ { l q } . \\end{align*}"}
{"id": "2292.png", "formula": "\\begin{align*} - \\Delta v _ e ^ i = 0 , \\ v _ e ^ i ( x , 0 ) = - v _ p ^ { i - 1 } ( x , 0 ) , \\ v _ e ^ i ( x , \\infty ) = 0 , \\ u _ e ^ i = \\int _ x ^ \\infty v _ { e Y } ^ i ( \\theta , Y ) { \\rm d } \\theta , \\end{align*}"}
{"id": "4957.png", "formula": "\\begin{align*} Y ^ n _ t & = n ^ { \\alpha + \\frac 1 2 } \\int ^ t _ 0 [ ( t - \\eta _ n ( s ) ) ^ { \\alpha } - ( t - s ) ^ { \\alpha } ] \\sigma ( X ^ n _ { \\eta _ { n } ( s ) } ) \\ , d W _ s \\\\ & + n ^ { \\alpha + \\frac 1 2 } \\int ^ t _ 0 ( t - s ) ^ { \\alpha } [ \\sigma ( X ^ n _ { \\eta _ n ( s ) } ) - \\sigma ( X _ s ) ] \\ , d W _ s \\\\ & : = A ^ n _ t + Y ^ { n , 1 } _ t . \\end{align*}"}
{"id": "7747.png", "formula": "\\begin{align*} a ( w , z ; \\alpha ) = \\displaystyle \\int _ \\Omega \\mu \\nabla w \\cdot \\nabla z \\ , d \\Omega + \\int _ \\Omega \\big ( { \\bf b } \\cdot \\nabla w + \\sigma w \\big ) z \\ , d \\Omega , f ( z ; \\alpha ) = \\displaystyle \\int _ \\Omega f z \\ , d \\Omega , \\end{align*}"}
{"id": "8309.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } j c _ j = O \\left ( m v \\left ( \\frac { 1 } { m } \\right ) \\right ) , \\end{align*}"}
{"id": "2302.png", "formula": "\\begin{align*} \\zeta _ { 3 } ( x ) = \\left \\{ \\begin{array} { l l } 0 & 1 \\leq x \\leq \\frac { 3 } { 2 } , \\\\ 1 & x \\geq 2 . \\end{array} \\right . \\end{align*}"}
{"id": "6411.png", "formula": "\\begin{align*} \\left \\langle \\Phi _ 0 ( x ) ^ { ( 0 ) } ( \\alpha ) , e \\right \\rangle = \\varrho ( x ) [ \\langle \\alpha , e \\rangle ] - \\left \\langle \\alpha , \\nabla _ x ( e ) \\right \\rangle , \\quad \\end{align*}"}
{"id": "2207.png", "formula": "\\begin{align*} \\varrho ( r ) = \\int _ 0 ^ r \\ \\inf _ { d _ X ( x , x ^ * ) \\leq \\rho } C ( J f ( x ) ) d \\rho . \\end{align*}"}
{"id": "3706.png", "formula": "\\begin{align*} \\mu ( x _ 1 , x _ 2 , \\xi _ 1 , \\xi _ 2 ) = \\delta _ { ( 1 , 0 ) } ( \\xi _ 1 , \\xi _ 2 ) d x _ 1 d x _ 2 . \\end{align*}"}
{"id": "5729.png", "formula": "\\begin{align*} \\pi _ { [ a , i ] } \\cdot \\pi _ { [ i + 1 , b ] } & = \\pi _ { [ a - 1 , i - 1 ] } \\pi _ { [ i , b - 1 ] } + \\pi _ { [ a , i - 1 ] } \\pi _ { [ i , b - 1 ] } y _ i + \\pi _ { [ a , i ] } \\pi _ { [ i + 1 , b - 1 ] } y _ { b } . \\end{align*}"}
{"id": "8179.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\| \\chi _ { S ^ c _ { t ^ \\beta } } J U _ 0 f \\| = 0 \\end{align*}"}
{"id": "6191.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\omega _ { T _ { 0 } } = \\widehat { \\omega } _ { T _ { 0 } } + \\sqrt { - 1 } \\partial _ { B } \\overline { \\partial } _ { B } \\varphi _ { T _ { 0 } } \\geq 0 . \\end{array} \\end{align*}"}
{"id": "3793.png", "formula": "\\begin{align*} E r r ^ 0 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) : = \\sum _ { i = 1 , 2 } ( - 1 ) ^ i \\int _ { \\R ^ 3 } e ^ { i X ( t _ i ) \\cdot ( \\xi + \\eta ) + i \\mu t _ i | \\xi | + i \\mu _ 1 t _ i | \\eta | } { \\clubsuit K _ { k _ 1 , j _ 1 ; n _ 1 } ^ { \\mu _ 1 , i _ 1 } ( t _ i , \\eta , V ( t _ i ) ) \\cdot { } ^ 1 \\clubsuit K ( t _ i , \\xi , X ( t _ i ) , V ( t _ i ) ) } \\end{align*}"}
{"id": "1589.png", "formula": "\\begin{align*} - f ( x ^ 1 ) f '' ( x ^ 1 ) + 3 f '^ 2 ( x ^ 1 ) + 3 = 0 . \\end{align*}"}
{"id": "7907.png", "formula": "\\begin{align*} ( z \\partial _ z + \\mu - \\rho / z ) ( z ^ { - \\mu } z ^ \\rho \\alpha ) = 0 . \\end{align*}"}
{"id": "4969.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\in [ 0 , T ] } E [ | A ^ n _ t - \\widetilde { A } ^ n _ t | ^ 2 ] = 0 , \\end{align*}"}
{"id": "2023.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : p d e p d m p } \\frac { \\dd } { \\dd t } \\langle f _ t , \\varphi \\rangle = \\langle f _ t , a \\cdot \\nabla _ x \\varphi \\rangle + \\iint _ { E \\times E } \\lambda ( x , f _ t ) \\{ \\varphi ( y ) - \\varphi ( x ) \\} P _ { f _ t } ( x , \\dd y ) f _ t ( \\dd x ) , \\end{align*}"}
{"id": "8098.png", "formula": "\\begin{align*} X = p \\partial _ { 1 } + q \\partial _ { 2 } . \\end{align*}"}
{"id": "2922.png", "formula": "\\begin{align*} A x + e = F ^ { - 1 } G ( x ) + F ^ { - 1 } d , \\end{align*}"}
{"id": "72.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Y ' ( t ) = A ( t ) Y ( t ) , \\\\ Y ( 0 ) = , \\end{array} \\right . \\end{align*}"}
{"id": "8374.png", "formula": "\\begin{align*} 0 & = u v ^ T + u ' ( v ' ) ^ T , \\\\ 0 & = u v ^ T + u '' ( v ' ) ^ T = u v ^ T - u ' ( v ' ) ^ T . \\end{align*}"}
{"id": "5585.png", "formula": "\\begin{align*} \\Phi ^ \\tau _ x ( \\eta , \\ , h ) \\ = \\ \\big ( \\tau ^ { x , h ( x ) } \\eta , \\ , h + \\delta _ x \\big ) \\ , , \\end{align*}"}
{"id": "3413.png", "formula": "\\begin{align*} r = r \\cdot w _ 1 \\cdots w _ s = h _ s \\cdots h _ 1 \\cdot r , \\end{align*}"}
{"id": "7711.png", "formula": "\\begin{align*} \\frac { ( k - 1 ) ! S _ f ( x ) } { x ( \\log x ) ^ { k - 1 } } = \\sum _ { d \\le x } \\frac { g ( d ) } { d } + \\sum _ { j = 1 } ^ { k - 1 } ( - 1 ) ^ j \\binom { k - 1 } { j } ( \\log x ) ^ { - j } \\sum _ { d \\le x } \\frac { g ( d ) } { d } ( \\log d ) ^ j + O \\left ( ( \\log x ) ^ { - 1 } \\right ) . \\end{align*}"}
{"id": "502.png", "formula": "\\begin{align*} R P P ( w , n ) = B _ w ( n \\lambda ) \\end{align*}"}
{"id": "2363.png", "formula": "\\begin{align*} \\nu \\left ( f - l ( h _ \\sigma ) \\right ) = \\beta _ b + b \\gamma _ \\sigma . \\end{align*}"}
{"id": "395.png", "formula": "\\begin{align*} q _ A = \\frac { 2 } { 1 + \\frac { G ( x , y ) } { G ( x , x ) } } . \\end{align*}"}
{"id": "6206.png", "formula": "\\begin{align*} \\int f ( x ) d _ q x = F ( x ) , \\end{align*}"}
{"id": "8063.png", "formula": "\\begin{align*} \\operatorname { H e s s } _ { p } ( V , W ) = \\operatorname { H e s s } _ { p } ( W , V ) , \\operatorname { H e s s } _ { \\| X \\| ^ { 2 } } ( V , W ) = \\operatorname { H e s s } _ { \\| X \\| ^ { 2 } } ( W , V ) . \\end{align*}"}
{"id": "8336.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty | \\beta ( k ) - \\alpha ( k ) | ^ 2 w ' ( k ) < \\infty . \\end{align*}"}
{"id": "2918.png", "formula": "\\begin{align*} f _ i ( x ) = ( f \\circ S ^ { 1 - i } ) ( A x + e ) = f ( A S ^ { ( 1 - i ) k } x + e ) . \\end{align*}"}
{"id": "4504.png", "formula": "\\begin{align*} \\Delta | \\nabla u | = | k | ^ 2 | \\nabla u | - \\langle \\div k , \\nabla u \\rangle \\end{align*}"}
{"id": "3247.png", "formula": "\\begin{align*} { \\rm t r } ( A D ^ 2 u ) + \\langle \\beta , \\nabla u \\rangle - \\lambda u = - \\beta ^ k , \\lambda > 0 . \\end{align*}"}
{"id": "3685.png", "formula": "\\begin{align*} \\begin{aligned} d _ h ( a , b ) & \\leq d _ h ( a , \\tilde x ) + d _ h ( \\tilde x , \\tilde y ) + d _ h ( \\tilde y , b ) \\\\ & \\leq 3 \\eta + \\delta + 3 \\eta \\\\ & \\leq 2 \\delta \\end{aligned} \\end{align*}"}
{"id": "8397.png", "formula": "\\begin{gather*} y ^ 2 = x ^ 3 - 1 3 3 9 2 D ^ 2 x - 1 0 8 0 4 3 2 D ^ 3 \\end{gather*}"}
{"id": "2330.png", "formula": "\\begin{align*} \\dots \\nu _ { - 3 } \\nu _ { - 2 } \\nu _ { - 1 } & = \\dots \\psi ( \\nu _ { - 3 } ) \\psi ( \\nu _ { - 2 } ) \\psi ( \\nu _ { - 1 } ) \\\\ \\nu _ 0 \\nu _ 1 \\nu _ 2 \\dots & = \\psi ( \\nu _ { 0 } ) \\psi ( \\nu _ { 1 } ) \\psi ( \\nu _ { 2 } ) \\dots \\end{align*}"}
{"id": "8160.png", "formula": "\\begin{align*} & \\| P _ { \\delta } ( A _ { v t , m } ) e ^ { - i H _ 0 t } \\psi \\| = \\| P _ { \\delta } ( A _ { v t , m } ) e ^ { - i H _ 0 t _ n } \\chi _ { S _ R } \\psi \\| \\leq \\| \\chi _ { S _ R } e ^ { i H _ 0 t _ n } P _ { \\delta } ( A _ { v t , m } ) \\| _ \\textrm { o p } \\| \\psi \\| \\xrightarrow { t \\rightarrow \\infty } 0 \\end{align*}"}
{"id": "5751.png", "formula": "\\begin{align*} f = \\pi _ { i _ 1 } \\pi _ { i _ 2 } \\cdots \\pi _ { i _ d } \\end{align*}"}
{"id": "8795.png", "formula": "\\begin{align*} ( \\phi ^ * ) ^ { - 1 } ( \\mathbb { R } _ { > t } ) = \\left ( - \\frac { \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) } { 2 } , \\frac { \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) } { 2 } \\right ) \\end{align*}"}
{"id": "7635.png", "formula": "\\begin{align*} { B } ^ { \\prime } _ { { \\rm S T } } = \\{ t { t ^ { \\prime } _ 1 } \\ldots { t ^ { \\prime } _ n } , \\ n \\in \\mathbb { N } \\} . \\end{align*}"}
{"id": "3711.png", "formula": "\\begin{align*} \\bar { \\phi } _ { C _ 1 , C _ 2 } : = \\phi _ { C ' _ { n - 1 } , C ' _ n } \\circ \\cdots \\circ \\phi _ { C ' _ 1 , C ' _ 2 } \\circ \\phi _ { C ' _ 0 , C ' _ 1 } \\end{align*}"}
{"id": "168.png", "formula": "\\begin{align*} \\hat f ( k ) : = \\| k \\| _ 2 \\int _ 0 ^ 1 d \\alpha \\int _ { \\R ^ { 6 } } d p \\ , d q \\ , \\frac { 1 } { 1 + \\| p \\| _ 2 ^ 2 } \\cdot \\frac { 1 } { 1 + \\| q \\| _ 2 ^ 2 } \\cdot \\frac { \\| \\alpha k - p - q \\| _ 2 } { \\big [ 1 + \\| \\alpha k - p - q \\| _ 2 ^ 2 \\big ] ^ 2 } . \\end{align*}"}
{"id": "5485.png", "formula": "\\begin{align*} H ( x , y , t ) \\lesssim \\frac { 1 } { \\sqrt { \\mathrm { v o l } ( B ( x , r ) ) \\mathrm { v o l } ( B ( y , r ) ) } } e ^ { - \\frac { \\rho ^ 2 } { 4 t } - \\lambda _ 1 ( M ) t } = \\overline { H } ( \\rho , t ) , \\end{align*}"}
{"id": "7139.png", "formula": "\\begin{align*} ( p x _ 1 + q x _ 2 + r x _ 3 ) ^ { m _ 2 } = \\alpha x _ 1 ^ { m _ 1 } + q ^ { m _ 2 } x _ 2 ( x _ 1 + x _ 2 ) ^ { m _ 2 - 1 } + \\gamma x _ 3 ( x _ 1 + x _ 2 + x _ 3 ) ^ { m _ 3 - 2 } \\end{align*}"}
{"id": "3230.png", "formula": "\\begin{align*} \\Gamma ( T ^ * \\mathcal { U } ) = \\Gamma ( T ^ * \\mathcal { M } _ { \\mathrm { S p } } ( 2 m , \\alpha , L ) ) . \\end{align*}"}
{"id": "361.png", "formula": "\\begin{align*} n ^ { - q _ A } \\leq n ^ { - k } n ^ { K \\log ( D / \\delta ) / f ( n ) } = n ^ { - k } ( D / \\delta ) ^ { K ( \\log n ) / f ( n ) } = n ^ { - k } ( D / \\delta ) ^ { K / b ( n ) } . \\end{align*}"}
{"id": "428.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\Im H ( x + i y ) > 0 y = f ( x ) , \\ : x \\in ( \\alpha - \\varepsilon , \\alpha + \\varepsilon ) . \\end{align*}"}
{"id": "3300.png", "formula": "\\begin{align*} S ( s ) \\varphi ( x ) : = \\int _ \\Gamma \\mathcal { K } ( | x - y | , s ) \\varphi ( y ) d \\Gamma _ y x \\in \\R ^ d \\setminus \\Gamma \\end{align*}"}
{"id": "7699.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\tau _ k ( n ) = x \\ , P _ { k - 1 } ( \\log x ) + O ( x ^ { \\theta _ k + \\varepsilon } ) \\end{align*}"}
{"id": "6616.png", "formula": "\\begin{align*} S _ k ( c ; \\xi \\ , ( \\ell ) ) = { \\sum _ { a \\ , \\hbox { \\tiny m o d } \\ , k } } ^ { \\ ! \\ ! \\ ! * } e _ { k } ( - a h ^ 2 ) S _ { k _ 1 } \\big ( a \\bar k _ 2 , \\bar k _ 2 \\bar \\ell _ 2 c ; \\xi \\ , ( \\ell _ 1 ) \\big ) \\ , S _ { k _ 2 } \\big ( a \\bar k _ 1 , \\bar k _ 1 \\bar \\ell _ 1 c ; \\xi \\ , ( \\ell _ 2 ) \\big ) . \\end{align*}"}
{"id": "4105.png", "formula": "\\begin{align*} \\widetilde { C } ' ( m , p ) \\begin{bmatrix} \\widehat { m } \\\\ \\widehat { p } \\end{bmatrix} = - \\widetilde { C } ( m , p ) \\circ C ( \\widehat { m } , \\widehat { p } ) \\circ \\widetilde { C } ( m , p ) \\end{align*}"}
{"id": "2602.png", "formula": "\\begin{align*} e _ { j , k , l } = \\left ( \\dfrac { p ! } { j ! \\ , k ! \\ , l ! } \\right ) ^ { - 1 / 2 } \\sum _ { f \\in S _ p } e _ { i _ { f ( 1 ) } } \\otimes . . . \\otimes e _ { i _ { f ( p ) } } \\in \\mathcal H _ { p , 0 } \\ . \\end{align*}"}
{"id": "4770.png", "formula": "\\begin{align*} | \\mathcal K _ { \\kappa , 2 } | = \\left | \\mathcal { T } _ { \\kappa , 2 } \\right | + \\frac 1 2 \\left | \\mathcal S _ { \\kappa , 2 } \\setminus \\mathcal T _ { \\kappa , 2 } \\right | . \\end{align*}"}
{"id": "1358.png", "formula": "\\begin{align*} k = k _ 1 - k _ 2 + \\cdots + k _ p . \\end{align*}"}
{"id": "7409.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ N f ( n _ k \\alpha ) \\right | \\ll ( N \\log N ) ^ { 1 / 2 } ( \\log \\log N ) ^ { 3 / 2 + \\varepsilon } \\textrm { f o r a . e . } \\alpha , \\end{align*}"}
{"id": "3797.png", "formula": "\\begin{align*} E r r ^ 1 _ { i , i _ 1 , i _ 2 } ( t _ 1 , t _ 2 ) = \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } e ^ { i X ( s ) \\cdot ( \\xi + \\eta + \\sigma ) + i \\mu _ 2 s | \\sigma | + i \\mu s | \\xi | + i \\mu _ 1 s | \\eta | } \\big ( \\hat { V } ( s ) \\cdot ( \\xi + \\eta + \\sigma ) + \\mu _ 2 | \\sigma | + \\mu | \\xi | + \\mu _ 1 | \\eta | \\big ) ^ { - 1 } \\end{align*}"}
{"id": "5958.png", "formula": "\\begin{align*} b ( \\hat \\eta ) x \\Omega _ \\phi & = x \\hat \\eta \\\\ b ( \\hat \\eta ) \\eta & = \\eta \\otimes _ \\phi \\hat \\eta \\end{align*}"}
{"id": "2248.png", "formula": "\\begin{align*} & ( u _ p ^ n , v _ p ^ n , h _ p ^ n , g _ p ^ n ) ( x , 0 ) = ( - \\overline u _ e ^ n ( x ) , 0 , - \\overline h _ e ^ n ( x ) , 0 ) , \\\\ & ( u _ p ^ n , h _ p ^ n ) ( x , \\infty ) = ( 0 , 0 ) , \\ ( u _ p ^ n , h _ p ^ n ) ( 1 , y ) = ( u _ 0 ^ n , h _ 0 ^ n ) ( y ) . \\end{align*}"}
{"id": "1510.png", "formula": "\\begin{align*} \\Big ( x \\frac { d } { d x } \\Big ) _ { p , \\lambda } e ^ { x } = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( n ) _ { p , \\lambda } } { n ! } x ^ { n } = e ^ { x } \\phi _ { p , \\lambda } ( x ) . \\end{align*}"}
{"id": "2765.png", "formula": "\\begin{align*} \\Sigma : = ( X , U , F ) , \\end{align*}"}
{"id": "6498.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { c _ 1 } ( \\abs { v } ^ 2 + \\abs { u _ x } ^ 2 ) d x = o ( 1 ) . \\end{align*}"}
{"id": "786.png", "formula": "\\begin{align*} \\frac { \\beta ( 1 + ( 1 - 2 \\alpha ) r ^ m ) } { ( 1 - r ^ m ) ^ { 2 ( 1 - \\alpha ) + 1 } } & + \\frac { ( 1 - \\beta ) r ^ m } { ( 1 - r ^ m ) ^ { 2 ( 1 - \\alpha ) } } + \\sum _ { n = 2 } ^ { \\infty } \\prod _ { k = 0 } ^ { n - 2 } \\frac { k + 2 ( 1 - \\alpha ) } { k + 1 } \\phi _ { n } ( r ) \\\\ & < \\frac { 1 } { 4 ^ { 1 - \\alpha } } - \\phi _ { 1 } ( r ) . \\end{align*}"}
{"id": "7293.png", "formula": "\\begin{align*} \\theta _ k ( x ) & = { a } _ k | x | ^ \\frac { 2 k ( p - q ) } { 1 - q } \\theta _ 0 + { \\sf h } _ k ( x ) ( { a } _ k \\not = 0 ) , \\\\ { \\sf h } _ k ( x ) & = | x | ^ \\frac { 2 k ( p - q ) } { 1 - q } \\theta _ 0 \\sum _ { i = 1 } ^ { N ' } b _ i | x | ^ \\frac { 2 ( p - q ) i } { 1 - q } . \\end{align*}"}
{"id": "5825.png", "formula": "\\begin{align*} \\frac { y _ { n } } { x _ { n } } = \\prod _ { k \\leq n } \\frac { p _ k ^ 2 } { p _ k ^ 2 - \\chi ( p _ k ) } = \\left ( \\frac { p _ n ^ 2 } { p _ n ^ 2 - 1 } \\right ) \\prod _ { k \\leq n - 1 } \\frac { p _ k ^ 2 } { p _ k ^ 2 - 1 } , \\end{align*}"}
{"id": "2693.png", "formula": "\\begin{align*} \\mathrm { T r } _ { \\perp , r } ( A ) : = \\sum _ { \\alpha \\ge 1 } \\langle u _ { r , \\alpha } , A \\ , u _ { r , \\alpha } \\rangle , \\mathrm { T r } _ { \\perp , \\ell } ( A ) : = \\sum _ { \\alpha \\ge 1 } \\langle u _ { \\ell , \\alpha } , A \\ , u _ { \\ell , \\alpha } \\rangle \\end{align*}"}
{"id": "4537.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\pi } \\int _ { 0 } ^ { \\pi } \\frac { \\sin 2 m t _ { 1 } \\sin m t _ { 1 } \\sin 2 n t _ { 2 } \\sin n t _ { 2 } } { \\left ( 4 \\sin \\frac { t _ { 1 } } { 2 } \\sin \\frac { t _ { 2 } } { 2 } \\right ) ^ { 2 } } d t _ { 1 } d t _ { 2 } = \\frac { m n \\pi ^ { 2 } } { 4 } \\end{align*}"}
{"id": "5659.png", "formula": "\\begin{align*} \\sup _ { \\mathbb { R } ^ n } \\left | \\underline { u } ( x ) - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n a _ i x _ i ^ 2 \\right | \\leq C . \\end{align*}"}
{"id": "5871.png", "formula": "\\begin{align*} D _ 2 \\approx \\sum _ { k = - \\infty } ^ { M - 1 } \\bigg ( \\int _ { x _ k } ^ { x _ { k + 1 } } u ( s ) \\bigg ( \\int _ { s } ^ { \\infty } u \\bigg ) ^ { \\frac { q } { p - q } } d s \\bigg ) \\sum _ { i = - \\infty } ^ k 2 ^ { - i \\frac { q } { p - q } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \\frac { p q } { p - q } } . \\end{align*}"}
{"id": "6368.png", "formula": "\\begin{align*} { \\rm e r f } _ { \\kappa } \\ ! \\left ( x \\right ) \\ ! = \\ ! \\left ( \\ ! 1 + \\frac { 1 } { 2 } \\ , \\kappa \\ ! \\right ) \\ ! \\sqrt { 2 \\kappa } \\ , \\ , \\frac { \\Gamma \\ ! \\left ( \\frac { 1 } { 2 \\kappa } + \\frac { 1 } { 4 } \\right ) } { \\Gamma \\ ! \\left ( \\frac { 1 } { 2 \\kappa } - \\frac { 1 } { 4 } \\right ) } \\frac { 2 } { \\sqrt { \\pi } } \\int _ 0 ^ x \\ ! \\exp _ { \\kappa } ( - t ^ 2 ) \\ , d t \\ \\ , \\end{align*}"}
{"id": "4342.png", "formula": "\\begin{align*} T \\mathcal { M } ( p , q ) = \\R ( - \\partial _ s ) \\oplus T ( \\mathcal { M } ( p , q ) / \\R ) . \\end{align*}"}
{"id": "153.png", "formula": "\\begin{align*} t \\mapsto \\frac { p _ { d , t , \\mu , m ^ 2 } ( \\lambda ) } { t ^ 2 } \\end{align*}"}
{"id": "7500.png", "formula": "\\begin{align*} u _ t + ( - \\Delta ) ^ { d } u = | x | ^ { \\alpha } | u | ^ { p } + \\zeta ( t ) { \\mathbf w } ( x ) \\ \\quad \\mbox { f o r } ( x , t ) \\in \\mathbb { R } ^ { N } \\times ( 0 , \\infty ) , \\end{align*}"}
{"id": "5273.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ h & \\Big ( ( k _ 2 ( 0 ) + 1 ) k _ 1 ( j ) - ( k _ 1 ( 0 ) + 1 ) k _ 2 ( j ) \\Big ) d ^ { \\widehat { Q } _ j } \\\\ & - ( k _ 2 ( 0 ) + 1 ) ( 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) ) d ^ { \\widehat { Q } ^ { + r } } + ( k _ 1 ( 0 ) + 1 ) ( 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) ) d ^ { \\widehat { Q } ^ { + s } } . \\end{align*}"}
{"id": "1780.png", "formula": "\\begin{align*} z ^ * = \\sum _ { w \\in \\mathcal { W } \\setminus \\left ( \\{ z \\} \\cup \\{ x ^ t _ s : \\ ; s \\in \\mathbb { N } , \\ : t \\in A \\cup \\{ k \\} \\} \\right ) } y ^ * ( w ) \\widetilde { w } . \\end{align*}"}
{"id": "908.png", "formula": "\\begin{align*} P ( u ) \\cdot v = \\begin{bmatrix} f & h \\end{bmatrix} ^ T \\end{align*}"}
{"id": "25.png", "formula": "\\begin{align*} \\sum _ i ( \\alpha , h _ i ) _ { M , n } = \\Tilde { \\Phi } _ L ( \\beta ) ( \\sum _ j ( \\Tilde { \\sigma } _ j ^ { - 1 } ( \\xi ) ) ) - \\sum _ j ( \\Tilde { \\sigma } _ j ^ { - 1 } ( \\xi ) ) . \\end{align*}"}
{"id": "3120.png", "formula": "\\begin{align*} \\lambda b ( u , b _ { T } u _ { \\mathrm { n c } } ) & = a _ { \\mathrm { p w } } ( u , b _ { T } u _ { \\mathrm { n c } } ) = ( D ^ { m } u , D ^ { m } ( b _ T u _ { \\mathrm { n c } } ) ) _ { L ^ 2 ( T ) } \\le \\vert u - I u \\vert _ { H ^ m ( T ) } \\vert b _ T u _ { \\mathrm { n c } } \\vert _ { H ^ m ( T ) } . \\end{align*}"}
{"id": "6396.png", "formula": "\\begin{align*} \\Bigl ( \\rho \\wedge \\pi _ { { } _ S } = \\pi \\Bigr ) \\ \\Leftrightarrow \\ \\Bigl ( \\begin{array} { c } S \\cap V _ i = \\emptyset \\mbox { f o r e v e r y $ 1 \\leq i \\leq p $ , a n d } \\\\ S \\cap W _ j \\neq \\emptyset \\mbox { f o r e v e r y $ 1 \\leq j \\leq q $ } \\end{array} \\Bigr ) . \\end{align*}"}
{"id": "2320.png", "formula": "\\begin{align*} \\mathcal { R } ( \\psi ) = R _ { \\varphi _ 1 } R _ { \\varphi _ 1 } \\cdots R _ { \\varphi _ n } , \\end{align*}"}
{"id": "4109.png", "formula": "\\begin{align*} \\Phi '' ( \\vec p ) [ \\widehat { \\vec p } , \\cdot ] = F '' ( V ( \\vec p ) ) [ V ' ( \\vec p ) \\widehat { \\vec p } , V ' ( \\vec p ) \\ , \\cdot ] + F ' ( V ( \\vec p ) ) V '' ( \\vec p ) [ \\widehat { \\vec p } , \\cdot ] . \\end{align*}"}
{"id": "7263.png", "formula": "\\begin{align*} \\begin{cases} Z _ 2 ( r ) = a _ 1 r ^ { - ( n - 2 ) } + o ( r ^ { - ( n - 2 ) } ) r \\to 0 , \\\\ Z _ 2 ( r ) = a _ 2 + o ( 1 ) r \\to \\infty \\end{cases} \\end{align*}"}
{"id": "6941.png", "formula": "\\begin{align*} \\prod _ { k \\ge 1 } \\frac { \\mu _ k ^ 2 } { \\lambda _ k ^ 2 } = 0 , \\sum _ { k \\ge 1 } \\Bigg ( \\frac { \\lambda _ { k } ^ { 2 } } { \\mu _ { k } ^ { 2 } } - 1 \\Bigg ) = \\infty ; \\end{align*}"}
{"id": "2787.png", "formula": "\\begin{align*} f _ * = \\min _ { x \\in \\mathbb { R } ^ d } f ( x ) = \\min _ { i \\in \\mathcal { I } } \\big \\{ f _ i - \\tfrac { 1 } { 2 L } \\| g _ i \\| ^ 2 \\big \\} \\end{align*}"}
{"id": "1892.png", "formula": "\\begin{align*} W _ { [ n , j ] } = \\sum _ { k = i } ^ { n - j + i } \\sum _ { \\gamma \\in \\mathcal { P } _ { [ n , j , k ] } } w ( \\gamma ) = \\sum _ { k = i } ^ { n - j + i } W _ { [ k , i ] } A _ { [ n - k - 1 , j - i - 1 ] } ^ { ( i + 1 ) } , n \\geq j , \\end{align*}"}
{"id": "6582.png", "formula": "\\begin{align*} W = ( \\lambda _ 1 ) ^ { - 1 } ( W _ 1 ) \\cap ( \\lambda _ 2 ) ^ { - 1 } ( W _ 2 ) \\end{align*}"}
{"id": "346.png", "formula": "\\begin{align*} h _ t ( x , y ) = p _ t ( x , y ) - \\frac { 1 } { n } . \\end{align*}"}
{"id": "2860.png", "formula": "\\begin{align*} \\mu & = \\frac { \\kappa } { 1 - \\kappa } \\delta ^ { - 1 } \\ , \\| A ^ T A \\| _ 2 \\\\ L & = \\frac { 1 } { 1 - \\kappa } \\delta ^ { - 1 } \\ , \\| A ^ T A \\| _ 2 . \\end{align*}"}
{"id": "1720.png", "formula": "\\begin{align*} B _ { n + 1 } ( t , q ^ { \\frac { 1 } { 2 } } ) = B _ { n } ( t , q ^ { \\frac { 1 } { 2 } } ) \\cdot ( 1 - x y ^ { n } ) ^ { - 1 } , \\ \\ D _ { n + 1 } ( t , q ^ { \\frac { 1 } { 2 } } ) = D _ { n } ( t , q ^ { \\frac { 1 } { 2 } } ) \\cdot \\prod _ { k = 0 } ^ { n - 1 } ( 1 - ( q ^ { \\frac { 1 } { 2 } } ) ^ { 1 - n + 2 k } x y ^ { n } ) ^ { - 1 } . \\end{align*}"}
{"id": "5260.png", "formula": "\\begin{align*} t \\in \\tau : = [ 0 , 1 ] \\setminus \\{ t _ i \\ , | \\ , 1 \\le i \\le 2 ^ { | I | } - 1 \\} , \\end{align*}"}
{"id": "6902.png", "formula": "\\begin{align*} \\max _ { v \\in [ 0 , V _ F ] } \\rho _ { } ( v ) \\geq \\frac { \\int _ 0 ^ { V _ F } \\rho _ { } ( v ) d v } { V _ F } = \\frac { 1 } { V _ F } \\geq \\frac { 1 } { g _ 1 } . \\end{align*}"}
{"id": "5675.png", "formula": "\\begin{align*} e _ k ( \\tau _ 1 , \\tau _ 2 , \\ldots , \\tau _ n ) = 0 ( 1 \\le k \\le n ) . \\end{align*}"}
{"id": "873.png", "formula": "\\begin{align*} x _ i * x _ { j + m - 1 } * x _ { j + m - 2 } * \\cdots * x _ j & = ( x _ i * x _ { j + m - 1 } * x _ { j + m - 2 } * \\cdots * x _ { j + 1 } ) * ( x _ { j + m } * x _ { j + m - 1 } * \\cdots * x _ { j + 1 } ) \\\\ & = x _ i * x _ { j + m } * x _ { j + m - 1 } * \\cdots * x _ { j + 1 } \\end{align*}"}
{"id": "396.png", "formula": "\\begin{align*} c _ 1 & = n \\bmod 2 \\\\ c _ 2 & = n - c _ 1 \\bmod 3 \\\\ c _ 3 & = n - ( c _ 1 + c _ 2 ) \\bmod 4 \\\\ & \\ ; \\ ; \\vdots \\\\ c _ { k } & = n - ( c _ 1 + c _ 2 + \\cdots + c _ { k - 1 } ) \\bmod ( k + 1 ) . \\end{align*}"}
{"id": "1532.png", "formula": "\\begin{align*} - m \\lambda \\log ( \\tfrac { \\lambda } { 2 } ) & = \\frac { \\log ( L \\log ( m ) ) } { L \\log ( m ) } \\log \\left ( \\frac { 2 L m \\log ( m ) } { \\log ( L \\log ( m ) ) } \\right ) \\\\ [ 3 p t ] & \\leq \\frac { ( \\log ( 2 L \\log ( m ) ) ^ 2 } { L \\log ( m ) } + \\frac { \\log ( L \\log ( m ) ) } { L } \\leq C _ 4 + \\frac { \\log ( L \\log ( m ) ) } { L } \\end{align*}"}
{"id": "7419.png", "formula": "\\begin{align*} \\mathcal { L } = \\int _ 0 ^ T \\left [ \\bold { K - ( P + E ) + B + W } \\right ] d t , \\end{align*}"}
{"id": "3606.png", "formula": "\\begin{align*} & \\langle y , ( - u _ i , 1 , 0 , 0 ) \\rangle \\leq 0 , \\ i = 1 , \\ldots , m , \\ , \\langle y , - e _ { s + 1 } + e _ { s + 3 } \\rangle \\leq 0 , \\\\ & \\langle y , - e _ i \\rangle \\leq 0 , \\ , i = 1 , \\ldots , s , \\ , \\langle y , - e _ { s + 3 } \\rangle \\leq 0 , \\\\ & \\langle y , ( \\gamma _ j , 0 , 0 , 0 ) - d _ j e _ { s + 2 } + e _ { s + 3 } \\rangle \\leq 0 , \\ , \\langle y , e _ { s + 2 } \\rangle \\leq 1 , \\ , \\langle y , - e _ { s + 2 } \\rangle \\leq - 1 . \\end{align*}"}
{"id": "1279.png", "formula": "\\begin{align*} \\Psi ( & E _ { j + 1 } ( x , d - 1 ) ) - \\Psi ( E _ j ( x , d - 1 ) ) = \\\\ & = 2 \\sqrt { d - 1 } \\left ( \\csc \\left ( \\dfrac { \\pi } { 2 j + 4 } \\right ) - 1 \\right ) - 2 \\sqrt { d - 1 } \\left ( \\cot \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) - 1 \\right ) \\\\ & = 2 \\sqrt { d - 1 } \\left ( \\csc \\left ( \\dfrac { \\pi } { 2 j + 4 } \\right ) - \\cot \\left ( \\dfrac { \\pi } { 2 j + 2 } \\right ) \\right ) \\\\ & = f _ j \\ , \\sqrt { d - 1 } \\ , . \\end{align*}"}
{"id": "6493.png", "formula": "\\begin{align*} u = 0 \\ \\ \\ \\ ( c _ 2 , L ) y = 0 \\ \\ \\ \\ ( 0 , c _ 1 ) \\cup ( c _ 2 , L ) . \\end{align*}"}
{"id": "205.png", "formula": "\\begin{align*} \\hat { H } _ { 2 } = g ^ { n } _ { 2 } ( Z ^ { n } _ { 2 } , M _ 0 , M _ 2 ) . \\end{align*}"}
{"id": "2796.png", "formula": "\\begin{align*} \\frac { 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left ( 2 h _ i - h _ i ^ 2 \\frac { - \\kappa } { 1 - \\kappa } \\right ) } \\end{align*}"}
{"id": "3755.png", "formula": "\\begin{align*} \\times \\varphi _ { j , l } ( v , { \\omega } ) d { \\omega } d v d s \\big | \\lesssim \\sup _ { s \\in [ 0 , t ] } \\sup _ { g \\in L ^ 2 , \\| g \\| _ { L ^ 2 } = 1 } 2 ^ { 2 m + k + 2 l + \\tilde { c } ( m , k , l ) } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\big | \\hat { f } ( s , \\xi , v ) \\big | \\varphi _ k ( \\xi ) \\psi _ { \\leq \\tilde { c } ( m , k , l ) + \\epsilon M _ t } ( \\tilde { v } \\times \\tilde { \\xi } ) \\big | \\overline { \\hat { g } ( \\xi ) } \\big | \\varphi _ j ( v ) d v d \\xi \\end{align*}"}
{"id": "4706.png", "formula": "\\begin{align*} ( - 1 ) ^ k \\sum _ { j = - k } ^ k ( - 1 ) ^ j p ( n - j ( 3 j + 1 ) / 2 ) = \\widetilde P _ k ( n ) . \\end{align*}"}
{"id": "893.png", "formula": "\\begin{align*} \\norm { f _ m } ^ 2 = \\norm { g _ 0 } ^ 2 + \\norm { g _ 1 } ^ 2 + \\cdots + \\norm { g _ m } ^ 2 \\leq \\norm { r h } ^ 2 \\end{align*}"}
{"id": "5851.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } \\bigg ( \\int _ 0 ^ { x _ k } f ^ r v \\bigg ) ^ { \\frac { q } { r } } \\int _ { x _ k } ^ { x _ { k + 1 } } u ( t ) d t \\bigg ) ^ { \\frac { 1 } { q } } \\leq \\mathfrak { C ' } \\bigg ( \\sum _ { k = - \\infty } ^ { M - 1 } 2 ^ k \\bigg ( \\int _ { x _ k } ^ { x _ { k + 1 } } f \\bigg ) ^ p \\bigg ) ^ { \\frac { 1 } { p } } \\end{align*}"}
{"id": "4022.png", "formula": "\\begin{align*} \\kappa ( z ) \\triangleq \\log \\phi ( z ) = \\sum _ { j = 1 } ^ \\infty \\kappa _ j \\frac { z ^ j } { j ! } , \\end{align*}"}
{"id": "6913.png", "formula": "\\begin{align*} \\widehat { \\mathsf { F D P } } ( t ) : = \\frac { 1 + \\# \\{ j : W _ j \\leq - t \\} } { \\# \\{ j : W _ j \\geq t \\} \\vee 1 } , \\end{align*}"}
{"id": "6550.png", "formula": "\\begin{align*} \\lim _ { z \\to \\infty } A ( z ) = \\infty . \\end{align*}"}
{"id": "7592.png", "formula": "\\begin{align*} Y = \\exp ( \\lambda _ 0 M _ 0 ) \\exp ( \\lambda _ 1 M _ 1 ) \\exp ( \\lambda _ 2 M _ 2 ) \\ , . \\end{align*}"}
{"id": "27.png", "formula": "\\begin{align*} \\mathfrak { X } _ L : = \\{ y \\in L ; ~ < x , y > _ L \\in \\mathcal { O } ~ \\forall x \\in \\lambda _ { \\rho } ( \\mathfrak { p } _ L ) \\} . \\end{align*}"}
{"id": "1760.png", "formula": "\\begin{align*} \\Omega ( \\alpha , A , z + a f _ k ) = \\{ z + a f _ k \\} \\cup \\bigcup _ { n = 1 } ^ \\infty \\Omega ( \\alpha _ n , A _ n , x ^ k _ n + a _ n f _ { p ( n ) } ) . \\end{align*}"}
{"id": "8313.png", "formula": "\\begin{align*} \\frac { 1 } { \\zeta ( s ) } \\sum _ { n = 1 } ^ { \\infty } \\frac { c _ k ( n ) } { n ^ s } = \\sum _ { d | k } \\mu ( \\frac { k } { d } ) d ^ { 1 - s } , \\end{align*}"}
{"id": "5130.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , \\begin{pmatrix} 1 & 2 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
{"id": "2668.png", "formula": "\\begin{align*} & ( c _ i ^ 2 - c _ i ) = 0 i > \\ell \\\\ & c _ i c _ j = 0 i > \\ell , j > i . \\end{align*}"}
{"id": "6942.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } ^ \\infty \\left ( \\frac { \\mu _ k ^ 2 } { \\lambda _ { k + 1 } ^ 2 } - 1 \\right ) = \\infty . \\end{align*}"}
{"id": "3342.png", "formula": "\\begin{align*} Q _ { 3 , \\tilde { q } , \\tilde { p } } = \\underbrace { \\sigma _ 2 ^ { \\lambda ( 1 ) } \\sigma _ 1 ^ { \\lambda ( 2 ) } \\sigma _ 2 ^ { \\lambda ( \\frac { \\tilde { q } - 3 } { 2 } ) } \\sigma _ 1 ^ { \\lambda ( \\frac { \\tilde { q } - 1 } { 2 } ) } } _ { P } \\sigma _ 2 ^ { - \\lambda ( \\frac { \\tilde { q } - 1 } { 2 } ) } \\sigma _ 1 ^ { - \\lambda ( \\frac { \\tilde { q } - 3 } { 2 } ) } . . . \\sigma _ 2 ^ { - \\lambda ( 2 ) } \\sigma _ 1 ^ { - \\lambda ( 1 ) } \\end{align*}"}
{"id": "3904.png", "formula": "\\begin{align*} \\overline { \\varepsilon } \\ ! = \\ ! \\frac { 1 } { 2 ^ { K + 2 } } \\left | a ^ { p / 2 } - b ^ { p / 2 } \\right | = \\frac { 1 } { 2 ^ { K + 2 } } \\left | \\frac { a ^ p - b ^ p } { a ^ { p / 2 } + b ^ { p / 2 } } \\right | & \\geq \\frac { 1 } { 2 ^ { K + 2 } } \\left | \\frac { a ^ { \\lfloor p \\rfloor } - b ^ { \\lfloor p \\rfloor } } { a ^ { \\lceil p / 2 \\rceil } + b ^ { \\lceil p / 2 \\rceil } } \\right | \\triangleq \\varepsilon > 0 . \\end{align*}"}
{"id": "4707.png", "formula": "\\begin{align*} p ( n - 1 ) + p ( n - 2 ) - p ( n ) & = \\widetilde P _ 1 ( n ) \\\\ p \\left ( n - \\frac { k ( 3 k - 1 ) } { 2 } \\right ) + p \\left ( n - \\frac { k ( 3 k + 1 ) } { 2 } \\right ) & = \\widetilde P _ { k - 1 } ( n ) + \\widetilde P _ k ( n ) , \\ k \\geqslant 2 . \\end{align*}"}
{"id": "536.png", "formula": "\\begin{align*} \\tau ^ * ( p ^ m ) = 2 \\end{align*}"}
{"id": "8224.png", "formula": "\\begin{align*} \\phi _ { n } ^ { \\pm } ( x ) = \\frac { e ^ { i \\left [ \\left ( \\varphi \\pm 2 n \\pi \\right ) x / a \\pm \\theta \\right ] } } { \\sqrt { a } } ; E _ { n } ^ { \\pm } = \\frac { \\hslash ^ { 2 } } { 2 m } \\left ( \\frac { 2 n \\pi \\pm \\varphi } { a } \\right ) ^ { 2 } + \\frac { \\beta \\hslash ^ { 4 } } { 3 m } \\left ( \\frac { 2 n \\pi \\pm \\varphi } { a } \\right ) ^ { 4 } , \\end{align*}"}
{"id": "7262.png", "formula": "\\begin{align*} \\| u _ k ( t ) \\| _ \\infty = ( T - t ) ^ { - 3 k } ( k \\in \\N ) . \\end{align*}"}
{"id": "3307.png", "formula": "\\begin{align*} \\gamma ^ + D ( \\partial _ t ) \\psi = \\frac 1 2 \\psi + K ( \\partial _ t ) \\psi \\end{align*}"}
{"id": "5632.png", "formula": "\\begin{align*} \\mathcal { I } _ n = \\underbrace { \\iint _ { Q _ T } Z \\frac { B _ n } { A _ n } ( A - A _ n ) [ \\nabla \\cdot ( m \\nabla \\varphi _ n ) ] } _ { \\mathcal { I } _ { n , 1 } } + \\underbrace { \\iint _ { Q _ T } Z ( B _ n - B ) [ \\nabla \\cdot ( m \\nabla \\varphi _ n ) ] } _ { \\mathcal { I } _ { n , 2 } } . \\end{align*}"}
{"id": "5384.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | \\leq m } \\langle a _ { 1 , \\alpha } - a _ { 2 , \\alpha } , ( D ^ \\alpha v _ 1 ) v _ 2 \\rangle = 0 \\mbox { f o r a l l } v _ 1 , v _ 2 \\in C ^ \\infty _ c ( \\Omega ) . \\end{align*}"}
{"id": "7679.png", "formula": "\\begin{align*} \\phi _ { k + 1 } : = \\phi _ { k } , \\ \\psi _ { k + 1 } : = \\frac { 1 } { 2 } \\psi _ { k } , \\ \\gamma _ { k + 1 } : = \\gamma _ { k } , \\ y _ { k + 1 } : = \\overline { y } _ { k + 1 } , \\ Z _ { k + 1 } : = \\overline { Z } _ { k + 1 } , \\end{align*}"}
{"id": "5020.png", "formula": "\\begin{align*} R ^ { n , 3 } _ \\tau & = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) \\Xi ^ { n , 1 } _ s d s \\\\ & = n ^ { 2 \\alpha + 1 } \\sum _ { i = 0 } ^ { M - 1 } \\int _ { \\tau _ i } ^ { \\tau _ { i + 1 } } \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) \\Xi ^ { n , 1 } _ s d s \\\\ & = : R ^ { n , M , 1 } _ \\tau + R ^ { n , M , 2 } _ \\tau , \\end{align*}"}
{"id": "8324.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { n \\in \\Z } f _ n e ^ { 2 \\pi i n x } \\ , . \\end{align*}"}
{"id": "758.png", "formula": "\\begin{align*} \\varphi ( x ) = ( 1 + \\lVert x \\rVert ^ 2 ) ^ 8 , U ( x ) = 8 T \\left ( 8 T e ^ { \\alpha T } \\bar { r } ^ 2 \\bar { a } ^ 2 + 3 + \\bar { r } \\right ) + e ^ { - \\alpha T } \\lVert x \\rVert ^ 2 , \\bar { U } ( x ) = 0 , \\end{align*}"}
{"id": "6273.png", "formula": "\\begin{align*} \\int \\dfrac { x ^ \\alpha } { ( - x ; q ) _ \\infty } L _ n ^ { \\alpha } ( x ; q ) d _ q x = \\dfrac { x ^ { \\alpha + 1 } } { [ n ] _ q ( - x ; q ) _ \\infty } L _ { n - 1 } ^ { \\alpha + 1 } ( x ; q ) . \\end{align*}"}
{"id": "5196.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i \\in I } 2 r s d _ i + 2 \\sum _ { i \\in I ( \\Gamma ) } s a _ i - 2 s k _ 1 ( \\Gamma ) + 2 \\sum _ { i \\in I ( \\Gamma ) } r b _ i - 2 r k _ 2 ( \\Gamma ) & = 2 r s | I | - 2 r s . \\end{aligned} \\end{align*}"}
{"id": "454.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\theta } \\arg \\left ( \\Phi ( r e ^ { i \\theta } ) \\right ) > 0 r = r _ { 0 } \\theta = h _ { \\nu } ( r _ { 0 } ) . \\end{align*}"}
{"id": "8202.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } \\psi ( x ) = \\sqrt { \\frac { 2 } { a } } \\cos ( \\frac { n \\pi x } { a } ) , $ n o d d $ \\\\ \\psi ( x ) = \\sqrt { \\frac { 2 } { a } } \\sin ( \\frac { n \\pi x } { a } ) , $ n e v e n $ \\end{array} \\right . . \\end{align*}"}
{"id": "1803.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k \\tilde d ( x ^ { ( i + 1 ) } , z ^ { ( i + 1 ) } ; x ^ { ( i ) } , z ^ { ( i ) } ) \\leq d ( x ^ \\star , z ^ \\star ; x ^ { ( 0 ) } , z ^ { ( 0 ) } ) . \\end{align*}"}
{"id": "6469.png", "formula": "\\begin{align*} \\rho _ \\mathcal { A } ( x _ i , X _ \\lambda ) = \\varrho ( x _ i ) + X _ \\lambda \\end{align*}"}
{"id": "5079.png", "formula": "\\begin{align*} E [ M ^ { n , 1 } _ { s _ 1 } M ^ { n , 2 } _ { s _ 1 } M ^ { n , 1 } _ { s _ 2 } M ^ { n , 2 } _ { s _ 2 } ] = E [ M ^ { n , 1 } _ { s _ 1 } M ^ { n , 2 } _ { s _ 1 } M ^ { n , 2 } _ { s _ 2 } E [ M ^ { n , 1 } _ { s _ 2 } | \\mathcal { F } _ { \\eta _ n ( s _ 2 ) } ] ] = 0 . \\end{align*}"}
{"id": "3780.png", "formula": "\\begin{align*} J ^ { m ; p , q } _ { k , j ; n , l } ( t , x , \\zeta ) : = \\int _ 0 ^ { t } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ { \\pi } 2 ^ { 3 k + 2 n } ( 1 + 2 ^ k | y \\cdot \\tilde { \\zeta } | ) ^ { - N _ 0 ^ 3 } ( 1 + 2 ^ { k + n } ( | y \\cdot \\tilde { \\zeta } _ 1 | + | y \\cdot \\tilde { \\zeta } _ 2 | ) ) ^ { - N _ 0 ^ 3 } f ( s , x - y + ( t - s ) \\omega , \\zeta ) \\end{align*}"}
{"id": "765.png", "formula": "\\begin{align*} w _ m = \\frac { [ T _ m - \\alpha ] ^ + } { \\sum _ { k = 1 } ^ { M } [ { T _ k } - \\alpha ] ^ + } , m = 1 , 2 , \\ldots , M \\end{align*}"}
{"id": "4841.png", "formula": "\\begin{align*} \\tau ^ \\mathrm { a n } ( M ' / S , \\mathbb { 1 } ) = \\tau ^ \\mathrm { a n } \\big ( M / S , \\mathrm { I n d } ^ G _ H \\mathbb { 1 } \\big ) \\ ; . \\end{align*}"}
{"id": "5838.png", "formula": "\\begin{align*} \\sup _ { N \\leq k \\leq M } a _ k \\bigg ( \\sum _ { i = N } ^ k b _ i \\bigg ) ^ \\alpha \\approx \\sup _ { N \\leq k \\leq M } a _ k b _ k ^ \\alpha . \\end{align*}"}
{"id": "1785.png", "formula": "\\begin{align*} \\widetilde { u } _ n & \\in \\left ( Q _ 1 ( \\alpha - 1 , A _ n , u _ n + a _ n f _ { q ( n ) } ) \\right ) ^ { ( \\alpha ) } \\\\ & \\subseteq \\left ( \\bigcap _ { j = 1 } ^ \\infty K ( \\alpha - 1 , A _ j , u _ j + a _ j f _ { q ( j ) } ) \\right ) ^ { ( \\alpha ) } = K ^ { ( \\alpha ) } . \\end{align*}"}
{"id": "3450.png", "formula": "\\begin{align*} \\mathrm { T C M } _ { q } ( X ^ { n } ) = \\mathrm { E } \\left [ ( X - \\mathrm { T C E } _ { q } ( X ) ) ^ { n } | X > x _ { q } \\right ] , \\end{align*}"}
{"id": "3901.png", "formula": "\\begin{align*} \\inf _ { B _ 1 ( 0 ) } u \\geq \\lim _ { j \\to + \\infty } \\Phi _ \\rho ( \\mu _ { j + 1 } , h ^ j _ 1 ) \\geq C _ 2 \\Phi _ \\rho ( - p _ 1 , 2 ) , C _ 2 = C ^ { - \\frac { \\kappa } { p _ 1 ( \\kappa - 1 ) } } ( 2 ^ { \\beta } \\kappa ^ { \\nu } ) ^ { - \\frac { 1 } { p _ 1 } \\sum _ j \\frac { j } { \\kappa ^ j } } , \\end{align*}"}
{"id": "4901.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n - 1 } { \\lambda _ i } ( A ) \\det ( ( B v ) ^ * ( B v ) ) = \\det ( B ^ * A B ) , \\end{align*}"}
{"id": "6133.png", "formula": "\\begin{gather*} ( A _ 1 - \\{ x ' _ 1 , \\dots , x ' _ j \\} ) + \\{ y ' _ 1 , \\dots , y ' _ j \\} \\ , \\\\ ( A _ 2 - \\{ y ' _ 1 , \\dots , y ' _ j \\} ) + \\{ x ' _ 1 , \\dots , x ' _ j \\} \\ . \\end{gather*}"}
{"id": "1846.png", "formula": "\\begin{align*} \\mathbf { v } _ { k } ( z ) : = \\begin{cases} ( \\phi _ { 0 } ^ { ( k ) } , \\ldots , \\phi _ { p - k - 1 } ^ { ( k ) } , - \\sum _ { j = k } ^ { p } a _ { 0 } ^ { ( j ) } \\phi _ { j - k } ^ { ( k ) } , \\ldots , - \\sum _ { j = 1 } ^ { p } a _ { k - 1 } ^ { ( j ) } \\phi _ { j - 1 } ^ { ( k ) } ) , & 0 \\leq k \\leq p , \\\\ [ 0 . 5 e m ] ( - a _ { k - p } ^ { ( p ) } \\phi _ { 0 } ^ { ( k ) } , - \\sum _ { j = p - 1 } ^ { p } a _ { k - p + 1 } ^ { ( j ) } \\phi _ { j - p + 1 } ^ { ( k ) } , \\ldots , - \\sum _ { j = 1 } ^ { p } a _ { k - 1 } ^ { ( j ) } \\phi _ { j - 1 } ^ { ( k ) } ) , & k \\geq p + 1 . \\end{cases} \\end{align*}"}
{"id": "7890.png", "formula": "\\begin{align*} ( [ \\alpha ] _ { \\vec s } , [ \\beta ] _ { \\vec s ^ \\prime } ) = \\begin{cases} \\int _ { D _ { I _ { \\vec s } } } \\alpha \\cup \\beta , & \\vec s = - \\vec s ^ \\prime ; \\\\ 0 , & \\end{cases} \\end{align*}"}
{"id": "2450.png", "formula": "\\begin{align*} & x _ { i + 1 } \\cdot x _ i = y _ { i } \\cdot y _ { i + 1 } = 0 , \\\\ & ( y _ { i + 1 } x _ { i + 1 } ) ^ { n - ( n - 1 ) \\delta _ { i , - 1 } } = - ( x _ i y _ i ) ^ { n - ( n - 1 ) \\delta _ { i , 0 } } , \\end{align*}"}
{"id": "8676.png", "formula": "\\begin{align*} \\begin{cases} \\dot u = R _ { \\varphi ( u \\wedge v ) } ( u ) \\\\ \\dot v = R _ { \\varphi ( u \\wedge v ) } ( v ) \\end{cases} \\end{align*}"}
{"id": "2395.png", "formula": "\\begin{align*} \\langle a , b \\rangle _ { l ^ 2 _ n } = \\frac { \\pi } { n } \\sum _ { i = 1 } ^ { n - 1 } a _ i b _ i , \\qquad \\| a \\| _ { l ^ p _ n } = \\begin{cases} \\Big ( \\frac { \\pi } { n } \\sum \\limits _ { i = 1 } ^ { n - 1 } | a _ i | ^ p \\Big ) ^ { \\frac { 1 } { p } } , \\quad & 1 \\le p < \\infty , \\\\ \\sup \\limits _ { 1 \\le j \\le n - 1 } | a _ i | , \\quad & p = \\infty , \\end{cases} \\end{align*}"}
{"id": "3298.png", "formula": "\\begin{align*} \\| \\cdot \\| _ { - 1 / 2 , \\Gamma } : = \\| \\cdot \\| _ { H ^ { - 1 / 2 } ( \\Gamma ) } \\| \\cdot \\| _ { 1 / 2 , \\Gamma } : = \\| \\cdot \\| _ { H ^ { 1 / 2 } ( \\Gamma ) } . \\end{align*}"}
{"id": "7207.png", "formula": "\\begin{align*} u ^ 0 ( x , t ) : = \\frac { 1 } { 2 } \\{ f ( x + t ) + f ( x - t ) \\} + \\frac { 1 } { 2 } \\int _ { x - t } ^ { x + t } g ( y ) d y \\end{align*}"}
{"id": "4207.png", "formula": "\\begin{align*} A \\Phi _ { \\nu , \\nu ' } = ( 2 | \\nu | + n ) \\Phi _ { \\nu , \\nu ' } . \\end{align*}"}
{"id": "8112.png", "formula": "\\begin{align*} \\lvert g ( t - s , x - y ) - g ( t - s , x ' - y ) \\rvert ^ 2 z ^ 2 & \\leq C ( t - s ) ^ { - d } z ^ 2 \\Bigl \\lvert e ^ { - \\frac { \\lvert x - y \\rvert ^ 2 } { 2 ( t - s ) } } - e ^ { - \\frac { \\lvert x ' - y \\rvert ^ 2 } { 2 ( t - s ) } } \\Bigr \\rvert ^ 2 \\\\ & \\leq C ( t - s ) ^ { - \\frac d 2 q } z ^ q \\Bigl \\lvert e ^ { - \\frac { \\lvert x - y \\rvert ^ 2 } { 2 ( t - s ) } } - e ^ { - \\frac { \\lvert x ' - y \\rvert ^ 2 } { 2 ( t - s ) } } \\Bigr \\rvert ^ q \\\\ & = C \\lvert g ( t - s , x - y ) - g ( t - s , x ' - y ) \\rvert ^ q z ^ q , \\end{align*}"}
{"id": "6723.png", "formula": "\\begin{align*} - \\operatorname { D i v } \\sigma & = u & & \\Omega , \\\\ \\nabla \\cdot y & = 0 & & \\Omega , \\\\ y & = 0 & & \\Gamma , \\end{align*}"}
{"id": "8475.png", "formula": "\\begin{align*} U & = ( n - k + 1 ) y _ k + \\frac { L ( n - k ) ( n - k + 1 ) } { 2 } \\\\ & \\leq ( n - 1 ) L + \\frac { L ( n - 2 ) ( n - 1 ) } { 2 } \\\\ & = \\frac { L n ( n - 1 ) } { 2 } < U , \\end{align*}"}
{"id": "7882.png", "formula": "\\begin{align*} I _ { \\vec s } : = \\{ i : s _ i \\neq 0 \\} \\subseteq \\{ 1 , \\ldots , n \\} . \\end{align*}"}
{"id": "8683.png", "formula": "\\begin{align*} d V _ { B H } = ( 1 - \\| \\beta \\| ^ 2 _ { \\alpha } ) ^ { \\frac { n + 1 } { 2 } } d V _ { \\alpha } \\end{align*}"}
{"id": "7162.png", "formula": "\\begin{align*} ( g _ { 3 1 } y _ 1 + g _ { 3 2 } y _ 2 + g _ { 3 3 } y _ 3 ) \\{ ( g _ { 1 1 } + g _ { 3 1 } ) y _ 1 + ( g _ { 1 2 } + g _ { 3 2 } ) y _ 2 + ( g _ { 1 3 } + g _ { 2 3 } + g _ { 3 3 } ) y _ 3 \\} ^ { b - 1 } \\\\ = g _ { 3 2 } ( g _ { 1 2 } + g _ { 2 3 } ) ^ { b - 1 } y _ 2 ( y _ 1 + y _ 2 ) ^ { b - 1 } + \\alpha y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { a - 1 } , \\end{align*}"}
{"id": "8117.png", "formula": "\\begin{align*} \\lambda _ { k + 2 } ( ( 0 , \\infty ) ) = \\lambda ( ( z _ { k + 2 } , z _ { k + 1 } ] ) = \\overline \\lambda ( z _ { k + 2 } ) - \\overline \\lambda ( z _ { k + 1 } ) \\leq 2 . \\end{align*}"}
{"id": "4801.png", "formula": "\\begin{align*} \\mathcal { N } ' = \\phi ^ { - 1 } \\big ( \\mathbb { A } \\big ( \\rho ( \\tfrac 1 5 \\sigma ^ * ) \\big ) \\big ) . \\end{align*}"}
{"id": "6283.png", "formula": "\\begin{align*} v ( x ) = g ( x ) \\int _ { 0 } ^ { x } \\frac { 1 } { g ( u ) } d _ q u , x \\in I \\end{align*}"}
{"id": "1067.png", "formula": "\\begin{align*} \\mathrm { e } ^ { - z t } B R ( z , A ) & = \\mathrm { e } ^ { - z t } \\int _ 0 ^ t \\mathrm { e } ^ { z s } \\tilde { B } T ( s , A ) \\ , \\mathrm { d } s + B R ( z , A ) T ( t , A ) . \\end{align*}"}
{"id": "4586.png", "formula": "\\begin{align*} w _ { a b } \\stackrel { \\mathrm { d } } { = } N ^ { - 1 / 2 } \\times \\begin{cases} \\chi _ \\mathrm { o d } , & a \\ne b \\\\ \\chi _ \\mathrm { d } , & a = b , \\end{cases} \\end{align*}"}
{"id": "6861.png", "formula": "\\begin{align*} a _ 1 b ^ * - g _ 1 c ^ * = a _ 1 g _ 0 - g _ 1 a _ 0 . \\end{align*}"}
{"id": "8692.png", "formula": "\\begin{align*} J '' ( \\gamma _ 0 , y ) = \\int \\limits _ { t _ 0 } ^ { t _ 1 } 2 \\omega ( t , y ( t ) , \\dot { y } ( t ) ) d t , \\end{align*}"}
{"id": "4283.png", "formula": "\\begin{align*} D ( \\phi _ 0 , \\gamma _ 0 ) = \\coprod _ { ( a ' , [ b _ 0 ' ] ) } D _ { ( a ' , [ b _ 0 ' ] ) } \\end{align*}"}
{"id": "2263.png", "formula": "\\begin{align*} & \\int _ 0 ^ r \\left [ ( - \\phi '' - \\frac { z } { 2 } \\phi ' + \\frac { z } { 2 } e _ \\sigma \\psi ' ) \\cdot \\phi + ( - \\psi '' - \\frac { z } { 2 } \\psi ' + \\frac { z } { 2 } e _ \\sigma \\phi ' ) \\cdot \\psi \\right ] { \\rm { d } } z \\\\ & = \\int _ 0 ^ r | ( \\phi ' , \\psi ' ) | ^ 2 + \\frac { 1 } { 4 } \\int _ 0 ^ r | ( \\phi , \\psi ) | ^ 2 - \\frac { 1 } { 2 } \\int _ 0 ^ r ( e _ \\sigma + z e _ \\sigma ' ) \\psi \\phi . \\end{align*}"}
{"id": "2233.png", "formula": "\\begin{align*} z _ s = e ^ { - \\pi t + \\pi \\sigma i } , \\cos \\frac { \\pi s } { 2 } = \\frac { e ^ { \\frac { \\pi } { 2 } \\left ( t - \\sigma i \\right ) } } { 2 } \\left ( 1 + z _ s \\right ) , \\quad \\sin \\frac { \\pi s } { 2 } = \\frac { e ^ { \\frac { \\pi } { 2 } \\left ( t - \\sigma i \\right ) } i } { 2 } \\left ( 1 - z _ s \\right ) . \\end{align*}"}
{"id": "6975.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } \\left ( 1 - \\frac { \\mu _ k ^ 2 } { \\lambda _ k ^ 2 } \\right ) = \\infty , \\end{align*}"}
{"id": "135.png", "formula": "\\begin{align*} \\psi : = ( C ^ 3 - { \\bf 1 } ^ \\epsilon _ 0 \\| C ^ 3 \\| _ { L ^ 1 } ) . \\end{align*}"}
{"id": "7541.png", "formula": "\\begin{align*} \\bar { G } ( z ) : = \\psi ' ( \\psi ^ { - 1 } z ) \\bar { g } ( \\psi ^ { - 1 } ( z ) ) + \\frac { 1 } { 2 } \\psi '' ( \\psi ^ { - 1 } ( z ) ) ( \\psi ' ( \\psi ^ { - 1 } ( z ) ) ) ^ { - 2 } \\int _ { M } v ^ 2 \\rm d \\mu . \\end{align*}"}
{"id": "8432.png", "formula": "\\begin{align*} d u \\left ( t \\right ) & + \\nu A u \\left ( t \\right ) d t + \\lambda ( r ( t ) ) u \\left ( t \\right ) d t = \\left ( { f \\left ( { t , r ( t ) , u ( t ) } \\right ) + g ( r ( t ) ) } \\right ) d t \\\\ & + \\varepsilon \\sum \\limits _ { k = 1 } ^ \\infty { \\left ( { h _ k ( r ( t ) ) + \\sigma _ k \\left ( { t , r ( t ) , u \\left ( t \\right ) } \\right ) } \\right ) } d W _ k \\left ( t \\right ) , t > s , \\end{align*}"}
{"id": "5470.png", "formula": "\\begin{align*} g _ { r , x } ( z ) & = \\left \\lbrace \\begin{matrix} g ( x ) & \\mathrm { d i s t } ( x , z ) \\leq \\frac { r } { 2 } \\\\ \\Phi _ x ^ { - 1 } \\left ( \\chi _ r ( \\mathrm { d i s t } ( x , z ) ) \\Phi _ x ( g ( z ) ) \\right ) & \\frac { r } { 2 } \\leq \\mathrm { d i s t } ( x , z ) \\leq r \\\\ g ( z ) & \\end{matrix} \\right . . \\end{align*}"}
{"id": "623.png", "formula": "\\begin{align*} f ( x , y , n , z ) ^ { - \\alpha } & = \\sum _ { m = 0 } ^ \\infty n ^ { - 2 m } \\sum _ { k + j = m } \\frac { F ' _ { k , j } ( x , y ) } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { \\alpha + j } } \\\\ & = \\sum _ { m = 0 } ^ { N - 1 } n ^ { - 2 m } \\sum _ { j = 0 } ^ m \\frac { F ' _ { m - j , j } ( x , y ) } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { \\alpha + j } } + n ^ { - 2 N } \\frac { G \\cdot ( x ^ 2 + y ^ 2 ) ^ N } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { \\alpha } } , \\end{align*}"}
{"id": "3425.png", "formula": "\\begin{align*} \\hat \\delta _ j \\cdot ( I _ 3 , n _ j ) = 1 , n _ j \\in \\Z . \\end{align*}"}
{"id": "4203.png", "formula": "\\begin{align*} \\norm { g _ { \\le \\iota } ^ { ( 1 ) } } _ p ^ p \\lesssim 2 ^ { - \\varepsilon \\iota } \\sum _ { \\ell = - 1 } ^ \\iota ( 2 ^ { \\ell ( d _ 1 - d _ 2 ) + \\iota ( d _ 2 - d _ 1 ) } ) ^ { p / q } \\norm { F ^ { ( \\iota ) } } _ { L ^ 2 _ s } ^ p \\norm { f } _ p ^ p \\end{align*}"}
{"id": "5341.png", "formula": "\\begin{align*} W ^ { s , p } ( \\Omega ) \\vcentcolon = \\{ \\ , u \\in W ^ { m , p } ( \\Omega ) \\ , ; \\ , \\partial ^ { \\alpha } u \\in W ^ { \\sigma , p } ( \\Omega ) \\forall | \\alpha | = m \\} , \\end{align*}"}
{"id": "8612.png", "formula": "\\begin{align*} \\bar { G } ( 0 ) = & - \\bar { C } ( \\bar { A } ) ^ { - 1 } \\bar { B } + \\bar { D } , \\\\ = & \\ \\bar { C } ( ( J - R ) Q ) ^ { - 1 } ( J - R ) \\bar { C } ^ T + \\bar { D } , \\\\ = & \\ \\bar { C } Q ^ { - 1 } \\bar { C } ^ T + \\bar { D } . \\end{align*}"}
{"id": "7225.png", "formula": "\\begin{align*} ( x , t ) \\in D _ { n } : = D _ { \\rm I n t } \\cap \\{ x > 0 \\} \\cap \\{ t - x > l _ n R \\} , l _ n : = \\sum _ { i = 0 } ^ n \\left ( \\frac { 1 } { 2 } \\right ) ^ i . \\end{align*}"}
{"id": "2596.png", "formula": "\\begin{align*} \\begin{cases} a = p _ 1 + p _ 2 - u - v - 2 s \\ , \\ \\ b = q _ 1 + q _ 2 - u - v + s \\ , \\ \\ \\mbox { f o r ( \\ref { p s i - 1 c a s e } ) } \\ , \\\\ a = p _ 1 + p _ 2 - u - v + s \\ , \\ \\ b = q _ 1 + q _ 2 - u - v - 2 s \\ , \\ \\ \\mbox { f o r ( \\ref { p s i - 2 c a s e } ) } \\ , \\end{cases} \\end{align*}"}
{"id": "7429.png", "formula": "\\begin{align*} E _ m ( t ) = E _ { m , 1 } ( t ) + E _ { m , 2 } ( t ) + E _ { m , 3 } ( t ) . \\end{align*}"}
{"id": "7480.png", "formula": "\\begin{align*} P _ { n _ { k + 1 } } = T _ { h \\left ( n _ { k + 1 } \\right ) } \\cdots T _ { h \\left ( n _ { k + 1 } + M + 1 \\right ) } T _ { l _ { M } } \\cdots T _ { l _ { 1 } } P _ { n _ { k } } . \\end{align*}"}
{"id": "5587.png", "formula": "\\begin{align*} \\forall j \\in \\{ 1 , \\ , \\ldots , \\ , k - 1 \\} \\ell _ j \\ = \\ d ( x _ j , \\ , x _ { j + 1 } ) \\ = \\ \\min _ { i > j } \\ , d ( x _ j , \\ , x _ i ) \\ , . \\end{align*}"}
{"id": "2420.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb P \\Big ( \\sup _ { t , x } | u ^ n ( t , x ) - u ( t , x ) | \\ge 1 \\Big ) = \\mathbb P \\Big ( \\sup _ { t , x } | u ^ n ( t , x ) - u ( t , x ) | ^ { 2 p } \\ge 1 \\Big ) = 0 . \\end{align*}"}
{"id": "8934.png", "formula": "\\begin{align*} g ( x , Y , Z ) = u , g _ x ( x , Y , Z ) = p . \\end{align*}"}
{"id": "4021.png", "formula": "\\begin{align*} P _ { W | X } ( 0 | x ) = \\begin{cases} 1 & z _ x > \\gamma \\\\ \\tau & z _ x = \\gamma \\\\ 0 & z _ x < \\gamma \\end{cases} , \\end{align*}"}
{"id": "1990.png", "formula": "\\begin{align*} \\bar A ( x , s ) & = A ( x _ 0 + \\Theta x , s ) \\\\ \\bar f ( x , s ) & = \\Phi \\Theta ^ 2 f ( x _ 0 + \\Theta x ) \\\\ \\bar Q ( x ) & = \\Phi ^ 2 \\Theta ^ 2 Q ( x _ 0 + \\Theta x ) . \\end{align*}"}
{"id": "8646.png", "formula": "\\begin{align*} - \\widehat { H } ' _ { n H } ( t ) = - e ^ { - \\alpha ( t ) } \\widehat { H } _ 1 + \\left ( \\frac { d \\alpha } { d t } ( t ) + 2 \\beta ( t ) \\right ) \\widehat { H } _ 2 + e ^ { \\alpha ( t ) } \\left ( \\frac { d \\beta } { d t } ( t ) - \\beta ^ 2 ( t ) - \\omega ^ 2 ( t ) \\right ) \\widehat { H } _ 3 - e ^ { \\frac { k \\alpha ( t ) } 2 } c ( t ) \\widehat { H } _ 4 , \\end{align*}"}
{"id": "6173.png", "formula": "\\begin{align*} K _ { \\overline { M } } ^ { T } = \\psi ^ { \\ast } ( K _ { M } ^ { T } ) + \\sum _ { i } a _ { i } [ V _ { i } ] _ { B } , \\end{align*}"}
{"id": "4656.png", "formula": "\\begin{align*} f = \\lambda \\tilde R _ \\lambda f - \\tilde R _ \\lambda q f . \\end{align*}"}
{"id": "2614.png", "formula": "\\begin{align*} a = p + q - n - m - 2 k \\ , \\ \\ 0 \\le n \\le p - k \\ , \\ \\ 0 \\le m \\le q - k \\ , \\end{align*}"}
{"id": "1076.png", "formula": "\\begin{align*} R ( \\mu , A _ 2 ) & - R ( \\mu , A _ 1 ) \\\\ & = \\left ( 1 + ( w - x ) R ( \\lambda , A _ 2 ) \\right ) ^ { - 1 } \\left ( R ( \\lambda , A _ 2 ) - R ( \\lambda , A _ 1 ) \\right ) \\left ( 1 + ( x - w ) R ( \\mu , A _ 1 ) \\right ) , \\end{align*}"}
{"id": "6428.png", "formula": "\\begin{align*} \\Bar { \\Psi } ^ { ( 1 ) } - \\Xi _ 1 ^ { ( 1 ) } = [ Q , H ^ { ( 1 ) } ] , \\ , \\ ; H ^ { ( 1 ) } \\colon S _ \\mathbb { K } ^ { 2 } ( \\mathfrak { g } [ 1 ] ) \\longrightarrow \\mathfrak { X } _ { - 2 } ( E ) [ 1 ] . \\end{align*}"}
{"id": "4616.png", "formula": "\\begin{align*} c _ 1 ( \\alpha , h ) = r s \\ ; \\ ; \\ ; \\ ; \\ ; c _ 2 ( \\alpha , h ) = t u \\end{align*}"}
{"id": "4127.png", "formula": "\\begin{align*} A ^ { i } : = I - G ^ { i + 1 } d _ { V } ^ { i } . \\end{align*}"}
{"id": "1131.png", "formula": "\\begin{align*} C ( \\lambda w ) = C \\left ( \\lambda \\sqrt { w ^ 2 } \\right ) = \\ln \\left [ \\delta \\cdot \\cosh \\left ( z _ 0 \\lambda \\sqrt { w ^ 2 } \\right ) + \\frac { 1 - \\delta } { 1 - \\lambda ^ 2 w ^ 2 / q ^ 2 } \\right ] , \\end{align*}"}
{"id": "5846.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { k = N } ^ M \\bigg ( \\sum _ { i = N } ^ k x _ i b _ i \\bigg ) ^ q a _ k \\bigg ) ^ { \\frac { 1 } { q } } \\leq C \\bigg ( \\sum _ { k = N } ^ M x _ k ^ p \\bigg ) ^ { \\frac { 1 } { p } } \\end{align*}"}
{"id": "8526.png", "formula": "\\begin{align*} a _ { m + 1 } = \\frac { 4 ^ { - m - 1 } ( - 1 ) ^ m } { ( 2 m + 1 ) a _ m a _ { m - 1 } \\cdots a _ 1 } \\int _ { - 1 } ^ 1 \\phi _ m ( t ) \\frac { t ^ 2 } { 2 } d t . \\end{align*}"}
{"id": "1102.png", "formula": "\\begin{align*} g ^ f _ { k \\rho } ( k X , k X ) = k g ^ f _ \\rho ( X , X ) . \\end{align*}"}
{"id": "8678.png", "formula": "\\begin{align*} \\sigma _ { B H } ( x ) = \\frac { v o l ( B ^ n ( 1 ) ) } { v o l \\left \\lbrace ( y ^ i ) \\in T _ x M : F ( x , y ) < 1 \\right \\rbrace } . \\end{align*}"}
{"id": "319.png", "formula": "\\begin{align*} \\sum _ { 2 < p \\le 2 3 } \\frac { 1 } { m _ p ^ 2 } { \\rm d } ( { \\rm L } _ { p } ) & = \\frac { 1 } { \\mu _ 7 ^ 2 } \\sum _ { 2 < p \\le 7 } { \\rm d } ( { \\rm L } _ { p } ) + \\frac { 1 } { \\mu _ { 1 9 } ^ 2 } \\sum _ { 7 < p \\le 1 9 } { \\rm d } ( { \\rm L } _ { p } ) + \\frac { 1 } { \\mu _ { 2 3 } ^ 2 } { \\rm d } ( { \\rm L } _ { 2 3 } ) = 0 . 3 9 0 1 2 6 \\cdots , \\end{align*}"}
{"id": "2814.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\big \\{ \\| \\nabla f ( x _ i ) \\| ^ 2 \\big \\} \\leq { } \\frac { 2 L \\ , \\big [ f ( x _ 0 ) - f _ * \\big ] } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left ( 2 h _ i - h _ i ^ 2 \\right ) } \\ , , \\ , \\forall h _ i \\in ( 0 , 2 ) . \\end{align*}"}
{"id": "8627.png", "formula": "\\begin{align*} \\mathbb { Y } : = \\{ ( I , R ) \\ | \\ 0 < I \\leq 1 , 0 \\leq R \\leq 1 - I \\} \\times \\mathbb { X } \\times \\mathbb { R } \\end{align*}"}
{"id": "7189.png", "formula": "\\begin{align*} & ( u ' \\ast _ { g _ 1 g _ 2 , g _ 2 } w _ 1 ) \\ast _ { g _ 2 , g _ 1 g _ 2 } a \\intertext { b y L e m m a \\ref { l 3 . 4 } , } \\in & u ' \\ast _ { g _ 1 g _ 2 , g _ 2 } ( w _ 1 \\ast _ { g _ 2 , g _ 1 g _ 2 } a ) + O ' _ { g _ 1 g _ 2 , g _ 2 } ( M ^ 1 ) \\\\ \\subseteq & O '' _ { g _ 1 g _ 2 , g _ 2 } ( M ^ 1 ) + O ' _ { g _ 1 g _ 2 , g _ 2 } ( M ^ 1 ) \\\\ = & O _ { g _ 1 g _ 2 , g _ 2 } ( M ^ 1 ) \\end{align*}"}
{"id": "6253.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { x } f ( t ) h ( t / q ) \\left ( - \\frac { q ( 1 + q ) + t } { q ^ 2 ( 1 - q ) t } u ( t / q ) + \\frac { 1 } { q ( 1 - q ) ^ 2 t } \\right ) y ( t ) \\ , d _ q t \\\\ & = \\frac { f ( x ) } { q ( 1 - q ) } \\sum _ { n = 0 } ^ { \\infty } ( - q ) ^ n \\left ( - 1 + q ^ 2 + q ^ n x \\right ) y ( q ^ n x ) = H ( x ) . \\end{align*}"}
{"id": "971.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\tau \\left ( \\theta ^ { k } \\right ) = \\overline { e } _ { k + 1 } , \\ \\ 0 \\leq k \\leq n - 1 \\\\ H ^ { * } \\left ( \\overline { e } _ { k } \\right ) = H ^ { k - 1 } , \\ \\ 1 \\leqslant k \\leqslant n . \\end{array} \\right . \\end{align*}"}
{"id": "2173.png", "formula": "\\begin{align*} d ( \\mu \\boxplus \\nu ) _ { } = p _ { \\mu \\boxplus \\nu } \\ , d \\lambda . \\end{align*}"}
{"id": "1336.png", "formula": "\\begin{align*} f _ \\infty ( x , v ) = \\frac { 1 } { Z } e ^ { - \\Phi ( x ) } , \\end{align*}"}
{"id": "8469.png", "formula": "\\begin{align*} \\left [ \\frac { 2 \\sqrt { 2 L } } { 3 } \\right ] ^ d - \\left [ \\frac { 2 \\sqrt { 2 L } } { 3 } - \\frac { L ^ 2 } { 1 2 b ^ 2 } \\right ] ^ d & \\leq d \\frac { L ^ 2 } { 1 2 b ^ 2 } \\left [ \\frac { 2 \\sqrt { 2 L } } { 3 } \\right ] ^ { d - 1 } \\\\ & = \\frac { d L ^ { \\frac { d + 3 } { 2 } } } { 1 2 b ^ 2 } \\left [ \\frac { \\sqrt { 8 } } { 3 } \\right ] ^ { d - 1 } . \\end{align*}"}
{"id": "237.png", "formula": "\\begin{align*} \\delta ^ { ( s ) } u = f ^ { ( 1 ) } ( u , u ^ { \\phi ^ s } ) \\end{align*}"}
{"id": "523.png", "formula": "\\begin{align*} L _ { t } ( \\dot { x } , x , t ] = L _ { n s } ( \\dot { x } , x , t ) - \\frac { \\partial \\Phi _ { n u l l } ( x , t ) } { \\partial t } \\ , \\end{align*}"}
{"id": "4750.png", "formula": "\\begin{align*} u _ b ^ { ( n + 1 ) } = \\frac { 1 } { 2 ^ g } \\sum _ { b _ 1 + b _ 2 = b } v _ { b _ 1 } ^ { ( n ) } t _ { b _ 2 } ^ { ( n ) } s _ b ^ { ( n + 1 ) } = \\frac { 1 } { 2 ^ g } \\sum _ { b _ 1 + b _ 2 = b } t _ { b _ 1 } ^ { ( n ) } t _ { b _ 2 } ^ { ( n ) } . \\end{align*}"}
{"id": "3635.png", "formula": "\\begin{align*} u ( x _ 1 , x ' ) = u ( y _ 1 , x ' ) + \\int _ { y _ 1 } ^ { x _ 1 } \\frac { \\partial u } { \\partial x _ 1 } ( t , x ' ) \\dd t . \\end{align*}"}
{"id": "5662.png", "formula": "\\begin{align*} F l _ n = \\{ V _ { \\bullet } = ( V _ 1 \\subset V _ 2 \\subset \\cdots \\subset V _ n = \\C ^ n ) \\mid \\dim _ { \\C } V _ i = i \\ ( 1 \\le i \\le n ) \\} . \\end{align*}"}
{"id": "2661.png", "formula": "\\begin{align*} d = \\frac { a _ i ^ 2 - a _ i } { b _ i ^ 2 - b _ i } . \\end{align*}"}
{"id": "4977.png", "formula": "\\begin{align*} E \\left [ Z ^ n _ t - \\widetilde { Z } ^ n _ t \\right ] & = n ^ { 2 \\alpha + 1 } E \\left [ \\left | \\int ^ t _ 0 ( t - s ) ^ { \\alpha } \\left ( \\sigma ' ( X _ s ) - \\Psi ^ n _ s \\right ) \\left ( X ^ n _ { s } - X _ s \\right ) \\ , d W _ s \\right | ^ 2 \\right ] \\\\ & \\leq n ^ { 2 \\alpha + 1 } \\int ^ t _ 0 ( t - s ) ^ { \\alpha } \\left ( E [ ( \\sigma ' ( X _ s ) - \\Psi ^ n _ s ) ^ 4 ] \\right ) ^ { \\frac 1 2 } \\left ( E [ ( X ^ n _ { s } - X _ s ) ^ 4 ] \\right ) ^ { \\frac 1 2 } \\ , d s \\\\ & \\leq C n ^ { - \\beta ( 2 \\alpha + 1 ) } . \\end{align*}"}
{"id": "8400.png", "formula": "\\begin{gather*} \\# \\mathcal { Q } _ N ^ 0 ( D ) / \\Gamma _ 0 ( N ) = \\left ( n _ D ( N ) + n _ { \\frac { D } { N ^ 2 } } ( 1 ) \\right ) \\# \\mathcal { Q } _ 1 ^ 0 ( D ) / \\Gamma _ 0 ( 1 ) \\end{gather*}"}
{"id": "4565.png", "formula": "\\begin{align*} \\mathbf { c } + \\left ( \\Z ^ { 1 + n } \\cap C \\right ) = \\Z ^ { 1 + n } \\cap C ^ \\circ , \\end{align*}"}
{"id": "943.png", "formula": "\\begin{align*} p ( 0 ) = 0 , p ' ( \\rho ) > 0 \\ \\forall \\ \\rho > 0 , \\liminf _ { \\rho \\rightarrow 0 } \\frac { p ' ( \\rho ) } { \\rho } > 0 , p ( \\rho ) \\sim _ { \\rho \\rightarrow \\overline { \\rho } - } | \\overline { \\rho } - \\rho | ^ { - \\beta } , \\mbox { f o r s o m e } \\beta > 5 / 2 , \\end{align*}"}
{"id": "3364.png", "formula": "\\begin{align*} F _ m ( \\varphi ) ( ( x _ j , y _ j ) _ { 1 \\leq j \\leq K } ) = \\left ( \\sum _ { j = 1 } ^ K a _ { i , j } x _ j , \\sum _ { j = 1 } ^ K b _ { i , j } y _ j \\right ) _ { 1 \\leq i \\leq L } \\end{align*}"}
{"id": "6018.png", "formula": "\\begin{align*} g _ { j + 1 } ( \\lambda ) = { \\mu _ j } g _ j ( \\lambda ) \\neq 0 . \\end{align*}"}
{"id": "1042.png", "formula": "\\begin{align*} \\psi ( t + s ) = \\| \\tilde { B } T ( t + s , A ) \\| _ q \\leq \\| \\tilde { B } T ( t , A ) \\| _ q \\| T ( s , A ) \\| _ \\infty = \\psi ( t ) \\phi ( s ) . \\end{align*}"}
{"id": "4076.png", "formula": "\\begin{align*} P _ 0 ( D _ j ) = \\sum \\limits _ { k = 0 } ^ m P _ 0 ( D _ j \\cap D ' _ k ) . \\end{align*}"}
{"id": "7866.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| P _ k ( \\alpha _ { \\pi ( t ) } \\phi _ n ( T ) \\alpha _ { \\pi ( t ^ { - 1 } ) } - \\phi _ n ( \\alpha _ { \\pi ( t ) \\oplus \\pi ( t ) } T \\alpha _ { \\pi ( t ^ { - 1 } ) \\oplus \\pi ( t ^ { - 1 } ) } ) ) P _ k \\| = 0 . \\end{align*}"}
{"id": "381.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } n s P _ s ( o , o ) e ^ { - \\frac { s } { t _ { 1 } } } \\mathrm { d } s \\le t _ 1 ^ { 2 } + \\sum _ { i = 2 } ^ { n } \\beta _ i ^ { - 2 } . \\end{align*}"}
{"id": "4628.png", "formula": "\\begin{align*} c _ i ( \\alpha , g ) = c _ i ( \\alpha , h ) \\mbox { f o r } i = 1 , 2 , \\end{align*}"}
{"id": "5219.png", "formula": "\\begin{align*} A _ { I , \\mathrm { s y m } } : = \\Q [ t _ { \\alpha , \\beta , d } \\ , | \\ , 0 \\le \\alpha \\le r - 2 , 0 \\le \\beta \\le s - 2 , d \\in \\Z _ { \\ge 0 } ] / \\mathrm { I d e a l } ( I ) , \\end{align*}"}
{"id": "1914.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( q ) } _ { [ n , j ] } & : = \\{ \\gamma + q : \\gamma \\in \\mathcal { S } _ { [ n , j ] } \\} , \\\\ \\widehat { \\mathcal { S } } ^ { ( q ) } _ { [ n , j ] } & : = \\{ \\gamma - q : \\gamma \\in \\widehat { \\mathcal { S } } _ { [ n , j ] } \\} , \\end{align*}"}
{"id": "5779.png", "formula": "\\begin{align*} E ( G / H ) = E ( ( G / G ^ 0 ) / ( H / A ) ) \\neq 0 \\mod p \\end{align*}"}
{"id": "2573.png", "formula": "\\begin{align*} \\mathbf S _ { 3 1 } = - \\mathbf S _ { 1 3 } = - \\dfrac { 1 } { 2 } ( C _ 3 ^ { 1 2 2 } + 3 C _ 3 ^ { 2 1 1 } + 2 C _ 3 ^ { ( 1 ) } - C _ 2 ^ { 2 1 } ) \\ . \\end{align*}"}
{"id": "6685.png", "formula": "\\begin{align*} k = h = \\dfrac { 1 } { p - 1 } \\end{align*}"}
{"id": "1957.png", "formula": "\\begin{align*} ( x _ { 1 } , \\ldots , x _ { p } ) \\cdot ( y _ { 1 } , \\ldots , y _ { p } ) : = ( x _ { 1 } \\ , y _ { 1 } , \\ldots , x _ { p } \\ , y _ { p } ) . \\end{align*}"}
{"id": "2273.png", "formula": "\\begin{align*} J _ 2 = & \\int _ 0 ^ r - \\frac { z ^ 3 } { 2 } ( \\bar \\psi \\bar \\psi ' \\cdot \\phi + \\bar \\psi \\bar \\phi ' \\cdot \\psi ) { \\rm { d } } z \\\\ \\leq & \\Vert z ( \\phi , \\psi ) \\Vert _ { L ^ 2 } \\cdot \\Vert z ( \\bar \\phi ' , \\bar \\psi ' ) \\Vert _ { L ^ 2 } \\cdot \\frac { 1 } { 2 } \\Vert z \\bar \\psi \\Vert _ { L ^ \\infty } , \\end{align*}"}
{"id": "3550.png", "formula": "\\begin{align*} \\delta = \\begin{cases} 0 , & , \\\\ 1 , & . \\end{cases} \\end{align*}"}
{"id": "4135.png", "formula": "\\begin{align*} D = P _ { \\Upsilon } d _ { V } A = \\left ( \\begin{array} { c c c c c c } P _ { \\Upsilon } d & 0 & 0 & 0 & \\cdots & 0 \\\\ P _ { \\Upsilon } d T d & P _ { \\Upsilon } d & 0 & 0 & \\cdots & 0 \\\\ P _ { \\Upsilon } ( d T ) ^ { 2 } d & P _ { \\Upsilon } d T d & P _ { \\Upsilon } d & 0 & \\cdots & 0 \\\\ P _ { \\Upsilon } ( d T ) ^ { 3 } d & P _ { \\Upsilon } ( d T ) ^ { 2 } d & P _ { \\Upsilon } d T d & P _ { \\Upsilon } d & \\cdots & 0 \\\\ & \\cdots & \\cdots & \\cdots & & \\end{array} \\right ) . \\end{align*}"}
{"id": "8347.png", "formula": "\\begin{align*} \\widetilde { \\varphi } = { w _ 1 } ^ + \\tilde { f } _ + + { w _ 1 } ^ - \\tilde { f } _ - . \\end{align*}"}
{"id": "5642.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\nabla \\cdot ( m \\nabla \\varphi ) = 1 \\R ^ d \\end{align*}"}
{"id": "1753.png", "formula": "\\begin{align*} Z ( \\overline \\delta , \\overline \\mu , \\beta ) = \\frac { \\beta } { \\sqrt { \\overline \\mu - \\frac { 1 } { 2 } ( 1 - \\beta ) } } \\frac { _ 3 \\big ( \\frac { 1 } { 2 } ( \\sqrt { \\beta } + \\frac { 1 } { \\sqrt { \\beta } } ) + \\overline \\delta \\overline \\mu \\ , | \\ , \\frac { 1 } { \\sqrt { \\beta } } , \\sqrt { \\beta } , \\overline \\delta \\big ) } { _ 3 \\big ( \\sqrt { \\beta } \\ , | \\ , \\frac { 1 } { \\sqrt { \\beta } } , \\sqrt { \\beta } , \\overline \\delta \\big ) } , \\end{align*}"}
{"id": "8600.png", "formula": "\\begin{align*} \\dot { x } ( t ) = & A x ( t ) + B u ( t ) , \\\\ y ( t ) = & \\ ; C x ( t ) + D u ( t ) , \\end{align*}"}
{"id": "3242.png", "formula": "\\begin{align*} & \\nu _ p ( \\{ ( a , r ) \\in \\omega \\times \\R _ + : \\ c _ d ' r | \\nabla f ( a ) | + C A r ^ 2 > \\kappa \\} ) \\\\ & = \\frac { c _ { d , p } } { \\kappa ^ p } \\int _ { \\omega } \\left ( \\sqrt { \\tfrac { C A } { c _ d '^ 2 } \\kappa + \\left ( \\tfrac 1 2 | \\nabla f ( a ) | \\right ) ^ 2 } + \\tfrac 1 2 | \\nabla f ( a ) | \\right ) ^ p d a \\ , . \\end{align*}"}
{"id": "3715.png", "formula": "\\begin{align*} w & : = v _ 1 \\Uparrow u = v _ 2 \\Uparrow u , \\\\ w ' & : = v ' _ 1 \\Uparrow u ' = v ' _ 2 \\Uparrow u ' , \\\\ y _ 1 & : = v _ 1 \\Uparrow x _ 1 = v ' _ 1 \\Uparrow x _ 1 , \\\\ y _ 2 & : = v _ 2 \\Uparrow x _ 2 = v ' _ 2 \\Uparrow x _ 2 . \\end{align*}"}
{"id": "2472.png", "formula": "\\begin{align*} w \\cdot \\lambda : = w \\left ( \\lambda + \\rho \\right ) - \\rho . \\end{align*}"}
{"id": "7201.png", "formula": "\\begin{align*} \\langle f , I ^ { \\circ } ( w _ 1 , z _ 2 ) u _ { k + \\frac { j _ 2 } { T } } w _ 2 \\rangle = 0 . \\end{align*}"}
{"id": "5194.png", "formula": "\\begin{align*} e _ 1 = \\frac { 2 \\sum a _ i + ( k _ 1 + k _ { 1 2 } - 1 ) ( r - 2 ) } { r } \\in \\Z , ~ ~ e _ 2 = \\frac { 2 \\sum b _ i + ( k _ 2 + k _ { 1 2 } - 1 ) ( s - 2 ) } { s } \\in \\Z , \\end{align*}"}
{"id": "6249.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { x } f ( t ) h ( t / q ) \\left ( - \\frac { q ( 1 + q ) + t } { q ^ 2 ( 1 - q ) t } u ( t / q ) + \\frac { 1 } { q ( 1 - q ) ^ 2 t } \\right ) y ( t ) d _ q t = H ( x ) - \\lim _ { n \\rightarrow \\infty } H ( q ^ n x ) . \\end{align*}"}
{"id": "3363.png", "formula": "\\begin{align*} F _ m ( \\varphi ) : \\bigoplus _ { j = 1 } ^ K V _ { n _ j } ^ m \\to \\bigoplus _ { i = 1 } ^ L V _ { \\ell _ i } ^ m \\end{align*}"}
{"id": "8255.png", "formula": "\\begin{align*} \\left ( \\frac { 2 \\beta \\hslash ^ { 2 } } { 3 a ^ { 2 } } \\right ) ^ { 2 } . \\left ( \\frac { \\lambda ^ { 2 } - 1 } { \\lambda } \\right ) ^ { 2 } = 1 . \\end{align*}"}
{"id": "179.png", "formula": "\\begin{align*} 0 \\leq \\mathcal { X } \\in C _ \\textup { b } ^ \\infty ( \\R ^ N ) \\qquad \\mathcal { X } ( x ) = \\begin{cases} 0 & | x | \\leq \\frac { 1 } { 2 } , \\\\ 1 & | x | \\geq 1 . \\end{cases} \\end{align*}"}
{"id": "5293.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Xi _ { 0 } } e ^ { W / \\hbar } \\Omega & = 1 + ( y _ 0 - \\tfrac 1 8 y _ 2 ^ 2 ) \\hbar ^ { - 1 } + O ( \\hbar ^ { - 2 } ) \\\\ \\int _ { \\Xi _ { 1 } } e ^ { W / \\hbar } \\Omega & = y _ 1 \\hbar ^ { - 1 } + O ( \\hbar ^ { - 2 } ) \\\\ \\int _ { \\Xi _ { 2 } } e ^ { W / \\hbar } \\Omega & = y _ 2 \\hbar ^ { - 1 } + O ( \\hbar ^ { - 2 } ) \\end{aligned} \\end{align*}"}
{"id": "2391.png", "formula": "\\begin{align*} 1 = \\| C _ { \\sigma _ 2 } \\psi \\| _ { L ^ 2 ( \\widehat { \\R ^ 2 } \\times \\widehat { \\R } ) } = \\| r T \\psi \\| _ { L ^ 2 ( \\widehat { \\R ^ 2 } \\times \\widehat { \\R } ) } = r \\| T \\psi \\| _ { L ^ 2 ( \\widehat { \\R ^ 2 } \\times \\widehat { \\R } ) } = r . \\end{align*}"}
{"id": "8829.png", "formula": "\\begin{align*} f _ t \\circ ( \\phi _ 1 ^ * * \\phi _ 2 ^ * ) ( x ) = \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } f _ { T ( t _ 1 , t _ 2 ) } \\circ ( 1 _ { ( - J _ 1 ( t _ 1 ) , J _ 1 ( t _ 1 ) ) } * 1 _ { ( - J _ 2 ( t _ 2 ) , J _ 2 ( t _ 2 ) ) } ) ( x ) d t _ 1 d t _ 2 \\end{align*}"}
{"id": "7357.png", "formula": "\\begin{align*} \\frac { \\partial K ( \\delta , r ) } { \\partial \\delta } & = \\frac { 2 } { ( 1 + r ) ^ 2 } \\left ( \\frac { 1 - r ^ 2 } { 2 } - \\frac { k _ { \\phi } ( r ) } { k ' _ { \\phi } ( r ) } \\right ) \\\\ & = \\frac { 2 } { ( 1 + r ) ^ 2 } \\left ( \\frac { 1 - r ^ 2 } { 2 } - \\frac { r } { \\psi ( - r ) } \\right ) \\\\ & < 0 \\end{align*}"}
{"id": "7260.png", "formula": "\\begin{align*} u _ t = \\Delta u - | u | ^ { q - 1 } u . \\end{align*}"}
{"id": "6341.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { N } ^ s _ { p ( x , . ) } u ( x ) = \\int _ { \\Omega } \\dfrac { | u ( x ) - u ( y ) | ^ { p ( x , y ) - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { N + s p ( x , y ) } } d y , ~ ~ \\forall x \\in \\R ^ N \\setminus \\Omega , \\end{aligned} \\end{align*}"}
{"id": "6995.png", "formula": "\\begin{align*} L = L ^ 0 + \\hat L , \\end{align*}"}
{"id": "2727.png", "formula": "\\begin{align*} \\langle H _ N \\rangle _ { \\psi _ \\mathrm { g s } } \\ge \\ ; & E _ { 2 - \\mathrm { m o d e } } + \\frac { c \\lambda } { N } \\big \\langle ( \\mathcal { N } _ 1 - \\mathcal { N } _ 2 ) ^ 2 \\big \\rangle _ { \\psi _ \\mathrm { g s } } + E ^ \\mathrm { B o g } \\\\ & - C _ \\Lambda o _ N ( 1 ) - \\frac { C } { ( \\mu _ { 2 M _ \\Lambda + 2 } - \\mu _ + ) } ^ { 1 / 2 } , \\end{align*}"}
{"id": "8061.png", "formula": "\\begin{align*} \\frac { f ^ { 2 } } { 2 } \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) = \\operatorname { g r a d } p \\end{align*}"}
{"id": "5990.png", "formula": "\\begin{align*} f ( t ) = \\lambda u ( t ) + \\Gamma ( \\beta + 1 ) ( 1 + H ( t - r ) ) . \\end{align*}"}
{"id": "1104.png", "formula": "\\begin{align*} G _ { ( t , \\rho ) } \\left ( \\frac { \\partial } { \\partial z } , \\frac { \\partial } { \\partial z } \\right ) = t ^ 2 G _ { ( 1 , \\rho ) } \\left ( \\frac { \\partial } { \\partial z } , \\frac { \\partial } { \\partial z } \\right ) . \\end{align*}"}
{"id": "8266.png", "formula": "\\begin{align*} k \\tan ( k a / 2 ) + k ' \\tanh ( k ' a / 2 ) = 0 , \\end{align*}"}
{"id": "5251.png", "formula": "\\begin{align*} ( I ^ { Q ^ { - } } _ j , k _ 1 ^ { Q ^ { - } } ( j ) , k _ 2 ^ { Q ^ { - } } ( j ) ) = \\begin{cases} ( I ^ { Q } _ j , k _ 1 ^ { Q } ( j ) , k _ 2 ^ { Q } ( j ) ) , ~ ~ \\\\ ( I ^ { Q } _ { j + 1 } , k _ 1 ^ { Q } ( j + 1 ) , k _ 2 ^ { Q } ( j + 1 ) ) , ~ ~ . \\end{cases} \\end{align*}"}
{"id": "4808.png", "formula": "\\begin{align*} n ^ { - 1 } \\sum _ { k = 1 } ^ n \\zeta _ { n . k } ^ 2 & = n ^ { - 1 } \\sum _ { k = 1 } ^ n ( [ \\mathcal Z _ n ] _ k - Q ( u _ { n , k } ) ) ^ 2 = \\int _ 0 ^ 1 ( Q ^ * _ n ( u ) - Q _ n ( u ) ) ^ 2 d u \\\\ & \\le 2 \\int _ 0 ^ 1 ( Q ^ * _ n ( u ) - Q ( u ) ) ^ 2 d u + 2 \\int _ 0 ^ 1 ( Q ( u ) - Q _ n ( u ) ) ^ 2 d u . \\end{align*}"}
{"id": "2688.png", "formula": "\\begin{align*} H _ N = \\sum _ { m , n \\ge 1 } h _ { m n } a ^ * _ m a _ n + \\frac { \\lambda } { 2 ( N - 1 ) } \\sum _ { m , n , p , q \\ge 1 } w _ { m n p q } a ^ * _ m a ^ * _ n a _ p a _ q , \\end{align*}"}
{"id": "3343.png", "formula": "\\begin{align*} s _ 2 = { } ^ { t } s _ 1 ^ { - 1 } \\end{align*}"}
{"id": "6004.png", "formula": "\\begin{align*} [ x _ 0 , \\dots , x _ n ] ( f g ) = \\sum _ { k = 0 } ^ n [ x _ 0 , \\dots , x _ k ] f [ x _ k , \\dots , x _ n ] g . \\end{align*}"}
{"id": "5133.png", "formula": "\\begin{align*} Z _ g & = \\# ^ g S ^ n \\times S ^ { n + 1 } , \\\\ Z _ { g , 1 } & = Z _ g \\setminus \\operatorname { i n t } D ^ { 2 n + 1 } , \\end{align*}"}
{"id": "2206.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\inf _ { d _ X ( x , x ^ * ) \\leq \\rho } C ( J f ( x ) ) d \\rho = \\infty . \\end{align*}"}
{"id": "590.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\abs { \\frac { H _ n ( 1 - s ) } { H _ n ( s ) } } = 1 . \\end{align*}"}
{"id": "3764.png", "formula": "\\begin{align*} H ^ { S ; U ; m , i } _ { j , n , l , r } ( t , x , \\zeta ) = \\int _ 0 ^ t \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } ( t - s ) \\big ( E ( s , x + ( t - s ) \\omega ) + \\hat { v } \\times B ( s , x + ( t - s ) \\omega ) \\big ) \\cdot \\nabla _ v \\big ( \\frac { m _ { U } ( v , \\omega ) \\varphi _ { j , n } ^ { i ; r } ( v , \\zeta ) \\varphi _ { l ; r } ( \\tilde { v } + \\omega ) } { 1 + \\hat { v } \\cdot \\omega } \\big ) \\end{align*}"}
{"id": "5349.png", "formula": "\\begin{align*} 0 = ( q _ 1 - q _ 2 ) ( u _ f , u _ g ^ * ) . \\end{align*}"}
{"id": "1352.png", "formula": "\\begin{align*} \\Gamma _ p : = \\{ \\tau - \\tau _ 1 + \\cdots - \\tau _ p = 0 , \\ , \\ , k - k _ 1 + \\cdots - k _ p = 0 \\} , \\end{align*}"}
{"id": "6138.png", "formula": "\\begin{align*} r = | A ' _ 1 | = | A ' _ 1 \\cap H _ 2 | + | A ' _ 1 \\cap H _ 4 | \\leq r _ 2 + r _ 4 = | A ' _ 2 \\cap H _ 2 | + | A ' _ 2 \\cap H _ 4 | = | A ' _ 2 | = r . \\end{align*}"}
{"id": "3155.png", "formula": "\\begin{align*} \\Gamma \\left ( \\Delta _ { \\mathfrak { a } } \\right ) \\left ( \\rho \\right ) = \\int _ { \\mathbb { S } } \\left \\vert \\rho \\left ( \\mathfrak { e } _ { \\Psi } \\right ) \\right \\vert ^ { 2 } \\mathfrak { a } _ { + } \\left ( \\mathrm { d } \\Psi \\right ) , \\rho \\in E _ { 1 } \\ . \\end{align*}"}
{"id": "7295.png", "formula": "\\begin{align*} \\tilde { v } _ 1 ( \\xi , t ) & = v _ 1 ( \\xi , t ) x \\in \\overline { B } _ { { \\sf R } _ { \\sf m i d } } \\times [ 0 , T - \\delta ] , \\\\ | \\tilde { v } _ 1 ( x , t ) | & \\leq ( T - t ) ^ { { \\sf d } _ 1 } ( 1 + | \\xi | ^ 2 ) ^ \\frac { \\gamma } { 2 } ( x , t ) \\in \\R ^ n \\times [ 0 , T ] , \\\\ \\tilde { v } _ 1 ( x , t ) & = 0 ( x , t ) \\in \\R ^ n \\times [ T - \\tfrac { 1 } { 2 } \\delta , T ] . \\end{align*}"}
{"id": "6252.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { x } f ( t ) h ( t / q ) \\left ( - \\frac { q ( 1 + q ) + t } { q ^ 2 ( 1 - q ) t } u ( t / q ) + \\frac { 1 } { q ( 1 - q ) ^ 2 t } \\right ) y ( t ) \\ , d _ q t \\\\ & = \\frac { f ( x ) } { c q ( 1 - q ) } \\sum _ { n = 0 } ^ { \\infty } ( - q ) ^ n \\left ( c - 1 + q ^ 2 + q ^ n x \\right ) y ( q ^ n x ) = H ( x ) . \\end{align*}"}
{"id": "2201.png", "formula": "\\begin{align*} \\mathbf { f } = \\psi \\circ f \\circ \\varphi ^ { - 1 } : \\varphi ( W ) \\rightarrow \\psi ( V ) \\end{align*}"}
{"id": "3229.png", "formula": "\\begin{align*} h ^ { - 1 } ( \\mathcal { D } _ x ) \\cap T _ { ( E _ * , \\psi ) } ^ * \\mathcal { M } _ { \\textnormal { S p } } ( 2 m , \\alpha , L ) = \\bigcup _ { i = 1 } ^ { 2 m } \\tilde { N } _ i ( E _ * ) \\end{align*}"}
{"id": "8512.png", "formula": "\\begin{align*} \\dim ( C ) + \\dim ( C ^ \\perp ) = n \\omega . \\end{align*}"}
{"id": "1640.png", "formula": "\\begin{align*} \\mathcal { X } \\cap [ \\emptyset , N ] = \\bigcup _ { j < \\tilde { j } } ( \\mathcal { X } \\cap [ B _ j , N ] ^ * ) . \\end{align*}"}
{"id": "6020.png", "formula": "\\begin{align*} \\Lambda = ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ m ) \\end{align*}"}
{"id": "4787.png", "formula": "\\begin{align*} \\{ [ V _ { H _ f } / N _ G ( f ) ] \\} = \\{ B N _ G ( f ) \\} \\L ^ { d _ f } - \\sum _ { f ' } \\{ [ V _ { H _ { f ' } } / N _ G ( f ' ) ] \\} , \\end{align*}"}
{"id": "6527.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { \\infty } \\frac { 1 } { \\alpha _ i } \\sum \\limits _ { k : k \\geq 0 , k \\leq \\alpha _ i } \\mu ( k ) k \\leq c _ { \\alpha } , \\end{align*}"}
{"id": "7900.png", "formula": "\\begin{align*} E : = c _ 1 ( T _ X ( - \\log D ) ) + \\sum _ { \\vec s , k } \\left ( 1 - \\frac 1 2 \\deg ( \\tilde { T } _ { \\vec s , k } ) - \\# \\{ i : s _ i < 0 \\} \\right ) t _ { \\vec s , k } \\tilde { T } _ { \\vec s , k } \\end{align*}"}
{"id": "357.png", "formula": "\\begin{align*} S ( \\delta _ { } ) = o ( 1 ) . \\end{align*}"}
{"id": "7010.png", "formula": "\\begin{align*} \\langle \\chi , \\Lambda ( \\rho _ { i _ 1 } ) \\rangle = 1 , \\ , \\langle \\chi , \\Lambda ( \\rho _ { i _ 2 } ) \\rangle = 0 , \\ , \\ldots , \\ , \\langle \\chi , \\Lambda ( \\rho _ { i _ s } ) \\rangle = 0 . \\end{align*}"}
{"id": "5006.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } n ^ { 2 \\alpha + 1 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' ( X _ s ) ) ^ 2 \\Psi ^ { n , 2 } _ s d s \\\\ & = \\frac 1 { 2 \\alpha + 2 } \\int _ 0 ^ { \\tau \\wedge t _ 1 \\wedge t _ 2 } ( t _ 1 - s ) ^ { \\alpha } ( t _ 2 - s ) ^ { \\alpha } ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s , \\end{align*}"}
{"id": "4940.png", "formula": "\\begin{align*} u _ { \\l , o ' } ( \\cdot , t ) = \\widehat { u } ( \\cdot , t ) . \\end{align*}"}
{"id": "366.png", "formula": "\\begin{align*} \\frac { \\displaystyle \\sum _ { m = 0 } ^ \\infty [ p ^ \\# _ m ( o , o ) - \\pi ( o ) ] } { \\displaystyle \\sum _ { m = 0 } ^ \\infty [ \\tilde p ^ \\# _ m ( A , A ) - \\pi ( A ) ] } \\ge 1 - C \\frac { \\log ( D / \\delta ) } { f ( n ) } \\end{align*}"}
{"id": "5634.png", "formula": "\\begin{align*} \\mathcal { J } _ { 2 , 1 } = - 2 \\iint _ { Q _ T } \\nabla \\cdot ( p ( m \\nabla p ) ) \\Delta p = - 2 \\iint _ { Q _ T } m | \\nabla p | ^ 2 \\Delta p + p \\Delta p \\left [ \\nabla \\cdot ( m \\nabla p ) \\right ] . \\end{align*}"}
{"id": "2905.png", "formula": "\\begin{align*} \\textup { c o l u m n $ i $ o f $ A $ } = S ^ { ( i - 1 ) k } b 1 \\leq i \\leq n , \\end{align*}"}
{"id": "8922.png", "formula": "\\begin{align*} \\mathcal L = \\mathcal L [ u ] : = w ^ { i j } [ D _ { i j } - D _ { p _ k } A _ { i j } ( \\cdot , u , D u ) D _ k ] , \\end{align*}"}
{"id": "2597.png", "formula": "\\begin{align*} \\begin{aligned} & 0 \\le u \\le \\min \\{ p _ 1 , q _ 2 \\} \\ , \\ \\ 0 \\le v \\le \\min \\{ p _ 2 , q _ 1 \\} \\ , \\\\ & \\begin{cases} 0 \\le s \\le \\min \\{ p _ 1 - u , p _ 2 - v \\} \\ , \\ \\ \\mbox { f o r ( \\ref { p s i - 1 c a s e } ) } \\ , \\\\ 0 \\le s \\le \\min \\{ q _ 1 - v , q _ 2 - u \\} \\ , \\ \\ \\mbox { f o r ( \\ref { p s i - 2 c a s e } ) } \\ . \\end{cases} \\end{aligned} \\end{align*}"}
{"id": "5065.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { \\tau \\in [ 0 , T ] } \\| L ^ { n , 2 } _ \\tau + L ^ { n , 3 } _ \\tau \\| _ 2 = 0 . \\end{align*}"}
{"id": "1839.png", "formula": "\\begin{align*} \\begin{aligned} q _ { n } ( z ) & : = \\det ( z I _ { n } - H _ { n } ) , \\\\ q _ { n , k } ( z ) & : = \\det ( z I _ { n - k } - H _ { n } ^ { [ k ] } ) , 1 \\leq k \\leq p . \\end{aligned} \\end{align*}"}
{"id": "241.png", "formula": "\\begin{align*} ( X ^ k ) _ { i j } = X ^ k _ { i j } : = ( X _ i ) _ { j k } = ( X _ i ^ t ) _ { k j } ; \\end{align*}"}
{"id": "5820.png", "formula": "\\begin{align*} { \\bf { a } } = ( a _ 1 = a , a _ 2 = a , \\ldots , a _ k = a ) , \\end{align*}"}
{"id": "879.png", "formula": "\\begin{align*} \\sigma _ 2 * \\sigma _ 1 * \\sigma _ 2 = \\sigma _ 1 . \\end{align*}"}
{"id": "263.png", "formula": "\\begin{align*} \\Lambda ^ { \\nu } = ( \\Lambda _ 1 ^ { \\nu } , \\ldots , \\Lambda _ n ^ { \\nu } ) , \\ \\ \\nu \\geq 1 \\end{align*}"}
{"id": "4360.png", "formula": "\\begin{align*} f = \\sum \\nolimits _ { 0 \\le s } a _ s g ^ s = \\sum \\nolimits _ { 0 \\le s } b _ s \\phi ^ s , b _ s = a _ s G ^ s . \\end{align*}"}
{"id": "4647.png", "formula": "\\begin{align*} & \\lim _ { k \\to \\infty } \\left ( \\left [ \\prod _ { j = k } ^ { k + n - 1 } w ^ j \\right ] \\left ( x _ { k + n } - l ( x ) \\right ) + \\left [ \\prod _ { j = 0 } ^ { n - 1 } w ^ j \\right ] \\left ( x _ { n } - l ( x ) \\right ) \\right ) \\\\ & = \\left [ \\prod _ { j = 0 } ^ { n - 1 } w ^ j \\right ] \\left ( x _ { n } - l ( x ) \\right ) . \\end{align*}"}
{"id": "8355.png", "formula": "\\begin{align*} \\begin{array} { c c } \\Lambda ( g ) ^ { - 1 } \\tilde { f } ^ { | u \\rangle } ( \\Lambda ( g ) p ) = \\tilde { f } ^ { | u ' \\rangle } ( p ) = \\frac { u ' } { u ' \\cdot p } , \\ , \\ , \\ , u ' = \\Lambda ( g ) ^ { - 1 } u , \\\\ g \\in S L ( 2 , \\mathbb { C } ) , \\end{array} \\end{align*}"}
{"id": "1825.png", "formula": "\\begin{align*} m ( z ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { A _ { n } } { z ^ { n + 1 } } \\end{align*}"}
{"id": "8054.png", "formula": "\\begin{align*} 2 g ( \\nabla _ { X } X , W ) = - g ( \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) , W ) \\end{align*}"}
{"id": "7547.png", "formula": "\\begin{align*} \\cos ^ 2 ( \\alpha _ k ^ * ) | e _ k | + \\sin ^ 2 ( \\alpha _ k ^ * ) | e _ k | = | e _ k | . \\end{align*}"}
{"id": "4985.png", "formula": "\\begin{align*} \\Theta ^ n _ s : = \\sigma ( X _ s ) \\left ( \\int _ 0 ^ s \\psi _ { n , 1 } ( u , s ) d W _ u \\right ) + X ^ n _ { \\eta _ n ( s ) } - X ^ n _ s . \\end{align*}"}
{"id": "8410.png", "formula": "\\begin{gather*} \\left ( { \\varrho _ m ( \\gamma , \\upsilon ) } \\hat h | _ { k - \\frac 1 2 } ( \\gamma , \\upsilon ) \\right ) ( \\tau ) = O ( 1 ) \\end{gather*}"}
{"id": "7254.png", "formula": "\\begin{align*} l _ { \\rho } \\left ( \\Gamma , \\sum _ { n = 1 } ^ { \\infty } a _ n \\mu ^ n \\right ) \\leq \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } n a _ n \\right ) l _ { \\rho } ( \\Gamma , \\mu ) . \\end{align*}"}
{"id": "4812.png", "formula": "\\begin{align*} \\sum _ { k = n / 2 + 1 } ^ { n - 1 } \\int _ { ( k - 1 ) / n } ^ { k / n } ( Q ( u ) - q _ { n , k } ) ^ 2 d u \\le \\frac { \\pi } { 6 n } \\sum _ { k = n / 2 + 1 } ^ { n - 1 } \\frac { 1 } { ( n - k ) ^ 2 } \\le \\frac { \\pi } { 6 n } \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m ^ 2 } = \\frac { \\pi ^ 3 } { 3 6 n } . \\end{align*}"}
{"id": "7939.png", "formula": "\\begin{align*} \\mathbb K _ \\pm : = \\left \\{ f \\in \\mathbb L \\otimes \\mathbb Q | \\left \\{ i \\in \\{ 1 , \\ldots , m \\} | D _ i \\cdot f \\in \\mathbb Z \\right \\} \\in \\mathcal A _ \\pm \\right \\} . \\end{align*}"}
{"id": "4947.png", "formula": "\\begin{align*} S = A \\times T _ 1 \\times \\dots \\times T _ m \\end{align*}"}
{"id": "717.png", "formula": "\\begin{align*} \\| u \\| _ { k , \\alpha } = \\sup _ { x \\in M } \\left \\| \\left . \\Psi ^ * u \\right | _ { B _ \\delta ( x ) } \\right \\| _ { k , \\alpha , B _ \\delta ( 0 ) \\times [ 0 , T ] } . \\end{align*}"}
{"id": "8380.png", "formula": "\\begin{align*} F ( k - 1 , 2 , 2 ) = 4 \\left \\lceil \\frac { k } { 4 } \\right \\rceil . \\end{align*}"}
{"id": "6339.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { N } ^ s _ 2 u ( x ) = \\int _ { \\Omega } \\dfrac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 s } } d y , ~ ~ \\forall x \\in \\R ^ N \\setminus \\Omega . \\end{aligned} \\end{align*}"}
{"id": "2976.png", "formula": "\\begin{align*} T _ m ( d ) = \\sum _ { Q \\in \\Gamma \\backslash \\mathcal Q _ { d ^ 2 } } \\chi _ d ( Q ) \\begin{dcases} \\int _ { C _ Q } F _ { m , Q } ( z , s ) y ^ { - 1 } | d z | & d > 0 , \\\\ \\int _ { C _ Q } i \\partial _ z F _ { m , Q } ( z , s ) \\ , d z & d < 0 . \\end{dcases} \\end{align*}"}
{"id": "7514.png", "formula": "\\begin{align*} E _ { x \\leftrightarrow y } = E _ { x \\rightarrow y } \\land E _ { y \\rightarrow x } , \\end{align*}"}
{"id": "3625.png", "formula": "\\begin{align*} v ( \\phi , w ) \\leq v ( \\phi ' , w ) & v ( \\Delta ^ \\neg ( \\phi \\rightarrow \\phi ' ) , w ) = ( 1 , 0 ) , \\\\ v ( \\phi , w ) > v ( \\phi ' , w ) & v ( \\Delta ^ \\neg ( \\phi ' \\rightarrow \\phi ) \\wedge { \\sim } \\Delta ^ \\neg ( \\phi \\rightarrow \\phi ' ) , w ) = ( 1 , 0 ) . \\end{align*}"}
{"id": "960.png", "formula": "\\begin{align*} N ( \\alpha R ) = | N ( \\alpha ) | , \\forall \\alpha \\in R . \\end{align*}"}
{"id": "1307.png", "formula": "\\begin{align*} W ^ { n , p } ( \\O ) : = \\{ f \\in L ^ p ( \\O ) \\mid D ^ \\alpha f \\in L ^ p ( \\O ) , \\ , 0 \\leq | \\alpha | \\leq n \\} , \\end{align*}"}
{"id": "5278.png", "formula": "\\begin{align*} \\# Z \\left ( H _ 2 | _ { [ 0 , 1 ] \\times \\Phi _ p } \\right ) = \\# Z \\left ( \\hat H _ { 2 , p } \\right ) . \\end{align*}"}
{"id": "6043.png", "formula": "\\begin{align*} \\lambda _ i = r \\kappa _ i + s \\end{align*}"}
{"id": "563.png", "formula": "\\begin{align*} \\mathbb { E } ( X _ i X _ j ) = \\sum \\limits _ { \\substack { 0 \\leq l \\leq i \\\\ \\gcd ( l , i ) \\ { \\rm i s } \\ k - { \\rm f r e e } } } C _ { \\alpha } ( i , l ) T _ k ( l , i , j , \\alpha ) , \\end{align*}"}
{"id": "6010.png", "formula": "\\begin{align*} D _ f ( x ) = \\left ( f ( x ) , f ' ( x ) , \\ldots , f ^ { ( j ) } ( x ) , \\ldots , f ^ { ( n ) } ( x ) \\right ) \\end{align*}"}
{"id": "3291.png", "formula": "\\begin{align*} w ( y , d ) : = \\int _ D Q _ { - A , q } e ^ { i k z \\cdot d } u _ { A , q } ( z , y ) d z . \\end{align*}"}
{"id": "740.png", "formula": "\\begin{align*} & \\left \\langle \\gamma _ j ( w ) , \\gamma _ i ( z ) \\right \\rangle \\\\ & \\quad = \\big ( K _ 2 \\partial _ i \\bar { \\partial } _ j K _ 3 \\big ) ( z , w ) + \\big ( K _ 3 \\partial _ i \\bar { \\partial } _ j K _ 2 \\big ) ( z , w ) - \\big ( \\partial _ i K _ 2 \\bar { \\partial } _ j K _ 3 \\big ) ( z , w ) - \\big ( \\partial _ i K _ 3 \\bar { \\partial } _ j K _ 2 \\big ) ( z , w ) . \\end{align*}"}
{"id": "3502.png", "formula": "\\begin{align*} \\overline { G } _ { ( 1 ) } ( u ) = \\frac { \\exp ( - u ) } { 1 + \\exp ( - u ) } \\end{align*}"}
{"id": "3317.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\varrho ^ { 2 j } \\left \\langle \\varphi _ j , K ( \\partial _ t ^ { \\Delta t } ) \\varphi ( t _ j ) \\right \\rangle \\geq C _ 2 ( \\sigma ) \\sum _ { j = 0 } ^ \\infty \\varrho ^ { 2 j } \\| ( \\partial _ t ^ { \\Delta t } ) ^ \\eta \\varphi _ j \\| ^ 2 _ X , \\end{align*}"}
{"id": "4063.png", "formula": "\\begin{align*} ( \\Psi \\otimes \\cdots \\otimes \\Psi ) \\rho ( u ) & = \\varepsilon ( u ) ( \\Psi \\otimes \\cdots \\otimes \\Psi ) , \\\\ \\rho ( u ) ( \\Psi ^ * \\otimes \\cdots \\otimes \\Psi ^ * ) & = \\varepsilon ( u ) ( \\Psi ^ * \\otimes \\cdots \\otimes \\Psi ^ * ) \\end{align*}"}
{"id": "2328.png", "formula": "\\begin{align*} P Q ^ k P ^ { - 1 } = P \\begin{pmatrix} M ^ k & \\vec { 0 } \\\\ e _ k ^ { \\top } & 1 \\end{pmatrix} P ^ { - 1 } = \\begin{pmatrix} M ^ k & \\vec { 0 } \\\\ f _ k ^ { \\top } & 1 \\end{pmatrix} \\end{align*}"}
{"id": "2287.png", "formula": "\\begin{align*} p _ e ^ { 1 , a } = - \\frac { 1 - \\sigma ^ 2 } { 2 } \\left ( | u _ e ^ 1 | ^ 2 + | v _ e ^ 1 | ^ 2 \\right ) \\end{align*}"}
{"id": "1695.png", "formula": "\\begin{align*} y ( z x ) = - ( - 1 ) ^ { \\vert x \\vert ( \\vert y \\vert + \\vert z \\vert ) } ( x y ) z . \\end{align*}"}
{"id": "8998.png", "formula": "\\begin{align*} P _ k ( m , x ) & = \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ { j - k } x ^ { j } \\binom { j + m - k } { m - k } { m + 1 \\brack j + m - k + 1 } \\\\ & = \\sum _ { j = m - k } ^ { m } ( - 1 ) ^ { j - m } x ^ { j + k - m } \\binom { j } { m - k } \\frac { m ! } { j ! } \\\\ & \\times B _ { j } \\left ( H _ m ^ { ( 1 ) } , - H _ m ^ { ( 2 ) } , \\ldots , ( - 1 ) ^ { j - 1 } ( j - 1 ) ! H _ m ^ { ( j ) } \\right ) . \\end{align*}"}
{"id": "1188.png", "formula": "\\begin{align*} \\min _ { v \\in V } J ( v ) \\ \\ J ( v ) : = \\frac { 1 } { q } \\int _ { \\R ^ { d } } | v | ^ { q } d \\rho - L ( v ) . \\end{align*}"}
{"id": "5595.png", "formula": "\\begin{align*} \\theta ( s ) = \\begin{cases} 1 , & \\ 0 \\leq s < t , \\\\ \\displaystyle { \\frac { t - s } { \\delta } + 1 } , & \\ t \\leq s < t + \\delta , \\\\ 0 , & \\ s > t + \\delta , \\end{cases} \\end{align*}"}
{"id": "551.png", "formula": "\\begin{align*} \\sum \\limits _ { 1 \\leq n \\leq N } f _ k ( n ) f _ k ( n + 1 ) = \\sum \\limits _ { 1 \\leq n \\leq N } \\sum _ { w _ 1 \\mid n } \\frac { g _ k ( w _ 1 ) } { w _ 1 } \\sum _ { w _ 2 \\mid ( n + 1 ) } \\frac { g _ k ( w _ 2 ) } { w _ 2 } . \\end{align*}"}
{"id": "3815.png", "formula": "\\begin{align*} | { } _ { } ^ { \\kappa } E r r ^ { \\mu , 4 ; l } _ { k , j ; n } ( t _ 1 , t _ 2 ) | \\lesssim \\sum _ { a = 1 , 2 } \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } f ( t _ a , X ( t _ a ) - y , v ) \\mathcal { K } _ { k , j , n } ^ { l , \\kappa } ( y , v , V ( t _ a ) ) d y d v \\end{align*}"}
{"id": "7378.png", "formula": "\\begin{align*} \\Delta \\left ( \\sum _ { i = 1 } ^ 3 \\frac { \\partial U _ { 0 } ( y ) } { \\partial y _ i } \\right ) - \\left ( \\sum _ { i = 1 } ^ 3 \\frac { \\partial U _ { 0 } } { \\partial y _ i } \\right ) + p U _ 0 ( y ) ^ { p - 1 } \\left ( \\sum _ { i = 1 } ^ 3 \\frac { \\partial U _ 0 ( y ) } { \\partial y _ i } \\right ) = 0 \\end{align*}"}
{"id": "6825.png", "formula": "\\begin{align*} N ( t ) = \\int _ 0 ^ { + \\infty } J _ v ( V _ F , g ) \\ , p ( t , V _ F , g ) \\ , d g = \\int _ 0 ^ { + \\infty } \\bigl ( - g _ L V _ F + g ( V _ E - V _ F ) \\bigr ) \\ , p ( t , V _ F , g ) \\ , d g . \\end{align*}"}
{"id": "2065.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : T i n f t y t p h i k } T _ { \\infty , t } \\Phi _ k ( \\nu ) = \\big \\langle \\overline { S } _ t ( \\nu ) ^ { \\otimes k } , \\varphi _ k \\big \\rangle , \\end{align*}"}
{"id": "4462.png", "formula": "\\begin{align*} \\left \\{ ~ \\begin{aligned} & \\theta _ t + \\mathbf { u } \\cdot \\nabla \\theta = 0 , \\\\ & \\mathbf { u } = \\nabla ^ \\perp \\Psi , \\\\ & \\mathbf { A } \\Psi = \\theta , \\end{aligned} \\right . \\end{align*}"}
{"id": "4678.png", "formula": "\\begin{align*} h _ 1 \\pi _ K ( b _ 1 ) = b _ 4 h _ 4 h _ 4 \\in U ^ - _ { u \\i , > 0 } b _ 4 \\in B _ { K , \\ge 0 } ^ + . \\end{align*}"}
{"id": "3941.png", "formula": "\\begin{align*} \\tilde { I } _ 4 & \\le \\frac { c } { ( \\prod _ { l = 1 } ^ d h _ l ^ * ) ^ 2 } \\frac { 1 } { T _ n ^ 2 } \\int _ 0 ^ { T _ n } \\int _ { D } ^ { T _ n } e ^ { - \\rho s ' } e ^ { \\rho \\Delta _ n } d t d s ' \\\\ & \\le \\frac { c } { T _ n ( \\prod _ { l = 1 } ^ d h _ l ^ * ) ^ 2 } e ^ { - \\rho D } \\end{align*}"}
{"id": "2603.png", "formula": "\\begin{align*} \\mathcal H _ 1 \\to \\mathcal H _ { p , 0 } : w = ( z _ 1 , z _ 2 , z _ 3 ) \\mapsto w \\otimes . . . \\otimes w = \\ ! \\ ! \\sum _ { j + k + l = 1 } \\sqrt { \\dfrac { p ! } { j ! \\ , k ! \\ , l ! } } z _ 1 ^ j z _ 2 ^ k z _ 3 ^ l \\ , e _ { j , k , l } \\ . \\end{align*}"}
{"id": "7218.png", "formula": "\\begin{align*} ( x , t ) \\in D : = D _ { \\rm I n t } \\cap \\{ x > 0 \\} \\cap \\{ t - x > R \\} . \\end{align*}"}
{"id": "8813.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\phi _ n ^ * ( x ) = \\phi ^ * ( x ) \\end{align*}"}
{"id": "1730.png", "formula": "\\begin{align*} G ( z \\ , | \\ , \\omega _ 1 , \\widetilde \\omega _ 1 , \\omega _ 2 ) = \\exp \\Bigg ( \\int _ C \\frac { - e ^ { ( z + \\overline \\omega _ 1 ) s } } { ( e ^ { \\omega _ 1 s } - 1 ) ( e ^ { \\widetilde \\omega _ 1 s } - 1 ) ( e ^ { \\omega _ 2 s } - 1 ) } \\frac { d s } { s } \\Bigg ) , \\end{align*}"}
{"id": "3710.png", "formula": "\\begin{align*} \\{ H ( e _ 1 ) \\mid e _ 1 \\in E ^ - ( v _ 1 ) \\} = \\{ H ( e _ 2 ) \\mid e _ 2 \\in E ^ - ( v _ 2 ) \\} . \\end{align*}"}
{"id": "2404.png", "formula": "\\begin{align*} & \\quad \\| \\{ \\Sigma _ n ( \\mathbb U ( t ) ) - \\Sigma _ n ( U ( t ) ) \\} e _ l \\| ^ 2 = \\sum _ { m = 1 } ^ { n - 1 } \\left | \\left ( \\sigma ( \\mathbb U _ m ( t ) ) - \\sigma ( U _ m ( t ) ) \\right ) e _ l ( m ) \\right | ^ 2 \\\\ & \\le \\frac { C } { n } \\sum _ { m = 1 } ^ { n - 1 } \\left | \\mathbb U _ m ( t ) - U _ m ( t ) \\right | ^ 2 = \\frac { C } { n } \\left \\| \\mathbb U ( t ) - U ( t ) \\right \\| ^ 2 , \\quad \\forall ~ l \\in \\Z _ n . \\end{align*}"}
{"id": "5342.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { s , p } ( \\Omega ) } \\vcentcolon = \\left ( \\| u \\| _ { W ^ { m , p } ( \\Omega ) } ^ p + \\sum _ { | \\alpha | = m } [ \\partial ^ { \\alpha } u ] _ { W ^ { \\sigma , p } ( \\Omega ) } ^ p \\right ) ^ { 1 / p } . \\end{align*}"}
{"id": "2837.png", "formula": "\\begin{align*} 2 \\big \\langle g _ { 1 } , g _ { 0 } \\big \\rangle = \\| g _ 0 \\| ^ 2 + \\| g _ 1 \\| ^ 2 - \\| g _ 0 - g _ 1 \\| ^ 2 , \\end{align*}"}
{"id": "7438.png", "formula": "\\begin{align*} U _ t = { \\mathcal { A } } _ m U , U ( 0 ) = U _ 0 , \\end{align*}"}
{"id": "5611.png", "formula": "\\begin{align*} \\frac { ( x + y + z ) ^ 3 } { x y z } = n \\end{align*}"}
{"id": "6966.png", "formula": "\\begin{align*} \\left | 1 - \\frac { z - \\mu _ k ^ 2 } { z - \\lambda _ k ^ 2 } \\right | = \\left | \\frac { \\lambda _ k ^ 2 - \\mu _ k ^ 2 } { z - \\lambda _ k ^ 2 } \\right | \\le \\frac 2 R ( \\lambda _ k ^ 2 - \\mu _ k ^ 2 ) \\forall z : \\ | z | > R . \\end{align*}"}
{"id": "6414.png", "formula": "\\begin{align*} \\left [ \\sum _ { j = 1 } ^ n Q ^ j _ k \\xi ^ k \\frac { \\partial } { \\partial x _ j } , X \\right ] = \\xi ^ k \\left [ \\sum _ { j = 1 } ^ n Q ^ j _ k \\frac { \\partial } { \\partial x _ j } , X \\right ] . \\end{align*}"}
{"id": "8403.png", "formula": "\\begin{gather*} ( \\gamma , \\upsilon ) ( \\gamma ' , \\upsilon ' ) = ( \\gamma \\gamma ' , ( \\upsilon \\circ \\gamma ' ) \\upsilon ' ) . \\end{gather*}"}
{"id": "4964.png", "formula": "\\begin{align*} \\kappa _ 5 = \\sum _ { k = 1 } ^ { \\infty } \\int _ 0 ^ 1 \\int _ { 0 } ^ { 1 } [ ( y + k ) ^ \\alpha - ( y + k - z ) ^ \\alpha ] [ ( y + k ) ^ \\alpha - k ^ \\alpha ] d z d y . \\end{align*}"}
{"id": "371.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\delta ^ 2 } p ^ \\# _ m ( o , y ) & \\le \\int _ 0 ^ { 1 0 \\delta ^ 2 } p _ t ( o , y ) \\mathrm { d } t + \\delta ^ 2 \\P ( N \\le 2 \\delta ^ 2 ) \\\\ & \\le O ( 1 / f ( n ) ) + \\delta ^ 2 \\exp ( - c \\delta ^ 2 ) , \\end{align*}"}
{"id": "7742.png", "formula": "\\begin{align*} \\Psi ^ * ( \\lambda _ * H ^ { ( 1 ) } ( \\mu _ N ) - \\lambda _ * \\Psi ( - 1 ) ) \\cong \\Psi ^ * \\Psi _ * \\binom { N } { 2 } \\cong * . \\end{align*}"}
{"id": "1709.png", "formula": "\\begin{align*} ( \\mathcal { O } _ C ( n ) ) = \\beta - n \\delta , \\ ( \\mathcal { O } _ x ) = - \\delta \\ . \\end{align*}"}
{"id": "8141.png", "formula": "\\begin{align*} \\varphi _ n = \\varphi _ { n ; \\mathrm { o u t } } + \\varphi _ { n ; \\mathrm { i n } } + \\varphi _ { n ; \\mathrm { s u r } } \\end{align*}"}
{"id": "784.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ { \\infty } \\prod _ { k = 0 } ^ { n - 2 } \\frac { | E - D + E k | } { k + 1 } \\phi _ { n } ( r ) = ( 1 - E ) ^ { \\frac { D - E } { E } } - \\phi _ { 1 } ( r ) , \\end{align*}"}
{"id": "8065.png", "formula": "\\begin{align*} g ( \\operatorname { g r a d } ( f ^ { 2 } ) , V ) g ( \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) , W ) = g ( \\operatorname { g r a d } ( f ^ { 2 } ) , W ) g ( \\operatorname { g r a d } ( \\| X \\| ^ { 2 } ) , V ) \\end{align*}"}
{"id": "6999.png", "formula": "\\begin{align*} \\sum W ( U , f ) \\int _ U Q \\omega = \\int _ { \\Delta ^ \\perp } f ^ * \\alpha \\end{align*}"}
{"id": "8631.png", "formula": "\\begin{align*} \\widehat { H } _ { { \\textrm { C K } } } ( t ) : = \\exp { ( - 2 \\gamma t ) } \\frac { \\widehat { p } ^ 2 } { 2 } + \\omega ^ 2 \\exp { ( 2 \\gamma t ) } \\frac { \\widehat { x } ^ 2 } { 2 } , \\gamma \\in \\mathbb { R } , \\ , \\ , \\omega \\in \\mathbb { R } , \\end{align*}"}
{"id": "4036.png", "formula": "\\begin{align*} \\sup _ { \\delta < | t | \\leq \\lambda } \\left \\lvert \\frac { \\phi ( s + \\mathrm { i } t ) } { \\phi ( s ) } \\right \\rvert & = \\sup _ { \\delta < | t | \\leq \\lambda } \\left \\lvert \\prod _ { j = 1 } ^ { w _ n } \\frac { \\tilde { \\phi } _ j ( s + \\mathrm { i } t ) } { \\tilde { \\phi } _ j ( s ) } \\right \\rvert \\cdot 1 \\\\ & \\leq c _ 2 ^ { w _ n } \\\\ & = o ( n ^ { - 1 / 2 } ) , \\end{align*}"}
{"id": "5723.png", "formula": "\\begin{align*} y _ i \\pi _ { [ a , b ] } = y _ { b + 1 } \\pi _ { [ a , b ] } ( a \\le i \\le b ) \\end{align*}"}
{"id": "461.png", "formula": "\\begin{align*} \\Phi _ { \\mu } ( \\eta _ { \\mu } ( z ) ) = z \\end{align*}"}
{"id": "5111.png", "formula": "\\begin{align*} \\Lambda ^ { ( 3 ) } _ { n , \\delta } & = \\exp \\left ( i \\mu n ^ { \\alpha + \\frac 1 2 } \\int _ 0 ^ { t - \\delta } ( t - s ) ^ \\alpha \\xi ^ n _ s d W _ s + i \\lambda n ^ { \\alpha + \\frac 1 2 } \\int _ { t - \\delta } ^ { \\eta _ n ( t ) } \\psi _ { n , 1 } ( s , \\eta _ n ( t ) ) d W _ s \\right ) \\\\ & \\times \\left [ \\exp \\left ( i \\mu n ^ { \\alpha + \\frac 1 2 } \\int _ { t - \\delta } ^ t ( t - s ) ^ \\alpha \\xi ^ n _ s d W _ s \\right ) - 1 \\right ] \\end{align*}"}
{"id": "874.png", "formula": "\\begin{align*} s * ^ { \\alpha _ p } s _ { a _ p } * ^ { \\alpha _ { p - 1 } } s _ { a _ { p - 1 } } * ^ { \\alpha _ { p - 2 } } \\cdots * ^ { \\alpha _ 1 } s _ { a _ 1 } = t * ^ { \\beta _ q } s _ { b _ q } * ^ { \\beta _ { q - 1 } } s _ { b _ { q - 1 } } * ^ { \\beta _ { q - 2 } } \\cdots * ^ { \\beta _ 1 } s _ { b _ 1 } \\end{align*}"}
{"id": "4223.png", "formula": "\\begin{align*} \\sum _ { n = k ^ { \\ell _ 0 } + 1 } ^ { N } f ( n ) \\leq \\sum _ { \\ell = \\ell _ 0 } ^ { \\ell _ 1 } k ^ { \\ell + 1 } f ( k ^ { \\ell } ) \\leq k ^ { \\ell _ 0 + 1 } f ( k ^ { \\ell _ 0 } ) + k ^ 2 \\sum _ { \\ell = \\ell _ 0 } ^ { \\ell _ 1 - 1 } k ^ \\ell f ( k ^ { \\ell + 1 } ) . \\end{align*}"}
{"id": "3916.png", "formula": "\\begin{align*} { } \\Delta _ n \\underset { \\sim } { < } ( \\frac { \\log T _ n } { T _ n } ) ^ { \\frac { \\bar { \\beta } _ 3 } { 2 \\bar { \\beta } _ 3 + d - 2 } ( \\frac { 1 } { \\beta _ 1 } + \\frac { 1 } { \\beta _ 2 } ) } \\log T _ n \\mbox { f o r } \\beta _ 2 < \\beta _ 3 , \\\\ \\Delta _ n \\underset { \\sim } { < } ( \\frac { 1 } { T _ n } ) ^ { \\frac { \\bar { \\beta } _ 3 } { 2 \\bar { \\beta } _ 3 + d - 2 } ( \\frac { 1 } { \\beta _ 1 } + \\frac { 1 } { \\beta _ 2 } ) } \\mbox { f o r } \\beta _ 2 = \\beta _ 3 . { } \\end{align*}"}
{"id": "5946.png", "formula": "\\begin{align*} \\lambda _ k = \\left ( { b + c } \\right ) \\left ( { d + 1 } - 2 k \\right ) \\end{align*}"}
{"id": "1956.png", "formula": "\\begin{align*} \\psi _ { 0 } ( z ) & = \\frac { 1 } { z - a _ { 0 } ^ { ( 0 ) } - \\sum _ { j = 1 } ^ { p } \\sum _ { k = 0 } ^ { j } a _ { - k } ^ { ( j ) } \\ , \\phi _ { j - k - 1 } ^ { ( 1 ) } ( z ) \\ , \\beta _ { k - 1 } ^ { ( 1 ) } ( z ) } \\\\ \\psi _ { j } ( z ) & = \\psi _ { 0 } ( z ) \\ , \\phi _ { j - 1 } ^ { ( 1 ) } ( z ) 1 \\leq j \\leq p , \\end{align*}"}
{"id": "6965.png", "formula": "\\begin{align*} \\left | 1 - \\frac { z - \\mu _ k ^ 2 } { z - \\lambda _ k ^ 2 } \\right | = \\left | \\frac { \\lambda _ k ^ 2 - \\mu _ k ^ 2 } { z - \\lambda _ k ^ 2 } \\right | \\le C ( K ) ( \\lambda _ k ^ 2 - \\mu _ k ^ 2 ) \\forall z \\in K . \\end{align*}"}
{"id": "8657.png", "formula": "\\begin{align*} \\sup \\left \\{ \\int _ 0 ^ \\infty f g \\ , d r : \\ g \\geq 0 \\ , , \\ \\underline \\nu _ \\beta ( g ) \\leq 1 \\right \\} = \\beta \\ , \\overline \\mu _ \\beta ( f ) \\ , , \\end{align*}"}
{"id": "8933.png", "formula": "\\begin{align*} \\mathcal { U } = \\{ ( x , g ( x , y , z ) , g _ x ( x , y , z ) ) | \\ ( x , y , z ) \\in \\Gamma \\} , \\end{align*}"}
{"id": "6354.png", "formula": "\\begin{align*} D ^ s u : = \\dfrac { u ( x ) - u ( y ) } { | x - y | ^ s } . \\end{align*}"}
{"id": "2283.png", "formula": "\\begin{align*} - \\Delta v _ e ^ 1 = 0 , v _ e ^ 1 ( x , 0 ) = - v _ p ^ 0 ( x , 0 ) , v _ e ^ 1 ( x , \\infty ) = 0 . \\end{align*}"}
{"id": "1110.png", "formula": "\\begin{align*} D ^ * _ X U = D ^ * _ U X = \\frac { 1 } { t } U - \\frac { 1 } { t } U = 0 . \\end{align*}"}
{"id": "980.png", "formula": "\\begin{align*} H = V _ { \\phi } ^ { - 1 } \\operatorname { d i a g } \\left \\{ \\theta _ { 0 } , \\theta _ { 1 } , \\cdots , \\theta _ { n - 1 } \\right \\} V _ { \\phi } \\end{align*}"}
{"id": "6727.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 } \\left \\| \\dfrac { G ( z + t h ) - G ( z ) } { t } - G ' ( z , h ) \\right \\| = 0 , \\end{align*}"}
{"id": "2092.png", "formula": "\\begin{align*} \\dfrac { \\log _ 2 \\left ( N _ { X _ { H , V } } ( p n , n ) \\right ) } { p n ^ 2 } \\geq \\dfrac { \\log _ 2 \\left ( \\max \\limits _ { v \\in \\{ 0 , \\dots , p - 1 \\} ^ n } \\Pi _ { j = 0 } ^ { p - 1 } N _ { v ^ j } \\right ) } { p n } \\end{align*}"}
{"id": "4083.png", "formula": "\\begin{align*} E ( X _ 3 ^ 2 ) = \\varepsilon ^ 2 \\big [ P ( \\boldsymbol { X } = \\mathbf { s } ( 3 ) ) + P ( \\boldsymbol { X } = \\mathbf { s } ( 6 ) ) \\big ] . \\end{align*}"}
{"id": "1724.png", "formula": "\\begin{align*} \\frac { s \\ , e ^ { z s } } { e ^ s - 1 } = \\sum _ { n = 0 } ^ { \\infty } B _ n ( z ) \\frac { s ^ n } { n ! } . \\end{align*}"}
{"id": "397.png", "formula": "\\begin{align*} m _ 1 & = \\left \\lceil \\dfrac { n } { 2 } \\right \\rceil \\\\ m _ 2 & = \\left \\lceil \\dfrac { n - m _ 1 } { 3 } \\right \\rceil \\\\ m _ 3 & = \\left \\lceil \\dfrac { n - ( m _ 1 + m _ 2 ) } { 4 } \\right \\rceil \\\\ & \\ ; \\ ; \\vdots \\\\ m _ { \\ell } & = \\left \\lceil \\dfrac { n - ( m _ 1 + m _ 2 + \\cdots + m _ { \\ell - 1 } ) } { \\ell + 1 } \\right \\rceil . \\end{align*}"}
{"id": "5676.png", "formula": "\\begin{align*} x _ 1 ^ { d + 1 } + x _ 2 ^ { d + 1 } + \\cdots + x _ i ^ { d + 1 } = ( x _ 1 ^ d + x _ 2 ^ d + \\cdots + x _ i ^ d ) x _ { i + 1 } ( d = 1 , 2 , \\ldots ) . \\end{align*}"}
{"id": "6343.png", "formula": "\\begin{align*} \\widehat { \\varPhi } _ { x } ( t ) : = \\widehat { \\varPhi } ( x , t ) = \\int _ { 0 } ^ { t } \\widehat { \\varphi } _ { x } ( \\tau ) d \\tau ~ ~ ~ ~ t \\geqslant 0 . \\end{align*}"}
{"id": "3733.png", "formula": "\\begin{align*} I _ { j , l } ^ { 1 ; 1 } ( t , x ) = \\int _ { \\R ^ 3 } \\int _ { | y - x | \\leq t } \\frac { \\varphi _ { m ; - 1 0 M _ t } ( | y - x | ) } { | y - x | } \\frac { \\hat { v } _ 2 \\omega _ 3 - \\hat { v } _ 3 \\omega _ 2 } { 1 + \\hat { v } \\cdot \\omega } S f ( t - | y - x | , y , v ) \\varphi _ { j , l } ( v , \\omega ) d y d v \\end{align*}"}
{"id": "232.png", "formula": "\\begin{align*} \\Gamma ^ { k ( s ) } _ { i j } - \\Gamma ^ { k ( s ) } _ { j i } = L ^ { k ( s ) } _ { i j } ( \\Lambda ^ { ( s ) } ) . \\end{align*}"}
{"id": "2342.png", "formula": "\\begin{align*} \\nu _ Q ( q Q ) - ( \\epsilon ( Q ) - \\gamma ) \\geq \\nu ( f ) = \\nu ( r ) . \\end{align*}"}
{"id": "7103.png", "formula": "\\begin{align*} P ^ E _ t = \\begin{pmatrix} e ^ { - D ^ { E } _ { t , - } D ^ { E } _ { t , + } } & e ^ { \\frac { - D ^ { E } _ { t , - } D ^ { E } _ { t , + } } { 2 } } \\cdot \\frac { ( 1 - e ^ { - D ^ { E } _ { t , - } D ^ { E } _ { t , + } } ) } { D ^ { E } _ { t , - } D ^ { E } _ { t , + } } \\cdot D ^ { E } _ { t , - } \\\\ e ^ { - \\frac { D ^ { E } _ { t , + } D ^ { E } _ { t , - } } { 2 } } \\cdot D ^ { E } _ { t , + } & 1 - e ^ { - D ^ { E } _ { t , + } D ^ { E } _ { t , - } } \\end{pmatrix} \\end{align*}"}
{"id": "3612.png", "formula": "\\begin{align*} \\rho ( I _ { d , s } ) = \\rho _ { i c } ( I _ { d , s } ) = \\rho _ { i c } ( ( I _ { d , s } ) ^ \\vee ) = \\frac { d ( s - d + 1 ) } { s } = \\frac { d } { \\widehat { \\alpha } ( I _ { d , s } ) } = \\frac { s - d + 1 } { \\widehat { \\alpha } ( ( I _ { d , s } ) ^ \\vee ) } . \\end{align*}"}
{"id": "7073.png", "formula": "\\begin{align*} n _ 1 = - 2 l _ 1 + 2 l _ 2 + n _ 2 . \\end{align*}"}
{"id": "2790.png", "formula": "\\begin{align*} \\min _ { 0 { } \\leq { } i { } \\leq { } N } \\| \\nabla f ( x _ i ) \\| ^ 2 { } \\leq { } \\frac { 2 L \\ , \\big ( f ( x _ 0 ) - f _ * \\big ) } { 1 + \\sum \\limits _ { i = 0 } ^ { N - 1 } \\left [ 2 h _ i - h _ i ^ 2 \\frac { - \\kappa } { 2 \\min \\left ( 1 , \\frac { 1 } { h _ i } \\right ) - ( 1 + \\kappa ) } \\right ] } \\end{align*}"}
{"id": "5060.png", "formula": "\\begin{align*} \\| N ^ { n , 5 } _ \\tau \\| _ 2 \\le C \\frac 1 n \\sum _ { j = 0 } ^ { \\lfloor n T \\rfloor } \\sum _ { k = j + 1 } ^ { \\infty } k ^ { 2 \\alpha - 2 } \\to 0 , \\end{align*}"}
{"id": "936.png", "formula": "\\begin{align*} \\langle \\nabla _ i \\nabla _ j \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { i j } ^ l , \\nabla _ i \\nabla _ j \\Psi _ t - \\nabla _ l \\Psi _ t \\Gamma _ { i j } ^ l \\rangle = \\frac { 1 } { 2 t } \\Big ( 1 + O ( t ) \\Big ) + \\sum _ l \\Gamma _ { i j } ^ l \\Gamma _ { i j } ^ l . \\end{align*}"}
{"id": "2669.png", "formula": "\\begin{align*} & ( b _ i ^ 2 - b _ i ) = 0 k < i \\\\ & b _ i b _ j = 0 k < i < j . \\end{align*}"}
{"id": "3849.png", "formula": "\\begin{align*} f ^ { a , ( x _ 0 , y _ 0 ) , \\Pi ^ { \\mu , \\nu } } = \\int _ 0 ^ 1 & \\Big ( \\partial _ a f \\big ( x _ 0 + \\xi ( y _ 0 - x _ 0 ) , \\Pi ^ { \\mu , \\nu } _ \\xi , x _ 1 + \\xi ( y _ 1 - x _ 1 ) , . . . , x _ { m [ a ] } + \\xi ( y _ { m [ a ] } - x _ { m [ a ] } ) \\big ) \\\\ - \\partial _ a f \\big ( x _ 0 , \\mu , & x _ 1 , . . . , x _ { m [ a ] } \\big ) \\Big ) ( 1 - \\xi ) ^ { n - 1 } d \\xi . \\end{align*}"}
{"id": "7355.png", "formula": "\\begin{align*} \\left \\vert \\frac { g ' ( z ) } { f ' ( z ) } \\right \\vert \\leq \\beta + \\frac { 1 - \\beta ^ 2 } { 1 - r ^ 2 } \\frac { r } { m ( r ) } = : h ( \\beta , r ) . \\end{align*}"}
{"id": "3292.png", "formula": "\\begin{align*} w ( y , d ) = - \\int _ D \\Phi ( z , y ) Q _ { - A , q } u _ { - A , q } ( z , d ) d z , \\end{align*}"}
{"id": "1489.png", "formula": "\\begin{align*} e ^ { x ( e ^ { t } - 1 ) } = \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n } ( x ) \\frac { t ^ { n } } { n ! } , ( \\mathrm { s e e } \\ [ 6 - 1 2 , 1 4 - 2 0 ] ) . \\end{align*}"}
{"id": "4825.png", "formula": "\\begin{align*} \\tau ^ \\mathrm { a n / t o p } ( M / S , \\rho ) = \\tau ^ \\mathrm { a n / t o p } ( M / S , F _ \\rho ) \\ ; . \\end{align*}"}
{"id": "6997.png", "formula": "\\begin{align*} f _ h ( v _ I ) = x _ I , \\end{align*}"}
{"id": "1166.png", "formula": "\\begin{align*} T _ { \\Xi } ( \\theta ) : = \\left \\{ h \\in D : h = \\lim _ { n \\to \\infty } \\frac { \\theta _ n - \\theta } { t _ n } \\ \\right \\} . \\end{align*}"}
{"id": "411.png", "formula": "\\begin{align*} \\Gamma _ { \\alpha , \\beta } = \\{ x + i y \\in \\mathbb { C } : y \\ge \\max \\{ \\beta , \\alpha | x | \\} \\} . \\end{align*}"}
{"id": "6721.png", "formula": "\\begin{align*} q _ { n , m } = \\begin{cases} q ^ + _ { m - 2 } \\cdot q ^ + _ { m - 4 } \\cdots q ^ + _ { n + 2 } \\cdot q ^ + _ n , & n < m \\\\ q ^ - _ { m } \\cdot q ^ - _ { m + 2 } \\cdots q ^ - _ { n - 4 } \\cdot q ^ - _ { n - 2 } , & n > m \\\\ , & n = m . \\end{cases} \\end{align*}"}
{"id": "3543.png", "formula": "\\begin{align*} \\pi = \\bigoplus _ { i = 0 } ^ { n } c _ i \\sigma _ i , \\end{align*}"}
{"id": "8662.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\underline f > \\tau \\} } ( r ) g ( r ) \\ , d r & = \\int _ { a _ \\tau } ^ \\infty g ( r ) \\ , d r \\leq a _ \\tau ^ { - p + 1 } \\sup _ { \\alpha > 0 } \\alpha ^ { p - 1 } \\int _ \\alpha ^ \\infty g ( r ) \\ , d r \\\\ & = ( p - 1 ) \\int _ 0 ^ \\infty \\ 1 _ { \\{ \\underline f > \\tau \\} } ( r ) \\ , \\frac { d r } { r ^ p } \\ \\overline \\nu _ p ( g ) \\ , . \\end{align*}"}
{"id": "7694.png", "formula": "\\begin{align*} L _ { \\mathcal { R } } ( f ) ( x ) = \\int _ { \\Theta } f ( \\psi _ { \\theta } ( x ) ) d p ( \\theta ) , \\end{align*}"}
{"id": "8938.png", "formula": "\\begin{align*} | \\psi | ( x , u , p ) = \\frac { f ( x ) } { f ^ * \\circ Y ( x , u , p ) } , \\end{align*}"}
{"id": "627.png", "formula": "\\begin{align*} J _ \\alpha ^ 0 ( s ) \\sim a _ 0 ( s ) n ^ { \\alpha - 2 s } + \\sum _ { m = 0 } ^ { M - 1 } b _ m ( s ) n ^ { - 2 m } + O ( n ^ { - 2 M - 2 s + 1 } ) . \\end{align*}"}
{"id": "8765.png", "formula": "\\begin{align*} \\phi ^ * ( x ) : = \\inf \\{ t \\geq 0 \\mid \\mu ( \\phi ^ { - 1 } ( \\mathbb { R } _ { > t } ) ) \\leq 2 | x | \\} . \\end{align*}"}
{"id": "5749.png", "formula": "\\begin{align*} \\pi _ i \\cdot \\pi _ { [ a , b ] } = ( b - i + 1 ) \\pi _ { [ a - 1 , b ] } + ( i - a + 1 ) \\pi _ { [ a , b + 1 ] } . \\end{align*}"}
{"id": "8978.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\lambda _ \\delta ( \\Omega _ k , \\Sigma _ k ) = \\lambda ( \\Omega , \\Sigma ) . \\end{align*}"}
{"id": "7737.png", "formula": "\\begin{align*} \\sum _ { k = \\tau _ { s _ 1 } } ^ { \\tau _ { s _ 1 + 1 } - 1 } { A _ { i l _ 1 } ( k ) } \\geq \\frac { 1 } { n ^ 2 } , \\ , \\ , \\sum _ { k = \\tau _ { s _ 2 } } ^ { \\tau _ { s _ 2 + 1 } - 1 } { A _ { l _ 1 l _ 2 } ( k ) } \\geq \\frac { 1 } { n ^ 2 } , \\ , \\ , \\ldots , \\ , \\ , \\sum _ { k = \\tau _ { s _ { r } } } ^ { \\tau _ { s _ { r } + 1 } - 1 } { A _ { l _ { r - 1 } j } ( k ) } \\geq \\frac { 1 } { n ^ 2 } . \\end{align*}"}
{"id": "2495.png", "formula": "\\begin{align*} g ( R ( U , V ) W , S ) = g ( R ^ \\nu ( U , V ) W , S ) + g ( T _ U W , T _ V S ) - g ( T _ V W , T _ U S ) , \\end{align*}"}
{"id": "149.png", "formula": "\\begin{align*} m ^ 2 _ t : = m ^ 2 + \\frac { 1 } { t } \\quad \\Rightarrow \\lim _ { t \\downarrow 0 } m _ t = + \\infty . \\end{align*}"}
{"id": "10.png", "formula": "\\begin{align*} \\tau x = x ^ q \\tau , ~ ~ \\forall x \\in \\Omega . \\end{align*}"}
{"id": "8201.png", "formula": "\\begin{align*} \\psi ( - a / 2 ) = \\psi ( a / 2 ) = 0 , \\end{align*}"}
{"id": "5647.png", "formula": "\\begin{align*} \\begin{cases} \\limsup _ { | x | \\rightarrow \\infty } | x | ^ { \\min \\{ \\beta , n \\} - 2 } \\left | u ( x ) - \\left ( \\frac { 1 } { 2 } x ^ T A x + b \\cdot x + c \\right ) \\right | < \\infty , & \\ \\beta \\neq n , \\\\ \\limsup _ { | x | \\rightarrow \\infty } | x | ^ { n - 2 } ( \\ln | x | ) ^ { - 1 } \\left | u ( x ) - \\left ( \\frac { 1 } { 2 } x ^ T A x + b \\cdot x + c \\right ) \\right | < \\infty , & \\ \\beta = n . \\end{cases} \\end{align*}"}
{"id": "5955.png", "formula": "\\begin{align*} \\sum _ k \\norm { \\omega _ k \\circ \\sigma ^ \\phi _ t } \\norm { V _ { - t } \\zeta _ k } = \\sum _ k \\norm { \\omega _ k } \\norm { \\zeta _ k } , \\end{align*}"}
{"id": "6389.png", "formula": "\\begin{align*} \\pi \\wedge \\pi _ { { } _ S } = \\pi \\wedge \\pi _ { { } _ { \\Phi ( S ) } } , \\ \\ \\forall \\ , S \\subseteq \\{ 1 , \\ldots , k \\} . \\end{align*}"}
{"id": "6144.png", "formula": "\\begin{align*} n l _ h ( u , v ) : = ( \\nabla \\times u ) \\times v , \\end{align*}"}
{"id": "3913.png", "formula": "\\begin{align*} \\hat { \\pi } _ { h , T } ( x ) = \\frac { 1 } { T \\prod _ { l = 1 } ^ d h _ l } \\int _ 0 ^ T \\prod _ { m = 1 } ^ d K ( \\frac { x _ m - X _ u ^ m } { h _ m } ) d u . \\end{align*}"}
{"id": "3417.png", "formula": "\\begin{align*} \\Upsilon = \\{ ( g _ { i , j } ) \\in \\Gamma ( \\sqrt { - 3 } ) : g _ { 1 , 1 } \\equiv 1 \\bmod 3 \\} . \\end{align*}"}
{"id": "5347.png", "formula": "\\begin{align*} 0 = ( \\Lambda _ { L , q _ 1 } - \\Lambda _ { L , q _ 2 } ) [ f ] [ g ] = ( B _ { L , q _ 1 } - B _ { L , q _ 2 } ) ( u _ f , u _ g ^ * ) = ( q _ 1 - q _ 2 ) ( u _ f , u _ g ^ * ) \\end{align*}"}
{"id": "4263.png", "formula": "\\begin{align*} \\min \\limits _ { h = 2 , \\dots , k } \\left | \\xi _ 1 - \\xi _ h \\right | = \\left | \\xi _ 1 - \\xi _ 2 \\right | = 2 \\sin { \\pi \\over k } \\end{align*}"}
{"id": "2593.png", "formula": "\\begin{align*} A _ { 1 2 } \\psi _ { ( a , b ) } = A _ { 3 2 } \\psi _ { ( a , b ) } = A _ { 1 3 } \\psi _ { ( a , b ) } = 0 \\ . \\end{align*}"}
{"id": "2110.png", "formula": "\\begin{align*} ( - L z _ s , z _ s ) _ V & = \\frac { 1 } { 2 } \\sum _ { e \\in E } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum _ { u , v \\in e } ( z _ s ( u ) - z _ s ( v ) ) ^ 2 \\\\ & = \\frac { 1 } { 2 } \\sum _ { e \\in \\partial S } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } 2 | e \\setminus S | | S \\cap e | | V | ^ 2 \\\\ & \\le \\frac { 1 } { 4 } \\delta _ E ( \\max ) | \\partial S | | V | ^ 2 . \\end{align*}"}
{"id": "636.png", "formula": "\\begin{align*} & F _ { 1 , 0 } ( x , y ) = 0 , \\ F _ { 1 , 1 } ( x , y ) = \\frac { 2 } { 3 } \\pi ^ 2 \\Bigl ( x ^ 4 + y ^ 4 \\Bigr ) , \\\\ & \\widetilde { F } _ { 1 , 0 } ( x , y ) = 0 , \\ \\widetilde { F } _ { 1 , 1 } ( x , y ) = \\frac { 2 } { 3 } \\pi ^ 2 \\Bigl ( x ^ 2 + y ^ 2 \\Bigr ) ^ 2 . \\end{align*}"}
{"id": "265.png", "formula": "\\begin{align*} \\Lambda _ i ^ { \\nu t } A _ i \\Lambda _ i ^ { \\nu } = B - \\pi ^ { \\nu } C ^ { \\nu } _ i , \\end{align*}"}
{"id": "115.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\tilde p _ { d , t , \\mu } ( \\lambda ) \\underset { \\mu \\rightarrow \\infty } { = } o ( \\mu ^ { - 1 } ) . \\end{align*}"}
{"id": "7625.png", "formula": "\\begin{align*} \\sum _ { e \\in E _ i } t '' _ e = 2 n _ i \\ , . \\end{align*}"}
{"id": "2381.png", "formula": "\\begin{align*} a _ i = a _ { \\theta 0 } \\left ( \\partial _ i L ( h _ \\theta ) \\right ) \\end{align*}"}
{"id": "4956.png", "formula": "\\begin{align*} Y ^ n _ t = n ^ { \\alpha + \\frac 1 2 } ( X ^ n _ t - X _ t ) . \\end{align*}"}
{"id": "7719.png", "formula": "\\begin{align*} \\pi _ { S } ^ T ( k + 1 ) A _ { S \\bar { S } } ( k ) + \\pi _ { \\bar { S } } ^ T ( k + 1 ) A _ { \\bar { S } } ( k ) = \\pi _ { \\bar { S } } ^ T ( k ) \\end{align*}"}
{"id": "1860.png", "formula": "\\begin{align*} ( A _ { 0 } ( z ) , \\ldots , A _ { p - 1 } ( z ) ) = \\frac { ( 1 , \\ldots , 1 ) } { ( 0 , \\ldots , 0 , z - a _ { 0 } ^ { ( 0 ) } ) + ( A _ { 0 } ^ { ( 1 ) } ( z ) , \\ldots , A _ { p - 2 } ^ { ( 1 ) } ( z ) , - \\sum _ { j = 1 } ^ { p } a _ { 0 } ^ { ( j ) } A _ { j - 1 } ^ { ( 1 ) } ( z ) ) } . \\end{align*}"}
{"id": "4803.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , 1 ] } \\big | W _ t - W ^ { ( n ) } _ t - ( W _ 1 - W ^ { ( n ) } _ 1 ) t \\big | = o _ P ( n ^ { - 1 / 2 + p / 4 } ) . \\end{align*}"}
{"id": "4028.png", "formula": "\\begin{align*} F _ n ( - x ) & = Q ( x ) \\exp \\bigg \\{ - a _ 0 \\frac { x ^ 3 } { \\sqrt { n } } + a _ 1 \\frac { x ^ 4 } { n } \\\\ & - O \\left ( \\frac { x ^ 5 } { n ^ { 3 / 2 } } \\right ) + O \\left ( \\frac { x } { \\sqrt { n } } \\right ) \\bigg \\} . \\end{align*}"}
{"id": "3845.png", "formula": "\\begin{align*} A _ 0 = & \\emptyset , A _ 1 = \\Big \\{ ( 1 ) \\Big \\} , A _ 2 = \\Big \\{ ( 1 , 1 ) , ( 1 , 2 ) \\Big \\} , \\\\ A _ 3 = & \\Big \\{ ( 1 , 1 , 1 ) , ( 1 , 1 , 2 ) , ( 1 , 2 , 1 ) , ( 1 , 2 , 2 ) , ( 1 , 2 , 3 ) \\Big \\} , \\\\ A _ 4 = & \\Big \\{ ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 2 ) , ( 1 , 1 , 2 , 1 ) , ( 1 , 1 , 2 , 2 ) , ( 1 , 1 , 2 , 3 ) , ( 1 , 2 , 1 , 1 ) , ( 1 , 2 , 1 , 2 ) , \\\\ & ( 1 , 2 , 1 , 3 ) , ( 1 , 2 , 2 , 1 ) , ( 1 , 2 , 2 , 2 ) , ( 1 , 2 , 2 , 3 ) , ( 1 , 2 , 3 , 1 ) , ( 1 , 2 , 3 , 2 ) , ( 1 , 2 , 3 , 3 ) , ( 1 , 2 , 3 , 4 ) \\Big \\} , \\end{align*}"}
{"id": "666.png", "formula": "\\begin{align*} \\left | \\left . \\int u _ { 1 , t } d \\nu ^ 2 _ { t } \\right | _ { t = \\bar s } ^ { t = s ' } \\ , \\right | & \\le C \\varepsilon ( 1 + \\sqrt { s ' - s } ) \\leq C \\varepsilon ( 1 + s ' - s ) . \\end{align*}"}
{"id": "5417.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s m - \\frac { ( - \\Delta ) ^ s m _ 1 } { \\gamma _ 1 ^ { 1 / 2 } } m & = 0 \\quad \\Omega , \\\\ m & = 0 \\quad \\Omega _ e . \\end{align*}"}
{"id": "7984.png", "formula": "\\begin{align*} I _ { D _ { Y , - } , d } = \\frac { 1 } { \\prod _ { i \\in I _ - , D _ i \\cdot d > 0 } ( \\bar D _ i + ( D _ i \\cdot d ) z ) } [ \\textbf { 1 } ] _ { ( - D _ i \\cdot d ) _ { i \\in I _ - } } . \\end{align*}"}
{"id": "8746.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t \\psi ( t ) = A _ { 1 } ( t ) \\psi ( t ) , \\end{array} \\end{align*}"}
{"id": "4663.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } t ^ { \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| u ( t , \\cdot ) - A \\Psi _ t \\| _ { q , h } & = \\lim _ { t \\to \\infty } t ^ { \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| u ( t + 1 , \\cdot ) - A \\Psi _ { t + 1 } \\| _ { q , h } \\\\ & = \\lim _ { t \\to \\infty } t ^ { \\frac { d - 2 \\delta } { \\alpha } ( 1 - \\frac { 1 } { q } ) } \\| \\tilde P _ t \\left ( \\tilde P _ 1 f - A \\Psi _ 1 \\right ) \\| _ { q , h } = 0 . \\end{align*}"}
{"id": "8333.png", "formula": "\\begin{align*} c U + U ^ * b = 0 \\ , . \\end{align*}"}
{"id": "1438.png", "formula": "\\begin{align*} \\prod _ { v \\in { { \\mathfrak { M } } } _ K } | \\xi | _ v = 1 \\ \\ \\ \\ \\xi \\in K \\setminus \\{ 0 \\} \\enspace . \\end{align*}"}
{"id": "6666.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\xi _ 1 ( i ) = \\left \\{ \\begin{array} { l l } 1 , & | i - h _ 0 | < \\frac { h _ 0 } { 4 } , \\\\ 0 , & | i - h _ 0 | > \\frac { h _ 0 } { 2 } , \\end{array} \\right . & & \\xi _ 2 ( i ) = \\left \\{ \\begin{array} { l l } 1 , & | i - g _ 0 | < \\frac { g _ 0 } { 4 } , \\\\ 0 , & | i - g _ 0 | > \\frac { g _ 0 } { 2 } . \\end{array} \\right . \\end{array} \\end{align*}"}
{"id": "4905.png", "formula": "\\begin{align*} \\mathrm { q a d j } ( A ) A = A \\mathrm { q a d j } ( A ) = \\det ( A ) E . \\end{align*}"}
{"id": "7148.png", "formula": "\\begin{align*} ( g _ { 1 1 } y _ 1 + g _ { 1 2 } y _ 2 + g _ { 1 3 } y _ 3 ) ^ a = g _ { 1 1 } ^ a y _ 1 ^ a + g _ { 1 2 } ^ a y _ 2 ( y _ 1 + y _ 2 ) ^ { a - 1 } + \\gamma y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { b - 1 } , \\end{align*}"}
{"id": "6701.png", "formula": "\\begin{align*} J _ { w , \\sigma , \\lambda } \\circ \\phi ( \\sigma , \\lambda ) = \\phi ( w \\sigma , w \\lambda ) \\circ J _ { w , \\sigma , \\lambda } \\end{align*}"}
{"id": "1347.png", "formula": "\\begin{align*} \\phi ( x , v ) = e ^ { \\chi ( | x | ^ 2 + | v ^ 2 | ) } . \\end{align*}"}
{"id": "3966.png", "formula": "\\begin{align*} \\iota ( r ( e ) , z ) \\ , e = e \\ , \\iota ( s ( e ) , z ) \\quad \\end{align*}"}
{"id": "8369.png", "formula": "\\begin{align*} F ( k , s , t ) & \\leq F ^ * ( k , s , t ) , \\\\ F ( k , s , t ) & \\leq F ( k + 1 , s , t ) , \\\\ 2 \\ , F ( k , 2 , 2 u ) & = F ( k + 1 , 2 , 2 u + 1 ) , \\\\ 2 \\ , F ^ * ( k , 2 , 2 u ) & = F ^ * ( k + 1 , 2 , 2 u + 1 ) . \\end{align*}"}
{"id": "5101.png", "formula": "\\begin{align*} & E \\left [ \\sigma ' ( X _ { \\eta _ n ( s _ 1 ) - \\delta } ) M _ { s _ 1 } ^ { n , 3 } \\sigma ' ( X _ { \\eta _ n ( s _ 2 ) - \\delta } ) M _ { s _ 2 } ^ { n , 3 } \\right ] \\\\ & = E \\left [ \\sigma ' ( X _ { \\eta _ n ( s _ 1 ) - \\delta } ) M _ { s _ 1 } ^ { n , 3 } \\sigma ' ( X _ { \\eta _ n ( s _ 2 ) - \\delta } ) E [ M _ { s _ 2 } ^ { n , 3 } | \\mathcal { F } _ { \\eta _ n ( s _ 2 ) - \\delta } ] \\right ] = 0 . \\end{align*}"}
{"id": "7800.png", "formula": "\\begin{align*} \\nu ( g ) = \\left \\{ \\begin{array} { c c c } 1 & \\mbox { i f } & g \\ \\mbox { i s o r i e n t a t i o n p r e s e r v i n g } \\\\ - 1 & \\mbox { i f } & g \\ \\mbox { i s o r i e n t a t i o n r e v e r e s i n g } . \\end{array} \\right . \\end{align*}"}
{"id": "2039.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : p o l y n o m i a l s o b o l e v } \\| \\mu - \\nu \\| _ { H ^ { - s } } ^ 2 & = \\int _ { \\R ^ { 2 d } } \\Phi _ s ( x - y ) ( \\mu ^ { \\otimes 2 } - \\mu \\otimes \\nu ) ( \\dd x , \\dd y ) \\\\ & \\qquad \\qquad + \\int _ { \\R ^ { 2 d } } \\Phi _ s ( x - y ) ( \\nu ^ { \\otimes 2 } - \\nu \\otimes \\mu ) ( \\dd x , \\dd y ) \\\\ & = \\int _ { \\R ^ { 2 d } } \\Phi _ { s } ( x - y ) ( \\mu - \\nu ) ^ { \\otimes 2 } ( \\dd x , \\dd y ) , \\end{align*}"}
{"id": "7858.png", "formula": "\\begin{align*} | \\varphi ( S ) - \\varphi ( \\alpha _ u S \\alpha _ u ^ * ) | \\leq 2 \\| S \\| \\varphi ( \\sum _ { t \\not \\in F } p _ t ) ^ { 1 / 2 } + \\| S \\| \\varphi ( 1 - \\sum _ { t \\in F } \\alpha _ { t ^ { - 1 } } ( p _ t ) ) ^ { 1 / 2 } = 3 \\| S \\| \\varphi ( \\sum _ { t \\not \\in F } p _ t ) ^ { 1 / 2 } . \\end{align*}"}
{"id": "2076.png", "formula": "\\begin{align*} P ^ { \\nu } _ { s + t } ( x , \\cdot ) = \\int _ E P ^ { \\overline { S } _ t ( \\nu ) } _ { s } ( y , \\cdot ) P ^ { \\nu } _ { t } ( x , \\dd y ) . \\end{align*}"}
{"id": "8954.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } i \\partial _ t u + \\Delta u & = \\pm | u | ^ { \\frac { 4 } { d - 2 } } u ( t , x ) \\in \\R \\times \\R ^ d , \\\\ u ( 0 ) & = u _ 0 \\in M ^ { 1 + \\varepsilon } _ { 4 , 2 } ( \\R ^ d ) . \\end{array} \\right . \\end{align*}"}
{"id": "977.png", "formula": "\\begin{align*} t ^ { - 1 } ( f ( x ) g ( x ) ) = \\overline { f } \\otimes \\overline { g } . \\end{align*}"}
{"id": "3871.png", "formula": "\\begin{align*} \\inf _ { X \\not = Y } \\frac { \\langle | X + Y | ^ { p - 2 } ( X + Y ) - | X | ^ { p - 2 } X , Y \\rangle } { ( | X | + | Y | ) ^ { p - 2 } | Y | ^ 2 } > 0 . \\end{align*}"}
{"id": "1806.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m i n i m i z e } & f ( x ) + h ( x ) \\\\ \\mbox { s u b j e c t t o } & A x = b . \\end{array} \\end{align*}"}
{"id": "8620.png", "formula": "\\begin{align*} d i m _ x f ^ { - 1 } ( f ( x ) ) = d i m \\mathcal { O } _ { X _ y , x } + t r . d e g \\kappa ( x ) / \\kappa ( y ) . \\end{align*}"}
{"id": "8242.png", "formula": "\\begin{align*} k _ { \\pm } = \\frac { 1 } { \\sqrt { 4 \\beta \\hslash ^ { 2 } / 3 } } \\left ( \\sqrt { \\frac { \\sqrt { ( 1 + | a | ) ^ { 2 } + | b | ^ { 2 } } + ( 1 + | a | ) } { 2 } } \\pm i \\sqrt { \\frac { \\sqrt { ( 1 + | a | ) ^ { 2 } + | b | ^ { 2 } } - ( 1 + | a | ) } { 2 } } \\right ) \\end{align*}"}
{"id": "1699.png", "formula": "\\begin{align*} & \\frac { e ^ { - t _ 0 | \\xi | ^ 2 } - e ^ { - t _ 0 ( | \\xi - \\eta | ^ 2 + | \\eta | ^ 2 ) } } { | \\xi - \\eta | ^ 2 + | \\eta | ^ 2 - | \\xi | ^ 2 } = t _ 0 e ^ { - t _ 0 | \\xi | ^ 2 } \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k + 1 } \\frac { \\big ( t _ 0 ( | \\xi - \\eta | ^ 2 + | \\eta | ^ 2 - | \\xi | ^ 2 ) \\big ) ^ { k - 1 } } { k ! } \\\\ = & t _ 0 e ^ { - t _ 0 | \\xi | ^ 2 } + t _ 0 e ^ { - t _ 0 | \\xi | ^ 2 } \\sum _ { k = 2 } ^ \\infty ( - 1 ) ^ { k + 1 } \\frac { \\big ( t _ 0 ( | \\xi - \\eta | ^ 2 + | \\eta | ^ 2 - | \\xi | ^ 2 ) \\big ) ^ { k - 1 } } { k ! } : = t _ 0 e ^ { - t _ 0 | \\xi | ^ 2 } + G ( \\xi - \\eta , \\eta ) . \\end{align*}"}
{"id": "2660.png", "formula": "\\begin{align*} g _ \\alpha ( p ) & = p ^ { 3 n - k - \\ell - 9 } + ( n - \\ell - 1 ) ( p - 1 ) ( p + 1 ) p ^ { 2 n - k - 7 } \\\\ & + ( n - k - 1 ) ( p - 1 ) p ^ { 2 n - \\ell - 6 } - ( n - \\ell - 1 ) p ^ { n - 4 } ( p - 1 ) \\\\ & + ( n - k - 2 ) ( n - \\ell - 1 ) ( p - 1 ) ^ 2 p ^ { n - 4 } \\\\ & + \\left ( ( n - \\ell - 2 ) + \\binom { n - \\ell - 2 } { 2 } \\right ) ( p - 1 ) ( p - 2 ) ( p - 3 ) p ^ { n - 4 } . \\end{align*}"}
{"id": "5479.png", "formula": "\\begin{align*} \\mathrm { d i s t } ( h _ d ( x ) , f ( x ) ) \\leq 2 C ' \\sup _ { x \\in M } \\abs { \\Phi ( x ) } = D . \\end{align*}"}
{"id": "3583.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { y \\leq x } g ( y ) \\quad g ( x ) = \\sum \\limits _ { y \\leq x } \\mu ( y , x ) \\cdot f ( y ) \\end{align*}"}
{"id": "1112.png", "formula": "\\begin{align*} D _ { \\frac { \\partial } { \\partial \\sigma } } \\frac { \\partial } { \\partial \\sigma } = \\frac { 1 } { \\sigma } \\frac { \\partial } { \\partial \\sigma } \\mbox { o r } - \\frac { 3 } { \\sigma } \\frac { \\partial } { \\partial \\sigma } . \\end{align*}"}
{"id": "8881.png", "formula": "\\begin{align*} | A | \\times | A _ { i j | k l } | = \\begin{vmatrix} | A _ { i | k } | & | A _ { i | l } | \\\\ | A _ { j | k } | & | A _ { j | l } | \\end{vmatrix} . \\end{align*}"}
{"id": "1392.png", "formula": "\\begin{align*} z & = \\chi ( t / \\tilde { T } ) \\left ( W _ t ^ \\gamma T [ u _ 0 , \\cdots , u _ 0 ] - \\frac { f } { \\bigtriangleup + i \\gamma } \\right ) \\\\ & \\qquad - i \\chi ( t / \\tilde { T } ) \\int _ 0 ^ t W _ { t - s } ^ \\gamma \\left ( \\mathfrak { F } ( v ) + \\mathcal { H L } _ B [ z , v , \\cdots , v ] \\right ) \\ , d s . \\end{align*}"}
{"id": "3351.png", "formula": "\\begin{align*} \\widehat { \\varphi } _ { i , j } ( z ) = [ \\lambda \\in \\sigma ( w _ { i , j } ( z ) ) : | \\lambda | = 1 ] . \\end{align*}"}
{"id": "2659.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in S _ { 3 , 2 } } g _ \\alpha ( p ) & = \\frac { 1 } { 6 ( p - 1 ) } \\bigg ( 6 ( n - 1 ) p ^ { 2 n - 5 } + ( n - 1 ) ( n - 2 ) ( n - 3 ) p ^ { n } \\\\ & - 3 ( n - 1 ) ( n - 2 ) ( n - 4 ) p ^ { n - 1 } + ( n - 1 ) ( n - 2 ) ( 5 n - 2 4 ) p ^ { n - 2 } \\\\ & - 3 ( n - 1 ) ( n - 3 ) ( n - 4 ) p ^ { n - 3 } \\bigg ) . \\end{align*}"}
{"id": "2131.png", "formula": "\\begin{align*} H ( z ) = a + b z + \\int _ \\mathbb { R } \\frac { 1 + s z } { z - s } \\ , d \\rho ( s ) , z \\in \\mathbb { C } ^ + , \\end{align*}"}
{"id": "81.png", "formula": "\\begin{align*} \\bigcup _ { m \\in \\N } & \\phi _ m \\big ( [ 0 , T ] \\times \\phi _ m ( s _ m '' , p _ m ) \\big ) \\\\ & = \\Big ( \\bigcup _ { m < m _ \\epsilon } \\phi _ m \\big ( [ 0 , T ] \\times \\phi _ m ( s _ m '' , p _ m ) \\big ) \\Big ) \\cup \\Big ( \\bigcup _ { m \\geq m _ \\epsilon } \\phi _ m \\big ( [ 0 , T ] \\times \\phi _ m ( s _ m '' , p _ m ) \\big ) \\Big ) \\end{align*}"}
{"id": "1854.png", "formula": "\\begin{align*} \\mathcal { D } _ { [ n , j ] } ^ { ( q ) } & : = \\{ \\gamma + q : \\gamma \\in \\mathcal { D } _ { [ n , j ] } \\} q \\in \\mathbb { Z } _ { \\geq 0 } , \\\\ \\widehat { \\mathcal { D } } _ { [ n , j ] } ^ { ( q ) } & : = \\{ \\gamma - q : \\gamma \\in \\widehat { \\mathcal { D } } _ { [ n , j ] } \\} q \\in \\mathbb { Z } _ { \\geq 0 } . \\end{align*}"}
{"id": "1253.png", "formula": "\\begin{align*} P ( T , x ) & = \\prod _ { j = 1 } ^ { l ( T ) } \\mathcal { G } _ j ^ { n ( T , j ) } = \\prod _ { j = 1 } ^ { l ( T ) } \\left ( \\dfrac { W _ { l ( T ) + 1 - j } } { W _ { l ( T ) - j } } \\right ) ^ { n ( T , j ) } = \\dfrac { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ { l ( T ) + 1 - j } ^ { n ( T , j ) } } { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ { l ( T ) - j } ^ { n ( T , j ) } } . \\end{align*}"}
{"id": "8912.png", "formula": "\\begin{align*} - \\tilde u '' _ i + \\tilde \\lambda \\tilde u _ i = \\tilde u _ i ^ { p - 1 } , \\tilde u _ i \\ge 0 \\end{align*}"}
{"id": "7683.png", "formula": "\\begin{align*} P ( x ) : = \\frac { 1 } { 2 } \\norm { g ( x ) } ^ 2 + \\frac { 1 } { 2 } \\norm { \\left [ - X ( x ) \\right ] _ { + } } _ { \\mathrm { F } } ^ 2 . \\end{align*}"}
{"id": "4617.png", "formula": "\\begin{align*} c _ 1 ( \\alpha , g ) = s t \\ ; \\ ; \\ ; \\ ; \\ ; c _ 2 ( \\alpha , g ) = u r \\end{align*}"}
{"id": "7534.png", "formula": "\\begin{align*} & \\bigg \\| 1 _ { \\{ | T _ { n } - 1 | \\le 1 \\} } \\sup _ { t \\in [ 0 , 1 ] } | X _ { n } ( t ) - B ( t ) | \\bigg \\| _ { L ^ { \\frac { p } { 2 } } } \\\\ \\le & \\bigg \\| 1 _ { \\{ | T _ { n } - 1 | \\le 1 \\} } \\sup _ { t \\in [ 0 , 1 ] } | X _ { n } ( t ) - B ( T _ { k } ) | \\bigg \\| _ { L ^ { \\frac { p } { 2 } } } + \\bigg \\| 1 _ { \\{ | T _ { n } - 1 | \\le 1 \\} } \\sup _ { t \\in [ 0 , 1 ] } | B ( T _ { k } ) - B ( t ) | \\bigg \\| _ { L ^ { \\frac { p } { 2 } } } \\\\ = : & I _ 1 + I _ 2 . \\end{align*}"}
{"id": "8641.png", "formula": "\\begin{align*} \\widehat { H } _ { n H } ( t ) : = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\left ( { \\widehat { p } } _ i ^ 2 + \\omega ^ 2 ( t ) \\widehat { x } _ i ^ 2 \\right ) + c ( t ) { U _ { n H } } ( { \\widehat { x } } _ 1 , \\ldots , { \\widehat { x } } _ n ) , \\end{align*}"}
{"id": "7387.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } c _ n = 0 , \\ \\ \\int _ { \\mathbb { R } ^ N } \\sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \\frac { \\partial U _ { h _ n , i } } { \\partial h _ n } \\left ( \\psi _ n + c _ n \\sum _ { i = 1 } ^ 4 U _ { h _ n , i } ^ { p - 1 } \\frac { \\partial U _ { h _ n , i } } { \\partial h _ n } \\right ) ~ d y = 0 . \\end{align*}"}
{"id": "6015.png", "formula": "\\begin{align*} M = \\left ( \\mu _ 0 , \\mu _ 1 , \\ldots , \\mu _ n \\right ) \\end{align*}"}
{"id": "3501.png", "formula": "\\begin{align*} g _ { 1 } ( u ) = \\frac { \\exp ( - u ) } { [ 1 + \\exp ( - u ) ] ^ { 2 } } , \\end{align*}"}
{"id": "8335.png", "formula": "\\begin{align*} \\rho _ g ( a ) = \\mathcal { U } _ g a \\mathcal { U } _ g ^ { - 1 } . \\end{align*}"}
{"id": "2027.png", "formula": "\\begin{align*} \\label [ I e q ] { e q : g e n e r a l L 2 } L ^ { ( 2 ) } \\varphi _ 2 ( z _ 1 , z _ 2 ) = \\lambda ( z _ 1 , z _ 2 ) \\iint _ { E \\times E } \\{ \\varphi _ 2 ( z _ 1 ' , z _ 2 ' ) - \\varphi _ 2 ( z _ 1 , z _ 2 ) \\} \\Gamma ^ { ( 2 ) } ( z _ 1 , z _ 2 , \\dd z _ 1 ' , \\dd z _ 2 ' ) . \\end{align*}"}
{"id": "2511.png", "formula": "\\begin{align*} \\begin{array} { l l } g ( ( \\nabla _ U A ) _ { X _ j } X _ j , V ) & = g ( \\nabla _ U g ( X _ j , X _ j ) H ' - g ( \\mathcal { H } \\nabla _ U X _ j , X _ j ) H ' \\\\ & - g ( X _ j , \\mathcal { H } \\nabla _ U X _ j ) H ' , V ) . \\end{array} \\end{align*}"}
{"id": "7518.png", "formula": "\\begin{align*} P ' ( f p ) = \\sum _ { x \\in N _ { s o r t e d } } \\sum _ { y \\in S _ x } f _ { f p } ( m _ { x } , \\gamma _ { x } , k _ { x } ) f _ { f p } ( m _ { y } , \\gamma _ { y } , k _ { y } ) . \\end{align*}"}
{"id": "6636.png", "formula": "\\begin{align*} \\left \\| \\rho _ { p , \\ell } ( \\beta _ h ) \\phi _ w ( ( \\cdot ) ^ { - 1 } , e ) - \\int _ { X _ { p , \\ell } } \\phi _ w \\ , d \\mu _ { p , \\ell } \\right \\| _ { L ^ 2 ( Q _ { \\ge R } ) } & \\ll _ { w , p , \\ell } \\left \\| \\omega _ R \\right \\| _ { L ^ { 2 s } ( X _ { p , \\ell } ) } \\\\ & = \\mu _ { p , \\ell } ( \\Omega _ R ) ^ { 1 / ( 2 s ) } \\\\ & \\le m _ \\infty ( Q _ { \\ge R } ) ^ { 1 / ( 2 s ) } \\ll _ s R ^ { - 1 / s } . \\end{align*}"}
{"id": "6175.png", "formula": "\\begin{align*} \\frac { d } { d t } g ^ { T } ( x , t ) = - ( R i c ^ { T } ( x , t ) - \\varkappa g ^ { T } ( x , t ) ) \\end{align*}"}
{"id": "8719.png", "formula": "\\begin{align*} h = ( 1 - b ^ 2 ) ^ { \\frac { 3 } { 2 } } ( x ^ { 1 } \\dot { x } ^ { 2 } - x ^ { 2 } \\dot { x } ^ { 1 } ) + \\lambda \\left ( \\sqrt { ( \\dot { x } ^ { 1 } ) ^ 2 + ( \\dot { x } ^ { 2 } ) ^ 2 } + b \\cos ( t + c ) \\dot { x } ^ 1 + \\sin ( t + c ) \\dot { x } ^ 2 \\right ) . \\end{align*}"}
{"id": "4211.png", "formula": "\\begin{align*} \\hat g ( \\zeta ) = \\int _ { \\R ^ { 2 n } } g ( z ) e ^ { - i \\langle \\zeta , z \\rangle _ { \\R ^ { 2 n } } } \\ , d z , \\zeta \\in \\R ^ { 2 n } . \\end{align*}"}
{"id": "3972.png", "formula": "\\begin{align*} j ( a ^ { * } ) ( \\sigma ) = j ( a ) ( \\sigma ^ { - 1 } ) ^ { * } j ( a * b ) ( \\sigma ) = \\sum _ { q ( \\tau ) \\in G ^ { r ( \\sigma ) } } j ( a ) ( \\tau ) \\ , \\alpha _ { \\tau } \\bigl ( j ( b ) ( \\tau ^ { - 1 } \\sigma ) \\bigr ) \\end{align*}"}
{"id": "2298.png", "formula": "\\begin{align*} \\mathcal { E } _ h ^ { ( n ) } = & ( \\chi - 1 ) f _ h ^ { ( n ) } + \\mathcal { \\tilde E } _ h ^ { ( n ) } , \\end{align*}"}
{"id": "4588.png", "formula": "\\begin{align*} \\partial _ t m _ t ( z ) = - m _ t \\partial _ z m _ t ( z ) , m _ 0 ( z ) = m ( z ) , \\end{align*}"}
{"id": "2756.png", "formula": "\\begin{align*} \\gamma : = & \\gamma ( n , \\sigma , p ) = \\int _ { B _ { 1 } ^ { c } } \\frac { d y } { | y | ^ { n + \\sigma p } } = \\frac { | \\mathbb { S } ^ { n - 1 } | } { \\sigma p } , \\\\ \\tau : = & \\tau ( n , \\sigma , p , s ) = \\int _ { B _ { 1 } ^ { c } } \\frac { d y } { | y | ^ { n + \\sigma p + s ( p - 1 ) } } = \\frac { | \\mathbb { S } ^ { n - 1 } | } { \\sigma p + s ( p - 1 ) } . \\end{align*}"}
{"id": "7016.png", "formula": "\\begin{align*} \\dot { x } _ i & = v _ { i } \\left ( \\cos { \\psi _ i } - \\sin { \\psi _ i } \\tan { \\beta _ i } \\right ) \\\\ \\dot { y } _ i & = v _ { i } \\left ( \\sin { \\psi _ i } + \\cos { \\psi _ i } \\tan { \\beta _ i } \\right ) \\\\ \\dot { \\psi } _ i & = \\frac { v _ { i } } { l _ r } \\tan { \\beta _ i } \\\\ \\dot { \\beta } _ i & = \\omega _ i \\\\ \\dot { v } _ { i } & = a _ { i } , \\end{align*}"}
{"id": "5439.png", "formula": "\\begin{align*} \\frac { 1 } { p _ 2 } = \\frac { \\theta } { p _ 1 } + \\frac { 1 } { r _ 2 } , \\end{align*}"}
{"id": "1561.png", "formula": "\\begin{align*} \\mathcal { F } ( x , z ) = \\frac { v o l ( B ^ n ( 1 ) ) } { v o l ( D ^ n _ x ( 1 ) ) } . \\end{align*}"}
{"id": "6716.png", "formula": "\\begin{align*} \\phi _ { \\sigma ' } = 0 \\sigma ' \\in \\widehat { M } \\vert \\sigma ' \\vert > \\vert \\sigma \\vert . \\end{align*}"}
{"id": "5279.png", "formula": "\\begin{align*} \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 1 ( i ) } { r + k _ 1 ( 0 ) } a ^ { \\widehat { P } ^ { C o n t , + r } } + & \\frac { 1 + \\sum _ { i = 0 } ^ h k _ 2 ( i ) } { s + k _ 2 ( 0 ) } b ^ { \\widehat { P } ^ { C o n t , + s } } \\\\ & = 1 + \\sum _ { j = 1 } ^ h \\left ( \\frac { k _ 1 ( j ) } { k _ 1 ( 0 ) + k _ 1 ( j ) } a ^ { \\widehat { P } ^ { C o n t } _ j } + \\frac { k _ 2 ( j ) } { k _ 2 ( 0 ) + k _ 2 ( j ) } b ^ { \\widehat { P } ^ { C o n t } _ j } \\right ) . \\end{align*}"}
{"id": "6880.png", "formula": "\\begin{align*} | \\epsilon ( t ) | = | N ( t ) - N ( B ( t ) , C ( t ) ) | & \\leq \\int _ { \\mathbb { R } } | N ( B ( t ) + y , C ( t ) ) - N ( B ( t ) , C ( t ) ) | p _ { t , 0 } ( y ) d y \\\\ & \\leq \\frac { 1 } { V _ F } \\int _ { \\mathbb { R } } | y | p _ { t , 0 } ( y ) d y = \\frac { 1 } { V _ F } e ^ { - t } \\int _ { \\mathbb { R } } | y | p _ { } ( y ) d y . \\end{align*}"}
{"id": "4898.png", "formula": "\\begin{align*} \\dot { h } ( t ) = R \\tilde { u } ( t ) , \\dot { R } = 0 , \\end{align*}"}
{"id": "4071.png", "formula": "\\begin{align*} A = \\bigcup \\limits _ { l = 0 } ^ m D ' _ l \\cup D _ { n + 1 } . \\end{align*}"}
{"id": "4051.png", "formula": "\\begin{align*} \\hat { \\beta } _ { \\alpha } = \\hat { \\beta } _ { O L S } + \\alpha ( Y ^ { \\prime } H _ { 1 } Y , \\ldots , Y ^ { \\prime } H _ { k } Y ) ^ { \\prime } \\end{align*}"}
{"id": "4862.png", "formula": "\\begin{align*} P _ { i } = N _ { i } ^ { 1 / 9 - 2 \\varepsilon } , Q _ { i } = N _ { i } ^ { 8 / 9 + \\varepsilon } \\end{align*}"}
{"id": "8529.png", "formula": "\\begin{align*} \\psi _ { m + 1 } ( t ) = 2 ( t ^ 2 + 1 ) \\phi _ m ( t ) + 4 a _ { m + 1 } \\theta _ m ( t ) ; \\\\ \\theta _ { m + 1 } ( t ) = 2 ( t ^ 2 - 1 ) \\phi _ m ( t ) + 4 a _ { m + 1 } \\psi _ m ( t ) . \\end{align*}"}
{"id": "1401.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} B & { \\bf 0 } \\\\ { \\bf 0 } & C \\end{matrix} \\right ] , B = \\left [ \\begin{matrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ \\epsilon & 0 & 1 \\end{matrix} \\right ] , \\end{align*}"}
{"id": "1073.png", "formula": "\\begin{align*} R ( w + i y , A ) = R ( x + i y , A ) \\sum _ { k = 0 } ^ \\infty R ( x + i y , A ) ^ k ( x - w ) ^ k , \\end{align*}"}
{"id": "8299.png", "formula": "\\begin{align*} \\hat { H } \\phi _ { n } ^ { \\pm } ( x ) = E _ { n } \\phi _ { n } ^ { \\pm } ( x ) , \\end{align*}"}
{"id": "6176.png", "formula": "\\begin{align*} \\frac { d } { d t } \\varphi = \\log \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } + \\varphi _ { \\alpha \\overline { \\beta } } ) - \\log \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } ) + \\kappa \\varphi - F . \\end{align*}"}
{"id": "3525.png", "formula": "\\begin{align*} L _ { 2 } ^ { \\ast \\ast } & = \\frac { \\xi _ { p } ( 3 + \\xi _ { p } ^ { 2 } + 3 | \\xi _ { p } | ) \\exp ( - | \\xi _ { p } | ) - \\xi _ { q } ( 3 + \\xi _ { q } ^ { 2 } + 3 | \\xi _ { q } | ) \\exp ( - | \\xi _ { q } | ) } { 2 F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } + \\frac { 8 F _ { Y _ { ( 2 ) } } ( \\xi _ { p } , \\xi _ { q } ) } { F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "6380.png", "formula": "\\begin{align*} \\nu _ d = \\mu _ d * N ( 0 , 1 / d ^ 2 ) , \\end{align*}"}
{"id": "8012.png", "formula": "\\begin{align*} H _ { ( Z _ + , D _ { Z , + } ) , f _ + , \\vec d _ + } = H _ { ( Z _ - , D _ { Z , - } ) , f _ - , \\vec d _ - } \\end{align*}"}
{"id": "4017.png", "formula": "\\begin{align*} \\log M ^ * ( n , \\epsilon ) = n C - \\sqrt { n V _ { \\epsilon } } Q ^ { - 1 } ( \\epsilon ) + O ( \\log n ) . \\end{align*}"}
{"id": "7801.png", "formula": "\\begin{align*} \\widetilde { H } _ 0 ( \\phi ) : = \\nu ( \\widetilde { H } ( \\phi ) _ 0 ) \\widetilde { H } ( \\phi ) _ 0 , \\end{align*}"}
{"id": "3726.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { t _ 2 } \\tilde { V } ( s ) \\cdot K ( s , X ( s ) , V ( s ) ) d s , \\tilde { V } ( s ) : = V ( s ) / | V ( s ) | . \\end{align*}"}
{"id": "94.png", "formula": "\\begin{align*} \\dot \\kappa _ t = \\frac { 1 } { t } - \\frac { \\chi _ t } { t ^ 2 } . \\end{align*}"}
{"id": "3737.png", "formula": "\\begin{align*} \\big ( ( \\hat { v } + \\omega ) \\times B \\big ) _ 1 = ( \\hat { v } + \\omega ) _ 2 B _ 3 - ( \\hat { v } + \\omega ) _ 3 B _ 2 , \\big ( ( \\hat { v } + \\omega ) \\times B \\big ) _ 2 = - ( \\hat { v } + \\omega ) _ 1 B _ 3 + ( \\hat { v } + \\omega ) _ 3 B _ 1 \\end{align*}"}
{"id": "6597.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { x \\in V _ 1 } d ^ { J ' } ( x ) \\frac { p ^ 2 n ^ 2 } { 1 2 } - \\sum _ { x \\in V _ 1 } \\frac { p ^ 2 n ^ 2 } { 1 0 0 } d ^ { J ' } ( x ) \\ge \\frac { 1 } { 4 } \\sum _ { x \\in V _ 1 } d ^ { J ' } ( x ) \\frac { p ^ 2 n ^ 2 } { 1 2 } = \\frac { 1 } { 2 4 } | J ' | p ^ 2 n ^ 2 . \\end{align*}"}
{"id": "4523.png", "formula": "\\begin{align*} \\tilde g = 2 d \\tau d u + ( | \\nabla u | ^ { - 2 } + a _ 0 ^ 2 + \\rho ^ { - 2 } b _ 0 ^ 2 ) d u ^ 2 + 2 a _ 0 d u d \\rho + 2 b _ 0 d u d \\theta + d \\rho ^ 2 + \\rho ^ 2 d \\theta ^ 2 \\end{align*}"}
{"id": "40.png", "formula": "\\begin{align*} \\Tilde { \\mathcal { N } } ( f ) ( v _ n ) = \\operatorname { N } _ { n + 1 , n } ( f ( v _ { n + 1 } ) ) . \\end{align*}"}
{"id": "6818.png", "formula": "\\begin{align*} \\left \\langle u , v \\right \\rangle _ { H ^ 2 _ { \\frac { 1 } { a } } ( 0 , 1 ) } = \\int _ { 0 } ^ { 1 } \\frac { f v } { a } \\ , d x \\Longleftrightarrow \\int _ { 0 } ^ { 1 } \\biggl ( \\frac { u v } { a } + u '' v '' \\biggr ) d x = \\int _ { 0 } ^ { 1 } \\frac { f v } { a } \\ , d x , \\end{align*}"}
{"id": "2282.png", "formula": "\\begin{align*} - \\Delta v _ e ^ 1 + \\sigma \\Delta g _ e ^ 1 = 0 . \\end{align*}"}
{"id": "2050.png", "formula": "\\begin{align*} \\dd X _ t & = b ( X _ t ) \\dd t + \\sigma \\dd B _ t \\\\ \\dd Y _ t & = b ( Y _ t ) \\dd t + \\sigma ( I _ d - 2 e _ t e _ t ^ \\mathrm { T } ) \\dd B _ t , \\end{align*}"}
{"id": "3497.png", "formula": "\\begin{align*} L _ { 1 } ^ { \\ast } = \\frac { \\Gamma \\left ( ( m + 1 ) / 2 \\right ) \\sqrt { m } \\left [ \\xi _ { p } ^ { 2 } \\left ( 1 + \\frac { \\xi _ { p } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } - \\xi _ { q } ^ { 2 } \\left ( 1 + \\frac { \\xi _ { q } ^ { 2 } } { m } \\right ) ^ { - ( m - 1 ) / 2 } \\right ] } { \\Gamma ( m / 2 ) ( m - 1 ) \\sqrt { \\pi } F _ { Y } ( \\xi _ { p } , \\xi _ { q } ) } , \\end{align*}"}
{"id": "2826.png", "formula": "\\begin{align*} \\begin{aligned} Z ( x _ i ) = f _ i - \\frac { \\mu } { 2 } \\| x _ i \\| ^ 2 \\nabla Z ( x _ i ) = g _ i - \\mu x _ i . \\end{aligned} \\end{align*}"}
{"id": "3256.png", "formula": "\\begin{align*} \\Vert u ^ { \\infty } \\Vert _ { \\mathcal { F } } ^ 2 : = \\sum _ { ( \\ell _ 1 , m _ 1 ) \\in \\Gamma } \\sum _ { ( \\ell _ 2 , m _ 2 ) \\in \\Gamma } \\Big ( \\frac { 2 \\ell _ 1 + 1 } { e k a } \\Big ) ^ { 2 \\ell _ 1 } \\Big ( \\frac { 2 \\ell _ 2 + 1 } { e k a } \\Big ) ^ { 2 \\ell _ 2 } \\left \\vert \\mu _ { ( \\ell _ 1 , m _ 1 ; \\ell _ 2 , m _ 2 ) } \\right \\vert ^ 2 . \\end{align*}"}
{"id": "3907.png", "formula": "\\begin{align*} \\widetilde a _ i = \\widetilde W _ c ( \\mu , \\nu _ { i / I } ) - \\widetilde W _ c ( \\mu , \\nu _ { ( i - 1 ) / I } ) \\forall i \\in \\mathcal I , \\end{align*}"}
{"id": "3772.png", "formula": "\\begin{align*} \\sum _ { i = 3 , 4 } \\big \\| T _ { k , j ; n , l , r } ^ { \\mu , m , i } ( \\mathfrak { m } , E ) ( t , x , \\zeta ) + \\hat { \\zeta } \\times T _ { k , j ; n , l , r } ^ { \\mu , m , i } ( \\mathfrak { m } , B ) ( t , x , \\zeta ) \\big \\| _ { L ^ \\infty _ x } \\lesssim \\| \\mathfrak { m } ( \\cdot , \\zeta ) \\| _ { \\mathcal { S } ^ \\infty } \\big [ 2 ^ { ( 1 - 1 9 . 5 \\epsilon ) M _ { t ^ { \\star } } } \\end{align*}"}
{"id": "2142.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { 0 } \\frac { 1 + s ^ { 2 } } { ( x - s ) ^ { 2 } } \\ , d \\sigma _ { \\mu _ { n } } ( s ) & = n b _ n ^ 2 \\int _ { - \\infty } ^ 0 \\frac { 1 + s ^ 2 } { ( x - b _ n s ) ^ 2 } \\ ; d \\sigma _ \\mu ^ - ( s ) \\\\ & \\leq \\int _ { - \\infty } ^ { - 1 } \\frac { 2 n s ^ { 2 } } { ( x / b _ n - s ) ^ { 2 } } \\ , d \\sigma _ { \\mu } ^ - ( s ) . \\end{align*}"}
{"id": "635.png", "formula": "\\begin{align*} b _ 0 ( s ) = \\widetilde { b } _ 0 ( s ) = V _ \\alpha ( s ) ^ { - 1 } \\zeta ( \\Delta , s ) . \\end{align*}"}
{"id": "5635.png", "formula": "\\begin{align*} \\mathcal { I } _ { 2 , 1 } - \\iint _ { Q _ T } [ \\nabla m \\cdot \\nabla p ] | \\nabla p | ^ 2 = \\iint _ { Q _ T } m \\Delta p | \\nabla p | ^ 2 . \\end{align*}"}
{"id": "8048.png", "formula": "\\begin{align*} s = - 1 - l , l \\geq 0 s = - \\frac { H } { 2 \\pi i } - \\frac { m } { d } - \\bar { f } , m \\geq 1 . \\end{align*}"}
{"id": "310.png", "formula": "\\begin{align*} \\sum _ { 0 \\le i \\le k } \\frac { { \\rm d } ( { \\rm L } _ { A _ { ( i ) } } ) } { 1 + v _ { i } } = \\sum _ { 0 \\le i \\le k } c _ i d _ i \\ \\le \\ \\sum _ { 0 \\le i \\le k } c _ i ( D _ i - D _ { i - 1 } ) = r _ q \\ , { \\rm d } ( { \\rm L } _ n ) \\sum _ { 0 \\le i \\le k } \\frac { \\sqrt { v _ { i + 1 } } - \\sqrt { v _ { i } } } { 1 + v _ { i } } . \\end{align*}"}
{"id": "8100.png", "formula": "\\begin{align*} Z = f ( p \\partial _ { 1 } + q \\partial _ { 2 } ) , \\end{align*}"}
{"id": "2785.png", "formula": "\\begin{align*} \\begin{aligned} \\tfrac { \\sigma _ 0 L } { h _ 0 B } = 1 - \\tfrac { 1 - \\kappa h _ 0 ^ 2 } { 2 \\left [ 2 - ( 1 + \\kappa ) h _ 0 \\right ] } = \\tfrac { \\kappa h _ 0 ^ 2 - 2 ( 1 + \\kappa ) h _ 0 + 3 } { 2 \\left [ 2 - ( 1 + \\kappa ) h _ 0 \\right ] } \\ , \\ , , \\tfrac { \\alpha _ 0 } { B } = \\tfrac { 1 - \\kappa h _ 0 } { 2 - ( 1 + \\kappa ) h _ 0 } \\end{aligned} \\end{align*}"}
{"id": "4006.png", "formula": "\\begin{align*} u _ t = \\sum _ { h = 1 } ^ { H } M ^ { [ h ] } w _ { t - h } , \\end{align*}"}
{"id": "7437.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ { m , 3 } ( t ) - \\delta \\int _ 0 ^ L u _ t \\overline { w } _ x d x = c ( m - 1 ) \\int _ 0 ^ L \\abs { \\omega _ x } ^ 2 d x + \\frac { c m } { 2 } \\int _ 0 ^ { L } \\int _ 0 ^ \\infty \\sigma ' ( s ) \\abs { \\eta _ x } ^ 2 d s d x . \\end{align*}"}
{"id": "2809.png", "formula": "\\begin{align*} \\begin{aligned} - \\kappa \\left ( 1 + \\kappa \\right ) h ^ 3 + \\left [ 3 \\kappa + \\left ( 1 + \\kappa \\right ) ^ 2 \\right ] h ^ 2 - 4 \\left ( 1 + \\kappa \\right ) h + 4 = 0 \\end{aligned} \\end{align*}"}
{"id": "1259.png", "formula": "\\begin{align*} n ( T , l ( T ) + 1 - j ) = c _ { l ( T ) - j } n ( T , l ( T ) - j ) , \\end{align*}"}
{"id": "4687.png", "formula": "\\begin{align*} & ( - 1 ) ^ { k - 1 } \\left ( C _ m ( n ) - \\sum _ { j = - k } ^ k ( - 1 ) ^ j u _ m \\big ( n - j ( 3 j - 1 ) / 2 \\big ) \\right ) = \\sum _ { j = 0 } ^ n C _ m ( j ) \\ , \\widetilde { P } _ k ( n - j ) . \\end{align*}"}
{"id": "6164.png", "formula": "\\begin{align*} \\mathbb { Z } _ { r } & \\rightarrow \\mathbb { C } ^ { 2 } \\\\ \\gamma & \\mapsto \\begin{bmatrix} e ^ { 2 \\pi i \\frac { a _ { 1 } } { r _ { 1 } } } & 0 \\\\ 0 & e ^ { 2 \\pi i \\frac { a _ { 2 } } { r _ { 2 } } } , \\end{bmatrix} \\end{align*}"}
{"id": "8293.png", "formula": "\\begin{align*} k \\left [ \\tan ( k a / 2 ) + \\cot ( k a / 2 ) \\right ] = k ' \\left [ \\tanh ( k ' a / 2 ) - \\coth ( k ' a / 2 ) \\right ] , \\end{align*}"}
{"id": "8961.png", "formula": "\\begin{align*} & \\sum _ { N _ 0 \\geq 1 } \\sum _ { N _ 1 \\geq \\ldots \\geq N _ { \\frac { d + 2 } { d - 2 } } \\geq 1 } \\big | \\int _ 0 ^ T \\int _ { \\R ^ d } v _ { N _ 0 } ( t , x ) \\prod _ { j = 1 } ^ { \\frac { d + 2 } { d - 2 } } u _ { N _ j } ^ { ( j ) } ( t , x ) d x d t \\big | \\\\ & \\lesssim \\| v \\| _ { Y ^ { - s } } \\prod _ { j = 1 } ^ { \\frac { d + 2 } { d - 2 } } \\| u ^ { ( j ) } \\| _ { X ^ s ( [ 0 , T ] ) } . \\end{align*}"}
{"id": "850.png", "formula": "\\begin{align*} \\mathbf { J \\left ( \\boldsymbol { \\theta } \\right ) = \\mathbb { E } } \\left \\{ \\left [ \\frac { \\partial \\ln p \\left ( \\boldsymbol { y } \\mid \\boldsymbol { \\theta } \\right ) } { \\partial \\boldsymbol { \\theta } } \\right ] \\left [ \\frac { \\partial \\ln p \\left ( \\boldsymbol { y } \\mid \\boldsymbol { \\theta } \\right ) } { \\partial \\boldsymbol { \\theta } } \\right ] ^ { T } \\right \\} , \\end{align*}"}
{"id": "7674.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underset { x \\in \\Re ^ n } { \\mbox { m i n i m i z e } } & f ( x ) \\\\ \\mbox { s u b j e c t t o } & g ( x ) = 0 , \\ : X ( x ) \\in \\S ^ d _ + , \\end{array} \\end{align*}"}
{"id": "7642.png", "formula": "\\begin{align*} M = \\begin{pmatrix} f _ 1 ^ 1 & \\ldots & f _ { n + m } ^ 1 \\\\ f _ 1 ^ 2 & \\ldots & f _ { n + m } ^ 2 \\\\ \\vdots & \\ddots & \\vdots \\\\ f _ 1 ^ n & \\ldots & f _ { n + m } ^ n \\\\ \\end{pmatrix} \\end{align*}"}
{"id": "2752.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } ( - \\Delta ) ^ { \\sigma } _ { p } v _ { j } ( x ) = - 1 . \\end{align*}"}
{"id": "283.png", "formula": "\\begin{align*} G ^ 1 ( R _ { \\pi } ) : = \\textup { k e r } ( G ( R _ { \\pi } ) \\rightarrow G ( k ) ) \\end{align*}"}
{"id": "2111.png", "formula": "\\begin{align*} ( - L y _ s , y _ s ) _ V & = \\frac { 1 } { 2 } \\sum \\limits _ { e \\in E } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } \\sum \\limits _ { u , v \\in e } \\left ( y _ s ( u ) - y _ s ( v ) \\right ) ^ 2 \\\\ & = \\frac { 1 } { 2 } \\sum \\limits _ { e \\in \\partial S } \\frac { \\delta _ E ( e ) } { | e | ^ 2 } 2 | S \\cap e | | e - S | \\frac { 1 } { | S | } \\\\ & \\le \\frac { 1 } { 4 } \\sum \\limits _ { e \\in \\partial S } \\delta _ E ( e ) \\frac { 1 } { | S | } \\\\ & \\le \\frac { 1 } { 4 } \\delta _ E ( \\max ) \\frac { | \\partial S | } { | S | } . \\end{align*}"}
{"id": "525.png", "formula": "\\begin{align*} L _ { n s } ( \\dot { x } , x , t ) = \\frac { 1 } { C _ 3 \\dot { x } } \\ , \\end{align*}"}
{"id": "1016.png", "formula": "\\begin{align*} \\lim _ { n _ 1 , \\dots , n _ k \\to \\infty } \\mu ( A _ 0 \\cap T ^ { - n _ 1 } A _ 1 \\cap T ^ { - n _ 1 - n _ 2 } A _ 2 \\cap \\cdots \\cap T ^ { - n _ 1 - n _ 2 - \\cdots - n _ k } A _ k ) = \\mu ( A _ 1 ) \\mu ( A _ 2 ) \\cdots \\mu ( A _ k ) . \\end{align*}"}
{"id": "4620.png", "formula": "\\begin{align*} \\bar c _ i ( \\alpha , g ) = \\bar c _ i ( \\alpha , h ) . \\end{align*}"}
{"id": "6973.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 1 } ^ { \\infty } \\Bigg ( \\frac { z - \\mu _ { k } ^ { 2 } } { z - \\lambda _ { k } ^ { 2 } } \\Bigg ) = 1 - \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { a _ k } { \\lambda _ { k } ^ { 2 } - z } \\end{align*}"}
{"id": "5156.png", "formula": "\\begin{align*} \\begin{pmatrix} n _ 1 - 2 & 1 & 0 & 0 & \\ldots & 0 & 0 & 0 \\\\ 2 & n _ 2 & - 2 & 0 & \\ldots & 0 & 0 & 0 \\\\ 0 & - 1 & n _ 3 & 1 & \\ldots & 0 & 0 & 0 \\\\ 0 & 0 & 2 & n _ 4 & \\ldots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ldots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & 0 & \\ldots & n _ { 2 k - 2 } & - 2 & 0 \\\\ 0 & 0 & 0 & 0 & \\ldots & - 1 & n _ { 2 k - 1 } & 1 \\\\ 0 & 0 & 0 & 0 & \\ldots & 0 & 2 & n _ { 2 k } - 2 \\\\ \\end{pmatrix} . \\end{align*}"}
{"id": "5073.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 1 } ^ { \\infty } \\int _ 0 ^ 1 \\int _ { 0 } ^ { 1 } [ ( y + k ) ^ \\alpha - ( y + k - z ) ^ \\alpha ] [ ( k + y ) ^ \\alpha - k ^ \\alpha ] d z d y \\right ) \\int _ 0 ^ \\tau \\gamma _ s ( \\sigma ' \\sigma ) ^ 2 ( X _ s ) d s . \\end{align*}"}
{"id": "7150.png", "formula": "\\begin{align*} ( g _ { 2 1 } y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 ) \\{ ( g _ { 1 1 } + g _ { 2 1 } ) y _ 1 + g _ { 2 2 } y _ 2 + g _ { 2 3 } y _ 3 \\} ^ { b - 1 } \\\\ = g _ { 2 3 } ^ { b } y _ 3 ( y _ 1 + y _ 2 + y _ 3 ) ^ { b - 1 } . \\end{align*}"}
{"id": "8505.png", "formula": "\\begin{align*} t r ( A ( t , \\delta ) ^ { 9 } ) = t ^ { 9 } - 9 t ^ { 7 } \\delta + 2 7 t ^ { 5 } \\delta ^ 2 - 3 0 t ^ { 3 } \\delta ^ 3 + 9 t \\delta ^ 4 \\in S ^ 9 , \\ ; \\forall t , \\delta \\in R . \\end{align*}"}
{"id": "6058.png", "formula": "\\begin{align*} f ( x ) = x ^ 2 - 2 x = x ( x - \\lambda ) \\end{align*}"}
{"id": "3223.png", "formula": "\\begin{align*} \\lambda _ i \\psi ( v _ i , v _ j ) = \\psi ( \\lambda _ i v _ i , v _ j ) = \\psi ( \\Phi v _ i , v _ j ) = - \\psi ( v _ i , \\Phi v _ j ) = - \\lambda _ j \\psi ( v _ i , v _ j ) . \\end{align*}"}
{"id": "7460.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\abs { \\omega _ x } ^ 2 d x = o ( 1 ) \\int _ 0 ^ { \\infty } \\int _ 0 ^ L \\sigma ( s ) \\abs { \\eta _ x } ^ 2 d x d s = o ( 1 ) , \\end{align*}"}
{"id": "5648.png", "formula": "\\begin{align*} | D ^ m ( f ( x ) - 1 ) | \\leq c _ 0 | x | ^ { - \\beta - m } , \\quad \\forall \\ | x | > r _ 0 , \\ m = 0 , 1 , 2 , 3 \\end{align*}"}
{"id": "3140.png", "formula": "\\begin{align*} E \\doteq \\bigcap \\limits _ { A \\in \\mathcal { U } } \\{ \\rho \\in \\mathcal { U } ^ { \\ast } : \\rho ( \\mathfrak { 1 } ) = 1 , \\ ; \\rho ( | A | ^ { 2 } ) \\geq 0 \\} \\ . \\end{align*}"}
{"id": "2846.png", "formula": "\\begin{align*} \\begin{aligned} x _ i { } & = { } \\frac { U } { L } \\sum \\limits _ { j = i } ^ { N - 1 } h _ j \\\\ g _ i { } & = { } U \\\\ f _ i { } & = { } \\Delta - \\frac { U ^ 2 } { 2 L } \\sum \\limits _ { j = 0 } ^ { i - 1 } h _ j \\Big ( 2 - h _ j \\frac { - \\kappa } { 1 - \\kappa } \\Big ) \\end{aligned} \\end{align*}"}
{"id": "2803.png", "formula": "\\begin{align*} \\begin{aligned} h _ * = \\arg \\max _ { 0 < h { } \\leq { } \\bar { h } ( \\kappa ) } 2 h - h ^ 2 \\frac { - \\kappa } { - ( 1 + \\kappa ) + 2 \\min \\left ( 1 , \\frac { 1 } { h } \\right ) } \\end{aligned} \\end{align*}"}
{"id": "6904.png", "formula": "\\begin{align*} \\xi ( s ) - \\mu = e ^ { s - t } ( \\xi ( t ) - \\mu ) . \\end{align*}"}
{"id": "6538.png", "formula": "\\begin{align*} \\sup \\limits _ { m \\in \\N } \\sum \\limits _ { i = 0 } ^ { m - 1 } \\P \\{ \\ell ^ { ( A ) } _ i > m - i \\} < \\infty \\end{align*}"}
{"id": "6609.png", "formula": "\\begin{align*} \\big | \\left < \\rho _ { S , \\mathcal { O } } ( \\beta ^ * * \\beta ) \\phi _ { x , r } , \\psi _ i \\right > \\big | = \\big | \\left < \\phi _ { x , r } , \\rho _ { S , \\mathcal { O } } ( \\beta ^ * * \\beta ) \\psi _ i \\right > \\big | = \\lambda _ i \\big | \\left < \\phi _ { x , r } , F _ \\pi \\right > \\big | . \\end{align*}"}
{"id": "3139.png", "formula": "\\begin{align*} \\mathcal { U } ^ { + } \\doteq \\{ A \\in \\mathcal { U } : A = \\mathrm { g } _ { \\pi } ( A ) \\} \\subseteq \\mathcal { U } . \\end{align*}"}
{"id": "3597.png", "formula": "\\begin{align*} \\langle a , u _ i \\rangle \\geq n \\mbox { f o r } i = 1 , \\ldots , m \\mbox { a n d } \\langle a , \\gamma _ j \\rangle \\leq r d _ j - 1 \\mbox { f o r s o m e } 1 \\leq j \\leq p . \\end{align*}"}
{"id": "3008.png", "formula": "\\begin{align*} S \\left ( W ^ C _ { \\bullet , \\bullet } ; p \\right ) = \\frac { 2 } { ( - ( K _ S + \\Omega ) ) ^ 2 } \\int _ 0 ^ \\tau h ( u ) d u . \\end{align*}"}
{"id": "6747.png", "formula": "\\begin{align*} H ^ { 0 } \\left ( X , K _ X \\right ) = H ^ { 0 } \\left ( \\mathbb { P } ^ 1 \\times C , K _ { \\mathbb { P } ^ 1 \\times C } \\right ) \\oplus \\bigoplus _ { \\chi \\ne \\chi _ { 0 0 0 } } { H ^ { 0 } \\left ( \\mathbb { P } ^ 1 \\times C , K _ { \\mathbb { P } ^ 1 \\times C } + L _ { \\chi } \\right ) } , \\end{align*}"}
{"id": "7529.png", "formula": "\\begin{align*} T ( x ) = \\begin{cases} x ( 1 + 2 ^ \\gamma x ^ \\gamma ) & x \\in [ 0 , \\frac { 1 } { 2 } ) , \\\\ 2 x - 1 & x \\in [ \\frac { 1 } { 2 } , 1 ] , \\end{cases} \\end{align*}"}
{"id": "1899.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\mathcal { U } _ { [ n , j ] } } w ( \\gamma ) = \\sum _ { k = j } ^ { n - 1 } \\sum _ { \\gamma \\in \\mathcal { U } _ { [ n , j , k ] } } w ( \\gamma ) = \\sum _ { k = j } ^ { n - 1 } a _ { - j } ^ { ( j ) } \\ , B _ { [ k - 1 , j - 1 ] } ^ { ( 1 ) } W _ { [ n - k - 1 , 0 ] } . \\end{align*}"}
{"id": "2094.png", "formula": "\\begin{align*} N _ { X _ { \\tilde { H } , V _ Y } } ( n , n ) = \\sum _ { k = 1 } ^ { n } \\sum _ { i _ \\ell | i _ 1 + \\dots + i _ k = n } N _ { X _ { H , V _ Y } } ( i _ \\ell , n ) \\leq \\sum _ { k = 1 } ^ n \\binom { n + k - 1 } { n } N _ { X _ { H , V _ Y } } ( n , n ) = \\binom { 2 n } { n + 1 } N _ { X _ { H , V _ Y } } ( n , n ) . \\end{align*}"}
{"id": "4545.png", "formula": "\\begin{align*} \\Vert J _ { 3 } \\Vert _ { p } = \\mathcal { O } \\left ( \\left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \\right ) \\left ( \\frac { \\omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \\frac { \\omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \\right ) \\right ) . \\end{align*}"}
{"id": "6921.png", "formula": "\\begin{align*} f - q | _ g = & \\ - L ^ 2 ( x ) p | _ { L ^ 2 ( y z ) } + L ^ 2 ( x p | _ { L ^ 2 ( y ) z } ) \\\\ \\equiv & \\ - L ^ 2 ( x ) p | _ { L ^ 2 ( y ) z } + L ^ 2 ( x ) p | _ { L ^ 2 ( y ) z } \\\\ \\equiv & \\ 0 . \\end{align*}"}
{"id": "7952.png", "formula": "\\begin{align*} D _ + = \\bar { D } _ { i _ - } D _ - = \\bar { D } _ { i _ + } . \\end{align*}"}
{"id": "6467.png", "formula": "\\begin{align*} \\left ( \\varrho _ a ( x ) - \\varrho _ b ( x ) \\right ) [ f ] = [ Q ^ { ( 1 ) } , H ( x ) ^ { ( - 1 ) } ] = [ Q ^ { ( 1 ) } , \\iota _ { \\beta ( x ) } ] [ f ] = \\rho ( \\beta ( x ) ) [ f ] . \\end{align*}"}
{"id": "6982.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } \\left ( \\frac { \\mu _ k ^ 2 } { \\lambda _ { k + 1 } ^ 2 } - 1 \\right ) = \\infty . \\end{align*}"}
{"id": "5818.png", "formula": "\\begin{align*} \\beta ( 2 ) = \\sum _ { n \\geq 0 } \\frac { ( - 1 ) ^ { n } } { ( 2 n + 1 ) ^ 2 } = 0 . 9 1 5 9 6 5 5 9 4 1 \\ldots , \\end{align*}"}
{"id": "4516.png", "formula": "\\begin{align*} & \\partial _ 2 ( | \\nabla u | ^ { - 2 } A _ { 1 1 } ) - \\partial _ 1 ( | \\nabla u | ^ { - 2 } A _ { 1 2 } ) \\\\ = & | \\nabla u | ^ { - 2 } ( \\partial _ 2 A _ { 1 1 } - \\partial _ 1 A _ { 1 2 } ) + A _ { 1 1 } \\partial _ 2 | \\nabla u | ^ { - 2 } - A _ { 1 2 } \\partial _ 1 | \\nabla u | ^ { - 2 } \\\\ = & | \\nabla u | ^ { - 2 } ( 2 A _ { 1 2 } k _ { 1 3 } - 2 A _ { 1 1 } k _ { 2 3 } ) + 2 A _ { 1 1 } | \\nabla u | ^ { - 2 } k _ { 2 3 } - 2 A _ { 1 2 } | \\nabla u | ^ { - 2 } k _ { 1 3 } \\\\ = & 0 . \\end{align*}"}
{"id": "5244.png", "formula": "\\begin{align*} \\langle \\sigma _ 1 ^ { r } \\sigma _ { 1 2 } \\rangle ^ { \\mathbf { s } , o } = \\langle \\sigma _ 2 ^ { s } \\sigma _ { 1 2 } \\rangle ^ { \\mathbf { s } , o } = - 1 . \\end{align*}"}
{"id": "5856.png", "formula": "\\begin{align*} \\int _ 0 ^ { x _ { k } } \\bigg ( \\int _ 0 ^ t w \\bigg ) ^ { - \\frac { p } { p - r } } w ( t ) V _ r ( 0 , t ) ^ { \\frac { p r } { p - r } } d t & = \\sum _ { i = - \\infty } ^ k \\int _ { x _ { i - 1 } } ^ { x _ i } \\bigg ( \\int _ 0 ^ t w \\bigg ) ^ { - \\frac { p } { p - r } } w ( t ) V _ r ( 0 , t ) ^ { \\frac { p r } { p - r } } d t \\\\ & \\lesssim \\sum _ { i = - \\infty } ^ k 2 ^ { - i \\frac { r } { p - r } } V _ r ( x _ { i - 1 } , x _ i ) ^ { \\frac { p r } { p - r } } . \\end{align*}"}
{"id": "6472.png", "formula": "\\begin{align*} u = y = 0 \\Gamma \\times ( 0 , \\infty ) , \\end{align*}"}
{"id": "2981.png", "formula": "\\begin{align*} T _ m ( d ) = \\frac 1 2 \\sum _ { \\substack { Q \\in \\Gamma _ \\infty \\backslash \\mathcal Q _ { d ^ 2 } \\\\ Q = [ a , b , c ] , a \\neq 0 } } \\chi _ d ( Q ) \\int _ { C _ Q } e ( m x ) \\phi _ { 2 , m } ( y , s ) \\ , d z . \\end{align*}"}
{"id": "3948.png", "formula": "\\begin{align*} | k ( t , s ) | & \\le \\prod _ { l = 1 } ^ d \\left \\| K _ { h _ l ^ * } \\right \\| _ { \\infty } \\left \\| P _ { \\tilde { w } _ 3 - s } g ( X _ s ) - \\pi ( g ) \\right \\| _ { L ^ 1 } \\\\ & \\le \\frac { c } { ( \\prod _ { l = 1 } ^ d h _ l ^ * ) ^ 2 } e ^ { - \\rho ( \\tilde { w } _ 3 - s ) } , \\end{align*}"}
{"id": "7992.png", "formula": "\\begin{align*} h | _ { D ^ \\vee _ 1 } = h _ 2 : D ^ \\vee _ 1 \\rightarrow \\mathbb C \\end{align*}"}
{"id": "8823.png", "formula": "\\begin{align*} f _ t \\left ( \\int _ { \\Omega } ^ { } \\alpha ( \\omega ) d \\omega \\right ) = 0 \\leq \\int _ { \\Omega } ^ { } f _ { T ( \\omega ) } \\circ \\alpha ( \\omega ) d \\omega . \\end{align*}"}
{"id": "1174.png", "formula": "\\begin{align*} ( f * \\phi _ \\sigma ) ( x ) = \\int _ { \\R ^ d } \\frac { \\phi _ { \\sigma } ( x - y ) } { \\phi _ { \\sigma } ( y ) } f ( y ) \\phi _ { \\sigma } ( y ) d y . \\end{align*}"}
{"id": "4623.png", "formula": "\\begin{align*} c _ 1 ( \\alpha , h ^ { - 1 } ) = c _ 2 ( \\alpha , g ^ { - 1 } ) = t ^ { - 1 } \\ , \\ , \\mbox { a n d } \\ , \\ , c _ 1 ( \\alpha , g ^ { - 1 } ) = c _ 2 ( \\alpha , h ^ { - 1 } ) = r ^ { - 1 } \\end{align*}"}
{"id": "6288.png", "formula": "\\begin{align*} & \\int \\dfrac { x \\sin ( x ; q ) } { ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\ , _ 2 \\phi _ 1 ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 , q ^ 2 ; 0 ; q ^ 2 , q ) d _ q x \\\\ & = \\frac { 1 } { ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 ; q ^ 2 ) _ \\infty } \\left ( x \\cos ( q ^ { \\frac { - 1 } { 2 } } x ; q ) \\ , _ 2 \\phi _ 1 ( \\frac { - x ^ 2 } { q } ( 1 - q ) ^ 2 , q ^ 2 ; 0 ; q ^ 2 , q ) - \\dfrac { q \\sin ( x / q ; q ) } { 1 - q } \\right ) , \\end{align*}"}
{"id": "3528.png", "formula": "\\begin{align*} \\overline { G } _ { ( 2 ) } ( u ) = \\frac { 1 } { 4 ( t - 1 ) ( t - 2 ) } ( 1 + 2 u ) ^ { - ( t - 2 ) } . \\end{align*}"}
{"id": "6180.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l c c } \\frac { \\partial } { \\partial t } \\omega = - R i c ^ { T } ( \\omega ) & \\mathrm { o n } & ( 0 , T _ { 0 } ) \\times M _ { \\operatorname { r e g } } , \\\\ \\omega ( 0 ) = \\omega _ { 0 } & \\mathrm { o n } & M . \\end{array} \\right . \\end{align*}"}
{"id": "793.png", "formula": "\\begin{align*} | g ( z ^ m ) | + \\sum _ { k = N } ^ { \\infty } | b _ k | | z | ^ k \\leq d ( 0 , \\partial { \\Omega } ) \\end{align*}"}
{"id": "7104.png", "formula": "\\begin{align*} [ P ^ E _ t ] - \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
{"id": "7759.png", "formula": "\\begin{align*} { \\bf T } _ j ( \\alpha ^ * ) = [ T _ j ^ 1 ( \\alpha ^ * ) , \\ldots , T _ j ^ m ( \\alpha ^ * ) ] ^ T = \\displaystyle \\sum _ { k = 1 } ^ { \\mu _ j } Q _ j ^ k ( \\alpha ^ * ) { \\bf r } _ j ^ k j = 1 , \\ldots , L \\end{align*}"}
{"id": "8488.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n g ^ n _ i \\frac { 1 } { n } \\rightarrow \\int _ 0 ^ 1 g ( x ) d x = U , \\end{align*}"}
{"id": "7029.png", "formula": "\\begin{align*} k _ 1 + k _ 2 = g = h + 1 = h _ 0 + h _ 1 + 1 \\mathrlap , \\end{align*}"}
{"id": "7827.png", "formula": "\\begin{align*} \\mathbb S ( \\Gamma ) = \\{ f \\in \\ell ^ \\infty \\Gamma \\mid f - R _ t ( f ) \\in c _ 0 ( \\Gamma ) , \\ t \\in \\Gamma \\} \\subset \\ell ^ \\infty \\Gamma . \\end{align*}"}
{"id": "6983.png", "formula": "\\begin{align*} 0 < C \\leq \\prod \\limits _ { k = N } ^ { \\infty } \\Bigg ( \\frac { \\lambda _ { N } ^ { 2 } + \\mu _ { k } ^ { 2 } } { \\lambda _ { N } ^ { 2 } + \\lambda _ { k } ^ { 2 } } \\Bigg ) \\leq 1 \\end{align*}"}
{"id": "7971.png", "formula": "\\begin{align*} H _ { ( \\mathbb P ^ 2 , H + H ) } ( y ) = e ^ { \\frac { H \\log y } { 2 \\pi i } } \\sum _ { d \\geq 0 } y ^ d \\frac { 1 } { \\Gamma ( 1 + \\frac { H } { 2 \\pi i } + d ) } [ \\textbf { 1 } ] _ { d , d } , \\end{align*}"}
{"id": "8606.png", "formula": "\\begin{align*} \\mathcal { F } ( \\tilde A , \\tilde B , \\tilde C , \\tilde D ) = & { \\| A - \\tilde A \\| } _ F ^ 2 + { \\| B - \\tilde B \\| } _ F ^ 2 \\\\ & + { \\| C - \\tilde C \\| } _ F ^ 2 + { \\| D - \\tilde D \\| } _ F ^ 2 . \\end{align*}"}
{"id": "8963.png", "formula": "\\begin{align*} \\delta ^ \\ast = \\frac { 2 - n } { 2 } , \\end{align*}"}
{"id": "5954.png", "formula": "\\begin{align*} ( \\alpha ( \\xi ) | \\alpha ( \\zeta ) ) & = L ( \\alpha ( \\xi ) \\otimes \\Omega _ \\phi ) ^ \\ast L ( \\alpha ( \\zeta ) \\otimes \\Omega _ \\phi ) \\\\ & = L ( U ( \\xi \\otimes \\Omega _ \\phi ) ) ^ \\ast L ( U ( \\zeta \\otimes \\Omega _ \\phi ) ) \\\\ & = L ( \\xi \\otimes \\Omega _ \\phi ) ^ \\ast U ^ \\ast U L ( \\zeta \\otimes \\Omega _ \\phi ) \\\\ & = ( \\xi | \\zeta ) . \\end{align*}"}
{"id": "4679.png", "formula": "\\begin{align*} ( { } ^ J G _ { v , { \\bf w } , > 0 } ) ^ \\sharp \\dot { s } ^ \\sharp _ 0 \\dot { x } ^ \\sharp = { } ^ { { I ^ ! } ^ \\flat } \\ ! \\ ! G _ { v ^ \\sharp ( s _ 0 x ) ^ \\sharp , { \\bf w ^ \\sharp s ^ \\sharp _ 0 x ^ \\sharp } , > 0 } . \\end{align*}"}
{"id": "8617.png", "formula": "\\begin{align*} { \\| A - \\hat A \\| } _ F ^ 2 + { \\| B - \\hat B \\| } _ F ^ 2 + { \\| C - \\hat C \\| } _ F ^ 2 + { \\| D - \\hat D \\| } _ F ^ 2 = 0 . 6 4 3 0 . \\end{align*}"}
{"id": "2211.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\frac { \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } + \\varphi _ { \\alpha \\overline { \\beta } } ) } { \\det ( g _ { \\alpha \\overline { \\beta } } ^ { T } ) } = e ^ { - \\kappa \\varphi + F } . \\end{array} \\end{align*}"}
{"id": "2335.png", "formula": "\\begin{align*} F _ p = L ( h _ \\theta ) + \\sum _ { i \\in I } \\partial _ i L ( h _ \\theta ) Q _ \\theta ^ i \\end{align*}"}
{"id": "7656.png", "formula": "\\begin{align*} f \\colon \\mathbb { P } ^ { n - 1 } \\hookrightarrow \\mathbb { P } = \\mathbb { P } \\left ( \\mathsf { M a t } _ { n + m , s + 1 } ( \\mathbb C ) \\right ) , [ \\alpha ] \\mapsto [ A _ { \\alpha } ] . \\end{align*}"}
{"id": "3826.png", "formula": "\\begin{align*} \\clubsuit \\widetilde { K _ { k _ 1 , j _ 1 ; n _ 1 } ^ { \\mu _ 1 , i _ 1 } } ( s , \\xi , v , V ( s ) ) : = \\clubsuit { K _ { k _ 1 , j _ 1 ; n _ 1 } ^ { \\mu _ 1 , i _ 1 } } ( s , \\xi - \\eta , V ( s ) ) + \\big [ \\mathcal { F } _ { x \\rightarrow \\xi } [ { } _ { } ^ z T _ { k _ 1 , n _ 1 } ^ { \\mu _ 1 } ( B ) ] ( s , \\xi , V ( s ) ) \\mathbf { 1 } _ { n _ 1 \\geq - 2 M _ t / 1 5 } \\end{align*}"}
{"id": "7282.png", "formula": "\\begin{align*} u ( x , t ) = - { \\sf U } _ \\infty ( x ) - \\theta ( x ) - \\Theta _ J ( x ) \\end{align*}"}
{"id": "3961.png", "formula": "\\begin{align*} j ( a ) ( \\gamma \\beta ^ { - 1 } ) = \\lim _ { n } j ( f _ { n } ) ( \\gamma \\beta ^ { - 1 } ) . \\end{align*}"}
{"id": "4809.png", "formula": "\\begin{align*} I _ 2 - I _ 1 & = 2 ( f ( v _ 0 ) - f ( a ) ) \\int _ a ^ b f ( v ) \\ , d v + ( f ^ 2 ( a ) - f ^ 2 ( v _ 0 ) ) ( b - a ) \\\\ & = ( f ( v _ 0 ) - f ( a ) ) \\biggl [ 2 \\int _ a ^ b f ( v ) \\ , d v - ( f ( v _ 0 ) + f ( a ) ) ( b - a ) \\biggr ] \\ge 0 \\end{align*}"}
{"id": "8856.png", "formula": "\\begin{align*} \\dfrac { \\partial \\lambda _ i } { \\partial \\eta } = - \\dfrac { f _ \\eta ( \\lambda _ i ) } { f _ \\lambda ( \\lambda _ i ) } , \\ \\ \\frac { \\partial ^ 2 \\lambda _ i } { \\partial \\eta ^ 2 } = - \\frac { f _ { \\eta \\eta } ( \\lambda _ i ) + 2 f _ { \\lambda \\eta } ( \\lambda _ i ) \\lambda _ { i , \\eta } + f _ { \\lambda \\lambda } ( \\lambda _ i ) \\lambda _ { i , \\eta } ^ 2 } { f _ \\lambda ( \\lambda _ i ) } . \\end{align*}"}
{"id": "5303.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { C o n t } ( Q ^ 0 ) | _ { b _ l = 0 } & = - ( h r + 1 - b _ { I \\setminus \\{ a _ l \\} } ) \\mathrm { C o n t } ( Q ) \\\\ \\mathrm { C o n t } ( Q ^ j ) | _ { b _ l = 0 } & = ( r - b _ { I _ j } + | Q _ j | ) \\mathrm { C o n t } ( Q ) . \\end{aligned} \\end{align*}"}
{"id": "4401.png", "formula": "\\begin{align*} A _ { \\mu } ^ p = \\{ f : R \\cdot \\mathbb { D } \\rightarrow \\mathbb { C } : f \\mbox { h o l o m o r p h i c w i t h } \\Vert f \\Vert _ p < \\infty \\} . \\end{align*}"}
{"id": "8199.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { c c } A & B \\\\ C & D \\end{array} \\right ) . \\end{align*}"}
{"id": "2153.png", "formula": "\\begin{align*} q ( s ) = ( 1 2 / 7 ) \\left [ s ^ { 2 } I _ { [ - 1 , 0 ] } ( s ) + s ^ { 3 } I _ { [ 0 , 1 ] } ( s ) \\right ] . \\end{align*}"}
{"id": "680.png", "formula": "\\begin{align*} \\partial _ s \\left ( e ^ { f _ o } Q \\right ) - \\Delta ^ { f _ o } _ { g _ o } \\left ( e ^ { f _ o } Q \\right ) = 0 , \\end{align*}"}
{"id": "1687.png", "formula": "\\begin{align*} j ( i ) = \\begin{cases} ( i + 1 ) / 2 & \\hbox { i f $ i $ i s o d d ; } \\\\ i / 2 & \\hbox { i f $ i $ i s e v e n , } \\end{cases} \\end{align*}"}
{"id": "559.png", "formula": "\\begin{align*} \\mathbb { V } ( \\overline { S } _ n ) = \\frac { 1 } { n ^ 2 } \\sum _ { 1 \\leq i \\leq n } \\mathbb { E } ( { X _ i } ^ 2 ) + \\frac { 2 } { n ^ 2 } \\sum _ { 1 \\leq i \\textless j \\leq n } \\mathbb { E } ( X _ i X _ j ) - \\frac { 1 } { n ^ 2 } \\mathbb { E } ^ 2 \\Big ( \\sum _ { 1 \\leq i \\leq n } { X _ i } \\Big ) . \\end{align*}"}
{"id": "7815.png", "formula": "\\begin{align*} \\rho _ { \\mu _ t g } ( \\gamma \\cup \\delta ) = \\rho _ { \\mu _ t g } ( \\gamma ) \\star \\rho _ { \\mu _ t g } ( \\delta ) , \\end{align*}"}
{"id": "6900.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow ( T ^ * ) ^ { - } } N ( t ) = + \\infty . \\end{align*}"}
{"id": "6797.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 u } { \\partial t ^ 2 } + \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\Biggl ( a \\frac { \\partial ^ 3 u } { \\partial t \\partial x ^ 2 } \\Biggr ) + \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\Biggl ( a \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } \\Biggr ) = 0 , \\end{align*}"}
{"id": "1200.png", "formula": "\\begin{align*} g ( x ) = c ( x , y ) - g ^ c ( y ) . \\end{align*}"}
{"id": "4167.png", "formula": "\\begin{align*} \\varphi _ k ^ \\lambda ( z ) = ( 2 \\pi ) ^ { n / 2 } \\lambda ^ { n / 2 } \\sum _ { | \\nu | _ 1 = k } \\Phi ^ \\lambda _ { \\nu , \\nu } ( z ) . \\end{align*}"}
{"id": "2749.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } \\lim _ { i \\rightarrow \\infty } \\Psi _ { i } ( x , R ) = c _ { n , \\sigma p } \\lim _ { R \\rightarrow \\infty } \\lim _ { i \\rightarrow \\infty } \\int _ { B ^ { c } _ { R } ( 0 ) } \\frac { u ^ { p - 1 } _ { i } ( y ) } { | y | ^ { n + \\sigma p } } d y \\geq 0 . \\end{align*}"}
{"id": "2397.png", "formula": "\\begin{align*} \\mathbb E \\big [ \\sup _ { t } \\| O ( t ) \\| _ { l ^ \\infty _ n } ^ p \\big ] = \\mathbb E \\big [ \\sup _ { t , x } \\left | o ^ n ( t , x ) - o ^ n ( 0 , 0 ) \\right | ^ { p } \\big ] \\le C ( p , T ) . \\end{align*}"}
{"id": "1446.png", "formula": "\\begin{align*} & P _ { \\ell } ( z ) = \\left [ \\dfrac { 1 } { ( n - 1 ) ! ^ r } \\right ] \\circ { \\rm { E v a l } } _ z \\circ \\mathcal { T } _ { \\bold { c } } \\bigcirc _ { j = 1 } ^ { n - 1 } B ( \\theta _ t + j ) \\left ( t ^ { \\ell } \\prod _ { i = 1 } ^ m ( t - \\alpha _ i ) ^ { r n } \\right ) \\enspace , \\\\ & P _ { \\ell , i , s } ( z ) = \\psi _ { i , s } \\left ( \\dfrac { P _ { \\ell } ( z ) - P _ { \\ell } ( t ) } { z - t } \\right ) \\ \\ 1 \\le i \\le m , 0 \\le s \\le r - 1 \\enspace , \\end{align*}"}
{"id": "7488.png", "formula": "\\begin{align*} \\int _ { ( 0 , \\infty ) } \\frac { 1 } { \\pi ( 1 + x ^ 2 ) ( t + x ^ 2 ) } \\ , \\dd x & = \\frac { 1 } { \\pi ( 1 - t ) } \\int _ { ( 0 , \\infty ) } \\left ( - \\frac { 1 } { 1 + x ^ 2 } + \\frac { 1 } { t + x ^ 2 } \\right ) \\ , \\dd x \\\\ & = \\frac { 1 } { \\pi ( 1 - t ) } \\left ( - \\frac { \\pi } { 2 } \\right ) + \\frac { 1 } { \\pi t ( 1 - t ) } \\int _ { ( 0 , \\infty ) } \\frac { 1 } { 1 + \\Big ( \\frac { x } { \\sqrt { t } } \\Big ) ^ 2 } \\ , \\dd x \\end{align*}"}
{"id": "5372.png", "formula": "\\begin{align*} \\norm { P } _ { m , s } \\vcentcolon = \\sum _ { | \\alpha | \\leq m } \\| a _ \\alpha \\| _ { s - | \\alpha | , - s } = \\sum _ { | \\alpha | \\leq m } \\| a _ \\alpha \\| _ { s , | \\alpha | - s } \\end{align*}"}
{"id": "2560.png", "formula": "\\begin{align*} \\rho ( g ) : \\mathcal B ( \\mathcal H ) \\to \\mathcal B ( \\mathcal H ) \\ , \\ A \\mapsto A ^ g = \\rho ( g ) A \\rho ( g ) ^ { - 1 } \\ , \\ \\forall g \\in S U ( 3 ) \\ . \\end{align*}"}
{"id": "1215.png", "formula": "\\begin{align*} z : = y _ 1 - y _ 2 , \\ \\ w : = \\dd \\frac { \\beta ( u ) } u \\ , ( y _ 1 - y _ 2 ) . \\end{align*}"}
{"id": "8134.png", "formula": "\\begin{align*} & \\hat { \\eta } _ { x , p ; \\delta } ( \\xi ) = e ^ { - i x \\xi } \\hat { \\eta } _ \\delta ( \\xi - p ) \\end{align*}"}
{"id": "4686.png", "formula": "\\begin{align*} & ( - 1 ) ^ { k } \\left ( C _ m ( n ) - \\sum _ { j = 1 - k } ^ k ( - 1 ) ^ j u _ m \\big ( n - j ( 3 j - 1 ) / 2 \\big ) \\right ) \\geqslant 0 , \\end{align*}"}
{"id": "5290.png", "formula": "\\begin{align*} \\int _ { \\Xi ^ r _ j } x ^ k e ^ { x ^ r / \\hbar } d x = \\delta _ { j k } \\end{align*}"}
{"id": "7208.png", "formula": "\\begin{align*} \\| u \\| _ 1 : = \\sup _ { ( x , t ) \\in \\R \\times [ 0 , T ] } | u ( x , t ) | . \\end{align*}"}
{"id": "5842.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ M a _ k \\bigg ( \\sum _ { i = k } ^ M a _ i \\bigg ) ^ s b _ k \\approx \\sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \\bigg ( \\sum _ { i = k } ^ M a _ i \\bigg ) ^ { s + 1 } + \\bigg ( \\sum _ { k = N } ^ M a _ k \\bigg ) ^ { s + 1 } b _ N . \\end{align*}"}
{"id": "734.png", "formula": "\\begin{align*} \\mathcal G _ { \\rho } ( z ) = - \\frac { \\big ( \\partial \\bar { \\partial } \\log \\rho \\big ) ( z ) } { \\rho ( z ) ^ 2 } , z \\in \\Omega . \\end{align*}"}
{"id": "8791.png", "formula": "\\begin{align*} H ( p , G ) = \\tilde { Y } _ O ( P , G ) ^ { 2 / p ' } \\leq C ( P ) ^ { 2 / p ' } = B ( p ) \\end{align*}"}
{"id": "4694.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\big ( u _ 1 ( n ) - u _ 2 ( n ) \\big ) \\ , q ^ n = \\sum _ { k = 0 } ^ \\infty \\frac { q ^ { k } ( 1 - q ^ { k } ) } { ( q ; q ) ^ 2 _ k } = \\sum _ { n = 0 } ^ \\infty u _ 0 ( n ) \\ , q ^ n . \\end{align*}"}
{"id": "2700.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } | E ( N ) - E _ { 2 - \\mathrm { m o d e } } - E ^ \\mathrm { B o g } | = 0 . \\end{align*}"}
{"id": "1168.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } \\frac { \\mathsf { W } _ 2 \\big ( ( 1 + \\epsilon h ) \\mu , \\mu \\big ) } { \\epsilon } = \\| h \\mu \\| _ { \\dot { H } ^ { - 1 , 2 } ( \\mu ) } , \\end{align*}"}
{"id": "6334.png", "formula": "\\begin{align*} \\sin ( z ; q ) & : = \\frac { ( q ; q ) _ \\infty } { ( q ^ { 1 / 2 } ; q ) _ \\infty } z ^ { 1 / 2 } J ^ { ( 3 ) } _ { 1 / 2 } ( z ( 1 - q ) ; q ^ 2 ) = \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n \\frac { q ^ { n ^ 2 + n } z ^ { 2 n + 1 } } { \\Gamma _ q ( 2 n + 2 ) } , \\end{align*}"}
{"id": "8058.png", "formula": "\\begin{align*} X _ { 2 } ( h ) = 0 \\end{align*}"}
{"id": "3165.png", "formula": "\\begin{align*} \\mathit { \\Omega } _ { \\mathfrak { m } } \\left ( c _ { + } \\right ) \\doteq \\left \\{ \\omega \\in \\mathit { \\Omega } _ { \\mathfrak { m } \\left ( c _ { + } \\right ) } ^ { \\sharp } : e _ { ( \\cdot ) } \\left ( \\omega \\right ) = c _ { + } \\mathfrak { a } _ { + } \\right \\} \\subseteq E _ { 1 } \\ , \\end{align*}"}
{"id": "8671.png", "formula": "\\begin{align*} \\mathbf V = \\begin{pmatrix} F _ 1 \\\\ F _ 2 \\\\ F _ 3 \\end{pmatrix} \\cdot \\begin{pmatrix} G _ 1 & G _ 2 & G _ 3 \\end{pmatrix} \\end{align*}"}
{"id": "5758.png", "formula": "\\begin{align*} f = \\pi _ { i _ 1 } \\pi _ { i _ 2 } \\cdots \\pi _ { i _ d } = ( \\pi _ { i _ 1 } \\pi _ { i _ 2 } \\cdots \\pi _ { i _ { n - 1 } } ) \\cdot ( \\pi _ { i _ { n } } \\pi _ { i _ { n + 1 } } \\cdots \\pi _ { i _ d } ) . \\end{align*}"}
{"id": "2434.png", "formula": "\\begin{align*} T _ f = M _ f + \\sum _ { g \\prec f } t _ { g f } ( q ) M _ g , L _ f = M _ f + \\sum _ { g \\prec f } \\ell _ { g f } ( q ) M _ g , \\end{align*}"}
{"id": "7988.png", "formula": "\\begin{align*} h : = ( h _ 1 , \\ldots , h _ n ) : X ^ \\vee \\rightarrow \\mathbb C ^ n , \\end{align*}"}
{"id": "2542.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( L _ { Z _ 1 } g ) ( X _ 1 , Y _ 1 ) = a _ 1 a _ 3 a _ 5 e ^ { 2 x _ 1 } . \\end{align*}"}
{"id": "498.png", "formula": "\\begin{align*} \\Phi \\mapsto ( \\phi _ 1 , \\dots , \\phi _ n ) \\phi _ k = \\Phi ^ { - 1 } ( \\{ n - k + 1 , \\dots , n \\} ) . \\end{align*}"}
{"id": "7446.png", "formula": "\\begin{align*} \\mathcal { B } ( ( u , y ) , ( \\phi ^ 1 , \\phi ^ 2 ) ) = \\mathcal { L } ( \\phi _ 1 , \\phi _ 2 ) , \\forall \\ , ( \\phi ^ 1 , \\phi ^ 2 ) \\in H _ L ^ 1 ( 0 , L ) \\times H _ L ^ 1 ( 0 , L ) , \\end{align*}"}
{"id": "687.png", "formula": "\\begin{align*} \\partial _ s \\left ( e ^ { f _ o } v \\right ) = \\Delta _ { g _ o } ^ { f _ o } \\left ( e ^ { f _ o } v \\right ) \\end{align*}"}
{"id": "8231.png", "formula": "\\begin{align*} z = a + i b . \\end{align*}"}
{"id": "924.png", "formula": "\\begin{align*} E _ c ( \\Psi _ t ) ( 0 , h ) = v ' \\Longleftrightarrow \\begin{pmatrix} 0 \\\\ h \\end{pmatrix} = P _ c ( \\Psi _ t ) \\cdot v ' , \\end{align*}"}
{"id": "1877.png", "formula": "\\begin{align*} \\mathcal { D } _ { [ n , j ] } = \\bigcup _ { k = i } ^ { n - j + i } \\widetilde { \\mathcal { D } } _ { [ n , j , k ] } , \\end{align*}"}
{"id": "6847.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d b ( t ) } { d t } & = g _ 0 + { g _ 1 } N ( b ( t ) , c ( t ) ) - b ( t ) , \\\\ \\frac { d c ( t ) } { d t } & = 2 a _ 0 + 2 { a _ 1 } N ( b ( t ) , c ( t ) ) - 2 c ( t ) . \\end{aligned} \\end{align*}"}
{"id": "171.png", "formula": "\\begin{align*} \\| C _ { ( m ) } ( C _ { ( m ) } ^ 2 \\star C _ { ( m ) } ^ 2 ) \\| _ { L ^ 1 } \\leq \\frac { c } { m ^ 4 } { \\bf 1 } _ { d = 2 } + \\frac { c } { m } { \\bf 1 } _ { d = 3 } . \\end{align*}"}
{"id": "1824.png", "formula": "\\begin{align*} J = \\begin{pmatrix} b _ { 0 } & 1 & & & \\\\ a _ { 0 } & b _ { 1 } & 1 & \\\\ & a _ { 1 } & b _ { 2 } & 1 & \\\\ & & \\ddots & \\ddots & \\ddots \\end{pmatrix} . \\end{align*}"}
{"id": "5608.png", "formula": "\\begin{align*} ( x , y ) = ( y , z ) = ( z , x ) = ( x , P _ 1 P _ 2 ) = ( y , P _ 1 P _ 2 ) = 1 . \\end{align*}"}
{"id": "2093.png", "formula": "\\begin{align*} h ( X _ { H , V } ) \\geq \\lim \\limits _ { n \\to + \\infty } \\dfrac { \\log _ 2 \\left ( \\max \\limits _ { v \\in \\{ 0 , \\dots , p - 1 \\} ^ n } \\Pi _ { j = 0 } ^ { p - 1 } N _ { v ^ j } \\right ) } { p n } \\end{align*}"}
{"id": "5042.png", "formula": "\\begin{align*} Q ^ { n , 8 } _ \\tau = n ^ { 2 \\alpha + 1 } \\int ^ { \\tau } _ 0 \\gamma _ { \\eta _ n ( s ) } ( s - \\eta _ n ( s ) ) ^ { 2 \\alpha + 1 } ( \\sigma ' \\sigma ) ^ 2 ( X _ { \\eta _ n ( s ) } ) d s . \\end{align*}"}
{"id": "791.png", "formula": "\\begin{align*} \\frac { \\beta ( 1 + ( 1 - 2 \\alpha ) r ^ m ) } { ( 1 - r ^ m ) ^ { 2 ( 1 - \\alpha ) + 1 } } & + \\frac { ( 1 - \\beta ) r ^ m } { ( 1 - r ^ m ) ^ { 2 ( 1 - \\alpha ) } } + \\sum _ { n = 1 } ^ { \\infty } \\prod _ { k = 0 } ^ { 2 ( n - 1 ) } \\frac { k + 2 ( 1 - \\alpha ) } { k + 1 } r ^ { 2 n } \\\\ & = \\frac { 1 } { 4 ^ { 1 - \\alpha } } . \\end{align*}"}
{"id": "217.png", "formula": "\\begin{align*} N _ { \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } } ( x ^ n , b _ { \\mathcal { T } } ) \\coloneqq \\abs { \\left \\{ \\ y ^ { n } _ { \\mathcal { T } } : \\ \\mathcal { B } _ i ( y ^ { n } _ { i } ) = b _ i \\ f o r \\ i \\in \\mathcal { T } \\ \\wedge \\ y ^ { n } _ { \\mathcal { T } } \\in \\mathcal { T } ^ { n } _ { \\pi _ { \\bar { Y } _ { \\mathcal { T } } | \\bar { X } } } ( x ^ n ) \\right \\} } , \\end{align*}"}
{"id": "6549.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\prod _ { i = 1 } ^ m \\mu ( [ 0 , A ( m ) ^ { \\rho i } ] ) & \\leq z _ 0 - 1 + \\sum _ { m = z _ 0 } ^ \\infty \\prod _ { i = 1 } ^ m \\mu ( [ 0 , A ( m ) ^ { \\rho i } ] ) \\\\ & = z _ 0 - 1 + \\sum _ { m = 1 } ^ \\infty \\prod _ { i = 1 } ^ { m + z _ 0 - 1 } \\mu ( [ 0 , A ( m + z _ 0 - 1 ) ^ { \\rho i } ] ) \\\\ & = z _ 0 - 1 + \\sum _ { m = 1 } ^ \\infty \\prod _ { i = 1 } ^ { m + z _ 0 - 1 } \\mu ( [ 0 , \\tilde { A } ( m ) ^ { \\rho i } ] ) \\\\ & \\leq z _ 0 - 1 + \\sum _ { m = 1 } ^ \\infty \\prod _ { i = 1 } ^ { m } \\mu ( [ 0 , \\tilde { A } ( m ) ^ { \\rho i } ] ) . \\end{align*}"}
{"id": "6199.png", "formula": "\\begin{align*} \\frac { r _ { M _ { 1 } } } { a _ { M _ { 1 } } } = [ b _ { 1 } , \\cdot \\cdot \\cdot , ( b _ { i } - 1 ) , \\cdot \\cdot \\cdot , b _ { l } ] . \\end{align*}"}
{"id": "4376.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t \\} } \\left \\vert \\frac { \\tilde f _ 1 + \\tilde f _ 2 } { 2 } \\right \\vert ^ 2 + \\int _ { \\{ \\Psi < - t \\} } \\left \\vert \\frac { \\tilde f _ 1 - \\tilde f _ 2 } { 2 } \\right \\vert ^ 2 \\\\ = \\frac { \\int _ { \\{ \\Psi < - t \\} } | \\tilde f _ 1 | ^ 2 + \\int _ { \\{ \\Psi < - t \\} } | \\tilde f _ 2 | ^ 2 } { 2 } = G ( t ) , \\end{align*}"}
{"id": "7551.png", "formula": "\\begin{align*} & \\sum _ { i + j = n } \\mu _ i ( x _ 1 , x _ 2 , \\mu _ j ( x _ 3 , x _ 4 , x _ 5 ) ) \\\\ = & \\sum _ { i + j = n } \\mu _ i ( \\mu _ j ( x _ 1 , x _ 2 , x _ 3 ) , x _ 4 , x _ 5 ) + \\mu _ i ( x _ 3 , \\mu _ j ( x _ 1 , x _ 2 , x _ 4 ) , x _ 5 ) + \\mu _ i ( x _ 3 , x _ 4 , \\mu _ j ( x _ 1 , x _ 2 , x _ 5 ) ) , \\end{align*}"}
{"id": "4884.png", "formula": "\\begin{align*} n _ { i } = N _ { i } - 2 ^ { v _ { 1 } } - \\cdots - 2 ^ { v _ { k } } \\end{align*}"}
{"id": "5467.png", "formula": "\\begin{align*} \\chi ' ( b ) = a ^ { - 1 } \\chi ( b ) \\kappa ( b ) a \\kappa ( b ) ^ { - 1 } \\ , \\ , \\ , \\ , b \\in B . \\end{align*}"}
{"id": "3527.png", "formula": "\\begin{align*} \\overline { G } _ { ( 1 ) } ( u ) = \\frac { 1 } { 2 ( t - 1 ) } ( 1 + 2 u ) ^ { - ( t - 1 ) } \\end{align*}"}
{"id": "8209.png", "formula": "\\begin{align*} \\psi _ { A _ { k } } = A _ { k } \\left [ \\frac { \\cos ( k x ) } { \\cos ( k a / 2 ) } - \\frac { \\cosh ( k ' x ) } { \\cosh ( k ' a / 2 ) } \\right ] \\end{align*}"}
{"id": "5992.png", "formula": "\\begin{align*} e ^ { \\max } = \\max _ { j = 1 , \\dots , N } E ^ 2 ( t _ j ) \\end{align*}"}
{"id": "253.png", "formula": "\\begin{align*} \\delta ^ { ( s ) } w = \\alpha w ^ { \\phi ^ s } , \\end{align*}"}
{"id": "6523.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { \\infty } \\mu \\left ( \\left [ a _ i , \\infty \\right ) \\right ) < \\infty . \\end{align*}"}
{"id": "5036.png", "formula": "\\begin{align*} Q ^ { n , 2 } _ \\tau = Q ^ { n , 5 } _ \\tau + Q ^ { n , 6 } _ \\tau , \\end{align*}"}
{"id": "2648.png", "formula": "\\begin{align*} e ^ { \\Delta ' | y | ^ { 2 - \\delta _ k } + | y | } \\partial _ { r } p _ r ( y ) = \\tfrac 1 { \\sqrt { 4 \\pi } } r ^ { - 3 / 2 } \\left [ \\tfrac 1 t y ^ 2 e ^ { - \\frac { y ^ 2 } { 4 r } } - e ^ { - \\frac { y ^ 2 } { 4 r } } \\right ] e ^ { - \\frac { y ^ 2 } { 4 r } + \\Delta ' | y | ^ { 2 - \\delta _ k } + | y | } . \\end{align*}"}
{"id": "724.png", "formula": "\\begin{align*} v _ i = - g ^ { j k } u _ j h _ { k i } - \\frac { f ' ( u ) } { f ( u ) } v u _ i . \\end{align*}"}
{"id": "6889.png", "formula": "\\begin{align*} p ( t , v , g ) = \\rho ( t , v ) \\mathcal { G } ( g ; g _ { } ( t ) , a ( t ) ) . \\end{align*}"}
{"id": "1563.png", "formula": "\\begin{align*} \\frac { d ^ 2 x ^ i ( t ) } { d t ^ 2 } + { G } ^ i \\left ( \\gamma , \\frac { d \\gamma } { d t } \\right ) = 0 , 1 \\le i \\le n . \\end{align*}"}
{"id": "1255.png", "formula": "\\begin{align*} P ( T , x ) & = \\dfrac { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { W _ 0 ^ { n ( T , l ( T ) ) } \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) - 1 } W _ j ^ { n ( T , l ( T ) - j ) } } = \\dfrac { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) } W _ j ^ { n ( T , l ( T ) + 1 - j ) } } { \\displaystyle \\prod _ { j = 1 } ^ { l ( T ) - 1 } W _ j ^ { n ( T , l ( T ) - j ) } } . \\end{align*}"}
{"id": "1201.png", "formula": "\\begin{align*} g ( x ' ) - g ( x ) & \\le c ( x ' , y ) - g ^ c ( y ) - [ c ( x , y ) - g ^ c ( y ) ] \\\\ & = c ( x ' , y ) - c ( x , y ) \\\\ & = | x ' - y | ^ p - | x - y | ^ p \\\\ & \\le p ( | x - y | ^ { p - 1 } \\vee | x ' - y | ^ { p - 1 } ) | x - x ' | \\\\ & \\le C _ 3 \\big [ 1 + ( | x | \\vee | x ' | ) ^ { \\frac { 2 p } { \\beta } } \\big ] | x - x ' | , \\end{align*}"}
{"id": "2969.png", "formula": "\\begin{align*} F _ { m } ( z , s ) = \\sum _ { \\gamma \\in \\Gamma _ \\infty \\backslash \\Gamma } f _ m ( \\gamma z , s ) , \\end{align*}"}
{"id": "3466.png", "formula": "\\begin{align*} & \\mathrm { E } [ X ^ { n } | x _ { p } < X < x _ { q } ] \\\\ & = \\mu ^ { n } + n \\mu ^ { n - 1 } \\sigma \\mathrm { D T E } _ { ( p , q ) } ( Y ) + \\sum _ { i = 2 } ^ { n } \\binom { n } { i } \\mu ^ { n - i } \\sigma ^ { i } \\left [ L _ { 1 } + ( i - 1 ) \\frac { c _ { 1 } } { c _ { ( 1 ) } ^ { \\ast } } L _ { 2 } \\right ] , \\end{align*}"}
{"id": "1939.png", "formula": "\\begin{align*} W _ { j } ( z ) = R _ { j } ( z ) , A _ { j } ^ { ( q ) } ( z ) = S _ { j } ^ { ( q ) } ( z ) , B _ { j } ^ { ( q ) } ( z ) = T _ { j } ^ { ( q ) } ( z ) , \\end{align*}"}
{"id": "3586.png", "formula": "\\begin{align*} \\pi _ k = a _ 1 \\cdot \\mathtt { w } ^ + ( \\phi _ 1 ) + \\ldots + a _ s \\cdot \\mathtt { w } ^ + ( \\phi _ s ) \\geqslant c _ k \\end{align*}"}
{"id": "3773.png", "formula": "\\begin{align*} H _ { k , j ; n , l , r } ^ { \\mu , m , i ; l i n } ( t , x , \\zeta ) : = \\int _ { 0 } ^ t \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } \\int _ { \\mathbb { S } ^ 2 } \\varphi _ { m ; - 1 0 M _ t } ( t - s ) \\big [ ( t - s ) \\big ( \\mathfrak { K } ^ { \\mu , E } _ { k ; n } ( y , \\omega , v , \\zeta ) + \\hat { \\zeta } \\times \\mathfrak { K } ^ { \\mu , B } _ { k ; n } ( y , \\omega , v , \\zeta ) \\big ) + \\mathfrak { K } ^ { e r r ; \\mu , E } _ { k ; n } ( y , v , \\zeta ) \\end{align*}"}
{"id": "7245.png", "formula": "\\begin{align*} S _ { [ 0 , \\Lambda ] , s } = \\Bigl \\{ & \\mu = \\sum _ { j = 1 } ^ { s } c _ { j } \\delta _ { t _ { j } } : \\ , c \\in \\mathbb { C } ^ s 0 \\leq t _ { 1 } < t _ { 2 } < . . . < t _ { s } \\leq \\Lambda \\Bigr \\} \\ , . \\end{align*}"}
{"id": "2453.png", "formula": "\\begin{align*} X = \\left \\{ x \\in \\gg _ { 1 } \\mid \\left [ x , x \\right ] = 0 \\right \\} . \\end{align*}"}
{"id": "3330.png", "formula": "\\begin{align*} | S p ( 2 n , \\mathbb { Z } _ p ) | = \\prod _ { m = 1 } ^ n \\big [ ( p ^ { 2 m } - 1 ) p ^ { 2 m - 1 } \\big ] \\geq \\frac { 1 } { 2 ^ { 2 n } } \\prod _ { s = 1 } ^ { 2 l } p ^ { s } = \\dfrac { p ^ { 2 l ^ 2 + l } } { 2 ^ { 2 n } } \\end{align*}"}
{"id": "5789.png", "formula": "\\begin{align*} G / B = N ' \\rtimes P ' \\end{align*}"}
{"id": "3046.png", "formula": "\\begin{align*} \\Omega _ { \\bullet } ( \\sqrt { q } , \\sqrt { q } ^ { - 1 } ) = & \\sum _ { \\lambda \\in \\mathcal { P } } N _ { \\lambda } ( q , q ^ { - 1 } ) ^ { g - 1 } \\prod ^ k _ { j = 1 } H _ { \\lambda } ( \\Z _ j ; q , q ^ { - 1 } ) \\\\ \\Omega _ { \\ast } ( \\sqrt { q } , \\sqrt { q } ^ { - 1 } ) = & \\sum _ { \\lambda \\in \\mathcal { P } } N _ { \\lambda } ( q , q ^ { - 1 } ) ^ { 2 g - 2 } \\prod ^ k _ { j = 1 } H _ { \\lambda } ( \\Z _ j ; q , q ^ { - 1 } ) ^ 2 . \\end{align*}"}
{"id": "2316.png", "formula": "\\begin{align*} ( M _ \\varphi ) _ { i , j } = | \\varphi ( j ) | _ i i , j \\in \\{ 0 , 1 \\} , \\end{align*}"}
{"id": "7193.png", "formula": "\\begin{align*} \\langle f , I ^ { \\circ } ( w _ 1 , z ) w _ 2 \\rangle = 0 , \\mbox { f o r a n y } \\ f \\in ( M ^ 3 ( 0 ) ) ^ * , \\ w _ 1 \\in M ^ 1 , \\ w _ 2 \\in M ^ 2 ( 0 ) . \\end{align*}"}
{"id": "747.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty { \\mathcal R } _ 1 ( e _ n ) ( z ) { { \\mathcal R } _ 1 ( e _ n ) ( w ) } ^ * = \\mathbb K ^ { ( \\alpha , \\beta ) } ( z , w ) , \\ ; z , w \\in \\Omega _ 0 . \\end{align*}"}
{"id": "5353.png", "formula": "\\begin{align*} [ u ] _ { s , p , \\R ^ n } \\vcentcolon = \\left ( \\frac { C ( n , s , p ) } { 2 } \\int _ { \\R ^ n } \\int _ { \\R ^ n } \\frac { \\abs { u ( x ) - u ( y ) } ^ p } { \\abs { x - y } ^ { n + s p } } d x d y \\right ) ^ { 1 / p } \\end{align*}"}
{"id": "772.png", "formula": "\\begin{align*} { \\rm P r } \\left ( \\left . Z > \\lambda \\right | \\mathcal { H } _ 1 \\right ) = Q \\left ( \\frac { \\lambda - 1 - \\varrho \\bar { \\gamma } } { \\sqrt { ( 1 + 2 \\varrho \\bar { \\gamma } ) / M \\bar { N } } } \\right ) . \\end{align*}"}
{"id": "8680.png", "formula": "\\begin{align*} d V _ { \\max } = \\sigma _ { \\max } ( x ) d x , \\end{align*}"}
{"id": "1503.png", "formula": "\\begin{align*} \\begin{aligned} Y _ { \\lambda } ( x , p ) & = \\sum _ { n = 1 } ^ { \\infty } \\sum _ { k = 1 } ^ { n } ( k ) _ { p , \\lambda } \\frac { x ^ { n } } { n ! } \\\\ & = \\sum _ { n = 1 } ^ { \\infty } \\Big ( ( 1 ) _ { p , \\lambda } + ( 2 ) _ { p , \\lambda } + \\cdots + ( n ) _ { p , \\lambda } \\Big ) \\frac { x ^ { n } } { n ! } \\end{aligned} \\end{align*}"}
{"id": "8609.png", "formula": "\\begin{align*} \\inf _ { J , R , Q } & { \\| A - ( J - R ) Q \\| } _ F ^ 2 + { \\| B - ( R - J ) Q \\| } _ F ^ 2 , \\\\ & J ^ T = - J , Q = Q ^ T > 0 , R ^ T = R \\geq 0 . \\end{align*}"}