diff --git "a/data_tmp/process_17/tokenized_finally.jsonl" "b/data_tmp/process_17/tokenized_finally.jsonl"
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-{"id": "7273.png", "formula": "\\begin{align*} t _ b t _ c t _ a t _ b ( a ) & = t _ b t _ c ( b ) = c , \\\\ t _ b t _ c t _ a t _ b ( c ) & = t _ b t _ a t _ c t _ b ( c ) = t _ b t _ a ( b ) = a , \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} { \\bf E } _ v ( { \\rm e } ^ { - \\beta X _ t } ) = { \\bf E } _ v ( { \\rm e } ^ { - \\beta H _ { t + v } } ) = \\frac { \\psi _ { \\beta } ( v ) } { \\psi _ { \\beta } ( t + v ) } , t \\geq 0 , \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} K ^ * = \\left \\lceil \\frac { 2 e } { \\theta _ 0 } \\left ( \\sqrt { \\frac { 1 + \\mu ( v , x _ 0 ) } { \\mu ( v , x _ 0 ) } } - 1 \\right ) + 1 \\right \\rceil . \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} N _ G ( B ^ \\Sigma ) = N _ G ( N _ \\Sigma B ) = N _ D B \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} E _ { t } ( \\underline { m } ) = F _ t ( \\underline { m } ) + \\sum _ { m ' < m } C _ { m , m ' } F _ t ( \\underline { m } ' ) \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\deg ( ( H ( 0 , \\cdot ) , D ( r ) ) = 1 \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\Bigl ( \\frac { b x } a \\Bigr ) ^ 2 + \\frac { b x } a + \\frac { b ( a ^ 2 + b ^ 2 ) } { a ^ 2 } = 0 . \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} i \\mathcal { G } _ { \\alpha \\beta } ( x , y ) = \\frac { \\langle \\psi _ { 0 } | T [ \\hat { \\psi } _ { \\mathrm { H } \\alpha } ( x ) \\hat { \\psi } _ { \\mathrm { H } \\beta } ^ { \\dagger } ( y ) ] | \\psi _ { 0 } \\rangle } { \\langle \\psi _ { 0 } | \\psi _ { 0 } \\rangle } , \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} \\log r & \\sim ( r _ 1 + r _ 2 ) \\log ( r _ 1 + r _ 2 ) - r _ 2 \\log r _ 2 \\\\ & = r _ 1 \\log r _ 2 + r _ 1 \\log \\Big ( 1 + \\frac { r _ 1 } { r _ 2 } \\Big ) + r _ 2 \\log \\Big ( 1 + \\frac { r _ 1 } { r _ 2 } \\Big ) \\sim r _ 1 \\log r _ 2 . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} \\Delta \\begin{pmatrix} 1 & y \\\\ 0 & x \\end{pmatrix} = N _ K ( x ) ^ { - 1 } , y \\in K x \\in K ^ * _ + , \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\alpha _ { n } ^ { ( N ) } ( f ) & = \\sum _ { j = 1 } ^ { m _ { N } } P ( A _ { j } ^ { ( N ) } ) \\alpha _ { n , j } ^ { ( N - 1 ) } ( f ) , \\\\ \\alpha _ { n , j } ^ { ( N - 1 ) } ( f ) & = \\sqrt { n } \\left ( \\mathbb { E } _ { n } ^ { ( N - 1 ) } ( f | A _ { j } ^ { ( N ) } ) - \\mathbb { E } ( f | A _ { j } ^ { ( N ) } ) \\right ) . \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} \\mu _ 1 & = \\max _ { f : \\sum _ { v \\in V } \\deg v \\cdot f ( v ) ^ 2 = 1 } \\sum _ { h \\in H } \\biggl ( \\sum _ { v _ i h } f ( v _ i ) - \\sum _ { v ^ j h } f ( v ^ j ) \\biggr ) ^ 2 \\\\ & \\leq \\max _ { f : \\sum _ { v \\in V } \\deg v \\cdot f ( v ) ^ 2 = 1 } \\sum _ { h \\in H } \\biggl ( \\sum _ { v _ i \\in h : f ( v _ i ) > 0 } f ( v _ i ) - \\sum _ { v ^ j \\in h : f ( v ^ j ) < 0 } f ( v ^ j ) \\biggr ) ^ 2 , \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\tilde { S } _ { k , n } = \\sum _ { \\substack { ( i _ { 1 } , i _ { 2 } , . . . , i _ { k } ) \\in B \\\\ B \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | B | = k } } \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k } ; n ) , \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} A ( t ) = \\sum _ { k , l ; k > l } \\int _ 0 ^ t \\frac { \\mu _ k ( X ^ { 1 / 2 , \\mu } ( s ) ) - \\mu _ l ( X ^ { 1 / 2 , \\mu } ( s ) ) } { X ^ { 1 / 2 , \\mu } _ k ( s ) - X ^ { 1 / 2 , \\mu } _ l ( s ) } \\ , d s . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\xi ^ { [ N ] } & = \\sum _ { \\lambda \\in \\Lambda } f _ \\lambda \\otimes e _ \\lambda , & \\eta ^ { [ N ] } & = \\sum _ { \\lambda \\in \\Lambda } g _ \\lambda \\otimes e _ \\lambda , \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} M _ { n } ( x ; \\beta , c ) = ( - 1 ) ^ n M _ { n } ( - x - \\beta ; \\beta , c ^ { - 1 } ) , \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} P _ \\# ( T '' _ \\theta ) = T _ a + d L , \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\langle a , b , x \\ \\vline \\ \\{ a x b a ( x b ) ^ { - 1 } \\} ^ { - 1 } x = x b \\{ a x b a ( x b ) ^ { - 1 } \\} ^ { - 1 } ( a x b ) ^ { - 1 } x b , \\ [ x , a x b a ( x b ) ^ { - 1 } ] = 1 \\rangle , \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} m - s + 1 - n _ { 1 } = \\underbrace { n _ { 1 } + \\cdots + n _ { 1 } } _ { M _ { 1 } - 1 \\ ; \\mathrm { t i m e s } } + \\underbrace { n _ { 2 } + \\cdots + n _ { 2 } } _ { M _ { 2 } \\ ; \\mathrm { t i m e s } } + \\cdots + \\underbrace { n _ { \\ell } + \\cdots + n _ { \\ell } } _ { M _ { \\ell } \\ ; \\mathrm { t i m e s } } \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} 0 ^ { * _ R } _ { H ^ t _ K ( R / I ) } = \\{ \\eta \\in H ^ t _ K ( R / I ) \\mid c F ^ e _ R ( \\eta ) = 0 \\in H ^ t _ K ( R / I ^ { [ p ^ e ] } ) c \\in R ^ { \\circ } e \\gg 0 \\} . \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} N _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T = P ( \\lambda ) \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} a _ s q ^ N \\prod _ { k = 1 } ^ { s - 1 } \\left ( 1 + a _ k q ^ N \\right ) . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} R _ k ( \\alpha ) ~ = ~ \\sum _ { x \\in \\Z ^ d } e ^ { \\alpha \\cdot x } \\mu _ k ( x ) \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} A = & \\sum _ { x , y \\in \\Omega _ { \\sigma } : x \\sim y } w _ { x y } ( u ( x ) - u ( y ) ) \\\\ = & - \\sum _ { x , y \\in \\Omega _ { \\sigma } : x \\sim y } w _ { x y } ( u ( y ) - u ( x ) ) \\\\ = & - \\sum _ { y \\in \\Omega _ { \\sigma } } \\sum _ { x \\in \\Omega _ { \\sigma } : x \\sim y } w _ { x y } ( u ( y ) - u ( x ) ) = - A . \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} h _ 0 ( r ) = \\sum _ { i = 0 } ^ { p } \\alpha _ i \\psi _ i ( r ) \\quad h _ { j + 1 } ( r ) = - \\tfrac { 1 } { r } h _ j ' ( r ) , \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\varepsilon ^ { 2 s } ( - \\Delta ) ^ s u + V ( x ) u = | u | ^ { 2 _ s ^ \\ast - 2 } u + \\lambda f ( u ) \\ \\ \\ x \\in \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} A _ { E } ^ { \\ast } E = 0 . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} p _ x = p _ x ( \\beta , H , h _ x ) : = 1 - a ^ 2 ( \\beta , H , | h _ x | ) , x \\in \\Lambda . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} s _ { \\lambda } ( 1 , q , q ^ 2 , \\dots , q ^ { m - 1 } ) = q ^ { b ( \\lambda ) } \\prod _ { ( i , j ) \\in \\lambda } \\frac { [ m + c _ { i , j } ] _ q } { [ h _ { i , j } ] _ q } , \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} W ^ { ( 1 ) } = & \\frac { a + b _ 1 } { 2 } y ^ 5 + \\left ( - 9 a + 1 6 b _ 1 + \\frac { b _ 2 + b } { 2 } \\right ) y ^ 9 \\\\ & + \\left ( \\frac { 1 5 3 a - 5 5 3 b _ 1 - c } { 2 } - 7 b _ 2 - 8 b \\right ) y ^ { 1 3 } \\\\ & + ( 1 5 2 1 9 2 - 4 0 8 a - 1 8 5 9 2 b _ 1 - 3 3 9 b _ 2 + 6 0 b + 7 c ) y ^ { 1 7 } + \\cdots . \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} e = \\begin{cases} ( c , - d ) _ p ( - d , p ^ m ) _ p & \\mathrm { o r d } _ p ( d ) , \\\\ ( - d , p ^ m ) _ p & \\mathrm { o r d } _ p ( d ) . \\end{cases} \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} \\lambda ' = ( \\lambda ' ) ^ 1 \\wedge \\lambda ^ 2 + \\lambda ^ 1 \\wedge ( \\lambda ' ) ^ 2 , \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} \\langle f _ 1 , ( \\Lambda _ 2 - \\Lambda _ 1 ) f _ 2 \\rangle _ { H ^ { 1 / 2 } ( \\partial \\Omega ) , H ^ { - 1 / 2 } ( \\partial \\Omega ) } = \\frac 1 2 \\int _ \\Omega ( \\mu _ 1 - \\mu _ 2 ) \\langle D \\bar { F _ 1 } , D \\bar { F _ 2 } \\rangle d x , \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} M ( t ) & = \\sum _ { i = 1 } ^ { d } \\sum _ { k ; k \\neq i } \\int _ { 0 } ^ { t } \\frac { d W _ i ( s ) } { X ^ { 1 / 2 , \\mu } _ i ( s ) - X ^ { 1 / 2 , \\mu } _ k ( s ) } , & Z ( t ) & = \\exp \\left ( \\nu M ( t ) - \\frac { \\nu ^ 2 } { 2 } \\langle M \\rangle ( t ) \\right ) . \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} H _ { i , j } = \\tbinom { N - 1 + i } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N - 1 + j } { N - 1 } ^ { \\frac { 1 } { 2 } } ( 1 - r ) ^ N r ^ { i + j } , \\quad \\forall i , j \\in \\N , \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c | c c c c } B _ k ( \\lambda ) & - I _ m \\\\ B _ { k - 1 } ( \\lambda ) & \\lambda ^ \\ell I _ m & - I _ m \\\\ B _ { k - 2 } ( \\lambda ) & & \\lambda ^ \\ell I _ m & \\ddots \\\\ \\vdots & & & \\ddots & - I _ m \\\\ B _ 1 ( \\lambda ) & & & & \\lambda ^ \\ell I _ m \\end{array} \\right ] = \\left [ \\begin{array} { c | c } M _ 2 ^ \\ell ( \\lambda ) & L _ { k - 1 } ( \\lambda ^ \\ell ) ^ T \\otimes I _ m \\end{array} \\right ] , \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\left | \\nabla f _ { 0 } ( M ) \\right | = o ( d i s t ^ { - 1 } ( M , L ) ) , \\ ; o \\ ; \\ ; \\mathbb { R } ^ { 3 } , \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\widetilde { \\Omega } _ { n } \\stackrel { \\rm d e f } = \\bigcup _ { k = 0 } ^ { 2 ^ n } \\overline { B } _ { 2 \\Lambda _ { n } } ( M _ { k n } ) , \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} G = P ^ { \\top \\ ! } P \\succeq 0 , \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} S _ C ^ { o , h } ( i _ { \\kappa } , \\kappa , j ) = \\frac { w _ { j , i _ { \\kappa } } } { h } \\sum \\limits _ { k \\in [ h ] } \\Bigl ( \\pi _ { i _ { \\kappa } , k } ^ { o , \\kappa , h } ( n _ { \\kappa } ) n _ { \\kappa } + \\pi _ { i _ { \\kappa } , k } ^ { o , \\kappa , h } ( n _ { \\kappa } + 1 ) \\bigl ( n _ { \\kappa } + 1 \\bigr ) \\Bigr ) = C _ j ^ 0 - \\frac { o ( h ) } { h } , \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } = 2 ^ { k - 1 + \\nu ( N ) } \\left ( \\prod _ { p \\mid M } \\frac { p } { p + 1 } \\right ) \\frac { | D | ^ { k - 1 / 2 } \\Lambda ( f , D , k ) } { | c _ h ( | D | ) | ^ 2 } . \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} T _ i = ( - 1 ) ^ M \\cdot ( L _ i - \\mu _ M ) + \\frac { 2 } { 9 } . '' \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} ( \\alpha \\otimes 1 , \\theta ) _ Y = ( \\alpha , \\beta ) _ X \\neq 0 . \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} \\widetilde { e } ( M _ { 1 } , M _ { 1 } ) & = 2 p ^ { 2 } - p + 3 , \\\\ \\widetilde { e } ( C _ { p } ^ { 3 } , M _ { 1 } ) & = 2 p + 1 , \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\max \\{ p ( x ) \\mid x \\in \\overline { B ( x _ { 0 } , \\varepsilon ) } \\} = \\max \\{ p ( x ) \\mid x \\in B ( x _ { 0 } , \\varepsilon , \\delta , \\theta ) , \\left \\vert x - x _ { 0 } \\right \\vert = \\varepsilon \\} . \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ \\delta ( t ) = \\sigma ( t ) = \\sigma _ 0 + \\beta t , \\\\ x ( t ) = \\sigma _ { 0 } t + \\frac { 1 } { 2 } \\beta t ^ 2 , \\\\ w ( t ) = \\sqrt { \\rho _ - \\rho _ + } ( u _ - - u _ + ) t , \\end{array} \\right . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} a \\ell = b _ 1 m + c _ 2 n , b m = a _ 2 \\ell + c _ 1 n , c n = a _ 1 \\ell + b _ 2 m \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\mathbf h ( b ( z ) \\tilde { \\kappa } ( \\theta ) ) = v ^ { k / 2 + 1 / 4 } e ^ { i ( k + 1 / 2 ) \\theta } h ( z ) \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} N _ \\lambda & = \\{ x \\in G | x \\cdot y z = x y \\cdot z \\ \\forall \\ y , z \\in G \\} \\\\ N _ \\mu & = \\{ y \\in G | x \\cdot y z = x y \\cdot z \\ \\forall \\ x , z \\in G \\} \\\\ N _ \\rho & = \\{ z \\in G | x \\cdot y z = x y \\cdot z \\ \\forall \\ x , y \\in G \\} \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} w ( W ) = \\sigma ( X ) \\cdot w ( F ) \\in W ( \\Q ) \\ : . \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K ( n ) \\} \\\\ = C _ { n } ^ { K ( n ) } ( 2 ^ { K ( n ) } - 1 ) . \\end{array} \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} \\| H \\| _ { S _ 1 ( \\ell _ 2 ( \\N ) ) } = \\| T \\| _ { S _ 1 ( \\ell _ 2 ( \\N ^ N ) ) } . \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} { \\rm g w } _ t ( A ) = \\left ( \\begin{array} { c c } e ^ { t ( x + y ) / 2 } & e ^ { i \\theta } \\sqrt { e ^ { t ( x + r ) } + e ^ { t ( x - r ) } - e ^ { t ( x + y ) } - e ^ { t ( x - y ) } } \\\\ 0 & e ^ { t ( x - y ) / 2 } \\end{array} \\right ) \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\psi ( T , p ) = ( \\sigma , \\alpha ( ( T _ \\sigma , p _ \\sigma ) ) . \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} C ^ { - 1 } & = \\frac { 1 } { 2 } \\begin{pmatrix} 2 & 1 \\\\ 2 & 2 \\end{pmatrix} & C ( z ) & = \\begin{pmatrix} z ^ 2 + z ^ { - 2 } & - 1 \\\\ - z - z ^ { - 1 } & z + z ^ { - 1 } \\end{pmatrix} & \\widetilde { C } ( z ) & = \\frac { 1 } { z ^ { 3 } + z ^ { - 3 } } \\begin{pmatrix} z + z ^ { - 1 } & 1 \\\\ z + z ^ { - 1 } & z ^ { 2 } + z ^ { - 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} & R _ 1 : = L _ 1 L _ 2 \\cdots L _ { 1 2 } C _ { 2 , 1 } L ^ { - 1 } , \\\\ & R _ { i + 1 } : = { } _ { \\Phi ^ \\prime } ( R _ i ) . \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} y z \\frac { \\partial f _ d } { \\partial x } + x z \\frac { \\partial f _ d } { \\partial y } - x y \\frac { \\partial f _ d } { \\partial z } & = 0 \\\\ a x \\frac { \\partial f _ j } { \\partial x } + b y \\frac { \\partial f _ j } { \\partial y } + z \\frac { \\partial f _ j } { \\partial z } + y z \\frac { \\partial f _ { j - 1 } } { \\partial x } + x z \\frac { \\partial f _ { j - 1 } } { \\partial y } - x y \\frac { \\partial f _ { j - 1 } } { \\partial z } & = c _ 0 f _ j \\\\ c _ 0 f _ 0 & = 0 \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} \\sigma ( X ) = - 0 - 7 \\cdot 1 + 1 \\cdot 1 1 = 4 . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{gather*} M \\otimes N = M \\otimes _ k N , a \\cdot ( m \\otimes n ) = a ^ 1 m \\otimes a ^ 2 n . \\end{gather*}"}
-{"id": "7459.png", "formula": "\\begin{align*} B _ s B _ t = B _ { \\varphi ( s ) } B _ { \\varphi ( s ) } \\subseteq B _ { \\varphi ( s t ) } . \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} \\mathbb { G } [ \\mathcal { A } ] = ( \\mathbb { G } ( A _ { 1 } ) , \\dots , \\mathbb { G } ( A _ { m _ { 1 } } ) ) ^ { t } , & \\qquad \\mathbb { E } [ f | \\mathcal { A } ] = \\left ( \\mathbb { E } ( f | A _ { 1 } ) , \\dots , \\mathbb { E } ( f | A _ { m _ { 1 } } ) \\right ) ^ { t } , \\\\ \\mathbb { G } [ \\mathcal { B } ] = ( \\mathbb { G } ( B _ { 1 } ) , \\dots , \\mathbb { G } ( B _ { m _ { 2 } } ) ) ^ { t } , & \\qquad \\mathbb { E } [ f | \\mathcal { B } ] = \\left ( \\mathbb { E } ( f | B _ { 1 } ) , \\dots , \\mathbb { E } ( f | B _ { m _ { 2 } } ) \\right ) ^ { t } . \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} b _ { i j } = ( \\alpha _ i , \\alpha _ j ) \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\frac { F ( X + b Y , Y ) } { { \\Big ( \\prod \\limits _ { j = 0 } ^ { p - 1 } { \\big ( \\delta _ j ( X + b Y ) - \\gamma _ j Y \\big ) } \\Big ) } ^ { t } } = \\frac { F ( X , Y ) } { { \\Big ( \\prod \\limits _ { j = 0 } ^ { p - 1 } { \\big ( \\delta _ j X - \\gamma _ j Y \\big ) } \\Big ) } ^ { t } } , \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} G ( u ) = \\left \\{ \\begin{array} { l l } ( u + 1 ) / 2 & \\textrm { f o r } u > 0 \\ , , \\\\ ( u - 1 ) / 2 & \\textrm { f o r } u < 0 \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} u ( \\varphi U ) = p u ( U ) , u ( \\zeta ) = q , \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} & \\partial _ t \\delta u + \\delta u \\partial _ x u _ 1 + u _ 2 \\partial _ x \\delta u = - \\partial _ x ( ( \\rho _ 1 ) - ( \\rho _ 2 ) ) + \\rho _ 1 ^ { - 1 } \\partial _ x ( \\mu ( \\rho _ 1 ) \\partial _ x u _ 1 ) - \\rho _ 2 ^ { - 1 } \\partial _ x ( \\mu ( \\rho _ 2 ) \\partial _ x u _ 2 ) , \\\\ & \\partial _ t \\delta \\rho + \\partial _ x ( u _ 1 \\delta \\rho + \\rho _ 2 \\delta u ) = 0 , \\\\ & ( \\delta \\rho , \\delta u ) | _ { t = 0 } = ( 0 , 0 ) \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} \\| \\sigma _ r f \\| _ k \\le \\sum _ { l = 0 } ^ { \\infty } r ^ { k l } \\| \\sigma _ r P _ l \\| _ k = \\sum _ { l = 0 } ^ { \\infty } r ^ { k l } \\| \\sigma _ r P _ l \\| _ { H ^ 2 } \\le \\sum _ { l = 0 } ^ { \\infty } r ^ { k l } \\| P _ l \\| _ { H ^ 2 } \\\\ \\le \\biggl ( \\sum _ { l = 0 } ^ { \\infty } r ^ { 2 k l } \\biggr ) ^ { 1 / 2 } \\biggl ( \\sum _ { l = 0 } ^ { \\infty } \\| P _ l \\| _ { H ^ 2 } ^ 2 \\biggr ) ^ { 1 / 2 } = \\frac { \\| f \\| _ { H ^ 2 } } { ( 1 - r ^ { 2 k } ) ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\tilde { S } _ { k , l } = S _ { n - l , n - k } . \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} \\tilde K : = \\sum _ { i , j = 1 } ^ v A _ i ^ T A _ j k _ { i j } \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} M = \\begin{bmatrix} x & 0 & x ^ 3 \\\\ y & x ^ 2 & 0 \\\\ 0 & y ^ 2 & z ^ 3 \\\\ 0 & z ^ 2 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ \\infty \\frac { e ^ { - a y \\tau } } { e ^ { y \\tau } - 1 } f ( y ) \\ , d y \\thicksim \\sum \\limits _ { r = 1 } ^ \\infty \\zeta ( r + 1 , \\ , 1 + a ) r ! a _ r / \\tau ^ { r + 1 } . \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\int _ { \\real _ + } m G ( d m ) = \\int _ { \\real _ + } m g ( m ) d m < + \\infty . \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\mathfrak { Q } _ { T _ { \\max } } \\left ( x _ { T _ { \\max } - 1 } , \\xi _ { T _ { \\max } } \\right ) = \\left \\{ \\begin{array} { l } \\inf \\ ; { \\overline { f } } _ { T _ { \\max } } ( x _ { T _ { \\max } } , x _ { T _ { \\max } - 1 } , \\xi _ { T _ { \\max } } ) : = - \\mathbb { E } \\Big [ \\sum \\limits _ { i = 1 } ^ { n + 1 } \\xi _ { T _ { \\max } + 1 } ( i ) x _ { T _ { \\max } } ( i ) \\Big ] \\\\ X _ { T _ { \\max } } \\in X _ { T _ { \\max } } ( x _ { T _ { \\max } - 1 } , \\xi _ { T _ { \\max } } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\int _ { \\R ^ 3 } | P u ( x ) | ^ 2 | P v ( x ) | ^ 2 \\d x = \\int _ { \\R ^ 3 } | u ( x ) | ^ 2 | v ( x ) | ^ 2 \\d x , \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } V , N ) = \\breve { g } ( V , \\breve { J } N ) = 0 . \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} \\dim ( D ^ \\circ ) = \\frac { n ^ 2 } { 8 } + \\frac { n } { 2 } + \\frac { 3 s ^ 2 } { 8 } . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\left [ \\begin{smallmatrix} I _ n \\\\ & I _ n \\\\ & D & I _ n \\\\ D & & & I _ n \\end{smallmatrix} \\right ] = \\left [ \\begin{smallmatrix} Y & & & * \\\\ & Y & * \\\\ & & Y ^ { - 1 } \\\\ & & & Y ^ { - 1 } \\end{smallmatrix} \\right ] \\left [ \\begin{smallmatrix} C & & & S \\\\ & C & S \\\\ & - S & C \\\\ - S & & & C \\end{smallmatrix} \\right ] , \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} { } ' \\ < z | \\phi ' = \\sum b _ k { } ' \\ < z | z _ k \\ > ' = \\sum b _ k K ( z , z _ k ) = \\sum a _ k K ( z , z _ k ) = \\sum a _ k { } ' \\ < z | z _ k \\ > ' = { } ' \\ < z | I ( \\phi ) \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} D _ S ( f _ t ) = \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } v \\ , \\frac { 1 } { \\sigma f _ t } \\big [ \\sigma \\partial _ v f _ t + ( v - G ( M _ 1 ( t ) ) ) f _ t \\big ] ^ 2 \\ , . \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\mathcal E _ { 1 , \\boldsymbol { \\alpha } } ( \\tau ) : = - \\frac { \\sqrt { 3 } } { 4 } \\int _ { - \\overline { \\tau } } ^ { i \\infty } \\int _ { w _ 1 } ^ { i \\infty } \\frac { \\theta _ 1 ( \\boldsymbol { \\alpha } ; \\boldsymbol { w } ) + \\theta _ 2 ( \\boldsymbol { \\alpha } ; \\boldsymbol { w } ) } { \\sqrt { - i ( w _ 1 + \\tau ) } \\sqrt { - i ( w _ 2 + \\tau ) } } d w _ 2 d w _ 1 \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} I _ t = \\lim _ { | \\pi | \\rightarrow 0 } \\sum _ { ( u , v ) \\in \\pi } G _ { u v } . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} H = \\Big \\{ h ( t ) = \\int _ 0 ^ t \\dot { h } ( s ) d s , \\ t \\in [ 0 , T ] ; \\ h ( 0 ) = 0 , \\ \\dot { h } \\in L ^ 2 ( [ 0 , T ] ) \\Big \\} . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} f ( i ) & \\leq \\frac { 1 } { 2 } \\int _ { t = 0 } ^ 1 f \\left ( i - \\frac { t } { 2 } \\right ) + f \\left ( i + \\frac { t } { 2 } \\right ) d t \\\\ & = \\frac { 1 } { 2 } ( \\log ( i + 1 ) - \\log ( i ) ) . \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} & C _ { 3 , j } : = { } _ { r ^ { j - 1 } } ( C _ { 3 , 1 } ) , & & & \\\\ & b _ j : = r ^ { j - 1 } ( b _ 1 ) , & & d _ j : = r ^ { j - 1 } ( d _ 1 ) , & \\\\ & V _ { i , j } : = { } _ { r ^ { j - 1 } } ( V _ i ) & & W _ { i , j } : = { } _ { r ^ { j - 1 } } ( W _ i ) & \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} \\beta ' | _ { S _ 1 \\cup \\cdots \\cup S _ { k - 1 } } = \\beta | _ { S _ 1 \\cup \\cdots \\cup S _ { k - 1 } } \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} y _ a ( u ) = \\begin{cases} \\int _ { - \\infty } ^ { \\varrho ( u ) } \\frac { 1 } { a \\sqrt { \\pi } } e ^ { - ( v - \\frac { 1 } { a } ) ^ { 2 } / a ^ { 2 } } d v & u \\in ( 0 , 1 ) \\\\ 0 & u = 0 \\\\ 1 & u = 1 , \\end{cases} \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} \\left . + \\sum _ { k = 1 } ^ 4 \\sum _ { { \\substack { l = 1 \\\\ l \\neq k } } } ^ 4 \\delta _ k \\delta _ l \\log \\left | \\frac { 1 - \\overline { \\zeta } _ k \\zeta _ l } { \\zeta _ k - \\zeta _ l } \\right | \\right \\} \\left ( \\frac { 1 } { \\log r } \\right ) ^ 2 + o \\left ( \\left ( \\frac { 1 } { \\log r } \\right ) ^ 2 \\right ) , \\ ; \\ ; r \\to 0 . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\mathcal { A } _ N & = \\left \\{ \\dot { \\phi } \\in \\ell _ \\infty ( \\N ) \\ , : \\ , A ( N , \\dot { \\phi } ) \\in S _ 1 ( \\ell _ 2 ( \\N ) ) \\right \\} \\\\ \\mathcal { B } _ N & = \\left \\{ \\dot { \\phi } \\in \\ell _ \\infty ( \\N ) \\ , : \\ , B ( N , \\dot { \\phi } ) \\in S _ 1 ( \\ell _ 2 ( \\N ) ) \\right \\} \\\\ \\mathcal { C } _ N & = \\left \\{ \\dot { \\phi } \\in \\ell _ \\infty ( \\N ) \\ , : \\ , C ( N , \\dot { \\phi } ) \\in S _ 1 ( \\ell _ 2 ( \\N ) ) \\right \\} . \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} F _ { x _ 0 , \\lambda } ( x ) : = \\ln \\left [ \\frac { 3 2 \\lambda ^ 2 } { ( 4 + \\lambda ^ 2 | x - x _ 0 | ^ 2 ) ^ 2 } \\right ] , \\lambda > 0 , x _ 0 \\in \\ \\R ^ 2 , \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G ) = \\Bigg ( \\sum _ { i = 1 } ^ k \\big ( \\lambda ^ { ( p ) } ( G _ i ) \\big ) ^ { p / ( p - r ) } \\Bigg ) ^ { ( p - r ) / p } . \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{gather*} \\Phi _ 2 ( X , 5 4 0 0 0 ) = X \\big ( X ^ 2 - 2 8 3 5 8 1 0 0 0 0 X + 6 5 4 9 5 1 8 2 5 0 0 0 0 \\big ) . \\end{gather*}"}
-{"id": "6751.png", "formula": "\\begin{align*} \\hat { K } ( v , c _ 1 , c _ 2 ) ( t ) = c _ 1 + \\int _ 0 ^ t \\psi ( \\int _ 0 ^ \\tau v ( s ) \\ , d s + c _ 2 ) \\ , d \\tau \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} f _ \\phi ( U ( m , n ) ) = \\tilde { \\phi } ( m + n ) \\gamma ( U ( m , n ) ) = \\tilde { \\phi } ( m + n ) . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\tilde \\Psi _ \\ast ( H ^ 1 ( C ; \\mathbb Q ) ) = L . \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} d X _ t = f ( X _ { t - } ) \\ , d t + g ( X _ { t - } ) \\ , d L _ t . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{gather*} L ( f _ { 3 6 } , 1 ) = \\frac 1 { 3 \\cdot 2 ^ { 4 / 3 } } B ( 1 / 3 , 1 / 3 ) = \\frac { 1 } { 3 ^ { 5 / 4 } \\cdot 2 ^ { 5 / 6 } } b _ { \\Q ( \\sqrt { - 3 } ) } . \\end{gather*}"}
-{"id": "7729.png", "formula": "\\begin{align*} \\pi \\sqrt { \\frac { \\hat { T } } { 2 P e } } = \\frac { 1 } { \\hat { C } ( \\hat { T } ) + 1 } . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\psi _ 3 ( ( a b ) ^ 3 ) = ( b ^ a , b , b ) , \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} \\begin{cases} & \\bar h _ 1 L _ 1 \\leq \\bar h _ 2 + \\bar h _ 3 L _ 2 , \\ , \\ , \\ , \\bar p _ 1 L _ 2 \\leq \\bar p _ 2 + \\bar p _ 3 L _ 1 \\\\ & \\underbar h _ 1 l _ 1 \\geq \\underbar h _ 2 + \\underbar h _ 3 l _ 2 , \\ , \\ , \\ , \\underbar p _ 1 l _ 2 \\geq \\underbar p _ 2 + \\underbar p _ 3 l _ 1 . \\end{cases} \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) = 2 ^ { k + 1 - \\nu ( M ) } C ( N , M , \\chi ) \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } \\frac { | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 } { \\langle g , g \\rangle ^ 2 } , \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} \\left \\{ \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} h _ 1 ( \\lambda ) , \\ldots , \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} h _ p ( \\lambda ) \\right \\} \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} A + B = \\left ( \\begin{array} { c c c c c c c } 3 & 3 & 1 & 0 & 0 & 0 & 0 \\\\ 2 & 0 & 4 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 2 & 5 & 6 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\Delta ( \\alpha ) = \\alpha \\otimes \\alpha - q \\gamma ^ { * } \\otimes \\gamma , & \\Delta ( \\gamma ) = \\gamma \\otimes \\alpha + \\alpha ^ { * } \\otimes \\gamma . \\end{array} \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\Psi _ 1 ^ { \\min } ( p ^ e , p ^ s ) = \\delta _ { e = s } 2 . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} B _ { y ^ { \\ast } } ^ { \\varepsilon } ( x ^ { \\ast } ) : = \\big \\{ x \\in X \\ , : \\ , ( x , 0 _ { Y } ) \\in ( M ^ { \\varepsilon } F ) ( x ^ { \\ast } , y ^ { \\ast } ) \\big \\} . \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} K _ { \\tau } : = \\tau ( K ) , L _ { \\tau } : = \\hat \\tau ( L ) , L _ { \\tau , n } = \\hat \\tau ( L _ n ) , \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} C ( \\tilde { J } U , V ) = C ( U , \\tilde { J } V ) , B ( \\tilde { J } U , V ) = B ( U , \\tilde { J } V ) , C ( U , V ) = C ( V , U ) , \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} a _ j = - 2 \\pi \\lim _ { x \\to y _ j } \\psi ( x ) ( \\log | x - y _ j | ) ^ { - 1 } , j = 1 , \\dots , N . \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\widetilde { ( - 1 ) ^ d \\Sigma ^ d ( \\varphi ) } ( \\delta ^ d \\alpha ^ d \\eta ) = ( - 1 ) ^ d \\Sigma ^ d ( \\varphi ) \\circ \\delta ^ d \\alpha ^ d \\eta = 0 \\circ \\alpha ^ d \\eta = 0 , \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} P ( \\mathcal { A } ) \\cdot \\mathbf { P } _ { \\mathcal { B } | \\mathcal { A } } \\mathbb { E } [ f | \\mathcal { B } ] ) & = \\sum _ { j = 1 } ^ { m _ { 1 } } P ( A _ { j } ) \\sum _ { k = 1 } ^ { m _ { 2 } } P ( B _ { k } \\ ; | \\ ; A _ { j } ) \\mathbb { E } ( f | B _ { k } ) \\\\ & = \\sum _ { j = 1 } ^ { m _ { 1 } } \\sum _ { k = 1 } ^ { m _ { 2 } } P ( A _ { j } \\cap B _ { k } ) \\mathbb { E } [ f | B _ { k } ] = P ( f ) . \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} ( ( f \\circ g ) ( T _ { \\lambda , \\mu } ) ) ^ i = ( f ( T _ { \\lambda } ) ) ^ i \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b ) \\triangleq \\exp \\Bigl ( \\bigl ( \\mathcal { S } _ { M - 1 } \\log \\Gamma _ M \\bigr ) ( q \\ , | a , \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } \\log \\Gamma _ M \\bigr ) ( 0 \\ , | a , \\ , b ) \\Bigr ) . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} G _ { k } ( \\textbf { a } , \\textbf { v } ) = \\left ( \\begin{array} { c c c c } v _ { 1 } & v _ { 2 } & \\cdots & v _ { n } \\\\ v _ { 1 } a _ { 1 } & v _ { 2 } a _ { 2 } & \\cdots & v _ { n } a _ { n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ v _ { 1 } a _ { 1 } ^ { k - 1 } & v _ { 2 } a _ { 2 } ^ { k - 1 } & \\cdots & v _ { n } a _ { n } ^ { k - 1 } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} M ^ { ( 3 , 1 ) } \\otimes M ^ { ( 2 , 2 ) } & = M ^ { ( 2 , 1 , 1 ) } \\oplus M ^ { ( 2 , 1 , 1 ) } \\displaybreak [ 1 ] \\\\ M ^ { ( 3 , 1 ) } \\otimes M ^ { ( 3 , 1 ) } & = M ^ { ( 3 , 1 ) } \\oplus M ^ { ( 2 , 1 , 1 ) } \\displaybreak [ 1 ] \\\\ M ^ { ( 3 , 1 ) } \\otimes M ^ { ( 4 ) } & = M ^ { ( 3 , 1 ) } \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} \\Lambda _ { f , c } ( x , t ) = u ( x , t ) , \\ \\ \\ \\ ( x , t ) \\in \\partial \\Omega \\times \\R _ { + } . \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ ( M ^ { \\star } ) ^ { q / \\tau } _ { ( 1 / \\tau , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 ) } \\bigr ] = \\frac { \\Gamma ( 1 - q ) } { \\Gamma ( 1 - q / \\tau ) } \\ , { \\bf E } \\bigl [ ( M ^ { \\star } ) ^ q _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\bigr ] . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} a ( x , y , \\xi ) & = \\sum _ { | \\alpha | + | \\beta | < M } \\frac { \\partial _ x ^ { \\alpha } \\partial _ y ^ \\beta a ( v , v , \\xi ) [ - \\tau ( - w ) ] ^ \\alpha [ - w - \\tau ( - w ) ] ^ \\beta } { \\alpha ! \\beta ! } + r _ M ( x , y , \\xi ) \\ , , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} X \\cdot Y = \\frac { 1 } { r } , Y \\cdot Z = \\frac { 1 } { p } , X \\cdot Z = \\frac { 1 } { q } , X ^ 2 = \\frac { p } { q r } , Y ^ 2 = \\frac { q } { p r } , X ^ 2 = \\frac { r } { p q } . \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} \\psi _ \\zeta ( x ) = e ^ { i A ( 0 ) x } \\phi _ \\zeta ( x ) \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} \\Delta _ \\phi ^ 0 f ( x ) = f ( x ) , \\sum _ { k = 1 } ^ 0 x _ k = 0 , \\prod _ { k = 1 } ^ 0 x _ k = 1 . \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J \\setminus J _ { K ( n ) } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) = \\int _ 0 ^ T \\Phi _ { t } d B _ { t } . \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} B _ { k , l } = \\sum _ { a , b \\geq 0 , a - b = l - k } ( - 1 ) ^ { l - a } F _ { 1 } ^ { ( b ) } E _ { 1 } ^ { ( a ) } 1 _ { ( k , l ) } : M ^ { k , l } \\rightarrow M ^ { l , k } \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} { \\tilde z } _ 1 = - z _ 4 , \\ ; \\ ; \\ ; { \\tilde z } _ 2 = \\frac { 1 } { z _ 1 } - z _ 3 , \\ ; \\ ; \\ ; { \\tilde z } _ 3 = z _ 2 , \\ ; \\ ; \\ ; { \\tilde z } _ 4 = z _ 1 . \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} F ( q _ 1 - q _ 2 ) = ( P _ N U ( q _ 1 ) - P _ N U ( q _ 2 ) ) + P _ N ^ \\perp F ( q _ 1 - q _ 2 ) - P _ N ( B ( q _ 1 ) - B ( q _ 2 ) ) . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} p ^ * ( \\vartheta _ n ) = \\lambda _ { b } \\frac { \\Delta _ { \\{ b , b + 2 , \\dots , n \\} } } { \\Delta _ { \\{ b + 1 , \\dots , n \\} } } = p ^ * ( X _ { 0 , 0 } ) , \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} p _ { o u t } = \\textrm { P r } \\{ C _ { s } - \\max _ { k } C _ { e , k } < R \\} . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} \\zeta ( \\alpha , 0 , s ) = \\frac { \\Gamma ( 1 - s ) } { ( 2 \\pi ) ^ { 1 - s } } \\biggl ( i e \\left ( - \\frac { s } { 4 } \\right ) \\zeta ( 0 , - \\alpha , 1 - s ) - i e \\left ( \\frac { s } { 4 } \\right ) \\zeta ( 0 , \\alpha , 1 - s ) \\biggr ) . \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} B _ \\phi = \\begin{bmatrix} \\kappa _ d P _ d \\\\ & I _ n \\\\ & & \\ddots \\\\ & & & I _ n \\end{bmatrix} \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\breve { J } V = B V + C V , \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} C _ 2 ( 1 ) C _ 2 ( 0 ) C _ 1 ( 0 ) F ( B _ 0 ; p ) & = C _ 2 ( 1 ) C _ 2 ( 0 ) C _ 1 ( 0 ) F ( 1 , 1 , 6 ; 1 , 3 , 4 ; p ) \\\\ & = C _ 2 ( 1 ) C _ 2 ( 0 ) F ( 2 , 1 , 5 ; 1 , 3 , 4 ; p ) \\ ( = C _ 2 ( 1 ) C _ 2 ( 0 ) F ( B _ 1 ; p ) ) \\\\ & = C _ 2 ( 1 ) F ( 2 , 2 , 4 ; 1 , 3 , 4 ; p ) \\\\ & = C _ 2 ( 1 ) F ( 2 , 3 , 3 ; 1 , 3 , 4 ; p ) \\ ( = F ( B _ 2 ; p ) ) . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\exp ( - ( 2 W _ s + \\mu s ) ) \\ , d s \\sim \\frac { 1 } { 2 \\ , Z _ \\mu } , \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} S _ { 1 , m } \\left ( p , k \\right ) = \\sum _ { t = 0 } ^ { m - 1 } \\left ( - 1 \\right ) ^ { t } s \\left ( m , m - t \\right ) S \\left ( p + m - t , k + m \\right ) . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } u _ r = 1 - \\lim _ { r \\rightarrow \\infty } \\sum _ { i = 1 } ^ r v _ i = 0 . \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} \\frac { A ( x ) } { B ( x ) } = \\frac { 1 + a _ 1 + b + a _ 1 z } { 1 + b + a _ 1 z } \\cdot \\frac { y + z + 1 } { y + z } . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\Delta ^ { k } M _ { n } ( x ; \\beta , c ) = ( m - k + 1 ) _ k M _ { n - k } ( x ; \\beta + k , c ) , \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\tilde { T } _ { m , n } & = T _ { 2 m , 2 n } \\\\ & = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\varphi } \\left ( 2 ( m + n + \\chi ^ I ) \\right ) \\\\ & = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\dot { \\phi } \\left ( | m | + | n | + | I | \\right ) \\\\ & = \\sum _ { k = 0 } ^ N \\tbinom { N } { k } ( - 1 ) ^ k \\dot { \\phi } \\left ( | m | + | n | + k \\right ) \\\\ & = \\mathfrak { d } _ 1 ^ N \\dot { \\phi } \\left ( | m | + | n | \\right ) . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} \\bar h _ 3 \\bar p _ 3 & = a _ 2 b _ 1 \\Big \\{ l \\frac { \\chi _ 2 } { d _ 3 } ( a _ 1 - k \\frac { \\chi _ 1 } { d _ 3 } ) + k \\frac { \\chi _ 1 } { d _ 3 } ( b _ 2 - l \\frac { \\chi _ 2 } { d _ 3 } ) \\Big \\} ^ 2 \\\\ & = ( l \\frac { \\chi _ 2 } { d _ 3 } ) ^ 2 ( a _ 1 - k \\frac { \\chi _ 1 } { d _ 3 } ) ^ 2 a _ 2 b _ 1 + ( k \\frac { \\chi _ 1 } { d _ 3 } ) ^ 2 ( b _ 2 - l \\frac { \\chi _ 2 } { d _ 3 } ) ^ 2 a _ 2 b _ 1 + 2 l k \\frac { \\chi _ 1 \\chi _ 2 } { d _ 3 ^ 2 } a _ 2 b _ 1 . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} a ^ { \\sim \\mathcal { T } _ { 1 } \\left ( { \\mathbf { L } } \\right ) } = \\left \\{ \\begin{array} [ c ] { l } 1 a = 0 ^ { \\mathcal { T } _ { 1 } \\left ( { \\mathbf { L } } \\right ) } \\\\ 0 ^ { \\mathcal { T } _ { 1 } \\left ( { \\mathbf { L } } \\right ) } \\end{array} \\right . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} u ' & = - A u - B ( u , u ) + f ( t ) \\\\ & = - A v _ N - A \\bar u _ N - B ( \\bar u _ N + v _ N , \\bar u _ N + v _ N ) + \\bar F _ N + F _ { N + 1 } + \\tilde F _ { N + 1 } , \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} [ X \\xleftarrow p V \\xrightarrow s Y ; E ] \\circ _ { \\otimes } & [ Y \\xleftarrow q W \\xrightarrow t Z ; F ] \\\\ & : = [ ( X \\xleftarrow p V \\xrightarrow s Y ) \\circ ( Y \\xleftarrow q W \\xrightarrow s Z ) ; \\widetilde { q } ^ * E \\otimes \\widetilde { s } ^ * F ] , \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} f _ 2 ( M _ 1 \\otimes M _ 2 ) = A ^ * _ { \\mathcal { C R } ^ \\preceq } \\left ( f _ 1 ( M _ 1 ) f _ 1 ( M _ 2 ) \\right ) , \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} D _ { \\omega } : = \\bigcap _ { n = 0 } ^ \\infty \\left ( f ^ n _ \\omega \\right ) ^ { - 1 } ( \\mathbb B _ R ) \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\varphi ( u , v ) = \\langle \\tilde { P } ( u ) , \\tilde { Q } ( v ) \\rangle , \\quad \\forall u , v \\in \\Gamma ( G ) ^ N , \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} M ( m ) : = \\overrightarrow { \\bigotimes _ { r \\in \\mathbb { Z } } } \\left ( \\bigotimes _ { i \\in I } L ( Y _ { i , r } ) ^ { \\otimes u _ { i , r } ( \\widetilde { m } ) } \\right ) , \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} \\psi _ 3 ( h _ s ) = \\big ( a ^ { i e _ 1 - j ( e _ 1 + \\dots + e _ { p - ( s - 1 ) } ) } u _ 1 , a ^ { i ( e _ 1 + e _ 2 ) - j ( e _ 1 + \\dots + e _ { p - ( s - 2 ) } ) } & u _ 2 , \\dots , \\\\ & a ^ { - j ( e _ 1 + \\dots + e _ { p - s } ) } u _ p \\big ) , \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} \\frac { \\partial \\mathbb { P } ( t , \\rho ) } { \\partial t } = \\Pi ( \\rho ) ^ { \\frac { 1 } { 2 } } \\nabla _ \\rho \\cdot ( L ( \\rho ) \\mathbb { P } ( t , \\rho ) \\big ( d _ \\rho \\mathcal { F } ( \\rho ) + \\beta d _ \\rho \\log \\mathbb { P } ( t , \\rho ) \\big ) \\Pi ( \\rho ) ^ { - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} y ^ { \\alpha } \\left ( b ^ { - 1 } y ^ { \\alpha } \\cdot x \\right ) & = x \\left ( y ^ { \\alpha } \\right ) ^ { 2 } \\cdot b ^ { - 1 } \\\\ & = \\left ( y ^ { \\alpha } \\right ) ^ { 2 } x \\cdot b ^ { - 1 } = \\left ( y ^ { \\alpha } \\right ) ^ { 2 } \\cdot x b ^ { - 1 } = y ^ { \\alpha } \\left ( y ^ { \\alpha } \\cdot x b ^ { - 1 } \\right ) \\\\ & b ^ { - 1 } y ^ { \\alpha } \\cdot x = y ^ { \\alpha } \\cdot x b ^ { - 1 } \\\\ & y ^ { \\alpha } b ^ { - 1 } \\cdot x = y ^ { \\alpha } \\cdot b ^ { - 1 } x \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} k ( k - 1 ) \\alpha _ N ^ 2 / 2 \\ge 2 \\tau _ { d , k } ^ 2 ( u ) = 2 u f ( k , d , N ) \\ , , \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} t _ { s _ { 1 , g - 1 } } ^ { 5 ( g - 1 ) n } [ T _ { s _ { 1 , 1 } } , r ] = [ \\mathcal { V } ' _ 1 , \\mathcal { W } ' _ 1 ] [ \\mathcal { V } ' _ 2 , \\mathcal { W } ' _ 2 ] \\cdots [ \\mathcal { V } ' _ { \\left [ \\frac { | 5 n | } { 2 } \\right ] + 1 } , \\mathcal { W } ' _ { \\left [ \\frac { | 5 n | } { 2 } \\right ] + 1 } ] \\cdot t _ { d _ { g - 1 } } ^ { 1 0 ( g - 1 ) n } [ T _ d , r ] , \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} E _ { \\tau } ^ { \\phi ^ { p ^ n } = 1 } = E _ { \\tau } ^ { \\Gamma ^ { p ^ n } } = L _ { \\tau , n } ' : = \\hat { \\tau } ( L _ n ' ) . \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} P = \\prod _ { f \\in I } X _ f . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{gather*} \\begin{cases} \\mathcal M _ 2 \\mathcal M _ 3 & k \\equiv 1 , 5 \\mod 6 , \\\\ \\mathcal M _ 2 & k \\equiv 2 , 4 \\mod 6 , \\\\ \\mathcal M _ 3 & k \\equiv 3 \\mod 6 , \\\\ { \\mathcal O } _ F & k \\equiv 0 \\mod 6 , \\end{cases} \\end{gather*}"}
-{"id": "2340.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } B R ( 0 ) ^ { 1 / 2 } \\psi _ n & = \\lim _ { n \\rightarrow \\infty } \\frac { 2 B } { \\pi } \\left ( \\int _ 0 ^ \\infty R ( \\lambda ) \\ , d \\lambda \\right ) _ 0 \\psi _ n , \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} e _ { \\star } ^ { p q } = K _ 1 + K _ 2 + K _ 3 + K _ 4 \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} \\| g ( w , p ) \\| _ { L ^ { q ' } _ { t > T } W ^ { 2 , r ' } _ x } \\lesssim \\big \\Vert \\sum ^ N _ { \\mu , \\nu = 1 } \\lambda _ { \\mu \\nu } \\| u _ \\mu \\| _ { W ^ { 2 , r } _ x } \\| u _ \\nu | ^ { p + 1 } | u _ \\mu | ^ { p - 1 } \\| _ { L _ x ^ { \\frac { r } { 2 p } } } \\big \\Vert _ { L ^ { q ' } _ { t > T } } & \\\\ \\lesssim \\sum ^ N _ { \\mu , \\nu = 1 } \\big \\Vert \\| u _ \\mu \\| _ { W ^ { 2 , r } _ x } \\| u _ \\nu | ^ { p + 1 } | u _ \\mu | ^ { p - 1 } \\| _ { L _ x ^ { \\frac { r } { 2 p } } } \\big \\Vert _ { L ^ { q ' } _ { t > T } } , & \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} C = \\left ( \\begin{matrix} 0 & 0 & 0 & 1 \\cr 0 & 0 & 1 & 0 \\cr 0 & - 1 & 0 & 1 \\cr - 1 & 0 & 0 & 0 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n B _ { n , k } t ^ k = \\sum _ { k = 0 } ^ n C _ { n , k } ( 1 + t ) ^ k , \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} \\phi ^ { \\lambda } \\left ( d + \\sum _ { i = 0 } ^ \\ell \\psi _ i \\check f _ i d t + b d t \\right ) \\phi ^ { - \\lambda } \\coloneqq d + \\sum _ { i = 0 } ^ \\ell \\phi ^ { - \\langle \\lambda , \\check \\alpha _ i \\rangle } \\psi _ i \\check f _ i d t - \\lambda \\phi ^ { - 1 } d \\phi + \\phi ^ { \\lambda } b \\phi ^ { - \\lambda } d t , \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} \\gamma _ n ( z ) = \\int \\limits _ T \\bigg ( \\prod _ { 1 \\leq i < j \\leq n } ( t _ i ^ 2 - t _ j ^ 2 ) \\bigg ) \\Big ( \\prod _ { j = 1 } ^ n t _ j \\Big ) ^ { - z - n } \\ , d \\mathbf { t } , \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} \\Delta \\ , = \\ , \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + ( N - 1 ) \\ , \\frac { \\psi ' } { \\psi } \\ , \\frac { \\partial } { \\partial r } + \\frac 1 { \\psi ^ 2 } \\ , \\Delta _ { \\mathbb S ^ { N - 1 } } \\ , . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\lambda = \\limsup _ { n \\to \\infty } \\frac 1 n \\log \\lVert x _ n \\rVert . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} c ( t , X _ t ) = \\frac { \\overrightarrow { b _ t } + \\overleftarrow { b _ t } } { 2 } \\mbox { a n d } w ( t , X _ t ) = \\frac { \\overrightarrow { b _ t } - \\overleftarrow { b _ t } } { 2 } , \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} m + 1 - n _ { 1 } - n _ { i } = \\underbrace { n _ { 1 } + \\cdots + n _ { 1 } } _ { M _ { 1 } - 1 \\ ; \\mathrm { t i m e s } } + \\cdots + \\underbrace { n _ { i } + \\cdots + n _ { i } } _ { M _ { i } - 1 \\ ; \\mathrm { t i m e s } } + \\cdots + \\underbrace { n _ { \\ell } + \\cdots + n _ { \\ell } } _ { M _ { \\ell } \\ ; \\mathrm { t i m e s } } , \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} H _ 3 = \\left ( \\tbinom { \\alpha + \\beta + i } { \\alpha + \\beta } ^ { \\frac { 1 } { 2 } } \\tbinom { \\alpha + \\beta + j } { \\alpha + \\beta } ^ { \\frac { 1 } { 2 } } a _ { i + j } \\right ) _ { i , j \\in \\N } \\in S _ 1 ( \\ell _ 2 ( \\N ) ) . \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} \\tilde { a } _ { i j } & = a _ { i j } - \\ 1 { i j \\in \\vec { T } } + \\ 1 { j i \\in \\vec { T } } , \\ , \\ , i j \\in \\vec { E } . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} & \\Phi ( \\theta ) = \\left \\{ \\begin{matrix} \\cos \\theta \\cdot I + i \\sin \\theta \\cdot F , & 0 \\leq \\theta \\leq \\pi , \\\\ & & \\\\ ( \\cos \\theta + i \\sin \\theta ) \\cdot I , & \\pi \\leq \\theta \\leq 2 \\pi . \\end{matrix} \\right . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} h _ 0 ( R ) = 1 , \\ h _ { 1 } ( R ) = 0 , \\ h _ { 2 } ( R ) = 0 , \\ \\dots , \\ h _ { p } ( R ) = 0 . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} \\psi _ { \\varepsilon , \\zeta } ( x ) = \\psi _ \\zeta ( \\varepsilon ^ { - 1 } x ) . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} B = \\left ( \\begin{array} { c c } b _ 1 & b _ 2 / 2 \\\\ b _ 2 / 2 & b _ 3 \\end{array} \\right ) . \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} L r _ l ( q _ i ) - q _ i r _ l ( q _ i ) = q _ i , i = 1 , 2 , \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} R _ 4 = R _ 2 R _ 2 - R _ 1 R _ { 1 2 } + R _ { 1 1 2 } = ( R _ 4 + R _ { 2 2 } ) - ( R _ { 2 2 } + R _ { 1 1 2 } ) + R _ { 1 1 2 } . \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} n ( T ) & = n ^ + ( T ) - n ^ - ( T ) = 0 - 0 , \\\\ n ( C _ 2 ) & = n ^ + ( C _ 2 ) - n ^ - ( C _ 2 ) = ( g - 1 ) n - 0 , \\\\ n ( L ) & = n ^ + ( L ) - n ^ - ( L ) = 1 1 ( g - 1 ) n - 0 \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} f _ { j , p } ( z ) : = \\frac { 1 } { 2 p } \\sum _ { \\substack { n \\in \\Z \\\\ n \\equiv j \\pmod { 2 p } } } n q ^ { \\frac { n ^ 2 } { 4 p } } \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} | \\eta _ r ( x , y ) | = p ^ { - n _ 0 } . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\zeta _ M \\bigl ( s , \\ , w \\ , | \\ , a \\bigr ) = \\sum \\limits _ { k _ 1 , \\cdots , k _ M = 0 } ^ \\infty \\bigl ( w + k _ 1 a _ 1 + \\cdots + k _ M a _ M \\bigr ) ^ { - s } , \\ , \\ , \\Re ( s ) > M , \\ , \\Re ( w ) > 0 , \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} V ' _ I = \\sum _ { k = i _ r } ^ n ( - 1 ) ^ { k - i _ r } \\binom { k - 1 } { i _ r - 1 } V ' _ { I [ n - k ] } V ' _ k . \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} Q ^ { ( m ) } _ { 2 r } ( 1 ) = - Q ^ { ( m ) } _ { 2 r + 1 } ( 1 ) = \\frac { 1 } { q ^ r } \\ ; \\frac { \\prod \\limits _ { i = 1 } ^ { 2 r } ( q ^ m - q ^ i ) } { \\prod \\limits _ { i = 0 } ^ { r - 1 } ( q ^ { 2 r } - q ^ { 2 i } ) } , \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} b _ { \\lambda } b _ { ( s ) } & = \\psi _ { w _ { 0 } } y _ 1 ^ { \\lambda _ 1 + n - 1 } y _ 2 ^ { \\lambda _ 2 + n - 2 } \\cdots y _ n ^ { \\lambda _ n } \\psi _ { w _ { 0 } } y _ { \\min } \\psi _ { w _ { 0 } } y _ 1 ^ s y _ 1 ^ { n - 1 } y _ 2 ^ { n - 2 } \\cdots y _ n \\psi _ { w _ { 0 } } y _ { \\min } \\\\ & = \\psi _ { w _ { 0 } } y _ 1 ^ { \\lambda _ 1 + n - 1 } y _ 2 ^ { \\lambda _ 2 + n - 2 } \\cdots y _ n ^ { \\lambda _ n } \\sum _ { l _ 1 + \\cdots + l _ n = s } y _ 1 ^ { l _ 1 } \\cdots y _ n ^ { l _ n } \\psi _ { w _ { 0 } } y _ { \\min } \\\\ \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} e = \\epsilon = \\begin{cases} ( c , d ) _ p ( c , p ^ n ) _ p & d \\neq 0 \\mathrm { o r d } _ p ( c ) , \\\\ ( c , p ^ n ) _ p & d = 0 \\mathrm { o r d } _ p ( c ) . \\end{cases} \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} \\int _ { S } g = e ^ { - R } \\sigma _ { 4 } R ^ { 4 } , \\qquad \\int _ { S } \\tfrac { \\partial } { \\partial \\nu } g = - e ^ { - R } \\sigma _ { 4 } R ^ { 4 } , \\qquad \\int _ { S } \\Delta g = e ^ { - R } ( 1 - \\tfrac { 4 } { R } ) \\sigma _ { 4 } R ^ { 4 } . \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} & g ''' _ 0 ( x , a , b ) = a \\bigl ( 2 F ^ { 0 , 4 } + 1 2 a ^ 2 F ^ { 1 , 5 } + 2 4 a ^ 4 F ^ { 2 , 6 } + 1 6 a ^ 6 F ^ { 3 , 7 } \\bigr ) , \\\\ & \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\\\ & g _ 0 ^ { ( l ) } ( x , a , b ) = a \\sum _ { j = 0 } ^ l c _ { j , l } a ^ { 2 j } F ^ { j , l + j + 1 } ( x ; a , b ) , \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} B _ A ( \\lambda ; \\ , X ) = \\sum _ { J } f _ J ( X ) \\ , B _ { A ^ J } ( \\lambda ) \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} \\phi _ { \\breve { \\mathbf g } , p } ( x ) = p ^ { - 1 / 2 } \\varepsilon ( 1 / 2 , \\lambda _ p ) \\mathbf 1 _ { p \\Z _ p } ( x _ 1 ) \\mathbf 1 _ { \\Z _ p } ( x _ 4 ) \\mathbf 1 _ { p ^ 2 \\Z _ p } ( x _ 3 ) \\mathbf 1 _ { p ^ { - 1 } \\Z _ p ^ { \\times } } ( x _ 2 ) \\underline { \\chi } _ p ( x _ 2 ) . \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\varphi _ p ( h ) = \\begin{cases} \\xi _ 1 ( p ) ^ { - 1 } \\xi _ 1 ( a ) \\xi _ 2 ( d ) | a d ^ { - 1 } | ^ { 1 / 2 } & h \\in \\left ( \\begin{smallmatrix} a & \\ast \\\\ 0 & d \\end{smallmatrix} \\right ) K _ 0 ^ 1 , a , d \\in \\Q _ p ^ { \\times } , \\\\ 0 & i f h \\not \\in B ( \\Q _ p ) K _ 0 ^ 1 . \\end{cases} \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} p ( x \\star u , y \\star u ) = p ( x , y ) , \\forall u , x , y \\in E . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} \\operatorname { f i x } ( y , M ^ G ) & \\le \\sum _ { 2 l _ { i , - i } + l _ { - 1 } + l _ 1 = m } p _ { l _ { i , - i } } ( E _ { i , - i } ) p _ { l _ { - 1 } } ( E _ { - 1 } ) p _ { l _ 1 } ( E _ 1 ) \\\\ & < \\sum _ { 2 l _ { i , - i } + l _ { - 1 } + l _ 1 = m } c q ^ { l _ { i , - i } ( \\frac { k b } { 2 } - 3 l _ { i , - i } ) + l _ { - 1 } ( \\frac { k b } { 4 } - \\frac { 3 } { 2 } l _ { - 1 } + \\frac { 1 } { 2 } ) + l _ { 1 } ( \\frac { k b } { 4 } + s - \\frac { 3 } { 2 } l _ { 1 } + \\frac { 1 } { 2 } ) } \\\\ & < q ^ { \\frac { 1 } { 8 } m ( 2 n - 3 m ) + c ' m } \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} \\bar r _ 1 \\big ( a _ { 1 , \\inf } - k \\frac { \\chi _ 1 } { d _ 3 } \\big ) & = a _ { 0 , \\sup } - a _ { 2 , \\inf } \\big \\{ \\frac { b _ { 0 , \\inf } - b _ { 1 , \\sup } \\bar r _ 1 - l \\frac { \\chi _ 2 } { d _ 3 } \\bar r _ 2 } { b _ { 2 , \\sup } - l \\frac { \\chi _ 2 } { d _ 3 } } \\big \\} \\\\ & \\ , \\ , \\ , \\ , - k \\frac { \\chi _ 1 } { d _ 3 } \\big \\{ \\frac { a _ { 0 , \\inf } - a _ { 2 , \\sup } \\bar r _ 2 - k \\frac { \\chi _ 1 } { d _ 3 } \\bar r _ 1 } { a _ { 1 , \\sup } - k \\frac { \\chi _ 1 } { d _ 3 } } \\big \\} . \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} L ^ { \\Delta , \\sigma } _ { \\alpha , u } u = L ^ { \\Delta , \\sigma } _ { \\alpha } u , u \\in \\mathcal { D } ^ { \\Delta , \\sigma } _ { \\alpha , u } . \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\partial _ t w & = ( \\partial _ t \\rho ) ( - p ' ( \\rho ) + \\mu ' ( \\rho ) \\partial _ x u ) + \\mu ( \\rho ) \\partial _ t \\partial _ x u \\\\ & = - \\partial _ x ( u \\rho ) ( - p ' ( \\rho ) + \\mu ' ( \\rho ) \\partial _ x u ) + \\mu ( \\rho ) \\partial _ t \\partial _ x u \\\\ & = - \\rho \\partial _ x u ( - p ' ( \\rho ) + \\mu ' ( \\rho ) \\partial _ x u ) - u ( \\partial _ x w - \\mu ( \\rho ) \\partial _ x ^ 2 u ) + \\mu ( \\rho ) \\partial _ t \\partial _ x u . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} e ^ { - b _ 0 t } e ^ { - q t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) = \\bigl ( \\mathcal { S } _ N \\exp ( - x t ) \\bigr ) ( q \\ , | \\ , b ) \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{gather*} F _ \\mu \\in C ^ 2 ( \\R ^ { N \\times n } ) \\ F _ \\mu ( 0 ) = 0 \\ , ; \\\\ \\widetilde { \\nu } _ 1 | P | - \\widetilde { \\nu } _ 2 \\le F _ \\mu ( P ) \\le \\widetilde { \\nu } _ 3 | P | + \\widetilde { \\nu } _ 4 ; \\\\ \\widetilde { \\nu } _ 5 \\left ( 1 + | P | \\right ) ^ { - \\mu } | Q | ^ 2 \\leq D ^ 2 F _ \\mu ( P ) ( Q , Q ) \\leq \\widetilde { \\nu } _ 6 \\frac { 1 } { 1 + | P | } | Q | ^ 2 , \\end{gather*}"}
-{"id": "2900.png", "formula": "\\begin{align*} M \\cong ( T ^ n \\times P ^ { n } ) / \\approx , \\ ; \\ ; ( t _ 1 , p _ 1 ) \\approx ( t _ 2 , p _ 2 ) \\ ; \\ ; p _ 1 = p _ 2 , \\ ; t _ 1 t _ 2 ^ { - 1 } \\in \\Lambda ( F ( p _ 1 ) ) , \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} A = A _ 0 e _ 0 + A _ 1 e _ 1 + A _ 2 e _ 2 + A _ 3 e _ 3 , \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} S _ 0 = \\begin{cases} x _ { r - 1 } y _ r + x _ r y _ { r - 1 } + x _ r y _ r & \\\\ x _ { r - 1 } y _ { r - 1 } - \\beta x _ r y _ r & , \\end{cases} \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} P _ { \\epsilon , h } [ f ] ( x ) = \\frac { 1 } { h ^ \\epsilon } \\sum _ { 0 \\leq \\gamma \\leq \\epsilon } ( - 1 ) ^ { \\abs { \\gamma } + 1 } C ^ \\gamma _ \\epsilon f ( x + \\gamma h ) . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} 0 & = \\left ( - c _ { \\ell _ 0 } + 1 \\right ) \\frac { r _ { k c _ { \\ell _ 0 } } } { r _ { 0 c _ { \\ell _ 0 } } } + 1 + \\sum _ { i \\in I } \\frac { r _ { i c _ { \\ell _ 0 } } } { r _ { 0 c _ { \\ell _ 0 } } } - \\frac { r _ { k c _ { \\ell _ 0 } } } { r _ { 0 c _ { \\ell _ 0 } } } \\\\ & = - c _ { \\ell _ 0 } \\frac { r _ { k c _ { \\ell _ 0 } } } { r _ { 0 c _ { \\ell _ 0 } } } + 1 + \\frac { | I | } { c _ { \\ell _ 0 } - | I | } \\\\ & < \\frac { - c _ { \\ell _ 0 } } { c _ { \\ell _ 0 } - | I | } + 1 + \\frac { | I | } { c _ { \\ell _ 0 } - | I | } = 0 , \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} G _ 0 ( P _ 1 , P _ 2 ) = \\{ ( n _ 1 , n _ 2 ) \\in G ( P _ 1 ) \\times G ( P _ 2 ) : n _ 1 < \\beta ^ { - 1 } ( n _ 2 ) \\beta ( n _ 1 ) > n _ 2 \\} . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { j = 0 } ^ \\infty a _ j ( x - \\alpha ) ^ j \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} x _ { 1 } & = - 1 . 0 2 0 7 0 4 1 7 8 6 , x _ { 2 } = 2 . 0 5 3 2 7 1 8 9 8 3 , x _ { 3 } = - 1 . 0 3 2 5 6 7 7 1 9 7 \\\\ y _ { 1 } & = 9 . 1 2 6 5 6 9 3 1 4 0 , y _ { 2 } = 0 . 0 6 6 0 2 3 8 9 2 2 , y _ { 3 } = - 9 . 1 9 2 5 9 3 2 0 6 1 \\\\ h & = - 1 . 0 4 0 0 3 9 , \\omega = 0 . 3 1 2 0 1 3 \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} d X _ { t } + A ^ { \\lambda , \\delta , \\varepsilon } ( X _ { t } ) \\ , d t = B ^ { \\delta } ( X _ { t } ) \\ , d W _ { t } , X _ { 0 } = x _ { n } , \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} \\tilde F + \\delta = ( \\tilde f _ 1 + \\delta , \\dots , \\tilde f _ s + \\delta ) . \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} \\mathcal { O } \\left ( m b ^ m + \\min \\{ s , s ^ { \\ast } \\} \\ , b ^ m + \\sum _ { d = 1 } ^ { \\min \\{ s , s ^ { \\ast } \\} } ( m - w _ d ) b ^ { m - w _ d } \\right ) . \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} h _ { \\chi } ( \\rho ( \\overline { \\gamma } ) - 1 ) = h _ { \\chi \\otimes \\rho } ( 0 ) = 0 . \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} A _ { \\eta } \\leq ( 2 e ) ^ { - \\eta / 2 } \\prod _ { k = 0 } ^ { \\eta - 1 } \\left ( \\frac { \\pi e c } { k + \\frac { 1 } { 2 } } \\right ) ^ k \\sqrt { \\frac { e } { k + \\frac { 1 } { 2 } } } = ( 2 e ) ^ { - \\eta / 2 } B _ { \\eta } . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} { \\rm i n v c } ( \\sigma ) = { \\rm i n v } ( f _ 1 ( \\sigma ) ) \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\epsilon = s _ p ( h s ) \\epsilon ( h s , \\alpha _ m ) = \\begin{cases} ( c , - d ) _ p ( - d , p ^ m ) _ p & c d \\neq 0 , \\mathrm { o r d } _ p ( d ) , \\\\ ( - d , p ^ m ) _ p & c d \\neq 0 , \\mathrm { o r d } _ p ( d ) , \\\\ ( c , p ^ m ) _ p & d = 0 , \\\\ ( - d , p ^ m ) _ p & c = 0 . \\end{cases} \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} \\Sigma _ N & = \\sum _ { d \\mid v } \\mu ( d ) \\sum _ { ( a , w ) = 1 } e \\left ( \\frac { a d \\overline { m \\ell _ 1 \\ell _ 3 v } } { w } \\right ) \\frac { N } { d w } \\sum _ { | h | \\leq W ^ { 1 + \\epsilon } d / N } e \\left ( \\frac { a h } { w } \\right ) \\widehat { G } \\left ( \\frac { h N } { d w } \\right ) + O _ \\epsilon \\left ( Q ^ { - 1 0 0 } \\right ) , \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} X ' _ { \\Pi _ { \\imath } } X ' _ { \\Pi _ { \\jmath } } & = \\begin{cases} v ^ { - \\delta ( \\imath \\in { { } ^ { \\jmath } B } ) } X ' _ { \\Pi _ { \\jmath } } X ' _ { \\Pi _ { \\imath } } & \\ \\Xi ' _ { \\imath } > \\Xi ' _ { \\jmath } \\\\ X ' _ { \\Pi _ { \\jmath } } X ' _ { \\Pi _ { \\imath } } & \\ \\Xi ' _ { \\imath } = \\Xi ' _ { \\jmath } . \\end{cases} \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\sum _ { N < | k | \\leq M } C _ { k , l } ^ 2 e _ k ^ 2 ( x ) = \\sum _ { N < | k | \\leq M } C _ { k , l } ^ 2 = : c _ { M , N } . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} 2 \\sum _ { g , \\sigma } f _ 1 \\ast _ g ( x _ { i _ \\sigma ( 1 ) i _ \\sigma ( 2 ) } x _ { i _ \\sigma ( 3 ) i _ \\sigma ( 4 ) } ) = 1 8 f _ { 2 } \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\dot \\sigma _ i ( 0 ) = \\frac { 1 } { F ( \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) ) } \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} h ^ + ( G ) = \\max \\limits _ { \\vec x \\ne \\vec 0 } \\frac { 2 F _ 2 ^ L ( \\vec x ) } { G _ 2 ^ L ( \\vec x ) } = 1 - \\min \\limits _ { \\vec x \\ne \\vec 0 } \\frac { I ^ + ( \\vec x ) } { \\| \\vec x \\| } , \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\partial _ { t t } \\rho _ t = \\Delta _ { \\partial _ t \\rho _ t } \\Delta _ { \\rho _ t } ^ { \\dagger } \\partial _ t \\rho _ t + \\frac { 1 } { 2 } \\Delta _ { \\rho _ t } ( \\nabla \\Delta _ { \\rho _ t } ^ { \\dagger } \\partial _ t \\rho _ t ) ^ 2 . \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} & \\int _ { ( B _ r \\backslash B _ { \\rho / 2 } ) \\cap \\{ 0 < d _ U < \\kappa \\} } \\frac { V ( d _ U ( y ) ) } { d _ U ( y ) } \\frac { d y } { | x - y | ^ { n - 2 } \\varphi ( | x - y | ) } \\\\ & \\le \\sum _ { k = 1 } ^ M \\frac { 1 } { ( 2 ^ { - k } r ) ^ { n - 2 } \\varphi ( 2 ^ { - k } r ) } \\int _ { ( C _ { k - 1 } \\backslash C _ k ) \\cap \\{ 0 < d _ U < 6 \\cdot 2 ^ { - k } r \\} } \\frac { V ( d _ U ( y ) ) } { d _ U ( y ) } | \\nabla d _ U ( y ) | d y . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta _ f | \\nabla u | ^ 2 = | \\nabla ^ 2 u | ^ 2 + \\langle \\nabla \\Delta _ f u , \\nabla u \\rangle + R i c _ f ( \\nabla u , \\nabla u ) . \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\displaystyle \\Big ( \\Delta - \\frac { \\partial } { \\partial t } \\Big ) u ( x , t ) + q ( x , t ) u ^ \\alpha ( x , t ) = 0 , \\ \\ \\ \\alpha \\geq 1 \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\mathcal { H } _ I ^ s & = \\bigoplus _ { p \\in \\N ^ I } V _ I ^ p \\left ( \\bigcap _ { q \\in \\N ^ { I ^ c } } V _ { I ^ c } ^ q \\left ( \\mathcal { H } ^ s \\cap \\bigcap _ { i \\in I } V _ i ^ * \\right ) \\right ) \\\\ & = \\bigoplus _ { p \\in \\N ^ I } V _ I ^ p \\left ( \\bigcap _ { q \\in \\N ^ { I ^ c } } V _ { I ^ c } ^ q \\left ( \\bigoplus _ { n \\geq 0 } V _ { N + 1 } ^ n \\left ( V _ { N + 1 } ^ * \\right ) \\cap \\bigcap _ { i \\in I } V _ i ^ * \\right ) \\right ) . \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} \\begin{aligned} & x ' ( 1 ) = 0 \\\\ & w ' ( t ) = 0 , t \\geq 1 \\\\ & a ' _ i ( t ) = \\begin{cases} C _ i x ( t ) & i \\in \\mathcal K _ 2 \\\\ 0 & , \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} p = g ( f ( p ) ) \\qquad q = f ( g ( q ) ) . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} p _ G ^ 1 ( M ) & \\stackrel { \\eqref { e q u a t i o n : r e g 1 } } { = } | G | ^ { n ^ * - g - 1 } \\sum _ { h _ 1 , \\dots , h _ g \\in G } \\chi _ { } ( h _ 1 ^ 2 \\cdots h _ g ^ 2 ) \\\\ & \\stackrel { \\eqref { e q u a t i o n : r e g 2 } } { = } | G | ^ { n ^ * - g - 1 } \\sum _ { h _ 1 , \\dots , h _ g \\in G } \\sum _ { \\rho \\in \\widehat { G } } n _ { \\rho } \\chi _ { \\rho } ( h _ 1 ^ 2 \\cdots h _ g ^ 2 ) . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} f ^ { L } ( \\vec x ) & = \\sum _ { i = 0 } ^ { n - 1 } ( | x _ { \\sigma ( i + 1 ) } | - | x _ { \\sigma ( i ) } | ) f ( V _ { \\sigma ( i ) } ^ + , V _ { \\sigma ( i ) } ^ - ) \\\\ & = \\sum _ { i = 0 } ^ { n - 1 } \\int _ { | x _ { \\sigma ( i ) } | } ^ { | x _ { \\sigma ( i + 1 ) } | } f ( V _ { t } ^ + , V _ { t } ^ - ) d t \\\\ & = \\int _ 0 ^ { \\| x \\| _ \\infty } f ( V _ t ^ + , V _ t ^ - ) d t . \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} & \\sum _ { i \\neq j \\neq k \\neq l } ^ n \\eta _ 1 ( i ) \\eta _ 1 ( j ) \\eta _ 2 ( k ) \\eta _ 2 ( l ) \\cr & = \\sum _ { j \\neq k \\neq l } \\Big ( 0 - \\eta _ 1 ( j ) ^ 2 \\eta _ 2 ( k ) \\eta _ 2 ( l ) - \\eta _ 1 ( j ) \\eta _ 1 ( k ) \\eta _ 2 ( k ) \\eta _ 2 ( l ) - \\eta _ 1 ( j ) \\eta _ 2 ( k ) \\eta _ 1 ( l ) \\eta _ 2 ( l ) \\Big ) \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} X _ { 0 , 0 } = \\frac { L _ { 0 , 0 } } { L _ { 1 , 1 } } X _ { i , j } = \\frac { L _ { i , j } L _ { i + 1 , j + 1 } } { L _ { i + 1 , j } L _ { i , j + 1 } } . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { 0 < \\abs { \\gamma } \\leq d } \\xi _ \\gamma x ^ \\gamma , \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) = \\sum \\limits _ { p = 0 } ^ N ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ N \\exp \\bigl ( - ( b _ { k _ 1 } + \\cdots + b _ { k _ p } ) t \\bigr ) , \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} C = \\{ \\vec x \\in \\mathbb { R } ^ n \\big | F ( \\vec y ) \\leq F ( \\vec x ) , \\ , \\ , \\forall \\ , \\vec y \\in \\{ T _ i \\vec x : i \\in \\{ 1 , \\ldots , n \\} \\} \\} , \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} h ^ 0 ( S , H - E ) = h ^ 0 ( S , H ) - 1 = 2 \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\breve { J } W = \\breve { J } T W + \\breve { J } Q W . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{gather*} F _ { N + 1 } ( U _ { N + 1 } ' ) \\cdots F _ 2 ( U _ 2 ' ) F _ 1 ( U _ 1 ' ) E ( I ) = E ( I ' ) F _ { N + 1 } ( U _ { N + 1 } ) \\cdots F _ 2 ( U _ 2 ) F _ 1 ( U _ 1 ) . \\end{gather*}"}
-{"id": "5835.png", "formula": "\\begin{align*} k ( M / A ) + k ( M \\backslash A ^ c ) = k ( M ) + v ( M / A ) . \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} b _ s ^ { } = ( - 1 ) ^ { 2 n + 1 - m - s } q \\tau _ { 2 s - 2 n - 2 + m } ' \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} \\eta ( \\xi ) = 1 , \\ g ( X , \\xi ) = \\eta ( X ) , \\ \\eta ( \\phi X ) = 0 , \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} g \\Bigl ( \\frac { X + z ^ q } { X + z } \\Bigr ) = g \\Bigl ( \\frac { X + z + 1 } { X + z } \\Bigr ) = \\frac { A ( X ) } { B ( X ) } , \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} u _ { \\infty } ^ 2 = \\tilde { v } , \\ \\ \\ \\textrm { o n } \\ \\ ( t _ 1 , t _ 1 + \\kappa ) \\times \\R ^ d . \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} \\Big | \\int _ { \\mathbb R } \\varphi _ k ^ 3 \\xi _ { n _ k } \\nu _ { n _ k } ^ 2 d x - ( \\lambda - \\theta ) \\Big | \\leqq \\Big | \\int _ { \\mathbb R } \\chi _ k ^ 3 \\xi _ { n _ k } \\nu _ { n _ k } ^ 2 d x \\Big | + \\epsilon \\leqq \\| { \\bf { w } } _ { k } \\| ^ 3 _ { 1 \\times 1 } + \\epsilon = O ( \\epsilon ) . \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} \\lim \\limits _ { q \\rightarrow \\infty } \\eta _ { M , N } ( q \\ , | \\ , a , b ) = \\exp \\bigl ( - ( \\mathcal { S } _ N \\log \\Gamma _ M ) ( 0 \\ , | \\ , a , b ) \\bigr ) . \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\begin{cases} d u ^ { n } ( t ) + A u ^ { n } ( t ) \\ d t = B ( u ^ { n } ( t ) ) \\ d t , & t > 0 \\\\ u ( 0 ) = u _ { 0 } ^ { n } . \\end{cases} \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} | F \\chi _ { R _ i } ( l ) | \\le \\prod _ { j = 1 } ^ d \\frac { 1 } { \\max ( | \\pi M ^ { - \\frac { 1 } { d } } A \\ , \\rho ( l ) _ j | , 1 ) } = \\prod _ { j = 1 } ^ d \\frac { ( A \\pi ) ^ { - 1 } M ^ { \\frac { 1 } { d } } } { \\max ( | \\rho ( l ) _ j | , ( A \\pi ) ^ { - 1 } M ^ { \\frac { 1 } { d } } ) } . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} \\big ( I \\odot \\bar { I } \\big ) ^ { A } { } _ { B } = \\big ( I ^ { A } { } _ { R } \\bar { I } ^ { R } { } _ { B } + \\bar { I } ^ { A } { } _ { R } I ^ { R } { } _ { B } \\big ) = [ I , \\bar { I } ] ^ A { } _ { B } + 2 \\bar { I } ^ { A } { } _ { R } I ^ { R } { } _ { B } , \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} r = 1 . 6 = 1 + \\frac { 1 } { 1 + \\frac { 1 } { 1 + \\frac { 1 } { 2 } } } \\equiv [ 1 ; 1 , 1 , 2 ] \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} I _ { \\lambda } ( u , v ) = \\frac { 1 } { 2 } \\left ( \\| ( u , v ) \\| _ { E } ^ { 2 } - 2 \\int _ { \\mathbb { R } ^ { N } } \\lambda ( x ) u v \\ , \\mathrm { d } x \\right ) - \\int _ { \\mathbb { R } ^ { N } } \\left ( F _ { 1 } ( u ) + F _ { 2 } ( v ) \\right ) \\ , \\mathrm { d } x . \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} B _ 1 ( m ) = - p ^ { 1 - 3 m / 2 } ( - 1 ) ^ m \\chi _ { \\psi } ( p ^ m ) \\mathcal J _ 1 ( m ) = p ^ { 3 m / 2 } ( - 1 ) ^ { m + 1 } \\chi _ { \\psi } ( p ^ m ) ( 1 - p ^ { - 1 } ) p ^ { - m } . \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} Q _ i ^ k & = Q _ i ^ { k + 1 } 1 \\leq i \\leq d , \\\\ Q _ { i , i + 1 } ^ k & = Q _ { i , i + 1 } ^ { k + 1 } 1 \\leq i < d , \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( ( \\mathbb { P } _ { n } ^ { ( N ) } ( f ) - P ( f ) ) ^ { 2 } \\right ) = \\frac { \\sigma _ { f } ^ { 2 } } { n } ( \\Delta ( f ) + e ( f ) v _ { n } ) , \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} \\phi _ * ( \\overline { \\mathcal { G } } ) = \\overline { \\phi _ * ( \\mathcal { G } ) } = \\overline { \\mathcal { W } } \\subseteq \\overline { \\mathcal { F } } . \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{gather*} L ( G _ k ( q ) , k ) = \\ ( \\frac { - 1 } { \\sqrt { - 3 } } \\ ) ^ k \\mathcal G _ { k , ( 1 ; 3 ) } ^ \\ast \\big ( { - } 1 / \\sqrt { - 3 } \\big ) , \\end{gather*}"}
-{"id": "1359.png", "formula": "\\begin{align*} \\Vert ( x - y + z ) \\chi _ N \\Vert = \\Vert x - y + z \\Vert - \\Vert ( x - y + z ) \\chi _ P \\Vert \\le 1 + t - ( 1 - s ) = s + t . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd r } \\left ( - \\frac { e ^ { - r } \\chi _ { i + 1 } ( r ) } { r } \\right ) & = \\frac { \\dd } { \\dd r } \\left ( - \\psi _ { i + 1 } ( r ) r ^ { 2 i + 1 } \\right ) \\\\ & = r \\psi _ { i + 2 } ( r ) r ^ { 2 i + 1 } - \\psi _ { i + 1 } ( r ) ( 2 i + 1 ) r ^ { 2 i } \\\\ & = r ^ { 2 i } \\left ( r ^ 2 \\psi _ { i + 2 } ( r ) - ( 2 i + 1 ) \\psi _ { i + 1 } ( r ) \\right ) \\\\ & = r ^ { 2 i } \\psi _ { i } ( r ) \\\\ & = e ^ { - r } \\chi _ i ( r ) . \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} & K ^ { ( 1 ) } : \\mathfrak { h } ^ { ( 1 ) } _ + \\to \\mathfrak { h } ^ { ( 1 ) } , & & K ^ { ( 1 ) } ( x , y ) : = V ^ { ( 1 ) } ( x - y ) u _ 0 ( x ) u _ 0 ( y ) \\\\ & K ^ { ( 2 ) } : \\mathfrak { h } ^ { ( 2 ) } _ + \\to \\mathfrak { h } ^ { ( 2 ) } , \\qquad & & K ^ { ( 2 ) } ( x , y ) : = V ^ { ( 2 ) } ( x - y ) v _ 0 ( x ) v _ 0 ( y ) \\\\ & K ^ { ( 1 2 ) } : \\mathfrak { h } ^ { ( 2 ) } _ + \\to \\mathfrak { h } ^ { ( 1 ) } , & & K ^ { ( 1 2 ) } ( x , y ) : = V ^ { ( 1 2 ) } ( x - y ) u _ 0 ( x ) v _ 0 ( y ) . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} [ \\Gamma _ i , \\Gamma _ j ] = 1 , ( 1 \\leq i , j \\leq n ) , ( \\Gamma _ 0 \\Gamma _ i ) ^ 2 = ( \\Gamma _ i \\Gamma _ 0 ) ^ 2 , ( 1 \\leq i \\leq n ) , \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { - \\infty } ^ { \\infty } x u ^ 2 ( t , x ) d x = - 6 \\left ( I _ 3 ( u _ 0 ) + \\int _ { - \\infty } ^ { \\infty } \\Big ( G ( u ) - \\frac { F ( u ) } { 3 } \\ , \\Big ) ( t , x ) d x \\right ) , \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} 2 n + 4 \\le e ( G ) & = e ( G [ W \\cup S ] ) + e _ G ( U , S ) \\le ( 3 ( w + s ) - 6 ) + ( 2 ( n - w ) - 4 ) \\\\ & = 2 n + w + 3 s - 1 0 = 2 n + ( 1 1 - 2 s ) + 3 s - 1 0 = 2 n + s + 1 , \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} T = \\sum _ { \\lambda \\in \\Lambda } f _ \\lambda \\odot g _ \\lambda , \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} g _ \\lambda = \\Big ( r _ { u _ 1 } , r _ { u _ 2 } , \\ldots , r _ { u _ { k - 1 } } , \\lambda , 1 , \\frac { \\lambda } { r _ { u _ { k + 1 } } } , \\ldots , \\frac { \\lambda } { r _ { u _ n } } ) \\Big ) \\in \\mathrm { P G L ( \\mathbf { 1 } ) } \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot p \\\\ \\dot v \\end{bmatrix} = \\begin{bmatrix} 0 & 1 \\\\ 0 & - b / m \\end{bmatrix} \\begin{bmatrix} \\dot x \\\\ \\dot v \\end{bmatrix} + \\begin{bmatrix} 0 \\\\ 1 / m \\end{bmatrix} u + \\begin{bmatrix} I _ 2 & O _ { 2 , 3 } \\end{bmatrix} w , \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{gather*} t : = \\phi ( \\tau ) / \\phi _ 1 ( \\tau ) - 3 = \\eta ( \\tau / 3 ) ^ 3 / \\eta ( 3 \\tau ) ^ 3 . \\end{gather*}"}
-{"id": "1352.png", "formula": "\\begin{align*} \\alpha _ 2 + \\beta _ 5 + \\beta _ 6 = v _ { 2 } ^ * ( x _ 0 - u _ 0 ) \\le \\Vert x _ 0 - u _ 0 \\Vert < \\varepsilon . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} x P ( x , n ) = P ( x , n + 1 ) + B ( n ) P ( x , n ) + C ( n ) P ( x , n - 1 ) , \\ \\ \\forall n \\geq 1 . \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} \\ln \\frac { \\beta - h ( x , t ) } { \\beta - h ( y , t ) } & = \\int _ 0 ^ 1 \\frac { d \\ln ( \\beta - h ( \\gamma ( s ) , t ) ) } { d s } d s \\\\ & \\leq \\int _ 0 ^ 1 | \\dot { \\gamma } | \\frac { | \\nabla u | } { u ( \\beta - \\ln u / D ) } d s \\\\ & \\leq C ( \\delta ) \\Big ( \\frac { 1 } { \\sqrt { t - ( t _ 0 - T ) } } + \\sqrt { K } + \\lambda \\Big ) r . \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} \\int \\limits _ G F ( g ) \\ , d ' g & = \\int \\limits _ T \\int \\limits _ { { \\rm S y m } _ n ( \\R ) } f ( X + i h \\ , ^ t h ) \\det ( h ) ^ { - ( n + 1 ) } \\ , d X \\ , d h \\\\ & = \\int \\limits _ T \\int \\limits _ { { \\rm S y m } _ n ( \\R ) } f ( X + i h \\ , ^ t h ) \\det ( h \\ , ^ t h ) ^ { - \\frac { n + 1 } 2 } \\ , d X \\ , d h \\\\ & = \\int \\limits _ { P _ + } \\int \\limits _ { { \\rm S y m } _ n ( \\R ) } f ( X + i Y ) \\det ( Y ) ^ { - ( n + 1 ) } \\ , d X \\ , d Y . \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} n _ 1 M ^ { M d _ 1 } 2 ^ { M d _ 1 - 1 } > n _ 1 \\tfrac { \\binom { M d _ j } { M d _ 1 } } { \\binom { d _ j } { M d _ 1 } } , \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} \\delta = \\min _ { x \\in { \\cal E } _ 0 } \\mu ( x ) ~ > ~ 0 . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} D _ s X ( t ) = J _ s ( t ) \\Big [ X ( s ) + \\int _ 0 ^ s g ( r ) d W ( r ) + g ( s ) \\Big ( \\int _ s ^ t J _ s ( r ) ^ { - 1 } d W ( r ) - \\int _ s ^ t J _ s ( r ) ^ { - 1 } d r \\Big ) \\Big ] . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{gather*} f _ { 3 6 } ( \\tau ) = \\eta ( 6 \\tau ) ^ 4 = \\sum _ { m , n \\in \\Z } \\big ( 3 m + 1 - n \\sqrt { - 3 } \\big ) q ^ { ( 3 m + 1 ) ^ 2 + 3 n ^ 2 } = \\sum _ { m , n \\in \\Z } ( 3 m + 1 ) q ^ { ( 3 m + 1 ) ^ 2 + 3 n ^ 2 } . \\end{gather*}"}
-{"id": "2369.png", "formula": "\\begin{align*} \\rho _ { \\{ a \\} } ( \\Delta ) = \\frac { 1 } { 2 } \\log \\frac { 1 - \\Delta _ { \\min , \\{ a \\} } } { \\Delta - \\Delta _ { \\min , \\{ a \\} } } \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\sum _ { \\mu = 1 } ^ N \\| u _ \\mu ( t ) \\| ^ { 2 } _ { L ^ { 2 } _ x ( \\widetilde Q _ { x _ n } ) } \\geq C ( d ) \\delta ^ { 2 } _ 0 > 0 , \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\lim _ { | x | \\rightarrow \\infty } | x | \\left ( \\frac { \\partial \\hat { u } ( x , k ) } { \\partial | x | } - i k \\hat { u } ( x , k ) \\right ) = 0 . \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\left ( - 2 L _ { - 2 } + \\frac { \\kappa } { 2 } L _ { - 1 } ^ { 2 } + \\frac { \\tau } { 2 } \\sum _ { r = 1 } ^ { 3 } X _ { r } ( - 1 ) ^ { 2 } \\right ) e ^ { \\Lambda } = 0 \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} & \\min y _ 1 ^ 4 - y _ 2 ^ 4 - x y _ 1 \\\\ \\hbox { s u b j e c t t o } & \\\\ & y _ 1 ^ 3 - 2 y _ 2 ^ 3 \\leq 0 \\\\ & y _ 1 ^ 3 + 2 y _ 2 ^ 3 \\leq 0 \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} A \\cap B = \\emptyset , g ( A \\cup B ) \\subset A , h ( A \\cup B ) \\subset B . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} J ( u , v ) = \\lim _ n \\frac { 1 } { n } \\log G ( u _ n , v _ n ) . \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} ( \\nabla _ { U } g ) ( V , Z ) = B ( U , Z ) \\theta ( V ) + B ( U , V ) \\theta ( Z ) , \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\mathcal B ^ { ( k + 1 ) } : = \\left \\{ \\omega \\in \\mathcal \\mathcal B ^ { ( k ) } \\ , \\Big | \\ , \\begin{array} { c } \\forall z , w \\in \\mathbb B _ r \\textrm { w i t h } \\Vert z - w \\Vert < \\delta _ { k + 1 } : \\\\ \\sup _ n \\Vert f ^ n _ \\omega ( z ) - f ^ n _ \\omega ( w ) \\Vert < \\varepsilon _ { k + 1 } \\end{array} \\right \\} \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} \\eta = \\eta _ { \\gamma } = \\frac { \\pi } { 2 } \\frac { c } { \\gamma } + \\frac { 1 } { 2 } , \\gamma = 2 . 7 , \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} { \\widehat \\theta } _ { 1 , n } = { \\widehat \\theta } _ { 1 , n } ( X _ A ^ n ) = \\frac { 1 } { n } \\sum \\limits _ { t = 1 } ^ { n } X _ { A t } X _ { A t } ^ T . \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} \\tilde x _ k = x _ k , \\tilde y _ { i , j } = y _ { i , j } - a _ { i , j } ( x ) \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} K _ u ^ 2 ( h ) = H _ u ^ 2 ( h ) - ( h \\vert u ) u , \\forall h \\in L ^ 2 _ + . \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} \\abs { S _ { N , N } ^ { ( r ) } } & \\leq \\abs { S _ { N , N } ^ { ( r + 1 ) } } ( \\log N ) + \\sum _ { m = 1 } ^ { N - 1 } \\left | S _ { N , m } ^ { ( r + 1 ) } \\right | m ^ { - 1 } \\\\ & \\leq C ' N ^ k ( \\log N ) ^ { - \\alpha + k '' - r } , \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} Z _ { t } = Z _ { 0 } + \\int _ { 0 } ^ { t } G _ { s } \\ , d s + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } P ^ { \\ast } L ^ { b } P Z _ { s } \\ , d s + \\int _ { 0 } ^ { t } B ( P Z _ { s } ) \\ , d W _ { s } \\quad \\forall t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} f _ \\phi ( U ( m , n ) ) = \\tilde { \\phi } ( m + n ) , \\forall \\ , m , n \\in \\N , \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} P M I & = \\{ f \\in \\mathcal { M } | \\ f ( x ) ^ { r } \\leq f ( x ^ { r } ) r \\geq 1 x \\in ( 0 , \\infty ) \\} , \\\\ P M D & = \\{ f \\in \\mathcal { M } | \\ f ( x ) ^ { r } \\geq f ( x ^ { r } ) r \\geq 1 x \\in ( 0 , \\infty ) \\} . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} \\ln R = & i \\ln a _ n + 2 \\sum _ { j = 1 } ^ i \\ln \\Big ( \\frac { k ^ * - j + 1 } { n - k ^ * + j } \\Big ) \\\\ \\sim & - 2 A _ n \\frac { i ( i + 1 ) } { 2 n } + 2 ( 1 + \\sqrt { a _ n } ) \\frac { i } { n } + \\Theta ( \\frac { A _ n ^ 2 i ^ 3 } { n ^ 2 } ) \\\\ = & \\frac { - A _ n i ^ 2 } { n } + \\frac { - A _ n i } { n } + 2 ( 1 + \\sqrt { a _ n } ) \\frac { i } { n } + \\Theta ( \\frac { A _ n ^ 2 i ^ 3 } { n ^ 2 } ) . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} & M _ 2 \\doteq \\bigcap _ { n = 1 } ^ \\infty { \\rm R a n } ( \\pi ( S _ 2 ) ^ n ) \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} \\eta ^ n = P ^ n \\circ X _ 0 ^ { - 1 } \\to P \\circ X _ 0 ^ { - 1 } \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} \\sum _ { \\nu = 1 } ^ N \\beta _ { \\mu \\nu } X _ { \\mu \\nu } + \\sum _ { \\mu , \\nu = 1 } ^ N \\lambda _ { \\mu \\nu } X _ { \\mu \\nu } = \\sum _ { \\nu = 1 } ^ N \\beta _ { \\mu \\nu } Y _ { \\mu \\nu } + \\sum _ { \\mu , \\nu = 1 } ^ N \\lambda _ { \\mu \\nu } Y _ { \\mu \\nu } . \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} & a _ { m _ 0 } \\ge ( z ^ * _ { m _ 0 } ) ^ { p - 1 } = 1 > ( z ^ * _ { m _ 0 + 1 } ) ^ { p - 1 } = a _ { m _ 0 + 1 } \\\\ \\Leftrightarrow & \\frac { m _ 0 | v _ { m _ 0 } | } { \\sum _ { j = 1 } ^ { m _ 0 } \\abs { v _ j } - r } \\ge 1 > \\frac { m _ 0 | v _ { m _ 0 + 1 } | } { \\sum _ { j = 1 } ^ { m _ 0 } \\abs { v _ j } - r } \\\\ \\Leftrightarrow & \\sum _ { j = 1 } ^ { m _ 0 } ( \\abs { v _ j } - \\abs { v _ { m _ 0 + 1 } } ) > r \\ge \\sum _ { j = 1 } ^ { m _ 0 - 1 } ( \\abs { v _ j } - \\abs { v _ { m _ 0 } } ) \\\\ \\Leftrightarrow & A ( m _ 0 ) > r \\ge A ( m _ 0 - 1 ) . \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\lbrace { \\tilde z _ 1 } , { \\tilde z _ 3 } \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace { \\tilde z _ 2 } , { \\tilde z _ 4 } \\rbrace = 1 . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { x ^ a - f ^ a ( x ) } { x ^ a f ^ a ( x ) } = \\dfrac { 1 } { k ^ a } \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\theta ' \\ , : \\ , { \\mathcal F } \\ , \\longrightarrow \\ , ( E _ H ) \\ , = \\ , ( \\gamma ^ * F _ H ) \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} 0 & = \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\left ( f ( B Y _ j Y _ i ^ { - 1 } A ^ { - 1 } ) - f ( A Y _ i Y _ j ^ { - 1 } B ^ { - 1 } ) - f ( B A ^ { - 1 } ) - f ( A B ^ { - 1 } ) \\right ) . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\mu ( q ) = \\begin{cases} ( - 1 ) ^ m & j _ 1 = \\cdots = j _ m = 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} S _ p = \\left \\{ 0 \\leq f \\leq e _ p \\ ; : \\ ; \\min \\{ f , e _ p - f \\} \\leq e _ p - s _ p \\right \\} . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} \\phi ^ 2 ( X ) = - X + \\eta ( X ) \\xi , \\ \\phi \\xi = 0 , \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} f _ n = \\frac { 1 } { \\sqrt { 2 \\pi } } n ^ { - 3 / 2 } + o ( n ^ { - 3 / 2 } ) \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} { \\bf u } = T _ G { \\bf Q } T _ G \\left ( { \\bf f } - { \\bf D } p \\right ) . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} y ^ { k + 1 } & = P _ C ( x ^ k - \\lambda _ k g ( y ^ k ) ) . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\lambda = \\begin{pmatrix} \\lambda _ { 1 1 } & \\lambda _ { 1 2 } & \\lambda _ { 1 3 } \\\\ & \\lambda _ { 2 2 } & \\lambda _ { 2 3 } \\\\ & & \\lambda _ { 1 1 } \\\\ \\end{pmatrix} \\in M _ { m , m } ( q ) \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} Z _ n ( t ) = \\frac { 1 } { n } S ( [ n t ] ) , \\ ; t \\in \\R _ + \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { t = 1 } ^ { n - 1 } \\left [ ( - 1 ) ^ t \\cdot t \\sum \\limits _ { 1 \\leq j _ 1 \\leq \\cdots \\leq j _ { t + 1 } \\leq n } z ^ { ( d _ { j _ 1 } + \\cdots + d _ { j _ { t + 1 } } - 1 ) } \\right ] + z ^ { - 1 } } { ( 1 - z ) ^ n } . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} P _ 1 \\sqcup P _ 2 = U \\tilde { P } _ 1 Q _ 1 \\cdots Q _ { j - 2 } \\tilde { Q } _ { j - 1 } R _ j D . \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\gamma ^ a \\gamma ^ { ( a _ 1 } \\ldots \\gamma ^ { a _ { j } ) } - \\gamma ^ { ( a _ 1 } \\ldots \\gamma ^ { a _ { j } ) } \\gamma ^ a = 2 j \\ , \\omega ^ { a ( a _ 1 } \\gamma ^ { a _ 2 } \\ldots \\gamma ^ { a _ { j } ) } \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 _ n & D & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , D \\in \\R ^ n . \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} h ( R ) = 1 , \\ h ^ { ( 1 ) } ( R ) = 0 , \\ h ^ { ( 2 ) } ( R ) = 0 , \\ \\dots , \\ h ^ { ( p ) } ( R ) = 0 . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} \\chi ' : = m \\sum _ { \\sigma \\in V } { } ^ { \\sigma } \\chi \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} \\binom { k b + i } { k a } \\binom { k a + i } { k a } ^ { - 1 } \\geq \\left ( \\frac { b + a } { 2 a } \\right ) ^ { k a } \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} { \\mathfrak F } ( N ) = \\{ Y \\in { \\mathfrak F } ( M ) : \\ Y \\subset N \\} \\subset { \\mathfrak F } ( M ) . \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} & B _ 1 ( \\lambda ) : = P _ \\ell \\lambda ^ \\ell + P _ { \\ell - 1 } \\lambda ^ { \\ell - 1 } + \\cdots + P _ 1 \\lambda + P _ 0 , \\\\ & B _ j ( \\lambda ) : = P _ { \\ell j } \\lambda ^ \\ell + P _ { \\ell j - 1 } \\lambda ^ { \\ell - 1 } + \\cdots + P _ { \\ell ( j - 1 ) + 1 } \\lambda , \\mbox { f o r } j = 2 , \\hdots , k . \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\gamma _ { j _ { k } } = \\frac { w _ { j _ { k ' } , i } } { w _ { j _ k , i } } \\Bigl ( \\frac { y _ { \\sigma ( \\iota _ i ) } } { w _ { j _ { k ' } , i } ( 1 + \\lambda _ { \\ell _ { \\iota _ i } } / \\mu _ { i } ) } - \\frac { y _ { \\sigma ( \\iota _ { i ' } ) } } { w _ { j _ { k ' } , i ' } ( 1 + \\lambda _ { \\ell _ { \\iota _ { i ' } } } / \\mu _ { i ' } ) } \\Bigr ) , \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 \\cr - \\hat { x } _ { n } ^ { - 1 } & 1 \\end{pmatrix} L _ n ( t \\mid \\hat { y } _ n , \\hat { Y } _ n ) \\begin{pmatrix} 1 & 0 \\cr \\hat { x } _ { n - 1 } ^ { - 1 } & 1 \\end{pmatrix} = \\begin{pmatrix} A _ n & B _ n \\cr 0 & D _ n \\end{pmatrix} \\ ; . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} & \\phi \\left ( i _ 1 , j _ 1 , i _ 2 , j _ 2 \\right ) \\triangleq \\left ( 1 , \\cdots , i _ 1 - 1 , i _ 2 , \\cdots , j _ 2 , \\right . \\\\ & \\left . j _ 1 + 1 , \\cdots , i _ 2 - 1 , i _ 1 , \\cdots , j _ 1 , j _ 2 + 1 , \\cdots , N \\right ) . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ \\mu = \\int _ { \\mathcal G } \\langle M u , M v \\rangle _ E \\ , d \\nu _ { \\mathcal G } ( M ) , \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} \\bigl ( \\rho \\circ \\hat { \\tau } ( a _ 1 , \\cdots , a _ { \\ell - n } ) \\bigr ) ' = ( a _ { \\ell - n } , a _ { \\ell - n - 1 } , \\cdots , a _ 2 , a _ 1 ) . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\mathcal { C } _ M = \\left ( c ^ { \\ast } M ^ { a _ 3 } \\right ) ^ { m - 1 } \\left ( a _ 3 = \\frac { 2 } { n } a _ 1 \\right ) . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} C = \\frac { 2 \\pi } { e ^ { \\kappa } \\Gamma ( 1 - \\beta ^ 2 ) } , \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{gather*} \\frac { { \\rm d } X } { Y ^ 2 } ( \\tau ) = 2 \\pi { \\rm i } \\cdot 3 f _ { 2 7 } ( \\tau ) { { \\rm d } \\tau } . \\end{gather*}"}
-{"id": "4624.png", "formula": "\\begin{align*} R i c ( \\omega _ i ( t ) ) = - \\omega _ i ( t ) + t \\omega _ i . \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} h ( K ) = \\lim _ { r \\to \\infty } r ^ { - 1 } \\log \\theta _ r ( K ) . \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} \\gamma _ { \\pm } = \\arg { ( 1 \\pm \\sqrt { - x ^ 2 / c ^ 2 } ) } . \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\Omega _ { n , j } ' = \\{ \\omega \\in \\Omega \\ , | \\ , \\forall i = 1 , \\dots N , \\ , \\exists k _ i \\ , : \\ , T ^ { k _ i } \\omega \\in U _ { i } \\} \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} \\mathbf { b } = \\begin{bmatrix} \\mathbf { c } \\\\ \\mathbf { u } \\end{bmatrix} . \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\mathfrak { d } _ j ^ m \\dot { \\phi } ( n ) = \\sum _ { k = 0 } ^ m \\tbinom { N } { k } ( - 1 ) ^ k \\dot { \\phi } ( n + j k ) . \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\frac { ( - z q ; q ) _ i } { ( q ; q ) _ i } - \\frac { ( - z q ; q ) _ { i - 1 } } { ( q ; q ) _ { i - 1 } } = \\frac { q ^ i ( - z ; q ) _ i } { ( q ; q ) _ i } , \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} \\dot { u } = X _ { \\mathcal { H } } ( u ) , \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha ^ { - } ( \\phi ) = \\lim _ { n \\to \\infty } \\frac { W ^ { - } ( n , \\phi ) } { n } \\\\ \\alpha ^ { + } ( \\phi ) = \\lim _ { n \\to \\infty } \\frac { W ^ { + } ( n , \\phi ) } { n } . \\end{aligned} \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{gather*} G _ k ( q ) = \\sum _ { \\mathfrak J \\in S _ 2 } \\tilde \\chi ^ k ( \\mathfrak J ) q ^ { N ( \\mathfrak J ) } . \\end{gather*}"}
-{"id": "3815.png", "formula": "\\begin{align*} \\sigma \\left ( \\frac { L ^ N ( r , \\pi \\boxtimes \\chi , \\varrho _ { 5 } ) } { ( 2 \\pi i ) ^ { 2 k + 3 r } G ( \\chi ) ^ 3 \\langle F , F \\rangle } \\right ) = \\frac { L ^ N ( r , { } ^ \\sigma \\pi \\boxtimes { } ^ \\sigma \\ ! \\chi , \\varrho _ { 5 } ) } { ( 2 \\pi i ) ^ { 2 k + 3 r } G ( { } ^ \\sigma \\ ! \\chi ) ^ 3 \\langle { } ^ \\sigma \\ ! F , { } ^ \\sigma \\ ! F \\rangle } . \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} ( x , y ) = \\langle x , A y \\rangle . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} \\Psi _ { \\Upsilon , M } ( - 2 \\delta - w _ 2 ) \\Psi _ { \\Upsilon , M } ( w _ 2 ) = \\frac { M ^ { - 2 \\delta } + M ^ { - 2 \\delta ( 1 - \\Upsilon ) } - M ^ { - \\Upsilon w _ 2 - 2 \\delta } - M ^ { \\Upsilon w _ 2 - 2 \\delta ( 1 - \\Upsilon ) } } { w _ 2 ^ 2 ( 2 \\delta + w _ 2 ) ^ 2 ( \\Upsilon \\log M ) ^ 2 } . \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} X _ { 1 } = \\frac { 1 } { \\sqrt { 2 } } H , \\ \\ X _ { 2 } = \\frac { 1 } { \\sqrt { 2 } } ( E + F ) , \\ \\ X _ { 3 } = \\frac { i } { \\sqrt { 2 } } ( E - F ) , \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( t ) = \\ , & - \\frac { \\beta } { m - 1 } ( \\tau + t ) ^ { \\alpha - 1 } + \\frac { C ^ { m - 1 } m } { a ^ 2 ( m - 1 ) ^ 2 } ( \\tau + t ) ^ { \\alpha m - 2 \\beta } \\\\ \\geq \\ , & \\frac { C ^ { m - 1 } m } { 2 a ^ 2 ( m - 1 ) ^ 2 } ( \\tau + t ) ^ { \\alpha m - 2 \\beta } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\tilde { \\phi } ( d ( x _ 1 , y _ 1 ) , . . . , d ( x _ N , y _ N ) ) , \\qquad \\forall \\ x , y \\in X , \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\mathbf { u } | _ { y = 0 } = \\mathbf { 0 } , \\theta | _ { y = 0 } = \\theta ^ \\ast ( t , x ) , ( \\partial _ y h _ 1 , h _ 2 ) | _ { y = 0 } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} A _ r & = \\begin{pmatrix} 3 . 0 2 9 1 & 0 . 0 7 9 6 & 1 . 9 0 7 3 \\\\ 0 . 0 7 9 6 & 1 . 9 8 2 1 & - 0 . 5 4 6 7 \\\\ 1 . 9 0 7 3 & - 0 . 5 4 6 7 & 3 . 2 1 9 2 \\end{pmatrix} , \\\\ B _ r & = \\begin{pmatrix} 0 . 9 2 6 3 & 1 . 1 4 1 4 \\\\ - 1 . 0 4 5 8 & 0 . 3 1 1 9 \\\\ - 0 . 3 5 3 3 & 0 . 9 4 0 6 \\\\ \\end{pmatrix} , \\\\ C _ r & = \\begin{pmatrix} 1 . 0 1 8 5 & 1 . 0 4 1 6 & - 0 . 5 3 3 4 \\\\ 0 . 7 6 9 6 & 1 . 3 9 4 7 & 0 . 7 3 1 5 \\end{pmatrix} . \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} \\textbf { s u p p } \\ , U ( t ) = \\textbf { s u p p } \\ , u ( t ) \\forall t > t _ 0 . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} \\delta _ 1 = l _ { 1 , b } , \\delta _ 2 = l _ { 1 , b - 1 } + l _ { 2 , b } , \\dots , \\delta _ n = a l _ { 0 , 0 } . \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} \\Pr ( R ) = \\frac { | Z | } { 1 2 | Z | } + \\frac { ( 2 \\times 2 ) | Z | ^ 2 + ( 9 \\times 6 ) | Z | ^ 2 } { { 1 2 } ^ 2 | Z | ^ 2 } = \\frac { 3 5 } { 7 2 } . \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} n ( T ) & = n ^ + ( T ) - n ^ - ( T ) = 0 - 0 , \\\\ n ( C _ 2 ) & = n ^ + ( C _ 2 ) - n ^ - ( C _ 2 ) = 1 - 0 , \\\\ n ( L ) & = n ^ + ( L ) - n ^ - ( L ) = 1 2 - 1 \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} N | A \\sqcap M | B : = V | C \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { c , d } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\overline { E } _ { a , b } \\overline { E } _ { c , d } } E _ { c , d } E _ { a , b } + ( q _ b - q _ b ^ { - 1 } ) E _ { a , d } E _ { c , b } , & M = N = 1 ; \\\\ \\sum _ { t = 0 } ^ { \\min ( M , N ) } q _ { b } ^ { \\frac { t ( t - 1 ) } { 2 } } ( q _ b - q _ b ^ { - 1 } ) ^ t [ t ] _ q ! E _ { c , b } ^ { ( t ) } E _ { c , d } ^ { ( N - t ) } E _ { a , b } ^ { ( M - t ) } E _ { a , d } ^ { ( t ) } , & . \\end{cases} \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} A _ \\phi = \\begin{bmatrix} - \\kappa _ { d - 1 } P _ { d - 1 } + \\kappa _ d \\beta _ { d - 1 } P _ d & \\alpha _ { d - 2 } I _ n & & & \\\\ - \\kappa _ { d - 1 } P _ { d - 2 } + \\kappa _ { d } \\gamma _ { d - 1 } P _ d & \\beta _ { d - 2 } I _ n & \\alpha _ { d - 3 } I _ n \\\\ - \\kappa _ { d - 1 } P _ { d - 3 } & \\gamma _ { d - 2 } I _ n & \\beta _ { d - 3 } I _ n & \\ddots \\\\ \\vdots & & \\ddots & \\ddots & \\alpha _ 0 I _ n \\\\ - \\kappa _ { d - 1 } P _ 0 & & & \\gamma _ 1 I _ n & \\beta _ 0 I _ n \\end{bmatrix} . \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { k } x _ j ^ { \\downarrow } \\leq \\prod _ { j = 1 } ^ { k } y _ j ^ { \\downarrow } , 1 \\leq k \\leq m \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} P _ i ^ { \\epsilon _ i } & = V _ i t _ { b _ i } ^ { - \\epsilon _ i } \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} c ( n ^ 2 | D | ) = \\frac { c ( n _ N ^ 2 | D | ) } { c ( | D | ) } \\frac { c ( n _ D ^ 2 | D | ) } { c ( | D | ) } c ( n _ 0 ^ 2 | D | ) = a _ { \\chi } ( n _ N ) a _ { \\chi } ( n _ D ) c ( n _ 0 ^ 2 | D | ) . \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} v \\partial _ x f + G \\bigg ( \\frac { \\alpha } { 1 + \\lambda \\psi * \\varrho } \\bigg ) \\partial _ v f = \\partial _ v ( v f + \\sigma \\partial _ v f ) \\ , . \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} a ' _ k = \\sum _ { i \\in I } \\ , Q _ k ( i ) \\ , a _ i . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\langle \\xi _ 1 , \\xi _ 2 \\rangle _ { S } : = { \\rm t r } ( \\xi _ 1 \\xi _ 2 ) \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} C _ 1 ( \\lambda ) = \\left [ \\begin{array} { c c c c } \\lambda P _ d + P _ { d - 1 } & P _ { d - 2 } & \\cdots & P _ 0 \\\\ \\hline - I _ n & \\lambda I _ n \\\\ & \\ddots & \\ddots \\\\ & & - I _ n & \\lambda I _ n \\\\ \\end{array} \\right ] = \\left [ \\begin{array} { c } M _ 1 ( \\lambda ) \\\\ \\hline L _ { d - 1 } ( \\lambda ) \\otimes I _ n \\end{array} \\right ] \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} [ g _ 1 , \\epsilon _ 1 ] [ g _ 2 , \\epsilon _ 2 ] = [ g _ 1 g _ 2 , \\epsilon ( g _ 1 , g _ 2 ) \\epsilon _ 1 \\epsilon _ 2 ] . \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} \\forall n \\in \\N _ { \\geq 2 } : 2 X _ { n + 1 } - X _ { n } & = 2 \\sum _ { k = 1 } ^ { n } \\frac { 1 } { 2 ^ { n + 1 - k } } Y _ { k } - \\sum _ { k = 1 } ^ { n - 1 } \\frac { 1 } { 2 ^ { n - k } } Y _ { k } \\\\ & = \\sum _ { k = 1 } ^ { n } \\frac { 1 } { 2 ^ { n - k } } Y _ { k } - \\sum _ { k = 1 } ^ { n - 1 } \\frac { 1 } { 2 ^ { n - k } } Y _ { k } \\\\ & = Y _ { n } . \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} \\mathcal { L } _ n : = \\mathcal { L } _ { \\Gamma _ n } \\hookrightarrow \\mathcal { O } _ { L _ n } \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\mathcal L = v _ 1 \\partial _ { v _ 1 } + v _ 2 \\partial _ { v _ 2 } - v _ 3 \\partial _ { v _ 3 } - v _ 4 \\partial _ { v _ 4 } . \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} \\sum _ i \\ell _ i \\lambda _ i + \\sum _ j m _ j \\mu _ j = 0 \\\\ \\sum _ i \\ell _ i + \\sum _ j m _ j = 0 . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} R _ j ^ t L _ j ^ t = L _ j ^ { t + 1 } R _ { j + 1 } ^ { t } , \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} & | I _ 3 | \\leq C \\frac { 1 } { \\| M _ 1 M _ 2 \\| } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { 2 n } } \\cdot 2 ^ { n } \\cdot \\| M _ 1 M _ 2 \\| \\cdot \\omega ( 2 ^ { n } \\| M _ 1 M _ 2 \\| ) = \\\\ & = C \\sum _ { n = 1 } ^ { \\infty } \\frac { \\omega ( 2 ^ { n } \\| M _ 1 M _ 2 \\| ) } { 2 ^ n } \\leq C \\omega ( \\| M _ 1 M _ 2 \\| ) . \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\lVert \\mathbf x \\rVert _ B : = \\lVert ( \\lVert x _ n \\rVert ) _ { n \\in \\Z } \\rVert _ B . \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\alpha _ m \\nu _ { \\delta } ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ m ) p ^ { - m / 2 } \\mathbf 1 _ { p ^ { - m } \\Z _ p } ( x ) \\int _ { \\Q _ p } \\psi ( - 2 x p ^ m y - \\delta p y ^ 2 ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) d y . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\frac { 1 } { p } \\sum _ { \\ell = 1 } ^ { p } f \\left ( \\frac { \\ell + r } { p } \\right ) \\leq \\frac { 1 } { p + 1 } \\sum _ { \\ell = 1 } ^ { p + 1 } f \\left ( \\frac { \\ell + r } { p + 1 } \\right ) . \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} E ^ { P } \\Big [ M ^ f _ { t \\wedge \\tau _ { \\lambda _ m } } \\prod _ { i = 1 } ^ l g _ i ( X _ { q _ i } ) \\Big ] = E ^ { P } \\Big [ M ^ { f } _ { s \\wedge \\tau _ { \\lambda _ m } } \\prod _ { i = 1 } ^ l g _ i ( X _ { q _ i } ) \\Big ] , \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} h _ { d j } ^ * ( z ) = 2 ^ { d - j } + \\sum _ { r = 1 } ^ { n + d - j } \\frac { 2 \\left ( h _ { d , j + 1 } ^ * \\right ) ^ { ( r ) } ( 1 ) + r \\left ( h _ { d , j + 1 } ^ * \\right ) ^ { ( r - 1 ) } ( 1 ) } { r ! } \\ , ( z - 1 ) ^ r , j = d - 1 , \\dots , 0 , \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} & \\tau '' _ k ( x _ { k } ^ { * } ) = \\frac { - 6 { \\omega _ { k } } } { ( 1 - 6 { x _ { k } ^ { * } } ^ 2 + { x _ { k } ^ { * } } ^ 4 ) } , \\\\ & \\tau _ k ( x _ { k } ^ { * } ) - \\tau _ k ( 0 ) = \\frac { 3 { x _ { k } ^ { * } } ^ 2 } { ( 1 - 6 { x _ { k } ^ { * } } ^ 2 + { x _ { k } ^ { * } } ^ 4 ) } { \\omega _ { k } } . \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\mathcal { K } ( \\Lambda _ 1 , \\Lambda _ 2 ) : = \\{ k _ { \\lambda } \\} _ { \\lambda \\in \\Lambda _ 1 } \\cup \\Big \\{ \\frac { G } { z - \\lambda } \\Big \\} _ { \\lambda \\in \\Lambda _ 2 } . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\lambda _ 2 ^ { ( n ) } = \\lambda _ 1 ^ { n } \\sum _ { k = 1 } ^ n \\lambda _ 1 ^ { k - 1 } a _ { i _ k } \\geq \\lambda _ 1 ^ { 2 n } a _ { i _ n } . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} \\xi = 1 \\mbox { i n $ B _ { \\left ( 1 - \\nu \\right ) R } $ } , \\xi = 0 \\mbox { o n $ \\partial B _ { R } $ } \\mbox { a n d } \\left | \\nabla \\xi \\right | \\leq \\frac { C } { \\nu R } \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} M C ( \\Phi ) ( \\Lambda \\oplus \\Pi ) = \\left ( \\ , \\Phi _ 1 ( \\Lambda ) + \\frac { 1 } { 2 } \\Phi _ 2 ( \\Pi , \\Pi ) \\ , \\right ) \\oplus \\Phi _ 1 ( \\Pi ) \\ ; . \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} & \\lim _ k A _ k = f _ 1 ( U ) \\ ; , \\lim _ k B _ k = f _ 2 ( U ) \\ ; , \\lim _ k C _ k = f _ 3 ( U ) \\ ; , \\lim _ k D _ k = f _ 4 ( U ) \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} 2 d x ^ { 2 } + 2 ( 1 - d D ) x - a d C = 0 . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} g _ d ( q ) = \\sum _ { d | r } \\mu ( r / d ) h _ r ( q ) \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} \\begin{pmatrix} Z ^ 3 A _ 3 & 0 & 0 \\\\ 0 & Z ^ 6 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} [ \\beta ] = \\sum _ { k = 1 } ^ n [ { \\jmath _ k } _ { ! * } \\left ( \\beta _ k | _ { S _ k } \\right ) ] . \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} & \\rho _ { 1 3 4 } ( S _ 1 ) = S _ 2 S _ 1 S _ 1 ^ * + S _ 1 S _ 2 S _ 2 ^ * \\\\ & \\rho _ { 1 3 4 } ( S _ 2 ) = S _ 2 ^ 2 S _ 1 ^ * + S _ 1 ^ 2 S _ 2 ^ * \\\\ & \\rho _ { 1 4 2 } ( S _ 1 ) = S _ 2 ^ 2 S _ 1 ^ * + S _ 1 ^ 2 S _ 2 ^ * \\\\ & \\rho _ { 1 4 2 } ( S _ 2 ) = S _ 2 S _ 1 S _ 1 ^ * + S _ 1 S _ 2 S _ 2 ^ * \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} \\left ( 1 - \\sum \\limits _ { i = 1 } ^ d \\ 1 _ { Q _ i ^ k } \\right ) \\lambda ^ { k + 1 } + \\left ( 1 - \\sum \\limits _ { i = 1 } ^ { d - 1 } \\ 1 _ { Q _ { i , i + 1 } ^ k } \\right ) u ^ { k + 1 } = \\sum \\limits _ { i = 1 } ^ d \\ 1 _ { Q _ i ^ k } u _ i + \\frac { 1 } { 2 } \\sum \\limits _ { i = 1 } ^ { d - 1 } \\ 1 _ { Q _ { i , i + 1 } ^ k } ( u _ i + u _ { i + 1 } ) , \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} { \\bf { C o v } } \\left [ V _ { \\varepsilon } ( \\psi ) , \\ , V _ { \\varepsilon } ( \\xi ) \\right ] = & \\begin{cases} - 2 \\ , \\log | e ^ { 2 \\pi i \\psi } - e ^ { 2 \\pi i \\xi } | , \\ , | \\xi - \\psi | \\gg \\varepsilon , \\\\ 2 \\left ( \\kappa - \\log \\varepsilon \\right ) , \\psi = \\xi , \\end{cases} \\\\ & + O ( \\varepsilon ) , \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} U ( L ) = \\left \\{ 0 \\right \\} \\cup \\{ a \\in L : a ^ { \\sim } = 0 \\} \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{gather*} m = R ^ 1 m _ { [ 0 ] } \\cdot \\varepsilon \\big ( R ^ 2 m _ { [ 1 ] } \\kappa \\big ) = \\varepsilon ( m _ { [ 1 ] } ) R m _ { [ 0 ] } \\cdot ( \\kappa ) . \\end{gather*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\Psi ( 1 - s ; \\widetilde { W } , \\widetilde { W ^ \\prime } , \\widehat { \\phi } ) = \\gamma ( s , \\pi \\times \\pi ^ \\prime , \\psi ) \\Psi ( s ; W , W ^ \\prime , \\phi ) , \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} \\tilde \\Sigma = \\Sigma + \\partial \\Omega _ 1 + \\ldots + \\partial \\Omega _ N \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\sum _ { | k | \\leq N } C _ { k , l } ^ 2 = \\frac 1 2 a _ N | l | ^ 2 \\mbox { w i t h } a _ N = \\sum _ { | k | \\leq N } \\frac 1 { | k | ^ { 2 ( \\gamma - 1 ) } } . \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} w '' ( t , x ) = - \\nabla \\mathcal { W } ( w ( t , \\cdot ) ) ( x ) , \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} \\mathcal { K } _ { t , \\mathcal { Q } ^ { \\flat } } = \\bigoplus _ { m \\in \\mathbb { B } ^ { \\xi , \\flat } } \\mathbb { Z } [ t ^ { \\pm 1 / 2 } ] E _ t ( m ) = \\bigoplus _ { m \\in \\mathbb { B } ^ { \\xi , \\flat } } \\mathbb { Z } [ t ^ { \\pm 1 / 2 } ] L _ t ( m ) \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} 1 + n = e ^ { N } \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} R = \\begin{pmatrix} R _ 1 \\\\ R _ 3 & R _ 2 \\\\ R _ 4 & R _ 5 & R _ 1 \\end{pmatrix} = \\begin{pmatrix} F _ d P _ 1 ^ T E _ e \\\\ E _ { k - 2 l _ 1 } P _ 5 ^ T E _ e & E _ { k - 2 l _ 1 } P _ 2 ^ T E _ { n - 2 k - 2 l _ 2 } \\\\ F _ d P _ 4 ^ T E _ e & F _ d P _ 3 ^ T E _ { n - 2 k - 2 l _ 2 } & F _ d P _ 1 ^ T E _ e \\end{pmatrix} . \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\gamma _ s ^ 2 : = \\sum _ { l \\in \\N } \\frac { 1 } { ( | k _ l | ^ s + 1 ) ^ { 2 - \\frac { 2 d } { p } } } = \\sum _ { k \\in \\Z ^ d } \\frac { 1 } { ( | k | ^ s + 1 ) ^ { 2 - \\frac { 2 d } { p } } } \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} C ^ { \\star } _ { \\Phi , d } = \\{ x + n x \\boldsymbol \\epsilon \\ ; | \\ ; x \\in C _ { \\Phi , d } , \\ ; n \\in \\mathbb Z , \\ ; - d \\le n \\le d \\} \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} A _ 0 ' ( z ) = E ( z ) A _ 0 ( z ) + 1 , \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{gather*} K _ { u ^ n } ^ 2 \\left ( ( u ^ n ) _ k ^ K \\right ) = \\sigma _ k ^ 2 ( u ^ n ) _ k ^ K , \\\\ K _ { u ^ n } \\left ( ( u ^ n ) _ k ^ K \\right ) = \\sigma _ k \\Psi ^ n \\cdot ( u ^ n ) _ k ^ K , \\end{gather*}"}
-{"id": "8578.png", "formula": "\\begin{align*} ( \\Gamma \\circ \\varphi ( g ' ) ) \\big ( ( g , q ) \\big ) & = \\Gamma \\big ( ( g ' g , q ) \\big ) \\\\ & = \\big ( \\alpha ' ( g ' g ) , \\alpha ( q ) \\big ) \\\\ & = \\big ( \\alpha ' ( g ' ) \\alpha ' ( g ) , \\alpha ( q ) \\big ) \\\\ & = \\varphi \\big ( \\alpha ' ( g ' ) \\big ) \\big ( ( \\alpha ' ( g ) , \\alpha ( q ) ) \\big ) \\\\ & = ( \\varphi \\big ( \\alpha ' ( g ' ) \\big ) \\circ \\Gamma ) \\big ( ( g , q ) \\big ) . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} M ^ n _ \\omega : = M _ { T ^ { n - 1 } \\omega } \\cdots M _ \\omega . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} Q _ i ' & = \\frac 1 2 \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\sum _ { q = 0 } ^ { p - 1 } ( Y _ i ^ { - 1 } B Y _ j ) ( Y _ i ^ { - 1 } A ^ { - 1 } B Y _ j ) ^ q ( Q _ j - Q _ i ) ( B Y _ j Y _ i ^ { - 1 } A ^ { - 1 } ) ^ { p - q - 1 } ( Y _ i ^ { - 1 } A ^ { - 1 } Y _ i ) \\\\ & + \\frac 1 2 \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\sum _ { q = 0 } ^ { p - 1 } ( Y _ i ^ { - 1 } A Y _ i ) ( Y _ j ^ { - 1 } B ^ { - 1 } A Y _ i ) ^ q ( Q _ j - Q _ i ) ( A Y _ i Y _ j ^ { - 1 } B ^ { - 1 } ) ^ { p - q - 1 } ( Y _ j ^ { - 1 } B ^ { - 1 } Y _ i ) . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} M = M _ 1 \\otimes \\cdots \\otimes M _ { k - 1 } \\otimes N _ 1 \\otimes \\cdots \\otimes N _ j \\otimes M _ { k + 1 } \\otimes \\cdots M _ n . \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} \\hat f ( x ) = M \\max ( \\langle x , e _ 1 \\rangle , \\dots , \\langle x , e _ d \\rangle , \\| x \\| - 1 ) , \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} { \\rm t r } \\ , ( \\xi ^ T \\nabla \\bar { f } ( S ) ) = { \\rm D } \\bar { f } ( S ) [ \\xi ] . \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} { S } : = \\{ x \\in X : \\Theta ( x ) > 0 \\} . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} \\P _ x ( \\tilde { X } _ k ( 1 ) = y ) ~ = ~ \\tilde \\mu _ k ( y - x ) \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{gather*} j = \\frac { ( t + 3 ) ^ 3 ( t + 9 ) ^ 3 \\big ( t ^ 2 + 2 7 \\big ) ^ 3 } { t ^ 3 \\big ( t ^ 2 + 9 t + 2 7 \\big ) ^ 3 } \\end{gather*}"}
-{"id": "6199.png", "formula": "\\begin{align*} \\widetilde { P } _ n = ( n - 1 ) ! S _ { n - 1 } , \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} I _ { h , T } \\mathbf { u } ( X ) - \\mathbf { u } ( X ) = \\sum _ { i \\in \\mathcal { I } ^ { s ' } } \\Phi _ { i , T } ( X ) ( \\mathbf { E } _ i ^ s ( X ) + \\mathbf { F } _ i ^ s ( X ) ) + \\sum _ { i \\in \\mathcal { I } } \\Phi _ { i , T } ( X ) \\mathbf { R } _ i ^ s ( X ) , ~ s = \\pm , \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} \\Pr \\left ( | \\hat { R } _ { n , m } - R _ { n , m } | \\geq \\frac { 4 \\sigma } { \\delta _ m \\sqrt { T _ m } } \\right ) \\leq \\pi _ 0 : = 6 \\cdot 1 0 ^ { - 5 } . \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} \\mu _ 1 & = \\max _ { f : \\sum _ { v \\in V } \\deg v \\cdot f ( v ) ^ 2 = 1 } \\sum _ { h \\in H } \\biggl ( \\sum _ { v _ i h } f ( v _ i ) - \\sum _ { v ^ j h } f ( v ^ j ) \\biggr ) ^ 2 \\\\ & = \\max _ { f : \\sum _ { v \\in V } \\deg v \\cdot f ( v ) ^ 2 = 1 } \\sum _ { h \\in H } \\biggl ( \\sum _ { v _ i \\in h : f ( v _ i ) > 0 } f ( v _ i ) - \\sum _ { v ^ j \\in h : f ( v ^ j ) < 0 } f ( v ^ j ) \\biggr ) ^ 2 . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\P _ { x + h ( s ) } ( \\tau _ { h ( s ) } > u _ 0 ) & = \\P _ { x } ( \\tau ^ { ( h ( s ) ) } _ { 0 } > u _ 0 ) \\\\ & \\geq \\P _ x ( \\tau ^ { ( h ( 0 ) ) } _ 0 > u _ 0 ) \\\\ & \\geq \\frac { \\P _ x ( \\tau ^ { ( h ( 0 ) ) } _ 0 > u _ 0 ) } { \\P _ x ( \\tau ^ { ( 0 ) } _ { u \\to - L u } > u _ 0 ) } \\P _ { x + h ( s ) } ( \\tau _ { u \\to h ( s ) - L u } > u _ 0 ) . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} \\textrm { K } _ { \\omega } ( x ) = - \\frac { \\psi '' ( r ) } { \\psi ( r ) } \\ , , \\textrm { R i c } _ { o } ( x ) = - ( N - 1 ) \\ , \\frac { \\psi '' ( r ) } { \\psi ( r ) } \\ , . \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} \\| T _ i \\vec x \\| _ \\infty = \\| \\vec x \\| _ \\infty , I ( T _ i \\vec x ) - I ( \\vec x ) = \\pm \\sum _ { j : \\{ j , i \\} \\in E } w _ { i j } 2 x _ j > 0 . \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} \\lambda \\big ( ( g , q _ 1 \\ast q _ 2 ) \\big ) & = g \\cdot \\partial ' ( q _ 1 \\ast q _ 2 ) \\\\ & = g \\cdot \\big ( \\partial ' ( q _ 1 ) \\ast \\big ( \\phi ( e _ { q _ 1 } ) \\cdot \\partial ' ( q _ 2 ) \\big ) \\big ) \\\\ & = g \\cdot \\partial ' ( q _ 1 ) \\ast \\big ( ( g \\phi ( e _ { q _ 1 } ) ) \\cdot \\partial ' ( q _ 2 ) \\big ) \\\\ & = \\lambda \\big ( ( g , q _ 1 ) \\big ) \\ast \\lambda \\big ( \\big ( g \\phi ( e _ { q _ 1 } ) \\big ) , q _ 2 \\big ) . \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} W ^ { ( 1 ) } = & \\frac { a + b _ 1 } { 2 } y ^ 5 + \\left ( - 1 2 a + 3 7 b _ 1 + \\frac { b _ 2 - b } { 2 } \\right ) y ^ 9 \\\\ & + \\left ( 1 3 8 a + 6 9 1 b _ 1 + 1 4 b _ 2 + 1 1 b + \\frac { b _ 3 + c } { 2 } \\right ) y ^ { 1 3 } \\\\ & + \\left ( - 1 0 1 2 a - 1 9 3 3 5 b _ 1 - 9 b _ 3 - 1 0 c - \\frac { 6 7 5 b _ 2 + 2 3 1 b + d } { 2 } \\right ) y ^ { 1 7 } + \\cdots . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\langle r ^ { - } _ { \\psi } ( \\alpha _ n ) \\mathbf h _ p , \\mathbf h _ p \\rangle = | p | _ p ^ { n / 2 } \\chi _ { \\psi } ( p ^ n ) \\int _ { \\Q _ p } \\mathbf 1 _ { p ^ { - n } \\Z _ p ^ { \\times } } ( x ) \\underline { \\chi } _ p ^ { - 1 } ( p ^ n x ) \\mathbf 1 _ { \\Z _ p ^ { \\times } } ( x ) \\underline { \\chi } _ p ( x ) d x = 0 , \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\nu \\left ( \\overline { { \\check { \\sigma } } } \\right ) : = \\int _ G \\overline { { \\check { \\sigma } } } d \\nu = \\int _ G \\overline { \\int _ { \\widehat { G } } \\gamma ( x ) d \\sigma ( \\gamma ) } d \\nu ( x ) = \\int _ { \\widehat G } \\widehat { \\nu } \\overline { d \\sigma } = : \\overline { \\sigma } ( \\widehat { \\nu } ) . \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\langle { \\bf u } ^ { \\tt M } , f \\rangle = \\int _ C f ( z ) \\Gamma ( - z ) \\Gamma ( \\beta + z ) ( - c ) ^ z \\ , d z , \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} v _ { \\infty } ( t ) = 0 \\ \\ \\textrm { f o r } \\ \\ t \\in ( - 5 / 4 , 0 ] . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} t _ { b } ^ { - 4 n } t _ { t _ { b } ^ { - 6 n } ( a ) } ^ { 4 n } t _ { b } ^ { - 4 n } t _ { t _ { b } ^ { - 2 n } ( c ) } ^ { 4 n } & = [ t _ { b } ^ { - 4 n } t _ { t _ { b } ^ { - 6 n } ( a ) } ^ { 4 n } , f _ 3 ] , \\\\ t _ { v } t _ { d } ^ { - 1 } t _ { t _ { b } ( v ) } t _ { d } ^ { - 1 } & = [ t _ { v } t _ { d } ^ { - 1 } , f _ 4 ] . \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} \\psi ( x ) = - \\frac 1 { 2 \\pi } \\sum _ { j = 1 } ^ N a _ j \\log | x - y _ j | . \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} \\lambda _ 1 m _ 1 + \\cdots + \\lambda _ k m _ k = 0 . \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } c _ { i _ p , j _ p } ^ { ' } z _ { i _ p , j _ p } \\prod \\limits _ { \\varepsilon _ { r p } = - 1 } z _ { i _ r , j _ r } = c _ { i _ p , j _ p } \\prod \\limits _ { \\varepsilon _ { r p } = 1 } z _ { i _ r , j _ r } , \\ ; s + 1 \\leq p \\leq l , \\ ; 1 \\leq r \\leq s , \\\\ z _ { i , j } = 0 , \\ ; ( i , j ) \\notin \\{ ( i _ 1 , j _ 1 ) , \\ldots , ( i _ l , j _ l ) \\} , \\end{array} \\right . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} g _ { \\nabla f _ p } ( \\nabla f _ p , \\dot \\alpha ( f ( p ) ) ) = d f _ p ( \\dot \\alpha ( f ( p ) ) ) = 1 . \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\widetilde { \\Phi } ^ T ( \\widetilde { D } ( ( i , r ) , ( i , r + 2 s r _ i ) ) ) = F _ t ( m _ { s , r } ^ { ( i ) } ) ^ T \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} { \\underline { \\mathfrak { Q } } } _ t ^ k ( x _ { t - 1 } , D _ { t - 1 } , \\xi _ t , D _ { t } ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; D _ { t - 1 } f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + \\mathcal { Q } _ { t + 1 } ^ k ( x _ { t } ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} \\left \\| u \\right \\| _ { L ^ 2 _ T } ^ 2 : = \\int _ 0 ^ T \\left \\| u ( t ) \\right \\| _ 2 ^ 2 d t < \\infty \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} ( U _ 2 ( \\lambda ) ^ { - T } \\oplus I _ { m _ 1 } ) \\ , \\mathcal { L } ( \\lambda ) \\ , ( U _ 1 ( \\lambda ) ^ { - 1 } \\oplus I _ { m _ 2 } ) \\begin{bmatrix} 0 \\\\ I _ n \\\\ - X ( \\lambda ) \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ Q ( \\lambda ) \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} & [ 1 1 2 9 , 4 3 1 ] + 3 ( - 2 1 \\times 1 1 2 9 + 5 5 \\times 4 3 1 ) [ 5 5 , 2 1 ] = [ 4 6 9 , 1 7 9 ] , & & Q ( [ 4 6 9 , 1 7 9 ] ) = 1 4 9 , \\\\ & [ 4 6 9 , 1 7 9 ] - 3 ( - 8 \\times 4 6 9 + 2 1 \\times 1 7 9 ) [ 2 1 , 8 ] = [ 2 8 , 1 1 ] , & & Q ( [ 2 8 , 1 1 ] ) = - 1 9 . \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} 4 \\pi k \\geq \\int _ { \\mathbb { R } ^ 3 } e ( A _ { x } , \\Phi _ { x } ) = 4 \\pi k _ x , \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} g _ 1 ( X ) & = \\frac { f _ 1 } { G C D ( f _ 1 , f _ 2 ) } = \\prod \\limits _ { b \\in D _ 1 } ( X + b ) , \\\\ g _ 2 ( X ) & = \\frac { f _ 2 } { G C D ( f _ 1 , f _ 2 ) } = \\prod \\limits _ { b \\in D _ 2 } ( X + b ) , \\\\ g _ 3 ( X ) & = G C D ( f _ 1 , f _ 2 ) = \\prod \\limits _ { b \\in D _ 3 } ( X + b ) . \\\\ \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\abs { Y } \\le q ^ { ( m + 1 ) ( n - \\delta ) } q ^ \\delta . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\mathfrak { r } _ { w _ i } = 1 \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} G ( z + 1 ) = \\Gamma ( z ) \\ , G ( z ) . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} | G | _ \\alpha = \\int _ { \\Gamma _ 0 } | G ( \\eta ) | e ^ { \\alpha | \\eta | } \\lambda ( d \\eta ) . \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } A ^ 3 + B ^ 2 - B + 1 = 0 \\\\ \\\\ \\xi ^ 3 = \\frac { 2 B ^ 2 - 3 B \\alpha + B - 4 } { 2 B ^ 2 + 3 B \\alpha + B - 4 } \\\\ \\end{array} \\right . . \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} T _ { m , n } & = f ( m _ 1 + n _ 1 ) + g ( m _ 2 + n _ 2 ) - f ( m _ 1 + n _ 1 + 2 ) - g ( m _ 2 + n _ 2 ) \\\\ & \\quad - f ( m _ 1 + n _ 1 ) - g ( m _ 2 + n _ 2 + 2 ) + f ( m _ 1 + n _ 1 + 2 ) + g ( m _ 2 + n _ 2 + 2 ) \\\\ & = 0 , \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} H _ { 1 , \\boldsymbol { \\alpha } } ( \\tau ) = \\int _ { \\R ^ 2 } \\cot \\left ( \\pi i w _ 1 + \\pi \\alpha _ 1 \\right ) \\cot \\left ( \\pi i w _ 2 + \\pi \\alpha _ 2 \\right ) e ^ { 2 \\pi i \\tau Q ( \\boldsymbol { w } ) } d w _ 1 d w _ 2 . \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} \\mathrm { G S O } ( V ) = \\{ h \\in \\mathrm { G O } ( V ) : \\det ( h ) = \\nu ( h ) ^ { m / 2 } \\} . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} h ( G ) = \\inf _ { \\vec x } \\sup _ { c \\in \\mathbb { R } } \\frac { I ( \\vec x ) } { \\sum _ { i = 1 } ^ n d _ i | x _ i - c | } . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} v ^ f : = \\lim _ { i \\to \\infty } \\frac { 1 } { F ( \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) ) } \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} & 2 \\sup _ { t \\in [ T _ 1 , T _ 2 ] } \\sum _ { \\mu , \\iota = 1 } ^ N \\left | \\int _ { \\R ^ 3 } \\int _ { \\R ^ 3 } j _ { u _ \\mu } ( t , x ) \\cdot \\nabla _ x \\psi ( x , y ) m _ { u _ \\iota } ( t , y ) \\ , d x d y \\right | \\\\ & \\leq C _ 1 \\sup _ { t \\in [ T _ 1 , T _ 2 ] } \\sum _ { \\mu = 1 } ^ N \\| u _ \\mu ( t ) \\| ^ 4 _ { H ^ 2 _ x } \\leq C _ 2 \\sum _ { \\mu = 1 } ^ N \\| u _ { \\mu , 0 } \\| ^ 4 _ { H ^ 2 _ x } < \\infty , \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} \\varphi _ k = \\sum _ { i = 1 } ^ { 4 ^ { k - 1 } } \\frac { i } { 4 ^ { k - 1 } } \\varphi _ { k , i } \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} P ^ * ( x ) = \\sum \\limits _ { n \\geq 1 } P _ n ^ * \\frac { x ^ n } { n ! } . \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} C = S \\cap Z ( s - 1 ) = Z ( G , F _ 3 , s - 1 ) \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\sup _ { N \\in \\mathbb { N } } \\mathbb { E } \\max _ { i = 1 , \\ldots , N } \\big | v ^ { i , N } _ 0 \\big | ^ 4 < + \\infty \\mbox { a n d } \\mathbb { E } \\left [ \\left \\vert \\langle S _ 0 ^ { N } , \\psi \\rangle - \\langle f _ 0 , \\psi \\rangle \\right \\vert \\wedge 1 \\right ] \\stackrel { N \\rightarrow \\infty } { \\longrightarrow } 0 , \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\begin{aligned} & A \\leqslant \\frac { V } { \\log Q } , Q = \\frac { c \\log N } { 2 V ^ 2 \\log V } \\qquad \\\\ & A \\log \\frac { c \\log N } { 2 A _ 0 ^ 2 \\log A _ 0 } = V + \\log \\frac { 2 \\| \\psi \\| _ { \\infty } } { I ( \\psi ) } + C _ f \\frac { A } { \\log A _ 0 } . \\end{aligned} \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} - p ^ { \\ast } ( x ^ { \\ast } ) = \\min { \\mathrm { ( R P ) } _ { { x ^ { \\ast } } } } = \\sup _ { u \\in U } F _ { u } ( \\bar { x } , 0 _ { u } ) - \\langle x ^ { \\ast } , \\bar { x } \\rangle = p ( \\bar { x } ) - \\langle x ^ { \\ast } , \\bar { x } \\rangle = \\sup { ( \\mathrm { O D P } ) _ { { x ^ { \\ast } } } } = - q ( x ^ { \\ast } ) . \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} \\chi _ { \\Lambda , A _ F } ( C ^ { \\bullet } , t ) = \\chi _ { \\Lambda , F } ( C ^ { \\bullet } , t ) : = [ P ^ { o d d } , \\phi _ t , P ^ { e v } ] \\in K _ 0 ( \\Lambda , F ) \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} E G - F ^ 2 = \\lambda ^ 2 \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} V _ 1 ( x ) & = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 ) } \\frac { \\Gamma \\left ( \\frac { s } { 2 } + \\frac { 1 } { 4 } \\right ) } { \\Gamma \\left ( \\frac { 1 } { 4 } \\right ) } \\frac { G _ 1 ( s ) } { s } \\pi ^ { - s / 2 } x ^ { - s } d s \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} \\dot { P } = A ^ * P + P A + C ^ * C , P ( 0 ) = G ^ * G , \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} 0 < \\inf _ { x \\in G } \\lvert \\gamma ( x ) \\rvert \\ ; \\frac { ( \\omega _ 2 \\circ \\phi ) ( x ) } { \\omega _ 1 ( x ) } \\sup _ { x \\in G } \\lvert \\gamma ( x ) \\rvert \\ ; \\frac { ( \\omega _ 2 \\circ \\phi ) ( x ) } { \\omega _ 1 ( x ) } < \\infty . \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} & e ^ { ( n - \\nu - 2 ) t } \\left ( \\frac { 2 ( n + 1 ) } { n } c _ 2 ( X ) - c _ 1 ( X ) ^ 2 \\right ) \\cdot [ \\omega ( t ) ] ^ { n - 2 } \\\\ & e ^ { ( n - \\nu - 2 ) t } \\left ( \\frac { 2 ( n + 1 ) } { n } c _ 2 ( X ) - c _ 1 ( X ) ^ 2 \\right ) \\cdot ( - ( 1 - e ^ { - t } ) 2 \\pi c _ 1 ( X ) + e ^ { - t } \\alpha ) ^ { n - 2 } \\\\ & \\to \\binom { n - 2 } { \\nu } ( 2 \\pi ) ^ \\nu \\left ( \\frac { 2 ( n + 1 ) } { n } c _ 2 ( X ) - c _ 1 ( X ) ^ 2 \\right ) \\cdot ( - c _ 1 ( X ) ) ^ \\nu \\cdot \\alpha ^ { n - \\nu - 2 } , \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} I ( x , y ) = \\{ u \\in X \\ , : \\ , d ( x , y ) = d ( x , u ) + d ( u , y ) \\} , \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} M _ { a } ( x ) = \\sum _ { n \\leq x } \\frac { \\mu ( n ) } { n } \\Delta _ { a } \\left ( \\frac { x } { n } \\right ) \\log \\frac { x } { e } + O _ { a } \\left ( \\log x \\right ) , \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} \\sigma ^ B ( t ) = u _ \\delta ^ { B } ( t ) = \\frac { 1 } { 2 } ( u _ + + u _ - ) + \\beta t , \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\mu _ t : = \\int _ 0 ^ t \\big [ \\Delta u _ r + \\mathrm { d i v } ( F ( u _ r ) ) \\big ] d r \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & K _ { 2 } ( s ) \\\\ & = ( 1 - p ( s ) \\sigma _ { z } ( s ) ) ^ { - 1 } \\left \\{ p ( s ) \\sigma _ { y } ( s ) + 2 \\left [ \\sigma _ { x } ( s ) + \\sigma _ { y } ( s ) p ( s ) + \\sigma _ { z } ( s ) K _ { 1 } ( s ) \\right ] \\right \\} P ( s ) \\\\ & \\ \\ + ( 1 - p ( s ) \\sigma _ { z } ( s ) ) ^ { - 1 } \\left \\{ Q ( s ) + p ( s ) \\left ( 1 , p ( s ) , K _ { 1 } ( s ) \\right ) D ^ { 2 } \\sigma ( s ) \\left ( 1 , p ( s ) , K _ { 1 } ( s ) \\right ) ^ { \\intercal } \\right \\} , \\end{array} \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta ^ q _ { 2 , 2 } ( \\delta ) \\bigr ] = \\exp \\Bigl ( \\frac { 1 } { 4 } \\int \\limits _ 0 ^ \\infty ( e ^ { - t q } - 1 ) e ^ { - ( \\delta - 1 / 2 ) t } \\ , { \\bf E } \\bigl [ e ^ { - t ^ 2 \\ , C _ 2 / 3 2 } \\bigr ] \\frac { d t } { t } \\Bigr ) . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} H _ { J } : = \\bigcap _ { j \\in { J } } H _ { j } . \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\begin{cases} 0 < \\underbar r _ 1 { - \\epsilon } \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 1 + \\epsilon \\cr 0 < \\underbar r _ 2 { - \\epsilon } \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 2 + \\epsilon , \\end{cases} \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\Phi _ { \\breve { \\mathbf g } _ p } ( \\beta _ m ) = - p ^ { - | 2 m - 1 | } , \\Phi _ { \\mathbf h _ p } ( \\beta _ m ) = p ^ { - | 3 m / 2 - 1 | } ( - 1 ) ^ { m + 1 } \\chi _ { \\psi } ( p ^ m ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} X _ 1 = 4 M \\sum _ { i = 1 } ^ t \\left ( \\pi _ i - \\frac { 1 } { 2 } \\right ) ^ 2 . '' \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} R ( r ) = C e ^ { \\eta { r } } r ^ { ( \\frac { 1 - \\tau } { 2 } ) } \\Big [ e ^ { - 2 \\eta { r } } r ^ { - ( 1 - \\tau + \\mathrm { c } ) } \\Big ] _ { - ( 1 + \\mathrm { c } E ^ { - 1 } ) } \\quad \\Bigg ( \\mathrm { c } = - \\Big ( \\frac { \\beta + \\eta ( 1 - \\tau ) } { 2 \\eta } \\Big ) \\Bigg ) , \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\mathcal N _ i : = \\{ \\vec { \\mathfrak w } \\in \\Omega ( x ) ^ { \\oplus N } : \\mathfrak V _ j \\vec { \\mathfrak w } = \\vec 0 , \\ \\forall i \\neq j \\} . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} f ^ { L } ( \\vec x ) = \\sum _ { i = 1 } ^ p \\lambda _ i f ( V _ i ^ + , V _ i ^ - ) \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} F _ \\varepsilon ( u ) : = \\int _ 0 ^ t e ^ { - t / \\varepsilon } \\bigg ( \\int _ { \\mathbb { R } ^ n } \\frac { \\varepsilon ^ 2 | u '' ( t , x ) | } { 2 } \\ , d x + \\mathcal { W } ( u ( t , \\cdot ) ) \\bigg ) \\ , d t , \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| y ^ { k + 1 } - y ^ { k } \\| = 0 . \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} { \\tilde \\theta _ n } = 2 n \\pi \\tilde f T + \\tilde \\varphi \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} \\mathcal { J } _ n = \\frac { 1 } { \\pi n } \\bigg ( 1 + \\int _ 0 ^ \\infty \\frac { 2 y ^ 3 } { ( y ^ 2 + \\pi ^ 2 n ^ 2 ) \\sinh ^ 2 y } d y \\bigg ) \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 5 x _ i v _ i \\ , \\longmapsto \\ , { } ^ t ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 ) , \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} w _ { l _ 1 , l _ 2 , k , n } ( d , e ) = \\begin{pmatrix} d _ { l _ 1 , k } \\\\ & e _ { l _ 2 , n - 2 k } \\\\ & & d _ { l _ 1 , k } \\end{pmatrix} , \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} [ { M ^ * _ \\epsilon } ^ { - 1 } \\Gamma _ { 0 } { M ^ * _ \\epsilon } , { M ^ * _ { \\epsilon ' } } ^ { - 1 } \\Gamma _ { 0 } { M ^ * _ { \\epsilon ' } } ] = 1 . \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} = ( U _ 2 ( \\lambda ) ^ T \\oplus I _ { m _ 1 } ) \\begin{bmatrix} 0 \\\\ Q ( \\lambda ) \\\\ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} d \\bar { s } ^ { 2 } = 2 \\bar { T } d t ^ { 2 } = d \\rho ^ { 2 } + \\rho ^ { 2 } d \\sigma ^ { 2 } = d \\rho ^ { 2 } + \\frac { \\rho ^ { 2 } } { 4 } ( d \\varphi ^ { 2 } + \\sin ^ { 2 } ( \\varphi ) d \\theta ^ { 2 } ) \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\bar { R } ^ { ' } ( X , Y , Z , W ) & = R ^ { ' } ( X , Y , Z , W ) - g \\big ( h ^ { ' } ( X , W ) , h ^ { ' } ( Y , Z ) \\big ) \\\\ & + g \\big ( h ^ { ' } ( X , Z ) , h ^ { ' } ( Y , W ) \\big ) , \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n ^ { - 1 } \\log \\Vert M ^ n _ \\omega v \\Vert = \\kappa _ i . \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} & \\int _ X | R i c ( \\omega ( t ) ) + \\omega ( t ) | ^ 2 _ { \\omega ( t ) } \\omega ( t ) ^ n \\\\ & = \\int _ X ( \\partial _ t S ( t ) ) \\omega ( t ) ^ n + \\int _ X ( S ( t ) + n ) \\omega ( t ) ^ n \\\\ & = \\partial _ t \\left ( \\int _ X S ( t ) \\omega ( t ) ^ n \\right ) + \\int _ X ( S ( t ) + 1 ) ( S ( t ) + n ) \\omega ( t ) ^ n , \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} \\delta \\tau ( g , n h ) = & g \\cdot \\tau ( n h ) - \\tau ( g n h ) + \\tau ( g ) \\\\ = & g \\cdot f ( n ) + g \\cdot \\tau ( h ) - f ( g n g ^ { - 1 } ) - \\tau ( g h ) + \\tau ( g ) = \\delta \\tau ( g , h ) \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\Big | \\sum _ { n _ 1 = M _ 0 } ^ { M _ 1 } A _ { n _ 1 , \\tilde { n } } \\big ( S _ { n _ 1 , \\tilde { n } } ^ { ( r ) } - S _ { n _ 1 - 1 , \\tilde { n } } ^ { ( r ) } \\big ) \\Big | \\leq \\abs { S _ { M _ 1 , \\tilde { n } } ^ { ( r ) } } + \\sum _ { n _ 1 = M _ 0 } ^ { M _ 1 - 1 } \\abs { S _ { n _ 1 , \\tilde { n } } ^ { ( r ) } } \\cdot \\abs { A _ { n _ 1 , \\tilde { n } } - A _ { n _ 1 + 1 , \\tilde { n } } } . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} ( \\bar { \\nabla } _ X \\phi ) ( Y ) = ( f _ 1 - f _ 3 ) [ g ( X , Y ) \\xi - \\eta ( Y ) X ] , \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} c ' = \\begin{cases} N / c & \\ : u _ 2 \\equiv \\frac { c u _ 2 + 2 } { N / c } \\pmod * 2 , \\\\ N / ( 2 c ) & \\ : 2 \\mid u _ 2 2 \\nmid \\frac { c u _ 2 + 2 } { N / c } , \\\\ 2 N / c & \\ : 2 \\nmid u _ 2 2 \\mid \\frac { c u _ 2 + 2 } { N / c } . \\end{cases} \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} u ( x ) = ( \\lambda _ x f , g ) , \\ \\ x \\in G , \\mbox { w h e r e } f , g \\in L ^ 2 ( G ) . \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} \\displaystyle \\eta = \\int _ 0 ^ { y ( \\xi , \\eta ) } h _ { 1 , 0 } ( \\xi , \\zeta ) d \\zeta . \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} \\mathcal E _ 1 ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * g ^ * : \\mathcal E _ 1 ( Y ) \\to \\mathcal E _ 1 ( X ) , \\\\ \\mathcal E _ 2 ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * g ^ * : \\mathcal E _ 2 ( Y ) \\to \\mathcal E _ 2 ( X ) . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} G _ i ( t ) = G ( z _ 1 ^ * , \\ldots , z _ { i - 1 } ^ * , t , z _ { i + 1 } ^ * , \\ldots z _ n ^ * ) = - | v _ i | + \\frac { r - \\sum _ { j \\ne i } z _ j ^ * ( | v _ j | - | v _ i | ) } { \\sum _ { j \\ne i } | z _ j ^ * | + t } . \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} B ( \\tilde { J } U , \\tilde { J } V ) = p B ( V , \\tilde { J } U ) + q B ( V , U ) , \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\tilde { x } = x + \\frac { 4 s } { 5 } y = t ^ 4 + \\frac { 4 s } { 5 } t ^ 5 + \\frac { 4 s ^ 2 } 5 t ^ 6 \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} 2 d = \\left ( a \\delta + \\sum b _ i \\deg ( \\psi | _ { E _ i } ) \\right ) . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} 0 = _ { q / 2 } \\Bigl ( 1 + \\frac 1 a \\Bigr ) , \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} F ( z ) & = e ^ { 2 h ( z ) } e ^ { \\bf h } F ( z ) e ^ { - { \\bf h } } , \\\\ e ^ { - { \\bf h } } F \\otimes x ( \\zeta ) e ^ { \\bf h } & = F \\otimes e ^ { 2 h ( \\zeta ) } x ( \\zeta ) . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\psi ( ( a b ^ i ) ^ p ) = \\big ( ( b ^ i ) ^ { a ^ { i e _ 1 } } , ( b ^ i ) ^ { a ^ { i ( e _ 1 + e _ 2 ) } } , \\dots , b ^ i , b ^ i \\big ) . \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} t _ { 1 } = T - \\int _ { - \\infty } ^ { - L _ { 1 } } \\frac { 1 } { F ( y ) } d y , \\ \\ t _ { 2 } = T - \\int _ { L _ { 1 } } ^ { \\infty } \\frac { 1 } { F ( y ) } d y , \\ \\ t ^ { \\ast } = t _ { 1 } \\vee t _ { 2 } . \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} t _ { b } t _ { v } ( a ) = t _ { b } t _ { a } t _ { c } t _ { b } t _ { c } ^ { - 1 } t _ { a } ^ { - 1 } ( a ) = t _ { b } t _ { a } t _ { c } t _ { b } ( a ) = c , \\\\ t _ { b } t _ { v } ( c ) = t _ { b } t _ { a } t _ { c } t _ { b } t _ { c } ^ { - 1 } t _ { a } ^ { - 1 } ( c ) = t _ { b } t _ { a } t _ { c } t _ { b } ( c ) = a . \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} & \\{ ( \\imath , - \\imath + 2 - 2 k ) \\mid k = 0 , \\dots , 2 n - 1 - \\imath \\ \\ \\imath = n + 1 , \\dots , 2 n - 1 \\} \\\\ & \\cup \\{ ( n , - n + \\frac { 3 } { 2 } - k ) \\mid k = 0 , \\dots , 2 n - 2 \\} \\\\ & \\cup \\{ ( \\imath , - \\imath + 1 - 2 k ) \\mid k = 0 , \\dots , 2 n - 2 - \\imath \\ \\ \\imath = 1 , \\dots , n - 1 \\} . \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\frac { d ( \\gamma _ i ( - \\delta ) , \\gamma _ i ( 0 ) ) - d ( \\gamma _ i ( - \\delta ) , p ) ) } { d _ i } = g _ { w _ \\infty } ( w _ \\infty , v ^ f ) . \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} \\begin{gathered} d X _ t = a X _ t d t + \\sigma \\sqrt { X _ t } \\circ d B _ t ^ H , t \\ge 0 , \\end{gathered} \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} X ^ { ( i ) } ( t ) ( \\omega ) = \\theta ^ { ( i ) } + \\int _ 0 ^ t b ^ { ( i ) } ( s , \\omega , X ( s ) ( \\omega ) ) d s + \\sum _ { j = 1 } ^ m \\int _ 0 ^ t \\sigma ^ { ( i , j ) } ( s , \\omega , X ( s ) ( \\omega ) ) d W ^ { ( j ) } ( s ) , \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} f ^ w ( x ) : = ( f ^ { i _ k } \\circ f ^ { i _ { k - 1 } } \\circ \\dots \\circ f ^ { i _ 1 } ) ( x ) . \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} \\eta _ { N , N } ( q \\ , | \\ , a , b ) = \\exp \\bigl ( - ( b _ 1 \\cdots b _ N / a _ 1 \\cdots a _ M ) \\log ( q ) + O ( 1 ) \\bigr ) , \\ ; q \\rightarrow \\infty . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} \\frac { d } { d t } X _ i \\cdot X _ i ^ { - 1 } = \\Omega _ i + \\frac 1 2 \\sum _ { j = 1 } ^ n \\gamma _ { i j } ( f ( X _ j X _ i ^ { - 1 } ) - f ( X _ i X _ j ^ { - 1 } ) ) . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} \\widetilde { C } ( z ) _ { 1 1 } & = \\sum _ { k \\geq 0 } ( - 1 ) ^ k ( z ^ { - 6 k - 2 } + z ^ { - 6 k - 4 } ) , & \\widetilde { C } ( z ) _ { 1 2 } & = \\sum _ { k \\geq 0 } ( - 1 ) ^ k z ^ { - 6 k - 3 } , \\\\ \\widetilde { C } ( z ) _ { 2 1 } & = \\sum _ { k \\geq 0 } ( - 1 ) ^ k ( z ^ { - 6 k - 2 } + z ^ { - 6 k - 4 } ) , & \\widetilde { C } ( z ) _ { 2 2 } & = \\sum _ { k \\geq 0 } ( - 1 ) ^ k ( z ^ { - 6 k - 1 } + z ^ { - 6 k - 5 } ) . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} \\partial _ y \\psi = \\hat h _ 1 ( t , x , \\psi ) , \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} \\tilde { J } ^ { 2 } = p \\tilde { J } + q I . \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} w ''' = \\Delta w \\ , . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} \\int _ { \\R ^ d \\times \\R ^ d } f ( x ) f ( y ) M ^ N _ T ( x , y ) d x d y & = \\sum _ { n \\leq N } \\left | \\int _ { \\R ^ d } f ( x ) e _ n ( x ) d x \\right | ^ 2 \\rightarrow \\sum _ { n \\geq 1 } | ( f , e _ n ) | ^ 2 = | f | _ 0 ^ 2 , \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} \\partial f _ n \\left ( M \\right ) = \\left ( f _ 1 X _ n - U _ n \\right ) \\left ( M \\right ) . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} X _ B : = \\bigg { \\{ } \\mathbf x = ( x _ n ) _ { n \\in \\Z } \\subset X : ( \\lVert x _ n \\rVert ) _ { n \\in \\Z } \\in B \\bigg { \\} } . \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} u _ t \\ , = \\ , \\Delta ( u ^ m ) \\textrm { i n } \\ ; \\ ; M \\times ( 0 , T ) \\ , . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{gather*} \\beta ^ { \\ss _ { 1 i } } = \\sum _ { j : j \\ne i } ^ n \\sigma _ { \\xi _ { 1 2 } } ( x _ { i j } ) \\sigma _ { \\eta + \\nu } ( x _ { i j } ^ + ) \\prod _ { l \\ne i , j } \\sigma _ \\mu ( x _ { j l } ) \\sigma _ \\mu ( x _ { l i } ) \\ , , \\\\ \\beta ^ { \\ss _ { 1 i } ^ + } = \\sum _ { j : j \\ne i } ^ n \\sigma _ { \\xi _ { 1 2 } } ( - x _ { i j } ^ + ) \\sigma _ { \\eta + \\nu } ( x _ { j i } ) \\prod _ { l \\ne i , j } \\sigma _ \\mu ( x _ { j l } ) \\sigma _ \\mu ( x _ { l i } ^ + ) \\ , . \\end{gather*}"}
-{"id": "8870.png", "formula": "\\begin{align*} \\big \\langle \\widetilde \\Lambda _ { v _ 1 } \\widetilde P _ { v _ 1 , \\rm R } \\widetilde F ( v _ 1 ) , \\widetilde P _ { v _ 1 , \\rm R } \\widetilde F ( v _ 1 ) \\big \\rangle = \\big \\langle \\Lambda _ { v _ 1 } P _ { v _ 1 , \\rm R } F ( v _ 1 ) , P _ { v _ 1 , \\rm R } F ( v _ 1 ) \\big \\rangle . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } V , V _ { 1 } ) = \\breve { g } ( V , \\breve { J } V _ { 1 } ) = 0 , \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} \\Vert z _ 1 - z _ 2 + z _ i \\Vert = \\Vert z _ i \\Vert \\le 1 \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\begin{aligned} & Q _ { i + 1 , j + 1 } ^ t = ( \\min [ Q _ { i + 1 , j } ^ t , W _ { i + 1 , j } ^ t ] - \\min [ Q _ { i , j } ^ t , W _ { i , j } ^ t ] ) + Q _ { i , j } ^ t , \\\\ & W _ { i , j } ^ { t + 1 } = ( \\min [ Q _ { i + 1 , j } ^ t , W _ { i + 1 , j } ^ t ] - \\min [ Q _ { i , j } ^ t , W _ { i , j } ^ t ] ) + W _ { i , j } ^ t . \\end{aligned} \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} S ( p ^ j , P ) = \\sum _ { \\sigma \\in \\{ 0 , 1 \\} ^ { k '' } } ( - 1 ) ^ \\sigma \\sum _ { ( x ' , x '' ) \\in \\Omega ^ \\sigma } \\exp \\big ( 2 \\pi i P ( x ' , p ^ \\sigma x '' ) / p ^ j \\big ) , \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\| t x + ( 1 - t ) y \\| ^ 2 = t \\| x \\| ^ 2 + ( 1 - t ) \\| y \\| ^ 2 - t ( 1 - t ) \\| x - y \\| ^ 2 , \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\bigcap _ { j = 0 } ^ \\infty B _ j \\supset B _ { \\frac { n - 1 } { n } R _ 0 } = : B _ \\infty . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} \\log G ( z \\ , | \\ , \\tau ) = \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ \\frac { 1 - z } { e ^ { t \\tau } - 1 } + ( 1 - z ) e ^ { - t \\tau } + ( z ^ 2 - z ) \\frac { e ^ { - t \\tau } } { 2 \\tau } + \\frac { 1 - e ^ { - t ( z - 1 ) } } { ( e ^ t - 1 ) ( 1 - e ^ { - t \\tau } ) } \\Bigr ] . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} f ( x , t ) = f ( t ) \\in L ^ 2 ( ( 0 , T ) ) \\quad \\forall T > 0 . \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} & \\Big ( A ^ { \\frac { 1 } { 2 } } \\big ( A ! _ \\lambda B \\big ) ^ { - 1 } A ^ { \\frac { 1 } { 2 } } \\Big ) ^ { \\frac { 1 } { 2 } } A ^ { - \\frac { 1 } { 2 } } ( A \\nabla _ \\lambda B ) A ^ { - \\frac { 1 } { 2 } } \\Big ( A ^ { \\frac { 1 } { 2 } } \\big ( A ! _ \\lambda B \\big ) ^ { - 1 } A ^ { \\frac { 1 } { 2 } } \\Big ) ^ { \\frac { 1 } { 2 } } - I \\\\ & = \\lambda ( 1 - \\lambda ) A ^ { - \\frac { 1 } { 2 } } ( A - B ) B ^ { - 1 } ( A - B ) A ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\bar { \\nabla } _ X \\xi = - ( f _ 1 - f _ 3 ) \\phi X , \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} \\psi _ { n + 1 } ( g ^ { p ^ n } ) = \\big ( ( ( a ^ { \\alpha } b ) ^ { k i } w _ 1 ) ^ { p ^ { n - 1 } } , ( ( a ^ { \\alpha } b ) ^ { k i } w _ 2 ) ^ { p ^ { n - 1 } } , \\dots ( ( a ^ { \\alpha } b ) ^ { k i } w _ p ) ^ { p ^ { n - 1 } } \\big ) , \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\mathcal { Q } _ { t } ( x _ { t - 1 } , 0 ) = 0 , \\ ; t = 2 , \\ldots , T _ { \\max } , \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} \\gamma \\left ( \\sum _ { n = N } ^ \\infty a _ n t ^ n \\right ) = \\lim _ { n \\to \\infty } \\gamma ( t ^ n ) ^ { a _ n } \\circ \\cdots \\circ \\gamma ( t ^ N ) ^ { a _ N } \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} \\det \\left ( t \\ , \\mathrm { I d } _ n - C _ a \\right ) = t ^ n - ( - 1 ) ^ { a - 1 } . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { x \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 3 } \\sin \\left ( \\frac { 1 } { x } \\right ) \\right ) - 0 } { x - 0 } = \\lim _ { x \\to 0 } \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 3 } \\sin \\left ( \\frac { 1 } { x } \\right ) \\right ) = D . N . E , \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } \\hat { Y } ( r ) = & p ( r ) \\hat { X } ( r ) + \\varphi ( r ) , \\\\ \\hat { Z } ( r ) = & K _ { 1 } ( r ) \\hat { X } ( r ) + ( 1 - p ( r ) \\sigma _ { z } ( r ) ) ^ { - 1 } [ p ( r ) \\sigma _ y ( r ) \\varphi ( r ) + p ( r ) \\varepsilon _ 2 ( r ) + \\nu ( r ) ] , \\end{array} \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} X ' _ g = R ^ * ( X _ g ) . \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} S _ { d , \\infty } ( n , W ) = \\binom { n - 1 - W d } { W - 1 } . \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} N _ { \\mathrm { c } m } = \\sum _ { n = 1 } ^ { m } \\mathcal { C } _ { n } ^ { m } \\left ( N _ { n } - N _ { \\mathrm { d } n } \\right ) , \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} E _ { b , c } { K _ b \\brack \\lambda _ b } = { K _ b ; - 1 \\brack \\lambda _ b } E _ { b , c } ; E _ { b , c } { K _ c \\brack \\lambda _ c } = { K _ c ; 1 \\brack \\lambda _ c } E _ { b , c } , \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} [ I , \\bar { I } ] ^ { A } { } _ { B } = & \\ , - ( I ^ { A } { } _ { R } \\bar { I } ^ { R } { } _ { B } - \\bar { I } ^ { A } { } _ { R } I ^ { R } { } _ { B } ) , \\\\ { \\big ( I \\odot \\bar { I } \\big ) _ o } ^ { A } { } _ { B } = & \\ , \\big ( I ^ { A } { } _ { R } \\bar { I } ^ { R } { } _ { B } + \\bar { I } ^ { A } { } _ { R } I ^ { R } { } _ { B } \\big ) _ o , \\\\ \\langle I , \\bar { I } \\rangle = & \\ , I ^ { R } { } _ { S } \\bar { I } ^ { S } { } _ { R } . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{gather*} x = \\sqrt [ 3 ] { \\lambda ( \\tau ) } , y = \\sqrt [ 3 ] { 1 - \\lambda ( \\tau ) } , x ^ 3 + y ^ 3 = 1 . \\end{gather*}"}
-{"id": "4202.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) : \\ , v \\in e } B ( v , e ) = \\sum _ { e \\in E ( G _ i ) : \\ , v \\in e } B _ i ( v , e ) = 1 . \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} G : = \\sum \\limits _ { i = 1 } ^ { s } { t _ i \\sum \\limits _ { Q \\in O r b _ \\sigma ( Q _ i ) } { Q } } , \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} \\norm { f - \\sum _ { j = 0 } ^ N \\mathrm { C } _ { \\eth } ^ j ( f ) } { L ^ 1 ( \\R _ + , r \\ , d r ) } & \\leq \\norm { f - \\sum _ { j = 0 } ^ N \\mathrm { C } _ { \\eth _ 0 } ( f ) } { L ^ 1 ( \\R _ + , r \\ , d r ) } \\\\ & + \\sum _ { j = 0 } ^ N \\norm { \\mathrm { C } _ \\eth ^ j ( f ) - \\mathrm { C } _ { \\eth _ 0 } ^ j ( f ) } { L ^ 1 ( \\R _ + , r \\ , d r ) } \\leq \\varepsilon \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} h _ { p , q : r , s } = \\frac { ( r p - s q ) ^ { 2 } - ( p - q ) ^ { 2 } } { 4 p q } , \\ \\ 0 < r < q , \\ 0 < s < p . \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\begin{aligned} { U } _ { F } ^ + . v & = \\span \\bigg \\{ \\prod _ { h + 1 \\leq b \\leq m + n } E _ { h , b } ^ { ( A _ { h , b } ) } \\ , \\prod _ { { \\epsilon _ a - \\epsilon _ b \\in \\Phi ^ + } \\atop a \\neq h } E _ { a , b } ^ { ( A _ { a , b } ) } \\ , . v \\ ; \\bigg | \\ ; A \\in P ( m | n ) \\bigg \\} \\\\ & = \\span \\bigg \\{ \\bigg ( \\prod _ { h + 1 \\leq b \\leq m + n } E _ { h , b } ^ { ( A _ { h , b } ) } \\bigg ) \\ ; v \\bigg | \\ ; A _ { h , b } \\in \\mathbb N \\bigg \\} . \\end{aligned} \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} & U _ { S W } ( \\theta , P _ { B S } , p _ { s } , \\rho _ { e } ) = \\mu T _ { s } - \\theta ( A P _ { B S } ^ { 2 } + B P _ { B S } ) \\\\ & ~ ~ ~ ~ ~ ~ = \\mu ( 1 - \\theta ) \\bigg [ \\log \\bigg ( 1 + \\frac { p _ { s } | h _ { s } | ^ { 2 } } { \\sigma _ { s } ^ { 2 } } \\bigg ) \\\\ & ~ ~ ~ ~ ~ ~ - \\log ( 1 + \\rho _ { e } ) \\bigg ] - \\theta ( A P _ { B S } ^ { 2 } + B P _ { B S } ) . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} ( T g ) ^ { i } = g ^ { - 1 } ( T ^ i ) T \\in Y ^ { \\lambda } . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( N ) } ( f ) = \\sum _ { j = 1 } ^ { m _ { N } } P ( A _ { j } ^ { ( N ) } ) \\mathbb { E } _ { n } ^ { ( N - 1 ) } ( f | A _ { j } ^ { ( N ) } ) , P ( f ) = \\sum _ { j = 1 } ^ { m _ { N } } P ( A _ { j } ^ { ( N ) } ) \\mathbb { E } ( f | A _ { j } ^ { ( N ) } ) . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} \\int _ { \\mathcal R _ j } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h = ( - 1 ) ^ j \\chi _ { \\psi } ( p ^ j ) p ^ { 3 j / 2 } \\int _ { \\mathcal R _ j } ( c d , p ^ j ) _ p d h . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} C ' _ r = \\sum _ s F ^ { ( m ) } _ r ( s ) B _ s . \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} r _ i : = \\frac { ( \\alpha _ i , \\alpha _ i ) } { 2 } \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} V ( s , t , \\omega ) = D _ s \\sigma ( t , \\omega , X ( t ) ( \\omega ) ) = g ( s ) \\ 1 _ { ( 0 , t ) } ( s ) \\Rightarrow \\int _ s ^ t V ( s , r , \\omega ) d W ( r ) = g ( s ) [ W ( t ) - W ( s ) ] . \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} S \\left ( p _ { 1 } + p _ { 2 } , k \\right ) = c _ { 1 } ^ { - p _ { 1 } } c _ { 2 } ^ { - p _ { 2 } } \\sum _ { j _ { 1 } = 0 } ^ { p _ { 1 } } \\sum _ { j _ { 2 } = 0 } ^ { p _ { 2 } } \\binom { p _ { 1 } } { j _ { 1 } } \\binom { p _ { 2 } } { j _ { 2 } } \\left ( - d _ { 1 } \\right ) ^ { p _ { 1 } - j _ { 1 } } \\left ( - d _ { 2 } \\right ) ^ { p _ { 2 } - j _ { 2 } } S _ { c _ { 1 } , d _ { 1 } } ^ { c _ { 2 } , d _ { 2 } , j _ { 2 } } \\left ( j _ { 1 } , k \\right ) . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c c } \\mathbb { E } \\left [ \\frac { 1 } { 2 } P ( \\tau ) \\hat { \\xi } _ { \\tau } ( \\tau ) ^ { 2 } | \\mathcal { F } _ { s } ^ { t } \\right ] ( \\omega _ { 0 } ) & = \\mathbb { E } \\left [ \\frac { 1 } { 2 } P ( s ) \\int _ { s } ^ { \\tau } \\sigma ( r ) ^ { 2 } d r | \\mathcal { F } _ { s } ^ { t } \\right ] ( \\omega _ { 0 } ) + o ( \\left \\vert \\tau - s \\right \\vert ) . \\end{array} \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} \\Theta ( \\pi _ v \\otimes \\chi _ D , \\psi _ v ^ D ) = \\begin{cases} \\tilde { \\pi } _ v ^ + & D \\in \\Q _ v ^ + ( \\pi _ v ) , \\\\ \\tilde { \\pi } _ v ^ - & D \\in \\Q _ v ^ - ( \\pi _ v ) . \\end{cases} \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\nabla ( W _ j ) = t ^ { - 1 } \\delta ( a _ j ) . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} I _ \\lambda \\leqq E _ { + \\infty } ( e _ k { \\bf { h } } _ { k , 1 } ) = e ^ 2 _ k E _ { + \\infty } ( { \\bf { h } } _ { k , 1 } ) < E _ { + \\infty } ( { \\bf { h } } _ { k , 1 } ) . \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\Lambda _ 1 = \\frac { \\theta ^ { n - 1 } } { 1 + ( 1 - 2 R ) q ^ { n - 1 } } \\frac { 1 + q ^ { n - 1 } } { 1 - q ^ { n - 1 } } , \\Lambda _ 2 = \\frac { q ^ { n - 1 } } { 1 + ( 1 - 2 R ) q ^ { n - 1 } } . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} B _ t : = \\sum _ { i , j } \\varepsilon _ { i j , t } X _ { i , t } \\frac { \\partial } { \\partial X _ { i , t } } \\wedge X _ { j , t } \\frac { \\partial } { \\partial X _ { j , t } } , \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} L ( 1 , \\pi _ p , \\mathrm { a d } ) = \\frac { p ^ 2 } { p ^ 2 - 1 } , L ( 1 / 2 , \\pi _ p ) = \\frac { 1 } { ( 1 + w _ p p ^ { - 1 } ) } = \\frac { p } { p + w _ p } = \\frac { p } { p - 1 } , \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} r _ { \\Omega , q } = \\max \\{ q ( x ) : \\ , \\ , x \\in \\Omega \\} . \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\delta = \\max \\lbrace \\beta ^ * ; ( \\langle b \\rangle + \\varepsilon ) g _ d ( h , r ) , \\rbrace , \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} \\nabla ^ k _ { \\ ! \\ ! D } \\sigma = 0 . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\left ( c _ + + ( - 1 ) ^ { | m | + | n | } c _ - \\right ) = \\left ( c _ + + ( - 1 ) ^ { | m | + | n | } c _ - \\right ) \\sum _ { k = 0 } ^ N \\tbinom { N } { k } ( - 1 ) ^ { k } = 0 . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} \\int \\varphi _ { p } \\frac { d } { d t } \\| u _ { \\geq p } \\| _ 2 ^ 2 d t = \\| u _ { \\geq p } ( T ) \\| _ 2 ^ 2 - \\int \\frac { d } { d t } \\varphi _ { p } \\| u _ { \\geq p } \\| _ 2 ^ 2 d t , \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} \\langle \\textbf { K } \\left [ f \\right ] ( h ) , h \\rangle & = \\int _ E \\textbf { K } \\left [ f \\right ] ( h ) ( x ) h ( x ) d x \\\\ & = \\int _ E \\int _ E \\sqrt { f ( x ) } K ( x , y ) \\sqrt { f ( y ) } h ( y ) h ( x ) d y d x \\\\ & = \\int _ E \\int _ E \\sqrt { \\frac { f ( x ) } { g ( x ) } } \\sqrt { g ( x ) } K ( x , y ) \\sqrt { g ( y ) } \\sqrt { \\frac { f ( y ) } { g ( y ) } } h ( y ) h ( x ) d y d x \\\\ & = \\langle \\textbf { K } \\left [ g \\right ] \\left ( l h \\right ) , l h \\rangle \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} \\begin{array} { c c } \\alpha _ { 1 } ( U , V ) & = g ( \\overset { \\mu _ { 0 } } { h } ( U , V ) , \\tilde { J } N ) , \\\\ \\alpha _ { 2 } ( U , V ) & = g ( \\overset { \\mu _ { 0 } } { h } ( U , V ) , \\tilde { J } E ) , \\\\ \\alpha _ { 3 } ( U , V ) & = g ( \\overset { \\mu _ { 0 } } { h } ( U , V ) , N ) , \\end{array} \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} \\dot { q } ^ { \\Lambda , N } _ t ( \\eta ) = \\int _ { \\Gamma _ 0 } \\dot { R } ^ { \\Lambda , N } _ t ( \\eta \\cup \\xi ) \\lambda ( d \\xi ) . \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} \\log _ p a = \\sum _ { j = 1 } ^ { + \\infty } \\frac { ( - 1 ) ^ { j - 1 } } { j } ( a - 1 ) ^ j = \\sum _ { j = 1 } ^ { + \\infty } \\frac { ( - 1 ) ^ { j - 1 } b ^ j p ^ j } { j } . \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} C _ k ( s ) = \\frac { \\pi ^ { n ( n + 1 ) / 2 } } { \\prod _ { m = 1 } ^ { n } ( m - 1 ) ! } 2 ^ { - n ( 2 n + 1 ) s + 3 n / 2 } \\ , \\frac { \\gamma _ n ( ( 2 n + 1 ) s - \\frac 1 2 + k ) } { \\prod _ { j = 1 } ^ n \\beta ( k - j , s ) } , \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\mathcal { K } ^ \\star _ \\alpha = \\{ k \\in \\mathcal { K } _ \\alpha : k ( \\varnothing ) = 1 \\ { \\rm a n d } \\ \\langle \\ ! \\langle G , k \\rangle \\ ! \\rangle \\geq 0 \\ { \\rm f o r } \\ { \\rm a l l } \\ G \\in B ^ \\star _ { \\rm b s } ( \\Gamma _ 0 ) \\} . \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\lambda = \\kappa _ 2 - \\kappa _ 1 + { \\i \\over 2 } \\Omega + \\i m '' \\omega _ 1 + \\i n '' \\omega _ 2 , m '' , n '' \\geq 0 \\ ; . \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} \\frac { G ( 1 + z \\ , | \\ , \\tau ) } { G ( 1 + z - k \\ , | \\ , \\tau ) } = \\prod \\limits _ { j = 0 } ^ { k - 1 } \\Gamma \\bigl ( \\frac { z - j } { \\tau } \\bigr ) . \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} C ^ + _ { \\partial _ { e x } \\Lambda } : = \\{ x \\in \\Lambda : x \\overset { \\mathbb { Z } ^ d } { \\longleftrightarrow } \\partial _ { e x } \\Lambda \\omega ^ + \\mathbb { P } _ { \\Lambda , + , \\beta , H } ^ { | \\vec { h } | } \\} . \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\nu _ k ^ * \\omega _ { \\log , I _ 1 , \\ldots , I _ k } ( a _ { I _ 1 } , \\ldots , a _ { I _ k } ) = F \\cdot \\widetilde { \\omega } _ { \\log , I _ 1 , \\ldots , I _ k } ( a _ { I _ 1 } , \\ldots , a _ { I _ k } ) . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} M _ { i j } ( t ) : = [ \\lambda _ { a , b } ( t ) ] _ { a , b \\in Q _ 1 ^ { ( i j ) } } , \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} ( S _ { 0 , \\alpha } ( t ) G ) ( \\eta ) = \\exp \\left ( - t \\Psi _ \\upsilon ( \\eta ) \\right ) G ( \\eta ) . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} E _ { + \\infty } ( { \\bf { h } } _ { n _ k } ) = E _ { + \\infty } ( { \\bf { h } } _ { k , 1 } ) + E _ { + \\infty } ( { \\bf { h } } _ { k , 2 } ) + O ( \\epsilon ) . \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} \\psi ( n ) : = \\sum _ { s = 1 } ^ m \\varphi ( 2 \\big \\lfloor ( s + 1 ) / 2 \\big \\rfloor - 1 , 4 s + 2 ) , \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} = \\sum _ { i = 1 } ^ n [ \\alpha _ 1 ( x _ 1 ) , \\cdots , \\alpha _ { i - 1 } ( x _ { i - 1 } ) , [ x _ i , y _ 1 , \\cdots , y _ { n - 1 } ] , \\alpha _ i ( x _ { i + 1 } ) , \\cdots \\alpha _ { n - 1 } ( x _ n ) ] . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} & \\omega < 0 ; \\\\ ( \\rho _ { 0 } , \\rho _ { 1 } , v _ { 0 } ) & = ( 1 4 . 5 9 2 1 0 3 9 1 \\cdot \\omega ^ { 2 } , - 1 . 3 0 2 5 7 7 8 7 7 \\cdot \\omega ^ { - 1 } , - 0 . 0 7 9 8 8 5 4 9 2 8 1 \\cdot \\omega ^ { - 3 } ) \\\\ & \\omega > 0 ; \\\\ ( \\rho _ { 0 } , \\rho _ { 1 } , v _ { 0 } ) & = ( 2 . 7 6 3 0 7 5 6 6 1 \\cdot \\omega ^ { 2 } , 1 . 2 2 7 8 6 3 2 1 7 \\cdot \\omega ^ { - 1 } , 0 . 5 5 0 5 4 8 7 7 8 5 \\cdot \\omega ^ { - 3 } ) \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} d ( a b x , x ) = d ( b x , a ^ { - 1 } x ) = 2 d ( a x , x ) = 2 d ( b x , x ) = 2 L ( { a , b } ) . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} t = \\frac { b } { \\langle \\mu , \\alpha \\rangle } - \\sum _ { k = 1 } ^ { d - 1 } \\frac { \\mu _ k } { \\langle \\mu , \\alpha \\rangle } x _ k , t = \\frac { c } { \\langle \\nu , \\alpha \\rangle } - \\sum _ { k = 1 } ^ { d - 1 } \\frac { \\nu _ k } { \\langle \\nu , \\alpha \\rangle } x _ k , \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} \\tilde { \\phi } ( m + n ) = c _ + + ( - 1 ) ^ { | m | + | n | } c _ - + \\left ( S ( m , n ) T \\right ) , \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\Lambda _ { j , n } ^ { ( k ) } ( x ; \\ell ) = ( - 1 ) ^ { k } \\det \\begin{pmatrix} C _ { k , n } ^ { \\ell } ( x ) & A _ { \\nu , n } ^ { j , \\ell } ( x ) \\\\ [ 3 m m ] D _ { k , n } ^ { \\ell } ( x ) & B _ { \\nu , n } ^ { j , \\ell } ( x ) \\end{pmatrix} , \\ \\ \\nu = 1 , 2 , \\ \\ k = 1 , 2 , \\ell = 1 , 2 . \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + \\ , { \\rm c o n s t } ) } \\ , F ( q \\ , | \\ , \\beta = 1 , \\lambda _ 1 , \\lambda _ 2 ) , \\ ; N \\rightarrow \\infty \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} & 2 ^ { r _ n } \\sup _ { s \\in [ 0 , T ] } | G _ p ( s ) - G _ p ^ { ( l ) , n } ( s ) | = o _ { \\mathbb { P } } ( 1 ) , \\\\ & 2 ^ { r _ n } \\sup _ { s , t \\in [ 0 , T ] , | t - s | \\leq | \\pi _ n | _ T } | G _ p ^ { ( l ) , n } ( t ) - G _ p ^ { ( l ) , n } ( s ) | = o _ { \\mathbb { P } } ( 1 ) . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} V \\big ( \\tau _ 1 ^ * \\cdots \\tau _ n ^ * \\big ) f = \\frac { 1 } { \\sigma \\big ( \\tau _ 1 \\dots \\tau _ n \\big ) } \\big ( D ^ n f \\big ) \\big ( V ( \\tau _ 1 ^ * ) , \\dots , V ( \\tau _ n ^ * ) \\big ) , \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} R ( \\rho ) = \\exp \\left ( - \\sum _ { i > 0 } v _ { i } L _ { i } \\right ) v _ { 0 } ^ { - L _ { 0 } } , \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} p ^ * ( Y _ { j , k } ) = 1 + p ^ * ( X _ { j , k } Y _ { j - 1 , k } ) = 1 + \\frac { \\Delta _ { I ( j + 1 , k ) } \\Delta _ { J ( j , k ) } } { \\Delta _ { I ( j , k - 1 ) } \\Delta _ { I ( j + 1 , k + 1 ) } } = \\frac { \\Delta _ { I ( j , k ) } \\Delta _ { J ( j + 1 , k ) } } { \\Delta _ { I ( j , k - 1 ) } \\Delta _ { I ( j + 1 , k + 1 ) } } , \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} \\alpha \\ge \\alpha _ 0 ( N , m , p , h ) > 0 \\ , , \\beta = \\frac { \\alpha ( m - 1 ) } 2 \\ , . \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} d = a , c = a ^ { - 1 } b a , ( a b ) ^ 2 = ( b a ) ^ 2 . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} B ( x , y ) = \\frac { \\Gamma ( x ) \\Gamma ( y ) } { \\Gamma ( x + y ) } . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} q _ n = A ( q _ { n - 1 } ) , n \\geq 1 . \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} A _ s & = a _ { 2 s , 1 } + a _ { 2 s , - 1 } + a _ { 2 s - 1 } , & A ' _ r & = a ' _ { 2 r , 1 } + a ' _ { 2 r , - 1 } + a ' _ { 2 r - 1 } , \\\\ B _ s & = a _ { 2 s , 1 } + a _ { 2 s , - 1 } + a _ { 2 s + 1 } , & B ' _ r & = a ' _ { 2 r , 1 } + a ' _ { 2 r , - 1 } + a ' _ { 2 r + 1 } , \\\\ C _ s & = q ^ { - s } ( a _ { 2 s , 1 } - a _ { 2 s , - 1 } ) , & C ' _ r & = \\frac { \\alpha _ { - 1 } } { \\beta _ r } a ' _ { 2 r , 1 } - \\frac { \\alpha _ { 1 } } { \\beta _ r } a ' _ { 2 r , - 1 } , \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} Q ( z , y ) ~ = ~ \\P _ z ( X ( t ) = y \\ ; \\ ; t < \\tau _ \\vartheta ) , x , y \\in E . \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} P = 2 \\frac { \\dot { \\rho } } { \\rho } , Q = \\frac { 2 \\omega v } { \\rho ^ { 2 } } , R = - \\frac { 4 } { \\rho ^ { 3 } } , \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\mathbb { P } [ \\sum _ { j = 1 } ^ k d ^ { e _ j } _ { i } = 1 ] = \\frac { 1 } { 2 } ( 1 - \\sigma ^ k ) \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} Q _ 2 ( \\lambda ) = \\begin{bmatrix} \\lambda ^ 2 P _ 6 + \\lambda P _ 5 + P _ 4 & \\lambda ^ 2 P _ { 1 0 } + \\lambda P _ 9 & 0 & \\lambda ^ 2 P _ 8 + \\lambda P _ 7 & I _ n \\\\ \\lambda P _ 1 + P _ 0 & 0 & - \\lambda ^ 2 I _ n & 0 & 0 \\\\ 0 & - I _ n & 0 & \\lambda ^ 2 I _ n & 0 \\\\ \\lambda P _ 3 + P _ 2 & 0 & I _ n & 0 & - \\lambda ^ 2 I _ n \\\\ \\lambda ^ 2 I _ n & 0 & 0 & - I _ n & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} \\tau = \\rho _ 1 \\circ \\cdots \\circ \\rho _ N . \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} S _ 1 = \\left ( P _ 1 ( 1 ) + P _ 3 ( 1 ) + \\frac { \\theta _ 2 } { 2 } \\widetilde { P _ 2 } ( 1 ) + o ( 1 ) \\right ) \\sum _ { q } \\Psi \\left ( \\frac { q } { Q } \\right ) \\frac { q } { \\varphi ( q ) } \\varphi ^ + ( q ) , \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} r Y _ { 2 } ( r ) + ( 2 \\sigma { r } + 1 + \\tau ) Y _ { 1 } ( r ) + \\Big [ \\Big ( \\sigma ^ { 2 } - ( \\alpha ^ { 2 } + \\gamma ) \\Big ) r + \\sigma ( 1 + \\tau ) + \\beta \\Big ] Y ( r ) = 0 . \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\tilde { \\varphi } \\left ( \\left [ \\left ( \\begin{smallmatrix} a & \\ast \\\\ 0 & a ^ { - 1 } \\end{smallmatrix} \\right ) , \\epsilon \\right ] g \\right ) = \\epsilon \\chi _ { \\psi } ( a ) \\mu ( a ) | a | _ p \\tilde { \\varphi } ( g ) = \\epsilon \\chi _ { \\psi } ( a ) \\chi _ { \\delta } ( a ) | a | ^ { 3 / 2 } _ p \\tilde { \\varphi } ( g ) \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} w _ M ( t ) = w ( x _ t , t ) . \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} \\widetilde b _ { d j } ^ * ( z ) = ( z ^ { - 1 } - 1 ) \\ , g _ { d j } ^ * ( z ) = : ( z ^ { - 1 } - 1 ) ^ { d - j } \\ , h _ { d j } ^ * ( z ^ { - 1 } ) , j = 0 , \\dots , d . \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} [ \\beta ] = \\sum _ { S \\subset X } [ { \\jmath _ S } _ { ! * } \\jmath _ S ^ * \\imath _ S ^ { ! * } \\beta ] \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} x \\wedge p ^ { \\prime } = x \\wedge p ^ { \\prime } \\wedge p = y \\wedge p ^ { \\prime } \\wedge p = y \\wedge p ^ { \\prime } \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} ( x , t ) \\simeq ( x ' , t ' ) \\ \\Leftrightarrow \\ x ' = - x , t ' = - t . \\ \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} w _ 1 & \\mapsto w _ 1 , \\\\ w _ 2 & \\mapsto r ^ 2 w _ 2 , \\\\ \\tilde { w } & \\mapsto r \\tilde { w } , \\ ; \\mbox { f o r } \\ ; \\tilde { w } = ( w _ 3 , \\ldots , w _ n ) , \\\\ w _ { n + 1 } & \\mapsto r ^ 2 w _ { n + 1 } , \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & A _ 5 & A _ 6 & B _ 6 & B _ 7 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} \\ I ^ { \\varepsilon } ( x ) = \\left \\{ \\begin{array} { l l } \\Big \\{ u \\in U \\ , : \\ , F _ { u } ( x , 0 _ { u } ) \\geq p ( x ) - \\varepsilon \\Big \\} , & \\mathrm { i f } p ( x ) \\in \\mathbb { R } , \\\\ \\ \\ \\ \\emptyset , & \\mathrm { i f } p ( x ) \\not \\in \\mathbb { R } . \\end{array} \\right . \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} P T _ u g + A _ u g ~ = ~ T _ u P g ~ = ~ T _ u ( g - \\ 1 ) ~ = ~ T _ u g - \\ 1 \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} c _ 5 \\sum _ { i = 1 } ^ n \\lambda _ i ( p ) \\leq | p | \\leq c _ 6 \\sum _ { i = 1 } ^ n \\lambda _ i ( p ) \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} D ( f g ) & = \\sum _ { j , k , l } \\partial _ j ( f _ k g _ l ) e _ j e _ k e _ l = \\sum _ { j , k , l } ( \\partial _ j f _ k g _ l + f _ k \\partial _ j g _ l ) e _ j e _ k e _ l \\\\ & = ( D f ) g + \\sum _ { j , k , l } f _ k \\partial _ j e _ j e _ k g _ l e _ l = ( D f ) g + \\left ( \\sum _ { j , k } f _ k \\partial _ j e _ j e _ k \\right ) g , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} x _ { n + 1 } = A _ n x _ n , n \\in \\Z . \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\nu _ c = ( - 1 ) s \\left ( \\begin{array} { c c } 1 & - c p \\\\ 0 & 1 \\end{array} \\right ) s . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} \\varphi ( r ) = A e ^ { - 2 \\eta { r } } r ^ { - ( 1 + \\tau + \\mathrm { a } ) } , \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\inf _ { x _ t \\in \\mathbb { R } ^ n } - \\mathbb { E } [ \\sum \\limits _ { i = 1 } ^ { n + 1 } \\xi _ { t + 1 } ( i ) x _ { t } ( i ) ] \\\\ x _ t \\in X _ t ( x _ { t - 1 } ^ k , { \\tilde \\xi } _ { t } ^ k ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } W , U ) = \\breve { g } ( W , \\breve { J } U ) \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} d _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x ( b ) + d _ 2 ~ ^ { A B R } D _ b ^ \\alpha x ( b ) = 0 , \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} P \\big ( \\big \\{ \\alpha \\in \\Omega \\colon \\lambda _ m \\in V ( \\alpha ) \\cup V ' ( \\alpha ) \\big \\} \\big ) = 0 . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} d x ( t ) = a ( t ) d t + \\sum _ { i = 1 } ^ v b _ i ( t ) d M _ i , \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} u _ t & = \\nabla \\cdot \\left ( m \\ , U ^ { m - 1 } \\nabla u \\right ) \\qquad \\mbox { i n $ \\R ^ n \\times ( 0 , \\infty ) $ } \\\\ u ( x , 0 ) & = u _ 0 ( x ) \\qquad \\forall x \\in \\R ^ n \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} W ^ { ( 1 ) } - W ^ { ( 3 ) } = \\sum _ { i = 0 } ^ { k - 1 } b _ i ( 1 + 1 4 y ^ 4 + y ^ 8 ) ^ { 3 k - 1 - 3 i } ( y ^ 4 ( 1 - y ^ 4 ) ^ 4 ) ^ i f ( y ) \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} A _ r & = \\begin{pmatrix} 1 . 8 9 6 5 & 0 . 0 2 3 7 & 0 . 7 7 7 8 \\\\ 0 . 0 2 3 7 & 3 . 1 5 5 4 & 1 . 8 0 0 9 \\\\ 0 . 7 7 7 8 & 1 . 8 0 0 9 & 3 . 1 7 8 4 \\end{pmatrix} , \\\\ B _ r & = \\begin{pmatrix} - 0 . 2 6 7 7 & 1 . 1 8 2 0 \\\\ 1 . 5 1 2 4 & 0 . 2 0 4 9 \\\\ - 0 . 7 7 5 9 & 1 . 2 1 5 5 \\end{pmatrix} , \\\\ C _ r & = \\begin{pmatrix} 0 . 8 7 2 6 & 0 . 1 5 0 3 & - 0 . 0 6 3 0 \\\\ 0 . 3 3 2 1 & 0 . 0 6 8 0 & 1 . 3 1 2 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} h ( x ) = \\begin{cases} Q ( x ) , & x \\ge 0 \\\\ Q ^ { - 1 } ( x ) & x < 0 . \\end{cases} \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} y ^ { k + 1 } & = P _ C ( x ^ k - \\lambda A y ^ k ) . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} \\sum _ { v _ i h } f ( v _ i ) = \\sum _ { v ^ j h } f ( v ^ j ) . \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} \\mathcal M _ i = \\Omega ( x ) \\vec { \\mathfrak u _ i } \\ \\ \\ \\ \\mathcal N _ i = \\vec { \\mathfrak w _ i } \\Omega ( x ) . \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big ( L ( \\rho ) ^ { \\dd } \\big ( \\dot h - L ( h ) L ( \\rho ) ^ { \\dd } \\dot \\rho \\big ) \\Big ) = \\Big ( \\nabla _ G L ( \\rho ) ^ { \\dd } \\dot \\rho \\circ \\nabla _ G L ( \\rho ) ^ { \\dd } \\big ( \\dot h - L ( h ) L ( \\rho ) ^ { \\dd } \\dot \\rho \\big ) \\Big ) . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} H : { \\mathcal Z } ( { { \\mathcal G } } ( { \\mathcal X } ) ) [ 1 ] \\to { \\mathcal C } ^ \\infty ( { \\mathcal X } ) [ 1 ] , \\ \\ \\ H = \\sum _ { i \\in S } ( \\phi ^ * \\chi _ i ) \\tau _ i ^ * \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\begin{array} { c c c } S ( \\alpha ) = \\alpha ^ { * } , & S ( \\alpha ^ { * } ) = \\alpha , & S ( \\gamma ) = - q \\gamma , \\\\ & S ( \\gamma ^ { * } ) = - q ^ { - 1 } \\gamma ^ { * } . \\end{array} \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} t _ b t _ c t _ a t _ b \\cdot t _ { \\beta } t _ { \\gamma } t _ { \\alpha } t _ { \\beta } ( a ) & = t _ b t _ c t _ a t _ b ( a ) = c , \\\\ t _ b t _ c t _ a t _ b \\cdot t _ { \\beta } t _ { \\gamma } t _ { \\alpha } t _ { \\beta } ( \\alpha ) & = t _ b t _ c t _ a t _ b ( \\gamma ) = \\gamma \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} [ P ( t ) ] = \\left [ \\begin{pmatrix} x \\cdot t \\\\ y \\cdot t \\\\ 1 \\end{pmatrix} \\right ] = \\left [ \\begin{pmatrix} x \\\\ y \\\\ 1 / t \\end{pmatrix} \\right ] . \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} \\varphi _ n ( x , t ) = \\frac { 1 } { ( r _ n ^ 2 - t ) ^ { \\frac { d } { 2 } } } \\exp { \\bigg \\{ - \\frac { | x | ^ 2 } { 4 ( r _ n ^ 2 - t ) } \\bigg \\} } . \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\mathcal { Q } _ { t } ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , 1 ) = \\mathbb { E } _ { \\xi _ t , D _ t } \\Big [ \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , 1 , \\xi _ t , D _ t ) | D _ { t - 1 } = 1 , \\xi _ { [ t - 1 ] } \\Big ] \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} \\psi _ g ^ \\omega ( V _ x \\otimes V _ { y } ) & : = \\mu _ g ( x , y ) \\psi _ h ( V _ x \\otimes V _ { y } ) , \\\\ \\phi ^ \\omega ( g , h ) ( V _ x ) & : = \\gamma _ { g , h } ( x ) \\phi ( g , h ) ( V _ x ) , \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { B _ { r _ { k + 1 } } } v \\leq \\sup _ { B _ { \\tau _ 2 r _ { k } } } v & \\leq ( 1 - \\mu ) \\left ( \\hat { C } M r _ { k } ^ { \\beta } - M r _ { k } ^ { \\alpha } \\right ) + M r _ { k } ^ { \\alpha } \\\\ & \\leq \\hat { C } M r _ { k + 1 } ^ { \\beta } \\cdot \\frac { 1 - \\mu } { \\tau _ 1 ^ { \\beta } } + \\mu M r _ { k } ^ { \\beta } \\\\ & \\leq \\hat { C } M r _ { k + 1 } ^ { \\beta } \\left ( \\frac { 1 - \\mu } { \\tau _ 1 ^ { \\beta } } + \\frac { \\mu } { \\hat { C } \\tau _ 1 ^ { \\beta } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} T _ { \\max } ( \\alpha _ 1 ) = \\max _ { \\alpha _ 2 > \\alpha _ 1 } T ( \\alpha _ 2 , \\alpha _ 1 ) = \\exp \\left ( - \\alpha _ 1 - \\delta ( \\alpha _ 1 ) \\right ) / \\langle a \\rangle . \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} x = \\frac { 1 + x ( 2 ) } { 2 } v _ 1 + \\sum _ { i = 2 } ^ 3 \\frac { x ( i + 1 ) - x ( i ) } { 2 } v _ i + \\frac { 1 - x ( 4 ) } { 2 } v _ 4 . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { D _ \\varepsilon } \\theta _ k ^ 2 \\leq C _ { \\min } ( \\alpha , \\beta ) \\varepsilon ^ 2 \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} \\sigma _ s \\colon R [ [ t ] ] & \\rightarrow R [ [ t ] ] \\\\ a ( t ) & \\mapsto a ( t ) ^ { \\sigma _ { s } } : = a ( s ( t ) ) \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} \\frac { \\hat { \\vec x } } { \\norm { \\hat { \\vec x } } } = \\vec 1 _ { S ^ * } - \\vec 1 _ { V \\setminus S ^ * } \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\begin{pmatrix} - a _ 1 + q a & - b _ 2 - q b & c \\end{pmatrix} = p \\begin{pmatrix} - a ' _ 1 & - b ' _ 2 & c ' \\end{pmatrix} \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\mathcal W _ { \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) = A _ { \\chi } ( B ) \\det ( Y ) ^ { ( k + 1 ) / 2 } e ^ { 2 \\pi \\sqrt { - 1 } \\mathrm { T r } ( B Z ) } , \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} | J | \\le \\frac { 2 e ( \\overline { G } [ X _ 1 , X _ 2 , X _ 3 ] ) } { \\gamma ^ { 1 / 8 } n } = 2 \\gamma ^ { 1 / 8 } n . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\vec { \\mathfrak u _ i } = \\sum _ { j = 0 } ^ { \\ell _ i } \\partial _ x ^ j \\vec u _ { j i } ( x ) \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} ( a b ^ i ) ^ p = a ^ p ( b ^ i ) ^ { a ^ { p - 1 } } \\dots ( b ^ i ) ^ a b ^ i = b _ { p - 1 } ^ i \\dots b _ 1 ^ i b _ 0 ^ i , \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} \\binom { m + 1 } { m + 1 - ( s + r ) } \\binom { s + r } { s + r - a _ { 1 } } \\cdots \\binom { s + r - a _ { 1 } - \\cdots - a _ { i - 1 } } { s + r - a _ { 1 } - \\cdots - a _ { i - 1 } - a _ { i } } = \\frac { ( m + 1 ) ! } { ( m + 1 - s - r ) ! a _ { 1 } ! \\cdots a _ { i } ! s ! } . \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\widetilde \\nabla = d + \\left ( \\sum _ { i = 0 } ^ \\ell \\psi _ i \\check f _ i + b \\right ) d t \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} { m _ { { v _ { n + 1 } } \\nwarrow } } & = { m _ { \\swarrow { v _ { n + 1 } } } } { m _ { \\uparrow { v _ { n + 1 } } } } \\\\ { m _ { \\swarrow { v _ n } } } & = \\int d { { \\tilde \\theta } _ { n + 1 } } \\sum \\limits _ { { s _ { n + 1 } } } { { m _ { { v _ { n + 1 } } \\nwarrow } } } \\psi \\left ( { { v _ n } , { v _ { n { \\rm { + 1 } } } } } \\right ) \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} T _ x P ^ n ~ = ~ P ^ n T _ x + \\sum _ { k = 1 } ^ n P ^ { k - 1 } A _ x P ^ { n - k } , \\forall n \\geq 1 , \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} w = \\prod _ { { \\epsilon _ a - \\epsilon _ b \\in \\Phi ^ + } \\atop a \\neq h } E _ { b , a } ^ { ( A _ { b , a } ) } \\bigg ( \\prod _ { h + 1 \\leq b \\leq m + n } E _ { b , h } ^ { ( A _ { b , h } ) } \\ , . \\mathfrak { m } _ \\lambda \\bigg ) \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} ( n + 1 ) \\sum _ { k = 0 } ^ { N } \\tbinom { N } { k } ( - 1 ) ^ { k } \\mathfrak { d } _ 1 \\dot { \\phi } ( n + k ) & = \\sum _ { k = 0 } ^ N \\tbinom { N } { k } ( - 1 ) ^ k \\dot { \\psi } ( n + k ) \\\\ & + N \\sum _ { k = 0 } ^ { N - 1 } \\tbinom { N - 1 } { k } ( - 1 ) ^ k \\mathfrak { d } _ 1 \\dot { \\phi } ( n + k + 1 ) , \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} d _ { 0 } ( M ) = \\frac { 1 } { \\left | B _ { \\frac { 1 } { 8 } \\cdot 2 ^ { n - 1 } } ( M ) \\right | } \\int \\limits _ { B _ { \\frac { 1 } { 8 } \\cdot 2 ^ { n - 1 } } ( M ) } d _ { 2 } ( \\widetilde { M } ) \\ , d m _ { 3 } ( \\widetilde { M } ) , \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} J _ { \\delta } \\setminus \\{ ( \\imath _ 0 , r _ 0 + \\frac { 1 } { 2 } ) , ( \\imath _ 0 - 1 , r _ 0 ) , ( \\imath _ 0 , r _ 0 - \\frac { 1 } { 2 } ) \\} = J _ { \\beta ( \\imath _ 0 , r _ 0 ) ( \\delta ) } \\setminus \\{ ( \\imath _ 0 - 1 , r _ 0 + \\frac { 1 } { 2 } ) , ( \\imath _ 0 , r _ 0 ) , ( \\imath _ 0 - 1 , r _ 0 - \\frac { 1 } { 2 } ) \\} . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} u _ t = \\nabla \\cdot \\left ( m U ^ { m - 1 } \\mathcal { A } \\left ( \\nabla u , u , x , t \\right ) + \\mathcal { B } \\left ( u , x , t \\right ) \\right ) , \\left ( m > 1 \\right ) \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} S _ 1 = 0 . \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} & \\mathcal O _ t ( v + \\bar v ) = 0 , & t \\in \\mathbb N . \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} \\Omega _ p ( \\nu _ { \\gamma } \\beta _ m \\nu _ { \\delta } ) = \\begin{cases} \\frac { p - G ( \\gamma , p ) } { p ( p - 1 ) } & m = 1 , \\ , \\gamma \\delta \\equiv 1 \\pmod p , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} Y ^ n _ t ( x ) \\stackrel { d } = ( \\sqrt { x } + \\xi \\sqrt { t } ) ^ 2 + Y ^ { n - 1 } _ t , \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\sum _ { e : \\ , v \\in e } B ( v , e ) = \\frac { \\sum _ { e : \\ , v \\in e } \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G ) x _ v ^ p } = 1 . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} V _ k : = \\mathrm { I n d } _ { \\widehat { H } _ n ^ k } ^ { H _ n } ( \\Bbbk ) \\cong H _ n \\otimes _ { \\widehat { H } _ n ^ k } \\Bbbk . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( t ) = - \\frac { \\beta } { m - 1 } \\ , ( T - t ) ^ { - \\alpha - 1 } + \\frac { C ^ { m - 1 } m } { a ^ 2 ( m - 1 ) ^ 2 } \\ , ( T - t ) ^ { - \\alpha m + 2 \\beta } \\geq \\frac { C ^ { m - 1 } m } { 2 a ^ 2 ( m - 1 ) ^ 2 } \\ , ( T - t ) ^ { - \\alpha m + 2 \\beta } \\ , ; \\end{aligned} \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\alpha _ { j } : = \\frac { | \\rho _ { j } ( 1 ) | ^ 2 } { \\cosh { \\pi t _ j } } . \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\left | e ^ { i ( x - y ) \\cdot ( A ( 0 ) - A ( \\frac { \\varepsilon x + \\varepsilon y } { 2 } ) ) } - 1 \\right | ^ 2 = 4 \\sin ^ 2 \\left [ \\frac { ( x - y ) \\cdot ( A ( 0 ) - A ( \\frac { \\varepsilon x + \\varepsilon y } { 2 } ) ) } { 2 } \\right ] . \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} H _ 2 & = x _ 1 ^ 2 x _ 2 + b _ { 1 0 } x _ 1 x _ 3 ^ 2 + b _ { 1 1 } x _ 1 x _ 2 x _ 3 + b _ { 1 2 } x _ 1 x _ 2 ^ 2 + b _ 0 ( x _ 2 , x _ 3 ) \\mbox { ; } \\\\ H _ 3 & = c _ { 1 2 } x _ 1 x _ 2 ^ 2 + c _ { 0 0 } x _ 3 ^ 3 + c _ { 0 1 } x _ 2 x _ 3 ^ 2 + c _ { 0 2 } x _ 2 ^ 2 x _ 3 + c _ { 0 3 } x _ 2 ^ 3 \\mbox { . } \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} & \\sigma ( r ) _ s = \\sigma _ { ( k - 1 ) T / 2 ^ r } , s \\in [ ( k - 1 ) T / 2 ^ r , k T / 2 ^ r ) , \\\\ & C ( r ) _ t = \\int _ 0 ^ t \\sigma _ s ( r ) d s . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} C = \\frac { 1 } { 1 - \\alpha } = e ^ { \\lambda t _ 0 } = e ^ { - \\log ( 1 - p _ { m i n } s _ * e ^ { - 2 p _ { \\max } s _ * } ) } = \\frac { 1 } { 1 - p _ { m i n } s _ * e ^ { - 2 p _ { \\max } s _ * } } > 1 . \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} ( a b ) ^ 3 = ( y ^ { - 3 } ) ^ g . \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( a ^ { i j } u _ i ) _ j = f ~ ~ & & \\mbox { i n } ~ ~ \\Omega ; \\\\ & u = g ~ ~ & & \\mbox { o n } ~ ~ \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} \\beta ( m , s ) = B \\Big ( \\Big ( n + \\frac 1 2 \\Big ) s + \\frac 1 4 + \\frac { m } 2 , \\Big ( n + \\frac 1 2 \\Big ) s + \\frac 1 4 - \\frac { m } 2 \\Big ) . \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} & \\ll q ^ \\varepsilon \\sum _ { l < M ^ { 1 - \\Upsilon } } \\frac { 1 } { l } \\sum _ { M ^ { 1 - \\Upsilon } < n < X ( \\log q ) ^ 2 } \\frac { 1 } { n } \\sum _ { c \\equiv 0 ( q ) } \\frac { ( c , n , l ^ 2 ) ^ { 1 / 2 } \\tau ( c ) } { c ^ { 1 / 2 } } \\cdot \\frac { l \\sqrt { n } } { c } \\\\ & \\ll M ^ { 1 - \\Upsilon } q ^ { - 1 + \\varepsilon } \\ll M ^ { - ( 1 - \\Upsilon ) } . \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} \\tilde \\mu _ k ( x ) ~ = ~ \\exp ( \\langle \\alpha _ k , x \\rangle ) \\mu _ k ( x ) , x \\in \\Z ^ d , \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} \\alpha _ n = p ^ n w \\rho _ { - 2 n } w , w = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\bigg ( ( b - a ) \\bigg ( 1 - \\frac { \\beta - f ( a ) } { f ( b ) - f ( a ) } \\bigg ) \\bigg ) . \\bigg ( ( b - a ) \\bigg ( 1 - \\frac { \\beta - g ( a ) } { g ( b ) - g ( a ) } \\bigg ) \\bigg ) = \\beta . \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\Phi _ \\mu ( r ) : = \\int _ 0 ^ r \\int _ 0 ^ s ( 1 + t ) ^ { - \\mu } \\ , d t \\ , d s , \\ ; r \\geq 0 , \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} f _ 1 X _ n - U _ n = \\partial f _ n . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} \\alpha _ { n } ^ { ( N - 1 ) } ( f 1 _ { A } ) - \\mathbb { E } ( f | A ) \\alpha _ { n } ^ { ( N - 1 ) } ( A ) = \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( f 1 _ { A } ) - \\mathbb { E } ( f | A ) \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A ) . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} Q _ { \\alpha _ 2 \\alpha _ 1 } ( t + s ; \\mathbf { B } ) = Q _ { \\alpha _ 2 \\alpha } ( t ; \\mathbf { B } ) Q _ { \\alpha \\alpha _ 1 } ( s ; \\mathbf { B } ) . \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} \\widetilde { c } _ { j i } ( - 2 k ) & = \\widetilde { c } _ { j i } ( - 4 n + 2 + 2 k ) \\ \\ i \\in I \\setminus \\{ n \\} , & \\widetilde { c } _ { j n } ( - 2 k - 1 ) & = \\widetilde { c } _ { j n } ( - 4 n + 3 + 2 k ) . \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\left ( S ( m , n ) T \\right ) = \\left ( S ^ m ( S ^ * ) ^ n T ' \\right ) , \\quad \\forall m , n \\in \\N ^ N . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ { n - 1 } ( v - v ^ { - 1 } ) ^ 2 \\left ( v ^ { 2 ( n - i ) } F _ i K _ { \\varpi _ { i + 1 } } E _ i K _ { \\varpi _ i } + v ^ { - 2 ( n - i - 1 ) } F _ i K _ { - \\varpi _ i } E _ i K _ { - \\varpi _ { i + 1 } } \\right ) + \\\\ & ( v - v ^ { - 1 } ) ^ 2 \\left ( F _ n K _ { - \\varpi _ { n - 1 } } E _ n K _ { \\varpi _ n } + F _ n K _ { - \\varpi _ n } E _ n K _ { \\varpi _ { n - 1 } } \\right ) . \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} T _ { i j } ( \\psi ) : = \\partial _ { i j } \\big ( | x | ^ { - ( d - 2 ) } \\big ) * \\psi \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\sum \\limits _ { i , j = 1 } ^ { n } \\langle \\pi ( x _ { j } ^ { \\# } \\cdot x _ i ) \\xi _ i , \\xi _ j \\rangle = \\sum \\limits _ { i , j = 1 } ^ { n } \\langle \\pi ^ c ( z _ { j } ^ { \\# } \\cdot z _ i ) \\zeta _ i , \\zeta _ j \\rangle \\ge 0 \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} K & = \\{ ( { \\rm s e c } ( \\theta ) , \\theta ) : \\theta \\in ( 0 , \\frac { \\pi } { 4 } ) \\cup ( \\frac { 7 \\pi } { 4 } , 2 \\pi ) \\} \\bigcup \\{ ( \\sqrt { 2 } , \\theta ) : \\theta \\in ( \\frac { \\pi } { 4 } , \\frac { 3 \\pi } { 4 } ) \\cup ( \\frac { 5 \\pi } { 4 } , \\frac { 7 \\pi } { 4 } ) \\} \\\\ & \\bigcup \\{ ( - { \\rm s e c } ( \\theta ) , \\theta ) : \\theta \\in ( \\frac { 3 \\pi } { 4 } , \\frac { 5 \\pi } { 4 } ) \\} . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} & \\norm { \\hat x _ 0 - \\hat x _ * } \\leq R _ x , \\\\ & \\hat x _ i \\in \\hat x _ 0 + \\mathrm { s p a n } \\{ g _ 0 , \\hdots , g _ { i - 1 } \\} , i = 0 , \\dots , N , \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} b _ { s t } = \\begin{cases} 1 & \\ t = s ^ + , \\\\ c _ { i _ s i _ t } & \\ s < t < s ^ + < t ^ + , \\\\ - 1 & \\ t ^ + = s , \\\\ - c _ { i _ s i _ t } & \\ t < s < t ^ + < s ^ + , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } | \\tilde { v } ( t ) | ^ 2 d x + 2 \\int _ { t _ 1 } ^ { t } \\int _ { \\R ^ d } | \\nabla \\tilde { v } | ^ 2 d x d \\tau = \\int _ { \\R ^ d } | u _ { \\infty } ^ 2 ( t _ 1 ) | ^ 2 d x + 2 \\int _ { t _ 1 } ^ { t } \\int _ { \\R ^ d } u _ { \\infty } ^ 1 \\otimes \\tilde { v } : \\nabla \\tilde { v } d x d \\tau \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} \\kappa _ n ( \\delta ) \\triangleq ( - 1 ) ^ n \\frac { d ^ n } { d q ^ n } \\vert _ { q = 0 } \\log { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ^ q ( \\delta ) \\bigr ] . \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\tilde e _ k \\ast K = 2 \\pi { \\rm i } \\frac { k ^ \\perp } { | k | ^ 2 } \\tilde e _ k , \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} \\chi ( T _ { a / c } ) = \\chi \\ ! \\left ( 1 - a c \\frac { N } { \\gcd ( c ^ 2 , N ) } \\right ) . \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} \\varphi _ n ( w , u ) : = \\displaystyle \\int \\limits _ T f ( t , w ( t ) ) d \\mu ( t ) + n \\int \\limits _ T \\| w ( t ) - u \\| ^ p d \\mu ( t ) + \\| x _ 0 - u \\| ^ p + \\delta _ B ( u ) , \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} \\psi _ 3 ( z ) = ( [ a , b ] , [ a , b ] , [ a , b ] ) . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { n - \\delta + 1 } { n - r \\brack \\delta - 1 } A ' _ r = c ^ { n - \\delta + 1 } \\sum _ { s = 0 } ^ n { n - s \\brack n - \\delta + 1 } A _ s . \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\delta _ 0 ( t ) : = \\frac { \\zeta ( t ) } { m - 1 } \\frac { \\eta ' ( t ) } { \\eta ( t ) } \\ , , \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } \\big ( \\big | R - R ( r ) \\big | + \\big | R ( r ) - R _ n ( r ) \\big | + \\big | R _ n ( r ) - R _ n \\big | > \\delta \\big ) = 0 ~ ~ \\forall \\delta > 0 . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} a \\cdot _ \\phi m = \\phi ( a ) m , \\forall a \\in A , \\ \\forall m \\in M . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} U = u ^ 1 + u ^ 2 + \\cdots + u ^ { k } = \\sum _ { i = 1 } ^ { k } u ^ i . \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\mu ( L F ^ \\theta ) = \\int _ { \\Gamma _ 0 } ( L ^ \\Delta k _ \\mu ) ( \\eta ) e ( \\theta ; \\eta ) \\lambda ( d \\eta ) . \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} P = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , A = \\begin{pmatrix} 1 & - 1 \\\\ 8 & 2 \\\\ 4 & 2 \\\\ 2 & 4 \\end{pmatrix} , b = \\begin{pmatrix} - 3 \\\\ 1 1 \\\\ 7 \\\\ 5 \\end{pmatrix} . \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} ( x _ 0 , u x _ 0 ) _ { y _ 0 } & \\leqslant ( u x _ 0 , x _ 0 ) _ { y } + | y - y _ 0 | \\leqslant 1 0 1 d \\delta \\hbox { a n d } \\\\ ( x _ 0 , u x _ 0 ) _ { u y _ 0 } & = ( x _ 0 , u ^ { - 1 } x _ 0 ) _ { y _ 0 } \\leqslant ( u ^ { - 1 } x _ 0 , x _ 0 ) _ z + | z - y _ 0 | \\leqslant 1 0 1 d \\delta . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} f = d _ N + c , \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} r U _ { 2 } ( r ) + 2 \\lambda { U _ { 1 } ( r ) } + \\Big [ - ( \\alpha ^ { 2 } + \\gamma ) r + \\beta + \\Big ( \\lambda ( \\lambda - 1 ) - \\delta \\Big ) r ^ { - 1 } \\Big ] U ( r ) = 0 . \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 - 3 x + b \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} M _ { ( \\tau , \\lambda , \\lambda ) } \\overset { { \\rm i n \\ , l a w } } { = } \\int _ { - 1 / 2 } ^ { 1 / 2 } | 1 + e ^ { 2 \\pi i \\psi } | ^ { 2 \\lambda } \\ , M _ { \\beta } ( d \\psi ) , \\ ; \\tau = 1 / \\beta ^ 2 > 1 . \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta ^ { - q } _ { 1 , 0 } \\bigl ( ( \\tau , 1 + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) \\bigr ) \\bigr ] = \\tau ^ { - \\frac { q } { \\tau } } \\frac { \\Gamma ( - \\frac { q } { \\tau } + 1 + \\lambda _ 1 + \\lambda _ 2 + \\frac { 1 } { \\tau } ) } { \\Gamma ( 1 + \\lambda _ 1 + \\lambda _ 2 + \\frac { 1 } { \\tau } ) } , \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 1 } \\| \\tilde { G } ^ \\rho - G \\| _ { o p } = 0 \\ , . \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} \\mathbf K ( t ) = \\int _ 0 ^ 1 \\frac { \\mathrm { d } s } { \\sqrt { ( 1 - s ^ 2 ) ( 1 - s ^ 2 t ^ 2 ) } } \\mathbf E ( t ) = \\int _ 0 ^ 1 \\sqrt { \\frac { 1 - s ^ 2 t ^ 2 } { 1 - s ^ 2 } } \\ , \\mathrm { d } s , 0 \\leq s \\leq 1 . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\epsilon ( \\chi ) \\sum _ { n } \\frac { \\overline { \\chi } ( n ) } { n ^ { 1 / 2 } } V \\left ( \\frac { n } { q } \\right ) & = \\frac { 1 } { 2 } \\varphi ^ + ( q ) + O ( q ^ { 1 - \\epsilon } ) . \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\mathrm { C o v } \\left ( \\mathbb { G } ^ { ( N ) } ( f ) , \\mathbb { G } ^ { ( N ) } ( g ) \\right ) & = \\mathrm { C o v } \\left ( \\mathbb { G } ( f ) , \\mathbb { G } ( g ) \\right ) - \\sum _ { k = 1 } ^ { N } \\Phi _ { k } ^ { ( N ) } ( f ) ^ { t } \\cdot \\mathbb { V } \\left ( \\mathbb { G } [ \\mathcal { A } ^ { ( k ) } ] \\right ) \\cdot \\Phi _ { k } ^ { ( N ) } ( g ) . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\pi _ { P _ x } ( e ) & = \\pi _ { P _ x } ( x + z + h + f ( z + h ) - x - z - f ( z ) - L ( h ) ) \\\\ & = \\pi _ { P _ x } ( h + f ( z + h ) - f ( z ) - L ( h ) ) \\\\ & = h - \\pi _ { P _ x } ( f ( z + h ) ) - \\pi _ { P _ x } ( f ( z ) ) - \\pi _ { P _ x } ( L ( h ) ) \\\\ & = h - h \\\\ & = 0 , \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} f _ r ( x ) = \\sum _ { j = 1 } ^ { n } \\ , \\ , c _ j ( x + p _ j ) ^ r . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} \\begin{cases} \\dot q _ i = p _ i , \\\\ \\dot p _ i = - \\frac 1 N \\sum _ { j = 1 } ^ N \\nabla W ( p _ i - p _ j ) + u _ { i } , \\end{cases} \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\tau ( a , b , c ) = \\left ( \\frac { a } { 2 } , \\frac { b } { 2 } , \\frac { c } { 2 } + \\frac { a b } { 6 } \\right ) . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\alpha _ I = 2 \\cdots i _ 1 1 \\cdot ( i _ 1 + 2 ) \\cdots ( i _ 1 + i _ 2 ) \\cdot ( i _ 1 + 1 ) \\cdots \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\sup _ { k \\geq 1 } \\sup _ { | y | \\leq R } \\sup _ { | \\xi | \\leq R ^ { - 1 } } | q _ k ( y , \\xi ) | = 0 , \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} | \\Gamma ( k + i t ) | ^ 2 \\asymp \\exp ( - \\pi | t | ) \\begin{cases} \\Gamma ^ 2 ( k ) \\exp ( | t | ( \\pi - g _ 1 ( | t | / ( k - 1 ) ) ) , & 0 \\le | t | \\le k - 1 , \\\\ | t | ^ { 2 k - 1 } \\exp ( | t | g _ 2 ( ( k - 1 ) / | t | ) ) , & k - 1 < | t | . \\\\ \\end{cases} \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } B _ { t } \\circ d B _ { t } = \\displaystyle \\frac { 1 } { 2 } B _ { T } ^ 2 = \\displaystyle \\lim _ { \\substack { n \\longrightarrow + \\infty } } \\sum _ { \\substack { j \\in J \\setminus J _ { K ( n ) } } } \\frac { B _ { t _ { j + 1 } } + B _ { t _ { j } } } { 2 } ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} I _ 1 : { \\mathcal C } ^ \\infty ( { { \\mathcal G } } [ { \\mathcal X } ] ) [ 1 ] \\to { \\mathcal C } ^ \\infty ( { { \\mathcal G } } ) [ 1 ] , \\ \\ \\ I _ 1 = \\sum _ { i _ 1 , i _ 2 \\in S } ( s ^ * \\chi _ { i _ 1 } ) ( t ^ * \\chi _ { i _ 2 } ) \\ , \\hat { \\sigma } _ { i _ 1 , i _ 2 } ^ * \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} \\frac { 1 + x } { 2 } & = F _ { 1 , 1 } ( x ) \\\\ & \\geq F _ { 1 , 0 } ( x ) = \\exp \\left ( \\frac { x \\log x } { x - 1 } - 1 \\right ) \\\\ & \\geq F _ { 0 , 0 } ( x ) = \\sqrt { x } . \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} q _ 0 ^ { \\Lambda , N } ( \\eta ) = \\int _ { \\Gamma _ 0 } R ^ { \\Lambda , N } _ 0 ( \\eta \\cup \\xi ) \\lambda ( d \\xi ) , \\eta \\in \\Gamma _ { 0 } . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} q ( k ) & = \\prod _ { j = 1 } ^ { s ^ { \\ast } } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\frac { k z _ j ^ 0 \\bmod b ^ m } { b ^ m } \\right ) \\right ) \\prod _ { j = s ^ { \\ast } + 1 } ^ { s } \\left ( 1 + \\gamma _ j \\ , \\omega ( 0 ) \\right ) . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\delta _ { J } ( \\imath , r ) : = \\begin{cases} 1 & \\ ( \\imath , r ) \\in J ' , \\\\ 0 & \\ ( \\imath , r ) \\notin J ' . \\end{cases} \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\Pr ( R ) = \\frac { | Z | } { 8 | Z | } + \\frac { ( 4 \\times 2 ) | Z | ^ 2 + ( 3 \\times 4 ) | Z | ^ 2 } { 8 ^ 2 | Z | ^ 2 } = \\frac { 7 } { 1 6 } . \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\mathcal { C } _ 1 ' : = \\left \\{ w _ 1 \\ ; \\Bigg | \\ ; \\Re ( w _ 1 ) = - \\frac { c } { \\log ^ { 3 / 4 } \\big ( 2 + | \\Im ( w _ 1 + 2 i t ) | \\big ) } \\right \\} , \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{gather*} c _ p ( n ) = \\frac { 1 } { p 2 ^ p } \\Bigl [ \\bigl ( \\zeta ( p , 1 + \\lambda _ 1 ) + \\zeta ( p , 1 + \\lambda _ 2 ) \\bigr ) \\Bigl ( \\frac { B _ { p + 1 } ( q ) - B _ { p + 1 } } { p + 1 } \\Bigr ) - \\zeta ( p ) n + \\zeta ( p ) \\times \\\\ \\times \\Bigl ( \\frac { B _ { p + 1 } ( n + 1 ) - B _ { p + 1 } } { p + 1 } \\Bigr ) - \\zeta ( p , 2 + \\lambda _ 1 + \\lambda _ 2 ) \\Bigl ( \\frac { B _ { p + 1 } ( 2 q - 1 ) - B _ { p + 1 } ( q - 1 ) } { p + 1 } \\Bigr ) \\Bigr ] , \\end{gather*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\mu ( \\gamma ) = \\int ^ { + \\infty } _ { 1 } ( \\rho ^ { - \\gamma } - 1 ) ^ { p - 1 } ( \\rho ^ { Q - 1 - \\gamma ( p - 1 ) } - \\rho ^ { p s - 1 } ) L ( \\rho ) d \\rho . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} P ( V _ n = k ) \\geq & P ( V _ n = 0 ) \\frac { \\beta _ n ^ k } { k ! } \\left ( 1 - \\frac { \\frac { k ^ 2 \\beta } { \\lambda _ n ( 1 - \\beta ) } } { k } \\right ) ^ k \\\\ \\geq & P ( V _ n = 0 ) \\frac { \\beta _ n ^ k } { k ! } \\left ( 1 - \\frac { k ^ 2 \\beta } { \\lambda _ n ( 1 - \\beta ) } \\right ) . \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} \\frac { 1 - \\int _ { U _ - } p ( t , | x - y | ) d x } { t } \\ge \\frac { 1 - \\int _ { B ( y , R ) } p ( t , | x - y | ) d x } { t } = \\frac { 1 } { t } \\left ( 1 - \\P ^ y ( X _ t \\in B ( y , R ) ) \\right ) = \\frac { \\P ^ 0 ( | X _ t | \\ge R ) } { t } . \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} ( \\tilde { \\mathsf { E } } - \\tilde { K } + \\tilde { R } Q \\tilde { K } ) Z ( K ) = 0 \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} ( ( \\lambda _ { i _ 0 j _ 0 } - \\lambda _ { j _ 0 k _ 0 } + \\lambda _ { i _ 0 k _ 0 } ) \\mu _ { j _ 0 } ^ { i _ 0 j _ 0 k _ 0 } - 2 \\lambda _ { i _ 0 j _ 0 } \\ , \\mu _ { k _ 0 } ^ { i _ 0 k _ 0 j _ 0 } ) \\wedge d F _ { i _ 0 } \\wedge d F _ { j _ 0 } \\wedge d F _ { k _ 0 } = 0 . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} \\rho _ k : = \\frac { J ( { \\rm E x p } _ { k } ( 0 , 0 , 0 ) ) - J ( { \\rm E x p } _ { k } ( \\xi _ k , \\eta _ k , \\zeta _ k ) ) } { \\hat { m } _ { k } ( 0 , 0 , 0 ) - \\hat { m } _ { k } ( \\xi _ k , \\eta _ k , \\zeta _ k ) } \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} - \\Delta v _ \\mu ( x ) = - \\mu ^ 2 \\sum _ { j } ^ N a _ j \\mathcal { G } _ \\mu ( x - y _ j ) . \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\overline { T } = \\{ \\tau \\in \\{ - , 0 , + \\} ^ n \\mid \\tau \\le \\rho \\rho \\in T \\} . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} H _ { 1 , \\boldsymbol { \\alpha } } ( i v ) = 2 \\lim _ { r \\to \\infty } \\sum _ { \\substack { \\boldsymbol { n } \\in \\boldsymbol { \\alpha } + \\Z ^ 2 \\\\ \\lvert n _ j - \\alpha _ j \\rvert \\leq r } } M _ 2 \\left ( \\sqrt { 3 } ; \\sqrt { \\frac { v } { 2 } } \\left ( \\sqrt { 3 } \\left ( 2 n _ 1 + n _ 2 \\right ) , n _ 2 \\right ) \\right ) e ^ { 2 \\pi Q ( \\boldsymbol { n } ) v } . \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\mathrm { d i m } ( \\mathrm { n u l l } ( \\Phi _ N ) ) = ( b - k + 1 ) s r , \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} | E ( J _ { G ' } ( U ^ * , d , 2 \\epsilon ) ) | = \\big | \\{ u _ { i , j } u _ { i ' , j ' } : u _ i u _ { i ' } \\in E ( J _ { G } ( U , d , 2 \\epsilon ) ) \\big | \\leq \\sum d _ { F } ( x ) d _ { F } ( x ' ) \\stackrel { \\rm ( A 3 ) _ { \\ref { l e m : r a n d o m m a t c h i n g b e h a v e s r a n d o m } } } { \\leq } \\epsilon n ^ 2 . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) = c _ + + ( - 1 ) ^ { | m | + | n | } c _ - + \\left ( S ^ { m + \\chi ^ I } ( S ^ * ) ^ { n + \\chi ^ I } T ' \\right ) . \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} ( p , q ) = ( 2 , 2 ) , ( 2 , \\infty ) , ( \\infty , \\infty ) . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} \\frac { \\log ( \\left \\| X Q ^ { \\frac { 1 } { 2 p } } \\right \\| _ { S ^ { 2 p } _ n } ^ { 2 p } ) - \\log ( \\left \\| X Q ^ { \\frac { 1 } { 2 } } \\right \\| _ { H S } ^ 2 ) } { 1 - p } & \\leq \\frac { 2 p \\log ( \\mathrm { t r } ( Q ) ^ { - \\frac { 1 } { 2 p ' } } ) + ( 2 p - 2 ) \\log ( \\left \\| X Q ^ { \\frac { 1 } { 2 } } \\right \\| _ { H S } ) } { 1 - p } \\\\ & = \\log ( \\mathrm { t r } ( Q ) ) - \\log ( \\left \\| X Q ^ { \\frac { 1 } { 2 } } \\right \\| _ { H S } ^ 2 ) \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\Tilde { e } _ { i } ^ { F } & = \\mathrm { K R } ^ { \\prime - 1 } \\circ ( \\mathrm { I d } , \\Tilde { e } _ { i } ^ { \\pm P } ) \\circ \\mathrm { K R } ^ { \\prime } , \\\\ \\Tilde { f } _ { i } ^ { F } & = \\mathrm { K R } ^ { \\prime - 1 } \\circ ( \\mathrm { I d } , \\Tilde { f } _ { i } ^ { \\pm P } ) \\circ \\mathrm { K R } ^ { \\prime } , \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} \\prod _ { ( i , j ) \\in F _ k } X _ { i , j } = \\frac { L _ { 1 , b - k + 1 } } { L _ { 1 , b - k + 2 } } \\frac { L _ { 2 , b - k + 2 } } { L _ { 2 , b - k + 3 } } \\cdots \\frac { L _ { a , n - k } } { L _ { a , n - k + 1 } } = \\frac { \\prod \\limits _ { i = 1 } ^ a L _ { i , b - k + i } } { \\prod \\limits _ { i = 1 } ^ a L _ { i , b - k + i + 1 } } . \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} B = \\biggl ( N _ D - N _ G - \\sum _ { \\begin{subarray} { l } \\widetilde { \\Sigma } < D \\\\ \\vert \\widetilde { \\Sigma } \\rvert = 2 \\end{subarray} } N _ { \\widetilde { \\Sigma } } \\biggr ) B \\subseteq B ^ G + \\sum _ { \\begin{subarray} { l } \\widetilde { \\Sigma } < D \\\\ \\vert \\widetilde { \\Sigma } \\rvert = 2 \\end{subarray} } B ^ { \\widetilde { \\Sigma } } \\subseteq B , \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\widetilde { Y } _ { i , r } \\widetilde { A } _ { j , s } ^ { - 1 } = t ^ { \\beta ( i , r ; j , s ) } \\widetilde { A } _ { j , s } ^ { - 1 } \\widetilde { Y } _ { i , r } , \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} M ( \\lambda ) F ( \\lambda ) = G ( \\lambda ) N ( \\lambda ) \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} \\ell _ Y ( \\Theta ) = - \\sum _ { ( i , j ) \\in \\Omega } \\xi ( s _ { ( i , j ) } X _ i ^ \\top \\Theta X _ j ) \\ , , \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} G _ { k , N } ^ \\chi ( Z _ 1 , Z _ 2 , r ; Q _ \\tau ) : = \\pi ^ { 2 r + 4 - 2 k } \\times ( \\mathfrak { p } ^ \\circ _ { \\ell , m } \\otimes \\mathfrak { p } ^ \\circ _ { \\ell , m } ) ( E _ { k , N } ^ \\chi ( Z _ 1 , Z _ 2 , 1 - \\frac { k - r } 2 ; Q _ \\tau ) ) . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} B ( v , e ) & = \\begin{dcases} \\displaystyle \\frac { \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G ) x _ v ^ p } , & ~ v \\in e , \\\\ 0 , & , \\end{dcases} \\\\ [ 2 m m ] w ( e ) & = \\frac { r \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G ) } . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { V } N = - A _ { N } U + \\tau ( U ) N , \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} \\psi _ j ^ l ( x ) = e ^ { \\zeta _ 1 ^ { k _ l , t _ { k _ l } } \\cdot x } + \\int _ { \\partial \\Omega } G ( x - y , \\zeta _ 1 ^ { k _ l , t _ { k _ l } } ) ( \\Lambda _ { q _ j } - \\Lambda _ 0 ) \\psi _ j ^ l ( y ) d \\sigma ( y ) , x \\in \\partial \\Omega , \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} ( \\mathsf { G } _ { [ + \\infty , t _ 0 ] } f ) ( t , x ) = \\int \\limits ^ { \\infty } _ { t _ 0 } d s \\int d ^ 2 z \\ , \\mathsf { G } ( t - s , x - z ) f ( s , z ) \\ , . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} T = \\{ T ^ { 1 1 } , T ^ { 1 2 } , \\ldots , T ^ { 2 1 } , T ^ { 2 2 } , \\ldots , T ^ { m n } \\} \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} V _ { \\sigma ( i ) } ^ \\pm : = \\{ j \\in V : \\pm x _ j > | x _ { \\sigma ( i ) } | \\} , \\ ; \\ ; \\ ; \\ ; i = 0 , 1 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} m = \\prod _ { i \\in I , r \\in \\mathbb { Z } } Y _ { i , r } ^ { u _ { i , r } ( m ) } \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} \\sum _ { n = m } ^ \\infty W _ \\alpha ( n , m ; x ) \\dfrac { z ^ n } { n ! } = \\dfrac { e ^ { z x } } { m ! } \\left ( \\dfrac { e ^ { \\alpha z } - 1 } { \\alpha } \\right ) ^ m . \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} \\begin{cases} \\mu \\ ! \\left ( \\frac { m } { m ' } \\right ) \\chi _ 2 \\ ! \\left ( \\frac { m } { m ' } \\right ) \\left ( \\frac { m } { m ' } \\right ) ^ { - s } & m ' \\mid m , \\chi _ 1 ' = \\chi _ 1 \\chi _ 2 ' = \\chi _ 2 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\dot { u } = B _ u ( u ) - i J H _ u ( u ) \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } } { d t ^ { 2 } } \\gamma = \\nabla U ( \\gamma ) = ( \\frac { 1 } { m _ { 1 } } \\frac { \\partial U } { \\partial \\mathbf { a } _ { 1 } } , \\frac { 1 } { m _ { 2 } } \\frac { \\partial U } { \\partial \\mathbf { a } _ { 2 } } , \\frac { 1 } { m _ { 3 } } \\frac { \\partial U } { \\partial \\mathbf { a } _ { 3 } } ) \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\Phi \\circ \\mathbb P F ^ k ( \\omega , [ v ] ) & \\le \\log \\Vert M ^ n _ \\omega \\Vert \\\\ & \\le \\log \\Vert M ^ N _ \\omega \\Vert \\Vert M ^ { n - N } _ { \\sigma ^ N ( \\omega ) } \\Vert \\\\ & \\le - \\log 2 . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} \\overline { \\Omega } _ { \\xi } ^ { \\mathrm { t w } , \\flat } ( \\beta ) = \\begin{cases} ( \\imath , r ) & \\ \\imath \\leq n , \\\\ ( 2 n - \\imath , r ) & \\ \\imath > n . \\end{cases} \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} F ( s ) = m s - ( m - 1 ) + ( m - 1 ) ( 1 - s ) ^ { 1 + \\beta } , \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} ( h \\cdot f ) ( x ) = \\sum _ h h _ 2 f ( S ^ { - 1 } ( h _ 1 ) x ) . \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} \\alpha + _ r \\beta = \\kappa \\bigl ( \\eta _ X ( \\alpha ) + _ r \\eta _ X ( \\beta ) \\bigr ) \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ^ j } { \\partial \\mathbf { x } ^ j } ( g _ { Y , z _ t } ( \\mathbf { x } ) _ { a b } - \\delta _ { a b } ) \\right | \\leq \\frac { 1 } { 1 0 0 } | \\mathbf { x } | ^ { 2 - j } \\ , \\ ; { \\rm f o r } \\ ; \\ , | \\mathbf { x } | \\leq 2 \\ ; \\ , { \\rm a n d } \\ ; \\ , j = 0 , 1 . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} \\Delta ( l ) < q ^ { \\left ( \\sum _ { 1 \\le i \\le l } n - ( m - 2 l + 2 i - 2 ) \\right ) + ( m - 2 l ) ( \\frac { k a } { 2 } + s ) } = q ^ { \\frac { n m } { 2 } + ( \\frac { m } { 2 } - l ) s - l ( m - l - 1 ) } . \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\langle M _ \\phi ( U ( m , n ) ) \\delta _ y , \\delta _ x \\rangle & = \\tilde { \\phi } ( d ( x _ 1 , y _ 1 ) , . . . , d ( x _ N , y _ N ) ) \\langle U ( m , n ) \\delta _ y , \\delta _ x \\rangle \\\\ & = \\tilde { \\phi } ( m + n ) \\langle U ( m , n ) \\delta _ y , \\delta _ x \\rangle . \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} \\frac { k } { d + 1 - k } - \\frac { d + 1 } { k ( d + 1 - k ) } [ k - ( r - \\lambda ) ] = \\frac { d + 1 } { k ( k - 1 ) } \\lambda - 1 \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G [ S ] ) } w ( e ) = \\frac { r } { \\lambda ^ { ( p ) } ( G [ S ] ) } \\sum _ { e \\in E ( G [ S ] ) } \\prod _ { u \\in e } x _ u = \\frac { \\lambda ^ { ( p ) } ( G [ S ] ) } { \\lambda ^ { ( p ) } ( G [ S ] ) } = 1 . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\varphi ' ( t ) y ( t ) \\ , \\d t = - \\int _ 0 ^ 1 \\varphi ( t ) y ' ( t ) \\ , \\d t \\qquad \\forall \\varphi \\in C ^ \\infty _ c ( 0 , 1 ) . \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} u ( t ) = e ^ { t \\Delta } u _ 0 - \\int ^ t _ 0 e ^ { ( t - \\tau ) \\Delta } \\mathbb { P } \\ \\textrm { d i v } ( u \\otimes u ) ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} h m _ 0 = \\epsilon ( h ) m _ 0 , \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} \\ell _ \\infty = \\| u _ \\infty ^ K \\| ^ 2 _ { L ^ 2 } ( 2 Q - | ( u \\vert 1 ) | ^ 2 ) . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\sigma ^ { 2 } _ { f } = \\mathbb { V } ( f ( X ) ) = P ( f ^ { 2 } ) - P ( f ) ^ { 2 } , \\quad \\sigma _ { \\mathcal { F } } ^ { 2 } = \\sup _ { f \\in \\mathcal { F } } \\sigma ^ { 2 } _ { f } . \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} [ M ( m ) ] = [ L ( m ) ] + [ L ( m ' ) ] . \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle { \\inf } \\ ; \\mathbb { E } _ { \\xi _ 2 , \\ldots , \\xi _ { T _ { \\max } } , D _ 2 , \\ldots , D _ { T _ { \\max } } } \\Big [ \\displaystyle { \\sum _ { t = 1 } ^ { T } } \\ ; f _ { t } ( x _ { t } , x _ { t - 1 } , \\xi _ t ) \\Big ] \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) \\mbox { a . s . } , \\ ; x _ { t } \\ ; { \\overline { \\mathcal { F } } } _ t \\mbox { - m e a s u r a b l e , } t = 1 , \\ldots , T _ { \\max } , \\end{array} \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} N = M t , \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} M _ { ( a , x ) } \\overset { { \\rm i n \\ , l a w } } { = } 2 ^ { - \\bigl ( 2 ( x _ 1 + x _ 2 ) - ( a _ 1 + a _ 2 ) \\bigr ) / a _ 1 a _ 2 } \\ , L \\ , X _ 1 \\ , X _ 2 \\ , X _ 3 . \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} { \\bf E } ( { \\rm e } ^ { - \\lambda X _ t } ) = { \\rm e } ^ { - t \\Phi ( \\lambda ) } . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} { \\Omega _ { \\max } } = \\left [ { \\max \\left ( { \\left | { \\hat h _ { \\max } ^ { { \\rm { r o u g h } } } } \\right | - \\beta , 0 } \\right ) , \\left | { \\hat h _ { \\max } ^ { { \\rm { r o u g h } } } } \\right | + \\beta } \\right ] \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} a ^ \\omega _ { X _ g , X _ h , X _ k } : = \\omega ( g , h , k ) a _ { X _ g , X _ h , X _ k } , & & c _ { X _ g , X _ h } ^ \\omega : = c _ { X _ g , X _ h } , \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\pi ^ c ( w ) : = \\left \\{ \\begin{array} { l l } s \\pi ^ { s c s } ( s w ) & \\mbox { i f } \\ \\ell ( s w ) < \\ell ( w ) \\\\ \\pi ^ { s c } ( w _ { \\langle s \\rangle } ) & \\mbox { i f } \\ \\ell ( s w ) > \\ell ( w ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} { \\tilde \\theta _ n } = 2 n \\pi \\tilde f T + \\tilde \\varphi + \\varphi _ { n , B } ^ { { \\rm { C F S K } } } - \\varphi _ { n , A } ^ { { \\rm { C F S K } } } \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\overline { \\mathcal { L } _ j } = \\bigcup _ { i \\in [ d ] } \\overline { \\mathcal { L } _ j ^ { ( i ) } } \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} \\mu _ { t } ( x ) = \\mu _ t ( z , y ) = d ^ { g _ { z , t } } ( x , \\partial ( B \\times Y ) ) ^ \\alpha \\sup _ { x ' \\in B ^ { g _ { z , t } } ( x , \\frac { 1 } { 4 } d ^ { g _ { z , t } } ( x , \\partial ( B \\times Y ) ) ) } \\frac { | \\eta _ t ( x ) - \\mathbf { P } ^ { g _ { z , t } } _ { x ' x } ( \\eta _ t ( x ' ) ) | _ { g _ { z , t } ( x ) } } { d ^ { g _ { z , t } } ( x , x ' ) ^ \\alpha } , \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} \\lim \\limits _ { z \\rightarrow 0 } \\ , \\Bigl [ z \\ , \\Gamma _ 2 ( z \\ , | \\ , a _ 1 , a _ 2 ) \\Bigr ] = 1 . \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\begin{cases} n _ { t t } - ( i D ) ^ 2 n = 0 , x , t \\in \\R ^ + \\\\ n ( x , 0 ) = n _ t ( x , 0 ) = 0 , \\\\ n ( 0 , t ) = h ( t ) . \\end{cases} \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} n \\int \\limits _ T \\ell ( x _ \\infty ^ n ( t ) - y ^ n _ \\infty ) d \\mu ( t ) + \\ell ( y _ \\infty ^ n - x _ 0 ) + \\sum \\limits _ { i = 1 } ^ { \\infty } \\eta _ i \\ell ( y _ \\infty ^ n - y ^ n _ i ) \\\\ = \\inf \\limits _ { u \\in X } \\left ( n \\int \\limits _ T \\ell ( x _ \\infty ^ n ( t ) - u ) d \\mu ( t ) + \\ell ( u - x _ 0 ) + \\sum \\limits _ { i = 1 } ^ { \\infty } \\eta _ i \\ell ( y _ \\infty ^ n - y ^ n _ i ) \\right ) . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} G ( E ( t , \\beta ; x ) , z ) = \\dfrac { e ^ { x z } } { ( 1 + \\beta ( e ^ z - 1 ) ) ^ t } . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} e _ 0 : = e ( A , \\Phi ) ( y _ 0 ) \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} | U ( t ) _ { u , v } | = 1 . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} \\langle \\frac { ( u _ 3 ) _ k ^ i - 2 ( u _ 3 ) _ k ^ { i - 1 } + ( u _ 3 ) _ k ^ { i - 2 } } { \\delta _ k ^ 2 } , \\varphi _ 3 \\rangle + \\langle \\sigma _ k ^ i , E \\varphi \\rangle = \\langle \\mathcal L _ k ^ i , \\varphi \\rangle . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\Big \\| a _ N ^ { - 1 } \\big ( f _ M U _ N \\psi _ N ^ { \\operatorname { M F } } \\big ) _ { j k } - \\Phi ^ { \\operatorname { g s } } _ { j k } \\Big \\| _ { L ^ 2 ( \\mathbb { R } ^ 3 ) ^ { \\otimes _ { \\operatorname { s y m } } j } \\otimes L ^ 2 ( \\mathbb { R } ^ 3 ) ^ { \\otimes _ { \\operatorname { s y m } } k } } = 0 . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\widehat { \\mathcal { Q } } ^ 1 = \\{ ( j , r - 1 ) \\to ( i , r ) \\mid i \\ \\ j \\ \\ \\mathcal { Q } \\} . \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} R ( \\chi ) = \\sum _ { n \\leqslant N } r ( n ) a _ f ( n ) \\chi ( n ) . \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\sup _ { x \\in B \\times Y } | \\hat { \\eta } _ i ( x ) | _ { \\hat { g } ( x ) } + \\sup _ { x = ( z , y ) \\in B \\times Y } \\Biggl ( \\sup _ { x ' \\in B \\times Y } \\frac { | \\hat \\eta _ i ( x ) - \\mathbf { P } ^ { \\hat { g } _ { z , i } } _ { x ' x } ( \\hat \\eta _ i ( x ' ) ) | _ { \\hat { g } ( x ) } } { d ^ { { g } _ { i } } ( x , x ' ) ^ \\alpha } \\Biggr ) \\leq C . \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} | | \\mathbf h _ p | | ^ 2 = \\int _ { \\Q _ p } \\mathbf h _ p ( x ) \\overline { \\mathbf h _ p ( x ) } d x = \\int _ { \\Z _ p ^ { \\times } } d x = ( \\Z _ p ^ { \\times } ) = 1 - p ^ { - 1 } . \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K ( n ) = n ^ r , \\\\ \\ \\ \\ \\ \\ \\ \\forall r \\in ( 0 , 1 ) , \\displaystyle \\lim _ { \\substack { n \\rightarrow + \\infty } } \\displaystyle \\frac { K ( n ) } { n } = 0 \\} \\\\ \\end{array} \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\rightarrow 1 } S _ \\alpha ( \\rho ) = S ( \\rho ) . \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} Q _ 1 & : = \\left \\{ q : q < \\frac { \\alpha } { 2 } ( u _ 1 + u _ 2 ) \\right \\} , \\\\ Q _ i & : = \\left \\{ q : \\frac { \\alpha } { 2 } ( u _ { i - 1 } + u _ i ) < q < \\frac { \\alpha } { 2 } ( u _ i + u _ { i + 1 } ) \\right \\} , 1 < i < d , \\\\ Q _ d & : = \\left \\{ q : q > \\frac { \\alpha } { 2 } ( u _ { d - 1 } + u _ d ) \\right \\} , \\\\ Q _ { i , i + 1 } & : = \\left \\{ q : q = \\frac { \\alpha } { 2 } ( u _ i + u _ { i + 1 } ) \\right \\} . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} { \\tilde \\theta _ n } = 2 n \\pi \\tilde f T + \\tilde \\varphi \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} \\sum _ { e \\ , \\cup \\{ v _ e \\} \\in E ( G ^ { r + 1 } ) } w ' ( e \\cup \\{ v _ e \\} ) = 1 , ~ \\sum _ { e \\ , \\cup \\{ v _ e \\} : \\ , v \\in e \\ , \\cup \\{ v _ e \\} } B ' ( v , e \\cup \\{ v _ e \\} ) = 1 , \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} C _ n ( \\omega _ n ^ j ) = \\begin{cases} \\binom { 2 j } { j } , & j < n , \\\\ \\frac { 1 } { n + 1 } \\binom { 2 n } { n } , & j = n . \\end{cases} \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} K _ k ( z , z ' ) : = e ^ { \\beta _ k F ( z , z ' ) } \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\ln \\left ( \\prod _ { j = 1 } ^ i \\lambda _ j ^ { ( k ) } ( b b ^ * ) \\right ) . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} F ( ( z _ 1 , z _ 2 ) , ( z _ 1 ' , z _ 2 ' ) ) : = F _ 1 ( z _ 1 , z _ 1 ' ) F _ 2 ( z _ 2 , z _ 2 ' ) \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} \\tilde { g } ( \\tilde { J } U , \\tilde { J } V ) = p \\tilde { g } ( U , \\tilde { J } V ) + q \\tilde { g } ( U , V ) , \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} \\begin{pmatrix} * & * & X \\\\ ( P X ) ^ t & * & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , P \\in \\operatorname { M a t } _ { n \\times n } , \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} R _ 2 P _ 2 = E _ { k - 2 l _ 1 } P _ 2 ^ T E _ { n - 2 k - 2 l _ 2 } P _ 2 = 0 . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} c _ { 0 0 } & = 0 & c _ { 0 1 } & = 2 b _ { 1 0 } c _ { 1 2 } & c _ { 0 2 } & = 2 b _ { 1 1 } c _ { 1 2 } \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} \\varphi \\left ( a \\wedge ^ { { \\mathbf { L } } } b \\right ) = \\varphi \\left ( a \\right ) = f \\left ( a ^ { \\prime { \\mathbf { L } } } \\right ) = f \\left ( a ^ { \\prime { \\mathbf { L } } } \\right ) \\wedge ^ { \\mathcal { T } _ { i } \\left ( { \\mathbf { L } } \\right ) } b = \\varphi \\left ( a \\right ) \\wedge ^ { \\mathcal { T } _ { i } \\left ( P \\left ( { \\mathbf { L } } \\right ) \\right ) } \\varphi \\left ( b \\right ) \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{gather*} \\sum _ { \\mathfrak I \\in \\Sigma } \\tilde \\chi ^ k ( \\mathfrak I ) N ( \\mathfrak I ) ^ { - s } = \\frac 1 { 1 - ( - 2 ) ^ k 4 ^ { - s } } \\sum _ { \\mathfrak I \\in S _ 2 } \\tilde \\chi ^ k ( \\mathfrak I ) N ( \\mathfrak I ) ^ { - s } = L \\big ( \\chi ^ k , s - k / 2 \\big ) . \\end{gather*}"}
-{"id": "2248.png", "formula": "\\begin{align*} F _ { \\Upsilon , M } ( x ) = \\frac { 1 } { 2 \\pi i } \\int \\limits _ { \\alpha - i \\infty } ^ { \\alpha + i \\infty } \\Psi _ { \\Upsilon , M } ( w ) x ^ { - w } d w , \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} U ( \\lambda ) L ( \\lambda ) V ( \\lambda ) = \\begin{bmatrix} I _ s & 0 \\\\ 0 & P ( \\lambda ) \\end{bmatrix} . \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\Bigl ( \\sum _ i n _ i [ X \\xleftarrow { f _ i } M _ i \\xrightarrow { g _ i } Y ] \\Bigr ) & \\circ \\Bigl ( \\sum _ j m _ j [ Y \\xleftarrow { h _ j } N _ j \\xrightarrow { k _ j } Z ] \\Bigr ) \\\\ & : = \\sum _ { i , j } n _ i m _ j [ ( X \\xleftarrow { f _ i } M _ i \\xrightarrow { g _ i } Y ) \\circ ( Y \\xleftarrow { h _ j } N _ j \\xrightarrow { k _ j } Z ) ] . \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} \\varphi _ { 1 } ( r ) + \\Big [ 2 \\eta + ( 1 + \\tau + \\mathrm { a } ) r ^ { - 1 } \\Big ] \\varphi ( r ) = 0 , \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} \\int _ X | R i c ( \\tilde \\omega _ \\epsilon ) + \\tilde \\omega _ \\epsilon | ^ 2 _ { \\tilde \\omega _ \\epsilon } \\tilde \\omega _ \\epsilon ^ n & = \\int _ X | 2 n \\epsilon \\omega _ { i _ 0 , \\epsilon } | ^ 2 _ { \\tilde \\omega _ \\epsilon } \\tilde \\omega _ \\epsilon ^ n \\\\ & \\le 4 n ^ 3 \\int _ X \\tilde \\omega _ \\epsilon ^ n \\\\ & \\to 4 n ^ 3 ( 2 \\pi ) ^ n \\int _ X c _ 1 ( K _ X ) ^ n \\\\ & = 0 \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} B ( \\bar u _ N , \\bar u _ N ) & = \\sum _ { m , j = 1 } ^ N t ^ { - \\mu _ m - \\mu _ j } B ( \\xi _ m , \\xi _ j ) \\\\ & = \\sum _ { n = 1 } ^ { N } \\frac 1 { t ^ { \\mu _ n } } \\Big ( \\sum _ { \\stackrel { 1 \\le m , j \\le N , } { \\mu _ m + \\mu _ j = \\mu _ n } } B ( \\xi _ m , \\xi _ j ) \\Big ) + \\frac 1 { t ^ { \\mu _ { N + 1 } } } \\sum _ { \\stackrel { 1 \\le m , j \\le N , } { \\mu _ m + \\mu _ j = \\mu _ { N + 1 } } } B ( \\xi _ m , \\xi _ j ) + h _ { N + 1 , 2 } , \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} F _ { 4 , 1 8 } ( \\tau ) = \\left ( \\begin{smallmatrix} 3 ( u + 1 / u ) \\\\ 2 ( u - 1 / u ) \\\\ 0 \\\\ - 2 ( u - 1 / u ) \\\\ - 3 ( u + 1 / u ) \\end{smallmatrix} \\right ) X ^ 3 Y ^ 3 + \\ldots \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} [ M ( m ) ] = \\sum _ { m ' } P _ { m , m ' } ( 1 ) [ L ( m ' ) ] . \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow 0 } g ' _ 0 ( x ; a , b ) & = \\lim _ { x \\downarrow 0 } g _ 1 ( x ; a , b ) + 4 a ^ 3 \\int _ { - 1 } ^ 1 \\int _ { 0 } ^ s f ''' ( A ) \\ , d u \\ , s \\ , d s \\\\ & = 4 a f '' ( A ) + \\frac { 8 a ^ 3 } { 3 } f ''' ( A ) . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ { p } ) ^ { s } u = \\omega ( x ) | u | ^ { p - 2 } u , \\ , \\ , x \\in \\Omega , \\\\ u ( x ) = 0 , \\ , \\ , x \\in \\mathbb { R } ^ { N } \\setminus \\Omega , \\end{cases} \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} \\exists \\delta \\in \\R ^ + : \\forall x \\in \\C : \\left ( | x - c | < \\delta \\Rightarrow f ( x ) = f ( c ) ) \\right . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} \\gamma ( h , - ) = - \\gamma ( h , + ) , \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} & \\mathfrak { d } _ 2 ^ { m + 1 } \\dot { \\phi } ( n ) \\\\ & = \\frac { \\Gamma ( N + \\frac { 1 } { 2 } + m ) } { \\Gamma ( N + \\frac { 1 } { 2 } ) } \\int _ 0 ^ 2 \\cdots \\int _ 0 ^ 2 ( 1 + n + t _ 1 + \\cdots + t _ m ) ^ { - N - \\frac { 1 } { 2 } - m } \\ , d t _ 1 \\cdots d t _ m . \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} L ( \\lambda \\mid \\hat { x } _ n , \\hat { X } _ n ) M ( \\lambda ; t \\mid \\hat { u } _ n , \\hat { U } _ n ) = M ( \\lambda ; t \\mid \\hat { u } _ { n + 1 } , \\hat { U } _ { n + 1 } ) L ( \\lambda \\mid \\hat { y } _ n , \\hat { Y } _ n ) \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\textrm { V e c } ( C D E ) = ( E ^ T \\otimes C ) \\textrm { V e c } ( D ) . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} A = \\sum _ { u \\in M } u s ( u ) \\langle u | \\ : \\ : \\rangle \\ : , \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} \\partial ( \\zeta _ { L _ { \\infty } / K } ) = - C _ { L _ { \\infty } / K } - U ' _ { L _ { \\infty } / K } + M _ { L _ { \\infty } / K } \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} D x + 3 D ^ 2 y - 2 z = 0 \\\\ D ^ 2 y + 5 D z = 0 \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} & \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } ( Y ( n ) > 2 N _ \\varepsilon K _ \\varepsilon ) \\\\ & ~ ~ ~ ~ ~ \\leq \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } ( \\Omega ( n , \\varepsilon ) ) + \\limsup _ { n \\rightarrow \\infty } \\sum _ { l = 1 , 2 } \\mathbb { P } \\big ( \\sum _ { i : t _ { i , n } ^ { ( l ) } \\leq T } \\big ( \\Delta _ { i , n } ^ { ( l ) } X ^ { ( l ) } \\big ) ^ 2 > K _ \\varepsilon \\big ) < \\delta . \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} D _ { n + 2 } ^ 1 ( n + 2 ; 2 ) D _ n ^ 2 - D _ { n + 1 } ^ 2 D _ { n + 1 } ^ 1 ( n + 1 ; 2 ) & = D _ { n + 1 } ^ 1 D _ n ^ 3 , \\\\ D _ { n + 1 } ^ 0 ( 1 ; 2 ) D _ { n + 2 } ^ 0 - D _ { n + 2 } ^ 0 ( 1 ; 2 ) D _ { n + 1 } ^ 0 & = - D _ { n + 2 } ^ { - 1 } ( 1 ; 2 ) D _ { n + 1 } ^ 1 , \\\\ D _ { n + 2 } ^ { - 1 } ( n + 2 ; 2 ) D _ { n + 1 } ^ 2 - D _ { n + 3 } ^ { - 1 } ( n + 3 ; 2 ) D _ { n } ^ 2 & = D _ { n + 2 } ^ { - 1 } ( 1 ; 2 ) D _ { n + 1 } ^ 1 . \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} \\begin{array} { l l } I _ 0 = ( - \\infty , - 2 , 5 ] , & \\\\ I _ j = ( - 2 . 5 + j - 1 , - 2 . 5 + j ] , & j = 1 , \\dots , 5 , \\\\ I _ 6 = ( 2 . 5 , \\infty ) , & \\end{array} \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{gather*} \\mathfrak { M } ( - l \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\frac { \\Gamma \\bigl ( 1 + l / \\tau \\bigr ) } { \\Gamma ^ { - l } \\bigl ( 1 - 1 / \\tau \\bigr ) } \\ , \\prod \\limits _ { j = 0 } ^ { l - 1 } \\frac { \\Gamma \\bigl ( 2 + \\lambda _ 1 + \\lambda _ 2 + ( l + j + 2 ) / \\tau \\bigr ) } { \\Gamma \\bigl ( 1 + ( j + 1 ) / \\tau \\bigr ) } \\star \\\\ \\star \\frac { 1 } { \\Gamma \\bigl ( 1 + \\lambda _ 1 + ( j + 1 ) / \\tau \\bigr ) \\Gamma \\bigl ( 1 + \\lambda _ 2 + ( j + 1 ) / \\tau \\bigr ) } , \\end{gather*}"}
-{"id": "6297.png", "formula": "\\begin{align*} T ( \\kappa , \\kappa ' ) = \\frac { \\kappa - \\kappa ' } { \\langle b \\rangle } e ^ { - \\kappa } ; \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} \\tau = t , \\quad \\xi = x , \\eta = \\psi ( t , x , y ) , \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} \\frac { \\partial U _ { L } } { \\partial P _ { B S } } = \\frac { \\mu ( 1 - \\theta ) \\frac { \\xi \\theta \\| \\mathbf { h } \\| ^ { 2 } | h _ { s } | ^ { 2 } } { ( 1 - \\theta ) \\sigma _ { s } ^ { 2 } } } { 1 + \\frac { \\xi \\theta \\| \\mathbf { h } \\| ^ { 2 } | h _ { s } | ^ { 2 } } { ( 1 - \\theta ) \\sigma _ { s } ^ { 2 } } P _ { B S } } - \\lambda \\theta \\| \\mathbf { h } \\| ^ { 2 } = 0 . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\tilde { \\phi } ( m + n + 2 k \\chi ^ J ) = \\lim _ { k \\to \\infty } \\sum _ { I \\subset [ N ] } f _ \\phi ^ I ( U ( m + k \\chi ^ J , n + k \\chi ^ J ) ) = f _ \\phi ^ \\varnothing ( U ( m , n ) ) , \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} s _ { \\widetilde { \\lambda } _ h + 1 } s _ { \\widetilde { \\lambda } _ h + 2 } \\cdots s _ { \\widetilde { \\lambda } _ h + a } d _ M = s _ { \\widetilde { \\lambda } ^ + _ h - 1 } s _ { \\widetilde { \\lambda } ^ + _ h - 2 } \\cdots s _ { \\widetilde { \\lambda } ^ + _ h - b } d _ { M ^ + _ { h , k } } . \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} Q ^ * ( x ) : = \\sup _ { 0 \\leq t \\leq T } \\sup _ { 0 < r \\leq D } Q _ { t , r } ( x ) . \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} V ^ { r , s } = \\bigoplus _ { j = 0 } ^ { \\min \\{ r , s \\} } \\bigoplus _ { [ \\mu , \\nu ] \\in \\Pi ( r - j , s - j ) } V ^ { [ \\mu , \\nu ] } \\otimes \\Delta ^ { [ \\mu , \\nu ] } \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} \\| u \\| _ { * } = \\sup _ { y \\in \\R ^ { N } } \\Big ( \\sum _ { j = 1 } ^ { k } \\frac { 1 } { ( 1 + \\lambda | y - \\xi _ { j } | ) ^ { \\frac { N - 2 } 2 + \\tau } } \\Big ) ^ { - 1 } \\lambda ^ { - \\frac { N - 2 } { 2 } } | u ( y ) | \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\gamma _ j e _ j & \\mapsto \\sum _ { j = 1 } ^ { \\frac { k a } { 4 } } \\gamma _ { j } e _ { j + \\frac { k a } { 4 } } + \\sum _ { j = \\frac { 3 k a } { 4 } + 1 } ^ { k a } \\gamma _ { j } e _ { j - \\frac { k a } { 4 } } . \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} ( 1 + \\lambda | y - \\xi _ { j } | ) ^ { \\frac { N + 2 } { 2 } + \\tau - 4 } \\le C \\lambda ^ { \\frac { N + 2 } { 2 } + \\tau - 4 } = C \\lambda ^ { - \\frac { N - 1 } 2 } \\lambda ^ { N - 2 } . \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} \\nabla ( t ^ { \\overline { \\nu } } ) = \\nu { t ^ { \\overline { \\nu - 1 } } } , \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} \\mu _ { k , t } ( x ) = d ^ { g _ t } ( x , \\partial ( B \\times Y ) ) ^ k | ( \\nabla ^ { k , g _ t } g _ t ^ \\bullet ) ( x ) | _ { g _ t ( x ) } . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\ell ' = b ' d ' . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} a _ n = \\frac { ( - 1 ) ^ n } { ( n + 1 ) ^ \\alpha } , \\quad \\forall n \\in \\N . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k + 1 } \\sum _ i 2 ^ { L - i } \\binom { L } { i } \\binom { N } { n + K - i } & = \\sum _ i 2 ^ { L - i } \\binom { L } { i } \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k + 1 } \\binom { N } { n + k - i } \\\\ & = \\sum _ i 2 ^ { L - i } \\binom { L } { i } \\binom { N - 1 } { n - i } , \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} g '' ( z _ 0 ) = 2 q ( z _ 0 ) . \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} ( \\overline { B } + i ) ( a _ \\epsilon e ) & = ( \\overline { B ' } + i ) ( a _ \\epsilon x ) \\\\ & = [ \\overline { B ' } , a _ \\epsilon ] e + a _ \\epsilon ( \\overline { B ' } + i ) e \\\\ & = c ( \\nabla a _ \\epsilon ) e + a _ \\epsilon ( \\overline { B ' } + i ) e . \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} & e ^ { ( n - \\nu - 2 ) t } \\int _ X ( S ( t ) ^ 2 + ( \\nu + 3 ) S ( t ) + n ) \\omega ( t ) ^ n \\\\ & = e ^ { ( n - \\nu - 2 ) t } \\int _ X S ( t ) ^ 2 \\omega ( t ) ^ n + ( \\nu + 3 ) L ( t ) + n e ^ { ( n - \\nu - 2 ) } [ \\omega ( t ) ] ^ n \\\\ & \\le e ^ { ( n - \\nu - 2 ) t } \\int _ X S ( t ) ^ 2 \\omega ( t ) ^ n + C _ 2 e ^ { - 2 t } , \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} ( \\tilde S _ { m , n } ^ s , z ) = \\{ h ( \\mathbf Y ( y ) - A ( x ) ) = 0 \\} . \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} \\log ^ + | u ( r e ^ { i \\theta } ) | = o ( r ) , r \\to \\infty , \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} f ^ { L } ( \\vec x ) = \\int _ 0 ^ { \\| x \\| _ \\infty } f ( V _ t ^ + ( \\vec x ) , V _ t ^ - ( \\vec x ) ) d t , \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} Y = \\tau ^ { 1 / \\tau } \\ , \\beta ^ { - 1 } _ { 1 , 0 } ( a = \\tau , b _ 0 = \\tau ) . \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} { x ' } ^ 3 + \\frac { c _ 2 ^ 2 + c _ 1 c _ 3 } { c _ 3 ^ 2 } x ' + \\frac { c _ 1 c _ 2 + c _ 0 c _ 3 } { c _ 3 ^ 2 } = 0 , \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | m _ { n + 1 } ( x , y ) - m _ n ( x , y ) \\big | & = \\bigg ( \\frac { 1 } { \\vartheta _ B ( n _ 0 ) } + \\sum _ { n = n _ 0 } ^ { N _ 2 - 1 } \\frac { 1 } { \\vartheta _ B ( n ) } - \\frac { 1 } { \\vartheta _ B ( n + 1 ) } \\bigg ) \\prod _ { j = 1 } ^ { k '' } \\log y _ j \\\\ & = \\bigg ( \\frac { 2 } { \\vartheta _ B ( n _ 0 ) } - \\frac { 1 } { \\vartheta _ B ( N _ 2 ) } \\bigg ) \\prod _ { j = 1 } ^ { k '' } \\log y _ j . \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , 1 , \\xi _ t , 0 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} h ^ d \\sum _ { l = 1 } ^ { + \\infty } \\beta _ l \\le \\langle b \\rangle + \\varepsilon , \\beta _ l : = \\sup _ { x \\in E _ l } \\beta ( x ) . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} w ( e _ 2 ) ^ p = \\alpha \\prod _ { v \\in e _ 2 } \\sum _ { f : \\ , v \\in f } w ( f ) \\leq \\alpha ( \\Delta \\cdot w ( e _ 2 ) ) ^ r , \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\dd Y _ t = \\left ( A Y _ t + b ( Y _ t ) \\right ) \\dd t + \\sigma ( Y _ t ) \\dd W _ t , \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} \\Omega ^ 1 ( M , \\partial M ) : = \\{ w \\in \\Omega ^ 1 ( M ) , i ^ * w = 0 \\} , \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} r _ { i j } ^ { ( \\ell ) } = \\begin{cases} 1 , & i \\in M _ { \\ell } j = c _ { \\ell } , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} v _ 1 \\cdot ( e _ 1 , v _ 2 ) = ( t _ { V _ 1 } ( v _ 1 ) , \\Phi ( v _ 1 ) v _ 2 ) \\ , , ( e _ 1 , v _ 2 ) \\cdot v _ 2 ' = ( e _ 1 , v _ 2 v _ 2 ' ) \\ ; . \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} P ^ n : = ( 0 , x ^ 1 ) ( 0 , x ^ 2 ) \\dots ( 0 , x ^ { 2 ^ { n - 1 } } ) ( 1 , x ^ { 2 ^ { n - 1 } } ) ( 1 , x ^ { 2 ^ { n - 1 } - 1 } ) \\dots ( 1 , x ^ 2 ) ( 1 , x ^ 1 ) . \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} G \\cdot f ^ { - 1 } ( B ) & = \\left \\{ g \\cdot x \\vert g \\in G , \\ , x \\in S , \\ , f ( x ) \\in B \\right \\} = \\left \\{ x \\in S \\vert f ( x ) \\in B \\right \\} = f ^ { - 1 } ( B ) . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} u _ 1 = & \\ \\frac { 1 } { 2 } ( d _ { 1 1 2 } - 1 , d _ { 2 1 2 } ) , & u _ 2 = & \\ u _ 1 \\ + \\ \\left ( 1 , 0 \\right ) , \\\\ u _ 3 = & \\ \\frac { 1 } { 2 } ( - 1 , 0 ) , & u _ 4 = & \\ u _ 3 \\ + \\ \\left ( 1 , 0 \\right ) , \\\\ u _ 5 = & \\ \\frac { 1 } { 3 } ( - d _ { 1 1 2 } - 1 , - d _ { 2 1 2 } ) , & u _ 6 = & \\ u _ 5 \\ + \\ \\left ( \\frac { 2 } { 3 } , 0 \\right ) . \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} ( U _ i ^ * U _ { i + 1 } L ) & = ( ( U _ i ^ * U _ { i + 1 } L ) ) \\\\ & = r _ { i + 1 } ( U _ i ^ * ) + r _ i ( U _ { i + 1 } ) + r _ i r _ { i + 1 } ( L ) \\\\ & = r _ i d _ { i + 1 } - r _ { i + 1 } d _ i + r _ i r _ { r + 1 } t . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\left ( \\frac { D } { \\pi _ v } \\right ) = \\chi _ D ( - 1 ) \\epsilon ( \\pi _ v , 1 / 2 ) \\epsilon ( \\pi _ v \\otimes \\chi _ D , 1 / 2 ) , \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} F ( x ) & = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 ) } \\frac { \\Gamma \\left ( \\frac { s } { 2 } + \\frac { 1 } { 4 } \\right ) \\Gamma \\left ( - \\frac { s } { 2 } + \\frac { 1 } { 4 } \\right ) } { \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } \\frac { G ( s ) } { s } x ^ { - s } d s . \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} S _ { \\lambda } : = S _ { \\{ 1 , \\ldots , \\lambda _ 1 \\} } \\times S _ { \\{ \\lambda _ 1 + 1 , \\ldots , \\lambda _ 1 + \\lambda _ 2 \\} } \\times \\cdots \\times S _ { \\{ \\lambda _ 1 + \\cdots + \\lambda _ { n - 1 } + 1 , \\ldots , d \\} } \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} X _ 5 \\left ( \\begin{array} { c | c | c | c | c } p _ + & p _ - & & p ' _ + & p ' _ - \\\\ q ' _ - & & q _ + & q _ - & \\\\ & r _ + & r _ - & & r ' _ - \\end{array} \\right ) = 0 \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} z ^ 2 _ { m i n } > 3 ( X _ u - x _ i ) ^ 2 - ( Y _ u - y _ i ) ^ 2 , \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} \\lambda _ n ( c ) \\sim \\widetilde { \\lambda _ n } ( c ) = \\frac { 1 } { 2 } \\exp \\left ( - \\frac { \\pi ^ 2 ( n + \\frac 1 2 ) } { 2 } \\int _ { \\Phi \\left ( \\frac { 2 c } { \\pi ( n + \\frac 1 2 ) } \\right ) } ^ 1 \\frac { 1 } { t ( \\mathbf E ( t ) ) ^ 2 } \\ , \\mathrm { d } t \\right ) . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} \\tau _ { j - 1 } ( y ) & \\in V _ { j } \\cap \\overline { B } _ { r - ( 2 + 3 6 C _ 1 ( m ) \\delta + 2 4 \\delta ) \\sum \\limits _ { k = j } ^ \\infty r _ k } ( x ) \\\\ & \\subset V _ j \\cap \\overline { B } _ { r - r _ j ( 2 + 3 6 C _ 1 ( m ) \\delta + 2 4 \\delta ) } ( x ) \\\\ & \\subset V _ j \\cap \\overline { B } _ { r - 2 r _ j ( 1 + 6 \\delta ) } ( x ) . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} \\mathrm { L H S } & = 1 + m v ( u + w ) + m ^ 2 u w v ^ 2 = 1 + m ^ 2 v ^ 2 + m ^ 2 u w v ^ 2 = 1 + m ^ 2 v ^ 2 ( 1 + u w ) \\\\ & = 1 + m ^ 2 v ^ 4 > m ^ 2 v ^ 4 = \\mathrm { R H S } . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\sigma = u _ 1 u _ 2 \\cdots u _ r \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} f ^ { n + 1 } _ \\omega ( z , w ) = \\begin{pmatrix} 1 / 2 ^ n & \\alpha _ n \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} z \\\\ w \\end{pmatrix} , \\qquad \\textrm { w i t h } \\alpha _ n = \\begin{cases} \\frac { \\alpha _ n } { 2 } & \\textrm { i f } f _ { T ^ n \\omega } = g \\\\ \\frac { \\alpha _ n } { 2 } + 1 & \\textrm { i f } f _ { T ^ n \\omega } = h . \\end{cases} \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\begin{pmatrix} a ' & - b ' & 0 \\\\ - a ' _ 1 & - b ' _ 2 & c ' \\end{pmatrix} \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} \\sum _ { m = g h } \\ , \\ , \\sum _ { \\substack { e \\pmod * { N / h } , \\\\ \\gcd ( e , g ) = 1 } } \\sum _ { \\substack { f \\pmod * { N } , \\\\ \\gcd ( f , h e , N ) = 1 } } \\chi _ 2 ( g ) \\chi _ 1 ( e ) \\chi _ 2 ( f ) \\Bigl ( \\frac { h } { g } \\Bigr ) ^ { \\ ! s } \\sum _ { \\gamma \\in h e + N \\Z } \\sum _ { \\substack { \\delta \\in f + N \\Z , \\\\ \\gcd ( \\gamma , \\delta ) = 1 } } \\frac { y ^ s } { | \\gamma z + \\delta | ^ { 2 s } } . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\sup _ { x \\in B \\times Y } | \\eta _ i ( x ) | _ { g _ { 0 , i } ( x ) } + \\sup _ { x = ( z , y ) \\in B \\times Y } \\Biggl ( \\sup _ { x ' \\in B \\times Y } \\frac { | \\eta _ i ( x ) - \\mathbf { P } ^ { g _ { z , i } } _ { x ' x } ( \\eta _ i ( x ' ) ) | _ { g _ { 0 , i } ( x ) } } { d ^ { g _ { 0 , i } } ( x , x ' ) ^ \\alpha } \\Biggr ) \\leq C . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} I _ \\lambda = \\inf \\{ E _ { \\mu _ 2 } ( \\xi , \\nu ) : ( \\xi , \\nu ) \\in H ^ 1 ( \\mathbb R ) \\times H ^ 1 ( \\mathbb R ) \\ ; \\ ; \\ ; \\ ; F ( \\xi , \\nu ) = \\lambda \\} \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} \\begin{aligned} d X _ { t } & = \\div ( a ^ { \\ast } \\phi ( a \\nabla X _ { t } ) ) \\ , d t + \\frac { 1 } { 2 } L ^ { b } X _ { t } \\ , d t + \\langle b \\nabla X _ { t } , d W _ { t } \\rangle , t \\in ( 0 , T ] , \\\\ X _ { 0 } & = x . \\end{aligned} \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} A ( x , \\tilde { x } ) & = \\alpha \\sum _ { i = 1 } ^ d ( x _ i - \\tilde { x } _ i ) ( a _ i ( x ) - a _ i ( \\tilde { x } ) ) , & B ( x , \\tilde { x } ) & = \\sum _ { i = 1 } ^ d ( x _ i - \\tilde { x } _ i ) ( b _ i ( x ) - b _ i ( \\tilde { x } ) ) . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} c _ { \\sigma } ( \\alpha _ { 1 } , \\ldots , \\alpha _ { q } , \\beta _ { 1 } , \\ldots , \\beta _ { p } ) = c ( g _ { 1 } , \\ldots , g _ { p + q } ) , \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} \\varepsilon _ { b , b ' } : = \\begin{cases} 1 & , \\\\ \\pi ^ { - 1 } & . \\end{cases} \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\frac { 1 } { 2 i \\pi } \\int _ { \\mathcal { C } ( \\sigma ^ 2 , \\varepsilon ) } ( z I - K _ v ^ 2 ) ^ { - 1 } d z = \\sum _ { \\substack { \\tilde { \\sigma } ^ 2 \\in \\ , \\Xi ^ K _ v \\\\ | \\tilde { \\sigma } ^ 2 - \\sigma ^ 2 | < \\varepsilon } } \\pi ^ { \\tilde { \\sigma } } , \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\Pr \\left ( { \\left . { { s _ n } } \\right | { \\bf { R } } } \\right ) = \\int { \\Pr \\left ( { \\left . { { s _ n } , { { \\tilde \\theta } _ n } , \\vartheta } \\right | { \\bf { R } } } \\right ) d { { \\tilde \\theta } _ n } } d \\vartheta \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} \\tilde { \\Psi } : u \\mapsto \\begin{cases} \\int _ { \\mathbb { T } ^ { d } } \\psi ( a \\nabla u ) \\ , d \\xi , & u \\in H ^ { 1 } ( \\mathbb { { T } } ^ { d } ) , \\\\ + \\infty , & u \\in L ^ { 2 } ( \\mathbb { { T } } ^ { d } ) \\setminus H ^ { 1 } ( \\mathbb { { T } } ^ { d } ) , \\end{cases} \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} p _ H ( x \\star u , y \\star u ) = p _ H ( x , y ) , \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} & \\sum _ { \\substack { n m > N \\\\ ( n , m ) = 1 } } \\frac { r ( n ) ^ 2 r ( m ) \\omega ( n ) \\omega ' ( m ) } { \\sqrt { m } } \\\\ & \\qquad \\leqslant \\prod _ p \\left ( 1 + r ( p ) ^ 2 \\omega ( p ) + \\frac { r ( p ) } { \\sqrt { p } } \\omega ' ( p ) \\right ) \\exp \\left ( - \\frac { C \\log N } { ( \\log \\log N ) ^ 3 } \\right ) \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} D _ 1 ( n ) = D ( n + 1 ) , D _ r ( r ) = r ! \\ ; ( r \\ge 1 ) , \\quad \\mbox { a n d } D _ r ( r + 1 ) = r ( r + 1 ) ! \\ ; ( r \\ge 2 ) . \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} e ^ { - t \\Lambda _ { C _ \\infty } ( a , b ) } : = s \\mbox { - } C _ \\infty \\mbox { - } \\lim _ n e ^ { - t \\Lambda _ { C _ \\infty } ( a _ n , b _ n ) } ( t \\geq 0 ) , \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} ( U _ 1 P ^ \\epsilon U _ 1 ^ { - 1 } ) W = ( U _ 1 P ^ \\epsilon U _ 1 ^ { - 1 } ) U _ 1 V U _ 2 = U _ 1 V ' U _ 2 = W ' . \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} J _ { \\delta } : = \\{ ( \\imath , r ) \\mid \\delta ( \\imath , r ) = 1 \\} & & & J _ { \\beta ( \\imath _ 0 , r _ 0 ) ( \\delta ) } : = \\{ ( i , r ) \\mid \\beta ( \\imath _ 0 , r _ 0 ) ( \\delta ) ( \\imath , r ) = 1 \\} . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} \\theta _ i ' = \\omega _ i + \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\sin ( \\theta _ j - \\theta _ i ) . \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} \\lambda _ i = \\dot { \\lambda } _ i + \\frac { k _ i } { h ^ \\vee } \\rho - \\Delta _ i \\delta \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} \\varphi _ { u , n } ( x ) = P ^ n T _ x A _ u V ( e ) , x \\in E . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} ~ ~ ^ { A B C } _ { a } \\nabla ^ \\alpha ( p ( t ) ~ ^ { A B R } \\nabla ^ \\alpha _ b x ( t ) ) + q ( t ) x ( t ) = \\lambda r ( t ) x ( t ) , ~ ~ t \\in \\mathbb { N } _ { a + 1 , b - 1 } , \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\widetilde { c } _ i ( X , Y ) = - \\widetilde { c } _ i ( X - 2 , Y - 2 ) + \\widetilde { c } _ i ( X , Y - 2 ) + \\widetilde { c } _ i ( X - 2 , Y ) . \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} D ( n ) = n ! \\left ( 1 - \\frac { 1 } { 1 ! } + \\frac { 1 } { 2 ! } + \\cdots + \\frac { ( - 1 ) ^ n } { n ! } \\right ) . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} B ( s , f ) & = T _ 1 ( s , f ) - T _ 2 ( s , f ) . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} N _ { \\mathrm { c } \\ , m + 1 } = N _ { m + 1 } - N _ { \\mathrm { d } \\ , m + 1 } - \\sum _ { s = 1 } ^ { m - 1 } \\left [ \\sum _ { n = s } ^ { m } \\binom { m + 1 } { m - n + 1 } N _ { \\mathrm { d } \\ , m - n + 1 } \\mathcal { C } _ { s } ^ { n } \\right ] \\left ( N _ { s } - N _ { \\mathrm { d } s } \\right ) \\\\ - \\binom { m + 1 } { 1 } N _ { \\mathrm { d } 1 } \\mathcal { C } _ { m } ^ { m } \\left ( N _ { m } - N _ { \\mathrm { d } m } \\right ) . \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} c _ { k + 1 } = c _ k - \\frac { \\theta _ k } { 1 - \\theta _ k } b _ k \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\| \\nabla \\times u _ { \\leq Q ( t ) } \\| _ { B ^ { 0 } _ { \\infty , \\infty } } d t < \\infty \\ , \\Lambda ( t ) = 2 ^ { Q ( t ) } , \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | h _ { n + 1 } ( x , y ) - h _ n ( x , y ) \\big | = \\abs { K ( x , y ) } \\prod _ { j = 1 } ^ { k '' } \\log \\abs { y _ j } , \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} p _ { 2 } ( x ) = m 2 ^ { 1 - s } g ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) ^ s [ g ( b ) - m g ( a ) ] . \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + V ( x ) u = \\lambda | u | ^ { p } u \\ \\ \\ \\ x \\in \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\partial _ t g + v \\partial _ x g + a ( t , x , v ) \\partial _ v g + ( c - 1 ) g - \\sigma \\partial _ { v v } g = U ( t , x , v ) \\ , . \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} \\eta = \\nu _ b \\circ \\nu _ { b - 1 } \\circ \\cdots \\circ \\nu _ 1 , \\qquad \\mbox { w h e r e } \\nu _ j = \\tau _ { 1 , j } \\circ \\tau _ { 2 , j } \\circ \\cdots \\circ \\tau _ { a , j } . \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} 2 \\sum _ { m = 2 } ^ \\infty \\frac { \\Lambda ( m ) } m \\biggl ( \\prod _ { p \\mid N } \\Phi _ n ( p ^ e , \\chi _ p ; m ) \\biggr ) \\ , g ( 2 \\log m ) . \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} r _ t ( \\eta ) = \\| k _ 0 \\| _ { \\alpha _ 1 } \\exp \\left ( ( \\alpha _ 1 + c t ) | \\eta | \\right ) , \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} u _ { t } ^ { \\varepsilon } + A \\Delta u ^ { \\varepsilon } = F \\left ( x , t ; u ^ { \\varepsilon } \\right ) + \\mathbf { P } _ { \\varepsilon } ^ { \\beta } u ^ { \\varepsilon } . \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} \\int _ { D } { h } ^ { \\ast } ( \\eta - { g } ^ { \\ast } \\eta ) \\wedge ( \\eta - { h } ^ { \\ast } \\eta ) = & \\int _ { D } h ^ { \\ast } d f _ { g } \\wedge ( \\eta - g ^ { \\ast } \\eta ) \\\\ = & \\int _ { D } d ( h ^ { \\ast } f _ { g } ( \\eta - g ^ { \\ast } \\eta ) ) \\\\ = & \\int _ { \\partial D } h ^ { \\ast } f _ { g } ( \\eta - g ^ { \\ast } \\eta ) = 0 \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} T ( \\alpha _ 2 , \\alpha _ 1 ; \\mathbf { B } ) = \\frac { \\alpha _ 2 - \\alpha _ 1 } { \\varpi ( \\alpha _ 2 ; \\mathbf { B } ) } , \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} Y ( x ; f ) : = \\sum _ { k \\leq x } \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } f ( \\gcd ( j , k ) ) = \\sum _ { n \\leq x } \\frac { ( f * \\phi ) ( n ) } { n } , \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} d X _ { t } + A ( X _ { t } ) \\ , d t \\ni B _ { t } ( X _ { t } ) \\ , d W _ { t } , X _ { 0 } = x , \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} \\left [ \\prod _ { i = 1 } ^ N \\frac { d _ i - 2 } { d _ i } \\right ] \\| T \\| _ { S _ 1 } + | c _ + | + | c _ - | \\leq \\| \\phi \\| _ { c b } \\leq \\| T \\| _ { S _ 1 } + | c _ + | + | c _ - | , \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\vert \\partial V _ t ^ + ( \\vec x ) \\vert & = \\sum _ { i < j } w _ { i j } ( \\chi _ { x _ i \\le t < x _ j } + \\chi _ { x _ j \\le t < x _ i } ) , \\\\ \\vert \\partial V _ t ^ - ( \\vec x ) \\vert & = \\sum _ { i < j } w _ { i j } ( \\chi _ { x _ i < - t \\le x _ j } + \\chi _ { x _ j < - t \\le x _ i } ) \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} { \\tilde z } _ 1 = z _ 1 , \\ ; \\ ; \\ ; { \\tilde z } _ 2 = z _ 2 , \\ ; \\ ; \\ ; { \\tilde z } _ 3 = z _ 3 , \\ ; \\ ; \\ ; { \\tilde z } _ 4 = z _ 4 . \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} F ^ { ( m + 1 ) } _ s ( r ) = q ^ { 2 s } F ^ { ( m ) } _ s ( r ) + ( q ^ m - q ^ { 2 s - 2 } ) F ^ { ( m ) } _ { s - 1 } ( r ) . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} C ( M ^ { 2 n } , P ^ k ) = \\cup _ { P _ { \\sigma } \\in P _ { \\mathfrak { S } } } \\stackrel { \\circ } { P _ { \\sigma } } . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} V ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) , W ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , W _ x ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , u ) = \\bar { Z } ^ { t , x ; \\bar { u } } ( s ) + \\Delta ( s ) \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} \\Delta ^ { ( x ) } : = \\{ ( u _ e ) _ { \\underline { e } = x } : u _ e > 0 , \\ , \\displaystyle \\sum _ { \\{ e : \\underline { e } = x \\} } u _ e = 1 \\} . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { ( 1 - c ) ^ { n + \\beta + z } M _ { n } ( z ; \\beta , c ) } { ( z - n + 1 ) _ n } = 1 , z \\in \\mathbb C \\setminus \\mathbb N . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} ( \\rho , u ) ( x , t ) = \\left \\{ \\begin{array} { l l } ( \\rho _ - , u _ - + \\beta t ) , \\ \\ \\ \\ - \\infty < x < u _ { - } t + \\frac { 1 } { 2 } \\beta t ^ { 2 } , \\\\ v a c u u m , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ u _ { - } t + \\frac { 1 } { 2 } \\beta t ^ { 2 } < x < u _ { + } t + \\frac { 1 } { 2 } \\beta t ^ { 2 } , \\\\ ( \\rho _ + , u _ + + \\beta t ) , \\ \\ \\ \\ u _ { + } t + \\frac { 1 } { 2 } \\beta t ^ { 2 } < x < \\infty . \\end{array} \\right . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} { } \\Phi _ { n } ( z ) + \\Psi _ { n } ( z ) = 2 z \\det ( z - \\mathcal { E } _ { [ 1 , n - 1 ] } ) . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} K ( A z , z '' ) = K ( z , A ^ * z '' ) = K ( z ' , A ^ * z '' ) = K ( A z ' , z '' ) , \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} C ( z ) _ { i j } & = \\begin{cases} z _ i + z _ i ^ { - 1 } & \\ i = j \\\\ [ c _ { i j } ] _ z & \\ i \\neq j \\end{cases} & D ( z ) _ { i j } & = \\delta _ { i j } [ r _ i ] _ z & B ( z ) & = D ( z ) C ( z ) . \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 0 } ^ { k a - 1 } \\binom { m k / n + i } { b k / n } \\binom { b k / n + i } { b k / n } ^ { - 1 } \\right ) ^ { \\frac { n } { k } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! - \\sum _ { \\substack { j | k \\\\ j < k } } \\left ( \\prod _ { i = 0 } ^ { j a - 1 } \\binom { m j / n + i } { b j / n } \\binom { b j / n + i } { b j / n } ^ { - 1 } \\right ) ^ { \\frac { n } { j } } . \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} f ' = { { \\left ( { { f _ { 1 , B } } { \\rm { + } } { f _ { 1 , A } } } \\right ) } \\mathord { \\left / { \\vphantom { { \\left ( { { f _ { 1 , B } } { \\rm { + } } { f _ { 1 , A } } } \\right ) } 2 } } \\right . \\kern - \\nulldelimiterspace } 2 } \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\varepsilon ^ 2 \\ , w _ { \\varepsilon } '''' - 2 \\varepsilon w _ { \\varepsilon } ''' + w _ { \\varepsilon } '' = \\Delta w _ { \\varepsilon } - k w _ { \\varepsilon } ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\lambda _ { j } ( n ) \\lambda _ { j } ( m ) = \\sum _ { d | ( m , n ) } \\lambda _ { j } \\left ( \\frac { n m } { d ^ 2 } \\right ) . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\Delta ( u _ { j , k } ) = \\sum _ { m = 0 } ^ { n } u _ { j , m } \\otimes u _ { m , k } . \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} ( t \\hat \\omega - R i c ( \\hat \\omega ) + \\frac { n + 1 } { 2 n } \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ( t ) ) ^ n = e ^ { \\varphi ( t ) } \\hat \\omega ^ n , \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} ( \\Delta \\otimes \\mathrm { i d } ) \\circ \\Delta = ( \\mathrm { i d } \\otimes \\Delta ) \\circ \\Delta , \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} \\frac 4 q + \\frac n { r } = \\frac n 2 . \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} F ( p ) : = \\psi \\big ( | p | \\big ) , p \\in \\R ^ { n \\times N } , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\Theta _ \\nu ( A , h _ 1 , N ; \\tau ) & = \\Theta _ \\nu ( A , h _ 2 , N ; \\tau ) , \\Theta _ \\nu ( A , - h , N ; \\tau ) = ( - 1 ) ^ \\nu \\Theta _ \\nu ( A , h , N ; \\tau ) , \\\\ \\Theta _ \\nu ( A , N - h , 2 N ; \\tau ) & = ( - 1 ) ^ \\nu \\Theta _ \\nu ( A , N + h , 2 N ; \\tau ) . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} T _ { \\mathrm { m a x } , 2 } = \\left ( \\frac { L _ { 2 } } { L _ { 1 } } \\right ) ^ { 2 } T _ { \\mathrm { m a x } , 1 } . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} ( A _ { \\lambda _ i } , \\Phi _ { \\lambda _ i } ) : = ( s _ { \\lambda _ i } ^ { \\ast } A , { \\lambda _ i } ^ { - 1 } s _ { \\lambda _ i } ^ { \\ast } \\Phi ) \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} X _ 2 \\leq \\frac { M } { N } \\left ( \\sum _ { r = 1 } ^ M \\binom { M } { r } Q _ r ( E _ N ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} R ^ j _ i ( \\mathbf { S } , \\delta ) = C ^ j _ { 0 i } \\mathsf { P } _ { 0 i } + C ^ j _ { 1 i } \\mathsf { P } _ { 1 i } \\ ; , \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} R _ j : = \\frac { n - 1 } { n } R _ 0 + \\Big ( \\frac { n - 1 } { n } \\Big ) ^ j \\frac { R _ 0 } { n } \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\omega ^ { - 1 } ( \\tau _ K ( \\omega \\circ \\chi ) ) = \\tau _ K ( \\chi ) \\cdot \\det _ { \\chi } ( \\kappa ( \\omega ) ) . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} B ^ \\Gamma = B ^ + \\cong B / B ^ - = B _ \\Gamma . \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} H _ { i , j } ' + R _ { i , j } = ( 1 + i ) ^ { \\alpha } ( 1 + j ) ^ { \\alpha } a _ { i + j - 1 } . \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u ^ 3 + z ^ 2 - z + 1 = 0 \\\\ ( 6 x ^ 2 y - 6 y ^ 3 ) z ^ 2 + ( 3 x ^ 3 + 3 x ^ 2 y - 2 7 x y ^ 2 - 3 y ^ 3 + 3 ) z - 1 2 x ^ 2 y + 1 2 y ^ 3 = 0 \\\\ x ^ 2 + 3 y ^ 2 - 1 = 0 \\\\ \\end{array} \\right . . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} f _ 1 = x ^ a - y ^ { b _ 1 } z ^ { c _ 2 } , f _ 2 = y ^ b - x ^ { a _ 2 } z ^ { c _ 1 } , f _ 3 = x ^ { a _ 1 } y ^ { b _ 2 } - z ^ c \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} \\lambda _ i f ( V _ i ^ + , V _ i ^ - ) = \\int _ { t _ { i - 1 } } ^ { t _ i } f ( V _ t ^ + ( \\vec x ) , V _ t ^ - ( \\vec x ) ) d t , i = 0 , \\ldots , p , \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} { \\bf E } [ M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } ^ { - n } ] = \\prod _ { k = 0 } ^ { n - 1 } \\frac { \\Gamma \\bigl ( 4 + \\lambda _ 1 + \\lambda _ 2 + n + k \\bigr ) } { \\Gamma \\bigl ( 2 + \\lambda _ 1 + k \\bigr ) \\Gamma \\bigl ( 2 + \\lambda _ 2 + k \\bigr ) \\Gamma \\bigl ( 1 + k \\bigr ) } . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\dot { x } = g ( t , x ) \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\int _ { \\Omega } | \\nabla U _ * | ^ 2 \\\\ = \\displaystyle \\int _ \\Omega | \\nabla u _ 0 | ^ 2 + \\int _ \\Omega | \\nabla V | ^ 2 - \\displaystyle 2 \\int _ \\Omega \\nabla V \\nabla u _ 0 \\\\ = \\displaystyle \\int _ \\Omega | \\nabla u _ 0 | ^ 2 + \\int _ \\Omega | \\nabla V | ^ 2 - \\displaystyle 2 \\int _ \\Omega u _ 0 ^ { 2 ^ * - 1 } V . \\\\ \\end{array} \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} \\varphi \\ , : \\ , ( E _ G ) \\ , = \\ , ( { \\rm a d } ( E _ G ) \\oplus ( E _ H ) ) / ( E _ H ) \\ , \\longrightarrow \\ , ( E _ G ) \\ , . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} r Y _ { 2 } ( r ) + ( 2 \\eta { r } + 1 + \\tau ) Y _ { 1 } ( r ) + \\Big [ \\eta ( 1 + \\tau ) + \\beta \\Big ] Y ( r ) = 0 . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} s _ p ( h s ) = \\begin{cases} ( c , - d ) _ p & c d \\neq 0 , \\mathrm { o r d } _ p ( d ) , \\\\ 1 & \\end{cases} x ( h s ) = \\begin{cases} - d & d \\neq 0 , \\\\ c & d = 0 , \\end{cases} x ( h \\beta _ m ) = \\begin{cases} - d p ^ m & d \\neq 0 , \\\\ c p ^ { - m } & d = 0 . \\end{cases} \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\zeta _ 1 ^ { ( 2 ) } \\\\ \\zeta _ 2 ^ { ( 2 ) } \\\\ \\zeta _ 1 ^ { ( 1 ) } \\end{array} \\right ) \\circ { \\rm g w } _ { \\infty } = \\frac { 1 } { 2 } \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end{array} \\right ) \\left ( \\begin{array} { c } \\lambda _ 1 ^ { ( 2 ) } \\\\ \\lambda _ 2 ^ { ( 2 ) } \\\\ \\lambda _ 1 ^ { ( 1 ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} - \\nabla \\cdot a \\cdot \\nabla + b \\cdot \\nabla \\equiv - \\sum _ { i , j = 1 } ^ d \\nabla _ i \\ , a _ { i j } ( x ) \\nabla _ j + \\sum _ { j = 1 } ^ d b _ j ( x ) \\nabla _ j , \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} g ( u ) ~ = ~ T _ u g ( e ) ~ = ~ \\tilde { g } ( e ) + G \\ 1 ( e ) + G A _ u g ( e ) ~ \\geq ~ g ( e ) + G A _ u g ( e ) . \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} P _ N w ( t ) & = e ^ { - t A _ N } P _ N w _ 0 + \\int _ 0 ^ t e ^ { - ( t - \\tau ) A _ N } ( P _ N \\xi + P _ N f ( \\tau ) ) d \\tau \\\\ & = e ^ { - t A _ N } P _ N w _ 0 + A _ N ^ { - 1 } ( P _ N \\xi - e ^ { - t A _ N } P _ N \\xi ) + \\int _ 0 ^ t e ^ { - ( t - \\tau ) A _ N } P _ N f ( \\tau ) d \\tau \\\\ & = e ^ { - t A } P _ N w _ 0 + A ^ { - 1 } ( P _ N \\xi - e ^ { - t A } P _ N \\xi ) + \\int _ 0 ^ t e ^ { - ( t - \\tau ) A } P _ N f ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} U _ { 0 , 2 } & \\ll \\frac { N ( \\log N ) ^ { \\nu - 1 } } { ( \\log y ) ^ { \\nu } } \\prod _ { \\substack { \\ell \\le x \\\\ \\ell ~ } } \\ ( 1 - \\frac { 1 } { \\ell } \\ ) ^ { - \\nu } \\\\ & \\ll \\frac { N ( \\log N ) ^ { \\nu - 1 } ( \\log x ) ^ { \\nu } } { ( \\log y ) ^ { \\nu } } . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\zeta = 1 / \\log \\log ( p / k ) . \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} u ^ { \\flat } _ { s t } : = \\delta u _ { s t } - A ^ 1 _ { s t } u _ s . \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} A ' _ s & = \\sum _ r F ^ { ( m + 1 ) } _ s ( r ) A _ r , \\\\ C ' _ s & = \\sum _ r F ^ { ( m ) } _ s ( r ) B _ r , \\\\ B ' _ s & = q ^ m \\sum _ r F ^ { ( m ) } _ s ( r ) C _ r . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} ( z ^ { - 1 } + 1 ) \\widetilde b _ { j k } ^ * ( z ) - \\sum _ { r = 0 } ^ { k - 1 } w _ { k , r + 1 } \\frac { \\widetilde b _ { j r } ^ * ( z ) } { z ^ { - 1 } - 1 } = ( z ^ { - 1 } - 1 ) ^ j h _ { j k } ^ * ( z ) , \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} Y ^ n _ t ( x ) \\stackrel { d } { = } \\bigl ( \\sqrt x + B ^ 1 _ t \\bigr ) ^ 2 + \\sum _ { i = 2 } ^ n \\bigl ( B ^ i _ t \\bigr ) ^ 2 \\stackrel { d } { = } ( \\sqrt x + \\xi \\sqrt t ) ^ 2 + Y ^ { n - 1 } _ t , t \\ge 0 , \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} S _ \\nu = \\{ M ^ n _ \\omega \\ , | \\ , \\omega \\in \\Omega , \\ , n \\in \\mathbb N \\} . \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} \\begin{pmatrix} \\chi _ 0 ( R ) & \\chi _ 1 ( R ) & \\dots & \\chi _ { p } ( R ) \\\\ \\chi _ 1 ( R ) & \\chi _ 2 ( R ) & \\dots & \\chi _ { p + 1 } ( R ) \\\\ \\vdots & \\vdots & & \\vdots \\\\ \\chi _ { p } ( R ) & \\chi _ { p + 1 } ( R ) & \\dots & \\chi _ { 2 p } ( R ) \\\\ \\end{pmatrix} \\begin{pmatrix} \\tilde \\alpha _ 0 \\\\ \\tilde \\alpha _ 1 \\\\ \\vdots \\\\ \\tilde \\alpha _ { p } \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\psi _ p ( z ) = \\sum _ { j = 0 } ^ \\kappa d _ { j , p } ( a z + b ) ^ j , \\forall p \\in \\{ 0 , . . . , \\kappa \\} , \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} \\partial _ { x _ j } ( I _ { h , T } \\mathbf { u } ( X ) - \\mathbf { u } ( X ) ) = \\sum _ { i \\in \\mathcal { I } ^ { s ' } } \\partial _ { x _ j } \\Phi _ { i , T } ( X ) ( \\mathbf { E } _ i ^ s ( X ) + \\mathbf { F } _ i ^ s ( X ) ) + \\sum _ { i \\in \\mathcal { I } } \\partial _ { x _ j } \\Phi _ { i , T } ( X ) \\mathbf { R } _ i ^ s ( X ) , ~ s = \\pm , \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} d a & = \\sum _ { x \\in X ^ { \\underline { n + 1 } } } a _ x \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i x _ 0 \\cdots \\hat { \\bar { x } } _ i \\cdots x _ n \\\\ & = \\sum _ { y \\in X ^ { \\underline { n } } } \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i \\sum _ { z \\in X } a ( y _ 0 \\cdots y _ { i - 1 } \\bar { z } y _ i \\cdots y _ { n - 1 } ) \\ : y . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} w _ 1 = \\frac { 1 } { 4 ( 1 + 4 ^ { 1 / ( p - 2 ) } ) ^ 2 } , ~ w _ 2 = \\frac { 4 ^ { 1 / ( p - 2 ) } } { 4 ( 1 + 4 ^ { 1 / ( p - 2 ) } ) ^ 2 } , ~ w _ 3 = \\frac { 4 ^ { 2 / ( p - 2 ) } } { 4 ( 1 + 4 ^ { 1 / ( p - 2 ) } ) ^ 2 } . \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} e ( G , N ) = \\sum _ { G _ { N } } \\dfrac { \\left \\lvert \\mathrm { A u t } ( G ) \\right \\rvert } { \\left \\lvert \\mathrm { A u t } _ { \\mathcal { B } r } ( G _ { N } ) \\right \\rvert } . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\sum _ { k = m } ^ N \\binom { k } { m } = \\binom { N + 1 } { m + 1 } , m \\leq N . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} ( \\gamma _ { a , w } ' ( t ) ) _ { a \\in Q _ 1 ^ { ( i j ) } } = - M _ { i j } ( t ) \\cdot ( \\gamma _ { a , w } ( t ) ) _ { a \\in Q _ 1 ^ { ( i j ) } } + ( F _ { a , w } ( t ) ) _ { a \\in Q _ 1 ^ { ( i j ) } } , \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} f ( 0 ) = \\sum _ { m \\mid N , q \\mid \\frac { N } { \\gcd ( m , N / m ) } } \\varphi ( \\gcd ( m , N / m ) ) = \\Psi _ 1 ( N , q ) \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\frac { \\varphi _ 2 ( x ) \\sigma _ 2 ( x ) } { x \\varphi _ 1 ( x ) \\sigma _ 1 ( x ) } = o ( 1 ) , \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\min _ { \\| \\vec x \\| _ p = 1 } \\{ r \\norm { \\vec x } - ( \\vec x , \\vec v ) \\} & = \\min _ { \\| \\vec u \\| _ p = 1 } \\{ r \\norm { \\vec u } - ( \\vec u , | \\vec v | ) \\} \\\\ & = \\min _ { \\vec w \\neq \\vec 0 } \\frac { r \\norm { \\vec w } - ( \\vec w , | \\vec v | ) } { \\| \\vec w \\| _ p } \\\\ & = \\min _ { \\norm { \\vec z } = 1 } \\frac { r - ( \\vec z , | \\vec v | ) } { \\| \\vec z \\| _ p } \\\\ & = \\min _ { { \\vec z } \\in B _ \\infty } G ( \\vec z ) , \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} X ^ x _ t = x + \\int _ 0 ^ t b \\bigl ( X ^ x _ s \\bigr ) \\ , d t + \\int _ 0 ^ t \\sigma \\bigl ( X ^ x _ s \\bigr ) \\ , d B _ s , X ^ x _ 0 = x , \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} S _ { a _ r } p _ i = \\frac 1 { 2 ^ { i } } p _ i , p _ i : = \\ell _ r ^ { ( r - i ) } , \\ell _ r ( x ) : = \\frac { 1 } { r ! } \\prod _ { j = 1 } ^ r ( x + j ) . \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} \\Omega _ { n , j } = \\{ \\omega \\in \\Omega \\ , | \\ , \\forall \\alpha \\in \\Omega , \\ , \\exists k \\ , : \\ , d _ { N _ n } ( f ^ n _ \\omega , f ^ n _ \\alpha ) < \\varepsilon _ j \\} . \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} C _ { i } = B _ i l o g _ { 2 } \\left ( 1 + \\dfrac { p _ { i } / L _ i } { N } \\right ) \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} S \\left ( p , k \\right ) = c ^ { - p } \\sum _ { j = 0 } ^ { p } \\binom { p } { j } \\left ( - d \\right ) ^ { p - j } S _ { c , d } \\left ( j , k \\right ) . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} A _ { k \\ldots l } ^ 2 + 2 = \\sum _ { i = k } ^ l ( A _ i ^ 2 + 2 ) . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} & C _ 1 : = \\frac { 1 } { 1 - s _ * \\beta } , & & \\lambda _ 1 = - \\frac { \\log ( 1 - s _ * \\beta ) } { 2 s _ * } \\\\ & C _ 2 : = \\frac { 1 } { 1 - \\epsilon \\delta ( s _ * - \\delta ) \\beta } , & & \\lambda _ 2 = - \\frac { \\log ( 1 - \\epsilon \\delta ( s _ * - \\delta ) \\beta ) } { 2 s _ * } \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\alpha ^ b = ( n + 1 , p + m - 1 - n ) , \\alpha ^ t = ( n + 3 - m , \\dots , 4 , 3 ) \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} { \\cal L _ \\gamma } : = \\int _ a ^ b F ( \\gamma ( t ) , \\dot \\gamma ( t ) ) d t , \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\dot x _ t = \\textrm { F } ( x _ t ) \\dot h _ t \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} \\dot { \\phi } = \\frac { 1 } { 2 r ^ 2 } ( a ^ 2 - 1 ) , \\ \\ \\dot { a } = 2 a \\phi , \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} g ( z , w ) = \\begin{pmatrix} 1 / 2 & 0 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} z \\\\ w \\end{pmatrix} , h ( z , w ) = \\begin{pmatrix} 1 / 2 & 1 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} z \\\\ w \\end{pmatrix} . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} \\widetilde { \\mathcal Q } ( M ; x , y , a , b ) = \\sum _ { A \\subseteq E } x ^ { r ( M / A ) } y ^ { r ^ * ( M \\backslash A ^ c ) } a ^ { s ( M / A ) } b ^ { s ( M \\backslash A ^ c ) } . \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} F ^ * = x + x F ^ * e ^ F + x ^ 2 F ' + x ^ 2 F ^ * F ' e ^ F . \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 0 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; f _ t ( x _ t , x _ { t - 1 } , \\xi _ { t j } ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ { t j } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} B _ i ( t ) = W _ i ( t ) - \\langle W _ i , \\nu M \\rangle ( t ) = W _ i ( t ) - \\nu \\sum _ { k ; k \\neq i } \\int _ { 0 } ^ { t } \\frac { d s } { X ^ { 1 / 2 , \\mu } _ i ( s ) - X ^ { 1 / 2 , \\mu } _ k ( s ) } \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} \\mathbb { V } _ { n } ^ { ( N ) } ( Y ) = \\left ( \\left ( \\mathbb { P } _ { n } ^ { ( N ) } ( f _ { i } f _ { j } ) \\right ) _ { i , j } \\right ) . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} z = \\sum _ { j = N } ^ \\infty k _ j t ^ j \\in F _ { p ^ m } ( \\ ! ( t ) \\ ! ) \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{gather*} \\beta = \\partial \\widehat { \\beta } , \\beta _ { \\Gamma } = \\partial \\widehat { \\beta } _ { \\Gamma } \\end{gather*}"}
-{"id": "9169.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , s ) : = \\Gamma _ { \\C } ( s ) \\Gamma _ { \\C } ( s + k ) \\Gamma _ { \\C } ( s - k + 1 ) L ( f \\otimes \\mathrm { A d } ( g ) , s ) , \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} - n ! \\ , R \\left | B ^ n _ R \\right | + \\sum _ { i = 0 } ^ { p } \\tilde \\xi _ { p , i } \\tilde \\alpha _ i = - R ^ { 2 p + 2 } . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} C _ n ( X ) = \\bigoplus _ { \\substack { u \\in X ^ + } } C _ n ( X ; u ) . \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} F _ { 4 , 1 4 } ( \\tau ) = \\left ( \\begin{smallmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 2 ( u - 1 / u ) \\\\ ( u + 1 / u ) \\end{smallmatrix} \\right ) X Y ^ 3 - \\left ( \\begin{smallmatrix} ( u + 1 / u ) \\\\ 2 ( u - 1 / u ) \\\\ 0 \\\\ 0 \\\\ 0 \\end{smallmatrix} \\right ) X ^ 3 Y + \\ldots \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} \\begin{aligned} & I _ 3 ( u _ 0 ) + \\int _ { - \\infty } ^ { \\infty } \\Big ( G ( u ) - \\frac { F ( u ) } { 3 } \\ , \\Big ) ( t , x ) d x \\\\ & \\geq \\frac 1 2 \\| \\partial _ x u ( t ) \\| _ 2 ^ 2 - c \\| u _ 0 \\| _ 2 ^ 4 \\ , \\| \\partial _ x u ( t ) \\| _ 2 ^ 2 \\geq \\frac 1 4 \\| \\partial _ x u ( t ) \\| _ 2 ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} d _ { G _ 1 , V ' _ i } ( v ) = n _ i p \\pm ( n _ i p ) ^ { 3 / 5 } \\enspace \\enspace d _ { G _ 1 , B } ( v ) = p | B | \\pm ( n _ i p ) ^ { 3 / 5 } = \\epsilon n _ i p \\pm ( n _ i p ) ^ { 3 / 5 } . \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\beta _ { M , N } ( a , \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } & \\prod \\limits _ { k = 0 } ^ \\infty \\beta _ { M - 1 , N } ( \\hat { a } _ i , \\ , b _ 0 + k a _ i , \\ , \\ , b _ 1 , \\cdots , b _ N ) , \\\\ \\beta _ { M , N } ( a , \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } & \\prod \\limits _ { n _ 1 , \\cdots , n _ M = 0 } ^ \\infty \\beta _ { 0 , N } ( b _ 0 + \\Omega , \\ , \\ , b _ 1 , \\cdots , b _ N ) . \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\varphi = \\begin{bmatrix} x _ 1 ^ { d _ 1 } & x _ 1 ^ { d _ 2 } & \\cdots & x _ 1 ^ { d _ n } \\\\ x _ 2 ^ { d _ 1 } & x _ 2 ^ { d _ 2 } & \\cdots & x _ 2 ^ { d _ n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ x _ n ^ { d _ 1 } & x _ n ^ { d _ n } & \\cdots & x _ n ^ { d _ n } \\\\ g _ 1 & g _ 2 & \\cdots & g _ n \\end{bmatrix} , \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} q ( B ^ D ) \\cap N _ D B = q ( B ^ D ) \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { b , a } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\bar E _ { a , b } \\bar E _ { b , a } } E _ { b , a } E _ { a , b } + ( q _ a - q _ a ^ { - 1 } ) ^ { - 1 } ( K _ { a , b } - K _ { a , b } ^ { - 1 } ) , & M = N = 1 ; \\\\ \\displaystyle \\sum _ { t = 0 } ^ { \\min ( M , N ) } E _ { b , a } ^ { ( N - t ) } \\begin{bmatrix} K _ { a , b } ; 2 t - M - N \\\\ t \\end{bmatrix} E _ { a , b } ^ { ( M - t ) } , & . \\end{cases} \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} W _ { \\infty , \\xi } ( u ( x ) t ( a ) k _ { \\theta } ) = e ^ { 2 \\pi \\sqrt { - 1 } \\xi x } a ^ { k + 1 / 2 } e ^ { - 2 \\pi \\xi a ^ 2 } e ^ { \\sqrt { - 1 } ( k + 1 / 2 ) \\theta } \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} H _ { \\mu \\iota } & : = u _ { \\mu } ( t , x ) \\nabla _ { y } \\overline { u _ { \\mu } ( t , y ) } + \\nabla _ { x } u _ { \\iota } ( t , x ) \\overline { u _ { \\iota } ( t , y ) } , \\\\ G _ { \\mu \\iota } & : = u _ { \\mu } ( t , x ) \\nabla _ { y } u _ { \\iota } ( t , y ) - \\nabla _ { x } u _ { \\mu } ( t , x ) u _ { \\iota } ( t , y ) . \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} \\mu ( L ) = \\mu _ { 1 } ( L _ 1 ) + \\mu _ { 2 } ( L _ 2 ) + \\ldots + \\mu _ { k } ( L _ k ) , \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} \\det ( A ) & = \\left ( - \\frac { r } { r _ 2 } \\right ) \\left ( - r \\right ) \\left ( - \\frac { r } { r _ 4 } \\right ) \\ldots \\left ( - r \\right ) . \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} A N _ + : = \\left \\{ b \\in A N _ { \\mathbb { R } } \\colon \\ , \\Delta _ i ^ { ( k ) } ( b ) > 0 , \\ , \\forall 1 \\leq i < k \\leq n \\right \\} . \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} \\ell ' = b _ 1 c + b _ 2 c _ 2 \\geq b ' _ 1 c ' + b ' _ 2 c ' _ 2 = \\ell ' \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} J ( u ) & = \\displaystyle \\int _ { \\Omega } \\frac { B ( x ) } { p ( x ) } | \\nabla u ( x ) | ^ { p ( x ) } d x + \\displaystyle \\int _ { \\Omega } \\frac { C ( x ) } { p ( x ) - 1 } | u ( x ) | ^ { p ( x ) - 1 } d x \\\\ & + \\displaystyle \\int _ { \\Omega } \\frac { A ( x ) } { p ( x ) } | u ( x ) | ^ { p ( x ) } d x + \\displaystyle \\int _ { \\Omega } \\frac { D ( x ) } { p ( x ) + 1 } | u ( x ) | ^ { p ( x ) + 1 } d x \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} \\nabla \\mathbf { u } ^ - ( \\widetilde { Y } _ i ) & = \\int ^ { \\tilde { t } _ i } _ 0 \\frac { d } { d t } \\nabla \\mathbf { u } ^ - ( Y _ i ( t , x ) ) d t + \\nabla \\mathbf { u } ^ - ( X ) . \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} H _ t ( s ) : = G ( s + i t ) G ( s - i t ) \\Gamma ( s + 1 + i t ) \\Gamma ( s + 1 - i t ) . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\xi : = ( [ \\epsilon ] - 1 ) / ( [ \\epsilon ^ { \\frac { 1 } { p } } ] - 1 ) \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\mathcal W _ { \\mathfrak R _ p \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) = \\mathcal W _ { \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) . \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} \\tilde \\ell = r _ 1 c + r _ 2 c _ 2 , b _ 1 r _ 2 = r _ 1 b _ 2 , \\tilde n = a _ 1 r _ 1 + a r _ 2 . \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} E \\big [ f ( \\Phi _ { 0 , t + s } ( y _ 0 ) ) | \\mathcal { F } _ s \\big ] = E \\big [ f ( \\Phi _ { s , t + s } \\circ \\Phi _ { 0 , s } ( y _ 0 ) ) | \\mathcal { F } _ s \\big ] = g _ { s , t , f } ( \\Phi _ { 0 , s } ( y _ 0 ) ) , \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} Q ( x ) = \\sum _ { l _ 1 , . . . , l _ N = 0 } ^ \\infty \\eta _ { l _ 1 } ^ - ( x _ 1 ) \\otimes \\cdots \\otimes \\eta _ { l _ N } ^ - ( x _ N ) \\otimes A e _ { ( l _ 1 , . . . , l _ N ) } , \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} h ( x _ 1 , x _ 2 ) = ( x _ 1 + 1 ) ( x _ 2 + 1 ) , x = ( x _ 1 , x _ 2 ) \\in \\Z _ + ^ 2 , \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} ( a ; q ) _ { k } = \\prod _ { n = 0 } ^ { k - 1 } ( 1 - a q ^ { n } ) \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} f ( x _ 0 ) = \\frac 1 \\gamma - | \\omega | - \\frac { 1 } { 4 \\gamma ^ 4 \\beta _ 0 } \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} \\partial _ t \\rho ^ N _ t = \\tilde \\L _ N ^ \\ast \\rho ^ N _ t , \\rho ^ N | _ { t = 0 } = \\rho ^ N _ 0 \\in C _ P ^ \\infty ( H _ N ) , \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K ( n ) \\} \\\\ = C _ { n } ^ { K ( n ) } ( 2 ^ { K ( n ) } - 1 ) . \\end{array} \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\lim _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } E | \\sum _ { \\substack { j \\in J _ { K } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) | ^ 2 = 0 . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} z = v \\left ( x ' , h \\left ( x ' , z , t \\right ) , t \\right ) \\big ( x ' = \\left ( x _ 1 , \\cdots , x _ { n - 1 } \\right ) \\big ) . \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} ( 1 - \\frac { \\epsilon } { 3 } ) ^ { 2 } & \\sum \\limits ^ { n } _ { \\substack { l = 0 } } ( - 1 ) ^ { l } \\binom { k + l } { k } S _ { k + l , n } \\leq P ( \\tilde { V } _ { n } = k ) \\\\ & \\leq ( 1 + \\frac { \\epsilon } { 3 } ) ^ { 3 } \\sum \\limits ^ { n } _ { \\substack { l = 0 } } ( - 1 ) ^ { l } \\binom { k + l } { k } S _ { k + l , n } \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} c _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x _ { \\lambda _ 2 } ( a ) + c _ 2 ~ ^ { A B R } D _ b ^ \\alpha x _ { \\lambda _ 2 } ( a ) = 0 , \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} V _ { \\delta , t } ( y ) : = \\frac { 1 } { 2 \\pi i } \\int \\limits _ { 3 - i \\infty } ^ { 3 + i \\infty } \\frac { H _ { t } ( s ) } { H _ { t } ( \\delta ) } \\zeta _ q ( 1 + 2 s ) y ^ { - s } \\frac { 2 s } { ( s - \\delta ) ( s + \\delta ) } \\ ; d s \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} T _ { A _ { n } } ( f ) & \\leqslant \\mathbb { E } \\left ( \\left \\vert \\mathbb { G } _ { n } ^ { ( N ) } ( f ) \\right \\vert ( d _ { 0 } + \\varepsilon ) 1 _ { A _ { n } } \\right ) \\\\ & = ( d _ { 0 } + \\varepsilon ) \\sqrt { \\mathbb { V } \\left ( \\mathbb { G } _ n ^ { ( N ) } ( f ) \\right ) } \\mathbb { E } \\left ( \\left \\vert \\mathcal { N } ( 0 , 1 ) \\right \\vert \\right ) \\leqslant \\sqrt { \\frac { 2 } { \\pi } } ( d _ { 0 } + \\varepsilon ) \\sigma _ { f } . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} T ( \\Gamma ; 2 , 1 ) & = \\# \\{ A \\subseteq E : n ( \\Gamma \\backslash A ^ c ) = 0 \\} \\\\ & = \\# \\{ A \\subseteq E : \\mbox { \\rm c o n n e c t e d c o m p o n e n t s o f } \\Gamma \\backslash A ^ c \\ : \\mbox { \\rm a r e t r e e s } \\} , \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} \\pi _ { \\rho } ^ { o , \\kappa , h } ( n ) = \\pi _ { \\rho } ^ { o , \\kappa , h } ( 0 ) \\frac { ( \\rho h - \\lfloor \\rho h \\rfloor ) h \\lambda _ { \\ell _ { \\kappa } } } { n \\mu _ { i _ { \\kappa } } } \\prod _ { n ' = 1 } ^ { n - 1 } \\frac { h \\lambda _ { \\ell _ { \\kappa } } } { n ' \\mu _ { i _ { \\kappa } } } ; \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} P _ { N _ { 0 } } = \\prod _ { N = 1 } ^ { N _ { 0 } } p _ { N } , M _ { N _ { 0 } } = \\prod _ { N = 1 } ^ { N _ { 0 } } m _ { N } , S _ { N _ { 0 } } = \\sum _ { N = 1 } ^ { N _ { 0 } } m _ { N } . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} I _ 3 ( B ) = 0 . \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} \\mathcal { F } _ + ^ { \\ , \\leqslant N } : = \\ ; \\bigoplus _ { L = 0 } ^ N \\ ; \\Bigg ( \\bigoplus _ { \\substack { n + m = L \\\\ n \\leqslant N _ 1 , \\ , m \\leqslant N _ 2 } } \\big ( \\mathfrak { h } ^ { ( 1 ) } \\big ) ^ { \\otimes _ { \\operatorname { s y m } } n } \\otimes \\big ( \\mathfrak { h } ^ { ( 2 ) } \\big ) ^ { \\otimes _ { \\operatorname { s y m } } m } \\Bigg ) . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} \\tau _ K ( \\chi ) = \\prod _ U \\tau _ { L ^ U } ( \\lambda _ U ) ^ { z _ U } . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} \\norm { F } _ { L ^ 2 \\bigl ( \\R \\setminus ( - 1 , 1 ) \\bigr ) } ^ 2 \\leq \\frac { 2 } { \\pi ^ { 1 / 2 } c ^ { 1 / 2 } } \\sum _ { k = 0 } ^ n \\frac { ( 2 c ) ^ k } { k ! } e ^ { - c } . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} R _ 0 = \\{ ( ( i - 1 , r + 2 ) , ( i - 1 , r + 4 s + 2 ) ) , ( ( i + 1 , r + 2 ) , ( i + 1 , r + 4 s + 2 ) ) \\} . \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} c _ \\pm = \\frac { 1 } { 2 } \\lim _ { n \\to \\infty } \\dot { \\phi } ( 2 n ) \\pm \\frac { 1 } { 2 } \\lim _ { n \\to \\infty } \\dot { \\phi } ( 2 n + 1 ) , \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} [ [ x _ 1 , \\cdots , x _ n ] , \\alpha ( y _ 1 ) , \\cdots , \\alpha ( y _ { n - 1 } ) ] \\\\ = \\sum _ { i = 1 } ^ n [ \\alpha ( x _ 1 ) , \\cdots , \\alpha ( x _ { i - 1 } ) , [ x _ i , y _ 1 , \\cdots , y _ { n - 1 } ] , \\alpha ( x _ { i + 1 } ) , \\cdots \\alpha ( x _ n ) ] , \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} & \\alpha _ { n } ^ { ( N ) } ( f ) = \\alpha _ { n } ^ { ( 0 ) } ( \\phi _ { ( 1 ) } \\circ . . . \\circ \\phi _ { ( N ) } f ) + \\digamma _ { n } ^ { ( N ) } ( f ) , \\\\ & \\digamma _ { n } ^ { ( N ) } ( f ) = \\sum _ { k = 1 } ^ { N } \\Gamma _ { n } ^ { ( k ) } ( \\phi _ { ( k + 1 ) } \\circ . . . \\circ \\phi _ { ( N ) } f ) . \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} f _ 1 = x _ { 1 2 } x _ { 3 4 } - x _ { 1 3 } x _ { 2 4 } + x _ { 1 4 } x _ { 2 3 } . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} J _ { 1 , 3 } = \\frac { \\log \\sqrt { 2 \\pi } } { 1 + a } x ^ { 1 + a } + \\zeta ( - a ) \\log \\sqrt { 2 \\pi } + O _ { a } ( x ^ a ) , \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} \\bar r _ 1 = \\frac { a _ { 0 , \\sup } - a _ { 2 , \\inf } \\underbar r _ 2 - k \\frac { \\chi _ 1 } { d _ 3 } \\underbar r _ 1 } { a _ { 1 , \\inf } - k \\frac { \\chi _ 1 } { d _ 3 } } , \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} \\mathbf { S } _ h ( \\Omega ) = & \\left \\{ \\mathbf { v } \\in [ L ^ 2 ( \\Omega ) ] ^ 2 \\ , : \\ , \\mathbf { v } | _ T \\in \\mathbf { S } _ h ( T ) ; \\right . \\\\ & \\left . ~ \\mathbf { v } | _ { T _ 1 } ( N ) = \\mathbf { v } | _ { T _ 2 } ( N ) ~ \\forall N \\in \\mathcal { N } _ h , ~ \\forall T _ 1 , T _ 2 \\in \\mathcal { T } _ h ~ \\textrm { s u c h ~ t h a t } ~ N \\in T _ 1 \\cap T _ 2 \\right \\} . \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} x _ + = { \\rm E x p } _ x ( \\xi ) . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} M _ 0 = \\{ m _ { s , r + 2 } ^ { ( n - 2 ) } , m _ { 2 s , r + 1 } ^ { ( n ) } \\} , \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} ( \\mu , \\lambda , \\kappa , \\nu ) = \\varepsilon ( \\tilde \\mu , \\tilde \\lambda , \\tilde \\kappa , \\tilde \\nu ) , \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\lvert \\nabla ^ j \\left ( \\varphi ^ { \\ast } g - g _ C \\right ) \\rvert _ C = O ( r ^ { \\nu - j } ) , \\quad \\forall j \\in \\mathbb { N } _ 0 . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} \\Gamma _ M ( k w \\ , | \\ , a ) = k ^ { - B _ { M , M } ( k w \\ , | \\ , a ) / M ! } \\ , \\prod \\limits _ { p _ 1 , \\cdots , p _ M = 0 } ^ { k - 1 } \\Gamma _ M \\Bigl ( w + \\frac { \\sum _ { j = 1 } ^ M p _ j a _ j } { k } \\ , \\Big | \\ , a \\Bigr ) . \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\zeta _ i ^ { ( k ) } = \\frac { 1 } { 2 } \\ell _ i ^ { ( k ) } \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\tau ( x ) - \\mathbf { P } _ { x ' x } ^ { g _ P } ( \\tau ( x ' ) ) = \\int _ 0 ^ \\ell \\frac { d } { d t } \\mathbf { P } ^ { g _ P } _ { \\gamma ( \\ell - t ) x } ( \\tau ( \\gamma ( \\ell - t ) ) ) \\ , d t = - \\int _ 0 ^ \\ell \\mathbf { P } ^ { g _ P } _ { \\gamma ( \\ell - t ) x } [ ( \\nabla _ { \\dot { \\gamma } ( \\ell - t ) } \\tau ) ( \\gamma ( \\ell - t ) ) ] \\ , d t . \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} W : = \\sum _ { i \\in I ^ 0 } \\vartheta _ { p _ i } . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} F [ u ] : = \\nabla ^ { 2 } u + L ( \\cdot , \\nabla u ) , \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} \\abs { h _ i } \\geq \\frac { 1 } { 2 \\sqrt { n } } \\abs { h } , \\ , i = 1 , \\cdots , n . \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\overline { d } _ { ( T _ n ) } ( B ) = \\varlimsup _ n \\ , \\frac { | B \\cap T _ n | } { | T _ n | } , \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} X _ 2 = \\sum _ { i = 0 } ^ K \\frac { ( \\nu _ i - t \\pi _ i ) ^ 2 } { t \\pi _ i } \\leq & \\sum _ { i = 0 } ^ K \\frac { \\pi _ i } { t } \\left ( \\sum _ { r = 1 } ^ M \\binom { M } { r } Q _ r ( E _ N ) \\right ) ^ 2 \\\\ & = \\frac { 1 } { t } \\left ( \\sum _ { r = 1 } ^ M \\binom { M } { r } Q _ r ( E _ N ) \\right ) ^ 2 \\sum _ { i = 0 } ^ K \\pi _ i = \\frac { M } { N } \\left ( \\sum _ { r = 1 } ^ M \\binom { M } { r } Q _ r ( E _ N ) \\right ) ^ 2 \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} D ( z , z ' ) : = F ( z , z ' ) - q _ z ^ * q _ { z ' } \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} N _ { \\mathrm { c } 3 } = N _ { 3 } - N _ { \\mathrm { d } 3 } - \\binom { 3 } { 2 } N _ { \\mathrm { d } 1 } \\left ( N _ { 2 } - N _ { \\mathrm { d } 2 } \\right ) + \\left [ \\binom { 3 } { 2 } \\binom { 2 } { 1 } N _ { \\mathrm { d } 1 } N _ { \\mathrm { d } 1 } - \\binom { 3 } { 1 } N _ { \\mathrm { d } 2 } \\right ] \\left ( N _ { 1 } - N _ { \\mathrm { d } 1 } \\right ) = 3 5 5 2 . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} \\widetilde H ^ k ( \\Gamma ) : = \\bigoplus _ { e \\in E } H ^ k ( 0 , L ( e ) ) , \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} \\pi _ { i j } ( s ) = ( \\pi ( s ) e _ j , e _ i ) , \\ \\ \\ s \\in G , \\ i , j = 1 , \\dots , d _ { \\pi } . \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} u _ 1 ( N ) & = \\sum _ { l = 1 } ^ { p _ 1 } a _ { l , 1 } E ( \\gamma _ { l , 1 } ) + \\sum _ { j = 1 } ^ q c _ j ( N ) E \\left ( \\frac { \\lambda _ j } { \\beta _ 1 } \\right ) \\\\ u _ i & = \\sum _ { l = 1 } ^ { p _ i } a _ { l , i } E ( \\gamma _ { l , i } ) , \\ i = 2 , \\dots , d \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\left | B ^ n _ R \\right | & = \\frac { 1 } { n ! } \\left \\{ R ^ n + n \\sum _ { i = 0 } ^ { p } \\tilde \\alpha _ i \\sum _ { j = 0 } ^ { p - i } \\frac { 2 ^ j ( p - i ) ! } { ( p - i - j ) ! } R ^ { 2 ( p - j ) - 1 } \\chi _ { i + j + 1 } ( R ) \\right \\} , \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} \\Theta _ 1 \\left ( 2 p , a , 2 p ; - \\frac { 1 } { \\tau } \\right ) & = - i ( - i \\tau ) ^ \\frac 3 2 ( 2 p ) ^ { - \\frac 1 2 } \\sum _ { k \\pmod { 2 p } } \\zeta _ { 2 p } ^ { k a } \\Theta _ 1 ( 2 p , k , 2 p ; \\tau ) , \\\\ \\Theta _ 1 \\left ( 6 p , a , 6 p ; - \\frac { 1 } { \\tau } \\right ) & = - i ( - i \\tau ) ^ \\frac 3 2 ( 6 p ) ^ { - \\frac 1 2 } \\sum _ { k \\pmod { 6 p } } \\zeta _ { 6 p } ^ { k a } \\Theta _ 1 ( 6 p , k , 6 p ; \\tau ) . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} X _ { t } = x - \\int _ { 0 } ^ { t } \\eta _ { s } \\ , d s + \\int _ { 0 } ^ { t } B _ { s } ( X _ { s } ) \\ , d W _ { s } , \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} m _ i > m _ { \\ast } \\quad r _ i : = R m _ i ^ { - 1 } < \\frac { r _ 0 } { 2 } , \\quad \\forall i \\geq i _ 0 . \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} c _ 1 ^ { } a _ 1 ^ m + \\dots + c _ r ^ { } a _ r ^ m = 1 \\enspace \\textup { f o r i n f i n i t e l y m a n y } \\enspace m \\ge 1 . \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} D _ { \\lambda } = ( A \\ln \\lambda ) = \\sum ^ { \\infty } _ { k = 0 } ( \\ln ( \\lambda ) A ) ^ { k } , \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} \\overline w ( t ) = w _ 0 + i \\int _ { 0 } ^ { t } e ^ { - i s ( \\Delta ^ 2 _ x - \\kappa \\Delta _ x ) } g ( w , p ) d s , \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} < f , g > = \\sum _ { t = a + 1 } ^ { b - 1 } r ( t ) f ( t ) g ( t ) . \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} \\frac { d } { d t } g ( \\rho + t \\sigma _ 1 ) = & - g ( \\rho + t \\sigma _ 1 ) \\frac { d } { d t } g ( \\rho + t \\sigma _ 1 ) ^ { \\dagger } g ( \\rho + t \\sigma _ 1 ) \\\\ = & - g ( \\rho + t \\sigma _ 1 ) \\frac { d } { d t } ( - \\Delta _ { \\rho + t \\sigma _ 1 } ) g ( \\rho + t \\sigma _ 1 ) \\\\ = & g ( \\rho + t \\sigma _ 1 ) \\Delta _ { \\sigma _ 1 } g ( \\rho + t \\sigma _ 1 ) , \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} a _ n ( t ) = ( - t ) _ n \\beta ^ n . \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} h ( \\cdot ) = \\sqrt n \\hat \\lambda \\| \\cdot \\| _ 1 , \\hat \\lambda \\ge 0 . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} 0 < \\frac { p _ i } { 1 - p _ 0 } < \\sum _ { n = 1 } ^ N \\frac { p _ n } { 1 - p _ 0 } = \\frac { 1 } { 1 - p _ 0 } \\sum _ { n = 1 } ^ N p _ n = \\frac { 1 } { 1 - p _ 0 } ( 1 - p _ 0 ) = 1 \\ : , \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\frac { d u ( t ) } { d t } = \\widetilde { q } u ^ \\alpha ( t ) \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} I ( \\vec x ) & = \\sum _ { i < j } w _ { i j } | x _ i - x _ j | , \\\\ \\hat { I } ( \\vec x ) & = \\sum _ { i < j } w _ { i j } \\left | | x _ i | - | x _ j | \\right | , \\\\ \\| \\vec x \\| _ \\infty & = \\max \\{ | x _ 1 | , \\ldots , | x _ n | \\} . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\alpha _ n ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ n ) p ^ { - n / 2 } \\mathbf 1 _ { \\Z _ p ^ { \\times } } ( p ^ n x ) \\underline { \\chi } _ p ^ { - 1 } ( p ^ n x ) = \\chi _ { \\psi } ( p ^ n ) p ^ { - n / 2 } \\mathbf 1 _ { p ^ { - n } \\Z _ p ^ { \\times } } ( x ) \\underline { \\chi } _ p ^ { - 1 } ( x ) . \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} & \\{ \\pi _ 2 ( i + 1 ) \\neq \\pi _ 2 ( i ) + 1 \\} , \\ o r \\\\ & \\{ \\pi _ 2 ( i ) = k \\ a n d \\ \\pi _ 1 ( k + 1 ) \\neq \\pi _ 1 ( k ) + 1 \\} . \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\bar { \\sigma } = \\frac { \\sigma + \\Sigma } { 2 } = \\frac { k \\lambda } { r } = \\frac { k ( k - 1 ) } { d } . \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} f _ \\phi ^ { [ N ] } ( U ( m , n ) ) = \\left ( S ( m , n ) T \\right ) . \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\sup \\limits _ { \\phi \\in K } \\sum _ { k = N + 1 } ^ { \\infty } \\| \\phi e _ k \\| ^ 2 = 0 . \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} r _ { \\alpha _ m ^ { k - 1 } , \\alpha _ m ^ k } = R _ { \\alpha _ m ^ { k } , m } - R _ { \\alpha _ m ^ { k - 1 } , m } = R _ { s ' , m } - R _ { s , m } . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} \\int \\limits _ { [ 0 , \\ , 1 ] ^ n } \\prod _ { i = 1 } ^ n s _ i ^ { \\lambda _ 1 } ( 1 - s _ i ) ^ { \\lambda _ 2 } \\ , \\prod \\limits _ { i < j } ^ n | s _ i - s _ j | ^ { - 2 / \\tau } d s _ 1 \\cdots d s _ n = \\prod _ { k = 0 } ^ { n - 1 } \\frac { \\Gamma ( 1 - ( k + 1 ) / \\tau ) \\Gamma ( 1 + \\lambda _ 1 - k / \\tau ) \\Gamma ( 1 + \\lambda _ 2 - k / \\tau ) } { \\Gamma ( 1 - 1 / \\tau ) \\Gamma ( 2 + \\lambda _ 1 + \\lambda _ 2 - ( n + k - 1 ) / \\tau ) } , \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} h _ { d j } ^ * ( 1 ) = 2 ^ { d - j } , j = 0 , \\dots , d . \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} a ( \\mathbf { u } _ h , \\mathbf { v } _ h ) = L ( v _ h ) , ~ ~ \\forall \\mathbf { v } _ h \\in \\mathbf { S } _ { h , 0 } ( \\Omega ) , \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\lambda - \\mu ^ { 1 / 2 } \\partial _ t ) u = f \\end{array} \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} ~ ~ ^ { A B C } _ { 0 } D ^ \\alpha f ( t ) = ~ ^ { A B R } _ { 0 } D ^ \\alpha f ( t ) - \\frac { B ( \\alpha ) } { 1 - \\alpha } f ( 0 ) E _ \\alpha \\left ( - \\frac { \\alpha } { 1 - \\alpha } t ^ \\alpha \\right ) . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 _ N ( M ) : = \\{ w \\in \\Omega ^ 1 ( M , \\partial M ) ; d w = 0 \\ \\mbox { a n d } \\ \\delta w = 0 \\} \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} V _ r ( a _ j : 0 \\leq j \\leq 2 ^ s ) \\leq \\sqrt { 2 } \\sum _ { i = 0 } ^ s \\bigg ( \\sum _ { j = 0 } ^ { 2 ^ { s - i } - 1 } \\big | a _ { ( j + 1 ) 2 ^ i } - a _ { j 2 ^ i } \\big | ^ 2 \\bigg ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) : \\ , v \\in e } B ' ( v , e ) = \\sum _ { e \\in E ( G ) : \\ , v \\in e } B ( v , e ) = 1 . \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} ( g \\gamma _ { j } ) ^ { t _ { j } } = \\tilde { g } ( 0 , 0 , s \\tilde { r _ { j } } ) = ( 0 , \\tilde { y } , \\tilde { z } + s \\tilde { r _ { j } } ) \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} S = \\{ g _ i \\mid 1 \\leq i \\leq n + \\delta \\} \\cup \\{ g ^ { - 1 } _ i \\mid 1 \\leq i \\leq n \\} . \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} F _ { 8 , 1 1 } = \\frac { 1 8 2 2 5 } { 1 6 } \\nu ( \\xi _ 4 ) \\chi _ 5 F _ { 8 , 1 3 } = \\frac { 5 4 6 7 5 } { 1 6 } \\nu ( \\xi _ 5 ) \\chi _ 5 , F _ { 8 , 1 5 } = - \\frac { 6 7 5 } { 4 } \\nu ( \\xi _ 6 ) \\chi _ 5 ^ 3 , \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\sigma _ { i j } ( \\mathbf { u } ) = \\lambda ( \\nabla \\cdot \\mathbf { u } ) \\delta _ { i , j } + 2 \\mu \\epsilon _ { i j } ( \\mathbf { u } ) , ~ ~ ~ \\textrm { w i t h } ~ ~ \\epsilon _ { i j } ( \\mathbf { u } ) = \\frac { 1 } { 2 } \\left ( \\frac { \\partial u _ i } { \\partial x _ j } + \\frac { \\partial u _ j } { \\partial x _ i } \\right ) \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) ( v ( x ) - v ( y ) ) } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d x d y = \\int _ { \\Omega } \\omega ( x ) | u ( x ) | ^ { p - 2 } u ( x ) v ( x ) d x \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\mathcal B _ 1 ^ + ( m ) : = \\{ h \\in \\mathcal B _ 1 ( m ) : d = 0 \\mathrm { o r d } _ p ( d ) \\} , \\mathcal B _ 1 ^ - ( m ) : = \\{ h \\in \\mathcal B _ 1 ( m ) : d \\neq 0 \\mathrm { o r d } _ p ( d ) \\} . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} \\gamma ( x y ) l _ { \\phi ( x y ) } = T l _ { x y } T ^ { - 1 } = T l _ x T ^ { - 1 } T l _ y T ^ { - 1 } = \\gamma ( x ) \\gamma ( y ) l _ { \\phi ( x ) \\phi ( y ) } . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} M ( \\mu ) = \\int _ { \\real ^ d \\times \\real _ + \\times \\real _ + } m \\mu \\big ( B ( x , r ) \\big ) \\ \\Phi ( d x , d r , d m ) . \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\chi ( M _ k ) = m \\chi ( M ) + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( \\chi ( \\Sigma ^ y _ \\ell ) - b ^ y _ { \\ell } ) \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} \\tilde { H } _ 1 & - ( x _ 1 x _ 3 x _ 4 - x _ 2 x _ 4 ^ 2 ) \\in K [ x _ 3 , x _ 4 , \\ldots , x _ n ] \\mbox { , } \\\\ \\tilde { H } _ 2 & - ( x _ 1 x _ 3 ^ 2 - x _ 2 x _ 3 x _ 4 ) \\in K [ x _ 3 , x _ 4 , \\ldots , x _ n ] \\mbox { , } \\\\ \\tilde { H } _ 3 & = \\tilde { H } _ 4 = \\cdots = \\tilde { H } _ n = 0 \\mbox { . } \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} K \\subset \\bigcup _ { i = 1 } ^ { N _ 1 } B _ { x _ i } ( \\epsilon ) , \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\delta _ \\sigma ( k ) = \\frac { \\sum _ { i = 1 } ^ n | \\sigma ^ { k + 1 } ( i ) - \\sigma ^ { k } ( i ) | } { 2 n } \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { p } \\chi _ { i + j } ( R ) \\tilde \\alpha _ i = \\delta _ { 0 , j } \\quad j = 0 , \\dots , p , \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} \\sin ( \\theta ) & = ( 1 - ( \\bar { \\mathbf { n } } \\cdot \\mathbf { n } ( Y ) ) ^ 2 ) ^ { 1 / 2 } \\\\ & \\leqslant ( 1 + ( 1 - 2 \\epsilon ^ 2 ) ^ { - 3 / 2 } ) \\kappa h \\left ( 4 - ( 1 + ( 1 - 2 \\epsilon ^ 2 ) ^ { - 3 / 2 } ) ^ 2 \\kappa ^ 2 h ^ 2 \\right ) ^ { 1 / 2 } \\\\ & \\leqslant ( 1 + ( 1 - 2 \\epsilon ^ 2 ) ^ { - 3 / 2 } ) \\kappa h ( 4 - \\bar { \\kappa } / 2 ) ^ { 1 / 2 } \\\\ & \\leqslant C \\kappa h , \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} \\Delta G ^ { \\epsilon } _ x ( y ) = 1 - p _ { \\epsilon } ( x , y ) . \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} X & : = H ^ { 1 } _ { 0 } ( \\Omega ) ^ { d } = \\{ v \\in H ^ { 1 } ( \\Omega ) ^ d : v = 0 \\ ; o n \\ ; \\partial \\Omega \\} , \\ ; Q : = L ^ { 2 } _ { 0 } ( \\Omega ) = \\{ q \\in L ^ { 2 } ( \\Omega ) : ( 1 , q ) = 0 \\} , \\\\ V & : = \\{ v \\in X : ( q , \\nabla \\cdot v ) = 0 \\ ; \\forall \\ ; q \\in Q \\} . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} ( a \\rhd b ) \\circ ( a \\lhd b ) = a \\circ b . \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} \\partial _ x w & = ( \\partial _ x \\rho ) ( - p ' ( \\rho ) + \\mu ' ( \\rho ) \\partial _ x u ) + \\mu ( \\rho ) \\partial _ x ^ 2 u . \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) ( 2 \\pi ) ^ { - \\frac { q } { \\tau } } \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) ( 2 \\pi ) ^ { - q } \\times \\\\ & \\times \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} J : = \\{ 1 , \\dots , \\ell ( w _ 0 ) \\} , \\ J _ f = \\{ j \\in J \\mid j ^ + = \\ell ( w _ 0 ) + 1 \\} \\ \\ J _ e : = J \\setminus J _ f . \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} G _ b & = F _ b \\big | _ { c \\leftarrow \\frac { F _ c | _ { b = 0 } } { d } } = ( a + 1 ) ^ 2 + \\left ( \\frac { a ^ 3 + a ^ 2 } { a d } \\right ) ^ 2 \\\\ & = ( a + 1 ) ^ 2 \\left \\{ 1 + \\left ( \\frac { a } { d } \\right ) ^ 2 \\right \\} = \\frac { ( a + 1 ) ^ 2 } { d ^ 2 } ( a ^ 2 + d ^ 2 ) , \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} \\lambda _ { 1 } & = - \\frac { q } { 2 } + \\sqrt { \\Delta } , ~ \\lambda _ { 2 } = - \\frac { q } { 2 } - \\sqrt { \\Delta } , \\\\ \\Delta & = \\frac { p ^ { 3 } } { 2 7 } + \\frac { q ^ { 2 } } { 4 } , ~ p = - \\frac { a ^ { 2 } } { 3 } + b , ~ q = \\frac { 2 a ^ { 3 } } { 2 7 } - \\frac { a b } { 3 } + c , \\\\ a & = 0 , ~ b = \\frac { 2 A X Y - B X } { Y \\| \\mathbf { h } \\| ^ { 2 } } , ~ c = - \\frac { 2 A X ^ { 2 } } { Y \\| \\mathbf { h } \\| ^ { 2 } } . \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} T = \\left ( \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\varphi } ( m + n + 2 \\chi ^ I ) \\right ) _ { m , n \\in \\N ^ N } \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\alpha f ( x ) + ( 1 - \\alpha ) g ( x ) = p _ { t } ( \\mu ; x ) \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} \\Theta _ 0 \\left ( 6 p , a , 6 p ; - \\frac { 1 } { \\tau } \\right ) & = ( - i \\tau ) ^ { \\frac 1 2 } \\frac { 1 } { \\sqrt { 6 p } } \\sum _ { k \\pmod { 6 p } } \\zeta _ { 6 p } ^ { k a } \\Theta _ 0 ( 6 p , k , 6 p ; \\tau ) , \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} \\vert ( D f ( z ) - D f ( w ) ) a \\vert & = \\vert ( D F ( z ) - D F ( w ) ) a \\vert \\\\ & = \\vert \\pi _ { P _ { F ( z ) } } ( D F ( z ) a ) - \\pi _ { P _ { F ( w ) } } ( D F ( w ) a ) \\vert \\\\ & \\leq \\vert \\pi _ { P _ { F ( z ) } } ( D F ( z ) a ) - \\pi _ { P _ { F ( w ) } } ( D F ( z ) a ) \\vert + \\vert \\pi _ { P _ { F ( w ) } } ( D F ( z ) a - D F ( w ) a ) \\vert \\\\ & \\leq \\sphericalangle \\left ( P _ { F ( z ) } , P _ { F ( w ) } \\right ) \\vert D F ( z ) a \\vert + \\vert \\pi _ { P _ { F ( w ) } } ( D F ( z ) a - D F ( w ) a ) \\vert . \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} \\sin ( \\theta ) & = ( 1 - ( \\bar { \\mathbf { n } } \\cdot \\mathbf { n } ( Y ) ) ^ 2 ) ^ { 1 / 2 } \\\\ & \\leqslant ( 1 + ( 1 - 2 \\epsilon ^ 2 ) ^ { - 3 / 2 } ) \\kappa h ( 4 - ( 1 + ( 1 - 2 \\epsilon ^ 2 ) ^ { - 3 / 2 } ) ^ 2 \\kappa ^ 2 h ^ 2 ) ^ { 1 / 2 } \\\\ & \\leqslant ( 1 + ( 1 - 2 \\epsilon ^ 2 ) ^ { - 3 / 2 } ) ( 4 - ( 1 + ( 1 - 2 \\epsilon ^ 2 ) ^ { - 3 / 2 } ) ^ 2 \\epsilon ^ 2 ) ^ { 1 / 2 } \\kappa h \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ m 2 x _ - ^ T ( s ) \\Sigma N _ { k } x _ - ( s ) u _ k ( s ) \\leq x _ - ^ T ( s ) \\Sigma x _ - ( s ) \\left \\| u ( s ) \\right \\| _ { 2 } ^ 2 + \\sum _ { k = 1 } ^ m x _ - ^ T ( s ) N _ k ^ T \\Sigma N _ { k } x _ - ( s ) . \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} W ^ { ( 1 ) } - W ^ { ( 3 ) } = & b _ 0 y + ( 7 8 b _ 0 + b _ 1 ) y ^ 5 + ( 1 6 8 8 b _ 0 + 3 2 b _ 1 + b _ 2 ) y ^ 9 \\\\ & + ( - 3 2 3 8 2 b _ 0 - 5 5 3 b _ 1 - 1 4 b _ 2 ) y ^ { 1 3 } \\\\ & + ( - 2 5 2 5 3 4 9 b _ 0 - 3 7 1 8 4 b _ 1 - 6 7 8 b _ 2 ) y ^ { 1 7 } + \\cdots . \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal { I } } \\left ( ( A _ i - X ) ^ T \\otimes ( \\partial _ { x _ j x _ k } \\Phi ^ { s } _ i ( X ) ) \\right ) + \\sum _ { i \\in \\mathcal { I } ^ { s ' } } \\left ( ( A _ i - \\overline { X } _ i ) ^ T \\otimes ( \\partial _ { x _ j x _ k } \\Phi ^ { s } _ i ( X ) ) \\right ) ( \\overline { M } ^ s - I _ 4 ) = 0 _ { 2 \\times 4 } . \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} \\partial _ t ( \\rho X ^ 2 ) = - \\partial _ x ( \\rho u X ^ 2 ) - 2 X \\partial _ x p ( \\rho ) + 2 \\rho f X . \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } & \\displaystyle - { \\rm d i v } \\ , ( B ( x ) | \\nabla u | ^ { p ( x ) - 2 } \\nabla u ) + ( A ( x ) | u | ^ { p ( x ) - 2 } + C ( x ) | u | ^ { p ( x ) - 3 } ) u = \\\\ & \\displaystyle ( b ( x ) | u | ^ { q ( x ) - 2 } - D ( x ) | u | ^ { p ( x ) - 1 } ) u \\quad \\mbox { i n } \\phantom { \\partial } \\Omega , \\\\ & u = 0 \\quad \\mbox { o n } \\ \\partial \\Omega \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} \\left \\langle ( x - \\alpha ) _ { j + 1 } p ( x ) , q ( x ) \\right \\rangle _ { \\lambda , j , \\ell } = & \\left \\langle ( x - \\alpha ) _ { j + 1 } { \\bf u } ^ { M } , p ( x ) q ( x ) \\right \\rangle \\\\ [ 2 m m ] = & \\left \\langle p ( x ) , ( x - \\alpha ) _ { j + 1 } q ( x ) \\right \\rangle _ { \\lambda , j , \\ell } , \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} U _ n ( t ) = A _ n ( t ) + \\int _ 0 ^ t f ( U _ n ( s ) ) d s + \\int _ 0 ^ t g ( U _ n ( s ) ) d W ( s ) , t \\in [ 0 , T ] \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} V _ r & \\ll \\sum _ { k = 0 } ^ { K } \\ ( N x ^ { - 1 / 2 } 2 ^ { - k / 2 } + N ^ { 1 / 2 } p ^ { 1 / 4 } x ^ { 1 / 2 } 2 ^ { k / 2 } ( \\log p ) ^ { 1 / 2 } + r N x ^ { - 1 } 2 ^ { - k } \\ ) \\\\ & \\ll N x ^ { - 1 / 2 } + N ^ { 1 / 2 } p ^ { 1 / 4 } x ^ { 1 / 2 } 2 ^ { K / 2 } ( \\log p ) ^ { 1 / 2 } + r N / x . \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} \\langle v _ { \\infty } ( 0 ) , \\varphi \\rangle = \\lim _ { n \\to \\infty } \\langle v _ { n } ( 0 ) , \\varphi \\rangle = 0 . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} \\langle g , ( \\Lambda _ q - \\Lambda _ { 0 } ) f \\rangle _ { { H ^ { \\frac 1 2 } ( \\partial \\Omega ) \\times H ^ { - \\frac 1 2 } ( \\partial \\Omega ) } } = \\int _ { \\Omega } q \\ , u ^ 0 _ g u _ f \\ , d x , \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} I _ P : = \\langle 1 _ { \\lambda } | \\lambda \\notin \\Pi ( P ) \\rangle \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\Theta ) = \\frac { p } { d N ^ d } \\displaystyle \\sum _ { \\tilde { e } \\in E _ N } ( \\delta _ { e _ 0 } - \\delta _ { \\tilde { e } } ) \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | H _ { n + 1 } f - H _ n f \\big | \\Big \\| _ { \\ell ^ p } & \\leq \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | h _ { n + 1 } - h _ n \\big | \\Big \\| _ { \\ell ^ 1 } \\| f \\| _ { \\ell ^ p } \\\\ & \\lesssim N _ 1 ^ { - k } \\big ( \\vartheta _ B ( N _ 2 ) - \\vartheta _ B ( N _ 1 ) \\big ) \\| f \\| _ { \\ell ^ p } , \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} \\mathcal Q ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = C _ 1 ^ { - 2 } | C _ 2 | ^ 2 \\mathcal P _ { \\chi } ( \\mathfrak R _ M \\mathbf F _ { \\chi } , \\mathbf Y _ M \\mathbf G ) , \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} u ^ i _ t = \\nabla \\cdot \\left ( m \\ , U ^ { m - 1 } \\nabla u ^ i \\right ) . \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\eta _ { r , R } : = \\eta _ { r } \\otimes 1 : { U } _ { q , R } ( m | n ) \\longrightarrow { S } _ { q , R } ( m | n , r ) . \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} V = D V + E V . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} ( v _ 1 , \\vartheta , \\partial _ \\eta w ) | _ { \\eta = 0 } = \\mathbf { 0 } , \\lim _ { \\eta \\rightarrow + \\infty } ( v _ 1 , \\vartheta , w ) ( \\tau , \\xi , \\eta ) = \\mathbf { 0 } . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} \\sum _ { j | k } \\mu ( k / j ) C _ n ( \\omega _ n ^ j ) & = \\sum _ { j | k } \\mu ( k / j ) \\binom { 2 j } { j } \\\\ & \\geq \\binom { 2 k } { k } - \\sum _ { \\substack { j | k \\\\ j < k } } \\binom { 2 j } { j } \\\\ & \\geq \\frac { 4 ^ k } { \\sqrt { \\pi ( k + 1 / 2 ) } } - ( 2 \\sqrt { k } - 1 ) \\frac { 4 ^ { k / 2 } } { \\sqrt { \\pi ( k / 2 ) } } \\geq 0 . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\pi _ x ( a \\circ b ) = \\lambda _ a ( \\pi _ { \\bar { a } \\circ x } ( b ) ) \\cdot \\pi _ x ( a ) , \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mathcal { P } _ { X _ { A } } ( A ^ { - } ( n ) ) = \\left [ \\lvert \\alpha ^ { - } ( \\phi ^ { - 1 } ) \\rvert - \\alpha ^ { - } ( \\phi ) \\right ] h _ { h o p } ( \\sigma _ { A } ) . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} \\nabla _ { 0 } ^ { \\nu } ( U Y ) ( t ) = \\sum _ { n = 0 } ^ { t } { \\nu \\choose n } [ \\nabla _ { 0 } ^ { \\nu - n } U ( t - n ) ] [ \\nabla ^ { n } Y ( t ) ] , \\quad ( \\nu > 0 , t \\in \\mathbf { Z } ^ { + } ) , \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} \\nu _ { X _ { A } | \\theta _ 1 = \\tau _ 1 } = \\nu _ { X _ { A } | \\theta = \\tau } = { \\cal N } ( { \\bf 0 } , { \\bf \\Sigma } _ { A \\tau _ 1 } ) , \\ \\tau \\in \\Lambda ( \\tau _ 1 ) . \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} \\displaystyle \\int \\limits _ T h _ n ( t , x ^ n _ \\infty ( t ) ) d \\mu ( t ) = & \\inf \\limits _ { w \\in \\textnormal { L } ^ p ( { T } , X ) } \\int \\limits _ T h _ n ( t , w ( t ) ) d \\mu ( t ) \\\\ = & \\int \\limits _ T \\inf \\limits _ { u \\in X } h _ n ( t , u ) d \\mu ( t ) , \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} \\mathcal { H } = \\bigoplus _ { I \\subset [ N ] } \\mathcal { H } _ I , \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} m _ 0 = \\min \\{ m \\in \\{ 1 , 2 , \\ldots , n \\} : A ( m ) > r \\} , \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} p _ 1 & = c _ d \\int _ { | y | < \\frac { 3 } { 2 } r } \\partial _ { y _ i } \\bigg ( \\frac { 1 } { | x - y | ^ { d - 2 } } \\bigg ) \\phi u ^ j \\partial _ { y _ j } u ^ i d y ; \\\\ p _ 2 & = c _ d \\int _ { \\frac { 3 } { 2 } r \\leq | y | < \\rho } \\partial _ { y _ i } \\bigg ( \\frac { 1 } { | x - y | ^ { d - 2 } } \\bigg ) \\phi u ^ j \\partial _ { y _ j } u ^ i d y . \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} \\left \\langle \\tau ^ { - 2 } \\alpha _ { 1 } , \\rho ^ { s } \\tau \\alpha _ { 2 } , \\sigma ^ { 2 } \\tau ^ { t _ { 3 } } \\alpha _ { 3 } \\right \\rangle \\cong M _ { 1 } \\ \\ t _ { 3 } = 0 , 1 , \\ s = 1 , \\delta , \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} \\overline { w _ I ( d _ i M _ I ; M _ I ) } = 1 . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\begin{aligned} & \\ , 2 K _ 1 ^ { \\frac { p - 1 } { p - 2 + m } } \\ , \\frac { C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } \\ , ( T - t ) ^ { - \\alpha m + \\beta } \\\\ \\leq & \\ , C ^ { \\frac { ( p - 1 ) ( m - 1 ) } { p - 2 + m } } \\left ( \\frac { C ^ { m - 1 } m } { 2 a ^ 2 ( m - 1 ) ^ 2 } \\right ) ^ { \\frac { p - 1 } { p - 2 + m } } ( T - t ) ^ { \\frac { - \\alpha p ( m - 1 ) + ( 2 \\beta - \\alpha m ) ( p - 1 ) } { p - 2 + m } } \\forall t \\in ( 0 , T ) \\ , . \\end{aligned} \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\int _ G \\overline { g \\star \\widetilde { g } } d \\nu = 0 ( g \\in L ^ 2 ( G , \\mathbb { C } ) ) . \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\triangleq \\lim \\limits _ { \\tau \\downarrow 1 } \\frac { \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) } { 2 \\pi } \\ , M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } , \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} F ( t ) [ h ] = h + \\int _ 0 ^ t \\nabla _ x b \\Big ( s , \\omega , X _ \\theta ( s ) ( \\omega ) \\Big ) F ( s ) [ h ] d s + \\int _ 0 ^ t \\nabla _ x \\sigma \\Big ( s , \\omega , X _ \\theta ( s ) ( \\omega ) \\Big ) F ( s ) [ h ] d W ( s ) , \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} \\mathcal I _ v ^ { \\sharp } ( \\mathbf h , \\mathbf g , \\pmb { \\phi } ) : = \\frac { \\mathcal I _ v ( \\mathbf h , \\mathbf g , \\pmb { \\phi } ) } { \\langle \\mathbf h _ v , \\mathbf h _ v \\rangle \\langle \\mathbf g _ v , \\mathbf g _ v \\rangle \\langle \\pmb { \\phi } _ v , \\pmb { \\phi } _ v \\rangle } = \\frac { \\mathcal I _ v ( \\mathbf h , \\mathbf g , \\pmb { \\phi } ) } { | | \\mathbf h _ v | | ^ 2 | | \\mathbf g _ v | | ^ 2 | | \\pmb { \\phi } _ v | | ^ 2 } , \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\mathcal { D } = \\left \\{ \\mu \\in \\mathcal { M } : \\int _ { \\Gamma _ 0 } \\Psi ( \\eta ) \\mu ^ { \\pm } ( d \\eta ) < \\infty \\right \\} . \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} 1 + 2 ^ { 0 } + 2 ^ { 1 } + 2 ^ { 2 } + \\cdots + 2 ^ { m - s - 2 } + 2 ^ { m - s - 1 } = 1 + \\sum _ { n = 0 } ^ { m - s - 1 } 2 ^ { n } = 1 + \\frac { 1 - 2 ^ { m - s } } { 1 - 2 } = 2 ^ { m - s } \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} \\Big \\langle \\ : \\chi _ { j k } \\ , \\boxtimes \\ , \\big ( u _ 0 ^ { \\otimes ( N _ 1 - j ) } \\otimes v _ 0 ^ { \\otimes ( N _ 2 - k ) } & \\big ) \\ ; , \\ ; \\chi _ { \\ell r } \\ , \\boxtimes \\ , \\big ( u _ 0 ^ { \\otimes ( N _ 1 - \\ell ) } \\otimes v _ 0 ^ { \\otimes ( N _ 2 - r ) } \\big ) \\ ; \\Big \\rangle \\ ; = \\\\ & = \\| \\chi _ { j k } \\| _ 2 ^ 2 \\ ; \\delta _ { j \\ell } \\ , \\delta _ { k r } \\ , , \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} \\mathcal F ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * g ^ * : \\mathcal F ( Y ) \\to \\mathcal F ( X ) , \\\\ \\mathcal H ^ { C h e r n } _ * ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * \\bigl ( c ( T _ g ) \\cap g ^ * \\bigr ) : \\mathcal H ^ { C h e r n } _ * ( Y ) \\to \\mathcal H ^ { C h e r n } _ * ( X ) . \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} & g _ 1 = 1 + 2 \\sum _ { n = 1 } ^ N \\epsilon ^ { n ( n + 1 ) } g ^ { - n } d _ { n + 1 } e ^ { n x } D _ { N - n } ^ 0 + o ( \\epsilon ^ { N ( N + 1 ) } ) , \\\\ & g _ 2 = 1 + \\sum _ { n = 1 } ^ N \\epsilon ^ { n ( n - 1 ) } g ^ { - n } d _ n e ^ { n x } D _ { N - n } ^ 2 + o ( \\epsilon ^ { N ( N - 1 ) } ) , \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} M \\cdot \\iota _ i & = M _ i \\iota _ i \\\\ M \\cdot p _ i & = p _ i M _ i ^ { - 1 } \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} \\mu : = \\mathbb { E } \\bigg [ \\sum _ { j \\in [ \\epsilon n _ i ] } W ' _ i \\bigg ] = \\epsilon \\sum _ { j \\in [ m _ i ] } f ( x _ { i , j } ) = \\epsilon q _ i \\sigma ^ 2 : = \\frac { 1 } { n _ i } \\sum _ { j \\in [ m _ i ] } ( f ( x _ { i , j } ) - \\epsilon q _ i / n _ i ) ^ 2 + \\sum _ { j \\in [ n _ i ] \\setminus [ m _ i ] } ( 0 - \\epsilon q _ i / n _ i ) ^ 2 . \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t \\left ( x _ { t - 1 } , \\xi _ t \\right ) = \\left \\{ \\begin{array} { l } \\inf \\ ; Q _ { t + 1 } \\left ( x _ { t } \\right ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ { t } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} | \\eta _ r ( y , x ) | < | y | = p ^ { - 2 m } < p ^ { - m } . \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} \\lambda _ i ( p ) : = \\sqrt { \\sigma _ i ( p ) } \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} - \\varepsilon ^ 2 \\Delta u + V ( x ) u = f ( x , u ) , \\ \\ x \\in \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} d ( x _ { i _ 0 } , x _ { i _ m } ) = d ( x _ { i _ 0 } , x _ { i _ 1 } ) + d ( x _ { i _ 1 } , x _ { i _ 2 } ) + \\dots + d ( x _ { i _ { m - 1 } } , x _ { i _ m } ) . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} ( \\rho , u ) ( x , t ) = \\left \\{ \\begin{array} { l l } ( \\rho _ - , u _ - + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x < x _ { 1 } ^ { A B } ( t ) , \\\\ ( \\rho _ * ^ { A B } , v _ * ^ { A B } + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ x _ { 1 } ^ { A B } ( t ) < x < x _ { 2 } ^ { A B } ( t ) , \\\\ ( \\rho _ + , u _ + + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x > x _ { 2 } ^ { A B } ( t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\begin{pmatrix} u & 0 & 0 \\\\ 0 & 0 & - u \\end{pmatrix} . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} 1 < \\sigma < \\min \\left ( 2 , q , q ' , \\frac { q ( d + 2 ) } { 2 d } , ( 2 q ' ) ' \\right ) = \\min \\left ( q ' , \\frac { q ( d + 2 ) } { 2 d } , ( 2 q ' ) ' \\right ) . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} y \\in \\mathcal { O } _ { L _ { \\tau , n } } [ G _ n ] _ { \\varphi } \\Longleftrightarrow \\phi ( y _ g ) = y _ { g \\phi _ n ^ { - 1 } } \\forall g \\in G _ n , \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} \\rho ( t ) & = \\rho _ { 0 } + \\rho _ { 1 } t + \\rho _ { 2 } t ^ { 2 } + . . . . . + \\\\ \\varphi ( t ) & = \\varphi _ { 0 } + \\varphi _ { 1 } t + \\varphi _ { 2 } t ^ { 2 } + . . . . . + \\\\ \\theta ( t ) & = \\theta _ { 0 } + \\theta _ { 1 } t + \\theta _ { 2 } t ^ { 2 } + . . . . . + \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} & A _ r Q ' + Q ' A _ r + A ' _ r Q + Q A ' _ r - C '^ T _ r C _ r - C ^ T _ r C ' _ r = 0 , \\\\ & A Y ' + Y ' A _ r + Y A ' _ r + C ^ T C ' _ r = 0 . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} h ( R ) = 1 ; \\mathcal { D } h ( R ) = 0 ; \\mathcal { D } ^ 2 h ( R ) = 0 ; \\dots ; \\mathcal { D } ^ p h ( R ) = 0 . \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} a ( x ) = I + c | x | ^ { - 2 } x \\otimes x , x \\in \\mathbb R ^ d \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\mathbb { E } ( \\nu _ i ) = t \\pi _ i = t \\frac { | \\mathcal { G } _ i | } { 2 ^ M } = \\frac { t } { 2 ^ M } \\gamma _ i . \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} \\Omega _ p ( \\beta _ m ) \\mathrm { v o l } ( \\Gamma _ 0 \\beta _ m \\Gamma _ 0 ) = \\begin{cases} p ^ { - 2 m } ( - w _ p ) ^ m ( 1 - p ^ { - 1 } ) & m > 0 , \\\\ p ^ { 2 m - 2 } ( - w _ p ) ^ m ( 1 - p ^ { - 1 } ) & m < 0 . \\end{cases} \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} G ( \\eta ) = G ^ { ( n ) } ( x _ 1 , \\dots , x _ n ) , { \\rm f o r } \\ \\ \\eta = \\{ x _ 1 , \\dots , x _ n \\} . \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} G ( K ) = \\langle x _ 1 , \\cdots , x _ n \\ \\vline \\ r _ 1 , \\cdots , r _ { n - 1 } \\rangle . \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} \\frac { \\mathrm d f ( x ) } { \\mathrm d x } = \\lim _ { n \\to \\infty } f _ n ^ \\prime ( x ) \\quad \\ x \\in [ a , b ] . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} \\begin{aligned} & \\Delta _ { \\frac { 1 } { d } ( 1 , p ) } ( k - ( r - 1 ) p ) + \\Delta _ { \\frac { 1 } { d } ( 1 , q ) } ( k - ( s - 1 ) q ) = \\frac { ( 1 - d ) ( r + s - 2 ) } { 2 d } + \\left \\{ \\frac { - k } { d } \\right \\} ( r + s - 1 ) \\end{aligned} \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} \\| I _ { h , T } u _ i - u _ i \\| _ { 0 , T } + h | I _ { h , T } u _ i - u _ i | _ { 1 , T } + h ^ 2 | I _ { h , T } u _ i - u _ i | _ { 2 , T } \\leqslant C h ^ 2 | u _ i | _ { 2 , T } , ~ ~ i = 1 , 2 , ~ ~ \\forall ~ T \\in \\mathcal { T } _ h ^ n . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left \\| { w } ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - k u ^ { * * } ( t ) - l v ^ { * * } ( t ) \\right \\| _ { C ^ 0 ( \\bar \\Omega ) } = 0 . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} L ( u _ { j } \\otimes u _ { j } , s ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { | \\rho _ { j } ( n ) | ^ 2 } { n ^ s } . \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} I _ { k , s , e } : = \\left \\{ ( A , B ) \\in G \\times G ' \\left | \\begin{array} { l } A \\subset B \\subset A ^ \\perp \\subset V , \\\\ \\dim B \\cap B ^ \\perp = m - k + e \\\\ \\end{array} \\right . \\right \\} . \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\mathbb { E } [ \\overline { V } ( p , f , \\pi _ n ) _ T | \\mathcal { S } ] & = n ^ { p / 2 - 1 } \\sum _ { i : t _ { i , n } \\leq T } | \\mathcal { I } _ { i , n } | ^ { p / 2 } \\mathbb { E } [ f ( | \\mathcal { I } _ { i , n } | ^ { - 1 / 2 } \\Delta _ { i , n } X ^ { t o y } ) | \\mathcal { S } ] \\\\ & = m _ { \\sigma \\sigma ^ * } ( f ) G _ p ^ { ( 1 ) , n } ( T ) \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} L ^ \\Delta _ { \\alpha \\alpha ' } k = L ^ \\Delta _ { \\alpha } k . \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\chi _ { k } * \\left ( \\prod _ { j = 0 } ^ { k } c _ { j } \\right ) = \\sum _ { l = 0 } ^ { k } \\left ( \\prod _ { j = 0 } ^ { l - 1 } c _ { j } \\right ) ( \\chi _ { k } * c _ { l } ) f _ { k } * \\left ( \\prod _ { j = l + 1 } ^ { k } c _ { j } \\right ) . \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} ( X , Y ) = E [ X Y ] , \\ \\ \\forall X , Y \\in L _ 2 . \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} f - \\lambda H _ \\dagger f & = h , \\\\ f - \\lambda H _ \\ddagger f & = h . \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} h _ n = d _ n e ^ { ( n - 1 ) x } ( e ^ x D _ { N - n } ^ 2 - 2 D _ { N - n + 1 } ^ 0 ) , n = 1 , \\ldots , N . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\alpha \\left ( v \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) \\alpha ^ { - 1 } = \\left ( \\alpha \\cdot v \\right ) \\alpha _ { 1 } ^ { a _ { 1 } b _ { 4 } - a _ { 3 } b _ { 3 } } \\alpha _ { 3 } ^ { a _ { 3 } b _ { 1 } - a _ { 1 } b _ { 2 } } , \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G ) ) = r ^ { 1 - r / p } \\max _ i \\{ \\alpha _ i ^ { - 1 / p } \\} . \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} B ( s , f ) & = \\frac { 1 } { 2 \\pi i } \\int \\limits _ { 3 - i \\infty } ^ { 3 + i \\infty } \\Gamma ( w ) B ( w + s , f ) X ^ w d w - \\frac { 1 } { 2 \\pi i } \\int \\limits _ { - \\delta _ 0 + \\delta _ 1 - i \\infty } ^ { - \\delta _ 0 + \\delta _ 1 + i \\infty } \\Gamma ( w ) B ( w + s , f ) X ^ w d w \\\\ & = : T _ 1 ( s , f ) - T _ 2 ( s , f ) . \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} d \\gamma = - \\sum _ { n \\geqslant 2 } ( - 1 ) ^ { \\binom { n + 1 } { 2 } + | \\gamma _ 1 | } \\gamma _ 1 \\cdots \\gamma _ n , \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} E ^ 2 = - \\frac { r } { p q } = - Z ^ 2 X ^ 2 = Y ^ 2 = 0 . \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} \\int _ { \\R ^ d \\times \\R ^ d } \\Delta _ x \\zeta ( t , x ) \\cdot \\nabla _ y \\zeta ( x , y ) \\Delta _ x \\psi ( x , y ) \\ , d x d y & \\\\ = ( \\Delta \\zeta ( t , \\cdot ) , ( - \\Delta ) ^ { \\frac { 1 - d } 2 } \\Delta \\zeta ( t , \\cdot ) ) \\geq 0 . \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} | \\widetilde { \\Gamma } _ n | = \\sum _ { s = 1 } ^ m \\varphi ( 2 s + 1 ) / 2 . \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} \\lambda _ { G / H } ( T _ H ( f ) ) \\xi ( y H ) & = \\int _ { G / H } T _ H f ( x H ) \\lambda _ { G / H } ( x H ) \\xi ( y H ) \\ , d ( x H ) \\\\ & = \\int _ { G / H } \\int _ H f ( x h ) \\xi ( x ^ { - 1 } y H ) \\ , d h d x . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} 2 \\Re \\int _ { \\R ^ d } \\kappa \\Delta u _ { \\mu } ( x ) [ \\Delta \\phi ( x ) \\bar u _ { \\mu } ( x ) + 2 \\nabla \\phi ( x ) \\cdot \\nabla \\bar u _ { \\mu } ( x ) ] \\ , d x \\\\ = \\int _ { \\R ^ d } \\kappa \\Delta ^ 2 \\phi ( x ) \\abs { u _ { \\mu } ( x ) } ^ 2 \\ , d x - 4 \\kappa \\int _ { \\R ^ d } \\nabla u _ { \\mu } ( x ) D ^ 2 \\phi ( x ) \\nabla \\bar u _ { \\mu } ( x ) \\ , d x . \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\rho ^ * ( g ) : = \\rho ( g ^ { - 1 } ) ^ T . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} h \\nu _ { \\gamma } \\beta _ m \\nu _ { \\delta } \\varpi _ p = \\left ( \\begin{array} { c c } p ^ { m - 2 } t ^ { - 1 } \\det ( h ) \\eta ( h ) ^ { - 1 } & x p ^ { - m } + \\gamma y p ^ { 1 - m } \\\\ 0 & p ^ { 1 - m } t \\eta ( h ) \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & 0 \\\\ \\delta - \\eta ( h ) ^ { - 1 } p ^ { 2 m - 2 } & 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} U _ { f , g } ( s ) = - F ' ( s ) G ( s - 1 ) + F ( s ) G _ { L } ( s ) , \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\Phi = ( \\tilde { \\rho } \\otimes \\tilde { \\alpha } ) \\circ \\phi , \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} \\sigma _ { j - 1 } \\left ( 1 - \\frac { r } { r _ j } \\right ) + \\sum _ { i = 1 , i \\neq j - 1 } ^ { 2 q } \\sigma _ i = - d _ { j - 1 } + \\frac { d } { r } + t - \\frac { e } { 2 q r _ j } \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\nu \\left ( \\frac { v } { x ^ m } \\right ) : = \\frac { a _ { \\min ( r , m - 1 ) } } { b ^ { m - \\min ( r , m - 1 ) } } + \\cdots + \\frac { a _ 1 } { b ^ { m - 1 } } + \\frac { a _ 0 } { b ^ m } \\in [ 0 , 1 ) . \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\omega = \\sum _ { I \\subset \\{ 1 , \\dots , m \\} \\atop | I | = q } \\lambda _ I \\hat { F } _ I d F _ { i _ 1 } \\wedge \\dots \\wedge d F _ { i _ q } , \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} \\big \\langle \\widetilde \\Lambda _ { v _ 1 } \\widetilde P _ { v _ 1 , \\rm R } \\widetilde F ( v _ 1 ) , \\widetilde P _ { v _ 1 , \\rm R } \\widetilde F ( v _ 1 ) \\big \\rangle & = \\alpha _ { v _ 1 } | \\widetilde f ( v _ 1 ) | ^ 2 = \\alpha _ { v _ 1 } | f ( v _ 1 ) | ^ 2 \\\\ & = \\big \\langle \\Lambda _ { v _ 1 } P _ { v _ 1 , \\rm R } F ( v _ 1 ) , P _ { v _ 1 , \\rm R } F ( v _ 1 ) \\big \\rangle \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\Omega _ { t } : = \\sum _ { i , j } \\varepsilon _ { i j , t } \\frac { { \\rm d } A _ { i , t } } { A _ { i , t } } \\wedge \\frac { { \\rm d } A _ { j , t } } { A _ { j , t } } . \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} x ( y ( x z ) ) & = ( ( x y ) x ) z , & ( x y ) ( z x ) & = ( x ( y z ) ) x , \\\\ ( x y ) ( z x ) & = x ( ( y z ) x ) , & ( ( x y ) z ) y & = x ( y ( z y ) ) , \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} \\sum _ { ( n , v w ) = 1 } G \\left ( \\frac { n } { N } \\right ) e \\left ( \\frac { b n \\overline { c \\ell m v } } { w } \\right ) & = \\mu ( w ) \\frac { N } { w } \\frac { \\varphi ( v ) } { v } \\\\ & + \\sum _ { d \\mid v } \\mu ( d ) \\frac { N } { d w } \\sum _ { | h | \\leq W ^ { 1 + \\epsilon } d / N } \\widehat { G } \\left ( \\frac { h N } { d w } \\right ) \\sum _ { ( a , w ) = 1 } e \\left ( \\frac { a b d \\overline { c \\ell m v } } { w } + \\frac { a h } { w } \\right ) \\\\ & + O ( Q ^ { - 1 0 0 } ) . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ Z _ { \\varepsilon } ( q \\beta ) Z ^ s _ { \\varepsilon } ( \\beta ) \\Bigr ] \\approx N ^ { 1 + q ^ 2 \\beta ^ 2 + ( 1 + \\beta ^ 2 ) s } \\ ; \\mathfrak { M } ( s \\ , | \\ , \\tau , \\lambda = - q \\beta ^ 2 ) , \\ , N \\rightarrow \\infty . \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} \\rho _ { 1 3 4 } ( S _ 1 ) ^ { 2 } & = ( S _ 2 S _ 1 S _ 1 ^ * + S _ 1 S _ 2 S _ 2 ^ * ) ( S _ 2 S _ 1 S _ 1 ^ * + S _ 1 S _ 2 S _ 2 ^ * ) \\\\ & = S _ 2 ( S _ 1 S _ 2 ) S _ 2 ^ * + S _ 1 ( S _ 2 S _ 1 ) S _ 1 ^ * \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} Z \\subseteq \\bigcap _ { n \\geq 1 } \\bigcup _ { j = 1 } ^ k \\overline { B _ { 1 0 m _ n ^ { - 1 / 2 } } ( x _ j ) } = \\bigcup _ { j = 1 } ^ k \\bigcap _ { n \\geq 1 } \\overline { B _ { 1 0 m _ n ^ { - 1 / 2 } } ( x _ j ) } = \\bigcup _ { j = 1 } ^ k \\lbrace x _ j \\rbrace . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\sigma _ { p , q } = \\frac { p + \\sqrt { p ^ { 2 } + 4 q } } { 2 } , \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} f ^ L + \\alpha g ^ L = ( f + \\alpha g ) ^ L = \\widehat { f + \\alpha g } \\ge \\hat { f } . \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} F ^ L ( \\prod _ { i = 1 } ^ l \\vec x ^ { ( i ) } ) = \\int _ 0 ^ { \\| \\vec x \\| _ \\infty } F ( V _ t ^ + , V _ t ^ - ) d t \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} T _ G { \\bf Q } D _ { { \\bf x } , t } ^ + { \\bf u } + T _ G { \\bf Q } T _ G { \\bf D } p = T _ G { \\bf Q } T _ G ( { \\bf f } ) . \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} \\lim _ { g \\to e } \\psi ( g ) \\chi _ j ( g ) = d _ j , \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} c ( \\xi ) = e ^ { 2 \\pi \\xi } W _ { \\mathbf h , \\xi } ( 1 ) . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} \\mathfrak { a } ^ + = \\{ { \\rm d i a g } ( a _ 1 , \\ldots , a _ n , - a _ 1 , \\ldots , - a _ n ) : a _ 1 > \\ldots > a _ n > 0 \\} . \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} h _ { d , j } ^ * ( z ) = \\frac 1 2 ( z + 1 ) ^ { d - j + 1 } , j = 0 , \\dots , d , \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( n a _ { n } \\right ) = - z \\frac { d } { d z } \\mathcal { A } \\left ( z \\right ) . \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq i \\leq n , Y _ i \\leq S } \\big | \\hat { G } _ n ( Y _ i ) - G _ C ( Y _ i ) \\big | = O \\biggl ( \\sqrt { \\frac { \\ln n } { n } } \\biggr ) . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} a _ { u } ( x , y ) = \\begin{cases} 0 , & \\\\ p ( x \\star u , y ) & \\end{cases} \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} ( \\hat { \\omega } ^ \\bullet _ t ) ^ { m + n } = \\lambda _ t ^ { 2 m + 2 n } \\Psi _ t ^ * ( e ^ F ( \\omega _ { \\mathbb { C } ^ m } + e ^ { - t } \\omega _ Y ) ^ { m + n } ) = \\delta _ t ^ { 2 n } e ^ { F _ t } \\omega _ P ^ { m + n } , \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\dot { a } ( \\zeta , t ) = 0 , \\dot { b } ( \\zeta , t ) = - 4 i \\zeta ^ 4 b ( \\zeta , t ) . \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} p _ 2 q ' = p _ 2 \\frac { p ' } { p ' - 1 } = p _ 2 \\frac { 2 - \\delta } { 2 - \\delta - p _ 1 } < 2 \\Leftrightarrow 2 ( p _ 1 + p _ 2 ) + ( 2 - p _ 2 ) \\delta < 4 . \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} v _ { t } ^ { \\varepsilon } + A \\Delta v ^ { \\varepsilon } - \\rho _ { \\varepsilon } v ^ { \\varepsilon } = e ^ { \\rho _ { \\varepsilon } \\left ( t - T \\right ) } F \\left ( x , t ; u ^ { \\varepsilon } \\right ) + \\mathbf { P } _ { \\varepsilon } ^ { \\beta } v ^ { \\varepsilon } , \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} n _ 1 & : = 1 + \\alpha ( q ^ 3 + 1 ) + \\beta ( q ^ 2 - q + 1 ) \\\\ n _ 2 & : = q ^ 5 - 3 q ^ 3 + 2 q ^ 2 - q - 1 - \\alpha ( q ^ 3 + 1 ) - \\beta ( q ^ 2 - q + 1 ) . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + { \\rm c o n s t } ) } \\ , { \\bf E } \\bigl [ M ^ q _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\bigr ] , \\ ; N \\rightarrow \\infty . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} U _ i ^ * \\delta _ y = \\frac { 1 } { \\sqrt { q _ i } } \\delta _ { \\omega _ y ( 1 ) } . \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\hat { \\bar { R } } ( X , Y , Z , W ) & = \\hat { R } ( X , Y , Z , W ) - g \\big ( \\hat { h } ( X , W ) , \\hat { h } ( Y , Z ) \\big ) \\\\ & + g \\big ( \\hat { h } ( X , Z ) , \\hat { h } ( Y , W ) \\big ) , \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} \\Big \\Vert \\sum _ { i \\in C _ k } z _ i \\Big \\Vert = \\vert C _ k \\vert . \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{gather*} T _ n = T _ { \\rm m a x } ( \\alpha ^ * _ { n - 1 } ) / 3 , \\alpha _ n ^ * = \\alpha ^ * _ { n - 1 } + c T _ n , \\\\ [ . 2 c m ] \\alpha _ n = \\alpha ^ * _ { n - 1 } + \\delta ( \\alpha ^ * _ { n - 1 } ) . \\end{gather*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\triangle _ \\beta \\left [ \\sum _ { 2 j + \\frac { k } { \\beta } + \\abs { \\sigma } < q } \\rho ^ { 2 j + \\frac { k } { \\beta } } \\left ( a _ { j , k } ^ \\sigma \\cos k \\theta + b _ { j , k } ^ \\sigma \\sin k \\theta \\right ) \\xi ^ \\sigma \\right ] = 0 . \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} G _ { \\eta _ { \\gamma } } = \\prod _ { n = n _ { \\gamma } } ^ { c } \\big ( 1 - \\lambda _ n ( \\frac { \\pi } { 2 } c ) \\big ) \\leq \\exp \\left ( - c \\big ( 1 - \\frac { \\pi } { 2 \\gamma } \\big ) \\log 2 \\right ) = \\exp ( - c \\ , a _ { \\gamma } ) . \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} S _ { 1 , } ( f ) = S _ { 1 , } ( f ) + P _ { 1 } [ f ] , S _ { 2 , } ( f ) = S _ { 2 , } ( f ) + P _ { 2 } [ f ] , \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} \\langle u _ k ' , v _ k ' \\rangle + \\| w _ k ' \\| ^ 2 = \\langle i s \\ , f _ k + g _ k - p _ k ( s ) ^ 2 u _ k , v _ k \\rangle + \\langle h _ k - q _ k ( s ) ^ 2 w _ k , w _ k \\rangle . \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} \\int \\varphi d \\pi _ * \\mu = \\int \\pi ^ * \\varphi d \\mu ~ \\textrm { f o r a n y } ~ \\varphi \\in C ( Y ) ~ \\textrm { a n d } ~ \\mu \\in M ( X ) . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} u ( 0 ) = c _ 1 , u ( 1 ) = c _ 1 + \\int _ 0 ^ 1 \\psi ( \\int _ 0 ^ \\tau v ( s ) \\ , d s + c _ 2 ) \\ , d \\tau \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} B _ { \\textnormal a } ( m , r ) = \\sum _ { i } \\binom { m } { i } \\binom { r } { i } \\binom { r + m - i } { m - i } . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} [ X \\xleftarrow p V \\xrightarrow s p t ; E ] \\circ _ { \\oplus } & [ p t \\xleftarrow q W \\xrightarrow t p t ; F ] = \\\\ & [ ( X \\xleftarrow p V \\xrightarrow s p t ) \\circ ( p t \\xleftarrow q W \\xrightarrow s p t ) ; ( p r _ 1 ) ^ * E \\oplus ( p r _ 2 ) ^ * F ] , \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} \\Omega ( A , B , f ) & = ( \\Omega A , \\Omega B \\oplus ( M \\otimes P ^ A _ 0 ) , { \\begin{pmatrix} 0 \\\\ M \\otimes i ^ A _ 1 \\end{pmatrix} } ) \\\\ & = ( \\Omega A , M \\otimes P ^ A _ 0 , M \\otimes i ^ A _ 1 ) \\oplus ( 0 , \\Omega B , 0 ) . \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} \\frac { 1 } { d } \\sum _ { j = 1 } ^ { d } F _ j ( v _ 1 ) \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} \\left | \\pi _ i - \\frac { 1 } { 2 } \\right | = \\frac { 1 } { 2 M } \\left | \\sum _ { j = ( i - 1 ) M + 1 } ^ { i M } e _ j \\right | \\leq \\frac { 1 } { 2 M } W ( E _ N ) \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N c ^ u _ { i } \\leq C _ { \\mathrm { U L } } , \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} x ^ { q ^ 2 + q - 1 } + A x ^ { q ^ 2 - q + 1 } + B x = a \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} \\delta _ 3 ^ { \\mathfrak { K } } ( t , x + w , y ; z ) - \\delta _ 3 ^ { \\mathfrak { K } } ( t , x , y ; z ) = \\int _ 0 ^ 1 \\left < w , \\nabla _ x \\delta _ 3 ^ { \\mathfrak { K } } ( t , x + \\theta w , y ; z ) \\right > d \\theta \\ , . \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} - 2 P _ 1 ( 1 ) P _ 3 ( 1 ) \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\sum _ n \\frac { \\chi ( n ) } { n ^ { \\frac { 1 } { 2 } } } F ( n ) = - P _ 1 ( 1 ) P _ 3 ( 1 ) \\sum _ { w \\mid q } \\varphi ( w ) \\mu ( q / w ) \\sum _ { \\substack { n \\equiv \\pm 1 ( w ) \\\\ ( n , q ) = 1 } } \\frac { 1 } { n ^ { \\frac { 1 } { 2 } } } F ( n ) . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} g _ i ( x ; a , b ) & = \\frac { f ^ { ( i ) } ( A + x + 2 a \\sqrt x ) - f ^ { ( i ) } ( A + x - 2 a \\sqrt x ) } { \\sqrt { x } } \\\\ & = \\frac { 1 } { \\sqrt { x } } f ^ { ( i ) } ( A + x + 2 a \\sqrt x s ) \\bigg | _ { s = - 1 } ^ { s = 1 } \\\\ & = \\frac { 1 } { \\sqrt { x } } \\int _ { - 1 } ^ 1 f ^ { ( i + 1 ) } ( A + x + 2 a \\sqrt x s ) 2 a \\sqrt x \\ , d s \\\\ & = 2 a \\int _ { - 1 } ^ 1 f ^ { ( i + 1 ) } ( A + x + 2 a \\sqrt x s ) \\ , d s , x > 0 . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\mathcal { H } _ I ^ s & = \\bigoplus _ { p \\in \\N ^ I } V _ I ^ p \\left ( \\bigcap _ { q \\in \\N ^ { I ^ c } } V _ { I ^ c } ^ q \\left ( \\bigoplus _ { n \\geq 0 } V _ { N + 1 } ^ n \\left ( V _ { N + 1 } ^ * \\cap \\bigcap _ { i \\in I } V _ i ^ * \\right ) \\right ) \\right ) \\\\ & = \\bigoplus _ { p \\in \\N ^ K } V _ K ^ p \\left ( \\bigcap _ { q \\in \\N ^ { I ^ c } } V _ { I ^ c } ^ q \\left ( \\bigcap _ { i \\in K } V _ i ^ * \\right ) \\right ) , \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} C : = \\sum _ { i = 1 } ^ k \\frac { 1 } { \\alpha _ i ^ { 1 / ( p - r ) } } = \\frac { 1 } { r } \\Bigg ( \\sum _ { i = 1 } ^ k \\big ( \\lambda ^ { ( p ) } ( G _ i ) \\big ) ^ { p / ( p - r ) } \\Bigg ) . \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} Z ( s ; f , \\phi ) ( g ) = \\Big ( \\prod _ v c _ v ( s ) \\Big ) \\phi . \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} \\frac { d } { d t } f ( K _ u ^ 2 ) H _ u u & = [ B _ u , f ( K _ u ^ 2 ) ] H _ u u + f ( K _ u ^ 2 ) ( [ B _ u , H _ u ] u + i \\bar { J } Q u ) + f ( K _ u ^ 2 ) H _ u ( B _ u u - i J H _ u u ) \\\\ & = B _ u f ( K _ u ^ 2 ) H _ u u + i \\bar { J } Q f ( K _ u ^ 2 ) u + i \\bar { J } f ( K _ u ^ 2 ) H _ u ^ 2 u \\\\ & = B _ u w _ k ^ K + i \\bar { J } Q u _ k ^ K + i \\bar { J } f ( K _ u ^ 2 ) ( K _ u ^ 2 u + Q u ) \\\\ & = B _ u w _ k ^ K + i \\bar { J } ( 2 Q + \\sigma _ k ^ 2 ) u _ k ^ K , \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} \\hat { \\omega } _ p ( t ( a ) r , 1 ) \\hat { \\phi } _ { \\mathbf h , p } ( \\beta ; 0 , 1 ) = \\omega _ p ( ( t ( a ) r , 1 ) ) \\phi _ { 1 , p } ( \\beta ) \\phi _ { 2 , p } ( ( 0 , 1 ) t ( a ) r ) , \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} T \\acute { N } = \\{ \\tilde { J } ( R a d T \\acute { N } ) \\oplus \\tilde { J } ( l t r ( T \\acute { N } ) ) \\} \\bot \\mu _ { 0 } \\bot R a d ( T \\acute { N } ) , \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\{ ( z _ 1 ^ * , \\ldots , z _ n ^ * ) \\in [ 0 , 1 ] ^ n \\big | \\ , \\ , \\ , z ^ * _ i = 1 , i = 1 , \\ldots , m _ 1 , \\ , \\ , \\ , \\ , \\ , \\ , z ^ * _ i = 0 , i = m _ 0 + 1 , \\ldots , n \\} . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\gamma ( s , \\pi \\times \\pi ^ \\prime , \\psi ) = \\sum _ { U _ n ( \\mathfrak { f } ) \\backslash G _ n ( \\mathfrak { f } ) } J _ { \\sigma , \\psi } ( g ) J _ { \\sigma ^ \\prime , \\psi ^ { - 1 } } ( g ) \\psi ( e _ 1 \\ , ^ t g ^ { - 1 } \\ , ^ t e _ n ) . \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} \\begin{aligned} k \\| u _ k \\| , | s | \\| u _ k \\| , \\| u _ k ' \\| & \\lesssim \\frac { s ^ 4 e ^ { \\R q _ k ( s ) } } { k | { p _ k ( s ) } | | { \\det M _ k ( s ) | } } \\| z _ k \\| , \\\\ \\mbox { a n d } \\| w _ k \\| & \\lesssim \\frac { s ^ 2 e ^ { \\R q _ k ( s ) } } { k | { \\det M _ k ( s ) } | } \\| z _ k \\| . \\end{aligned} \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} \\sigma ' ( c ) & = ( f \\circ \\rho \\circ \\sigma \\circ \\rho ^ { - 1 } ( P _ 1 ) , \\dots , f \\circ \\rho \\circ \\sigma \\circ \\rho ^ { - 1 } ( P _ n ) ) \\\\ & = ( h \\circ \\sigma ( \\rho ^ { - 1 } ( P _ 1 ) ) , \\dots , h \\circ \\sigma ( \\rho ^ { - 1 } ( P _ n ) ) ) \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{gather*} \\lim _ { n \\uparrow \\infty } v ( x _ n ) - f ( x _ n ) = \\inf _ x v ( x ) - f ( x ) , \\\\ \\lim _ { n \\uparrow \\infty } v ( x _ n ) - \\lambda H _ \\ddagger f ( x _ n ) - h ( x _ n ) \\geq 0 . \\end{gather*}"}
-{"id": "1956.png", "formula": "\\begin{align*} Z _ h ( x , \\theta ) = \\theta ^ { 1 + \\beta } Z _ h ( x ) . \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} m _ \\Gamma : = \\frac { 1 } { | \\Gamma | } \\int _ { \\Gamma } ^ { } u _ { 0 \\Gamma } d \\Gamma , \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} c \\leq \\max _ { t \\in [ 0 , 1 ] } I _ { \\lambda } ( \\gamma ( t ) ) \\leq \\max _ { t \\geq 0 } I _ { \\lambda } ( t u , t v ) = I _ { \\lambda } ( u , v ) . \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} \\psi _ p ( z ) = \\sum _ { j = 0 } ^ \\kappa d _ { j , p } z ^ j , \\forall p \\in \\{ 0 , . . . , \\kappa \\} , \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} 2 \\nabla ' s & = d \\rho - J ^ * \\lambda \\rho - i \\ , J ^ * ( d \\rho - J ^ * \\lambda \\rho ) , \\\\ 2 \\nabla '' s & = d \\rho + J ^ * \\lambda \\rho + i \\ , J ^ * ( d \\rho + J ^ * \\lambda \\rho ) , \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} L _ h ( 0 ) = \\frac { 1 } { ( p - 1 ) ! } f _ y ^ { ( p ) } ( 0 , 0 ) [ h ] ^ { p } + N _ C ( h ) . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} \\psi _ 3 ( ( a b ) ^ 3 ) = ( b _ 1 , b _ 0 , b _ 0 ) \\ \\ \\ \\ \\ \\ \\psi _ 3 ( y ^ { - 3 } ) = ( b _ 0 , b _ 1 , b _ 1 ) . \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} f = \\bigoplus _ { i = 1 } ^ { m + l - 1 } f _ i , \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } \\bar { Y } ^ { t , x ; \\bar { u } } ( s ) = W ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , \\\\ \\bar { Z } ^ { t , x ; \\bar { u } } ( s ) = V ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) , W ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , W _ x ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , \\bar { u } ( s ) ) , s \\in \\lbrack t , T ] \\end{array} \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} | \\underline { \\vec z _ { k + d } } | = \\bigg ( \\sum _ { i = k + 1 } ^ { k + d - 1 } \\nu _ i ^ 2 | \\vec b _ i | ^ 2 \\bigg ) ^ { 1 / 2 } = \\lambda _ k ' \\cdot 3 | \\underline { \\vec z _ { k + d - 1 } } | ^ { 1 + \\beta } ; \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} \\mathcal T _ 1 = \\coth ^ 2 ( \\sqrt { \\mu _ 2 } | D | ) - \\frac { 1 } { \\mu _ 2 | D | ^ 2 } , \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , M } ( x \\ , | \\ , a ) \\ , \\log ( x ) \\bigr ) ( q | b ) = \\log ( q ) \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , M } ( x \\ , | \\ , a ) \\ , \\bigr ) ( q | b ) + O ( q ) , \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} V ( D ) = \\begin{bmatrix} D & e ^ { - \\frac 1 2 D D ^ * } \\\\ \\\\ - e ^ { - \\frac 1 2 D ^ * D } & \\frac { I - e ^ { - D ^ * D } } { D ^ * D } D ^ * \\end{bmatrix} , V ( D ) ^ { - 1 } = \\begin{bmatrix} \\frac { I - e ^ { - D ^ * D } } { D ^ * D } D ^ * & - e ^ { - \\frac 1 2 D ^ * D } \\\\ \\\\ e ^ { - \\frac 1 2 D D ^ * } & D \\end{bmatrix} ; \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\rho _ 1 : = & ( 1 , 2 ) , ( 3 , 5 ) , ( 1 , 3 ) , ( 4 , 6 ) , ( 2 , 4 ) , ( 1 , 4 ) , ( 5 , 6 ) , ( 1 , 6 ) , \\\\ & ( 2 , 3 ) , ( 2 , 5 ) , ( 1 , 5 ) , ( 3 , 4 ) , ( 4 , 5 ) , ( 2 , 6 ) , ( 3 , 6 ) \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ \\infty \\abs { u _ l } ( x ) \\leq C _ q \\Lambda _ f \\abs { x } ^ { q + 2 } B _ 1 , \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & A _ 8 & A _ 9 & B _ 8 & B _ 9 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} q ^ { \\sum _ { 1 \\le i \\le l _ 3 } ( \\dim ( W _ { i - 1 } ^ \\perp ) + \\dim ( V _ 1 ) ) + l _ 3 s } = q ^ { \\sum _ { 1 \\le i \\le l _ 3 } ( \\frac { k b } { 2 } - ( l _ 2 + 2 i - 2 ) + \\frac { k b } { 4 } ) + l _ 3 s } = q ^ { l _ 3 ( \\frac { 3 } { 4 } k b - l _ 2 - l _ 3 + 1 + s ) } . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} \\mathrm { d i m } ( H _ { J } ) = 2 g - \\# { J } . \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} \\phi _ i ( x ) = y _ i + \\int _ 0 ^ 1 x \\ , \\phi _ { i + 1 } ( t x ) \\ , d t i = d - 1 , \\dots , 0 . \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} ( g _ { n + m } ( x ) ) ^ { - 1 } g _ n ( x ) = ( f _ { n + m } ( x ) ) ^ { - 1 } \\cdots ( f _ { n + 1 } ( x ) ) ^ { - 1 } \\in U _ { n + 1 } \\ ; \\mbox { f o r a l l $ x \\in X $ . } \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k \\right ) = \\frac { 1 } { k ! } \\sum _ { j = 0 } ^ { k } \\left ( - 1 \\right ) ^ { j } \\binom { k } { j } \\left ( a _ { 1 } \\left ( k - j \\right ) + b _ { 1 } \\right ) ^ { p _ { 1 } } \\left ( a _ { 2 } \\left ( k - j \\right ) + b _ { 2 } \\right ) ^ { p _ { 2 } } . \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\hbox { $ \\rho : C \\rightarrow E $ i s a v e c t o r b u n d l e m o r p h i s m o v e r t h e i d e n t i t y o f $ M $ } \\\\ R = ( R ^ C , R ^ E ) \\hbox { w i t h $ R ^ E \\in \\Gamma ( s ^ * E ^ \\vee \\otimes t ^ * E ) $ a n d $ R ^ C \\in \\Gamma ( s ^ * C ^ \\vee \\otimes t ^ * C ) $ } \\\\ \\Omega \\in \\Gamma ( ( s ^ { ( 2 ) } ) ^ * E ^ \\vee \\otimes ( t ^ { ( 2 ) } ) ^ * C ) \\end{array} \\right . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} \\rho _ 0 = - 2 ^ { 1 2 } \\cdot 3 ^ { - 2 } \\cdot 5 \\cdot 7 \\cdot ( v _ 2 ^ 2 - v _ 1 v _ 3 ) . \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} J ( t , x ; u ( \\cdot ) ) = Y ^ { t , x ; u } ( t ) . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\gamma _ { \\epsilon } = { m _ \\epsilon } ^ { - 1 } \\gamma _ 0 m _ \\epsilon . \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} u ( x ) = t ( - 1 ) \\cdot s \\cdot n ( - x ) \\cdot s x \\in \\Q _ v . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} R ( \\sigma ( \\omega ) ) & = f _ { \\omega _ 0 } ( R ( \\omega ) ) , \\\\ f _ { \\omega _ 0 } ^ { - 1 } ( R ( \\sigma ( \\omega ) ) ) & = R ( \\omega ) . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} I ( U _ * ) = I ( u _ 0 ) + k \\left [ A - \\frac { B _ 1 k ^ { N - 2 } } { r ^ { N - 2 } \\lambda ^ { N - 2 } } + \\frac { B _ 2 u _ 0 ( r ) } { \\lambda ^ { \\frac { N - 2 } { 2 } } } + O ( \\frac { 1 } { \\lambda ^ { \\frac { N - 2 } { 2 } ( 1 + \\delta ) } } ) \\right ] \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} E _ 3 = E _ 2 + \\Vert \\partial _ x ^ 2 \\rho _ 0 \\Vert _ { L ^ 2 } + \\Vert \\partial _ x ^ 2 u _ 0 \\Vert _ { L ^ 2 } . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} p _ i = \\left ( 2 ^ { \\frac { R . | I | } { B } } - 1 \\right ) . N . \\left ( \\dfrac { 4 \\pi d _ i f } { c } \\right ) ^ 2 = K d ^ 2 _ i \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} \\frac { ( k + 1 + ( 2 - \\theta _ 0 ) / \\theta _ 0 ) ^ 2 } { ( 2 - \\theta _ 0 ) ^ 2 } - \\frac { k + 1 + ( 2 - \\theta _ 0 ) / \\theta _ 0 } { 2 - \\theta _ 0 } < \\frac { 1 - \\theta _ { k + 1 } } { \\theta _ { k + 1 } ^ 2 } \\overset { \\eqref { a r e c t h e t a } } { = } \\frac { 1 } { \\theta _ k ^ 2 } \\leq \\frac { ( k + ( 2 - \\theta _ 0 ) / \\theta _ 0 ) ^ 2 } { ( 2 - \\theta _ 0 ) ^ 2 } . \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} \\alpha = \\alpha _ h u ^ h . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\Delta \\left ( x { \\bf u } ^ { \\tt M } \\right ) = \\left ( x ( c - 1 ) + \\beta c \\right ) { \\bf u } ^ { \\tt M } , \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} 0 < \\sum _ { i , j = 1 } ^ { N } a _ { i j } \\left ( x , t ; \\mathbf { p } ; \\mathbf { q } \\right ) \\xi _ { i } \\xi _ { j } \\le \\overline { M } \\left | \\xi \\right | ^ { 2 } \\ , \\xi \\in \\mathbb { R } ^ { N } , \\left ( \\mathbf { p } , \\mathbf { q } \\right ) \\in [ L ^ { 2 } \\left ( \\Omega \\right ) ] ^ { N } \\times \\left [ L ^ { 2 } \\left ( \\Omega \\right ) \\right ] ^ { N d } . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\Delta _ { \\alpha , \\varepsilon } ^ { D P } = \\mathbf { 1 } _ { \\lbrace S _ { D _ \\varepsilon } > s _ { \\alpha , \\varepsilon } \\rbrace } \\mathrm { w h e r e } S _ { D _ \\varepsilon } = \\sum _ { k = 1 } ^ { D _ \\varepsilon } ( y _ k ^ 2 - \\varepsilon ^ 2 ) , \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} x _ { \\sigma ( \\iota ) } = w _ { j _ k , i _ { \\iota } } \\bigl ( 1 + \\frac { \\ell ( i _ { \\iota } ) } { \\mu _ { i _ { \\iota } } } \\bigr ) \\Biggl ( \\frac { y _ { \\sigma ( \\iota _ { i ^ * _ k } ) } } { w _ { j _ k , i ^ * _ k } \\bigl ( 1 + \\frac { \\lambda _ { \\ell ( i ^ * _ k ) } } { \\mu _ { i ^ * _ k } } \\bigr ) } - \\frac { y _ { \\sigma ( \\iota ) } } { w _ { j _ k , i _ { \\iota } } \\bigl ( 1 + \\frac { \\lambda _ { \\ell ( i _ { \\iota } ) } } { \\mu _ { i _ { \\iota } } } \\bigr ) } \\Biggr ) ; \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} ( A \\mu ) ( d \\eta ) = - \\Psi ( \\eta ) \\mu ( d \\eta ) , ( B \\mu ) ( d \\eta ) = \\int _ { \\Gamma _ 0 } \\Xi ( d \\eta | \\xi ) \\mu ( d \\xi ) , \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\mathcal { T } _ 1 = 1 + \\frac { M ^ { - 2 ( 1 - \\Upsilon ) \\delta } - M ^ { - 2 \\delta } } { ( 2 \\delta \\Upsilon \\log M ) ^ 2 } + \\mathcal { O } \\left ( \\frac { \\log \\log q } { \\log q } M ^ { - 2 ( 1 - \\Upsilon ) \\delta } \\right ) . \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\beta _ i = \\frac { 1 } { w _ { j , i } ( 1 + \\frac { \\lambda _ { \\ell ( i ) } } { \\mu _ i } ) } , \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} p _ G ( M ) = \\sum _ { A \\subseteq E } ( - 1 ) ^ { | A ^ c | } p _ G ^ 1 ( M \\backslash A ^ c ) , \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\tilde { Q } _ { k } ( x , y ) ~ = ~ \\P _ x ( \\tilde { X } _ k ( n ) = y , \\ ; \\ ; n > 0 ) , x , y \\in E . \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} d X _ t = b ( X _ { t - } ) \\ , d t + f ( X _ { t - } ) \\ , d B _ t + g ( X _ { t - } ) \\ , d J _ t \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} \\bar u ( x ) \\in \\begin{cases} \\{ u _ i \\} & \\bar p ( x ) \\in Q _ i 1 \\leq i \\leq d , \\\\ [ u _ i , u _ { i + 1 } ] & \\bar p ( x ) \\in Q _ { i , i + 1 } 1 \\leq i < d . \\end{cases} \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} g ( T X , T Y ) = \\sum _ { \\lambda } ^ { } \\lambda ^ 2 g ( U ^ \\lambda X , U ^ \\lambda Y ) . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} R _ 1 ^ { A B } ( \\rho _ - , v _ - ) : \\left \\{ \\begin{array} { l l } \\xi = \\lambda _ 1 = v + \\beta t - \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } , \\\\ v - v _ - = - \\int _ { \\rho _ { - } } ^ { \\rho } \\frac { \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } } { \\rho } d \\rho , \\ \\ \\rho < \\rho _ { - } , \\end{array} \\right . \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) = T _ { n , m } ' . \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} \\begin{aligned} \\textstyle { \\max _ { K } } D ( u , w ) & A ^ { \\intercal } u = P ^ { \\intercal } w , \\\\ & ( u , w ) \\geq 0 \\\\ & e ^ { \\intercal } w = 1 , \\end{aligned} \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P ^ n \\Big ( \\| X _ { \\rho _ n + h _ n } - X _ { \\rho _ n } \\| \\geq \\epsilon \\Big ) = 0 . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ { k } \\frac { P _ j ( \\xi _ 1 + \\lambda ^ { - 1 } y ) } { \\lambda ^ { \\frac { N - 2 } 2 } } \\le & C \\sum _ { j = 2 } ^ { k } \\frac { 1 } { ( 1 + | y - \\lambda ( \\xi _ j - \\xi _ 1 ) | ) ^ { N - 2 } } \\\\ \\le & C \\sum _ { j = 2 } ^ { k } \\frac { 1 } { | \\lambda ( \\xi _ j - \\xi _ 1 ) | ^ { N - 2 } } \\\\ \\le & \\frac { C | \\ln k | ^ { \\sigma _ N } k ^ { N - 2 } } { \\lambda ^ { N - 2 } } \\le \\frac { C | \\ln k | ^ { \\sigma _ N } } { \\lambda ^ { \\frac { N - 2 } 2 } } , \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} L M ( s , f ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { s } } \\left ( \\sum _ { a b c ^ 2 = n } \\lambda _ f ( a ) \\lambda _ f ( b ) \\mu ( b ) \\mu ( b c ) ^ 2 F _ { \\Upsilon , M } ( b c ) \\right ) . \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} [ p t ] = \\sigma _ { 2 n - m , 2 n - 1 - m , \\dots , 2 n + 1 - 2 m } = j ^ * \\sigma _ { 2 n + 1 - m , 2 n - m , \\dots , 2 n + 2 - 2 m } ^ + , \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\nabla ^ { \\nu } [ b U ( t ) + c Y ( t ) ] = b \\nabla ^ { \\nu } U ( t ) + c \\nabla ^ { \\nu } Y ( t ) , \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} \\mathcal { I } \\cdot \\begin{pmatrix} \\gamma _ 1 \\\\ \\vdots \\\\ \\gamma _ M \\end{pmatrix} = \\mathbf { 0 } \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} \\min _ { 1 \\leq l \\leq m } v \\big ( a _ l - \\rho _ l ( a ) \\big ) = \\min _ { l \\neq k } v \\big ( a _ l - \\rho _ l ( a ) \\big ) \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} \\mathcal { H } = \\mathcal { H } ^ u \\oplus \\mathcal { H } ^ s , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\P _ { x + y } ( \\tau _ \\vartheta = + \\infty ) \\geq \\P _ { x + y } ( S ( n ) \\in { \\cal C } , \\forall n \\geq 0 ) > 0 \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} ( n + 1 ) \\sum _ { k = 0 } ^ m \\tbinom { m } { k } ( - 1 ) ^ k a _ { n + k } = & \\sum _ { k = 0 } ^ m \\tbinom { m } { k } ( - 1 ) ^ k ( n + k + 1 ) a _ { n + k } \\\\ & + m \\sum _ { k = 0 } ^ { m - 1 } \\tbinom { m - 1 } { k } ( - 1 ) ^ k a _ { n + k + 1 } . \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} \\dim ( H _ { J _ { k + 1 } } ) = \\dim ( H _ { J _ { k } } ) + \\dim ( H _ { k + 1 } ) - \\dim ( A [ \\ell ] ( \\overline { K } ) ) = 2 g - k + 2 g - 1 - 2 g = 2 g - ( k + 1 ) = 2 g - \\# { J _ { k + 1 } } . \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} \\P _ { \\alpha } ( X _ t \\in \\cdot | \\tau _ A > t ) = \\alpha . \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { r ^ { ( 1 ) } _ { i } } \\sum _ { q = 1 } ^ { r ^ { ( 1 ) } _ { i - 1 } } \\mathrm { m i n } \\{ n _ { p , i } , n _ { q , { i - 1 } } \\} & = \\sum _ { s = 1 } ^ { k } r ^ { ( s ) } _ { i - 1 } r _ { i } ^ { ( s ) } , & \\\\ \\sum _ { p = 1 } ^ { r _ { i } ^ { ( 1 ) } } ( \\sum _ { p > p ' > 0 } 2 \\mathrm { m i n } \\{ n _ { p , i } , n _ { p ' , i } \\} + n _ { p , i } ) & = \\sum _ { s = 1 } ^ { k } r ^ { ( s ) ^ { 2 } } _ { i } , & \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} { } S ^ { z } _ { n } = \\prod \\limits _ { j = n - 1 } ^ { 0 } S ^ { z } _ { \\alpha _ { n } } . \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} \\frac { \\Im w _ { n + 1 } } { 1 0 } = \\Re \\frac { 4 ( w _ 1 + 1 ) \\sum \\limits _ { j = 3 } ^ { n } | w _ j | ^ 2 + | w _ 1 | ^ 2 ( 2 w _ 2 + w _ 1 \\overline { w _ 2 } ) + 4 w _ 1 \\overline { w _ 2 } + 2 ( w _ 1 ) ^ 2 \\overline { w _ 2 } } { ( 2 + w _ 1 ) ( 2 + \\overline { w _ 1 } ) ( 2 0 - d _ n w _ 1 \\overline { w _ 1 } ) } . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} x _ { \\sigma ( \\iota ) } = \\frac { w _ { j _ { k ' } , i } } { w _ { j _ k , i } } \\Bigl ( \\frac { y _ { \\sigma ( \\iota _ i ) } } { w _ { j _ { k ' } , i } \\bigl ( 1 + \\frac { \\lambda _ { \\ell _ { \\iota _ i } } } { \\mu _ { i } } \\bigr ) } - \\frac { y _ { \\sigma ( \\iota _ { i ^ * _ { k ' } } ) } } { w _ { j _ { k ' } , i ^ * _ { k ' } } \\bigl ( 1 + \\frac { \\lambda _ { \\ell ( i ^ * _ { k ' } ) } } { \\mu _ { i ^ * _ { k ' } } } \\bigr ) } \\Bigr ) ; \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} B ' ( v , e ) = B ( v , e ) , ~ w ' ( e ) = w ( e ) . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} f _ { i + N } = g _ i , 1 \\le i \\le n . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} \\rho ( t , x , y ) = \\frac { 2 P ( t , x ) - h _ 1 ^ 2 ( t , x , y ) } { 2 R \\theta ( t , x , y ) } , \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} e ( N ( 1 ) ) & = \\sum _ { t = n - m - 1 } ^ { n - 1 } \\left [ ( - 1 ) ^ { n - 1 - t } \\cdot t \\cdot f _ { n - t - 1 } \\right ] + 1 \\\\ & = ( - 1 ) ^ m ( n - m - 1 ) f _ m + ( - 1 ) ^ { m - 1 } ( n - m ) f _ { m - 1 } + ( - 1 ) ^ { m - 2 } ( n - m + 1 ) f _ { m - 2 } + \\cdots + ( n - 1 ) f _ 0 + 1 . \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} ( u \\times v ) _ n = \\sum _ { k = 0 } ^ n \\binom { n } { k } u _ k v _ { n - k } . \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} \\vert \\tau ( y ) - \\tau _ k ( y ) \\vert & \\leq \\sum _ { i = k } ^ { \\infty } \\vert \\tau _ { i + 1 } ( y ) - \\tau _ i ( y ) \\vert \\\\ & \\leq \\frac 5 4 \\left ( 3 6 C _ 1 ( m ) + 2 4 \\right ) \\delta r _ k \\cdot \\sum _ { i = 0 } ^ \\infty 4 ^ { - i } \\\\ & = \\frac 5 3 \\left ( 3 6 C _ 1 ( m ) + 2 4 \\right ) \\delta r _ k . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} R _ 1 ^ { A B } : \\ \\ \\left \\{ \\begin{array} { l l } \\xi = \\lambda _ 1 ^ { A B } = v + \\beta t - \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } , \\\\ v - u _ - = - \\int _ { \\rho _ { - } } ^ { \\rho } \\frac { \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } } { \\rho } d \\rho , \\ \\ \\rho _ { * } ^ { A B } \\leq \\rho \\leq \\rho _ { - } . \\end{array} \\right . \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\begin{array} { c c c } M _ { W \\boxplus U } ( k ) = M _ { W } ( k ) + M _ { U } ( k ) , & M _ { W \\boxtimes U } ( k ) = \\sum _ { m + n = k } M _ { W } ( m ) M _ { U } ( n ) , & \\forall k \\in \\mathbb { Z } . \\end{array} \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} d ( \\underline { S } , f ) = ( b ( { \\mathbf { 0 } } ) \\Gamma _ { \\mathbf { 0 } , 0 } ( \\underline { S } ) , \\ldots , b ( { \\mathbf { i } } ) \\Gamma _ { \\mathbf { i } , \\vert \\mathbf { i } \\vert } ( \\underline { S } ) , \\ldots , b ( { \\mathbf { k } } ) \\Gamma _ { \\mathbf { k } , \\vert \\mathbf { k } \\vert } ( \\underline { S } ) ) . \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\alpha ( N , c ) = \\begin{cases} N & \\gcd ( c , N / c ) > 2 , \\\\ \\frac { N } { c } & \\gcd ( c , N / c ) = 1 , \\\\ \\frac { N } { 2 c } & \\gcd ( c , N / c ) = 2 \\frac { N } { 2 c } \\\\ \\frac { 2 N } { c } & \\gcd ( c , N / c ) = 2 \\frac { N } { 2 c } \\end{cases} \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} \\bigl \\langle u _ \\Gamma ' ( t ) , z _ \\Gamma \\bigr \\rangle _ { V _ \\Gamma ^ * , V _ \\Gamma } + \\int _ { \\Omega } ^ { } \\nabla \\mu ( t ) \\cdot \\nabla z d x + \\int _ { \\Gamma } ^ { } \\nabla _ \\Gamma \\mu _ \\Gamma ( t ) \\cdot \\nabla _ \\Gamma z _ \\Gamma d \\Gamma = 0 \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} N = 1 / \\varepsilon . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ 1 x _ { n } + x _ 1 \\sum _ { i = 2 } ^ { n - 1 } x _ i ^ 2 , \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} \\begin{aligned} & Q ( \\theta ) * E _ 1 = \\cos ( 2 \\theta ) E _ 1 + \\sin ( 2 \\theta ) E _ 2 , \\ & Q ( \\theta ) * E _ 2 = - \\sin ( 2 \\theta ) E _ 1 + \\cos ( 2 \\theta ) E _ 2 , \\\\ & Q ( \\theta ) * \\mathbb { E } _ 1 = \\cos ( 4 \\theta ) \\mathbb { E } _ 1 + \\sin ( 4 \\theta ) \\mathbb { E } _ 2 , \\ & Q ( \\theta ) * \\mathbb { E } _ 2 = - \\sin ( 4 \\theta ) \\mathbb { E } _ 1 + \\cos ( 4 \\theta ) \\mathbb { E } _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} | \\nabla ' s | & = \\tfrac 1 2 \\ , | d \\rho - J ^ * \\lambda \\rho | , \\\\ | \\nabla '' s | & = \\tfrac 1 2 \\ , | d \\rho + J ^ * \\lambda \\rho | . \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} \\lim _ { a \\to \\infty } h ' ( a ) = M \\sum _ { i = 1 } ^ { d + 1 } \\frac { \\gamma _ i } { M } \\log \\left ( \\frac { \\gamma _ i / M } { x _ i } \\right ) \\geq - M \\log \\left ( \\sum _ { i = 1 } ^ { d + 1 } x _ i \\right ) = 0 . \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} { \\bf P } ^ { i j } ( x ) = e ^ i _ { \\ ; k } e ^ j _ { \\ ; l } P ^ { k l } , \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) = \\frac { \\left ( r ! \\right ) ^ { p } \\prod _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\right ) ^ { p _ { s } } } { k ! } \\sum _ { i = 0 } ^ { r p + \\sigma } \\binom { r p + \\sigma - i } { r p + \\sigma - k } A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) . \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{gather*} E _ 2 ( \\tau ) : = 1 - 2 4 \\sum _ { n = 1 } ^ \\infty \\frac { n q ^ n } { 1 - q ^ n } \\end{gather*}"}
-{"id": "7061.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\langle \\dot { \\tilde { p } } _ k ( t ) , \\sigma _ k ( t ) \\rangle d t = \\int _ 0 ^ T \\langle E \\dot { \\tilde { u } } _ k ( t ) , \\sigma _ k ( t ) \\rangle d t - \\int _ 0 ^ T \\langle \\dot { \\tilde { e } } _ k ( t ) , \\sigma _ k ( t ) \\rangle d t . \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} \\begin{array} { l l } A ( x ^ i ) U ( x ^ i , \\xi ^ i ) u _ { \\xi ^ i } = ( \\kappa I + U ( x ^ i , \\xi ^ i ) ) ~ [ \\partial _ { x ^ i } { \\rm L o g } ~ ( U ( x ^ i , \\xi ^ i ) + \\kappa I ) ] u _ { \\xi ^ i } , \\end{array} \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} W ^ C _ { 1 0 6 } = & 1 + ( 3 5 2 4 5 + 2 d ) y ^ { 1 8 } + ( 4 1 6 2 6 2 + 1 2 8 c - 2 d ) y ^ { 2 0 } \\\\ & + ( 6 5 8 6 3 1 0 + 8 1 9 2 b - 8 9 6 c - 3 4 d ) y ^ { 2 2 } \\\\ & + ( 8 6 6 2 6 6 4 5 + 5 2 4 2 8 8 a - 1 0 6 4 9 6 b + 1 0 2 4 c + 3 4 d ) y ^ { 2 4 } + \\cdots , \\\\ W ^ S _ { 1 0 6 } = & a y ^ 5 + ( - 2 4 a - b ) y ^ 9 + ( 2 7 6 a + 2 2 b + c ) y ^ { 1 3 } \\\\ & + ( - 2 0 2 4 a - 2 3 1 b - 2 0 c - d ) y ^ { 1 7 } + \\cdots , \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} b _ 0 = \\frac { 1 } { \\theta _ 0 ^ 2 } \\frac { 1 } { 1 + \\sum _ { l = 0 } ^ { K - 1 } \\frac { \\theta _ { - 1 } ^ 2 } { \\theta _ { l } } \\prod _ { j = 1 } ^ { l } \\frac { 1 } { 1 - \\sigma ^ K _ { j } } } \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} L ( x ) ^ { q ^ i } = \\sum _ { k = 1 } ^ m c _ { k - i } ^ { q ^ i } \\ , x ^ { q ^ k } , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} I ( \\vec x ) & = \\sum _ { \\{ i , j \\} \\in E } w _ { i j } | x _ i - x _ j | , \\\\ \\| \\vec x \\| _ \\infty & = \\max \\{ | x _ 1 | , \\ldots , | x _ n | \\} , \\\\ F ( \\vec x ) & = \\frac { I ( \\vec x ) } { \\norm { \\vec x } } . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\ < z ' | \\lambda z \\ > = K ( z ' , \\lambda z ) = \\lambda ^ e K ( z ' , z ) = \\lambda ^ e \\ < z ' | z \\ > . \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} e = \\begin{cases} ( c , p ^ m ) _ p & d = 0 \\mathrm { o r d } _ p ( d ) \\\\ ( c , d ) _ p ( c , p ^ m ) _ p & d \\neq 0 \\mathrm { o r d } _ p ( d ) \\end{cases} \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta \\ ! \\left ( u ^ m \\right ) + u ^ p & B _ R \\times ( 0 , T ) \\ , , \\\\ u = 0 & \\partial B _ R \\times ( 0 , T ) \\ , , \\\\ u ( \\cdot , 0 ) = u _ 0 & B _ R \\ , . \\end{cases} \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} 2 g ' _ j f _ i = j g _ { i + j - 1 } = - 2 f ' _ i g _ j . \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} F ( P , a ) : = \\begin{cases} ( f ( P ) - b ( P ) , 2 w ( P ) ) & \\\\ ( - f ( P ) + b ( P ) , - 2 w ( P ) ) & \\end{cases} \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\omega \\cdot v & = q ^ { - i ( \\sum _ { k = 1 } ^ t n _ k ) } v , & & \\forall v \\in V ^ i . \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\sum _ { g , \\sigma } f _ 1 \\ast _ g ( x _ { i _ \\sigma ( 1 ) i _ \\sigma ( 2 ) } x _ { i _ \\sigma ( 3 ) i _ \\sigma ( 4 ) } ) = 9 f _ { 2 } \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } { A } _ { 1 , n } ^ { j , \\ell } ( x ) = 1 \\mbox { a n d } \\lim _ { n \\rightarrow \\infty } { B } _ { 1 , n } ^ { j , \\ell } ( x ) = 0 , \\ell = 1 , 2 . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\mathbf { D } \\mathbf { a } _ i = \\mathbf { e } _ i , \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} \\sigma _ t ( \\mu _ a ) = N _ a ^ { - i t } \\mu _ a \\quad \\sigma _ t ( e _ r ) = e _ r . \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} x _ n ' = x _ { n + N } \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} \\ell = \\mathcal { L } _ t ( \\Delta _ t ^ { - 1 } ( \\zeta , \\varphi ) ) . \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\mu ( \\gamma ) = \\int ^ { 1 } _ { 0 } \\phi ( \\rho ) d \\rho + \\int ^ { + \\infty } _ { 1 } \\phi ( \\rho ) d \\rho = I _ { 1 } + I _ { 2 } . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\sum _ { \\l \\leq L } \\vert x _ { \\l } \\vert ^ 2 & \\ll \\sum _ { \\l _ 1 \\leq L \\atop ( \\l _ 1 , r ) = 1 } \\vert \\mu _ f ( \\l _ 1 ) \\vert ^ 2 \\sum _ { \\l _ 2 \\leq L / \\l _ 1 \\atop ( \\l _ 2 , r ) = 1 } \\vert \\mu _ f ( \\l _ 2 ) \\vert ^ 2 \\\\ & \\ll \\sum _ { \\l _ 1 \\leq L \\atop ( \\l _ 1 , r ) = 1 } \\sum _ { \\l _ 2 \\leq L / \\l _ 1 \\atop ( \\l _ 2 , r ) = 1 } \\vert \\lambda _ f ( \\l _ 2 ) \\vert ^ 2 \\\\ & \\ll \\sum _ { \\l _ 1 \\leq L \\atop ( \\l _ 1 , r ) = 1 } ( L / \\l _ 1 ) \\ll L , \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\delta ( X ) = [ 1 \\otimes X + X \\otimes 1 , r ] , \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} & s _ { 1 , j } : = r ^ { j - 1 } ( s _ { 1 , 1 } ) , & & d _ j : = r ^ { j - 1 } ( d _ 1 ) , & \\\\ & V ' _ { i , j } : = { } _ { r ^ { j - 1 } } ( V ' _ i ) , & & W ' _ { i , j } : = { } _ { r ^ { j - 1 } } ( W ' _ i ) & \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} [ n ] _ q ! S _ n ( A ) = \\sum _ I \\tilde c _ I ( q ) \\tilde \\Theta ^ I ( q ) = \\sum _ I \\frac { [ n ] _ q ! } { [ i _ 1 ] _ q [ i _ 1 + i _ 2 ] _ q \\cdots [ i _ 1 + i _ 2 + \\cdots i _ r ] _ q } \\tilde \\Theta ^ I ( q ) \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} 2 ^ { k - \\nu ( N ) - 5 / 2 } \\zeta _ { \\Q } ( 2 ) ^ { - 1 } \\mu _ { N , M } c ( \\mathfrak d _ { \\xi } ) \\chi ( \\mathfrak f _ { \\xi } ) \\mathfrak f _ { \\xi } ^ { k - 1 / 2 } \\mathcal W _ { \\infty } \\times \\prod _ { p \\nmid N } \\sum _ { n = 0 } ^ { \\mathrm { m i n } ( \\mathrm { o r d } _ p ( b _ i ) ) } p ^ { n / 2 } \\Psi _ p \\left ( \\frac { 4 \\xi } { p ^ { 2 n } } ; \\alpha _ p \\right ) \\prod _ { p \\mid N } \\Psi _ p ( \\xi ; \\alpha _ p ) . \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} ( i v ) h = \\bar { T } - \\bar { U } = \\frac { 1 } { 2 } \\dot { \\rho } ^ { 2 } + \\frac { \\rho ^ { 2 } } { 8 } [ \\dot { \\varphi } ^ { 2 } + ( \\sin ^ { 2 } \\varphi ) \\dot { \\theta } ^ { 2 } ] + \\frac { \\omega ^ { 2 } } { 2 \\rho ^ { 2 } } - \\frac { U ^ { \\ast } } { \\rho } \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} \\log \\mathfrak { M } ( q = 0 \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\ , A ( t ) . \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} \\int \\limits _ { \\beta _ { k n } } \\ , \\Delta f _ 0 ( M ) \\ , d m _ 3 = C _ { k n } \\Lambda _ { n - 2 } \\omega ( \\Lambda _ { n - 2 } ) , \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\sum _ { n = N _ { k - 1 } } ^ { N _ k - 1 } \\abs { m _ { n + 1 } ( x ) - m _ n ( x ) } = \\big ( \\Psi _ { N _ { k - 1 } } ^ { - 1 } - \\Psi _ { N _ k } ^ { - 1 } \\big ) \\frac { \\log x } { \\psi ( x ) } . \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } y \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } w \\ , f _ t ( y , w ) = 1 \\ , , \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} \\hat { h } ( \\sigma ) ( \\gamma _ 1 , \\ldots , \\gamma _ p ) = - h ( s _ { V _ 1 } ( \\sigma ( 1 _ { t ( \\gamma _ 1 ) } , \\gamma _ 1 , \\ldots , \\gamma _ p ) ) ) \\cdot 0 _ { \\gamma _ 1 } , \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} X _ t ( \\omega ) = Y _ t ^ 2 ( \\omega ) \\mathbf { 1 } _ { \\{ t < \\tau ( \\omega ) \\} } . \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} { \\rm M a x } ( f ) \\cap U & = { \\rm M a x } _ 0 ( f ) = \\{ h = 0 \\} \\cap \\{ g < 0 \\} , \\\\ { \\rm M i n } ( f ) \\cap U & = { \\rm M i n } _ 0 ( f ) = \\{ h = 0 \\} \\cap \\{ g > 0 \\} , \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} { \\bf E } _ v ( { \\rm e } ^ { - \\beta X _ t } ) = \\frac { \\psi _ { \\beta } ( v ) } { \\psi _ { \\beta } ( t ) } = \\frac { v ^ { 1 - \\delta / 2 } \\ , I _ { \\delta / 2 - 1 } ( v \\sqrt { 2 \\beta } ) } { t ^ { 1 - \\delta / 2 } \\ , I _ { \\delta / 2 - 1 } ( ( t + v ) \\sqrt { 2 \\beta } ) } , \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{gather*} - \\Delta \\mu = 0 \\mbox { i n } Q , \\\\ \\tau \\partial _ t u - \\Delta u + { \\mathcal W } ' ( u ) = f \\mbox { i n } Q , \\end{gather*}"}
-{"id": "3670.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\Q _ { s , x } ( X _ t \\in B ) & = \\beta _ \\infty ( B ) \\\\ & = \\liminf _ { t \\to \\infty } \\Q _ { s , x } ( X _ t \\in B ) . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} \\psi _ 3 ( w _ i ) = \\big ( ( b ^ i ) ^ { a ^ { i e _ 1 } } , ( b ^ i ) ^ { a ^ { i ( e _ 1 + e _ 2 ) } } , \\dots , b ^ i , b ^ i \\big ) . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\eta = \\frac { p } { p _ 1 } \\mu , ~ \\xi = \\frac { p ( p _ 1 - r ) } { p _ 1 ( p - r ) } \\mu . \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} I ( U _ * + \\varphi ) = I ( U _ * ) + k O \\Bigl ( \\lambda ^ { - \\frac { N - 2 } 2 - 2 \\sigma } \\Bigr ) , \\ \\hbox { f o r s o m e $ \\sigma < 0 . $ } \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} u _ t - u _ { x x t } + 3 u u _ x = 2 u _ x u _ { x x } + u u _ { x x x } , \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} k _ 0 & = \\min \\{ k \\in \\N \\ , : \\ , \\omega _ x ( k ) \\in \\omega _ y \\} , \\\\ m _ 0 & = \\min \\{ m \\in \\N \\ , : \\ , \\omega _ y ( m ) \\in \\omega _ x \\} , \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} 0 = \\Delta v _ { j , k } - v _ { j , k - 1 } + \\sum _ { \\ell = 0 } ^ { k - 2 } t _ { j - k , j - \\ell } \\ , v _ { j , \\ell } . \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\P _ { x } ( X ( m ) = y ) ~ \\geq ~ \\delta ^ m ~ \\geq ~ \\delta ^ { \\kappa | y - x | } \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( a : N ) } ^ \\ast ( s , \\tau ) = \\sum _ { m , n \\in \\Z } \\frac { 1 } { \\ ( ( N m + a ) \\tau + n \\ ) ^ k } \\frac { \\Im ( \\tau ) ^ s } { | ( N m + a ) \\tau + n | ^ { 2 s } } , N , a \\in \\Z _ { > 0 } , \\end{gather*}"}
-{"id": "7611.png", "formula": "\\begin{align*} x _ 1 = - y _ 4 \\ ; , \\ ; \\ ; x _ 2 = y _ 1 - y _ 3 \\ ; , \\ ; \\ ; x _ 3 = y _ 2 \\ ; , \\ ; \\ ; x _ 4 = y _ 1 , \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} J [ w ] = K [ w ] w \\in W ^ { 1 , 1 } ( \\Omega , \\R ^ N ) , \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} & \\sum _ { \\substack { n m \\leqslant N \\\\ ( n , m ) = 1 } } \\frac { r ( n ) ^ 2 r ( m ) \\omega ( n ) \\omega ' ( m ) } { \\sqrt { m } } \\\\ & \\qquad = \\big ( 1 + O \\big ( N ^ { - 1 / ( \\log \\log N ) ^ 3 } \\big ) \\big ) \\prod _ p \\left ( 1 + r ( p ) ^ 2 \\omega ( p ) + \\frac { r ( p ) } { \\sqrt { p } } \\omega ' ( p ) \\right ) , \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} | \\Lambda _ { \\varepsilon } | = \\frac { 2 c } { \\pi } + \\frac { 1 } { \\pi ^ 2 } \\log \\left ( \\frac { 1 - \\varepsilon } { \\varepsilon } \\right ) \\log ( c ) + o ( \\log c ) . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} C ( N , M , \\chi ) = | \\chi ( 2 ) | ^ { - 2 } M ^ { 3 - k } N ^ { - 1 } \\prod _ { p \\mid N } ( p + 1 ) ^ 2 \\prod _ { p \\mid M } ( p + 1 ) . \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} E ( z ) & = e ^ { - 2 h ( z ) } e ^ { \\bf h } E ( z ) e ^ { - { \\bf h } } , \\\\ e ^ { - { \\bf h } } E \\otimes x ( \\zeta ) e ^ { \\bf h } & = E \\otimes e ^ { - 2 h ( \\zeta ) } x ( \\zeta ) . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\mathbb { P } [ \\mathbf { D } \\mathbf { x } = \\mathbf { e } _ i ~ | ~ w t ( \\mathbf { x } ) = k ] = 2 ^ { - m } \\left ( 1 - \\sigma ^ k \\right ) \\left ( 1 + \\sigma ^ k \\right ) ^ { m - 1 } . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} \\Vert b _ i - a _ i \\Vert = \\Vert a _ i \\chi _ { N _ i } \\Vert \\le 1 - \\Vert b _ i \\Vert < 2 \\rho , \\forall i \\in C ' . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} { \\theta _ { 1 , u } } \\left ( t \\right ) = - 2 \\pi \\Delta f \\left ( { t - n T } \\right ) + \\varphi _ { n , u } ^ { { \\rm { C F S K } } } + 2 \\pi f _ u ^ { { \\rm { R F } } } t + { \\varphi _ u } = 2 \\pi { f _ { 1 , u } } t + 2 \\pi \\Delta f n T + \\varphi _ { n , u } ^ { { \\rm { C F S K } } } + { \\varphi _ u } \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\| \\psi _ \\ell - \\psi \\| ^ 2 = ( \\psi _ \\ell - \\psi ) ^ * ( \\psi _ \\ell - \\psi _ m ) + ( \\psi _ \\ell - \\psi ) ^ * ( \\psi _ m - \\psi ) \\le \\| \\psi _ \\ell - \\psi \\| \\ , \\| \\psi _ \\ell - \\psi _ m \\| + ( \\psi _ \\ell - \\psi ) ^ * ( \\psi _ m - \\psi ) . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} y ^ T : = w ^ T \\begin{bmatrix} \\widehat { K } _ 2 ( \\lambda _ 0 ) ^ T \\\\ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} - d _ { j - 1 } + \\frac { d } { r } + t = \\frac { e } { 2 q r _ j } + \\sigma _ { j - 1 } \\left ( 1 - \\frac { r } { r _ j } \\right ) + \\sum _ { i = 1 , i \\neq j - 1 } ^ { 2 q } \\sigma _ i \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\mathfrak G ( a , \\mu ) : = \\int _ { \\Z _ p ^ { \\times } } \\psi ( a x ) \\mu ( x ) d x , \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} \\mathcal { A } ( \\mathbf { B } ) = \\{ ( \\alpha _ 1 , \\alpha _ 2 , t ) : - \\log \\omega < \\alpha _ 1 < \\alpha _ 2 , \\ t \\in [ 0 , T ( \\alpha _ 2 , \\alpha _ 1 ; \\mathbf { B } ) ) \\} . \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} \\begin{aligned} d X _ { t } & \\in - \\partial \\Psi ( X _ { t } ) \\ , d t + \\frac { 1 } { 2 } L ^ { b } X _ { t } \\ , d t + \\langle b \\nabla X _ { t } , d W _ { t } \\rangle , t \\in ( 0 , T ] , \\\\ X _ { 0 } & = x . \\end{aligned} \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} \\left ( h _ { d , j } ^ * \\right ) ^ { ( r ) } ( 1 ) = \\frac 1 2 \\left ( \\left ( h _ { d , j - 1 } ^ * \\right ) ^ { ( r ) } ( 1 ) - r \\ , \\left ( h _ { d , j } ^ * \\right ) ^ { ( r - 1 ) } ( 1 ) \\right ) , r = 1 , \\dots , j . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} R _ { 2 } ( r ) + \\frac { 2 m } { \\hbar ^ { 2 } } \\Big [ \\epsilon - \\frac { a } { r ^ { 2 } } + \\frac { b } { r } - c r ^ { \\rho } - \\frac { \\ell ( \\ell + 1 ) \\hbar ^ { 2 } } { 2 m r ^ { 2 } } \\Big ] R ( r ) = 0 , \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\widetilde T _ { C , d } : = \\begin{bmatrix} \\Delta & - 1 & - 1 / 2 ! & - 1 / 3 ! & \\dots & - 1 / d ! \\\\ & \\Delta & - 1 & - 1 / 2 ! & \\dots & - 1 / ( d - 1 ) ! \\\\ & & \\Delta & - 1 & & \\vdots \\\\ & & & \\ddots & \\ddots & \\vdots \\\\ & & & & \\Delta & - 1 \\\\ & & & & & \\Delta \\end{bmatrix} . \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\overline { \\bigcup _ { i \\geq n } \\Phi _ i ^ { - 1 } ( 0 ) } \\subseteq \\bigcup _ { j = 1 } ^ k \\overline { B _ { 1 0 m _ n ^ { - 1 / 2 } } ( x _ j ) } , \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} G ( \\rho ) X \\otimes f ( \\zeta ) G ( \\rho ) ^ { - 1 } = X \\otimes f ( \\rho ( \\zeta ) ) \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\mathcal I _ 1 ( n ) = \\int _ { \\mathcal A _ 1 ^ + ( n ) } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h + \\int _ { \\mathcal A _ 1 ^ - ( n ) } \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} 1 \\pm \\sqrt { - x ^ 2 / c ^ 2 } = 1 \\pm \\frac { x } { | c | } \\cos \\gamma \\mp i \\frac { x } { | c | } \\sin \\gamma . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} \\sigma ( t , \\omega , x ) = x + \\int _ 0 ^ t g ( s ) d W ( s ) \\quad \\textrm { a n d } b ( t , \\omega , x ) = 0 . \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} d ( X - Z ) = ( - \\eta - G ) \\ , d t + \\frac { 1 } { 2 } ( L ^ { b } X - P ^ { \\ast } L ^ { b } P Z ) \\ , d t + B ( X - P Z ) \\ , d W \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\nabla _ { U } \\psi = - \\varphi A _ { E } ^ { \\ast } U - \\tau ( U ) \\psi , \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} \\sigma ( \\beta \\alpha ) _ 2 ( i ) = ( \\beta \\alpha _ 2 ) \\sigma ^ { - 1 } ( i ) = \\beta _ 2 \\sigma ^ { - 1 } ( i ) = \\sigma ( \\beta ) _ 2 ( i ) = ( \\sigma ( \\beta ) \\tau ( \\alpha ) ) _ 2 ( i ) . \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow + \\infty } \\displaystyle \\int _ { \\Omega } A ( x ) ( | u _ { n } ( x ) | ^ { p ( x ) - 2 } u _ { n } ( x ) - | u _ { 0 } ( x ) | ^ { p ( x ) - 2 } u ( x ) ) ( u _ { n } ( x ) - u _ { 0 } ( x ) ) d x = 0 , \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} b _ { \\lambda } b _ { \\mu } & = \\psi _ { w _ { 0 } } y _ { \\lambda } \\psi _ { w _ { 0 } } y _ { \\min } \\psi _ { w _ { 0 } } y _ { \\mu } \\psi _ { w _ { 0 } } y _ { \\min } \\\\ & = ( - 1 ) ^ { \\tfrac { n ( n - 1 ) } { 2 } } \\psi _ { w _ { 0 } } ( y _ { \\lambda } \\psi _ { w _ { 0 } } y _ { \\mu } ) \\psi _ { w _ { 0 } } y _ { \\min } \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} & \\sigma _ 0 = \\{ ( x _ i ) \\in \\bold R ^ n \\mid x _ i > 0 ( 0 \\leq i \\leq n ) , \\sum x _ i < 1 \\} , \\\\ & \\tau _ 0 = \\{ ( x _ i ) \\in \\bold R ^ n \\mid x _ i > 0 ( 0 \\leq i \\leq n ) , \\sum x _ i = 1 \\} , \\\\ & \\tau _ i = \\{ ( x _ i ) \\in \\bold R ^ n \\mid x _ i > 0 ( 0 \\leq j \\leq n , j \\neq i ) , \\sum x _ i < 1 , x _ i = 0 \\} . \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} \\psi _ 3 ( x _ j ^ { a ^ s } ) = \\big ( b ^ { a ^ { j ( e _ 1 + \\dots + e _ { p - ( s - 1 ) } ) } } , \\dots , b , b , \\dots , b ^ { a ^ { j ( e _ 1 + \\dots + e _ { p - s } ) } } \\big ) , \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} E _ { k + 1 } = E _ { k } + \\Vert \\partial _ x ^ k \\rho _ 0 \\Vert _ { L ^ 2 } + \\Vert \\partial _ x ^ k u _ 0 \\Vert _ { L ^ 2 } . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\{ \\Z ( x ) , \\Z ( y ) \\} & = - 2 i ( x - y ) \\left ( ( I - x H _ u ^ 2 ) ^ { - 1 } ( I - y H _ u ^ 2 ) ^ { - 1 } ( u ) \\vert 1 \\right ) ^ 2 \\\\ & = - 2 i ( x - y ) \\left ( ( I - x H _ u ^ 2 ) ^ { - 1 } [ ( I - x H _ u ^ 2 ) + x H _ u ^ 2 ] ( I - y H _ u ^ 2 ) ^ { - 1 } ( u ) \\vert 1 \\right ) ^ 2 \\\\ & = - 2 i ( x - y ) \\left ( \\Z ( y ) + \\frac { x } { x - y } ( \\Z ( x ) - \\Z ( y ) ) \\right ) ^ 2 \\\\ & = - \\frac { 2 i } { x - y } ( x \\Z ( x ) - y \\Z ( y ) ) ^ 2 . \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} N _ { i , j } ^ k = \\sum _ \\ell \\frac { S _ { j , \\ell } S _ { i , \\ell } \\bar S _ { k , \\ell } } { S _ { 0 , \\ell } } \\ , . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} P ( Y ) = & Y ^ { 2 } [ \\beta _ { 0 } + \\beta _ { 1 } Y + \\ \\beta _ { 2 } Y ^ { 2 } + \\ \\beta _ { 3 } Y ^ { 3 } + \\ \\beta _ { 4 } Y ^ { 4 } ] \\\\ & + H [ \\alpha _ { 0 } + \\alpha _ { 1 } Y + \\alpha _ { 2 } Y ^ { 2 } ] = 0 \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\Delta ( l ) = \\sum _ { \\substack { 0 \\le l _ 1 \\le \\frac { l } { 2 } \\\\ 0 \\le l _ 2 \\le \\frac { m - 2 l } { 2 } } } \\Delta ( l , l _ 1 , l _ 2 ) < \\sum _ { \\substack { 0 \\le l _ 1 \\le \\frac { l } { 2 } \\\\ 0 \\le l _ 2 \\le \\frac { m - 2 l } { 2 } } } \\Lambda ( l _ 1 , l _ 2 , l , m ) q ^ { \\frac { m n } { 4 } + ( \\frac { 3 m } { 4 } - l ) s } < c q ^ { \\frac { m n } { 4 } + \\frac { 1 } { 2 } ( m ^ 2 - 2 l m + 2 l ^ 2 ) + ( \\frac { 3 m } { 4 } - l ) s } . \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} V \\tilde \\pi _ 1 ( S _ 2 ) V ^ * e _ { \\sigma _ 2 ^ k ( m _ { j '' } ) } & = V \\tilde \\pi _ 1 ( S _ 2 ) e _ { \\sigma _ 2 ^ k ( m _ { j ' } ) } = V e _ { \\sigma _ 2 ^ { k + 1 } ( m _ { j ' } ) } = e _ { \\sigma _ 2 ^ { k + 1 } ( m _ { j '' } ) } = \\tilde \\pi _ 2 ( S _ 2 ) e _ { \\sigma _ 2 ^ k ( m _ { j '' } ) } \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\kappa _ l = \\kappa - ( \\kappa - \\kappa ' ) l / n , l = 0 , 1 , \\dots , n . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} A ( x ) f ( x ) = \\lim _ { \\ell \\to \\infty } \\sum _ k \\alpha _ { \\ell k } f ( x + h _ { \\ell k } ) , \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} E \\big [ f ( \\Phi _ { 0 , t + s } ( y _ 0 ) ) | \\mathcal { F } _ s \\big ] = E \\big [ f ( \\Phi _ { 0 , t + s } ( y _ 0 ) ) | \\Phi _ { 0 , s } ( y _ 0 ) \\big ] , \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J \\setminus J _ { K ( n ) } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) = \\int _ { 0 } ^ { T } X ( t ) d t \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} B _ \\textnormal { s } ( m , r ) = \\sum _ { i } 2 ^ i \\binom { m } { i } \\binom { r } { i } . \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\triangle _ \\beta \\tilde { P } _ u = P _ f . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} \\begin{array} { c c c } - q A ^ { * } _ { 0 } = A _ { 2 } \\\\ A _ { 1 } ^ { * } = A _ { 1 } . \\end{array} \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} v _ p ( f ( x ) ) & = v _ p ( a _ l ) + l v _ p ( x - x _ j ) = v _ p ( a _ l ) + l v _ p ( ( x - i ) + ( i - x _ j ) ) \\\\ & = v _ p ( a _ l ) + l v _ p ( i - x _ j ) . \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} A _ r & = a _ { 2 r , 1 } + a _ { 2 r , - 1 } + a _ { 2 r - 1 } , & A ' _ s & = a ' _ { 2 s , 1 } + a ' _ { 2 s , - 1 } + a ' _ { 2 s - 1 } , \\\\ B _ r & = a _ { 2 r , 1 } + a _ { 2 r , - 1 } + a _ { 2 r + 1 } , & B ' _ s & = a ' _ { 2 s , 1 } + a ' _ { 2 s , - 1 } + a ' _ { 2 s + 1 } , \\\\ C _ r & = \\frac { \\alpha _ { - 1 } } { \\beta _ r } a _ { 2 r , 1 } - \\frac { \\alpha _ { 1 } } { \\beta _ r } a _ { 2 r , - 1 } , & C ' _ s & = q ^ { - s } ( a ' _ { 2 s , 1 } - a ' _ { 2 s , - 1 } ) , \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} z _ 1 & \\mapsto z _ 1 , \\\\ z _ 2 & \\mapsto r ^ 2 z _ 2 , \\\\ \\tilde { z } & \\mapsto r \\tilde { z } , \\ ; \\mbox { f o r } \\ ; \\tilde { z } = ( z _ 3 , \\ldots , z _ n ) , \\\\ z _ { n + 1 } & \\mapsto r ^ 2 z _ { n + 1 } . \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} f ( r ^ k ) & = r ^ k \\| \\vec x ^ { k + 1 } \\| _ { \\infty } - I ( \\vec x ^ { k + 1 } ) \\\\ & = r ^ k \\| \\vec x ^ { k + 1 } \\| _ { \\infty } - r ^ { k + 1 } \\| \\vec x ^ { k + 1 } \\| _ { \\infty } \\\\ & = \\| \\vec x ^ { k + 1 } \\| _ { \\infty } ( r ^ k - r ^ { k + 1 } ) \\to 0 k \\to + \\infty , \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} d f _ { p } ( v ) = g _ { \\dot \\gamma ( f ( p ) ) } ( { \\dot \\gamma ( f ( p ) ) } , v ) \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} { J ^ { \\varepsilon } } ( u ) = \\left \\{ x \\in p ^ { - 1 } ( \\mathbb { R } ) \\ , : \\ , f _ { u } ( x ) \\geq p ( x ) - \\varepsilon \\right \\} , \\ \\forall \\varepsilon \\geq 0 . \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} \\frac { d } { d t } g _ { t } ( z ) = \\frac { 2 } { g _ { t } ( z ) - B _ { t } } . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\sum _ { v \\in B } \\langle v | A v \\rangle = \\sum _ { v \\in B } \\sum _ { u \\in M } s ( u ) | \\langle v | u \\rangle | ^ 2 \\forall v \\in B \\ : . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} & \\lim _ { k \\to \\infty } \\| y ^ k - y ^ { k + 1 } \\| = \\lim _ { k \\to \\infty } \\| x ^ k - y ^ { k } \\| = 0 . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} \\mathcal { F } ^ { d e c } _ { r , D } ( C ) : = \\left \\{ \\theta : \\ \\forall \\varepsilon \\in ( 0 , 1 ) , \\ \\sum _ { k > D _ \\varepsilon } \\theta _ k ^ 2 \\leq C D _ \\varepsilon ^ { - 2 s } \\right \\} = \\left \\{ \\theta : \\sup _ { K \\in \\mathbb { N } ^ * } K ^ { 2 s } \\sum _ { k > K } \\theta _ k ^ 2 \\leq C \\right \\} \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} g ( x ) = \\begin{cases} x , & x < Q ^ { - 1 } ( - 1 ) \\ , \\ , x > 1 , \\\\ T _ 1 ( x ) & x \\in [ - 1 , 1 ] , \\\\ h T _ { - 1 } ^ { - 1 } h ^ { - 1 } ( x ) , & x \\in Q ^ { - 1 } ( [ - 1 , T _ { - 1 } ( 1 ) ] , \\\\ & \\\\ , & x \\in [ Q ^ { - 1 } ( T _ { - 1 } ( 1 ) ) , - 1 ] . \\end{cases} \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | M _ { n + 1 } f - M _ n f \\big | \\Big \\| _ { \\ell ^ p } \\leq C _ p N _ 1 ^ { - k } \\big ( \\vartheta _ B ( N _ 2 ) - \\vartheta _ B ( N _ 1 ) \\big ) \\| f \\| _ { \\ell ^ p } . \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} E _ 2 \\ , & = 1 + a _ 1 + a _ 1 b ^ 2 + b ^ 4 + a _ 1 ^ 3 k + a _ 1 ^ 4 k ^ 2 , \\\\ E _ 1 \\ , & = 1 + a _ 1 + a _ 1 b + a _ 1 b ^ 2 + a _ 1 b ^ 3 + b ^ 4 + a _ 1 ^ 3 k + a _ 1 ^ 3 b k + a _ 1 ^ 4 k ^ 2 , \\\\ E _ 0 \\ , & = k + a _ 1 k + a _ 1 ^ 3 b k + a _ 1 b ^ 2 k + b ^ 4 k + a _ 1 ^ 3 k ^ 2 + a _ 1 ^ 4 k ^ 3 , \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log S _ { d , k } ( n , w n ) = \\sigma _ { d , k } ( w ) . \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} \\Psi _ 1 ^ { \\min } ( N , q ) = \\delta _ { N = q } 2 ^ { \\omega ( N ) } . \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} V ~ = ~ h + G \\varphi \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} b \\circ \\beta _ { \\bar { b } \\circ x } ( \\rho _ { \\bar { b } } ( a ) ) = a \\cdot b \\cdot a ^ { - 1 } \\cdot \\beta _ x ( a ) . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} { \\left ( { \\mathop { \\sup } \\limits _ { u \\in U } { f _ { u } } } \\right ) ^ { \\ast } } ( { x ^ { \\ast } } ) = \\mathop { \\inf } \\limits _ { u \\in U } f _ { u } ^ { \\ast } ( { x ^ { \\ast } } ) , \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} A _ k \\ , x ^ { \\tilde { Y } } = 0 \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} x _ i = \\left \\{ \\begin{array} { l l l } 1 , & \\mbox { i f } \\lambda _ i + x _ { i + 1 } + x _ { i + 2 } + \\cdots \\not \\equiv 0 ( \\ ! \\ ! \\ ! \\ ! \\ ! \\mod l ) ; \\\\ 0 , & \\mbox { i f } \\lambda _ i + x _ { i + 1 } + x _ { i + 2 } + \\cdots \\equiv 0 ( \\ ! \\ ! \\ ! \\ ! \\ ! \\mod l ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} v _ t ( r , t ) = C \\zeta ' ( t ) G ^ { \\frac 1 { m - 1 } } - \\frac { C } { m - 1 } \\zeta ( t ) \\frac { \\eta ' ( t ) } { \\eta ( t ) } G ^ { \\frac 1 { m - 1 } - 1 } + \\frac { C } { m - 1 } \\zeta ( t ) \\frac { \\eta ' ( t ) } { \\eta ( t ) } G ^ { \\frac 1 { m - 1 } } \\ , ; \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} P = \\bigoplus _ { i \\in \\chi } V ^ { i } \\otimes M ^ { i } \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} v _ 1 \\cdot T ( e , v ) = v _ 1 \\cdot ( e , h ( e ) \\cdot 0 ^ { V _ 2 } + v ) = ( t _ { V _ 1 } ( v _ 1 ) , \\Psi ( v _ 1 ) \\cdot ( h ( e ) \\cdot 0 ^ { V _ 2 } + v ) ) . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} \\{ ( x , y ) \\mid z \\in R ^ n S ( g ) ( x ) = z z = S ( h ) ( y ) \\} . \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { * } \\leq \\Bigg ( o ( 1 ) + \\| h \\| _ { * * } + \\frac { \\sum _ { j = 1 } ^ { k } \\frac { 1 } { ( 1 + \\lambda | y - \\xi _ { j } | ) ^ { \\frac { N - 2 } 2 + \\tau + \\theta } } } { \\sum _ { j = 1 } ^ { k } \\frac { 1 } { ( 1 + \\lambda | y - \\xi _ { j } | ) ^ { \\frac { N - 2 } 2 + \\tau } } } \\Bigg ) . \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} P = \\begin{pmatrix} P _ 1 \\\\ P _ 3 & P _ 2 \\\\ P _ 4 & P _ 5 & P _ 1 \\end{pmatrix} , \\ R = \\begin{pmatrix} R _ 1 \\\\ R _ 3 & R _ 2 \\\\ R _ 4 & R _ 5 & R _ 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} \\mathcal { E } ( G ) = \\bigcup _ { i = 0 } ^ { m } \\mathcal { E } _ i ( G ) , \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} { } ^ \\sigma E _ { k - 2 m _ 0 , N } ^ \\chi ( Z , 0 ; h ) = E _ { k - 2 m _ 0 , N } ^ { { } ^ \\sigma \\ ! \\chi } \\left ( Z , 0 ; h _ \\tau \\right ) \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} n + m = r + 5 \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} \\chi ( x ) = 1 , \\ \\mbox { f o r } \\ | x | \\leq 1 \\ \\ \\mbox { a n d } \\ \\ \\chi ( x ) = 0 , \\ \\mbox { f o r } \\ | x | \\geq 2 \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} f ~ ~ v \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ' ( v ) = \\alpha _ v f ( v ) . \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} \\tau ( g ) = - \\int _ { D } \\eta \\wedge g ^ { \\ast } \\eta \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} & U _ { L } ( \\theta , P _ { B S } , p _ { s } , \\rho _ { e } ) = \\mu T _ { s } - E \\\\ & ~ ~ = \\mu ( 1 - \\theta ) \\bigg [ \\log \\bigg ( 1 + \\frac { p _ { s } | h _ { s } | ^ { 2 } } { \\sigma _ { s } ^ { 2 } } \\bigg ) - \\log ( 1 + \\rho _ { e } ) \\bigg ] \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - \\lambda \\theta P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } . \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} | ( f _ i \\pm \\varphi ) ( x ) - ( f _ i \\pm \\varphi ) ( y ) | = | f _ i ( x ) - f _ i ( y ) | \\leq d ( x , y ) , \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} g ' _ { 2 j } f _ 1 = j g _ { 2 j } = - f ' _ 1 g _ { 2 j } j \\geq 1 . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} \\Psi _ { \\Gamma } ( X ) = X + X ^ { 1 / 2 } \\sum _ { | t _ j | \\le T } \\frac { X ^ { i t _ j } } { 1 / 2 + i t _ j } + O \\left ( \\frac { X } { T } \\log ^ 2 X \\right ) , \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} { \\bf { C o v } } \\left [ V _ { \\varepsilon } ( u ) , \\ , V _ { \\varepsilon } ( v ) \\right ] = & \\begin{cases} - 2 \\ , \\log | u - v | , \\ , \\varepsilon \\ll | u - v | \\leq 1 , \\\\ 2 \\left ( \\kappa - \\log \\varepsilon \\right ) , \\ , u = v , \\end{cases} \\\\ & + O ( \\varepsilon ) , \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} U _ { B S } ( \\theta , \\lambda , P _ { B S } ) = \\theta ( \\lambda P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } - \\mathcal { F } ( P _ { B S } ) ) , \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} U _ { r } = \\frac { 1 } r V _ { r } + E _ r , \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} \\mathcal Q ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = C _ 1 ^ { - 2 } \\mathcal P ( \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) , \\mathbf Y _ M \\mathbf G ) , \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} \\omega _ p ( ( k , s _ p ( k ) ) , k ' ) \\phi _ { \\mathbf h , p } = \\phi _ { \\mathbf h , p } \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} S _ 1 & = \\sum _ { q } \\Psi \\left ( \\frac { q } { Q } \\right ) \\frac { q } { \\varphi ( q ) } \\ \\sideset { } { ^ + } \\sum _ { \\substack { \\chi ( q ) } } L \\left ( \\frac { 1 } { 2 } , \\chi \\right ) \\psi ( \\chi ) , \\\\ S _ 2 & = \\sum _ { q } \\Psi \\left ( \\frac { q } { Q } \\right ) \\frac { q } { \\varphi ( q ) } \\ \\sideset { } { ^ + } \\sum _ { \\substack { \\chi ( q ) } } \\left | L \\left ( \\frac { 1 } { 2 } , \\chi \\right ) \\psi ( \\chi ) \\right | ^ 2 . \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\begin{pmatrix} \\lambda _ { 1 1 } + \\lambda _ { 2 2 } & \\lambda _ { 2 3 } \\end{pmatrix} \\begin{pmatrix} A _ 1 & A _ 2 \\\\ A _ 5 & A _ 6 \\end{pmatrix} = 0 , \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} \\mathfrak { M } ( - n \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = \\prod _ { k = 0 } ^ { n - 1 } \\frac { \\Gamma \\bigl ( 2 + \\lambda _ 1 + \\lambda _ 2 + ( n + 2 + k ) / \\tau \\bigr ) \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) } { \\Gamma \\bigl ( 1 + \\lambda _ 1 + ( k + 1 ) / \\tau \\bigr ) \\Gamma \\bigl ( 1 + \\lambda _ 2 + ( k + 1 ) / \\tau \\bigr ) \\Gamma \\bigl ( 1 + k / \\tau \\bigr ) } . \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} \\frac { \\langle \\sqrt { f _ 1 } , - \\Delta \\sqrt { f _ 1 } \\rangle + \\langle \\sqrt { f _ 2 } , - \\Delta \\sqrt { f _ 2 } \\rangle } { 2 } = \\Big \\langle \\sqrt { \\frac { f _ 1 + f _ 2 } { 2 } } , - \\Delta \\sqrt { \\frac { f _ 1 + f _ 2 } { 2 } } \\Big \\rangle . \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} \\int _ { \\Gamma } ^ { } \\bar { y } _ \\Gamma d \\Gamma = 0 , \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } u _ { i 1 } ( x ) ^ 2 = \\sum _ { i = 1 } ^ { d } u _ { i 2 } ( x ) . \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} g _ \\beta = \\abs { z } ^ { 2 \\beta - 2 } d z ^ 2 + d \\xi ^ 2 , \\beta > 0 . \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } [ \\Pi , \\Pi ] = { \\rm d } \\Lambda \\ ; , \\ ; \\ ; \\ ; \\delta _ \\Pi ( \\Lambda ) = 0 \\ ; . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} \\lvert \\Phi \\rvert = m - \\frac { 1 } { \\emph { v o l } ( N ) } \\frac { \\| \\emph { d } _ A \\Phi \\| _ { L ^ 2 ( X ) } ^ 2 } { m \\rho } + o ( \\rho ^ { - 1 } ) . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\pi _ { \\rho } ^ { o , \\kappa , h } ( n ) = \\pi _ { \\rho } ^ { o , \\kappa , h } ( 0 ) \\prod _ { n ' = 1 } ^ { n } \\frac { h \\lambda _ { \\ell _ { \\kappa } } } { n ' \\mu _ { i _ { \\kappa } } } ; \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} \\det ( \\phi _ { n } ) & = ( - 1 ) \\det ( \\phi _ { n - 1 } ) + ( - 1 ) ^ { n + 1 } \\det ( X _ n ) \\\\ & = - \\det ( \\phi _ { n - 1 } ) + ( - 1 ) ^ { 2 n } \\\\ & = - \\det ( \\phi _ { n - 1 } ) + 1 , \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} \\sigma _ i = \\Omega _ i \\ ; \\ ; \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} \\begin{aligned} a _ n & : = 0 \\quad ; a _ { \\bar n } : = 2 ^ { - \\bar n } , \\\\ a _ { n + 1 } & : = \\min \\Big ( 2 a _ n , 2 ^ { - ( n - 1 ) } \\big ( k ( 2 ^ n ) - k ( 2 ^ { n - 1 } ) \\big ) \\Big ) \\quad n \\ge \\bar n . \\end{aligned} \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} \\mathrm { e v } _ { t = 1 } ( L _ { t } ( \\underline { m } ) ) = \\chi _ q ( L ( m ) ) . \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} [ t _ { s _ { h , 1 } } ^ { n } t _ { s _ { h , 2 } } ^ { 2 n } \\cdots t _ { s _ { h , k - 1 } } ^ { ( k - 1 ) n } , \\rho _ k ] ^ { - 1 } [ \\mathcal { X } _ 1 ^ \\prime , \\mathcal { Y } _ 1 ^ \\prime ] & = [ \\rho _ k , t _ { s _ { h , 1 } } ^ { n } t _ { s _ { h , 2 } } ^ { 2 n } \\cdots t _ { s _ { h , k - 1 } } ^ { ( k - 1 ) n } ] [ \\mathcal { X } _ 1 ^ \\prime , \\mathcal { Y } _ 1 ^ \\prime ] \\\\ & = [ \\rho _ k \\mathcal { X } _ 1 ^ \\prime , t _ { s _ { h , 1 } } ^ { n } t _ { s _ { h , 2 } } ^ { 2 n } \\cdots t _ { s _ { h , k - 1 } } ^ { ( k - 1 ) n } \\mathcal { Y } _ 1 ^ \\prime ] , \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} r - \\sum _ { j \\ne i } z _ j ^ * ( | v _ j | - | v _ i | ) & = r - \\sum _ { j = 1 } ^ { m _ 1 } ( | v _ j | - | v _ i | ) - \\sum _ { j = m _ 1 + 1 } ^ { m _ 0 } z _ j ^ * ( | v _ j | - | v _ i | ) \\\\ & = r - \\sum _ { j = 1 } ^ { m _ 1 } ( | v _ j | - | v _ { m _ 1 + 1 } | ) = r - A ( m _ 1 ) = 0 , \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X ^ { t , x ; u } ( s ) = & b ( s , X ^ { t , x ; u } ( s ) , Y ^ { t , x ; u } ( s ) , Z ^ { t , x ; u } ( s ) , u ( s ) ) d s \\\\ & + \\sigma ( s , X ^ { t , x ; u } ( s ) , Y ^ { t , x ; u } ( s ) , Z ^ { t , x ; u } ( s ) , u ( s ) ) d B ( s ) , \\\\ d Y ^ { t , x ; u } ( s ) = & - g ( s , X ^ { t , x ; u } ( s ) , Y ^ { t , x ; u } ( s ) , Z ^ { t , x ; u } ( s ) , u ( s ) ) d s + Z ^ { t , x ; u } ( s ) d B ( s ) , \\\\ X ^ { t , x ; u } ( t ) = & x , \\ Y ^ { t , x ; u } ( T ) = \\phi ( X ^ { t , x ; u } ( T ) ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} [ \\nabla ^ { k , g _ P } ( \\hat \\eta _ i ' ) ^ t ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( \\tilde { x } _ i , \\frac { i } { 4 } ) ) } \\leq C , \\\\ \\sum _ { j = 0 } ^ k \\lambda _ i ^ { - k + j - \\alpha } | ( \\nabla ^ { j , g _ P } ( \\hat \\eta _ i ' ) ^ t ) ( \\tilde { x } _ i ) | _ { g _ P ( \\tilde { x } _ i ) } \\leq C , \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} D _ { n , t } ^ 2 D _ { n + 2 } ^ 0 - D _ { n + 1 , t } ^ 2 D _ { n + 1 } ^ 0 = - 2 D _ { n + 2 } ^ { - 1 } ( 1 ; 2 ) D _ { n + 1 } ^ 1 , \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} g | _ { \\ell } [ \\gamma ] ( z ) = j ( \\gamma , z ) ^ { - \\ell } \\det ( \\gamma ) ^ { \\ell / 2 } g ( \\gamma z ) , \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} A \\in \\bigcup _ { \\substack { \\mu , \\nu \\in \\Lambda ( n , d ) \\\\ \\lambda = \\mu - \\nu } } A ^ { \\mu } _ { \\nu } . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\bar { R } ( X , Y ) Z & = f _ 1 \\big \\{ g ( Y , Z ) X - g ( X , Z ) Y \\big \\} + f _ 2 \\big \\{ g ( X , \\phi Z ) \\phi Y \\\\ & - g ( Y , \\phi Z ) \\phi X + 2 g ( X , \\phi Y ) \\phi Z \\big \\} + f _ 3 \\big \\{ \\eta ( X ) \\eta ( Z ) Y \\\\ & - \\eta ( Y ) \\eta ( Z ) X + g ( X , Z ) \\eta ( Y ) \\xi - g ( Y , Z ) \\eta ( X ) \\xi \\big \\} \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} \\big \\langle \\Lambda _ v P _ { v , \\rm R } F ( v ) , P _ { v , \\rm R } F ( v ) \\big \\rangle = 0 . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} F ( a ) = M ( a ) M ( a + 1 ) \\cdots M ( - 2 ) F ( - 1 ) \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} \\widetilde { J } : = I _ { \\mathrm { A } } \\times \\left ( [ 0 , - 2 ( 2 n - 3 ) ] \\cap \\frac { 1 } { 2 } \\mathbb { Z } \\right ) . \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} Y = g + \\varepsilon \\xi \\mathrm { o r } y _ k = \\vartheta _ k + \\varepsilon \\xi _ k \\forall k \\in \\mathbb { N } ^ { * } , \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} C _ a ^ * \\theta _ q = \\theta _ { R ^ t ( q ) } . \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} \\mathbf { A } \\mathbf { f } = \\mathbf { b } , \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{gather*} L \\big ( \\chi ^ k , k / 2 \\big ) = L ( G _ k ( q ) , k ) \\cdot \\begin{cases} 1 & k \\equiv \\pm 1 \\mod 6 , \\\\ \\dfrac 1 { 1 - ( - 3 ) ^ { k / 2 } 3 ^ { - k } } & k \\equiv \\pm 2 \\mod 6 , \\\\ \\dfrac 1 { 1 + 2 ( - 2 ) ^ k 4 ^ { - k } } & k \\equiv 3 \\mod 6 , \\\\ \\dfrac 1 { ( 1 - ( - 3 ) ^ { k / 2 } 3 ^ { - k } ) ( 1 + 2 ( - 2 ) ^ k 4 ^ { - k } ) } & k \\equiv 0 \\mod 6 . \\end{cases} \\end{gather*}"}
-{"id": "9517.png", "formula": "\\begin{align*} H ( z ) = \\frac { G ( z ) ( e ^ { - i a z / 2 } - 1 ) } { z A ( z ) } - \\sum _ n \\frac { G ( t _ n ) ( e ^ { - i a t _ n / 2 } - 1 ) } { t _ n A ' ( t _ n ) ( z - t _ n ) } . \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} \\theta _ { i j } & \\ , ( 1 \\leq i \\leq r _ 1 , \\ , 1 \\leq j \\leq r _ 2 ) , & \\sum _ { i = 1 } ^ { r _ 1 } u _ { i j } & \\ , ( 1 \\leq j \\leq r _ 2 ) , & \\sum _ { j = 1 } ^ { r _ 2 } u _ { i j } & \\ , ( 1 \\leq i \\leq r _ 1 ) . \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ R \\times \\left \\{ t \\right \\} } \\psi ^ 2 \\left ( u \\right ) \\xi ^ 2 \\ , d x \\leq \\left [ s _ 2 ^ 2 \\left ( \\log 2 \\right ) ^ 2 \\left ( \\frac { 1 - \\rho _ 0 } { 1 - \\frac { \\rho _ 0 } { 2 } } \\right ) + C \\left ( \\frac { s _ 2 \\log 2 } { \\nu ^ 2 } \\left ( \\frac { \\Lambda } { \\theta _ 0 ^ { \\ , \\beta } } \\right ) ^ { m - 1 } + \\left ( \\Lambda ^ { m + 1 } + \\Lambda ^ { 2 ( b - 1 ) } \\right ) 4 ^ { s _ 2 + 1 } R ^ { 2 - 2 \\epsilon } s _ 2 \\log 2 \\right ) \\right ] \\left | B _ R \\right | \\end{aligned} \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} t _ { t _ { c _ 1 } ( z _ 1 ) } ^ { - 1 } t _ { a _ 1 } t _ { z _ 1 } ^ { - 1 } t _ { a _ 1 } & = [ t _ { t _ { c _ 1 } ( z _ 1 ) } ^ { - 1 } t _ { a _ 1 } , \\phi _ 1 ] , \\\\ t _ { t _ { c _ 1 } ^ { - 2 m - 1 } ( z _ 1 ) } ^ { - 1 } t _ { a _ 1 } ^ { 2 m + 1 } t _ { c _ 1 } ^ { - 2 m - 1 } t _ { a _ 1 } & = [ t _ { t _ { c _ 1 } ^ { - 2 m - 1 } ( z _ 1 ) } ^ { - 1 } t _ { a _ 1 } ^ { 2 m + 1 } , \\phi _ 2 ] , \\\\ t _ { c _ 1 } ^ { - 2 m } t _ { a _ 1 } ^ { 2 m } & = [ t _ { c _ 1 } ^ { - 2 m } , \\phi _ 3 ] . \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} F ( Z ) = j ( g , I ) ^ k \\ , \\Phi ( g ) , \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( - \\Delta ) ^ { s / 2 } u & = f ~ ~ \\mbox { i n } ~ ~ \\Omega ; \\\\ u & = g ~ ~ \\mbox { i n } ~ ~ \\Omega ^ c , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\Phi ( u _ { * } , v _ { * } , 0 ) = \\frac { 1 } { ( u _ { * } ^ 2 + 4 ) ( v _ { * } ^ 2 + 4 ) } \\le \\frac { 1 } { 1 6 } < \\frac { 1 } { 1 2 } . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} A _ { 1 1 } ( N ) \\begin{bmatrix} B _ { b } \\\\ B _ { b - 1 } \\\\ \\vdots \\\\ B _ k \\end{bmatrix} = \\begin{bmatrix} Q _ { n + b } \\\\ Q _ { n + b - 1 } \\\\ \\vdots \\\\ Q _ { n + k } \\end{bmatrix} , \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} \\beta \\boxtimes \\alpha ( X ) = W ^ * _ { 1 2 } { U _ \\beta } _ { 1 3 } ( 1 \\otimes X ) { U _ \\beta } ^ * _ { 1 3 } W _ { 1 2 } , \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} N _ { w , G } = \\sum _ { \\chi \\in I r r ( G ) } N _ { w , G } ^ { \\chi } \\cdot \\chi \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\langle P ( x ) , Q ( y ) \\rangle , \\quad \\forall x , y \\in X , \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\alpha ( g ) . \\alpha _ A ( a ) = \\alpha _ A ( g . a ) \\quad \\mbox { f o r a l l $ a \\in A $ a n d $ g \\in G $ . } \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} \\widetilde \\nabla = d + p _ { - 1 } d x + \\sum _ { i = 1 } ^ \\ell \\widehat u _ i ( x ) \\alpha _ i d x . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} \\sum _ n \\frac { \\overline { \\chi } ( n ) } { n ^ { \\frac { 1 } { 2 } } } V \\left ( \\frac { T n } { q } \\right ) & = L \\left ( \\frac { 1 } { 2 } , \\overline { \\chi } \\right ) - \\epsilon ( \\overline { \\chi } ) \\sum _ n \\frac { \\chi ( n ) } { n ^ { \\frac { 1 } { 2 } } } F \\left ( \\frac { n } { T } \\right ) , \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\| f \\| _ \\infty & = \\| g \\| _ \\infty \\ge \\| g ( y _ 0 ) \\| = \\| V ( f ( \\phi ( y _ 0 ) ) ) \\| \\\\ & = \\| f ( \\phi ( y _ 0 ) ) \\| = \\| f \\| _ \\infty . \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\tilde { \\Gamma } ^ a ( \\delta ) ( x ) : = \\sup _ { \\{ t , s \\in [ 0 , 1 ] : | t - s | \\le \\delta , s \\sim _ x t \\} } | x _ t - x _ s | , \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} V _ 0 : = ( \\log q ) ^ { C } , \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} g _ 1 . \\omega ( g _ 2 , g _ 3 ) - \\omega ( g _ 1 g _ 2 , g _ 3 ) + \\omega ( g _ 1 , g _ 2 g _ 3 ) - \\omega ( g _ 1 , g _ 2 ) = 0 \\ ; \\mbox { f o r a l l $ g _ 1 , g _ 2 , g _ 3 \\in G $ } \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} \\begin{pmatrix} E P ^ T + P E ^ T \\end{pmatrix} _ { 2 i , 2 j } = 0 \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\vert x + \\tilde z + u _ x ( \\tilde z ) - y \\vert & = \\vert \\tilde z + u _ x ( \\tilde z ) + z ( y ) + u _ x ( z ( y ) ) \\vert \\\\ & = \\sqrt { \\vert \\tilde z - z ( y ) \\vert ^ 2 + \\vert u _ x ( \\tilde z ) - u _ x ( z ( y ) ) \\vert ^ 2 } \\\\ & \\leq \\sqrt { 1 + \\alpha ^ 2 } \\cdot \\vert \\tilde z - z ( y ) \\vert \\\\ & < r . \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} \\dfrac { \\log ^ k ( z + 1 ) } { k ! } = \\sum _ { n = k } ^ \\infty s ( n , k ) \\dfrac { z ^ n } { n ! } , \\dfrac { ( e ^ z - 1 ) ^ k } { k ! } = \\sum _ { n = k } ^ \\infty S ( n , k ) \\dfrac { z ^ n } { n ! } , | z | < 1 . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} \\begin{cases} \\underbar r _ 1 ^ { k + 1 } - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 1 ^ { k + 1 } + \\epsilon \\cr \\underbar r _ 2 ^ { k + 1 } - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 2 ^ { k + 1 } + \\epsilon \\end{cases} \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} \\begin{array} { c c c } q A _ { 2 } A _ { 0 } - q ^ { - 1 } A _ { 0 } A _ { 2 } = A _ { 1 } \\\\ q ^ { 2 } A _ { 1 } A _ { 0 } - q ^ { - 2 } A _ { 0 } A _ { 1 } = ( 1 + q ^ { 2 } ) A _ { 0 } \\end{array} \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\mathcal { L } ^ { ( \\theta , \\beta ) } E = F , \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} ( V _ 1 ) = ( U _ 1 ^ * ) ^ { r _ 2 } L ^ { r _ 2 } \\left ( P ^ * U _ 1 ^ * U _ 3 ^ * ( U _ 4 ) ^ * U _ 5 ^ * \\ldots U _ { 2 q + 1 } ^ * \\right ) . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} \\widetilde { m } \\leq \\widetilde { m } ' \\ \\ \\mathrm { e v } _ { t = 1 } ( \\widetilde { m } ) \\leq \\mathrm { e v } _ { t = 1 } ( \\widetilde { m } ' ) . \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\theta _ i ' = \\omega _ i + \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\sum _ { p = 1 } ^ \\infty a _ p \\left ( \\sin ( p ( \\theta _ j - \\theta _ i - \\alpha ) ) - \\sin ( p \\alpha ) \\right ) . \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} \\langle a ( t ) \\tilde { y } , \\tilde { y } \\rangle = \\langle q ( t ) ^ \\top \\Lambda ( t ) q ( t ) \\tilde { y } , \\tilde { y } \\rangle = \\langle \\Lambda ( t ) q ( t ) \\tilde { y } , q ( t ) \\tilde { y } \\rangle \\leq L | q ( t ) \\tilde { y } | ^ 2 = L | \\tilde { y } | ^ 2 . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { - \\infty } ^ { \\infty } x u ^ 2 ( t , x ) d x = - 6 \\ , I _ 3 ( u _ 0 ) . \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} H _ n = \\left ( ( 1 + i + j ) ^ { N - 1 } \\mathfrak { d } _ 1 ^ N \\dot { \\varphi } _ n ( i + j ) \\right ) _ { i , j \\in \\N } , \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} d X _ { t } + A ( X _ { t } ) \\ , d t \\ni B _ { t } ( X _ { t } ) \\ , d W _ { t } , X _ { 0 } = x . \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\| a _ n \\| \\left | \\ , \\frac { ( z + h ) ^ n - z ^ n } { h } - n z ^ { n - 1 } \\ , \\right | \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} A _ 1 ( n ) = \\begin{cases} p ^ { - 3 n / 2 } ( - 1 ) ^ n \\chi _ { \\psi } ( p ^ n ) ( 1 - p ^ { - 1 } ) ( 1 + p - p ^ n ) & n > 0 , \\\\ p ^ { 3 n / 2 } ( - 1 ) ^ n \\chi _ { \\psi } ( p ^ n ) ( 1 - p ^ { - 1 } ) p ^ { 1 - n } & n \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u = ( \\beta + 1 ) / 3 + \\sqrt { k } U \\ , , \\\\ v = ( \\beta + 1 ) / 3 \\gamma + \\sqrt { k } V \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} c _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x ( a ) + c _ 2 ~ ^ { A B R } D _ b ^ \\alpha x ( a ) = 0 , \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} \\mathcal { B } ( \\mathcal { C } ) = \\big \\{ \\frac { 1 } { 2 } ( A + A ^ \\top ) : A = ( \\tilde { A } | \\cdots | \\tilde { A } | O ) \\in \\R ^ { k \\times k } , \\tilde { A } \\in \\mathcal { C } \\big \\} \\ , , \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} A ^ { ( 1 ) } ( x _ 1 ) \\times \\cdots \\times A ^ { ( m ) } ( x _ m ) = ( A ^ { ( 1 ) } \\times \\cdots \\times A ^ { ( m ) } ) ( x ) . \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} \\phi _ { \\mathbf h , p } ( z ) = \\pmb { \\phi } _ p ( x _ 3 ) \\phi _ { \\breve { \\mathbf g } , p } ( X ) = p ^ { - 1 / 2 } \\varepsilon ( 1 / 2 , \\underline { \\chi } _ p ) \\mathbf 1 _ { p ^ { - 1 } \\Z _ p ^ { \\times } } ( x _ 1 ) \\mathbf 1 _ { p \\Z _ p } ( x _ 2 ) \\mathbf 1 _ { \\Z _ p } ( x _ 3 ) \\mathbf 1 _ { \\Z _ p } ( x _ 4 ) \\mathbf 1 _ { p ^ 2 \\Z _ p } ( x _ 5 ) \\underline { \\chi } _ p ( x _ 1 ) . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\| \\phi \\| _ { c b } = | c _ + | + | c _ - | + \\begin{cases} \\left ( 1 - \\frac { 1 } { d - 1 } \\right ) \\left \\| \\left ( 1 - \\frac { 1 } { d - 1 } \\tau \\right ) ^ { - 1 } H \\right \\| _ { S _ 1 } , & 3 \\leq d < \\infty , \\\\ \\| H \\| _ { S _ 1 } , & d = \\infty , \\end{cases} \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} T \\leq \\frac { 1 } { 4 \\| p \\| _ \\infty } C : = 2 \\| n _ 0 \\| _ { \\mathrm { T V } } , \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} X _ { \\mu \\nu } = Y _ { \\mu \\nu } . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} \\sum _ { } \\binom { k - a } { k - ( a + m ) } - \\sum _ { } \\binom { k - a } { k - ( a + m ) } = 0 \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} A ' _ r & = \\sum _ s F ^ { ( m + 1 ) } _ r ( s ) A _ s , \\\\ C ' _ r & = \\sum _ s F ^ { ( m ) } _ r ( s ) B _ s , \\\\ B ' _ r & = q ^ m \\sum _ s F ^ { ( m ) } _ r ( s ) C _ s . \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} \\lVert h _ m ( y ) - y \\rVert = \\lVert x _ m - y _ m \\rVert \\le \\epsilon , \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} \\left | L \\left ( \\frac { 1 } { 2 } , \\chi \\right ) \\right | ^ 2 & = 2 \\mathop { \\sum \\sum } _ { m , n } \\frac { \\chi ( m ) \\overline { \\chi } ( n ) } { ( m n ) ^ { \\frac { 1 } { 2 } } } V \\left ( \\frac { m n } { q } \\right ) , \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} M = O _ 1 \\left [ \\begin{array} { c c } \\Gamma & O \\\\ O & \\Gamma ^ { - 1 } \\end{array} \\right ] \\ , O _ 2 ^ T , \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} \\| \\omega \\| _ { L ^ { \\theta } ( \\Omega ) } = \\| \\lambda \\| _ { L ^ { \\theta } ( \\Omega ) } = \\left ( \\int _ { \\Omega } \\lambda ^ { \\theta } d x \\right ) ^ { 1 / \\theta } \\geq \\frac { C } { r _ { \\Omega , q } ^ { s p - Q / \\theta } } . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} ( \\tau _ { i , j } \\Pi ) _ { k , l } : = \\begin{cases} \\Pi _ { k , l } & , \\\\ \\displaystyle \\frac { ( \\Pi _ { i , j - 1 } + \\Pi _ { i - 1 , j } ) \\Pi _ { i + 1 , j } \\Pi _ { i , j + 1 } } { \\Pi _ { i , j } ( \\Pi _ { i + 1 , j } + \\Pi _ { i , j + 1 } ) } & . \\end{cases} \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\eta _ u = \\inf \\{ t \\geq 1 : ~ X ( t ) \\not \\in E \\star u \\} . \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} \\delta ( A , B ) = \\left ( \\sum _ { i = 1 } ^ { n } \\ , \\log ^ 2 \\ , \\lambda _ i \\ , ( A ^ { - 1 } B ) \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) ^ { \\mathrm { a l g } } = 2 ^ { 2 k + \\nu ( N / M ) } \\tilde { C } ( N , M , \\chi ) \\cdot \\frac { | D | ^ { k - 1 / 2 } \\Lambda ( f , D , k ) } { c ^ + ( f ) | c _ h ( | D | ) | ^ 2 } \\cdot \\frac { | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 } { \\langle g , g \\rangle ^ 4 } , \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\| f _ \\phi \\| \\leq \\| M _ \\phi \\| = \\| \\phi \\| _ { c b } , \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} a ( \\mathbf { u } _ h , \\mathbf { v } _ h ) = \\sum _ { T \\in \\mathcal { T } _ h } \\int _ T 2 \\mu \\epsilon ( \\mathbf { u } _ h ) : \\epsilon ( \\mathbf { v } _ h ) + \\lambda \\textrm { d i v } ( \\mathbf { u } _ h ) \\textrm { d i v } ( \\mathbf { v } _ h ) d X , ~ ~ \\textrm { a n d } ~ ~ L ( \\mathbf { v } _ h ) = \\sum _ { T \\in \\mathcal { T } _ h } \\int _ { T } \\mathbf { f } \\cdot \\mathbf { v } _ h d X . \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} { \\rm g w } _ { t _ 0 } ^ { - 1 } ( A N _ + ) \\cap { \\rm S y m } ^ { \\delta } ( n ) = A \\cup B , \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s _ { 1 } } u + V _ { 1 } ( x ) u = f _ { 1 } ( u ) , x \\in \\mathbb { R } ^ { N } . \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\bar D = \\frac { \\partial } { \\partial x _ 0 } - e _ 1 \\frac { \\partial } { \\partial x _ 1 } - e _ 2 \\frac { \\partial } { \\partial x _ 2 } . \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} \\breve { J } W = \\breve { J } T W + \\breve { J } Q W . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} b _ { 1 } b _ { 3 } b _ { 4 } + b _ { 2 } b _ { 4 } ( b _ { 3 } u _ { 2 } + b _ { 4 } u _ { 3 } ) & = 0 \\\\ b _ { 2 } b _ { 3 } ^ { 2 } + b _ { 2 } b _ { 4 } ( b _ { 3 } u _ { 2 } + b _ { 4 } u _ { 3 } ) & = b _ { 2 } ( b _ { 1 } b _ { 4 } - b _ { 2 } b _ { 3 } ) u _ { 3 } , \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\varphi ' _ { \\epsilon } : = \\psi _ { \\epsilon } \\ , , \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} A = \\begin{pmatrix} x & y & z \\\\ v & w & x \\end{pmatrix} \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} \\beta _ x ( a \\cdot b ) = a \\cdot \\beta _ x ( b ) \\cdot a ^ { - 1 } \\cdot \\beta _ x ( a ) . \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\sigma ( Q _ { 1 } ) = k \\cdot { Q _ { 1 } } = Q _ { 1 } + h _ { 2 , \\sigma } \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} d ' < \\frac { d } { n - 1 } \\sum _ { i = 1 } ^ { n - 1 } \\frac { r ' _ { i + 1 } } { r _ { i + 1 } } + \\sum _ { i = 1 } ^ { n - 1 } \\sigma _ i \\left ( r ' - r \\frac { r ' _ { i + 1 } } { r _ { i + 1 } } \\right ) . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle ( z _ i ^ * ) ^ { p - 1 } \\le \\frac { m _ 0 + \\gamma } { \\alpha + \\beta } | v _ i | , \\ , \\ , \\ , \\ , i = 1 , 2 \\ldots , m _ 0 , \\\\ \\displaystyle ( z _ i ^ * ) ^ { p - 1 } = \\frac { m _ 0 + \\gamma } { \\alpha + \\beta } | v _ i | , \\ , \\ , \\ , \\ , i = m _ 0 + 1 , \\ldots , n . \\end{cases} \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\langle \\ ! \\langle G , Q _ { \\alpha _ 2 \\alpha } ( t ) k \\rangle \\ ! \\rangle = \\langle \\ ! \\langle H _ { \\alpha \\alpha _ 2 } ( t ) G , k \\rangle \\ ! \\rangle . \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} ( I - \\Delta ) \\psi _ i ( r ) & = \\psi _ i ( r ) - \\psi _ i '' ( r ) - \\tfrac { n - 1 } { r } \\psi _ i ' ( r ) \\\\ & = \\tfrac { 2 i } { r } \\psi ' _ i ( r ) - \\tfrac { n - 1 } { r } \\psi _ i ' ( r ) \\\\ & = ( n - 1 - 2 i ) \\psi _ { i + 1 } ( r ) \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} u _ 1 = \\pm 0 . 5 6 4 2 u _ 2 = \\pm \\sqrt { 1 - u _ 1 ^ 2 } = \\pm 0 . 8 2 5 6 , \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} N < ( 2 - 2 + 2 ) ( n - 2 + 2 ) = 2 n . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mu _ t ( F ^ \\theta ) = ( L ^ * \\mu ^ { \\Lambda _ \\theta } _ t ) ( F ^ \\theta ) . \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\sigma - \\bigtriangleup u = 0 . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} | a | \\tau _ { | a | } [ | D | c o t h ( | D | ) \\tau _ { \\frac { 1 } { | a | } } \\nu ] = | D | c o t h \\Big ( \\frac { 1 } { | a | } | D | \\Big ) \\nu \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\cap } ^ { ( N ) } = \\left \\{ A : A = A _ { j _ { 1 } } ^ { ( 1 ) } \\cap A _ { j _ { 2 } } ^ { ( 2 ) } \\cap . . . \\cap A _ { j _ { N } } ^ { ( N ) } , j _ { k } \\leqslant m _ { k } , k \\leqslant N \\right \\} . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { k } d _ j ( A ^ t ) \\ge \\prod _ { j = 1 } ^ { k } d _ j ^ t ( A ) . \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} \\mathcal { D } & = [ 0 , 1 ] , \\ ; \\varphi ( x ) = x ^ { \\lambda _ 1 } ( 1 - x ) ^ { \\lambda _ 2 } \\ ; ( i n t e r v a l ) , \\\\ \\mathcal { D } & = [ - \\frac { 1 } { 2 } , \\ , \\frac { 1 } { 2 } ] , \\ ; \\varphi ( x ) = | 1 + e ^ { 2 \\pi i x } | ^ { 2 \\lambda } \\ ; ( c i r c l e ) , \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} M _ { 1 , 2 } & = O ( q ^ { 1 - \\epsilon } ) + \\frac { 2 P _ 3 ( 1 ) } { L } \\varphi ^ + ( q ) \\sum _ { \\substack { b \\leq y _ 2 \\\\ ( b , q ) = 1 } } \\frac { \\Lambda ( b ) P _ 2 [ b ] } { b } . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} h ( a ) \\circeq - \\log g ( a ) = - \\log \\Gamma ( a M + 1 ) + \\sum _ { i = 1 } ^ { d + 1 } \\log \\Gamma ( a \\gamma _ i + 1 ) - a \\sum _ { i = 1 } ^ { d + 1 } \\gamma _ i \\log x _ i . \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} M ( x ) = \\sum _ { n \\leq x } \\frac { \\mu ( n ) } { n } \\Delta \\left ( \\frac { x } { n } \\right ) \\log \\frac { x } { e } + O \\left ( ( \\log x ) ^ 2 \\right ) , \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} \\omega \\wedge \\omega = 0 . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} \\xi ^ { h } _ { L + \\theta , t } ( n ) = \\begin{cases} \\frac { 1 } { t ^ { h } _ { \\theta , s ^ * } ( n ) - \\tau ( t ^ { h } _ { \\theta , s ^ * } ( n ) ) } , & \\tau ( t ^ { h } _ { \\theta , s ^ * } ( n ) ) \\leq t < t ^ { h } _ { \\theta , s ^ * } ( n ) t ^ { h } _ { \\theta , s ^ * } ( n ) = \\min \\limits _ { \\begin{subarray} ~ k '' \\in \\mathbb { N } _ + \\end{subarray} } \\{ t ^ { h } _ { \\theta , k '' } ( n ) | t ^ { h } _ { \\theta , k '' } ( n ) > t \\} , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\begin{array} { l l } & \\sum _ { j + k = s } F _ j ( F _ k ( X ) , \\alpha ( y _ 1 ) , \\cdots , \\alpha ( y _ { n - 1 } ) ) \\\\ & = \\sum _ { i = 1 } ^ n \\sum _ { j + k = s } F _ j ( \\alpha ( x _ 1 ) , \\ldots , \\alpha ( x _ { i - 1 } ) , F _ k ( x _ i , y _ 1 , \\ldots , y _ { n - 1 } ) , \\alpha ( x _ { i + 1 } ) , \\ldots , \\alpha ( x _ { n } ) ) ) \\end{array} \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} s _ v \\left ( \\left ( \\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix} \\right ) \\right ) = \\begin{cases} ( c , d ) _ v & c d \\neq 0 , \\ , \\mathrm { o r d } _ v ( c ) , \\\\ 1 & , \\end{cases} \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} ( \\Sigma ' _ q ) & = \\left ( r - q - r _ 2 - r _ 4 - \\ldots - r _ { 2 q - 1 } \\right ) \\frac { ( - r ) ^ { 2 q - 1 } } { r _ 2 r _ 4 \\ldots r _ { 2 q - 1 } } . \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\Delta _ T ( s , \\alpha , f ) = \\int _ 0 ^ T f ( \\{ s _ 1 + t \\alpha _ 1 \\} , \\dots , \\{ s _ d + t \\alpha _ d \\} ) \\ , \\mathrm { d } t - T \\int _ { [ 0 , 1 ] ^ d } f ( x ) \\ , \\mathrm { d } x ( T > 0 ) . \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} A ( X ) = \\ , & ( 1 + a + b ) X ^ 3 + ( a + z + a z + b z ) X ^ 2 + ( 1 + a + k + a k + b k + z + a z + b z ) X \\\\ & + a + k + b k + a z + b z + k z + a k z + b k z , \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} { } \\rho _ { n } \\varphi _ { n + 1 } ^ { * } ( z ) = \\varphi _ { n } ^ { * } ( z ) - \\alpha _ { n } z \\varphi _ { n } ( z ) . \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\int _ { 0 } ^ T \\langle \\ddot { u } _ 3 ^ N ( t ) , \\varphi _ 3 \\rangle \\psi ( t ) d t + \\int _ { 0 } ^ T \\langle \\sigma ^ N ( t ) , E \\varphi \\rangle \\psi ( t ) d t = \\int _ 0 ^ T \\langle \\mathcal L ( t ) , \\varphi ( t ) \\rangle \\psi ( t ) d t , \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} \\frac { ( q ; q ) _ { \\infty } } { ( - q ; q ) _ { \\infty } } & = \\sum _ { k = - \\infty } ^ { \\infty } ( - 1 ) ^ { k } q ^ { k ^ 2 } , \\\\ \\frac { ( q ^ 2 ; q ^ 2 ) _ { \\infty } } { ( - q ; q ^ 2 ) _ { \\infty } } & = \\sum _ { k = - \\infty } ^ { \\infty } ( - 1 ) ^ { k } q ^ { 2 k ^ { 2 } - k } . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} B _ i ( x , y ) = \\{ w \\in A ( x , k - i ) \\ : \\ y \\in A ( w , i ) \\} . \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\vec z _ { k + d } = - \\vec z _ { k } + \\sum _ { i = k + 1 } ^ { k + d - 1 } \\mu _ i \\vec z _ i . \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { ( r - 1 ) / 2 } ( x _ { 2 i - 1 } y _ { 2 i } + x _ { 2 i } y _ { 2 i - 1 } ) + x _ r y _ r . \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} \\Delta ( l ) = \\sum _ { \\substack { 0 \\le l _ 1 \\le \\frac { l } { 2 } \\\\ 0 \\le l _ 2 \\le \\frac { m - 2 l } { 2 } } } \\Delta ( l , l _ 1 , l _ 2 ) . \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} f ( x , y ) + f ( y , z ) - f ( x , z ) & = \\langle A x , y - x \\rangle + \\langle A y , z - y \\rangle - \\langle A x , z - x \\rangle \\\\ & = - \\langle A y - A x , y - z \\rangle \\\\ & \\geq - \\| A x - A y \\| \\big \\| y - z \\big \\| \\\\ & \\geq - L \\| x - y \\| \\big \\| y - z \\big \\| \\\\ & \\geq - \\dfrac { L } { 2 } \\| x - y \\| ^ 2 - \\dfrac { L } { 2 } \\| y - z \\| ^ 2 \\\\ & = - c _ 1 \\| x - y \\| ^ 2 - c _ 2 \\| y - z \\| ^ 2 . \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} Q ^ { ( m ) } _ { 0 , 1 } ( i ) & = 1 \\intertext { f o r e a c h $ i \\in I $ w i t h $ \\Q _ i \\ne \\emptyset $ , } Q ^ { ( m ) } _ { 2 r } ( 0 , 1 ) & = \\mu ^ { ( m ) } _ { 2 r , 1 } + \\mu ^ { ( m ) } _ { 2 r , - 1 } , \\\\ Q ^ { ( m ) } _ { 2 r + 1 } ( 0 , 1 ) & = \\mu ^ { ( m ) } _ { 2 r + 1 } \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} K ( T ) & = \\bigcup \\{ \\mathcal { L } : \\mathcal { L } \\} \\\\ & = \\bigcup \\{ \\mathcal { R } : \\mathcal { R } \\} . \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} & \\norm { \\hat x _ 0 - \\hat x _ * } \\leq \\norm { x _ 0 - x _ * } , \\\\ & \\hat x _ i \\in \\hat x _ 0 + \\mathrm { s p a n } \\{ g _ 0 , \\hdots , g _ { i - 1 } \\} , { i = 0 , \\hdots , N } . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} V _ g { ( \\pi ( w ) f ) } ( z ) = e ^ { - 2 \\pi i ( z _ 2 - w _ 2 ) \\cdot w _ 1 } V _ { g } { f } ( z - w ) \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} B _ { n , k } = \\frac { 1 } { n + 1 } \\binom { 2 n + 2 } { n - k } \\binom { n + k } { n } . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} & \\left | V ^ { \\star } _ { k _ \\ell } ( x _ \\ell ) + V ^ { \\star } _ { k _ \\ell } ( - x _ \\ell ) - 2 \\ , V ^ { \\star } _ { k _ \\ell } ( 0 ) \\right | \\leq \\max _ { | x | \\leq x _ \\ell } \\left | V '' _ { k _ { \\ell } } ( x ) \\right | \\cdot x ^ 2 _ { \\ell } \\leq \\\\ & 2 C ' _ 4 \\frac { \\omega ( \\delta _ { k _ \\ell } ) } { \\lambda { k _ \\ell } ) \\delta ^ { 2 } _ { k _ \\ell } } \\cdot A ^ 2 _ { \\ell } \\delta ^ 2 _ { k _ \\ell } = C ' _ 4 \\omega ( \\delta _ { k _ \\ell } ) . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} c a b & \\equiv c b a \\\\ a b c & \\equiv a c b . \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} x _ 1 = q ( y _ 1 - y _ 2 ) \\ ; , \\ ; \\ ; x _ 2 = - \\frac { y _ 2 } { q ( y _ 2 - y _ 1 ) } \\ ; , \\ ; \\ ; x _ 3 = y _ 4 \\ ; , \\ ; \\ ; x _ 4 = - y _ 3 , \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\Theta = \\left ( \\begin{matrix} 1 & 0 & 0 & 0 \\\\ 0 & \\frac { 1 } { z - z ' } & 0 & 0 \\\\ 0 & 0 & z - z ' & 0 \\\\ 0 & 0 & 0 & 1 \\\\ \\end{matrix} \\right ) \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} L _ 1 ( X ) ^ t I _ { p , q } = I _ { p , q } L _ 1 ( X ) . \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} P '' ( x , n ) A _ 2 ( x ) + P ' ( x , n ) A _ 1 ( x ) + P ( x , n ) A _ 0 ( x ) = \\Lambda ( n ) P ( x , n ) \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} 2 \\alpha \\circ q + z \\alpha + [ d , \\alpha ] = 0 \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} H _ { J _ { k + 1 } } = H _ { J _ { k } } \\cap { H _ { k + 1 } } \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} L = \\sum _ { k = 0 } ^ { m - 1 } c _ k X ^ { q ^ k } , \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} k _ z ( z ' ) = e ^ { - z z ' } \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} b \\left [ ( b y ) ^ { - 1 } \\cdot y ^ { \\alpha } \\right ] = b ( b y ) ^ { - 1 } y ^ { \\alpha } \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} P _ { m + 1 } A _ m = A _ m P _ m \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\sigma ( x ) = V _ { \\Phi } ( x ) = - \\nabla \\cdot ( \\rho ( x ) \\nabla \\Phi ( x ) ) . \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq N ' \\leq N } \\abs { S _ { r , N ' } } & \\lesssim Q ^ { - 1 } N ( \\log N ) ^ { - \\alpha } \\sum _ { n = 1 } ^ { N - 1 } \\abs { \\theta } n ^ { d - 1 } \\\\ & \\lesssim Q ^ { - 1 } N ( \\log N ) ^ { - \\alpha } , \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} & \\sum _ { h \\in H } \\biggl ( \\sum _ { v _ i \\in h : f ( v _ i ) > 0 } f ( v _ i ) - \\sum _ { v ^ j \\in h : f ( v ^ j ) < 0 } f ( v ^ j ) \\biggr ) ^ 2 \\\\ & = \\sum _ { h \\in H } \\biggl ( \\frac { 1 } { \\sqrt { \\sum _ v \\deg v } } \\cdot \\big | h \\big | \\biggr ) ^ 2 \\\\ & = \\frac { \\sum _ { h \\in H } \\big | h \\big | ^ 2 } { \\sum _ { h \\in H } \\big | h \\big | } , \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} p _ { i } = \\left ( 2 ^ { \\frac { R . | I | } { B } } - 1 \\right ) . N . L _ i \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} \\mathfrak { f } e _ j = e _ j \\mathfrak { f } = 0 , \\mathfrak { f } ^ { \\dagger } e _ j = e _ j \\mathfrak { f } ^ { \\dagger } = 0 . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} \\Phi _ { 1 2 } : = \\frac { 1 } { 2 } ( \\sum _ { k ' \\in N ( k ) } \\nabla _ { k k ' } \\Phi _ 1 \\nabla _ { k k ' } \\Phi _ 2 ) _ { k = 1 } ^ n \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} v _ p ( S ( n , k ) ) = v _ p ( S ( a _ 0 + \\tilde { n } ( p - 1 ) , k ) ) = v _ p ( f _ { a _ 0 , k } ( \\tilde { n } ) ) = \\beta + l v _ p ( \\tilde { n } - \\tilde { x } _ 0 ) \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} \\Phi = \\left ( \\begin{matrix} 0 & 0 & \\cdots & & & & \\cdots & 0 \\\\ \\xi ^ 1 _ 1 & 0 & & & & & & \\vdots \\\\ \\vdots & \\vdots & \\ddots & & & & & \\\\ \\xi ^ { s _ 1 } _ 1 & & & & & & & \\\\ \\xi ^ 1 _ 2 & & & & & & & \\\\ \\vdots & & & & & & & \\\\ \\xi _ m ^ { s _ m } & 0 & & & & & & \\vdots \\\\ 0 & \\phi ^ 1 _ 1 & \\cdots & \\phi _ 1 ^ { s _ 1 } & \\phi ^ 1 _ 2 & \\cdots & \\phi _ m ^ { s _ m } & 0 \\\\ \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} \\ell _ i ^ { ( k ) } : = \\lambda _ 1 ^ { ( k ) } + \\cdots + \\lambda _ i ^ { ( k ) } . \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} X = \\begin{pmatrix} X _ 1 \\\\ X _ 2 & X _ 1 \\end{pmatrix} , \\ Q = \\begin{pmatrix} Q _ 1 \\\\ Q _ 2 & Q _ 1 \\end{pmatrix} , \\ P = \\begin{pmatrix} P _ 1 & 0 \\end{pmatrix} , \\ R = \\begin{pmatrix} 0 \\\\ R _ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\eta _ r ( b ^ { - 1 } t ^ 0 , b ^ { - 1 } t ^ { 2 n _ 0 } ) \\ ! - \\ ! \\eta _ r ( b ^ { - 1 } t ^ { 2 n _ 0 } , b ^ { - 1 } t ^ 0 ) = a \\eta _ s ( t ^ 0 , t ^ { 2 n _ 0 } ) \\ ! - \\ ! \\underbrace { a \\eta _ s ( t ^ { 2 n _ 0 } , t ^ 0 ) } _ { = 0 } = a t ^ { n _ 0 } s _ { n _ 0 } \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\frac { | { \\det M _ k ( s ) } | } { e ^ { \\R q _ k ( s ) } } & \\ge \\big | s q _ k ( s ) \\sin | p _ k ( s ) | - i | p _ k ( s ) | \\cos | p _ k ( s ) | \\big | \\\\ & - e ^ { - 2 \\R q _ k ( s ) } \\big | s q _ k ( s ) \\sin | p _ k ( s ) | + i | p _ k ( s ) | \\cos | p _ k ( s ) | \\big | , \\end{aligned} \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} \\mathcal { H } _ I = \\bigoplus _ { p \\in \\N ^ I } V _ I ^ p \\left ( \\bigcap _ { q \\in \\N ^ { I ^ c } } V _ { I ^ c } ^ q \\mathcal { W } _ I \\right ) , \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} \\gamma _ x : A \\times A \\rightarrow A , ( a , b ) \\mapsto a \\cdot _ x b : = b \\circ \\beta _ { \\bar { b } \\circ x } ( \\rho _ { \\bar { b } } ( a ) ) \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} N _ { \\mathrm { c } \\ , m + 1 } = \\sum _ { s = 1 } ^ { m + 1 } \\mathcal { C } _ { s } ^ { m + 1 } \\left ( N _ { s } - N _ { \\mathrm { d } s } \\right ) , \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} v ^ { \\pm 1 / 2 } \\mapsto t ^ { \\mp 1 / 2 } & & & X _ { ( i , r ) } \\mapsto \\underline { m _ { k ( i , r ) , r } ^ { ( i ) } } = \\underline { \\prod _ { s = 1 } ^ { k ( i , r ) } Y _ { i , r + 2 r _ i ( s - 1 ) } } , \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ M ^ q _ c \\bigr ] = \\frac { \\Gamma ( 1 - q ) \\ , \\Gamma ^ 2 ( 2 - q ) \\ , \\Gamma ( 4 - q ) } { \\Gamma ( 4 - 2 q ) \\ , \\Gamma ( 5 - 2 q ) } \\ , { \\bf E } \\bigl [ M ^ { q - 1 } _ c \\bigr ] \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ Y ^ { q / \\tau } ( 1 / \\tau ) \\bigr ] = \\frac { \\Gamma ( 1 - q ) } { \\Gamma ( 1 - q / \\tau ) } \\ , { \\bf E } \\bigl [ Y ^ q ( \\tau ) \\bigr ] . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} f _ { 3 6 } ( \\tau ) & = \\sum _ { m , n \\in \\Z \\atop { ~ m \\equiv n \\mod 2 } } \\big ( 3 m + 1 - n \\sqrt { - 3 } \\big ) q ^ { ( 3 m + 1 ) ^ 2 + 3 n ^ 2 } \\\\ & = \\sum _ { m , n \\in \\Z \\atop { m \\equiv n \\mod 2 } } ( 3 m + 1 ) q ^ { ( 3 m + 1 ) ^ 2 + 3 n ^ 2 } = \\eta ( 6 \\tau ) ^ 4 . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t u _ 1 = \\partial _ \\tau \\hat u _ 1 + \\partial _ t \\psi \\partial _ \\eta \\hat u _ 1 , \\partial _ x u _ 1 = \\partial _ \\xi \\hat u _ 1 + \\partial _ x \\psi \\partial _ \\eta \\hat u _ 1 , \\\\ & \\partial _ y u _ 1 = \\hat h _ 1 \\partial _ \\eta \\hat u _ 1 , \\partial _ { y } ^ 2 u _ 1 = \\hat h _ 1 \\partial _ \\eta ( \\hat h _ 1 \\partial _ \\eta \\hat u _ 1 ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} v _ { \\infty } = 0 \\ \\ \\textrm { o n } \\ \\ \\R ^ d \\times ( t _ 1 + 2 \\sigma , t _ 1 + \\kappa ) . \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} \\lambda _ n ( c ) = \\sup _ { V \\in \\mathcal { V } _ { n + 1 } ( c ) } \\inf _ { f \\in V \\setminus \\{ 0 \\} } \\frac { \\| f \\| ^ 2 _ { L ^ 2 ( - 1 , + 1 ) } } { \\| f \\| ^ 2 _ { L ^ 2 ( \\R ) } } , n \\geq 0 . \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} P ( X ) = \\left [ 4 ( Q + \\sigma _ 2 ^ 2 ) ( \\sigma _ 1 ^ 2 - \\sigma _ 2 ^ 2 ) - Q ^ 2 \\right ] - 2 X ( 3 Q + 2 \\sigma _ 2 ^ 2 ) - X ^ 2 . \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} \\lvert \\gamma ( x ) \\rvert \\ ; \\omega _ 2 ( \\phi ( x ) ) & = \\lim _ { \\alpha } \\frac { \\lvert \\gamma ( x ) \\rvert \\ ; \\| l _ { \\phi ( x ) } ( \\chi _ { { U _ \\alpha } } ) \\| _ { p , \\omega _ 2 } } { \\lambda _ 2 ( U _ \\alpha ) ^ { \\frac { 1 } { p } } } \\\\ & \\le \\lim _ \\alpha \\frac { \\| T \\| \\ ; \\| T ^ { - 1 } \\| \\ ; \\omega _ 1 ( x ) \\ ; \\| \\chi _ { U _ \\alpha } \\| _ { p , \\omega _ 2 } } { \\lambda _ 2 ( U _ \\alpha ) ^ { \\frac { 1 } { p } } } \\leq \\| T \\| \\ ; \\| T ^ { - 1 } \\| \\omega _ 1 ( x ) . \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} u _ k ( y ) & = c _ 2 ^ { - 1 } u ( y ) - m _ k V ( \\psi ( y ) ) \\geq ( m _ j - m _ k ) V ( \\psi ( y ) ) \\\\ & \\geq ( m _ j - M _ j + M _ k - m _ k ) V ( d _ D ( y ) ) \\geq - ( V ( r _ { j + 1 } / 2 ) ^ \\gamma - V ( r _ { k + 1 } / 2 ) ^ \\gamma ) V ( r _ j ) . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} A ( t ) & = \\sum _ { i = 1 } ^ { d } \\int _ { 0 } ^ { t } u _ { i 1 } ( X ^ { 1 / 2 , \\mu } ( s ) ) \\mu _ i ( X ^ { 1 / 2 , \\mu } ( s ) ) \\ , d s . \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} B ( \\alpha ) = B ( \\alpha _ 1 , . . . , \\alpha _ n ) = \\frac { \\prod _ { i = 1 } ^ { n } \\Gamma ( \\alpha _ i ) } { \\Gamma \\left ( \\sum _ { i = 1 } ^ { n } \\alpha _ i \\right ) } , \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} G ( s ) = a _ 0 \\prod _ { k = 1 } ^ { N } ( s ^ 2 - k ^ 2 ) , a _ 0 = ( - 1 ) ^ N ( N ! ) ^ { - 2 } . \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} \\Theta = \\left \\{ \\begin{pmatrix} \\sigma _ { 1 } ^ 2 & r \\sigma _ { 1 } \\sigma _ { 2 } \\\\ r \\sigma _ { 1 } \\sigma _ { 2 } & \\sigma _ { 2 } ^ 2 \\end{pmatrix} , c _ 1 \\leq \\sigma _ 1 ^ 2 , \\sigma _ 2 ^ 2 \\leq c _ 2 , \\ - d _ 1 \\leq r \\leq d _ 1 \\right \\} , \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { M , N } ( a , b ) ^ q \\bigr ] = \\eta _ { M , N } ( q \\ , | a , \\ , b ) , \\ ; \\Re ( q ) > - b _ 0 . \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} \\| \\psi \\| : = \\sqrt { \\ < \\psi , \\psi \\ > } \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} 1 = z _ 1 ^ * = z _ 2 ^ * = \\cdots = z ^ * _ { m _ 0 } > z _ { m _ 0 + 1 } ^ * \\ge \\cdots \\ge z ^ * _ n \\ge 0 . \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\Gamma = \\{ \\gamma \\in C ( [ 0 , 1 ] , X ) : \\gamma ( 0 ) = 0 \\ \\mbox { a n d } \\ \\gamma ( 1 ) = e \\} . \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\P _ x ( \\tau _ \\vartheta > n ) } / { \\P _ 0 ( \\tau _ \\vartheta > n ) } ~ = ~ V ( x ) . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} B _ { k , l } ( T ) = c _ { 2 , T ^ 1 } c _ { 1 , T ^ 2 } T \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\eta _ r ( x , y ) - \\eta _ r ( y , x ) = a \\eta _ s ( b x , b y ) - a \\eta _ s ( b y , b x ) \\ ; \\ , \\mbox { f o r a l l $ x , y \\in A $ . } \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} \\lambda & = \\frac 1 c \\norm { x } \\\\ \\Leftrightarrow \\frac { \\lambda } { \\norm { x } } & = \\frac 1 c . \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} \\mathcal F = \\{ u : M \\to \\mathbb R : u \\textrm { i s s m o o t h u p t o t h e b o u n d a r y a n d } \\ , \\int _ M u \\ , d M = 0 \\} . \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} L ( t ) & = e ^ { ( n - \\nu - 2 ) t } \\int _ X S ( t ) \\omega ( t ) ^ n \\\\ & = n e ^ { ( n - \\nu - 2 ) t } ( 2 \\pi c _ 1 ) \\cdot ( - 2 \\pi ( 1 - e ^ { - t } ) c _ 1 ( X ) + e ^ { - t } \\alpha ) ^ { n - 1 } \\\\ & = \\sum _ { k = 0 } ^ { \\nu - 1 } A _ k ( 1 - e ^ { - t } ) ^ k e ^ { ( k - \\nu - 1 ) t } , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} \\ell ( G ) = \\ell ( N ) + \\ell ( G / N ) \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G _ 1 ) = ( 1 6 ) ^ { 1 - 4 / p } ( 1 + 4 ^ { 1 / ( p - 2 ) } ) ^ { 2 ( p - 2 ) / p } . \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\mu _ n ( f ) = \\int f d \\mu _ n \\to \\int f d \\mu = \\mu ( f ) \\ , , \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} \\Xi : \\mathbb { F } _ { \\ell } [ \\lambda ] ^ { ( \\eta + 1 ) m \\times ( \\epsilon + 1 ) n } & \\longrightarrow \\mathbb { F } _ { d } [ \\lambda ] ^ { m \\times n } \\\\ M ( \\lambda ) & \\longrightarrow \\Xi [ M ] ( \\lambda ) = ( \\Lambda _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m ) M ( \\lambda ) ( \\Lambda _ { \\epsilon } ( \\lambda ^ \\ell ) \\otimes I _ n ) . \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} x _ 2 x _ 1 = q x _ 1 x _ 2 . \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} \\mu ( B _ r ( x ) ) = \\lim _ { i \\to \\infty } \\mu _ i ( B _ r ( x ) ) . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} \\varphi ( u , y ) = \\frac { 2 ( y ^ 4 + y ^ 2 + \\sqrt { y ^ 2 + 1 } \\sqrt { y ^ 2 + 1 } y ^ 2 ) u ^ 2 + 4 y ^ 2 + 1 } { ( u ^ 2 + 4 ) ^ 2 ( 1 + y ^ 2 ) ^ 3 } \\geq \\psi ( u , y ) \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} ( \\sigma \\star f ) \\ , ( \\gamma _ 1 , \\ldots , \\gamma _ { q } ) \\ , = \\ , \\sigma ( t ( \\gamma _ 1 ) ) \\cdot 0 _ { \\gamma _ 1 } ^ \\vee \\ , f ( \\gamma _ { 1 } , \\ldots , \\gamma _ { q } ) , \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} d X _ { t } + A ( X _ { t } ) \\ , d t \\ni B _ { t } ( X _ { t } ) \\ , d W _ { t } , X _ { 0 } = x , \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} & { \\rm D } \\bar { J } ( A _ r , B _ r , C _ r ) [ ( A ' _ r , B ' _ r , C ' _ r ) ] \\\\ = & 2 { \\rm t r } ( A _ r '^ T ( - Q P - Y ^ T X ) ) + 2 { \\rm t r } ( B _ r '^ T ( Q B _ r + Y ^ T B ) ) \\\\ & + 2 { \\rm t r } ( C _ r '^ T ( C _ r P - C X ) ) . \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} I _ { f _ j , g _ \\ell } ( \\tau ) : = & \\ \\int _ { - \\overline { \\tau } } ^ { i \\infty } \\int _ { w _ 1 } ^ { i \\infty } \\frac { f _ j ( w _ 1 ) g _ \\ell ( w _ 2 ) } { ( - i ( w _ 1 + \\tau ) ) ^ { 2 - \\kappa _ 1 } ( - i ( w _ 2 + \\tau ) ) ^ { 2 - \\kappa _ 2 } } d w _ 2 d w _ 1 . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} ( u _ 1 , u _ 2 , \\partial _ y h _ 1 , h _ 2 ) | _ { y = 0 } = \\mathbf { 0 } , \\theta | _ { y = 0 } = \\theta ^ \\ast ( t , x ) . \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} | y - y _ 0 | = \\inf _ { z \\in K } | y - z | . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} \\omega _ { \\ ! D } ( s _ { 1 } , \\dots , s _ { k } ) = \\omega _ { \\ ! D } ' ( s _ 1 , \\dots , s _ k ) \\cdot \\mathbf { \\Gamma } _ k . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} a { \\wedge } ^ { \\mathbf { L } \\boxplus \\mathbf { M } } b = \\left \\{ \\begin{array} [ c ] { l } a { \\wedge } ^ { \\mathbf { L } } b a , b \\in L \\\\ a { \\wedge } ^ { \\mathbf { M } } b a , b \\in M \\\\ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} - 4 n + 2 = - 2 h ^ { \\vee } , \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} B ( U , \\zeta ) = 0 . \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} \\rho ^ { * } = \\tau _ 1 , \\prescript { \\tau _ 1 } { } \\tau _ 2 , \\ldots , \\prescript { \\tau _ 1 \\ldots \\tau _ { m - 1 } } { } \\tau _ m \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} v = \\sqrt { \\dot { \\varphi } ^ { 2 } + ( \\sin ^ { 2 } \\varphi ) \\dot { \\theta } ^ { 2 } } \\dot { v } = \\frac { d } { d t } v = \\frac { 1 } { v } [ \\dot { \\varphi } \\ddot { \\varphi } + ( \\sin \\varphi \\cos \\varphi ) \\dot { \\varphi } \\dot { \\theta } ^ { 2 } + \\sin ^ { 2 } ( \\varphi ) \\dot { \\theta } \\ddot { \\theta } ] \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} | N _ * - N ( t ) | \\leq \\frac { p _ { \\max } } { 1 - L \\| n ( t ) \\| _ { T V } } \\| n ( t ) - n _ * \\| _ { \\mathrm { T V } } = \\frac { p _ { \\max } } { 1 - L } \\| n ( t ) - n _ * \\| _ { \\mathrm { T V } } \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ X ^ { \\frac { q } { \\beta } } \\bigr ] \\Gamma ( 1 - \\frac { q } { \\beta } ) = { \\bf E } \\bigl [ X ^ { \\frac { q } { \\beta } } \\bigr ] \\Gamma ( 1 - \\frac { q } { \\beta } ) \\Big | _ { \\beta = 1 } . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} R _ { L _ { \\infty } / K } : = T _ { L _ { \\infty } / K } + C _ { L _ { \\infty } / K } + U ' _ { L _ { \\infty } / K } - M _ { L _ { \\infty } / K } . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} P _ H \\ 1 ( x ) & = ~ G A _ x \\ 1 ( e ) = \\lim _ { n \\to \\infty } \\sum _ { k = 0 } ^ n P ^ k A _ x \\ 1 ( e ) = \\lim _ { n \\to \\infty } \\sum _ { k = 0 } ^ n P ^ k ( T _ x P - P T _ x ) \\ 1 ( e ) \\\\ & = ~ \\lim _ { n \\to \\infty } \\left ( \\sum _ { k = 0 } ^ n P ^ k T _ x P \\ 1 ( e ) - \\sum _ { k = 1 } ^ { n + 1 } P ^ { k } T _ x \\ 1 ( e ) \\right ) \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} \\displaystyle \\lim \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\mathbb { E } | \\sum _ { \\substack { j \\in J _ { K } } } \\Phi ( \\frac { B _ { t _ { j } } + B _ { t _ { j + 1 } } } { 2 } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) | ^ 2 = 0 \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\sigma _ \\lambda = j ^ * ( \\sigma ^ + _ { \\lambda ^ \\vee + 1 ^ m } ) ^ \\vee = j ^ * \\sigma _ { \\lambda + 1 ^ m } ^ + , \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\overset { { \\rm i n \\ , l a w } } { = } 2 \\pi \\ , 2 ^ { - \\bigl [ 3 ( 1 + \\tau ) + 2 \\tau ( \\lambda _ 1 + \\lambda _ 2 ) \\bigr ] / \\tau } \\ , \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) ^ { - 1 } \\ , L \\ , X _ 1 \\ , X _ 2 \\ , X _ 3 \\ , Y . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} V _ I V _ J = \\sum _ K c _ { I J } ^ K V _ K \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} \\alpha _ { 2 } \\alpha _ { 1 } = \\alpha _ { 1 } \\alpha _ { 2 } , \\ \\alpha _ { 3 } \\alpha _ { 1 } = \\alpha _ { 1 } \\alpha _ { 3 } , \\ \\alpha _ { 3 } \\alpha _ { 2 } = \\alpha _ { 1 } \\alpha _ { 2 } \\alpha _ { 3 } . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} f _ c ( x ) = W ( c \\circ x ^ { \\tilde W } ) \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} E = g ( \\partial _ u , \\partial _ u ) , F = g ( \\partial _ u , \\partial _ v ) , G = g ( \\partial _ v , \\partial _ v ) . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } \\Delta ( s ) = & p ( s ) \\left [ \\sigma ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) , \\bar { Y } ^ { t , x ; \\bar { u } } ( s ) , \\bar { Z } ^ { t , x ; \\bar { u } } ( s ) + \\Delta ( s ) , u ) \\right . \\\\ & \\left . - \\sigma ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) , \\bar { Y } ^ { t , x ; \\bar { u } } ( s ) , \\bar { Z } ^ { t , x ; \\bar { u } } ( s ) , \\bar { u } ( s ) ) \\right ] , \\ ; s \\in \\lbrack t , T ] . \\end{array} \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{gather*} \\frac { { \\rm d } X } { Y ^ 2 } ( q ) = 3 f _ { 2 7 } ( q ) \\frac { { \\rm d } q } q . \\end{gather*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\mathbf g _ p ( h \\varpi _ p ) = \\begin{cases} p ^ { 1 / 2 } \\xi _ 1 ( p ) ^ { - 2 } \\underline { \\chi } _ p ( h ) & h \\in K _ { 0 0 } , \\\\ 0 & h \\not \\in K _ { 0 0 } . \\end{cases} \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} A _ K : = C ^ * ( Y _ K \\boxtimes J _ K ) = p \\bigl ( J _ K \\ltimes C _ 0 ( X _ K ) \\bigr ) p \\cong J _ K ^ + \\ltimes C ( Y _ K ) , \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} Q ( x , x + n u ) ~ \\geq ~ Q ( 0 , n u ) ~ = ~ G ( 0 , n u ) / G ( n u , n u ) ~ \\geq ~ G ( 0 , n u ) / G _ S ( 0 , 0 ) . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathrm { d i v } ( | \\nabla u | ^ { p - 2 } \\nabla u ) & = 0 & & ~ ~ \\mbox { i n } ~ ~ \\Omega ; \\\\ u & = g & & ~ ~ \\mbox { o n } ~ ~ \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\partial _ t f _ t ( x , v ) = - v \\ , \\partial _ x f _ t ( x , v ) - \\partial _ v \\big \\{ \\big [ G ( M ( t , x ) ) - v \\big ] f _ t ( x , v ) \\big \\} + \\sigma \\ , \\partial _ { v v } f _ t ( x , v ) \\ , , \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} F _ t ( m ) & : = F _ t ( \\underline { m } ) & E _ t ( m ) & : = E _ t ( \\underline { m } ) & L _ t ( m ) & : = L _ t ( \\underline { m } ) \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} g ( s ) = \\inf \\{ s ' \\geq 0 : s + s ' \\in { \\cal T } _ { m a x } \\} . \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} f = ( f _ { i j } ) _ { i j } = \\sum _ { i } \\sum _ { j } f _ { i j } \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} \\int _ { \\mathcal A _ 1 ^ - ( n ) } \\mathbf h _ p ( h ) d h & = \\sum _ { j \\geq 0 } \\int _ { \\mathcal L _ { 2 j + 1 } } \\mathbf h _ p ( h ) d h = - p \\sum _ { j \\geq 0 } \\mathrm { v o l } ( \\mathcal L _ { 2 j + 1 } ) = - p \\sum _ { j \\geq 0 } p ^ { - 2 j - 1 } ( 1 - p ^ { - 1 } ) ^ 2 = \\\\ & = - ( 1 - p ^ { - 1 } ) ^ 2 \\sum _ { j \\geq 0 } p ^ { - 2 j } = \\frac { - ( 1 - p ^ { - 1 } ) ^ 2 } { 1 - p ^ { - 2 } } = \\frac { - ( 1 - p ^ { - 1 } ) } { 1 + p ^ { - 1 } } = \\frac { 1 - p } { p + 1 } . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} t r ( P \\ : U T U ^ { - 1 } ) = t r ( U ^ { - 1 } P U \\ : T ) \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\mu _ 1 & = \\max _ { f > 0 } \\frac { \\big | H \\big | \\cdot \\biggl ( \\sum _ { v \\in V } f ( v ) \\biggr ) ^ 2 } { \\big | H \\big | \\cdot \\sum _ { v \\in V } \\cdot f ( v ) ^ 2 } \\\\ & = \\max _ { f > 0 } \\frac { \\biggl ( \\sum _ { v \\in V } f ( v ) \\biggr ) ^ 2 } { \\sum _ { v \\in V } f ( v ) ^ 2 } \\\\ & = \\mu _ 1 ' , \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} \\begin{bmatrix} \\Lambda _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m & - ( \\Lambda _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m ) M ( \\lambda ) ( \\widehat { \\Lambda } _ \\epsilon ( \\lambda ^ \\ell ) \\otimes I _ n ) \\end{bmatrix} \\mathcal { L } ( \\lambda ) = \\begin{bmatrix} e _ { \\epsilon + 1 } \\\\ 0 \\end{bmatrix} ^ T \\otimes P ( \\lambda ) , \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} \\lambda : = \\limsup _ { n \\to \\infty } \\frac 1 n \\log \\lVert y _ n \\rVert > 0 . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} \\frac { d } { d t } Q _ { \\alpha _ 2 \\alpha _ 1 } ( t ; \\mathbf { B } ) = ( ( A ^ { \\Delta } _ \\upsilon ) _ { \\alpha _ 2 \\alpha _ 3 } + \\mathbf { B } _ { \\alpha _ 2 \\alpha _ 3 } ) Q _ { \\alpha _ 3 \\alpha _ 1 } ( t ; \\mathbf { B } ) , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\mathbf { B } = \\langle { B \\ , , \\ , \\wedge , \\vee \\ , , \\ , } ^ { \\prime } { , \\ , ^ { \\sim } , 0 \\ , , 1 } \\rangle \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\zeta _ i = \\frac { 2 } { 2 m - 2 i + 3 } \\pi . \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} \\alpha = 2 s + 1 \\ ; \\ ; \\ ; \\ ; \\beta = 2 t . \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} L _ { - 2 } v _ { \\Lambda } & = \\left ( \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { \\ell } H _ { i } ( - 1 ) ^ { 2 } + \\Lambda ( - 2 ) \\right ) v _ { \\Lambda } , \\\\ L _ { - 1 } ^ { 2 } v _ { \\Lambda } & = \\left ( \\Lambda ( - 1 ) ^ { 2 } + \\Lambda ( - 2 ) \\right ) v _ { \\Lambda } . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} & x _ 1 ^ 2 - m x _ 1 y _ 1 + { y _ 1 } ^ 2 - ( a ^ 2 - m a b + b ^ 2 ) \\\\ & = m j ( m j v ^ 4 - m ^ 2 j v ^ m w + m j v ^ 2 w ^ 2 + m v ^ 2 - 2 v w ) a ^ 2 \\\\ & + m j ( m j v ^ 2 w ^ 2 - m ^ 2 j v w ^ m + m j w ^ 4 + 2 v w - m w ^ 2 ) b ^ 2 \\\\ & - 2 m j ( m j v ^ m w - m ^ 2 j v ^ 2 w ^ 2 + m j v w ^ m + v ^ 2 - w ^ 2 ) a b . \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} F ( a ) = M ( a ) F ( a + 1 ) . \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} k ( a / c ) = \\begin{cases} \\frac { - a } { c } & \\gcd ( c , N / c ) > 2 , \\\\ \\frac { 1 } { c } & \\gcd ( c , N / c ) \\leq 2 . \\end{cases} \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} X ( t ) ~ = ~ \\begin{cases} S ( t ) , & , \\\\ \\vartheta & \\end{cases} \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} R _ { i , j } = \\begin{cases} ( 1 + j ) ^ { \\alpha } a _ { j - 1 } , & i = 0 , j > 0 , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\omega _ t = \\omega _ { \\mathbb { C } ^ m } + e ^ { - t } \\omega _ Y \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} C _ 2 ( \\lambda ) = \\left [ \\begin{array} { c | c c c } \\lambda P _ d + P _ { d - 1 } & - I _ m \\\\ P _ { d - 2 } & \\lambda I _ m & \\ddots \\\\ \\vdots & & \\ddots & - I _ m \\\\ P _ 0 & & & \\lambda I _ m \\end{array} \\right ] = \\left [ \\begin{array} { c | c } M _ 2 ( \\lambda ) & L _ { d - 1 } ( \\lambda ) ^ T \\otimes I _ m \\end{array} \\right ] , \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ n a _ j \\lambda _ { g _ j } \\right \\| _ { V N ( \\mathbb { F } _ { \\infty } ) } \\leq 2 ( \\sum _ { j = 1 } ^ n \\left | a _ j \\right | ^ 2 ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} P _ \\mu ( A ) = \\sum _ { \\sigma \\in S _ n } \\left ( \\prod _ { i = 1 } ^ { n } a _ { i \\sigma ( i ) } \\right ) \\mu ^ { \\ell ( \\sigma ) } \\ ; , \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} U e _ { 2 n } & = \\begin{cases} e _ { 2 n + 1 } & n \\ \\textrm { o d d } \\\\ e _ { 2 n - 3 } & n \\ \\textrm { e v e n } \\end{cases} \\\\ U e _ { 2 ^ k n - ( 2 ^ { k - 1 } + 1 ) } & = \\begin{cases} e _ { 2 ^ k n + ( 2 ^ k - 2 ^ { k - 1 } ) } & n \\ \\textrm { o d d } \\\\ e _ { 2 ^ k n - ( 2 ^ k + 2 ^ { k - 1 } ) } & n \\ \\textrm { e v e n } \\end{cases} \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} & E _ 1 ( F _ 3 ^ 2 + E _ 1 E _ 2 F _ 3 + E _ 1 ^ 2 F _ 4 ) + E _ 2 ^ 2 ( E _ 0 F _ 3 + E _ 2 F _ 1 + E _ 1 F _ 2 ) = 0 , \\\\ & E _ 1 ( F _ 1 ^ 2 + E _ 0 E _ 1 F _ 1 + E _ 1 ^ 2 F _ 0 ) + E _ 0 ^ 2 ( E _ 0 F _ 3 + E _ 2 F _ 1 + E _ 1 F _ 2 ) = 0 . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} P ( V _ n = 0 ) & \\sum _ { \\substack { A \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | A | = k - 1 } } \\prod _ { i \\in A } \\frac { b ( i ; n ) } { 1 - b ( i ; n ) } \\frac { \\beta _ n - \\sum _ { i \\in A } \\frac { b ( i ; n ) } { 1 - b ( i ; n ) } } { k } \\\\ & \\leq P ( V _ n = 0 ) \\ > \\frac { \\beta _ n } { k } \\sum _ { \\substack { A \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | A | = k - 1 } } \\prod _ { i \\in A } \\frac { b ( i ; n ) } { 1 - b ( i ; n ) } . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} & 2 ( \\nabla \\nabla F ( x ) \\nabla \\Phi _ 1 ( x ) , \\nabla \\Phi _ 2 ( x ) ) \\\\ = & \\nabla \\Phi _ 1 ( x ) \\nabla ( \\nabla F ( x ) \\cdot \\nabla \\Phi _ 2 ( x ) ) + \\nabla \\Phi _ 2 ( x ) \\nabla ( \\nabla F ( x ) \\cdot \\nabla \\Phi _ 1 ( x ) ) - \\nabla F ( x ) \\nabla ( \\nabla \\Phi _ 1 ( x ) \\cdot \\nabla \\Phi _ 2 ( x ) ) . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} f _ 1 g _ j = 0 j \\geq 1 . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } J _ b \\partial _ t \\zeta + \\mathcal L _ { \\mu _ 2 } \\partial _ x v - \\frac { \\epsilon } { \\gamma } \\partial _ x ( \\zeta v ) = 0 \\\\ J _ d \\partial _ t v + ( 1 - \\gamma ) J _ c \\partial _ x \\zeta - \\frac { \\epsilon } { 2 \\gamma } \\partial _ x ( v ^ 2 ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} \\chi _ { } = \\sum _ { \\rho \\in \\widehat { G } } n _ { \\rho } \\chi _ { \\rho } , \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} F _ 1 ^ c = q F _ 3 + x ^ k G , G ( x , y , z , 0 ) = g , \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} f ( y , x ) & = \\Big \\langle \\Big ( \\frac { 3 } { 2 } - \\| y \\| \\Big ) y , x - y \\Big \\rangle \\\\ & \\leq \\Big ( \\frac { 3 } { 2 } - \\| y \\| \\Big ) ( \\langle y , x - y \\rangle - \\langle x , x - y \\rangle ) \\\\ & = - \\Big ( \\frac { 3 } { 2 } - \\| y \\| \\Big ) \\| x - y \\| ^ 2 \\\\ & \\leq - \\frac { 1 } { 2 } \\| x - y \\| ^ 2 . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\begin{array} { c c c } [ h , e ] = 2 e & [ f , h ] = 2 f & [ f , e ] = h \\end{array} . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\| \\lambda \\psi \\| = | \\lambda | \\ , \\| \\psi \\| . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} \\partial _ t u _ t = \\Delta u _ t + \\mathrm { d i v } ( F ( u _ t ) ) + \\beta _ j \\nabla u _ t \\dot { Z } _ t ^ j \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\frac 1 2 F ( { \\vec x ^ * } ) = \\frac 1 2 \\max \\limits _ { \\vec x \\in \\mathbb { R } ^ n \\setminus \\{ \\vec 0 \\} } F ( \\vec x ) . \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ e ^ { - q C _ 2 } \\bigr ] = \\Bigl [ \\frac { 1 } { \\cosh \\sqrt { 2 q } } \\Bigr ] ^ 2 , \\ ; q > 0 , \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} \\langle V _ i , V _ j \\rangle = \\langle V e _ i , V e _ j \\rangle = e _ i ^ \\top \\ , V ^ \\top V \\ , e _ j \\ , . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} \\sum ^ N _ { \\mu = 1 } \\| ( - \\Delta ) ^ { \\frac { 1 } { 4 } } | u _ \\mu ( t , x ) | ^ 2 \\| ^ 2 _ { L ^ 2 ( ( T _ 1 , T _ 2 ) ; L _ x ^ 2 ) } \\lesssim \\sup _ { t \\in [ T _ 1 , T _ 2 ] } | \\mathcal { M } ( t ) | . \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\lambda _ { \\imath , \\jmath } & = ( \\varpi _ { \\imath } , \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } \\varpi _ { \\jmath } ) - ( \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } \\varpi _ { \\imath } , \\varpi _ { \\jmath } ) \\\\ & = ( \\varpi _ { \\imath } , \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } \\varpi _ { \\jmath } - \\varpi _ { \\jmath } ) - ( \\varpi _ { \\imath } , ( \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } ) ^ { - 1 } \\varpi _ { \\jmath } - \\varpi _ { \\jmath } ) . \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Phi _ \\mu ( t ) = \\frac { 1 } { \\mu - 1 } t + \\frac { 1 } { \\mu - 1 } \\frac { 1 } { \\mu - 2 } ( t + 1 ) ^ { - \\mu + 2 } - \\frac { 1 } { \\mu - 1 } \\frac { 1 } { \\mu - 2 } , \\ ; \\mu \\neq 2 , \\\\ & \\Phi _ 2 ( t ) = t - \\ln ( 1 + t ) , \\ ; t \\geq 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} x _ 2 x _ 1 = - x _ 1 x _ 2 . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} G _ \\bullet : = \\left ( \\left \\{ 0 \\right \\} \\subset G _ 1 \\subset G _ 2 \\subset . . . \\subset G _ { 2 n - 1 } \\subset W \\right ) , \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} k ( M \\backslash A ) - k ( M ) - f ( M \\backslash A ) + k ( M / A ^ c ) = \\\\ k ( ( M \\backslash A ) ^ * ) - k ( M ^ * ) - v ( ( M \\backslash A ) ^ * ) + k ( ( M / A ^ c ) ^ * ) = \\\\ k ( M ^ * / A ^ * ) - k ( M ^ * ) - v ( M ^ * / A ^ * ) + k ( M ^ * \\backslash ( A ^ * ) ^ c ) \\leq 0 , \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\alpha \\geq \\frac 1 { m - 2 p + 1 } , \\beta = \\frac { 1 + \\alpha ( m - 1 ) } 2 \\ , , \\tau \\geq 1 \\ , , \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} V _ { \\eta _ 2 } ^ \\vee ( 1 ) - V _ { \\eta _ 1 } ^ \\vee ( 1 ) = l _ e w _ e \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} [ p _ m , p _ n ] = m \\delta _ { m + n , 0 } \\ , \\delta , [ \\rho , p _ n ] = n \\ , p _ n , m , n \\in \\pm E . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} | z ^ a | = | z | ^ { \\Re { a } } \\exp ( - \\Im { a } \\arg { z } ) , \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} & - \\lim _ { r \\to \\infty } \\sum _ { \\substack { \\boldsymbol { n } \\in \\Z ^ 2 \\\\ \\lvert n _ j \\rvert \\leq r } } \\left ( \\frac { 1 } { i w _ 1 + \\alpha _ 1 + n _ 1 } \\right ) \\left ( \\frac { 1 } { i w _ 2 + \\alpha _ 2 + n _ 2 } \\right ) = - \\pi ^ 2 \\cot \\left ( \\pi \\left ( i w _ 1 + \\alpha _ 1 \\right ) \\right ) \\cot \\left ( \\pi \\left ( i w _ 2 + \\alpha _ 2 \\right ) \\right ) . \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} \\tau _ { t + 1 , m , n } \\tau _ { t - 1 , m , n } = \\tau _ { t , m + 1 , n } ^ { K _ 1 } \\tau _ { t , m - 1 , n } ^ { L _ 1 } + \\tau _ { t , m , n + 1 } ^ { K _ 2 } \\tau _ { t , m , n - 1 } ^ { L _ 2 } , \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} p ^ * ( L _ { i , b } ) = p ^ * ( X _ { i , b } L _ { i + 1 , b } ) & = \\left ( \\lambda _ { i - a } \\frac { A _ { i + 1 , b } A _ { i - 1 , b - 1 } } { A _ { i , b } A _ { i , b - 1 } } \\right ) \\left ( \\frac { A _ { i , b - 1 } } { A _ { i + 1 , b } } \\prod _ { \\substack { k = \\\\ i - a + 1 } } ^ { n } \\lambda _ k \\right ) \\\\ & = \\frac { A _ { i - 1 , b - 1 } } { A _ { i , b } } \\prod _ { k = i - a } ^ { n } \\lambda _ k . \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} & \\sigma ^ 2 : = \\frac { 1 } { n _ i } \\sum _ { j \\in [ m _ i ] } ( f ( x _ { i , j } ) - \\mu / n _ i ) ^ 2 + \\sum _ { j \\in [ n _ i ] \\setminus [ m _ i ] } ( 0 - \\mu / n _ i ) ^ 2 \\leq \\frac { 1 } { n _ i } \\left ( w q _ i ( w ^ { - 1 } - \\frac { \\epsilon q _ i } { n _ i } ) ^ 2 + ( n _ i - w q _ i ) ( \\frac { \\epsilon q _ i } { n _ i } ) ^ 2 \\right ) \\\\ & \\leq \\frac { q _ i } { w n _ i } - \\frac { 2 \\epsilon q _ i ^ 2 } { n _ i ^ 2 } + \\frac { \\epsilon ^ 2 q _ i ^ 2 } { n _ i ^ 2 } \\leq \\frac { 2 } { w n _ i } . \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} \\Delta Q _ 1 = - { \\rm d i v } \\ , ( V \\cdot \\nabla V ) . \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} \\langle f ( t ) , g ( t ) \\rangle = g ( t ) ' f ( t ) - f ( t ) ' g ( t ) . \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} y = t ^ 7 + s _ 1 t ^ 9 + s _ 2 t ^ { 1 1 } + s _ 3 t ^ { 1 6 } . \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\Phi _ { \\pmb { \\phi } _ p } ( \\alpha _ n ) = \\langle \\omega _ { \\overline { \\psi } _ p } ( \\alpha _ n ) \\pmb { \\phi } _ p , \\pmb { \\phi } _ p \\rangle \\Phi _ { \\pmb { \\phi } _ p } ( \\beta _ m ) = \\langle \\omega _ { \\overline { \\psi } _ p } ( \\beta _ m ) \\pmb { \\phi } _ p , \\pmb { \\phi } _ p \\rangle \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\mathcal { M } _ \\chi = \\left \\{ \\mu \\in \\mathcal { M } : \\int _ { \\Gamma _ 0 } \\chi ( | \\eta | ) \\mu ^ { \\pm } ( d \\eta ) < \\infty \\right \\} , \\mathcal { M } ^ { + } _ \\chi = \\mathcal { M } _ \\chi \\cap \\mathcal { M } ^ { + } , \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} e _ { \\infty } : = \\liminf _ { i \\to \\infty } m _ i ^ { - 1 } e ( A _ i , \\Phi _ i ) , \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} { \\mathbb Y } _ a { } ^ { A } { } _ { B } { \\mathbb X } ^ b { } ^ B { } _ { A } = \\delta _ a { } ^ { b } , { \\mathbb Z } _ a { } ^ { b } { } ^ A { } _ { B } { \\mathbb Z } _ c { } ^ { d } { } ^ B { } _ { A } = \\delta _ a { } ^ { d } \\delta _ c { } ^ { b } , { \\mathbb W } ^ { A } { } _ { B } { \\mathbb W } ^ { B } { } _ { A } = 1 , \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} \\abs { \\mathcal F } = \\sum _ { C \\in \\mathcal K } \\abs { C } . \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} \\forall i , j \\in [ n ] \\mbox { s u c h t h a t } i < j , \\ : f ( i ) > f ( j ) & \\Longleftrightarrow f ' ( i ) > f ' ( j ) , \\\\ f ( i ) = f ( j ) & \\Longleftrightarrow f ' ( i ) = f ' ( j ) , \\\\ f ( i ) < f ( j ) & \\Longleftrightarrow f ' ( i ) < f ' ( j ) . \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} \\mathrm { e v } _ { t = 1 } ( L _ { t } ( \\underline { m } ) ) = \\chi _ q ( L ( m ) ) . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} L ( f ' \\otimes g \\otimes g , s ) = L ( f , s - k ) L ( f \\otimes \\mathrm { A d } ( g ) , s - k ) . \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} T = \\{ \\vec t = ( t _ 1 , t _ 2 , \\ldots , t _ n ) \\in \\mathbb { R } ^ n : \\ , t _ i \\le 0 , \\ , \\ , i = 1 , 2 , \\ldots , m _ 0 \\} . \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} \\mathcal { L } f ( i ) = \\sum _ { j \\in V : j \\sim i } W _ { i j } ( f ( j ) - f ( i ) ) i \\in V . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} [ X \\xleftarrow { ( f _ 1 \\sqcup f _ 2 ) \\circ \\overline h } & ( M _ 1 \\sqcup M _ 2 ) \\times _ Y N \\xrightarrow { k \\circ \\overline { g _ 1 \\sqcup g _ 2 } } Z ] \\\\ & = [ X \\xleftarrow { ( f _ 1 \\sqcup f _ 2 ) \\circ ( \\widetilde h \\sqcup \\widetilde { \\widetilde h } ) } \\bigl ( M _ 1 \\times _ Y N \\bigr ) \\sqcup \\bigl ( M _ 2 \\times _ Y N \\bigr ) \\xrightarrow { k \\circ ( \\widetilde { g _ 1 } \\sqcup \\widetilde { g _ 2 } ) } Z ] . \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} \\alpha ^ { 2 s } = \\alpha _ 1 + \\frac { s } { l + 1 } \\delta + s \\epsilon , \\alpha ^ { 2 s + 1 } = \\alpha _ 1 + \\frac { s + 1 } { l + 1 } \\delta + s \\epsilon , \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} \\bar { B _ 1 } = \\frac { a _ { 0 , \\sup } ( b _ { 2 , \\inf } - \\frac { l \\chi _ 2 } { d _ 3 } ) + \\frac { l \\chi _ 1 } { d _ 3 } b _ { 0 , \\sup } } { ( a _ { 1 , \\inf } - \\frac { k \\chi _ 1 } { d _ 3 } ) ( b _ { 2 , \\inf } - \\frac { l \\chi _ 2 } { d _ 3 } ) - \\frac { l k \\chi _ 1 \\chi _ 2 } { d _ 3 ^ 2 } } \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} F \\left ( \\left [ \\frac { x } { y z } \\right ] \\right ) = \\left [ \\frac { x ^ p } { y ^ p z ^ p } \\right ] = 0 \\in H ^ 2 _ { G _ + } ( G ) _ { - p } . \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} U _ i \\delta _ x = \\frac { 1 } { \\sqrt { q _ i } } \\sum _ { \\substack { y \\in X _ i \\\\ \\omega _ y ( 1 ) = x } } \\delta _ y , \\forall x \\in X _ i . \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} f ( \\gamma ( b ) ) - f ( \\gamma ( a ) ) = f ( \\beta ( b ) ) - f ( \\alpha ( a ) ) = b - a \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} \\begin{cases} 0 < \\underbar r _ 1 ^ { n - 1 } \\leq \\underbar r _ 1 ^ n \\leq \\bar r _ 1 ^ n \\leq \\bar r ^ { n - 1 } _ 1 \\le \\bar A _ 1 \\cr 0 < \\underbar r _ 2 ^ { n - 1 } \\leq \\underbar r _ 2 ^ n \\leq \\bar r _ 2 ^ n \\leq \\bar r _ 2 ^ { n - 1 } \\le \\bar A _ 2 \\end{cases} \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} \\mathcal { W } = X _ { 0 , 0 } + X _ { a , b } + \\sum _ { i = 1 } ^ { a - 1 } \\sum _ { j = 1 } ^ b X _ { i , b } X _ { i , b - 1 } \\cdots X _ { i , j } + \\sum _ { j = 1 } ^ { b - 1 } \\sum _ { i = 1 } ^ a X _ { a , j } X _ { a - 1 , j } \\cdots X _ { i , j } . \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} ( T _ { \\rho } \\cdot ( T _ { \\mu } \\cdot X ) ) ( A , w ) = ( T _ { \\rho \\ast \\mu } \\cdot X ) ( A , w ) . \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { \\gamma } - \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } | x | - \\mu \\Big [ b | \\omega | + \\frac { 1 } { \\gamma } \\Big ( a - \\frac { 1 } { \\gamma ^ 2 } \\Big ) \\Big ] x ^ 2 , \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} R _ { B , o p t } ( N , t ) & \\geq 1 - \\frac { \\log \\left [ \\prod \\limits _ { k = 0 } ^ { 2 t } ( N - k ) \\right ] } { \\log N ! } \\\\ & > 1 - \\frac { ( 2 t + 1 ) \\log N } { ( N + \\frac { 1 } { 2 } ) \\log N - ( \\log e ) N } \\\\ & > 1 - \\frac { ( 2 t + 1 ) \\log N } { N ( \\log N - \\log e ) } \\\\ & > 1 - \\frac { 2 t + 1 } { N } \\left ( 1 + \\frac { 2 \\log e } { \\log N } \\right ) , \\\\ \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} \\kappa _ c ( u ) & = \\frac { - 2 ( | a | E R ^ { 1 / 2 } ) ^ { 3 / 2 } a ^ 3 E \\lambda _ v ^ 2 R ^ 2 } { a ^ 4 E ^ { 5 / 2 } \\lambda _ v ^ 2 R ^ { 5 / 2 } | a | | \\lambda _ v | ^ { 1 / 2 } } ( u , 0 ) \\\\ & = \\frac { - 2 | a | ^ { 3 / 2 } R ^ { 1 / 4 } } { a | a | | \\lambda _ v | ^ { 1 / 2 } } ( u , 0 ) = \\frac { - 2 | a | ^ { 1 / 2 } R ^ { 1 / 4 } } { a | \\lambda _ v | ^ { 1 / 2 } } ( u , 0 ) \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\sigma ^ K _ k & = \\frac { 1 - \\frac { \\theta _ k } { \\theta _ { k - 1 } } \\frac { 1 - \\theta _ 0 } { 1 - \\theta _ k } } { 1 + \\frac { \\theta _ { k - 1 } } { \\theta _ 0 \\mu } } , k \\in \\{ 1 , K - 1 \\} \\\\ \\sigma ^ K _ K & = \\frac { 1 - \\frac { \\theta _ { K - 1 } } { \\theta _ 0 } ( 1 - \\theta _ 0 ) } { 1 + \\frac { \\theta _ { K - 1 } } { \\theta _ 0 \\mu } } . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\tilde f ^ + _ w ( p ) = \\tilde f ^ + _ v ( \\tilde f ^ + _ u ( p ) ) > \\tilde f ^ + _ v ( 0 ) > \\tilde f ^ - _ v ( 0 ) + 1 > \\tilde f ^ - _ v ( \\tilde f ^ - _ u ( p ) ) + 1 = \\tilde f ^ - _ w ( p ) - 1 , \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} u _ t + 2 \\omega u _ x - u _ { x x t } + 3 u u _ x = 2 u _ x u _ { x x } + u u _ { x x x } , \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} u _ { t } + \\gamma \\bigtriangleup ^ 2 u - \\bigtriangleup u + f ( u ) = g ( \\textbf { z } , t ) , ( \\textbf { z } , t ) \\in \\Omega \\times J , \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} i ( \\sigma ) ( \\eta _ 1 , \\ldots , \\eta _ p ) = \\langle \\eta _ 1 , \\sigma ( \\gamma _ 1 , \\ldots , \\gamma _ p ) \\rangle \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} M = \\begin{pmatrix} Z _ 1 & X _ 1 & Y _ 1 \\\\ Y _ 2 & Z _ 2 & X _ 2 \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} 0 & \\leq \\varphi _ 1 ( p ) - \\lfloor \\varphi _ 1 ( p ) \\rfloor = \\varphi _ 1 ( p ) - 1 - \\lfloor \\varphi _ 1 ( p ) - \\psi ( p ) \\rfloor \\\\ & < \\varphi _ 1 ( p ) - 1 + 1 + \\psi ( p ) - \\varphi _ 1 ( p ) = \\psi ( p ) . \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N \\tbinom { N } { k } ( - 1 ) ^ k r ^ { i + j + k } = ( 1 - r ) ^ N r ^ { i + j } , \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} d ( { M } ^ { a _ 0 } _ { f } , q ) = f ( q ) - a _ 0 \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} q _ d ( k ) & = \\left [ \\prod _ { j = 1 } ^ { d - 1 } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\left \\{ \\frac { k Y _ j z _ j } { N } \\right \\} \\right ) \\right ) \\right ] \\left [ \\prod _ { j = d + 1 } ^ { s } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\left \\{ \\frac { k z _ j ^ 0 } { N } \\right \\} \\right ) \\right ) \\right ] . \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\left ( \\left [ \\prod _ { i = 1 } ^ { k + 1 } S _ i ^ { m _ i } ( S _ i ^ * ) ^ { n _ i } \\right ] \\left [ \\prod _ { i = 1 } ^ { k + 1 } \\left ( I - \\frac { \\tau _ i } { q _ i } \\right ) \\right ] T \\right ) . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} m _ n = \\frac { \\sum _ { i = 1 } ^ { E _ n } s _ i 2 ^ { ( n - 1 - s _ i ) / 2 } } { \\sum _ { i = 1 } ^ { E _ n } 2 ^ { ( n - 1 - s _ i ) / 2 } } . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} S _ { l } ( i , j ) = \\mathbf { 1 } { ( X _ i = Y _ j ) } \\big \\{ T _ { l 1 } ( i 1 , j 1 ) + R _ { l 1 } ( i , j 1 ) + C _ { l 1 } ( i 1 , j ) \\big \\} . \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\frac { ^ { \\ast } \\left ( ( q _ K - \\sigma ) e _ I \\right ) } { ^ { \\ast } ( - \\sigma e _ I ) } = { } ^ { \\ast } \\left ( ( 1 - q _ K \\sigma ^ { - 1 } ) e _ I \\right ) \\in \\Lambda ( \\mathcal { G } ) ^ { \\times } . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} T ( \\Gamma ; 1 , 2 ) & = \\# \\{ A \\subseteq E : r ( \\Gamma \\backslash A ^ c ) = r ( \\Gamma ) \\} . \\\\ & = \\# \\{ A \\subseteq E : \\mbox { \\rm c o n n e c t e d c o m p o n e n t s o f } \\Gamma / A \\ : \\mbox { \\rm a r e s i n g l e v e r t i c e s w i t h l o o p s } \\} . \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} | I ( x , y , z ) | = 1 , \\quad \\forall x , y , z \\in X . \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} G _ k ( q ) & = \\sum _ { \\mathfrak J \\in S _ 2 } \\tilde \\chi ^ k ( \\mathfrak J ) q ^ { N ( \\mathfrak J ) } + 3 \\sum _ { \\mathfrak J \\in \\Sigma } \\tilde \\chi ^ k ( ( - 2 ) \\mathfrak J ) q ^ { N ( ( - 2 ) \\mathfrak J ) } \\\\ & = \\sum _ { \\mathfrak J \\in S _ 2 } \\tilde \\chi ^ k ( \\mathfrak J ) q ^ { N ( \\mathfrak J ) } + 3 ( - 2 ) ^ k \\sum _ { \\mathfrak J \\in \\Sigma } \\tilde \\chi ^ k ( \\mathfrak J ) q ^ { 4 N ( \\mathfrak J ) } . \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } f _ { n _ k } = V \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} L _ t ( m ^ { ( i ) } _ { k , r } ) = F _ t ( m ^ { ( i ) } _ { k , r } ) = [ \\underline { W ^ { ( i ) } _ { k , r } } ] . \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q \\ , | \\ , a , b ) = \\prod \\limits _ { n _ 1 , \\cdots , n _ M = 0 } ^ \\infty \\eta _ { 0 , N } ( q \\ , | \\ , b _ 0 + \\Omega ) , \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\log r \\sim 2 r _ 1 \\log 2 r _ 1 - 2 r _ 1 \\log r _ 1 = 2 r _ 1 \\log 2 , \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} \\Omega ( \\Omega ^ n A , M \\otimes P ^ A _ { n - 1 } , M \\otimes i ^ A _ n ) & = ( \\Omega ^ { n + 1 } A , M \\otimes P ^ A _ { n } , M \\otimes i ^ A _ { n + 1 } ) . \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} 0 = _ { q / 2 } \\Bigl ( 1 + \\frac 1 { 1 + a + b } \\Bigr ) = _ { q / 2 } \\Bigl ( 1 + \\frac 1 { 1 + a ^ 2 + b ^ 2 } \\Bigr ) = _ { q / 2 } \\Bigl ( 1 + \\frac 1 { 1 + b } \\Bigr ) = _ { q / 2 } \\Bigl ( \\frac b { 1 + b } \\Bigr ) . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} \\left [ b / b ^ { \\alpha } y \\right ] y = b ( ( b ^ { \\alpha } y ) ^ { - 1 } y ) \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\rho v = \\frac { 2 U _ { \\nu } ^ { \\ast } } { \\omega } \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} u _ k = V ( \\psi ) \\left ( \\frac { u } { c _ 2 V ( \\psi ) } - m _ k \\right ) = \\frac { 1 } { c _ 2 } u - m _ k V ( \\psi ) \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} & \\delta ( \\jmath , r _ 0 + \\frac { 1 } { 2 } ) = \\delta ( \\jmath , r _ 0 - \\frac { 1 } { 2 } ) = \\delta _ { \\imath _ 0 \\jmath } \\ \\ \\jmath \\in I _ { \\mathrm { A } } , \\\\ & \\delta ( \\imath _ 0 - 1 , r _ 0 ) = 1 , \\ \\delta ( \\imath _ 0 + 1 , r _ 0 ) = 0 \\ ( \\delta ( 2 n , r _ 0 ) : = 0 ) . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} \\frac { ( C _ 1 ^ 2 + C _ 0 C _ 2 ) ( C _ 2 ^ 2 + C _ 1 C _ 3 ) } { ( C _ 1 C _ 2 + C _ 0 C _ 3 ) ^ 2 } = \\frac P Q , \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} M ^ { \\lambda } \\otimes M ^ { \\mu } = \\bigoplus _ { A \\in A ^ { \\lambda } _ { \\mu } } M ^ { A } \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} S ( \\lambda ) : = \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} R ( \\lambda ) \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} \\begin{aligned} & \\ , 2 K _ 1 ^ { \\frac { p - 1 } { p - 2 + m } } \\frac { C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } ( \\tau + t ) ^ { \\alpha m - \\beta } \\\\ \\leq & \\ , C ^ { \\frac { ( p - 1 ) ( m - 1 ) } { p - 2 + m } } \\left ( \\frac { C ^ { m - 1 } m } { 2 a ^ 2 ( m - 1 ) ^ 2 } \\right ) ^ { \\frac { p - 1 } { p - 2 + m } } ( \\tau + t ) ^ { \\frac { \\alpha p ( m - 1 ) + ( \\alpha m - 2 \\beta ) ( p - 1 ) } { p - 2 + m } } \\forall t > 0 \\ , . \\end{aligned} \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} I ' ( i , j ) = \\{ \\underbrace { b - j , \\dots , b - j + i - 1 } _ , \\underbrace { b + i , \\dots , n - 1 } _ \\} , \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} { \\bf { E } } \\Bigl [ \\Bigl ( \\int _ 0 ^ 1 s ^ { \\lambda _ 1 } ( 1 - s ) ^ { \\lambda _ 2 } \\ , M _ \\beta ( d s ) \\Bigr ) ^ n \\Bigr ] = \\int \\limits _ { [ 0 , \\ , 1 ] ^ n } \\prod _ { i = 1 } ^ n s _ i ^ { \\lambda _ 1 } ( 1 - s _ i ) ^ { \\lambda _ 2 } \\prod _ { i < j } ^ n | s _ i - s _ j | ^ { - 2 \\beta ^ 2 } d s _ 1 \\cdots d s _ n . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} b _ { n , h t ^ 2 k } = \\mathbf { 1 } _ { ( n , v ) = 1 } G \\left ( \\frac { n f } { N } \\right ) \\mathop { \\sum _ { \\ell _ 1 } \\sum _ { \\ell _ 3 } \\sum _ v } _ { \\substack { \\ell _ 1 \\ell _ 3 v = k \\\\ ( \\ell _ 1 \\ell _ 3 , v ) = 1 } } \\beta ( \\ell _ 1 ) \\gamma ( \\ell _ 3 ) \\alpha ( v ) G \\left ( \\frac { v t } { V } \\right ) G \\left ( \\frac { \\ell _ 1 } { L _ 1 } \\right ) G \\left ( \\frac { \\ell _ 3 } { L _ 3 } \\right ) \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} \\sigma ( \\beta \\alpha ) _ 2 ( i ) & = ( \\beta \\alpha ) _ 2 \\sigma ^ { - 1 } ( i ) = \\beta _ 4 ( \\alpha _ 2 ( \\tau ^ { - 1 } ( i ' ) ) ) + \\beta _ 4 ^ c ( \\beta _ 3 ( \\tau ^ { - 1 } ( i ' ) ) ) \\\\ & = \\beta _ 4 ( \\tau ( \\alpha ) _ 2 ( i ' ) ) + \\beta _ 4 ^ c ( \\sigma ( \\beta ) _ 3 ( i ' ) ) = ( \\sigma ( \\beta ) \\tau ( \\alpha ) ) _ 2 ( i ) . \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} \\| T \\| _ { S _ 1 } + | c _ + | + | c _ - | \\leq \\| \\xi ^ { [ N ] } \\| \\| \\eta ^ { [ N ] } \\| + \\| \\xi ^ { \\varnothing } \\| \\| \\eta ^ \\varnothing \\| = \\| f _ \\phi \\| \\leq \\| \\phi \\| _ { c b } . \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} ( \\phi ^ * f \\star g ) = \\phi ^ * ( f \\star \\phi _ * g ) . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) = \\mathcal C \\cdot \\mathcal I \\cdot \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 , \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} T _ { n , m } = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + \\chi ^ I ) , \\quad \\forall m , n \\in \\N ^ N , \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } ( 1 , \\ , \\tau ) = ( 1 , \\ , \\frac { 1 } { \\tau } ) \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\left ( r ! \\sum _ { j = 0 } ^ { r } \\frac { \\binom { b } { r - j } } { j ! } x ^ { j } \\frac { d ^ { j } } { d x ^ { j } } \\right ) ^ { p } = \\sum _ { k = 0 } ^ { r p } S _ { 1 , b , r } \\left ( p , k \\right ) x ^ { k } \\frac { d ^ { k } } { d x ^ { k } } . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | m _ { n + 1 } ( x , y ) - m _ n ( x , y ) \\big | = \\bigg ( \\frac { 1 } { \\vartheta _ B ( N _ 1 ) } - \\frac { 1 } { \\vartheta _ B ( N _ 2 ) } \\bigg ) \\prod _ { j = 1 } ^ { k '' } \\log y _ j . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\triangle _ \\beta u = \\triangle _ \\beta T + O ( d ^ { q - 2 } ) = 0 , \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} \\begin{aligned} G ( z ) = A ( z ) \\beta ( z ^ { - 1 } ) & = A ( z ) \\bigg ( 1 + z \\sum _ n \\frac { a _ n \\bar b _ n t _ n \\nu _ n } { z - t _ n } \\bigg ) \\\\ & = A ( z ) \\bigg ( 1 + \\sum _ n a _ n \\bar b _ n t ^ 2 _ n \\nu _ n \\Big ( \\frac { 1 } { z - t _ n } + \\frac { 1 } { t _ n } \\Big ) \\bigg ) . \\end{aligned} \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} K _ { 0 } = \\frac { - 2 } { \\rho _ { 0 } ^ { 2 } v _ { 0 } } ( \\omega - \\frac { 2 w _ { 0 } } { \\rho _ { 0 } v _ { 0 } } ) = J _ { \\theta } ( 1 + J _ { \\varphi } ^ { 2 } ) \\cos \\varphi _ { 0 } + ( J _ { \\varphi } J _ { \\theta } ^ { \\prime } - J _ { \\theta } J _ { \\varphi } ^ { \\prime } ) \\sin \\varphi _ { 0 } , \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\langle f e _ 1 , e _ 2 \\rangle _ { \\mathcal { E } ^ 0 } ( g ) & = \\mu ( g ) ^ { - 1 / 2 } \\langle f e _ 1 , g ( e _ 2 ) \\rangle _ { H ^ 0 ( E ) } \\\\ & = \\langle e _ 1 , f ^ * e _ 2 \\rangle _ { \\mathcal { E } ^ i } ( g ) . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\sigma _ \\nu ^ + \\cup \\sigma _ { 1 ^ m } ^ + = \\sum _ { \\beta } c _ { \\nu } ^ \\beta \\ , \\sigma _ \\beta ^ + , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} T _ x f _ n ~ \\geq ~ \\sum _ { k = 1 } ^ n P ^ { k - 1 } A _ x f _ n + f _ n , \\forall n \\geq 1 , \\ ; x \\in E , \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} T ^ { \\omega } = \\frac { \\omega ^ { 2 } } { 2 \\rho ^ { 2 } } . \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} 3 n - 6 = e ( T ) & = e ( T [ S \\cup \\{ u _ 1 , \\ldots , u _ 6 \\} ] ) + e _ T ( \\{ u _ 7 , \\ldots , u _ r \\} , S \\cup \\{ u _ 1 , \\ldots , u _ 6 \\} ) \\\\ & \\le e ( \\mathcal { T } _ { 1 1 } ) + ( 2 n - 4 ) = 2 7 + ( 2 n - 4 ) , \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} R = R _ { s } - R _ { e } = R _ { s } - \\log ( 1 + \\rho _ { e } ) . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} \\Phi _ * ( D _ a ) = a b + b a \\Phi _ * ( D _ b ) = ( a ^ 2 + a ^ 2 b - b ^ 3 ) ( 1 + b ) ^ { - 1 } . \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} ( ( r s ) t ) = ( r ( s t ) ) + ( - 1 ) ^ { | t | | s | } ( r ( t s ) ) , \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\Delta p = \\sum _ { k = 1 } ^ { n - 1 } \\Delta ^ k p ( \\cdot - k ) + \\Delta ^ n p ( \\cdot - n + 1 ) . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} F _ { u } ^ { \\ast } \\left ( { x , y _ { u } ^ { \\ast } } \\right ) = \\left \\{ \\begin{array} { l l } { f _ { u } ^ { \\ast } \\left ( { x ^ { \\ast } } \\right ) , } & \\mathrm { i f } \\ ; \\ ; { y _ { u } ^ { \\ast } } = 0 _ { u } ^ { \\ast } , \\\\ { + \\infty , } & \\mathrm { i f } \\ ; \\ ; y _ { u } ^ { \\ast } \\neq 0 _ { u } ^ { \\ast } . \\end{array} \\right . \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} \\Theta ( x ) = \\lim _ { i \\to \\infty } \\mu _ i ( B _ { r _ i } ( x ) ) . \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} f ( A ) = \\langle \\pi ( A ) \\xi , \\eta \\rangle , \\forall \\ A \\in \\mathcal { A } , \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} F = Q ^ 2 + \\pi ^ { r } G , \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\beta \\circ \\lambda ( e _ q ) & = \\beta ( \\varphi ( q ) ) \\\\ & = \\varphi \\circ \\alpha ( q ) \\\\ & = \\lambda ( e _ { \\alpha ( q ) ) } \\\\ & = \\lambda \\circ \\alpha ( e _ q ) . \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} A _ { \\mathbf { k } } ( z ) = 2 ^ { - n ( z - 1 ) } \\bigg ( \\prod _ { j = 1 } ^ n \\prod _ { i = 1 } ^ j \\frac { 1 } { z + k - 1 - j + 2 i } \\bigg ) \\bigg ( \\prod _ { j = 1 } ^ n \\prod _ { i = 0 } ^ { \\frac { k - k _ j } 2 - 1 } \\frac { z - ( k - 1 - j - 2 i ) } { z + ( k - 1 - j - 2 i ) } \\bigg ) . \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} \\partial _ { \\mathrm { e v e n } } : = A \\quad \\partial _ { \\mathrm { o d d } } : = B \\ , . \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} | W | = | w | + \\tau + \\sum _ { k = 1 } ^ \\tau \\lambda ( \\alpha + k ) \\le 2 n + n + \\sum _ { i = \\alpha + 1 } ^ n i ^ 2 \\le \\tau n ^ 2 + 3 n . \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} J ( u ) _ i & = \\begin{cases} 0 , & u _ i \\in A \\\\ 1 , & u _ i \\in B , \\end{cases} \\forall i \\in \\{ 1 , . . . , N \\} . \\\\ \\psi ( \\omega ) & = \\begin{cases} \\Psi ^ { - 1 } ( \\omega ) , & \\omega \\in A \\\\ \\Psi ^ { - 1 } \\circ f ^ { - 1 } ( \\omega ) , & \\omega \\in B . \\end{cases} \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} ( P u - u , v _ { h } ) = 0 \\ ; \\forall v _ { h } \\in V _ { h } . \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} 0 = A ( 0 ) \\leq A ( 1 ) \\leq A ( 2 ) \\leq \\cdots \\leq A ( n ) = \\| \\vec v \\| _ 1 . \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} i _ 2 ( C _ { H ^ F } ( y _ 1 ) ) & = i _ 2 ( S O _ \\frac { n } { 2 } ( q ^ 2 ) ) + | \\phi ^ { S O _ \\frac { n } { 2 } ( q ^ 2 ) } | \\\\ & = i _ 2 ( S O _ \\frac { n } { 2 } ( q ^ 2 ) ) + \\frac { | S O _ { \\frac { n } { 2 } } ( q ^ 2 ) | } { | S O _ { \\frac { n } { 2 } } ( q ) | } \\\\ & < q ^ { \\frac { n ^ 2 } { 8 } + c ' n } , \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} d _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x _ { \\lambda _ 2 } ( b ) + d _ 2 ~ ^ { A B R } D _ b ^ \\alpha x _ { \\lambda _ 2 } ( b ) = 0 . \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} g ( x ) ~ \\geq ~ g ( e ) G _ H \\ 1 ( x ) + \\lim _ n P _ H ^ n g ( x ) ~ \\geq ~ g ( e ) G _ H \\ 1 ( x ) ~ = ~ g ( e ) V ( x ) \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} { \\theta _ { 1 , A } } \\left ( t \\right ) = 2 \\pi { f _ { 1 , A } } t + 2 \\pi \\Delta f n T + \\varphi _ { n , A } ^ { { \\rm { C F S K } } } + { \\varphi _ A } \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} G ( L _ { \\boldsymbol { u } } A ( x ) , z ) = G ( A ( 0 ) , z ) G ( \\boldsymbol { u } , e ^ { z } - 1 ) e ^ { x z } . \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} \\forall j \\in C _ { \\Gamma , \\chi } : \\begin{cases} j , k ( j ) \\in R _ { \\Gamma , \\chi } & k ( j ) \\neq j , \\\\ j \\in R _ { \\Gamma , \\chi } \\iff \\chi ( V ^ { - 1 } U _ j ) = 1 & k ( j ) = j . \\end{cases} \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} ( \\rho _ + - \\rho _ - ) ( \\widehat { \\sigma ^ { B } } ) ^ { 2 } - 2 ( \\rho _ + u _ + - \\rho _ - u _ - ) \\widehat { \\sigma ^ { B } } + \\rho _ + u _ + ^ { 2 } - \\rho _ - u _ - ^ { 2 } - B ( \\frac { 1 } { \\rho _ + ^ \\alpha } - \\frac { 1 } { \\rho _ - ^ \\alpha } ) = 0 . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\mu _ p \\Bigl ( \\bigcup _ { r = 1 } ^ { 2 ^ q } \\Gamma _ r \\Bigr ) = 1 . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} \\gamma _ j \\ , T _ { j ' } ^ { \\frac { 1 } { m } } = \\zeta _ m ^ { \\delta _ { j j ' } } T _ { j ' } ^ { \\frac { 1 } { m } } , \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\binom { a n + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\alpha _ { s } n + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } \\right ) = \\frac { z \\sum _ { i = 0 } ^ { r p + \\sigma } A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) z ^ { r p + \\sigma - i } } { \\left ( z - 1 \\right ) ^ { r p + \\sigma + 1 } } . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} - \\varphi _ 2 \\nu \\big ( 1 \\otimes ( r _ 1 - r _ 2 ) \\big ) & = \\varphi _ 2 \\big ( \\alpha _ B \\pi ^ \\flat ( r _ 2 ) \\otimes \\frac { r _ 1 } { r _ 2 } \\big ) = \\alpha _ B \\pi ^ \\flat ( r _ 2 ) \\cdot ( u - 1 ) = \\alpha _ C \\varphi ^ \\flat ( r _ 2 ) \\cdot ( u - 1 ) \\\\ & = u \\alpha _ C \\varphi ^ \\flat ( r _ 2 ) - \\alpha _ C \\varphi ^ \\flat ( r _ 2 ) = \\alpha _ C \\varphi ^ \\flat ( r _ 1 ) - \\alpha _ C \\varphi ^ \\flat ( r _ 2 ) \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} Y \\overset { { \\rm i n \\ , l a w } } { = } Y ' = \\beta _ { 1 , 0 } ^ { - 1 } \\bigl ( \\tau = 1 , b _ 0 = 1 \\bigr ) . \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\Theta _ { t } ^ { - 1 } d \\Theta _ { t } = \\frac { \\tau } { 2 } \\sum _ { r = 1 } ^ { 3 } ( X _ { r } \\otimes \\rho _ { t } ( \\zeta ) ^ { - 1 } ) ^ { 2 } d t - \\sum _ { r = 1 } ^ { 3 } X _ { r } \\otimes \\rho _ { t } ( \\zeta ) ^ { - 1 } d B _ { t } ^ { ( r ) } . \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} \\varphi _ { Y } ^ { 2 } ( \\alpha _ { n } ^ { ( N ) } ) = { \\displaystyle \\sum \\limits _ { i = 1 } ^ { d } } { \\displaystyle \\sum \\limits _ { j = 1 } ^ { d } } \\left ( \\alpha _ { n } ^ { ( N ) } ( f _ { i } f _ { j } ) \\right ) ^ { 2 } , \\varphi _ { Y } ^ { 2 } ( \\mathbb { G } ^ { ( N ) } ) = { \\displaystyle \\sum \\limits _ { i = 1 } ^ { d } } { \\displaystyle \\sum \\limits _ { j = 1 } ^ { d } } \\left ( \\mathbb { G } ^ { ( N ) } ( f _ { i } f _ { j } ) \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} D ( n ) = ( n - 1 ) ( D ( n - 1 ) + D ( n - 2 ) ) \\quad ( n \\ge 2 ) \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} F _ { 4 , 9 } \\wedge F _ { 4 , 1 1 } \\wedge F _ { 4 , 1 3 } \\wedge F _ { 4 , 1 5 } \\wedge F _ { 4 , 1 7 } = - 2 8 6 6 5 4 4 6 4 0 \\ , \\chi _ { 5 } ^ 3 \\chi _ { 3 0 } ^ 2 \\ , . \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} Y ^ * = \\big \\{ Q | _ W : Q \\in Y \\big \\} , \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left \\| u \\left ( \\cdot , t \\right ) - \\frac { M _ 0 } { M } \\mathcal { B } _ { M } \\left ( \\cdot , t \\right ) \\right \\| _ { L ^ 1 } = 0 \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\dfrac { 1 } { - 2 H ( u , v ) } = \\dfrac { \\lambda ( u , v ) } { \\hat { H } ( u , v ) } \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { \\psi ( n ) \\varphi ( n ) } { n } < \\infty . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} \\Phi ( X ( t ) , A , p ( t ) ) : = \\| M ( t , A ) \\| _ { p ( t ) } \\ , . \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { ( \\phi * \\phi ) ( n ) } { n } & = \\frac { 1 } { \\zeta ^ { 2 } ( 2 ) } x \\log x + \\frac { x } { \\zeta ^ { 2 } ( 2 ) } \\left ( 2 \\gamma - 1 - 2 \\frac { \\zeta ' ( 2 ) } { \\zeta ( 2 ) } \\right ) + P ( x ) , \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} R _ { \\boldsymbol { k } } ( \\tau ) : = \\int _ 0 ^ { i \\infty } \\int _ { w _ 1 } ^ { i \\infty } \\frac { \\Theta _ 1 ( 2 p , k _ 1 , 2 p ; w _ 1 ) \\Theta _ 0 ( 6 p , k _ 2 , 6 p ; w _ 2 ) } { \\sqrt { - i ( w _ 1 + \\tau ) } ( - i ( w _ 2 + \\tau ) ) ^ \\frac 3 2 } d w _ 2 d w _ 1 . \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} & \\| \\mathbf { R } ^ s _ i \\| _ { 0 , T _ { \\ast } ^ s } \\leqslant C h ^ 2 | \\mathbf { u } | _ { 2 , T } , ~ ~ i \\in \\mathcal { I } ^ s , ~ s = \\pm , \\\\ & \\| \\mathbf { R } ^ s _ { i 1 } \\| _ { 0 , T _ { \\ast } ^ s } \\leqslant C h ^ 2 | \\mathbf { u } | _ { 2 , T } , ~ ~ \\| \\mathbf { R } ^ s _ { i 2 } \\| _ { 0 , T _ { \\ast } ^ s } \\leqslant C h ^ 2 | \\mathbf { u } | _ { 2 , T } , ~ ~ i \\in \\mathcal { I } ^ { s ' } , ~ s = \\pm . \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\left | \\ \\frac { ( z + h ) ^ n - z ^ n } { h } - n z ^ { n - 1 } \\ \\right | \\ & = \\ \\biggl | \\ \\sum _ { k = 1 } ^ n \\bigl ( \\ , ( z + h ) ^ { n - k } - z ^ { n - k } \\ , \\bigr ) z ^ { k - 1 } \\ \\biggr | \\\\ \\ & \\leq \\ 2 n r ^ { n - 1 } . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} \\eta _ { 2 , 2 } ( q \\ , | \\ , \\tau , b ) = \\frac { \\Gamma _ 2 ( q + b _ 0 \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( b _ 0 \\ , | \\ , \\tau ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 1 \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , \\tau ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 2 \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 2 \\ , | \\ , \\tau ) } \\frac { \\Gamma _ 2 ( q + b _ 0 + b _ 1 + b _ 2 \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( b _ 0 + b _ 1 + b _ 2 \\ , | \\ , \\tau ) } . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} B _ 1 ( m ) = - p ^ { 1 - 3 m / 2 } ( - 1 ) ^ m \\chi _ { \\psi } ( p ^ m ) \\int _ { \\mathcal B _ 1 ( m ) } ( c , p ^ m ) _ p \\chi _ { \\psi } ( c ) e ( h ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h , \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} ^ { C } L _ 1 x ( t ) = \\lambda r ( t ) x ( t ) , \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} \\begin{pmatrix} X \\\\ P & Y \\\\ Q & R & X \\end{pmatrix} \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} ( x _ 1 ^ { p ^ { e _ 1 } } , \\ldots , x _ d ^ { p ^ { e _ d } } ) ^ * = ( x _ 1 ^ { p ^ { e _ 1 } } , \\ldots , x _ d ^ { p ^ { e _ d } } ) ^ F \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\tau _ { N ' } : = N ' . \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} \\widehat { d } ( G ( A _ 1 , \\ldots , A _ m ) ) \\prec _ { \\log } \\left ( \\prod _ { j = 1 } ^ { m } \\widehat { d } ( A _ j ) \\right ) ^ { 1 / m } \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} \\langle s _ { V ^ \\vee } ( \\eta ) , c \\rangle = - \\langle \\eta , 0 _ \\gamma ^ V \\cdot c ^ { - 1 } \\rangle \\ ; \\ ; , \\langle t _ { V ^ \\vee } ( \\eta ) , c ' \\rangle = \\langle \\eta , c ' \\cdot 0 _ \\gamma ^ V \\rangle \\ ; \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} F _ t = \\bigoplus _ { 1 \\leq j _ 1 \\leq j _ 2 \\leq \\cdots \\leq j _ t \\leq n } S ( - t ) ^ { \\delta - ( d _ { j _ 1 } + \\cdots + d _ { j _ t } ) + n - 1 \\choose n - 1 } = \\bigoplus _ { 1 \\leq k _ 1 \\leq k _ 2 \\leq \\cdots \\leq k _ { n - t } \\leq n } S ( - t ) ^ { d _ { k _ 1 } + \\cdots + d _ { k _ { n - t } } - 1 \\choose n - 1 } , \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\frac { a _ { k + 1 } ( \\beta ) } { k ! } A ^ k < \\infty , \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} F ( \\alpha ; \\xi ; Z ) : = \\prod _ { x \\in V } \\Phi _ x ( \\alpha ; \\xi ; Z ) , \\ , F ( \\alpha ; Z ) : = F ( \\alpha ; 0 ; Z ) . \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} [ F _ - , p , F _ + ] [ E _ - , q , E _ + ] = [ F _ - ' , p ' q ' , E _ + ' ] \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} L ^ { 1 / \\tau } \\bigl ( \\frac { 1 } { \\tau } \\bigr ) & \\overset { { \\rm i n \\ , l a w } } { = } L ( \\tau ) , \\\\ X ^ { 1 / \\tau } _ i \\bigl ( \\frac { 1 } { \\tau } , \\tau \\lambda \\bigr ) & \\overset { { \\rm i n \\ , l a w } } { = } X _ i ( \\tau , \\lambda ) , \\ ; i = 1 , 2 , 3 , \\\\ M ^ { 1 / \\tau } _ { \\bigl ( a ( 1 / \\tau ) , \\ , x ( 1 / \\tau , \\tau \\lambda ) \\bigr ) } & \\overset { { \\rm i n \\ , l a w } } { = } M _ { \\bigl ( a ( \\tau ) , \\ , x ( \\tau , \\lambda ) \\bigr ) } . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} v _ p ( S ( n , k ) ) = \\beta + l v _ p ( n - a ) \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} f ( y , x ) & = \\bigg \\langle \\dfrac { y } { 1 + \\| y \\| ^ 2 } , x - y \\bigg \\rangle \\\\ & \\leq \\dfrac { 1 } { 1 + \\| y \\| ^ 2 } ( \\langle y , x - y \\rangle - \\langle x , x - y \\rangle ) \\\\ & \\leq - \\dfrac { 1 } { 2 } \\| x - y \\| ^ 2 \\\\ & = - \\gamma \\| x - y \\| ^ 2 , \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} \\sum _ { \\epsilon \\leq \\sigma , \\abs { \\sigma } < q } a _ \\sigma ( x ) Q ^ \\sigma _ \\epsilon h ^ { \\sigma - \\epsilon } = \\sum _ { \\epsilon \\leq \\sigma , \\abs { \\sigma } < q } a _ \\sigma ( x + h ) Q ^ \\sigma _ \\epsilon h ^ { \\sigma - \\epsilon } + O ( \\abs { h } ^ { q - \\abs { \\epsilon } } ) . \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial f ^ { ( k ) } } { \\partial \\varphi ^ { ( i ) } } \\right ) = - 2 t \\left ( \\begin{array} { c c c } e ^ { 2 t \\zeta _ 1 ^ { ( k ) } } & & \\\\ & \\ddots & \\\\ & & e ^ { 2 t \\zeta _ k ^ { ( k ) } } \\\\ \\end{array} \\right ) C ^ { ( k , i ) } , \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ] ^ { d - 1 } } f ( x ) e ^ { - 2 \\pi i \\langle n , x \\rangle } \\ , \\textrm { d } x = \\sum _ { j = 1 } ^ m \\frac { \\langle a _ j , n \\rangle } { 2 \\pi i | n | ^ 2 } \\int _ { A _ j } e ^ { - 2 \\pi i \\langle n , x \\rangle } \\ , \\textrm { d } x . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} & \\partial \\widetilde D _ { \\sigma _ 1 , \\tau } : a = d , \\\\ & \\partial \\widetilde D _ { \\sigma _ 2 , \\tau } : b a = d c , \\\\ & \\partial \\widetilde D _ { \\sigma _ 3 , \\tau } : c b a = d c b . \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Gamma _ 1 u ( t ) = u ( t ) = \\eta _ T ( t ) W _ 0 ^ t \\big ( u _ 0 ^ e , g \\big ) - i \\eta _ T ( t ) \\int _ 0 ^ t e ^ { i ( t - t ^ \\prime ) \\Delta } F ( u , n ) \\d t ' + i \\eta _ T ( t ) W _ 0 ^ t \\big ( 0 , q \\big ) , \\\\ \\Gamma _ 2 n ( t ) = n ( t ) = \\eta _ T ( t ) V _ 0 ^ t \\big ( \\phi _ { \\pm } , h \\big ) + \\frac 1 2 \\eta ( t ) ( n _ + + n _ - ) - \\frac 1 2 \\eta _ T ( t ) V _ 0 ^ t ( 0 , z ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { \\infty } \\gamma _ { i , \\pm r } ^ { \\pm } z ^ { \\pm r } = q _ i ^ { \\deg ( P _ i ) } \\frac { P _ i ( z q _ i ^ { - 1 } ) } { P _ i ( z q _ i ) } \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} R = \\frac { 1 } { N } \\sum _ { i , j } q ^ { - i j } K ^ i \\otimes K ^ j . \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} \\Sigma ( \\lambda , s ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { s } } \\sum _ { q = 1 } ^ { \\infty } \\frac { S ( 1 , n ^ 2 ; q ) } { q } \\varphi \\left ( \\lambda , \\frac { 4 \\pi n } { q } \\right ) , \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\sigma _ 0 ( t ) : = \\zeta ' ( t ) + \\frac { C ^ { m - 1 } m } { a ( m - 1 ) } ( N - 1 ) C _ 0 \\zeta ^ m ( t ) \\eta ( t ) + \\frac { \\zeta ( t ) } { m - 1 } \\frac { \\eta ' ( t ) } { \\eta ( t ) } \\ , , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\mathcal P _ { \\chi } ( \\mathfrak R _ M \\mathbf F _ { \\chi } , \\mathbf Y _ M \\mathbf G ) = C _ 3 ^ 2 M ^ { k - 1 } | \\langle ( \\mathrm { i d } \\otimes V _ M U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , ( \\mathrm { i d } \\otimes V _ M ) g \\times g \\rangle | ^ 2 , \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} x _ 1 ( t ) = a _ 1 + \\frac { 2 r } { | b - a | } ( a - b ) \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} f _ 2 \\circ u _ { 1 5 } & = g _ 1 \\circ e _ 1 \\\\ f _ 1 \\circ u _ { 2 5 } & = g _ 2 \\circ e _ 2 \\\\ ( f _ 1 - f _ 2 ) \\circ u _ { 3 5 } & = ( g _ 2 - g _ 1 ) \\circ e _ 3 \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { \\tilde { S } _ { k , n } } { \\sum \\limits _ { \\substack { I _ { k } ( n ) ^ { c } } } \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k } ; n ) } = 1 \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } V , \\xi ) = \\breve { g } ( V , \\breve { J } \\xi ) = 0 , \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} \\mbox { $ | \\eta _ r ( x , y ) | = p ^ { - m } $ i f $ r _ m \\not = 0 $ , w h i l e $ | \\eta _ r ( x , y ) | \\leq p ^ { - m - 1 } < p ^ { - m } $ i f $ r _ m = 0 $ , } \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} \\tilde { P } ( x ) = \\frac { a ' } { q ' } x ^ d + \\ldots + \\xi _ 1 x . \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} { { \\rm { e } } _ q ^ x } = \\frac { 1 } { \\left ( { 1 - \\left ( 1 - q \\right ) x } \\right ) _ q ^ \\infty } . \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} \\theta ( S ) = 2 \\theta _ 0 ( S ' ) \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} m ( D ) u ( x ) = \\frac 1 { ( 2 \\pi ) ^ { n / 2 } } \\int _ { \\R ^ n } e ^ { i x \\xi } m ( \\xi ) { \\mathcal F } u ( \\xi ) d \\xi \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} - ( k + 2 ) ( k - m + 1 ) S = G ( k , k ) , \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( 1 - \\gamma ) J _ c \\xi _ 0 - \\omega J _ b \\nu _ 0 = K r \\nu ^ 2 _ 0 , \\\\ \\mathcal L _ { + \\infty } \\nu _ 0 - \\omega J _ b \\xi _ 0 = 2 K r \\xi \\nu \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\| u _ \\mu - \\overline { u } \\| _ { L ^ p ( \\Omega , \\R ^ N ) } \\rightarrow 0 , \\ ; p < \\frac { n } { n - 1 } , \\\\ & u _ \\mu \\rightharpoondown \\overline { u } L ^ 2 ( \\Omega , \\R ^ N ) \\\\ & u _ \\mu \\rightarrow \\overline { u } \\end{aligned} \\right . \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} & \\frac { \\partial G ( \\vec z ^ * ) } { \\partial \\vec t } \\ge 0 , \\ , \\ , \\ , \\forall \\ , \\vec t \\in T \\\\ \\Leftrightarrow & \\begin{cases} \\displaystyle \\frac { \\partial G ( \\vec z ^ * ) } { \\partial z _ i } \\le 0 , \\ , \\ , \\ , \\ , i = 1 , 2 , \\ldots , m _ 0 , \\\\ \\displaystyle \\frac { \\partial G ( \\vec z ^ * ) } { \\partial z _ i } = 0 , \\ , \\ , \\ , \\ , i = m _ 0 + 1 , \\ldots , n . \\end{cases} \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} L _ { \\sigma } ^ { 2 } = \\ t h e \\ c l o s u r e \\ i n \\ [ L ^ { 2 } ( \\R ^ { d } ) ] ^ d \\ o f \\{ u \\in [ C _ { 0 } ^ { \\infty } ( \\R ^ { d } ) ] ^ d , \\ d i v u = 0 \\} \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{gather*} \\Phi _ 2 ( X , Y ) = X ^ 3 + Y ^ 3 - X ^ 2 Y ^ 2 + 1 4 8 8 \\big ( X Y ^ 2 + X ^ 2 Y \\big ) - 1 6 2 0 0 0 \\big ( X ^ 2 + Y ^ 2 \\big ) \\\\ \\hphantom { \\Phi _ 2 ( X , Y ) = } { } + 4 0 7 7 3 3 7 5 X Y + 8 7 4 8 0 0 0 0 0 0 ( X + Y ) - 1 5 7 4 6 4 0 0 0 0 0 0 0 0 0 \\end{gather*}"}
-{"id": "4946.png", "formula": "\\begin{align*} D ( \\omega \\cdot f ) = ( D \\omega ) \\cdot f + ( - 1 ) ^ { | \\omega | } \\omega \\cdot ( \\delta f ) \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} \\mathcal { R } _ x : = \\{ r \\in ( 0 , \\rho _ 0 ] : \\mu ( \\partial B _ r ( x ) ) > 0 \\} . \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} N _ 1 & = - \\left ( \\frac { \\theta _ 2 } { 2 } P _ 3 ( 1 ) \\widetilde { P _ 2 } ( 1 ) + o ( 1 ) \\right ) \\varphi ^ + ( q ) , \\\\ N _ 2 & = - \\left ( \\theta _ 2 \\int _ 0 ^ 1 P _ 2 ( u ) P _ 3 ( u ) d u + o ( 1 ) \\right ) \\varphi ^ + ( q ) . \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} V _ k ( \\lambda ) = \\begin{bmatrix} L _ k ( \\lambda ^ \\ell ) \\\\ e _ { k + 1 } ^ T \\end{bmatrix} \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} \\omega ( \\alpha ( g _ 1 ) , \\alpha ( g _ 2 ) ) = \\alpha _ A ( \\omega ( g _ 1 , g _ 2 ) ) \\quad \\mbox { f o r a l l $ g _ 1 , g _ 2 \\in G $ . } \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ n a _ n \\bar b _ n t _ n \\nu _ n = - 1 , \\\\ & \\sum _ n a _ n \\bar b _ n t _ n ^ k \\nu _ n = 0 , k = 2 , \\dots N , \\\\ & \\sum _ n a _ n \\bar b _ n t _ n ^ { N + 1 } \\nu _ n \\ne 0 . \\end{aligned} \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} \\binom { k + \\mu } { k } S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k + \\mu \\right ) = \\sum _ { j _ { 2 } = 0 } ^ { p _ { 2 } } \\sum _ { j _ { 1 } = 0 } ^ { p _ { 1 } } \\binom { p _ { 1 } } { j _ { 1 } } \\binom { p _ { 2 } } { j _ { 2 } } S _ { a _ { 1 } , 0 } ^ { a _ { 2 } , 0 , p _ { 2 } - j _ { 2 } } \\left ( p _ { 1 } - j _ { 1 } , \\mu \\right ) S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , j _ { 2 } } \\left ( j _ { 1 } , k \\right ) . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} \\int _ { x _ 1 , \\dots , x _ n = - \\infty } ^ \\infty \\ , \\prod _ { 1 \\leq j < k \\leq n } | x _ j - x _ k | ^ { 2 c } \\prod _ { j = 1 } ^ n e ^ { - x _ j ^ 2 / 2 } \\ , d x _ j = ( 2 \\pi ) ^ { n / 2 } \\prod _ { j = 1 } ^ n \\frac { \\Gamma ( 1 + j c ) } { \\Gamma ( 1 + c ) } . \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} \\theta ( \\mathbf G , \\phi _ { \\mathbf g ^ { \\sharp } } ) = \\theta ( \\mathbf G , 2 ^ { - 2 } \\omega ( t ( 2 ^ { - 1 } ) _ 2 , 1 ) \\phi _ { \\mathbf g } ) = 2 ^ { - 2 } \\tau ( t ( 2 ^ { - 1 } ) _ 2 ) \\theta ( \\mathbf G , \\phi _ { \\mathbf g } ) , \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} \\sum _ { h _ { } : i } \\gamma ( h _ { } ) = \\sum _ { h _ { } : i } \\gamma ( h _ { } ) \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} ( U _ i ^ * U _ { i + 1 } L ) & = ( U _ i ^ * ) ^ { \\otimes r _ { i + 1 } } \\otimes ( U _ { i + 1 } L ) ^ { \\otimes r _ i } \\\\ & = ( U _ i ^ * ) ^ { \\otimes r _ { i + 1 } } \\otimes ( U _ { i + 1 } ) ^ { \\otimes r _ i } \\otimes ( L ) ^ { \\otimes r _ i r _ { i + 1 } } \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} \\sum _ { i \\in [ 3 ] } e ( R [ U _ i ] ) \\le \\gamma ^ { 1 / 3 } m ^ 2 , | U _ i | = m / 3 \\pm \\gamma ^ { 1 / 7 } m \\enspace \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} L ( s , \\pi _ \\infty \\boxtimes \\chi _ \\infty , \\varrho _ { 2 n + 1 } ) = \\Gamma _ \\R ( s + \\varepsilon _ 0 ) \\prod _ { j = 1 } ^ n \\Gamma _ \\C ( s + k _ i - i ) , \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} \\| f \\| = \\left ( \\int _ { \\mathbb { C } _ I } e ^ { - | z | ^ 2 } | f _ I ( z ) | ^ 2 d \\sigma ( x , y ) \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} \\widetilde h [ f ] - h [ f ] & = \\alpha _ { v _ 0 } | f ( v _ 0 ) | ^ 2 - \\alpha _ { v _ 1 } | f ( v _ 1 ) | ^ 2 - \\alpha _ { v _ 2 } | f ( v _ 2 ) | ^ 2 = 0 , \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} \\iota _ { 1 \\leq 2 } ^ { ( 2 ) } [ 2 4 ] = ( S ^ { ( 2 ) } _ { 0 , 2 } ) ^ { 1 2 } \\iota _ { 1 \\leq 2 } ^ { ( 2 ) } ( S ^ { ( 2 ) } _ { 0 , 1 } ) ^ { - 1 2 } = \\Sigma ^ 8 \\iota _ { 1 \\leq 2 } ^ { ( 2 ) } \\Sigma ^ { - 6 } = \\Sigma ^ 2 \\iota _ { 1 \\leq 2 } ^ { ( 2 ) } \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} \\kappa _ L = \\frac { \\sum _ { H \\in H _ L } \\kappa _ H } { { \\rm c o d i m } L } , \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} T = B ^ v & \\cup \\{ u _ i : ( i , v ) \\notin \\R ^ v , 1 \\le i \\le \\ell \\} \\cup \\{ u _ i : ( i , v ) \\notin \\R ^ v , u _ i \\notin B ^ v , \\ell + 1 \\le i \\le \\ell + m \\} \\\\ & \\cup A \\cup S \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} \\mathbb { P } [ \\exists \\mathbf { x } \\in \\mathcal { F } ^ l ~ \\textrm { s . t . } ~ \\mathbf { D } \\mathbf { x } = \\mathbf { e } _ i ] \\ge \\frac { 1 } { 1 + \\frac { 1 } { \\mathbb { E } [ Y _ i ] } } . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} \\epsilon ( \\chi ) = \\frac { 1 } { q ^ { \\frac { 1 } { 2 } } } \\sum _ { h ( q ) } \\chi ( h ) e \\left ( \\frac { h } { q } \\right ) . \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} & \\left | D _ k \\right | \\leq C \\int \\limits _ { B _ { C _ 1 \\Lambda _ n } ( M _ { k , n - 2 } ) \\setminus \\widetilde { \\Omega } _ { n - 2 } } \\ , \\frac { \\omega ( \\Lambda _ { n } ) } { \\Lambda ^ { 2 } _ { n } } \\cdot \\frac { 1 } { \\Lambda _ n ( k - k _ 0 ) ^ 2 } \\ , d m _ 3 ( M ) \\leq \\\\ & \\leq C \\frac { \\omega ( 2 ^ { - n } ) } { ( k - k _ 0 ) ^ 2 } . \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\frac { \\varphi _ 2 ( J K ) \\sigma _ 2 ( J K ) } { J K \\varphi _ 1 ( J K ) \\sigma _ 1 ( J K ) } = o ( 1 ) , \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} & Q ( u ) : = \\int _ \\mathbb { T } | u | ^ 2 , \\\\ & M ( u ) : = \\int _ \\mathbb { T } \\bar { u } D u , D : = - i \\partial _ x . \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} F _ { 4 , 1 6 } ( \\tau ) = \\left ( \\begin{smallmatrix} 0 \\\\ 2 ( u + 1 / u ) \\\\ 3 ( u + 1 / u ) \\\\ ( u - 1 / u ) \\\\ 0 \\end{smallmatrix} \\right ) X Y ^ 3 + \\ldots \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} g _ W ( R ^ W ( \\sigma _ 1 , \\sigma _ 2 ) \\sigma _ 3 , \\sigma _ 4 ) = & \\sigma _ 1 ( g _ W ( \\nabla _ { \\sigma _ 2 } ^ W \\sigma _ 3 , \\sigma _ 4 ) ) - g _ W ( \\nabla _ { \\sigma _ 2 } ^ W \\sigma _ 3 , \\nabla _ { \\sigma _ 1 } ^ W \\sigma _ 4 ) \\\\ - & \\sigma _ 2 ( g _ W ( \\nabla _ { \\sigma _ 1 } ^ W \\sigma _ 3 , \\sigma _ 4 ) ) + g _ W ( \\nabla _ { \\sigma _ 1 } ^ W \\sigma _ 3 , \\nabla _ { \\sigma _ 2 } ^ W \\sigma _ 4 ) . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd R } | B _ R ^ n | \\stackrel { ? } { = } \\frac { \\bigl ( \\det [ \\chi _ { i + j + 1 } ( R ) ] _ { i , j = 0 } ^ { p } \\bigr ) ^ 2 } { ( 2 p ) ! \\ , R ^ { 2 } \\bigl ( \\det [ \\chi _ { i + j } ( R ) ] _ { i , j = 0 } ^ { p } \\bigr ) ^ 2 } . \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\overline { \\omega } : = ( \\omega _ 0 , \\ , \\omega _ { - d } ) : w _ 0 \\oplus \\Sigma ^ { - d } w _ { - d } \\rightarrow f . \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} q ( B ^ G ) \\cap N _ D B = q ( B ^ D ) \\cap N _ D B . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} \\xi ( t ) : = \\zeta ' ( t ) + \\frac { C ^ { m - 1 } m } { a ( m - 1 ) } ( N - 1 ) h \\zeta ^ m ( t ) \\eta ( t ) + \\frac { \\zeta ( t ) } { m - 1 } \\frac { \\eta ' ( t ) } { \\eta ( t ) } \\ , , \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} | a | = 1 . \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J _ { K } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) = 0 . \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} \\prod _ { 0 \\leq \\ell \\leq q , \\ell \\neq m } ( \\theta ^ { m } - \\theta ^ { \\ell } ) = \\sum _ { i = 0 } ^ { q } ( \\theta ^ { m } ) ^ { i } \\theta ^ { m ( q - i ) } = ( q + 1 ) \\theta ^ { q m } = \\theta ^ { q m } . \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} G ( - s ) = G ( s ) , G ( 0 ) = 1 , \\mathrm { a n d } G ( - N ) = \\ldots = G ( - 1 ) = 0 . \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} L _ { 1 } \\overline { x } ( t ) = \\overline { \\lambda } r ( t ) \\overline { x } ( t ) , \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} q = q _ F \\colon \\int _ C F \\to C \\colon ( c , x ) \\mapsto c \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\gamma _ 1 ^ { k - 1 } T _ u ^ { n - k } G \\varphi _ 1 \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} u = \\left ( \\begin{matrix} 0 & 1 \\cr - 1 & 0 \\end{matrix} \\right ) \\hbox { a n d } v = \\left ( \\begin{matrix} 0 & - 1 \\cr 1 & - 1 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\rho _ t + ( \\rho ( v + \\beta t ) ) _ x = 0 , \\\\ ( \\rho v ) _ t + ( \\rho ( v ( v + \\beta t ) + P ) _ x = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} m _ f ( x ) : = | f ( x ) | ^ 2 , j _ f ( x ) : = \\Im \\left [ \\overline { f } \\nabla f ( x ) \\right ] , \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} ( i + j + 1 ) ^ { N + 1 } \\mathfrak { d } _ 1 ^ { N + 2 } \\dot { \\phi } ( i + j ) = & ( i + j + 1 ) ^ { N } \\mathfrak { d } _ 1 ^ { N + 1 } \\dot { \\psi } ( i + j ) \\\\ & + ( N + 1 ) ( i + j + 1 ) ^ { N } \\mathfrak { d } _ 1 ^ { N + 1 } \\dot { \\phi } ( i + j + 1 ) , \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} \\mathcal A : = \\{ f _ i ( z ) = \\lambda _ 1 z + a _ i z ^ 2 \\} . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} \\mathbf { Q } _ { \\varepsilon } ^ { \\beta } u = \\frac { 1 } { T } \\sum _ { p \\in \\mathbb { N } } \\log \\left ( 1 + \\gamma ^ { - 1 } \\left ( T , \\beta \\right ) e ^ { \\overline { M } T \\mu _ { p } } \\right ) \\left \\langle u , \\phi _ { p } \\right \\rangle \\phi _ { p } \\quad u \\in L ^ { 2 } \\left ( \\Omega \\right ) . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} w ( t ) = \\sqrt { \\rho _ { + } \\rho _ { - } } ( u _ - - u _ + ) t , \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} g ( z , w ) = \\begin{pmatrix} 1 / 2 & 0 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} z \\\\ w \\end{pmatrix} , h ( z , w ) = \\begin{pmatrix} 1 / 2 & 1 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} z \\\\ w \\end{pmatrix} . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} d _ B ( \\pi _ 1 , \\pi _ 2 ) = \\lvert A ( \\pi _ 2 ) \\setminus A ( \\pi _ 1 ) \\rvert = \\lvert A ( \\pi _ 1 ) \\setminus A ( \\pi _ 2 ) \\rvert . \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\lim \\limits _ { A \\rightarrow 0 } \\int _ { x _ 1 ^ { A } } ^ { x _ 2 ^ { A } } \\rho _ * ^ { A } u _ * ^ { A } d x = ( \\sigma _ { 0 } ^ { B } + \\beta t ) w _ 0 ^ { B } t , \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} \\Delta \\bar w _ i = ( \\| A \\| ^ 2 - 4 H ^ 2 ) \\bar w _ i + 2 H \\langle A E _ i , \\star \\xi \\rangle - 2 g _ i \\langle A , \\nabla \\star \\xi \\rangle + \\langle E _ i , \\Delta \\star \\xi \\rangle , \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} \\varphi \\left ( e _ { i } \\right ) & = f _ { i } , & \\varphi ( f _ { i } ) & = e _ { i } , & \\varphi ( t _ i ) & = t _ i \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} { \\underline { \\mathfrak { Q } } } _ t ^ k ( x _ { t - 1 } ^ k , 1 , \\xi _ { t j } , 1 ) = \\left \\{ \\begin{array} { l } \\sup _ { \\lambda , \\mu } \\ ; \\lambda ^ T ( b _ { t j } - B _ { t j } x _ { t - 1 } ^ k ) + \\sum _ { i = 1 } ^ k \\mu _ i \\theta _ { t + 1 } ^ i \\\\ A _ { t j } ^ T \\lambda + \\sum _ { i = 1 } ^ k \\mu _ i \\beta _ { t + 1 } ^ i \\leq c _ { t j } , \\ ; \\sum _ { i = 1 } ^ k \\mu _ i = 1 , \\\\ \\mu _ i \\geq 0 , i = 1 , \\ldots , k . \\end{array} \\right . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} F _ j : = u _ { 1 j } f _ 1 + u _ { 2 j } f _ 2 + \\cdots + u _ { l j } f _ l , \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} L ( t \\mid \\hat { y } _ n , \\hat { Y } _ n ) \\begin{pmatrix} 1 \\\\ \\hat { x } _ { n - 1 } ^ { - 1 } \\end{pmatrix} \\propto \\begin{pmatrix} 1 \\\\ \\hat { x } _ { n } ^ { - 1 } \\end{pmatrix} \\ ; . \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} \\rho _ { 0 } ^ { 3 } v _ { 0 } ^ { 2 } = 4 \\mathfrak { S } _ { 0 } , \\frac { \\rho _ { 1 } } { \\rho _ { 0 } v _ { 0 } } = J _ { 2 } , \\rho _ { 0 } ( \\rho _ { 1 } ^ { 2 } - 2 h ) = J _ { 1 } \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\alpha \\ge \\alpha _ 0 ( N , m , p , h ) > 0 \\ , , \\beta = \\frac { \\alpha ( m - 1 ) } 2 \\ , , \\tau \\ge 0 , \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\nabla \\mathbf { u } ( Y _ i ( t , X ) ) \\right ) = \\left [ \\begin{array} { c } ( A _ i - X ) ^ T \\nabla ^ 2 u _ 1 \\\\ ( A _ i - X ) ^ T \\nabla ^ 2 u _ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | M _ { n + 1 } f - M _ n f \\big | \\Big \\| _ { \\ell ^ p } & \\leq \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | m _ { n + 1 } - m _ n \\big | \\Big \\| _ { \\ell ^ 1 } \\| f \\| _ { \\ell ^ p } \\\\ & \\lesssim \\frac { \\vartheta _ B ( N _ 2 ) - \\vartheta _ B ( N _ 1 ) } { \\vartheta _ N ( N _ 1 ) } \\| f \\| _ { \\ell ^ p } . \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} \\Delta = \\sum _ { i \\in I } \\kappa _ i f _ * \\widehat { H } _ i + \\sum _ { \\dim L = 1 , \\kappa _ L > 1 } \\left ( 2 - \\kappa _ L \\right ) f _ * D ( L ) . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t f ^ n _ t = - v \\partial _ x f ^ n _ t - G ( M ^ { n - 1 } ( t , x ) ) \\partial _ v f ^ n _ t + \\partial _ v ( v f ^ n _ t ) + \\sigma \\partial _ { v v } f ^ n _ t \\ , , \\\\ f ^ n _ 0 ( x , v ) = f _ 0 ( x , v ) \\ , , \\end{cases} \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\vec x ^ * = \\frac { ( \\vec v ) \\cdot \\vec z ^ * } { \\| \\vec z ^ * \\| _ p } , \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} M _ { ( n ) , s } ( t ) = & P _ n \\Big [ \\sigma ( \\cdot , X ( \\cdot ) ) \\Big ] ( s ) + \\int _ s ^ t P _ n \\Big [ U ( \\cdot , r ) \\Big ] ( s ) d r + \\int _ s ^ t P _ n \\Big [ V ( \\cdot , r ) \\Big ] ( s ) d W ( r ) \\\\ & + \\int _ s ^ t \\nabla _ x b ( r , X ( r ) ) M _ { ( n ) , s } ( r ) d r + \\int _ s ^ t \\nabla _ x \\sigma ( r , X ( r ) ) M _ { ( n ) , s } ( r ) d W ( r ) . \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\Delta _ { a } ( x ) & = \\frac { x ^ { \\frac 1 4 + \\frac { a } { 2 } } } { \\pi \\sqrt { 2 } } \\sum _ { n \\leq N } \\frac { \\sigma _ { a } ( n ) } { n ^ { \\frac { 3 } { 4 } + \\frac { a } { 2 } } } \\cos \\left ( 4 \\pi \\sqrt { n x } - \\frac { \\pi } { 4 } \\right ) + O \\left ( x ^ { \\frac 1 2 + \\varepsilon } N ^ { - \\frac 1 2 } \\right ) \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} ( p \\cdot C _ { 1 / L } ( \\nu ) ) \\cap \\Gamma = \\{ p \\} p \\in \\Gamma . \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\partial _ t \\Gamma _ { \\omega _ t } ^ { n - 2 , \\ , n } = - \\bar \\partial _ t \\omega _ t ^ { n - 1 } \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} \\mathfrak { d } _ 2 ^ { m + 1 } \\dot { \\phi } ( n ) & = \\mathfrak { d } _ 2 ^ { m } \\mathfrak { d } _ 2 \\dot { \\phi } ( n ) \\\\ & = ( - 1 ) ^ m \\int _ 0 ^ 2 \\cdots \\int _ 0 ^ 2 g ^ { ( m ) } ( n + t _ 1 + \\cdots + t _ m ) \\ , d t _ 1 \\cdots d t _ m , \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\Phi _ { m , n } ( t ) : = \\begin{cases} \\displaystyle \\sum _ { j = 1 } ^ { m } e _ j \\otimes \\Phi \\left ( \\tfrac { t ^ { n } _ { k } + t ^ { n } _ { k + 1 } } { 2 } \\right ) e _ j , & t \\in ( t _ k ^ n , t _ { k + 1 } ^ n ) , \\ , k = 0 , \\ldots , N _ n - 1 , \\\\ \\displaystyle \\sum _ { j = 1 } ^ { m } e _ j \\otimes \\Phi ( t ^ { n } _ k ) e _ j , & t = t _ k ^ n , \\ , k = 0 , \\ldots , N _ n , \\\\ \\end{cases} \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} t _ { v _ 1 } ^ { \\epsilon _ 1 } t _ { v _ 2 } ^ { \\epsilon _ 2 } \\cdots t _ { v _ n } ^ { \\epsilon _ n } [ \\mathcal { X } _ 1 , \\mathcal { Y } _ 1 ] [ \\mathcal { X } _ 2 , \\mathcal { Y } _ 2 ] \\cdots [ \\mathcal { X } _ h , \\mathcal { Y } _ h ] = \\mathrm { i d } . \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} c ( \\xi ) = c ( \\mathfrak d _ { \\xi } ) \\sum _ { \\substack { d \\mid \\mathfrak f _ { \\xi } , \\\\ d > 0 } } \\mu ( d ) \\chi _ { - \\xi } ( d ) \\chi ( d ) d ^ { k - 1 } a _ { \\chi } ( \\mathfrak f _ { \\xi } / d ) . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} w ( g _ { 1 } , \\cdots , g _ { r } ) = g _ { i _ { 1 } } ^ { n _ { 1 } } g _ { i _ { 2 } } ^ { n _ { 2 } } \\cdots g _ { i _ { \\ell } } ^ { n _ { \\ell } } , \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} h ( \\vartheta ) = 0 . \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} \\int _ { \\mathcal R _ { 2 t } } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h = p ^ { 3 t } \\mathrm { v o l } ( \\mathcal R _ { 2 t } ) = p ^ t ( 1 - p ^ { - 1 } ) ^ 2 . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} ( t _ 1 , \\dots , t _ n ) . \\theta _ q = t _ 1 ^ { M _ 1 ^ t ( q ) } \\cdots t _ n ^ { M _ n ^ t ( q ) } \\theta _ q . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( 1 - \\gamma _ 1 ) & + 1 5 & ( 1 - \\gamma _ 2 ) & + 8 4 & \\epsilon & < 1 , \\\\ 3 & ( 1 - \\gamma _ 1 ) & + 1 2 & ( 1 - \\gamma _ 2 ) & + 6 0 & \\epsilon & < 2 \\end{aligned} \\right . \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\int \\limits _ { \\left ( B _ { 2 ^ { - n + k + 3 } | \\Lambda | } ( M _ 0 ) \\setminus B _ { 2 ^ { - n + k + 2 } | \\Lambda | } ( M _ 0 ) \\right ) \\setminus \\widetilde { \\Omega } _ { n - 2 } } \\ , \\frac { | \\Phi _ n ( M ) | } { \\rho ^ 2 _ { M _ 0 } ( M ) } \\ , d m _ 3 ( M ) \\leq C \\cdot 2 ^ { n } \\cdot \\omega ( 2 ^ { - n } ) . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} C _ { k + 1 } \\begin{bmatrix} Q _ 1 & Q _ 2 V _ 2 \\end{bmatrix} = 0 \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} \\| u \\| _ { L _ x ^ { m ' } ( B ( r ) ) } \\leq C \\| u \\| ^ { \\theta } _ { L _ x ^ 2 ( B ( r ) ) } \\| u \\| _ { L _ x ^ 4 ( B ( r ) ) } ^ { 1 - \\theta } , \\ \\ \\theta = 4 / d . \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} v \\partial _ x f + G ( M ( x ) ) \\partial _ v f = \\partial _ v ( \\sigma \\partial _ v f + v f ) \\ , , M ( x ) : = \\frac { \\int _ { \\mathbb { T } _ L } \\ ! \\mathrm { d } y \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } w \\ , f ( y , w ) \\ , \\varphi ( x - y ) \\ , w } { \\int _ { \\mathbb { T } _ L } \\ ! \\mathrm { d } y \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } w \\ , f ( y , w ) \\ , \\varphi ( x - y ) } \\ , . \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\frac { ( 1 + z ) ( - z q ; q ) _ { i - 1 } } { ( q ; q ) _ { i - 1 } } + \\frac { q ^ i ( - z ; q ) _ i } { ( q ; q ) _ i } = \\frac { ( - z ; q ) _ i } { ( q ; q ) _ i } . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\lambda _ f ^ { \\ast } ( p ) = \\lambda _ f ( p ) - \\lambda _ { g } ( p ) / p \\hbox { a n d } \\lambda ^ \\ast _ { g } ( p ) = \\lambda _ { g } ( p ) - \\lambda _ f ( p ) / p . \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\lambda _ { G / H } \\circ q ( f ) \\xi ( y H ) = \\int _ G f ( g ) \\xi ( g ^ { - 1 } y H ) \\ , d g & = \\int _ { G / H } \\int _ H f ( x h ) \\xi ( h ^ { - 1 } x ^ { - 1 } y H ) \\ , d h d x \\\\ & = \\int _ { G / H } \\int _ H f ( x h ) \\xi ( x ^ { - 1 } y H ) \\ , d h d x \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} A ( x , \\tilde { x } ) = \\alpha \\sum _ { k , l ; k > l } \\{ ( x _ k - x _ l ) - ( \\tilde { x } _ k - \\tilde { x } _ l ) \\} \\left \\{ \\frac { 1 } { x _ k - x _ l } - \\frac { 1 } { \\tilde { x } _ k - \\tilde { x } _ l } \\right \\} \\leq 0 . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} F ( B _ k ; p ) = C _ k ( \\beta ^ { ( 1 ) } _ k - 2 ) \\cdots C _ k ( 1 ) C _ k ( 0 ) F ( B _ { k - 1 } ; p ) . \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} L = q ^ \\lambda \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} b _ s ^ { } & = ( - 1 ) ^ { 2 n + 1 - m - s } q ( j ^ * \\sigma _ { 2 n + 1 - m , 2 n - m , \\dots , 2 n + 3 - 2 m , 2 n + 2 - 2 m - p } ^ + ) ^ \\vee \\\\ & = ( - 1 ) ^ { 2 n + 1 - m - s } q \\sigma _ { 2 n - m , 2 n - 1 - m , \\dots , 2 n + 2 - 2 m , 2 n + 1 - 2 m - p } ^ \\vee , \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\lim _ { t \\to s ^ + } \\phi _ y ( t , x , s ) = q ( s , x , y ) \\ , . \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} f _ k ( x ) : = \\begin{cases} - \\frac { 1 } { 2 ^ k } \\left ( 1 - \\frac { d _ X ( x , \\omega _ j ) } { \\varepsilon _ k } \\right ) & \\textrm { i f } x \\in I _ { j , k } \\textrm { f o r } j = 1 , \\dots , a _ k ; \\\\ + \\frac { 1 } { 2 ^ k } \\left ( 1 - \\frac { d _ X ( x , \\omega _ j ) } { \\varepsilon _ k } \\right ) & \\textrm { i f } x \\in I _ { j , k } \\textrm { f o r } j = a _ k + 1 , \\dots , 2 a _ k ; \\\\ 0 & \\textrm { e l s e w h e r e } . \\end{cases} \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\sigma ( X ) = - 1 \\cdot 0 - 7 \\cdot ( g - 1 ) n + 1 \\cdot 1 1 ( g - 1 ) n = 4 ( g - 1 ) n . \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\mu = e _ { \\infty } \\mathcal { H } ^ 3 + \\nu , \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} \\frac { S ( z ) G ( z ) } { A ( z ) } = \\sum _ n \\frac { S ( t _ n ) G ( t _ n ) } { A ' ( t _ n ) ( z - t _ n ) } . \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} \\Gamma _ + = \\partial \\Omega _ + { \\Gamma } _ - = \\partial \\Omega _ - \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} r _ i ( x ) = v _ { m _ i i } ( x ) ^ { 2 \\ell _ i / m _ i - 1 } \\exp \\int \\frac { \\Re ( v _ { ( m _ i - 1 ) i } ( x ) ) } { ( m _ i / 2 ) v _ { m _ i i } ( x ) } d x , \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} \\| \\nabla \\phi ( t , y ( t ) ) \\| & \\leq \\sum \\limits _ { i = 0 } ^ { \\infty } 2 \\eta _ i ( t ) \\| y ( t ) - x _ i ( t ) \\| \\leq 4 \\frac { \\epsilon ( t ) } { \\lambda ( t ) } . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\mu ^ { ( J ) } \\coloneqq u _ 1 \\quad m ^ { ( J ) } \\coloneqq 2 \\left ( 2 ^ { J - 1 } - u _ 1 \\right ) = 2 m . \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} \\tilde y ( s ) - \\tilde y ( t ) = \\int _ t ^ s y ' ( r ) \\ , \\d r \\qquad \\forall t , s \\in [ 0 , 1 ] , \\ t < s . \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} D F = \\mu D \\bar F , \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ 4 \\widehat { J } _ i = I + \\mathcal { O } \\left ( | \\lambda - S _ \\infty | ^ 2 \\right ) . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} \\langle ( a b ^ i ) ^ p \\rangle = \\langle ( a b ^ j ) ^ p \\rangle ^ g \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } F _ 0 ( t ) = \\int _ { \\Omega } \\left | u ( x , t ) \\right | ^ p \\phi _ 0 ( x ) \\ , d x \\geq \\frac { | F _ 0 ( t ) | ^ p } { \\displaystyle \\left ( \\pi C ( t + R ) ^ 2 \\ln ( t + R ) \\right ) ^ { p - 1 } } = k [ \\ln ( t + R ) ] ^ { - ( p - 1 ) } ( t + R ) ^ { - 2 ( p - 1 ) } | F _ 0 ( t ) | ^ p \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} P _ l ( x ) = \\sum _ { \\abs { \\epsilon } < q + 2 } a _ \\epsilon x ^ \\epsilon . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} P _ 1 , R _ 1 ^ T & \\in M _ { l _ 2 , l _ 1 } ( q ) , \\\\ P _ 2 , R _ 2 ^ T & \\in M _ { n - 2 k - 2 l _ 2 , k - 2 l _ 1 } ( q ) , \\\\ R _ 2 P _ 2 & = 0 . \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} \\Omega \\triangleq \\sum _ { i = 1 } ^ M n _ i \\ , a _ i , \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} D = \\frac { \\partial } { \\partial x _ 0 } + e _ 1 \\frac { \\partial } { \\partial x _ 1 } + e _ 2 \\frac { \\partial } { \\partial x _ 2 } \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\zeta ( t ) : = ( T - t ) ^ { - \\alpha } [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } } \\ , , \\eta ( t ) : = [ - \\log ( T - t ) ] ^ { - \\beta } \\ , \\ , \\ ; \\ ; \\ ; t \\in [ 0 , T ) \\ , , \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} \\| u \\| _ { S _ 2 } = \\gamma \\sqrt { r m / n } , ( u ) \\le r , \\| u - v \\| _ { S _ 2 } \\ge ( \\gamma / 8 ) \\sqrt { r m / n } \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} C _ { \\Phi , d } = \\left \\{ | k | \\prod _ { i = 1 } ^ s | h _ i | ^ { e _ i } \\ , \\middle | \\ , k \\in K ( \\Phi ) , \\ ; e _ 1 \\dotsc e _ s \\in { \\mathbb Z } , \\ ; \\sum _ { r = 1 } ^ s \\lvert e _ r \\rvert < d \\right \\} \\subseteq \\Q ^ \\star _ { - 1 , 1 } \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\dbinom { n + k } { r } \\right ) = \\frac { z ^ { k + 1 } } { \\left ( z - 1 \\right ) ^ { r + 1 } } . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\log ( s ) = \\int \\limits _ 0 ^ \\infty \\bigl ( e ^ { - t } - e ^ { - t s } \\bigr ) \\frac { d t } { t } \\ , \\ , \\ , \\ , , \\ , \\ , \\Re ( s ) > 0 . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} & \\Psi ( \\theta ) = \\Phi ( \\theta ) ^ * = \\left \\{ \\begin{matrix} \\cos \\theta \\cdot I - i \\sin \\theta \\cdot F , & 0 \\leq \\theta \\leq \\pi , \\\\ & & \\\\ ( \\cos \\theta - i \\sin \\theta ) \\cdot I , & \\pi \\leq \\theta \\leq 2 \\pi . \\end{matrix} \\right . \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\int _ { B _ { \\lambda } \\setminus B _ { \\lambda ' } } K ( x ) { \\ : \\rm d } x = 0 , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} \\frac { c _ i ^ { n + 1 } - c _ i ^ n } { \\Delta T } = D \\frac { c ^ n _ { i + 1 } - 2 c ^ n _ i + c ^ n _ { i - 1 } } { ( \\Delta X ) ^ 2 } , \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\phi _ M & = P _ 0 \\cdot S _ + \\cdot S _ - \\\\ \\phi _ { M _ u } & = P _ 0 \\cdot ( S _ + R _ - + R _ + S _ - ) \\\\ \\phi _ { M _ { u v } } & = P _ 0 \\cdot R _ + \\cdot R _ - . \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} C _ { \\alpha , 1 } \\varepsilon ^ 2 \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } - \\sum _ { k = 1 } ^ { D _ \\varepsilon } \\theta _ k ^ 2 \\geq \\left ( 2 \\left ( 1 + \\sqrt { 2 C _ { \\min } ( \\alpha , \\beta ) } \\right ) \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } \\right ) \\varepsilon ^ 2 \\sqrt { - \\log ( 1 - \\beta ) } + 2 b _ { D _ { \\varepsilon } } ^ { - 2 } \\varepsilon ^ 2 \\sqrt { - \\log ( 1 - \\beta ) } . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\langle P ( x ) , Q ( y ) \\rangle = \\sum _ { j _ 1 , . . . , j _ { N - 1 } = 0 } ^ { \\infty } \\sum _ { \\substack { I \\subset [ N ] \\\\ N \\notin I } } ( - 1 ) ^ { | I | } \\tilde { \\phi } \\left ( \\vec { d } ( x , y ) + 2 ( j _ 1 , . . . , j _ { N - 1 } , 0 ) + 2 \\chi ^ I \\right ) . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dot { x } & = & f ( t , x , y ) \\\\ \\dot { y } & = & g ( t , y ) . \\end{array} \\right . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m _ 1 } \\eta _ j = \\sum _ { i = 1 } ^ { m _ 2 } \\epsilon _ i . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) = \\frac { 4 L ( 1 , \\pi , \\mathrm { a d } ) L ( 1 , \\tau , \\mathrm { a d } ) } { \\langle \\mathbf h , \\mathbf h \\rangle \\langle \\mathbf g , \\mathbf g \\rangle \\langle \\pmb { \\phi } , \\pmb { \\phi } \\rangle } \\left ( \\prod _ v \\mathcal I _ v ^ { \\sharp } ( \\mathbf h , \\mathbf g , \\pmb { \\phi } ) ^ { - 1 } \\right ) \\mathcal Q ( \\mathbf h , \\mathbf g , \\pmb { \\phi } ) . \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} K _ \\varepsilon ( t ) = h & + \\int _ 0 ^ t \\Big [ \\int _ 0 ^ 1 \\nabla _ x b \\big ( s , \\omega , X _ { x } ( s ) + \\xi [ X _ { x + \\varepsilon h } ( s ) - X _ { x } ( s ) ] \\big ) d \\xi \\Big ] K _ \\varepsilon ( s ) d s \\\\ & + \\int _ 0 ^ t \\Big [ \\int _ 0 ^ 1 \\nabla _ x \\sigma \\big ( s , \\omega , X _ { x } ( s ) + \\xi [ X _ { x + \\varepsilon h } ( s ) - X _ { x } ( s ) ] \\big ) d \\xi \\Big ] K _ \\varepsilon ( s ) d W ( s ) , \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\varphi ( \\lambda , x ) = \\frac { \\sinh \\beta } { \\pi } x ^ { \\lambda } \\exp ( i x \\cosh \\beta ) , \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} \\| A \\| = \\lambda _ 1 ( A ^ T A ) ^ { 1 / 2 } = \\underset { \\| x \\| = 1 } { \\sup } \\ , \\| A x \\| , \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} \\nabla _ { a } ^ { - \\nu } ( t - a + 1 ) ^ { \\overline { \\upsilon } } = \\frac { \\Gamma ( \\upsilon + 1 ) } { \\Gamma ( \\nu + \\upsilon + 1 ) } ( t - a + 1 ) ^ { \\overline { \\nu + \\upsilon } } \\quad ( \\forall { t } \\in \\mathbf { N } _ { a } ) , \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\left \\vert \\hat { \\varphi } ( s , \\omega _ { 0 } ) \\right \\vert = o ( \\left \\vert x ^ { \\prime } - \\bar { X } ^ { t , x ; \\bar { u } } ( s , \\omega _ { 0 } ) \\right \\vert ^ { 2 } ) , s \\in \\lbrack t , T ] . \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} P _ { n , r } : = | \\mathcal { P } _ { n , r } | = \\frac { n ! } { r ! } n ( n - 1 ) \\cdots ( n - r + 1 ) \\quad ( n \\ge r ) . \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} f ( x _ t ) = f ( x _ s ) + \\sum _ { \\varphi \\in \\mathcal { F } ; \\vert \\varphi \\vert \\leq k _ 0 } H ^ \\varphi _ { t s } \\ , \\big ( V ( \\varphi ^ * ) f \\big ) ( x _ s ) + O \\big ( | t - s | ^ { k _ 0 + 1 } \\big ) . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} L ( S _ \\epsilon ) = L ( S _ \\epsilon , x _ 0 ) = \\epsilon . \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K ( n ) = n ^ r , \\\\ \\ \\ \\ \\ \\ \\ \\forall r \\in ( 0 , 1 ) , \\displaystyle \\lim _ { \\substack { n \\rightarrow + \\infty } } \\displaystyle \\frac { K ( n ) } { n } = 0 \\} \\\\ \\end{array} \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} g ^ { \\ast } \\eta \\wedge ( g h ) ^ { \\ast } \\eta = & ( \\eta - d f _ { g } ) \\wedge h ^ { \\ast } ( \\eta - d f _ { g } ) \\\\ = & \\eta \\wedge h ^ { \\ast } \\eta - d f _ { g } \\wedge h ^ { \\ast } \\eta - \\eta \\wedge h ^ { \\ast } d f _ { g } + d f _ { g } \\wedge h ^ { \\ast } d f _ { g } \\\\ = & \\eta \\wedge h ^ { \\ast } \\eta - d ( f _ { g } \\wedge h ^ { \\ast } \\eta ) + f _ { g } \\omega + d ( \\eta \\wedge h ^ { \\ast } f _ { g } ) - h ^ { \\ast } f _ { g } \\omega + d ( f _ { g } \\wedge h ^ { \\ast } d f _ { g } ) . \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} A & : = \\begin{pmatrix} 3 & - 1 & 1 & 1 & - 1 \\\\ - 1 & 2 & 0 & 0 & 2 \\\\ 1 & 0 & 2 & 1 & 1 \\\\ 1 & 0 & 1 & 3 & 0 \\\\ - 1 & 2 & 1 & 0 & 4 \\end{pmatrix} , B : = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\\\ - 1 & 1 \\\\ 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\\\ C & : = \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} \\varphi \\circ A _ k ( x ) = d A _ k ( x _ 0 ) \\cdot \\varphi ( x ) \\forall k \\in K , x \\in U ) . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( N ) } ( A _ { j } ^ { ( N ) } ) = P ( A _ { j } ^ { ( N ) } ) , \\quad \\alpha _ { n } ^ { ( N ) } ( A _ { j } ^ { ( N ) } ) = 0 , \\quad 1 \\leqslant j \\leqslant m _ { N } , \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} f ^ { L } ( \\vec x ) = \\sum _ { i = 0 } ^ { n - 1 } ( | x _ { \\sigma ( i + 1 ) } | - | x _ { \\sigma ( i ) } | ) f ( V _ { \\sigma ( i ) } ^ + , V _ { \\sigma ( i ) } ^ - ) . \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\dim ( C _ { G L _ \\frac { k a } { 2 } } { ( j _ { \\frac { k a } { 4 } , \\frac { k a } { 2 } } ) } ) = \\frac { 1 } { 8 } k ^ 2 a ^ 2 , \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\omega ( \\nu _ n ) = \\nu _ n \\cdot \\phi _ n ^ { z ( \\omega ) } . \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} h ^ 0 ( S , ( H - E - E _ 1 ) = h ^ 0 ( S , H - E _ 1 ) - 1 = 1 \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\dot { \\varphi } = J _ { \\varphi } v , \\dot { \\theta } = J _ { \\theta } v , \\ \\ \\ \\ J _ { \\varphi } ^ { 2 } + \\sin ^ { 2 } ( \\varphi ) J _ { \\theta } ^ { 2 } = 1 \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} & \\frac { d u ( t ) } { d t } + A u ( t ) + B ( u ( t ) , u ( t ) ) = f ( t ) V ' ( 0 , \\infty ) , \\\\ & u ( 0 ) = u ^ 0 \\in H . \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\int _ r ^ { \\infty } \\frac { V ( s ) } { s \\varphi ( s ) } d s & = \\int _ r ^ \\infty V ( s ) d ( - \\mathcal { P } _ 1 ) ( s ) \\\\ & = V ( r ) \\mathcal { P } _ 1 ( r ) - \\lim _ { s \\rightarrow \\infty } V ( s ) \\mathcal { P } _ 1 ( s ) + \\int _ r ^ \\infty V ' ( s ) \\mathcal { P } _ 1 ( s ) d s \\\\ & \\le c _ 5 \\left ( \\frac { 1 } { V ( r ) } - \\lim _ { s \\rightarrow \\infty } \\frac { 1 } { V ( s ) } + \\int _ r ^ \\infty \\frac { V ' ( s ) } { V ( s ) ^ 2 } d s \\right ) = \\frac { 2 c _ 5 } { V ( r ) } , \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} c _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x _ { \\lambda _ 1 } ( a ) + c _ 2 ~ ^ { A B R } D _ b ^ \\alpha x _ { \\lambda _ 1 } ( a ) = 0 , \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } \\varpi _ { \\jmath } - \\varpi _ { \\jmath } & = - \\sum _ { \\jmath ' \\in { { } ^ { \\jmath } B } } \\alpha _ { \\jmath ' } , & ( \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } ) ^ { - 1 } \\varpi _ { \\jmath } - \\varpi _ { \\jmath } & = - \\sum _ { \\jmath ' \\in B ^ { \\jmath } } \\alpha _ { \\jmath ' } . \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} b \\nabla G _ { \\beta } ^ { a } u = G _ { \\beta } ^ { a } b \\nabla u + G _ { \\beta } ^ { a } R G _ { \\beta } ^ { a } u \\quad \\forall u \\in S . \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} f ( x ) = \\frac { x ^ { 2 \\theta } } { \\left ( ( 1 - x ) ^ 2 + x \\delta \\right ) ^ { 3 / 4 } } , \\delta = 2 \\sin ^ 2 ( \\gamma / 2 ) \\asymp \\frac { 1 } { 8 T ^ 2 } . \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} & \\left | A _ k \\right | \\leq C \\int \\limits _ { \\beta _ { k n } } \\ , \\frac { \\omega ( d ( M ) ) } { d ^ { 2 } ( M ) } \\cdot \\frac { 1 } { \\Lambda _ n { | k - k _ 0 | } ^ 2 } \\ , d m _ 3 ( M ) \\leq \\\\ & \\leq C \\omega ( 2 ^ { - n } ) \\cdot 2 ^ { - n } \\cdot \\frac { 1 } { \\Lambda _ n { | k - k _ 0 | } ^ 2 } \\leq C \\frac { \\omega ( 2 ^ { - n } ) } { ( k - k _ 0 ) ^ 2 } , \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} \\rho _ { 1 3 4 } ( S _ 1 ) ^ { 2 ( k + 1 ) } & = \\rho _ { 1 3 4 } ( S _ 1 ) ^ { 2 k } \\rho _ { 1 3 4 } ( S _ 1 ) ^ { 2 } \\\\ & = ( S _ 2 ( S _ 1 S _ 2 ) ^ k S _ 2 ^ * + S _ 1 ( S _ 2 S _ 1 ) ^ k S _ 1 ^ * ) ( S _ 2 ( S _ 1 S _ 2 ) S _ 2 ^ * + S _ 1 ( S _ 2 S _ 1 ) S _ 1 ^ * ) \\\\ & = S _ 2 ( S _ 1 S _ 2 ) ^ { + 1 } k S _ 2 ^ * + S _ 1 ( S _ 2 S _ 1 ) ^ { k + 1 } S _ 1 ^ * \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\mathbf { u } ( Y _ i ( t , X ) ) = \\left [ \\begin{array} { c } ( A _ i - X ) ^ T \\nabla ^ 2 u _ 1 ~ ( A _ i - X ) \\\\ ( A _ i - X ) ^ T \\nabla ^ 2 u _ 2 ~ ( A _ i - X ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\| f ( \\tau ) \\| _ { H ^ k ( \\Omega ) } : = \\left ( \\sum _ { \\alpha _ 1 + \\alpha _ 2 \\leq k } \\| \\partial ^ { \\alpha _ 1 } _ \\xi \\partial ^ { \\alpha _ 2 } _ \\eta f ( \\tau , \\cdot ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } \\right ) ^ \\frac { 1 } { 2 } < \\infty . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} K ( \\lambda ) = \\begin{bmatrix} 1 & - \\lambda & \\lambda ^ 2 \\end{bmatrix} \\mbox { a n d } N ( \\lambda ) = \\begin{bmatrix} \\lambda & 1 & 0 \\\\ 0 & \\lambda & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\rho _ t + ( \\rho u ) _ x = 0 , \\\\ ( \\rho u ) _ t + ( \\rho u ^ 2 + P ) _ x = \\beta \\rho , \\end{array} \\right . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} \\bigg | \\int \\frac { G _ 1 ( z ) H ( z ) \\overline { F _ 1 ( z ) } } { z - w } d \\nu ( z ) \\bigg | = o \\Big ( \\frac { 1 } { | w | } \\Big ) , \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ m 2 x _ + ^ T ( s ) \\Sigma ^ { - 1 } N _ { k } x _ + ( s ) u _ k ( s ) \\leq x _ + ^ T ( s ) \\Sigma ^ { - 1 } x _ + ( s ) \\left \\| u ( s ) \\right \\| _ { 2 } ^ 2 + \\sum _ { k = 1 } ^ m x _ + ^ T ( s ) N _ k ^ T \\Sigma ^ { - 1 } N _ { k } x _ + ( s ) , \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} B _ \\lambda ( s ) = \\alpha _ n \\int \\limits _ { \\mathfrak { a } ^ + } \\bigg ( \\prod _ { \\lambda \\in \\Sigma ^ + } \\sinh ( \\lambda ( H ) ) \\bigg ) f _ k ( Q _ n \\cdot ( \\exp ( H ) , 1 ) , s ) \\langle \\pi _ k ( \\exp ( H ) ) w _ k , w _ k \\rangle \\ , d H . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} [ y ^ U ] _ \\beta ^ T \\alpha = \\alpha y ^ T . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } & \\log \\P _ x ( X ( t ) = x \\star u ^ { \\star n } , \\ ; ) = \\\\ & \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\P _ { x \\star u ^ { \\star n } } ( X ( t ) = x , \\ ; ) ~ = ~ 0 , \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} \\Gamma _ { ( S , \\xi ) } ( t ) = \\phi _ { \\exp ( t \\zeta ) } ( S ) \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} \\Phi ( x , y ) : = L _ h ^ { - 1 } ( r ( x , h + y ) ) . \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} \\theta ( S ' ) = - 2 a + ( 1 - 2 b ) + 2 b = 1 - 2 a < 0 \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} T _ j ( K ) = \\sum _ { X / j < k \\leq K } e ^ { 2 \\pi i F ( j k ) } . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} w ( t ) u _ \\delta ( t ) = ( \\sigma _ { 0 } + \\beta t ) \\sqrt { \\rho _ { + } \\rho _ { - } } ( u _ - - u _ + ) t , \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} [ \\exists x \\phi ( x ) ] ^ { \\beta } & = [ \\phi ( x ) ] ^ { \\beta [ \\frac { n } { x } ] } \\\\ [ \\exists X \\psi ( X ) ] ^ { \\beta } & = [ \\psi ( X ) ] ^ { \\beta [ \\frac { N | A } { X } ] } \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} F _ 0 = \\sum \\limits _ { j = 1 } ^ { \\infty } \\alpha _ j e _ j . \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} w \\bullet \\lambda = w ( \\lambda - \\tau ) + \\tau , \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} \\omega & = \\frac { 2 w _ { 0 } } { \\rho _ { 0 } v _ { 0 } } - \\frac { 1 } { 2 } K _ { 0 } \\rho _ { 0 } ^ { 2 } v _ { 0 } \\\\ h & = \\frac { 1 } { 2 } \\rho _ { 1 } ^ { 2 } + \\frac { 1 } { 8 } \\rho _ { 0 } ^ { 2 } v _ { 0 } ^ { 2 } + 2 \\frac { w _ { 0 } ^ { 2 } } { \\rho _ { 0 } ^ { 4 } v _ { 0 } ^ { 2 } } - \\frac { w _ { 0 } K _ { 0 } + u _ { 0 } } { \\rho _ { 0 } } + \\frac { 1 } { 8 } K _ { 0 } ^ { 2 } \\rho _ { 0 } ^ { 2 } v _ { 0 } ^ { 2 } \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\exp \\left ( - \\int _ E h ( x ) \\Phi ( d x ) \\right ) \\right ] = \\exp \\left ( - \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\mathrm { T r } \\Big ( \\mathbf { K } \\left [ 1 - e ^ { - h } \\right ] ^ n \\Big ) \\right ) , \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\theta ( S ' ) = 1 - 2 b < 0 \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\binom { a n + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\alpha _ { s } n + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } \\right ) = \\frac { z } { \\left ( z - 1 \\right ) ^ { r p + \\sigma + 1 } \\left ( r ! \\right ) ^ { p } \\prod _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\right ) ^ { p _ { s } } } \\sum _ { k = 0 } ^ { r p + \\sigma } k ! S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) \\left ( z - 1 \\right ) ^ { r p + \\sigma - k } . \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} V _ { t + 1 } : = \\begin{cases} V _ t \\cup \\{ x \\} , & S _ x = 1 , \\\\ V _ t , & . \\end{cases} \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} \\check { \\varphi } ( l ) = \\int _ 0 ^ { \\infty } J _ l ( y ) \\varphi ( y ) \\frac { d y } { y } . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} Q ( \\lambda ) = \\varphi ( \\lambda ) \\left [ \\frac 1 \\lambda - \\frac 1 { \\lambda ^ 3 } + \\frac 3 { \\lambda ^ 5 } - \\frac { 1 5 } { \\lambda ^ 7 } \\right ] + \\int _ \\lambda ^ { + \\infty } \\frac { 1 0 5 \\varphi ( u ) } { u ^ 8 } d u . \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} U ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) u _ 0 = Q ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) u _ 0 . \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\lambda _ { \\imath , \\jmath } = ( \\varpi _ { \\imath } , \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } \\varpi _ { \\jmath } - \\varpi _ { \\jmath } ) - ( \\varpi _ { \\imath } , ( \\phi _ { \\mathcal { Q } } ^ { \\mathrm { t w } } ) ^ { - 1 } \\varpi _ { \\jmath } - \\varpi _ { \\jmath } ) . \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} \\prod _ { l = 0 } ^ { | u | } g ' _ { u _ l } ( g _ { u , l } ( b _ j ) ) . \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} \\mathcal { A } ( t , x ) = x \\mathcal { C } ( 1 - x + t x , x ) , \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\ell _ m ( \\hat { T } ) = \\frac { 1 } { \\hat { C } ( \\hat { T } ) + 1 } . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} F ( \\tau , z , \\tau ' ) = \\sum _ { \\substack { n , r , m \\in \\Z , \\\\ 4 n m - r ^ 2 > 0 } } A _ F ( n , r , m ) e ^ { 2 \\pi \\sqrt { - 1 } ( n \\tau + r z + m \\tau ' ) } . \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} A _ { 2 , n } ^ { j , \\ell } ( x ) = \\frac { ( c - 1 ) B _ { 1 , n - 1 } ^ { j , \\ell } ( x ) } { ( c + 1 ) ( n - 1 ) + \\beta c } , B _ { 2 , n } ^ { j , \\ell } ( x ) = A _ { 1 , n - 1 } ^ { j , \\ell } ( x ) + A _ { 2 , n } ^ { j , \\ell } ( x ) ( 1 - x ) . \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{gather*} { \\bf E } \\bigl [ M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] = \\frac { G ( 2 + \\lambda _ 1 ) } { G ( 2 - q + \\lambda _ 1 ) } \\frac { G ( 2 + \\lambda _ 2 ) } { G ( 2 - q + \\lambda _ 2 ) } \\frac { G ( 1 ) } { G ( - q + 1 ) } \\frac { G ( 4 - 2 q + \\lambda _ 1 + \\lambda _ 2 ) } { G ( 4 - q + \\lambda _ 1 + \\lambda _ 2 } . \\end{gather*}"}
-{"id": "4525.png", "formula": "\\begin{align*} & H _ 1 : = | { \\bf D ^ 1 } | \\cos ( \\theta _ 1 ) = { \\bf D ^ 1 } \\cdot E _ 1 , H _ 2 : = { \\bf D ^ 1 } | \\sin ( \\theta _ 1 ) = { \\bf D ^ 1 } \\cdot E _ 2 , \\\\ & L _ 1 : = | { \\bf D ^ 2 } | \\cos ( \\theta _ 2 ) = { \\bf D ^ 2 } \\cdot E _ 1 , \\quad \\ L _ 2 : = { \\bf D ^ 2 } | \\sin ( \\theta _ 2 ) = { \\bf D ^ 2 } \\cdot E _ 2 , \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} A ' \\nabla _ \\lambda B ' - A ' ! _ \\lambda B ' = \\lambda ( 1 - \\lambda ) ( A - B ) ( B ' \\nabla _ \\lambda A ' ) ^ { - 1 } ( A - B ) . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} \\eta _ { M , N } ( a _ i \\ , | \\ , a , b ) = \\exp \\bigl ( - ( \\mathcal { S } _ N L _ { M - 1 } ) ( 0 \\ , | \\ , \\hat { a } _ i , b ) \\bigr ) . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} \\begin{aligned} & Q _ { i , j } ^ t = F _ { i , j } ^ t + F _ { i - 1 , j } ^ { t + 1 } - F _ { i - 1 , j } ^ t - F _ { i , j } ^ { t + 1 } , \\\\ & W _ { i , j } ^ t = F _ { i , j } ^ t + F _ { i , j + 1 } ^ { t } - F _ { i - 1 , j } ^ t - F _ { i + 1 , j + 1 } ^ { t } , \\end{aligned} \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} \\bigl ( B ^ { \\Sigma } + B ^ { \\Sigma ' } \\bigr ) ^ - = ( I _ { G } B ) ^ - \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} & \\int \\limits _ { B _ { 2 ^ { - n + 3 } \\cdot | \\Lambda | } ( M _ 0 ) \\setminus \\widetilde { \\Omega } _ { n - 2 } } \\ , \\frac { | \\Phi _ n ( M ) | } { \\rho ^ 2 _ { M _ 0 } ( M ) } \\ , d m _ 3 ( M ) \\leq \\\\ & \\leq C \\cdot 2 ^ { 2 n } \\int \\limits _ { B _ { 2 ^ { - n + 3 } \\cdot | \\Lambda | } ( M _ 0 ) } \\ , | \\Phi _ n ( M ) | \\ , d m _ 3 ( M ) \\leq \\\\ & \\leq C \\cdot 2 ^ { n } \\omega ( 2 ^ { - n } ) . \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { f } ( x ) & : = \\sup \\left \\{ \\limsup _ { n \\rightarrow \\infty } R _ n ( \\lambda ) h _ n ( x _ n ) \\ , \\middle | \\ , x _ n \\rightarrow x \\right \\} , \\\\ \\underline { f } ( x ) & : = \\inf \\left \\{ \\liminf _ { n \\rightarrow \\infty } R _ n ( \\lambda ) h _ n ( x _ n ) \\ , \\middle | \\ , x _ n \\rightarrow x \\right \\} , \\end{aligned} \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} & \\ell \\left ( \\sum _ { i = 1 } ^ s \\left \\lfloor \\frac { n _ i + t \\lambda _ i } { m } \\right \\rfloor Q _ i + \\sum _ { i = s + 1 } ^ r \\left \\lfloor \\frac { t \\lambda _ i } { m } \\right \\rfloor Q _ i \\right ) \\\\ & - \\ell \\left ( \\sum _ { i = 1 } ^ s \\left \\lfloor \\frac { n _ i - c _ i + t \\lambda _ i } { m } \\right \\rfloor Q _ i + \\sum _ { i = s + 1 } ^ r \\left \\lfloor \\frac { t \\lambda _ i } { m } \\right \\rfloor Q _ i \\right ) = 0 \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} \\epsilon _ c & = m _ c I _ 2 + \\epsilon _ e ( \\sum _ { l = 1 } ^ L n _ l \\epsilon _ { a _ l } ) = \\begin{bmatrix} m _ c & \\sum _ { l = 1 } ^ L ( n _ l { r _ { a _ l } } ) \\lambda _ e \\\\ 0 & m _ c \\end{bmatrix} \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} X _ { \\ell _ i ^ { ( k ) } } = \\left \\{ \\ell _ i ^ { ( k ) } \\circ { \\rm g w } _ t ^ { - 1 } \\circ \\Delta _ t ^ { - 1 } , - \\right \\} _ t \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} B = B ^ G + ( B ^ { \\Sigma } + B ^ { \\Sigma ' } ) . \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} d \\Theta _ { t } = & ( d e ^ { { \\bf e } _ { t } } ) e ^ { { \\bf h } _ { t } } e ^ { { \\bf f } _ { t } } + e ^ { { \\bf e } _ { t } } ( d e ^ { { \\bf h } _ { t } } ) e ^ { { \\bf f } _ { t } } + e ^ { { \\bf e } _ { t } } e ^ { { \\bf h } _ { t } } ( d e ^ { { \\bf f } _ { t } } ) \\\\ & + ( d e ^ { { \\bf e } _ { t } } ) ( d e ^ { { \\bf h } _ { t } } ) e ^ { { \\bf f } _ { t } } + ( d e ^ { { \\bf e } _ { t } } ) e ^ { { \\bf h } _ { t } } ( d e ^ { { \\bf f } _ { t } } ) + e ^ { { \\bf e } _ { t } } ( d e ^ { { \\bf h } _ { t } } ) ( d e ^ { { \\bf f } _ { t } } ) . \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ 1 x _ { n } + \\sum _ { i = 2 } ^ { n - 1 } x _ i ^ 2 + x _ { 1 } ^ 3 ; \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} \\psi ^ N ( x ) : = \\inf _ { y \\in \\R ^ d } \\psi ( y ) + N \\theta _ 2 ( | x - y | ) , x \\in \\R ^ d . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} ( g ( \\mu ) A ) ( x ) = \\big ( - \\Delta _ \\mu \\big ) ^ { \\dd } ( A ( x ) - \\int _ { M } A ( y ) d y ) + \\int _ { M } A ( y ) d y . \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} M ^ { \\lambda } _ { d } \\circ M ^ { \\mu } _ { d ' } & = \\left ( \\C [ S _ { \\lambda } \\backslash S _ d ] \\otimes \\C [ S _ { \\mu } \\backslash S _ { d ' } ] \\right ) \\otimes _ { \\C [ S _ d \\times S _ { d ' } ] } \\C [ S _ { d + d ' } ] = \\C [ S _ { \\lambda } \\times S _ { \\mu } \\backslash S _ { d + d ' } ] = M ^ { ( \\lambda , \\mu ) } _ { d + d ' } \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d \\varphi ^ { 1 } \\left ( r \\right ) = & - \\mathrm { I I } \\left ( r \\right ) d r + \\nu ^ { 1 } \\left ( r \\right ) d B \\left ( r \\right ) , \\\\ \\varphi ^ { 1 } \\left ( T \\right ) = & \\varepsilon _ { 4 } \\left ( T \\right ) \\Gamma _ { 1 } ( T ) \\Gamma _ { 2 } ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} M _ k ' = \\begin{bmatrix} 1 & c _ { k ' } \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\int _ { \\mathbb R } \\nu ( x ) \\Big ( a - & \\frac { 1 } { \\gamma ^ 2 } \\coth ^ 2 ( \\sqrt { \\mu _ 2 } | D | ) \\Big ) \\partial _ x ^ 2 \\nu ( x ) d x \\\\ & = - \\Big ( a - \\frac { 1 } { \\gamma ^ 2 } \\Big ) \\| \\nu ' \\| ^ 2 + \\frac { 1 } { \\gamma ^ 2 } \\int _ { \\mathbb R } \\nu ( x ) [ 1 - \\coth ^ 2 ( \\sqrt { \\mu _ 2 } | D | ) ] \\partial _ x ^ 2 \\nu ( x ) d x \\\\ & \\geqq - \\Big ( a - \\frac { 1 } { \\gamma ^ 2 } \\Big ) \\| \\nu ' \\| ^ 2 , \\end{array} \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} \\begin{cases} \\underline u _ t \\leq \\underline u ( t ) ( a _ 0 ( t ) - a _ 1 ( t ) \\underline u ( t ) - a _ 2 ( t ) \\overline v ( t ) ) \\\\ \\overline v _ t \\geq \\overline v ( t ) ( b _ 0 ( t ) - b _ 1 ( t ) \\underline u ( t ) - b _ 2 ( t ) \\overline v ( t ) ) \\\\ \\end{cases} \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} { \\theta _ { 1 , B } } \\left ( t \\right ) = 2 \\pi { f _ { 1 , B } } t + 2 \\pi \\Delta f n T + \\varphi _ { n , B } ^ { { \\rm { C F S K } } } + { \\varphi _ B } \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\begin{array} { l } \\Gamma _ 1 ( u , n ) ( t ) = \\eta ( t ) W _ 0 ^ t \\big ( u _ 0 ^ e , g \\big ) - i \\eta ( t ) \\int _ 0 ^ t e ^ { i ( t - t ^ \\prime ) \\Delta } F ( u , n ) \\d t ^ \\prime + i \\eta ( t ) W _ 0 ^ t \\big ( 0 , q \\big ) , \\\\ \\Gamma _ 2 ( u , n ) ( t ) = \\eta ( t ) V _ 0 ^ t \\big ( \\phi _ { \\pm } , h \\big ) + \\frac 1 2 \\eta ( t ) ( n _ + + n _ - ) - \\frac 1 2 \\eta ( t ) V _ 0 ^ t ( 0 , z ) , \\end{array} \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\Lambda ( n ) = \\sum _ { \\atop { j , k > u } { j k = n } } \\Lambda ( k ) a _ j + \\sum _ { \\atop { j \\leq u } { j k = n } } \\mu ( j ) \\log ( k ) - \\sum _ { \\atop { j \\leq u ^ 2 } { j k = n } } b _ j , \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} R _ 2 ^ { A B } : \\ \\ \\left \\{ \\begin{array} { l l } \\xi = \\lambda _ 2 ^ { A B } = v + \\beta t + \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } , \\\\ u _ + - v = \\int _ { \\rho } ^ { \\rho _ { + } } \\frac { \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } } { \\rho } d \\rho , \\ \\ \\rho _ { * } ^ { A B } \\leq \\rho \\leq \\rho _ { + } . \\end{array} \\right . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} R _ V = t _ { V ^ \\vee } ^ \\vee : t ^ * C \\rightarrow V \\ ; , \\ ; \\ ; L _ V = s _ { V ^ \\vee } ^ \\vee : s ^ * C \\rightarrow V \\ ; . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} \\langle p ( s ) \\delta _ S , \\psi ( x ( s ) , t ( s ) ) \\rangle = \\int _ a ^ b p ( s ) \\psi ( x ( s ) , t ( s ) ) \\sqrt { { x ' ( s ) } ^ 2 + { t ' ( s ) } ^ 2 } d s , \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} f ( z , w ) & = ( z , w / 2 ) , \\\\ g ( z , w ) & = ( z + z w , w ) . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\Sigma ( s ) = \\zeta ( 2 s ) \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { s } } \\sum _ { q = 1 } ^ { \\infty } \\frac { S ( 1 , n ^ 2 ; q ) } { q } \\varphi \\left ( \\frac { 4 \\pi n } { q } \\right ) , \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ { n - 1 } ( v - v ^ { - 1 } ) ^ 2 \\left ( v ^ { 2 ( n + 1 - i ) } F _ i K _ { \\varpi _ { i + 1 } } E _ i K _ { \\varpi _ i } + v ^ { - 2 ( n - i ) } F _ i K _ { - \\varpi _ i } E _ i K _ { - \\varpi _ { i + 1 } } \\right ) + \\\\ & ( v ^ 2 - v ^ { - 2 } ) ^ 2 v ^ 2 F _ n K _ { - \\varpi _ n } E _ n K _ { \\varpi _ n } . \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\nu \\beta _ 1 \\nu ' ) \\mathbf h _ p ( x ) = C \\mathcal G \\left ( \\frac { - \\gamma } { p } \\right ) \\mathbf 1 _ { \\Z _ p } ( x ) \\psi \\left ( \\frac { x ^ 2 } { \\gamma p } \\right ) \\mathfrak G \\left ( \\frac { - 2 x } { \\gamma p } , \\underline { \\chi } _ p \\right ) . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { - \\theta ^ { - \\alpha _ { 0 } } R _ j ^ 2 } ^ { t } \\int _ { B _ { R _ j } } \\ , u ^ { \\ , 2 b } \\ , \\chi _ { _ { \\left [ u \\leq l _ j \\right ] } } f _ 3 ^ 2 \\eta _ j ^ 2 \\ , d x d \\tau \\leq C \\Lambda ^ { 2 \\left ( b - 1 \\right ) } \\theta _ 0 ^ { \\ , 2 } \\left ( \\int _ { - \\theta ^ { - \\alpha _ { 0 } } R _ j ^ 2 } ^ { t } \\left | A _ { l _ j , R _ j } ^ - ( \\tau ) \\right | ^ { \\frac { \\widehat { r } } { \\widehat { q } } } \\ , d \\tau \\right ) ^ { \\frac { 2 } { \\widehat { r } } \\left ( 1 + \\kappa \\right ) } \\end{aligned} \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\partial _ t S ( t ) = \\Delta _ { \\omega ( t ) } S ( t ) + | R i c ( \\omega ( t ) ) + \\omega ( t ) | ^ 2 _ { \\omega ( t ) } - ( S ( t ) + n ) . \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} \\frac { G ( z ) } { A ( z ) } \\bigg ( \\sum _ n \\frac { | c _ n | ^ 2 } { z - t _ n } + R ( z ) \\bigg ) = \\bigg ( \\sum _ n \\frac { c _ n \\mu _ n ^ { 1 / 2 } } { z - t _ n } \\bigg ) \\cdot \\bigg ( \\sum _ n \\frac { G ( t _ n ) \\bar c _ n } { A ' ( t _ n ) \\mu _ n ^ { 1 / 2 } ( z - t _ n ) } \\bigg ) , \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} \\langle \\tau _ p ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) \\breve { \\mathbf g } _ p , \\breve { \\mathbf g } _ p \\rangle = p ^ { 1 / 2 } \\xi _ 1 ( p ) ^ 2 \\int _ { K _ { 0 0 } } \\mathbf g _ p ( h \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } \\varpi _ p ) \\underline { \\chi } _ p ^ { - 1 } ( h ) d h = p \\underline { \\chi } _ p ( \\gamma ) \\mathrm { v o l } ( K _ { 0 0 } ) = \\frac { \\underline { \\chi } _ p ( \\gamma ) } { p + 1 } . \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} h ( x ) = \\begin{cases} f ( x ) & \\ f ( x ) \\neq y _ 0 \\\\ g ( x ) & \\end{cases} \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} d ( x _ i , x _ j ) = d ( x _ i , x _ { i + 1 } ) + \\dots + d ( x _ { j - 1 } , x _ j ) , \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{gather*} b _ 1 : = ( l _ { 2 1 } , d _ { 1 2 1 } , d _ { 2 2 1 } ) , b _ 2 : = ( l _ { 2 2 } , d _ { 1 2 2 } , d _ { 2 2 2 } ) , \\\\ a _ 1 : = ( - 1 , 0 , 0 ) , a _ 2 : = ( - 1 , 1 , 0 ) , \\\\ a _ 3 : = ( - 1 , d _ { 1 1 2 } , d _ { 2 1 2 } ) , a _ 4 : = ( - 1 , d _ { 1 1 2 } + 1 , d _ { 2 1 2 } ) . \\end{gather*}"}
-{"id": "7185.png", "formula": "\\begin{align*} \\prod _ { t = 1 } ^ { { s } } { \\lambda _ { h } - ( - 1 ) ^ { { \\bar h } + { \\bar i _ t } } \\lambda _ { i _ t } - a _ { t - 1 } \\cdots - a _ 1 \\brack a _ t } _ q \\neq 0 . \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} \\bar u ' ( t ) + A \\bar u ( t ) + B ( \\bar u ( t ) , \\bar u ( t ) ) & = \\frac { A \\xi _ 1 } t + \\sum _ { n = 2 } ^ { \\infty } \\frac 1 { t ^ { n } } \\Big \\{ - ( n - 1 ) \\xi _ { n - 1 } + A \\xi _ n + \\sum _ { k = 1 } ^ { n - 1 } B ( \\xi _ k , \\xi _ { n - k } ) \\Big \\} \\\\ & = \\sum _ { n = 1 } ^ \\infty \\phi _ n t ^ { - n } = f ( t ) . \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} M & = M _ 1 + M _ 2 , \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\sum _ { S \\subset X } [ { \\jmath _ S } _ { ! * } \\jmath _ S ^ * \\imath _ S ^ { ! * } \\beta ] = [ I _ { X _ 3 } ] \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} w ' : = \\min \\{ f ( x ) : M ^ \\top h ( x ) = 0 , g ( x ) \\leq 0 \\} \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} f ^ t = \\min \\{ x _ 1 + x _ 2 , 2 x _ 2 , 0 \\} - 3 \\min \\{ x _ 1 , x _ 2 \\} . \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} \\mathcal { A } \\big ( \\big \\{ \\omega _ f \\lambda _ f ( n ) \\big \\} ; q \\big ) = \\sum _ { f \\in H _ 2 ( q ) } \\omega _ f \\cdot \\lambda _ f ( n ) . \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} L _ I ( \\ell ^ { ( k ) } ) - \\ell _ j ^ { ( k ) } = \\sum _ { i \\in I } \\lambda _ i ^ { ( k ) } - \\left ( \\lambda _ 1 ^ { ( k ) } + \\cdots + \\lambda _ j ^ { ( k ) } \\right ) < - \\delta , \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} \\pi ^ { * _ { \\alpha } } = \\pi ^ { \\alpha } . \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { Q } } = s _ { i _ 1 } s _ { i _ 2 } \\cdots s _ { i _ { N } } \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} r Y _ { 2 + \\nu } ( r ) + ( \\nu { E } + 2 \\eta { r } + 1 + \\tau ) Y _ { 1 + \\nu } ( r ) + \\Big [ \\eta ( 2 \\nu { E } + 1 + \\tau ) + \\beta \\Big ] Y _ { \\nu } ( r ) = 0 , \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} 1 = [ \\nabla ^ { k , g _ P } \\eta _ i ] _ { C ^ { \\alpha } ( B ^ { g _ { P } } ( p _ i , \\delta _ i R _ i ) ) } > \\ ; & \\epsilon [ \\nabla ^ { k , g _ P } \\eta _ i ] _ { C ^ { \\alpha } ( B ^ { g _ { P } } ( p _ i , R _ i ) ) } + i [ \\nabla ^ { k - 1 , g _ P } L ^ { g _ P } \\eta _ i ] _ { C ^ { \\alpha } ( B ^ { g _ { P } } ( p _ i , R _ i ) ) } \\\\ & { + } \\sum _ { j = 1 } ^ k i R _ i ^ { - k + j - \\alpha } \\| \\nabla ^ { j , g _ P } \\eta _ i \\| _ { L ^ \\infty ( B ^ { g _ P } ( p _ i , \\delta _ i R _ i ) ) } . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} \\binom { a n + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\alpha _ { s } n + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } = \\frac { 1 } { \\left ( r ! \\right ) ^ { p } \\prod _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\right ) ^ { p _ { s } } } \\sum _ { k = 0 } ^ { r p + \\sigma } k ! S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) \\binom { n } { k } . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} E _ A = \\prod _ { \\alpha \\in \\Phi ^ + } E _ { \\alpha } ^ { ( A _ \\alpha ) } , \\ F _ A = \\prod _ { \\beta \\in \\Phi ^ - } E _ \\beta ^ { ( A _ \\beta ) } . \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\dot { z } ( t ) & = A z ( t ) , t \\ge 0 , \\\\ z ( 0 ) & = z _ 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} t _ { \\alpha , \\varepsilon } - \\mathbb { E } ( T _ { D _ \\varepsilon } ) \\geq 2 \\sqrt { \\Sigma x } + 2 \\max _ { l = 1 , . . . , D _ \\varepsilon } b _ l ^ { - 2 } \\varepsilon ^ 2 x . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} | \\langle \\vec { g } _ { i } , \\vec { g } _ { j } \\rangle | = \\frac { d } { k ( d + 1 - k ) } \\left | l - \\frac { 1 } { d } ( k ^ { 2 } - l ) \\right | = \\frac { d + 1 } { k ( d + 1 - k ) } \\abs * { l - \\frac { k ^ { 2 } } { d + 1 } } . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\xi _ k \\colon \\frac { M } { N _ 0 \\cap \\ldots \\cap N _ k } \\longrightarrow \\prod _ { i = 0 } ^ k \\frac { M } { N _ i } , \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} \\log G ( z \\ , | \\ , \\tau ) = \\frac { z ^ 2 } { 2 \\tau } \\log ( z ) - \\frac { z ^ 2 } { \\tau } \\left ( \\frac { 3 } { 4 } + \\frac { \\log \\tau } { 2 } \\right ) - \\frac { 1 } { 2 } \\left ( \\frac { 1 } { \\tau } + 1 \\right ) z \\log z + O ( z ) , \\ , \\ , z \\rightarrow \\infty , \\ , \\ , | \\arg ( z / \\tau ) | < \\pi . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} Z ( t ) = 1 + \\nu \\int _ 0 ^ t Z ( s ) \\ , d M ( s ) , \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} \\max \\{ p ( x ) \\mid x \\in \\overline { B ( x _ { 0 } , \\varepsilon ) } \\} = \\max \\{ p ( x ) \\mid x \\in B ( x _ { 0 } , \\varepsilon , \\delta , \\theta ) \\} . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} ( S ^ { ( 3 ) } _ { 0 , 3 } ) ^ 3 = \\Sigma ^ 3 ( S ^ { ( 3 ) } _ { 0 , 1 } ) ^ 3 = \\Sigma ^ 2 \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} \\lambda _ { \\rho } \\circ T _ H ( f ) = \\int _ G f ( g ) \\rho ( g ) \\ , d \\mu _ G \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} \\mathcal { L } ' _ y = 0 , \\ ; g ( y ) \\leq 0 , \\ ; \\lambda \\geq 0 , \\ ; \\langle \\lambda , g ( y ) \\rangle = 0 . \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} e \\ , : = \\ , ( \\overline { F } _ 1 \\wedge \\cdots \\wedge \\overline { F } _ n ) ^ { \\vee } \\otimes \\left ( ( d T _ 1 \\wedge \\cdots \\wedge d T _ { n + 1 } ) \\otimes 1 _ A \\right ) , \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\omega = F \\left ( \\sum _ { I : | I | = q } \\lambda _ I \\tfrac { d F _ { i _ 1 } } { F _ { i _ 1 } } \\wedge \\dots \\wedge \\tfrac { d F { i _ q } } { F _ { i _ q } } \\right ) = \\sum _ { I : | I | = q } \\lambda _ { I } \\hat { F } _ I d F _ I , \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} \\frac { F ( w ) } { H ( w ) } \\int \\frac { G _ 1 ( z ) H ( z ) \\overline { F _ 1 ( z ) } } { z - w } d \\nu ( z ) = \\int \\frac { G _ 1 ( z ) F ( z ) \\overline { F _ 1 ( z ) } } { z - w } d \\nu ( z ) \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} { \\bf E } [ \\beta _ { 1 , 0 } ( a , b , \\bar { b } ) ^ q ] = \\frac { \\sin ( \\frac { \\pi b _ 0 } { a } ) } { \\sin ( \\frac { \\pi ( q + b _ 0 ) } { a } ) } . \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\widetilde { M } ( t ) \\leq \\frac { \\widetilde { M } ( t _ 0 ) } { t _ 0 ^ { 1 / \\sigma } } t ^ { 1 / \\sigma } = C _ 0 t ^ { 1 / \\sigma } t \\geq t _ 0 > 0 . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} I _ B ( \\lambda , x ) : = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\Delta ) } \\frac { \\Gamma ( \\lambda - 1 / 2 + w / 2 ) } { \\Gamma ( \\lambda + 1 / 2 - w / 2 ) } \\Gamma ( 1 - s - w ) \\sin \\left ( \\pi \\frac { s + w } { 2 } \\right ) x ^ w d w \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} C \\left ( p \\right ) = \\left \\{ \\left ( x , y \\right ) : \\begin{array} [ c ] { c } x \\wedge p = y \\wedge p , x ^ { \\prime } \\wedge p = y ^ { \\prime } \\wedge p , x ^ { \\sim } \\wedge p = y ^ { \\sim } \\wedge p , \\\\ \\square x \\wedge p = \\square y \\wedge p , \\Diamond x \\wedge p = \\Diamond y \\wedge p , \\Diamond x ^ { \\prime } \\wedge p = \\Diamond y ^ { \\prime } \\wedge p \\end{array} \\right \\} \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} \\sigma _ s = \\begin{pmatrix} \\sigma _ s ^ { ( 1 ) } & 0 \\\\ \\rho _ s \\sigma _ s ^ { ( 2 ) } & \\sqrt { 1 - \\rho _ s ^ 2 } \\sigma _ s ^ { ( 2 ) } \\end{pmatrix} \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} \\{ f ( g \\cdot X ) = a \\} & = \\{ g \\cdot X \\in f ^ { - 1 } ( a ) \\} = \\{ X \\in g ^ { - 1 } \\cdot f ^ { - 1 } ( a ) \\} \\\\ & = \\{ X \\in f ^ { - 1 } ( a ) \\} = \\{ f ( X ) = a \\} . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { k _ n } { Y ^ { ( k _ n ) } _ s } - a Y ^ { ( k _ n ) } _ s \\ge \\frac { k _ n } { \\varepsilon } - a \\varepsilon , \\quad \\forall s \\in \\bigl ( \\tau ^ { ( k _ n ) } _ \\varepsilon , \\tau ^ { ( k _ n ) } \\bigr ) . \\end{gathered} \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} m ( x , y ) = x \\tau ( y ^ { - 1 } x ) ^ { - 1 } . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} \\varrho _ s ( \\xi ) = \\eta \\big ( Q _ { s + 1 } ^ { 3 d A } \\xi \\big ) . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} C _ { I S } : = \\inf \\frac { ( w ( \\partial \\Omega ) ) ^ 2 } { \\mu ( \\Omega ) } > 0 , \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} \\alpha _ 3 + \\beta _ 7 + \\beta _ 8 = v _ { 3 } ^ * ( x _ 0 - u _ 0 ) \\le \\Vert x _ 0 - u _ 0 \\Vert < \\varepsilon . \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} \\mathfrak { d } _ 2 ^ m \\dot { \\phi } ( n ) & = ( - 1 ) ^ m \\int _ 0 ^ 2 \\cdots \\int _ 0 ^ 2 f ^ { ( m ) } ( n + t _ 1 + \\cdots + t _ m ) \\ , d t _ 1 \\cdots d t _ m . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} G ( \\hat { \\vec z } ) - G ( \\vec z ^ * ) = \\frac { ( z _ i ^ * - z _ j ^ * ) ( | v _ i | - | v _ j | ) } { \\| \\vec z \\| _ p } < 0 . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} \\textrm { m e a s } ( S _ R ) \\ , = \\ , \\int _ { \\mathbb S ^ { N - 1 } } A ( R , \\theta ) \\ , d \\theta ^ 1 d \\theta ^ 2 \\ldots d \\theta ^ { N - 1 } \\ , , \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} Z ( x , z ) \\not \\subset Z ( g , f _ 3 ) \\subset Z ( f _ 1 , f _ 3 ) = Z ( f _ 1 , f _ 2 , f _ 3 ) \\cup Z ( x , z ) \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} \\omega _ n ( x , y ) : = \\sum _ { i = i _ 0 } ^ \\infty x _ i y _ { i + n } t ^ i \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} { a } _ { n } \\left ( \\mathcal { T } : X \\to Y \\right ) : = \\inf _ { \\left ( \\mathcal { A } \\right ) < n } \\left \\Vert \\mathcal { T } - \\mathcal { A } \\right \\Vert _ { X \\to Y } . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} \\mu ^ { ( J ) } \\coloneqq 2 ^ J - 1 - t _ K \\quad m ^ { ( J ) } \\coloneqq 2 \\left ( t _ K + 1 \\right ) = 2 m . \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} \\begin{cases} \\underbar r _ 1 ^ 1 - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 1 ^ 1 + \\epsilon \\cr \\underbar r _ 2 ^ 1 - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 2 ^ 1 + \\epsilon \\end{cases} \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} V ' _ I V ' _ J = \\sum _ K \\bar c _ { I J } ^ K V ' _ K \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\tau _ { K } ( \\chi ) = \\sum _ { u \\in U _ K ^ 0 / U _ K ^ { m _ { \\chi } } } \\chi ( u c _ { \\chi } ^ { - 1 } ) \\psi _ { K } ( u c _ { \\chi } ^ { - 1 } ) \\in \\Q ^ c , \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} \\pi _ n = \\big \\{ \\big ( t _ { i , n } ^ { ( 1 ) } \\big ) _ { i \\in \\mathbb { N } _ 0 } , \\big ( t _ { i , n } ^ { ( 2 ) } \\big ) _ { i \\in \\mathbb { N } _ 0 } \\big \\} , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} ( S ^ { ( 3 ) } _ { 1 , 2 } ) ^ n X _ { \\underline { m } } = \\begin{cases} \\Sigma ^ n x ^ { 2 + 2 n } , \\vert \\underline { m } \\vert = 1 , \\\\ \\Sigma ^ { n } x ^ { 1 + 2 n } , \\vert \\underline { m } \\vert = 2 . \\end{cases} \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} ( 1 - 4 y _ { * } ^ 4 ) ( u _ { * } ^ 2 - v _ { * } ^ 2 ) = - y _ { * } ^ 3 \\sqrt { 1 + y _ { * } ^ 2 } u _ { * } v _ { * } ( u _ { * } ^ 2 - v _ { * } ^ 2 ) , \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{gather*} L ( \\xi , s ) = \\prod _ { v ~ { \\rm n o n a r c h i m e d e a n } \\atop { \\xi _ v ~ { \\rm u n r a m i f i e d } } } \\frac { 1 } { 1 - \\xi _ v ( \\pi _ v ) ( N v ) ^ { - s } } . \\end{gather*}"}
-{"id": "8518.png", "formula": "\\begin{align*} & \\nabla \\mathbf { u } ^ - ( X ) ( A _ i - X ) = ( ( A _ i - X ) ^ T \\otimes I _ 2 ) \\textrm { V e c } ( \\nabla \\mathbf { u } ^ - ( X ) ) , \\\\ & \\nabla \\mathbf { u } ^ - ( \\widetilde { Y } _ i ) ( A _ i - \\widetilde { Y } _ i ) = ( ( A _ i - \\widetilde { Y } _ i ) ^ T \\otimes I _ 2 ) \\textrm { V e c } ( \\nabla \\mathbf { u } ^ - ( \\widetilde { Y } _ i ) ) . \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} | X ( t ) - \\tilde { X } ( t ) | ^ 2 = 2 \\int _ 0 ^ t A ( X ( s ) , \\tilde { X } ( s ) ) \\ , d s + 2 \\int _ 0 ^ t B ( X ( s ) , \\tilde { X } ( s ) ) \\ , d s , \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} \\begin{gathered} d Y _ t = \\frac { 1 } { 2 } \\biggl ( \\frac { k } { Y _ t } - a Y _ t \\biggr ) d t + \\frac { \\sigma } { 2 } d B _ t ^ H , Y _ 0 > 0 . \\end{gathered} \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} B _ 2 ( m ) = \\begin{cases} p ^ { 1 - 3 m / 2 } ( - 1 ) ^ { m + 1 } \\chi _ { \\psi } ( p ^ m ) ( 1 - p ^ { - 1 } ) p ^ { m } & m > 0 , \\\\ p ^ { 3 m / 2 } ( - 1 ) ^ { m + 1 } \\chi _ { \\psi } ( p ^ m ) ( 1 - p ^ { - 1 } ) ( 1 + p ^ { - 1 } - p ^ { - m } ) & m \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} g _ j ( x ) = 0 ; \\Delta _ { j } = \\widehat { \\Delta } _ j = 0 X ' = \\emptyset . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} d X _ t = b ( X _ { t - } ) \\ , d t + \\sigma ( X _ { t - } ) \\ , d L _ t , X _ 0 \\sim \\mu \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\bigcap _ { k \\geq 1 } D _ k ~ = ~ \\{ \\alpha \\in \\R ^ d : R ( \\alpha ) \\leq 1 \\} . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\omega _ { \\infty } ( \\tilde { k } _ { \\theta } , k ' ) \\phi _ { \\mathbf h , \\infty } = e ^ { - \\sqrt { - 1 } ( k + 1 / 2 ) \\theta } \\det ( \\mathbf k ) ^ { k + 1 } \\phi _ { \\mathbf h , \\infty } \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } y ' ( t ) = \\lambda ( t ) y ( t ) \\quad \\mbox { f o r } \\mathcal { L } ^ 1 \\mbox { - a . e . \\ } t \\in [ 0 , 1 ] , \\\\ y ( 0 ) = \\bar { y } . \\end{array} \\right . \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} \\dd Y _ t = b ^ n ( Y _ { t - } ) \\dd t + \\sigma ^ n ( Y _ { t - } ) \\dd W _ t + \\int _ E v ^ n ( y , Y _ { t - } ) ( p - q ) ( \\dd y , \\dd t ) , Y _ 0 = \\zeta \\sim \\eta , \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } X ( t ) d t = \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { i = 0 } ^ { n - 1 } X ( u _ { i } ) ( t _ { i + 1 } - t _ { i } ) \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} \\frac { 1 } { \\left ( { 1 - x } \\right ) _ q ^ \\infty } = \\sum \\limits _ { j = 0 } ^ \\infty { \\frac { { x ^ j } } { { \\left ( { 1 - q } \\right ) \\left ( { 1 - q ^ 2 } \\right ) \\ldots \\left ( { 1 - q ^ j } \\right ) } } } . \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i , j } q ^ { - i j } \\sum _ { a , b } q ^ { i a + j b } \\delta _ a \\otimes \\delta _ b & = \\frac { 1 } { N } \\sum _ { i , j , a , b } q ^ { a b } q ^ { - ( a - i ) ( b - j ) } \\delta _ a \\otimes \\delta _ b \\\\ & = \\frac { 1 } { N } \\sum _ { a , b } q ^ { a b } \\delta _ a \\otimes \\delta _ b \\sum _ { i , j } q ^ { - ( a - i ) ( b - j ) } \\\\ & = \\sum _ { a , b } q ^ { a b } \\delta _ a \\otimes \\delta _ b , \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} M ( \\lambda ) = \\Sigma _ { P } ^ { ( \\epsilon , \\eta ) } ( \\lambda ) + & \\left ( \\lambda \\begin{bmatrix} 0 \\\\ D ( \\lambda ) \\end{bmatrix} + B \\right ) \\left ( L _ \\epsilon ( \\lambda ^ \\ell ) \\otimes I _ n \\right ) \\\\ & + \\left ( L _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m \\right ) \\left ( \\lambda \\begin{bmatrix} 0 & - D ( \\lambda ) \\end{bmatrix} + C \\right ) , \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} C _ { } : = \\sum \\limits _ { \\ell = 1 } ^ L N _ \\ell C _ \\ell \\lesssim \\left \\{ \\begin{array} { l r } \\varepsilon ^ { - 2 } , & \\beta > \\gamma , \\\\ \\varepsilon ^ { - 2 } ( \\log \\varepsilon ) ^ 2 , & \\beta = \\gamma , \\\\ \\varepsilon ^ { - 2 - ( \\gamma - \\beta ) / \\alpha } , & \\beta < \\gamma . \\end{array} \\right . \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ( \\tau , b ) ^ k \\bigr ] & = \\prod \\limits _ { l = 0 } ^ { k - 1 } \\Bigl [ \\frac { \\Gamma \\bigl ( ( l + b _ 0 + b _ 1 ) / \\tau \\bigr ) \\ , \\Gamma \\bigl ( ( l + b _ 0 + b _ 2 ) / \\tau \\bigr ) } { \\Gamma \\bigl ( ( l + b _ 0 ) / \\tau \\bigr ) \\ , \\Gamma \\bigl ( ( l + b _ 0 + b _ 1 + b _ 2 ) / \\tau \\bigr ) } \\Bigr ] . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} \\mathcal I _ 2 ( n ) = \\int _ { \\mathcal A _ 2 ^ + ( n ) } \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} & K _ 1 : = | { \\bf D } | \\cos ( \\theta _ 3 ) = { \\bf D } \\cdot \\mathbb { E } _ 1 , K _ 2 : = | { \\bf D } | \\sin ( \\theta _ 3 ) = { \\bf D } \\cdot \\mathbb { E } _ 2 . \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} a _ r = \\inf _ { x \\in K _ { 2 r } ( 0 ) } a ( x ) . \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} \\Lambda ( n , d ) : = \\{ ( \\lambda _ 1 , \\ldots , \\lambda _ n ) \\in \\Z ^ { n } _ { \\geq 0 } | \\sum _ { i } \\lambda _ i = d \\} . \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} d _ { 1 } ( M ) = 2 ^ { n - 1 } , \\ ; M \\in \\sum _ { n } , \\ ; n \\in \\mathbb { Z } \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\mathcal T _ 2 \\nu = \\frac { \\gamma } { 1 - \\gamma } G _ 1 ( \\nu ) - \\mathcal M \\nu - \\frac { \\mu } { \\mu _ 2 \\gamma ^ 2 } \\nu \\in L ^ 2 ( \\mathbb R ) . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} m ^ { ( n ) } _ \\infty : = { \\bf E } ( I _ { \\infty } ^ n ) = \\begin{cases} { \\displaystyle \\frac { n ! } { \\prod _ { k = 1 } ^ n \\Phi ( k ) } } , & \\mbox { i f } \\ n < n ^ * , \\\\ + \\infty , & \\mbox { i f } \\ n \\geq n ^ * . \\end{cases} \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} f _ * ( A ' ) \\coloneqq \\{ b \\in B \\mid \\mbox { f o r a l l } a \\mbox { s u c h t h a t } f ( a ) = b , \\mbox { w e h a v e } a \\in A ' \\} \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{gather*} \\operatorname { H o m } ^ r ( M , N ) = \\operatorname { H o m } _ k ( M , N ) , h \\cdot \\varphi = h ^ 2 \\varphi \\big ( S ^ { - 1 } \\big ( h ^ 1 \\big ) { - } \\big ) . \\end{gather*}"}
-{"id": "4281.png", "formula": "\\begin{align*} & r _ { x _ 1 } = r _ { x _ 2 } = a \\\\ & r _ { y _ 1 } = r _ { y _ 2 } = b \\\\ & r _ { z _ 1 } = r _ { z _ 2 } = c \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} H ^ u _ n = H _ n \\star u \\in E \\star u \\subset E , \\forall n < { \\cal T } _ \\vartheta . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} M ( \\lambda ) = \\Psi ^ \\dagger _ { ( N _ 1 , N _ 2 ) } [ P ] ( \\lambda ) + \\Phi _ 1 ^ \\dagger [ X ^ T K _ 2 ] ( \\lambda ) ^ T + Y ^ T K _ 1 ( \\lambda ) , \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\gamma _ { n } ~ = ~ \\gamma _ 1 ^ n G \\varphi _ n ~ = ~ \\sum _ { k = 1 } ^ n \\gamma _ 1 ^ { k - 1 } T _ u ^ { n - k } G \\varphi _ 1 . \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{gather*} \\sum _ { \\mathfrak J \\in S _ 2 } \\tilde \\chi ^ k ( \\mathfrak J ) q ^ { N ( \\mathfrak J ) } = \\sum _ { m , n \\in \\Z \\atop { m \\equiv n \\mod 2 } } \\big ( 3 m + 1 - n \\sqrt { - 3 } \\big ) ^ k q ^ { ( 3 m + 1 ) ^ 2 + 3 n ^ 2 } , \\end{gather*}"}
-{"id": "3260.png", "formula": "\\begin{align*} \\varphi \\xi = p \\xi - v ( E ) \\xi , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} B ( x + m , y - m ) = B ( x , y ) \\prod _ { i = 0 } ^ { m - 1 } \\frac { x + i } { y - i - 1 } \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) \\cdot z = \\frac { a z + b } { c z + d } . \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} M = \\left [ \\begin{array} { c c } A & B \\\\ C & G \\end{array} \\right ] , \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} & \\widetilde { c } _ { 1 i } ( r - 2 ) + \\widetilde { c } _ { 1 i } ( r + 2 ) - \\widetilde { c } _ { 2 i } ( r ) = 0 \\ \\ i = 1 , 2 , \\\\ & \\widetilde { c } _ { 2 i } ( r - 1 ) + \\widetilde { c } _ { 2 i } ( r + 1 ) - \\widetilde { c } _ { 1 i } ( r - 1 ) - \\widetilde { c } _ { 1 i } ( r + 1 ) = 0 \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} | B _ R ^ n | \\stackrel { ? } { = } \\frac { 1 } { n ! \\ , R } \\frac { \\det \\left ( [ \\chi _ { i + j + 2 } ( R ) ] _ { i , j = 0 } ^ { p } \\right ) } { \\det \\left ( [ \\chi _ { i + j } ( R ) ] _ { i , j = 0 } ^ { p } \\right ) } . \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 0 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; - \\mathbb { E } [ \\xi _ { t + 1 } ^ T x _ t ] \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ { t j } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\triangleq \\lim \\limits _ { \\tau \\downarrow 1 } \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) \\ , M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} \\frac { | c ( | D | ) | ^ 2 } { \\langle h , h \\rangle } = 2 ^ { \\nu ( N ) } \\frac { ( k - 1 ) ! } { \\pi ^ k } | D | ^ { k - 1 / 2 } \\frac { L ( f , D , k ) } { \\langle f , f \\rangle } , \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} L ^ { W , n } : = \\sum _ { i = 1 } ^ n \\sigma ^ { ( i ) } _ z , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\sigma ^ { ( i ) } _ z : = \\mathbb { I } ^ { \\otimes ( i - 1 ) } \\otimes \\sigma _ z \\otimes \\mathbb { I } ^ { \\otimes ( n - i ) } , \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} ( \\lambda ^ { ( p ' ) } ( G ) ) ^ { p ' } \\leq \\frac { r ^ { p ' - r } } { \\alpha ' } = ( r m ) ^ { p ' - p } \\cdot ( \\lambda ^ { ( p ) } ( G ) ) ^ p , \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} f _ { n } ^ { \\ell } ( x ) = n - 1 - \\frac { n c ( \\beta + n - 1 ) A _ { 1 , n } ^ { j , \\ell } ( x ) } { ( 1 - c ) B _ { 1 , n } ^ { j , \\ell } ( x ) } . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} e ( G , N ) = \\frac { \\left \\lvert \\mathrm { A u t } ( G ) \\right \\rvert } { \\left \\lvert \\mathrm { A u t } ( N ) \\right \\rvert } e ' ( G , N ) . \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} { \\rm m u l t } _ \\lambda ( \\mathbf { m } ) = \\sum _ { \\sigma \\in S _ n } \\varepsilon ( \\sigma ) Q \\Big ( \\sum _ { j = 1 } ^ n ( \\ell _ 1 + 1 - \\ell _ j ) e _ j - \\sigma \\Big ( \\sum _ { j = 1 } ^ n j e _ j \\Big ) \\Big ) . \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} ( \\rho , u ) ( x , t ) = \\left \\{ \\begin{array} { l l } ( \\rho _ - , u _ - + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x < x ( t ) , \\\\ ( w ( t ) \\delta ( x - x ( t ) ) , u _ \\delta ( t ) ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x = x ( t ) , \\\\ ( \\rho _ + , u _ + + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x > x ( t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\begin{aligned} & \\big \\| \\partial ^ \\alpha ( f g ) ( \\tau , \\cdot ) \\big \\| _ { L ^ 2 ( \\Omega ) } \\lesssim \\| f ( \\tau ) \\| _ { \\mathcal { H } ^ k ( \\Omega ) } \\cdot \\| g ( \\tau ) \\| _ { \\mathcal { H } ^ k ( \\Omega ) } , \\end{aligned} \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} V ( f _ { 3 , 0 } , \\pi _ n ) _ T = 2 \\sum _ { i : t _ { i , n } ^ { ( 1 ) } \\leq T } ( \\Delta _ { i , n } ^ { ( 1 ) } X ^ { ( 1 ) } ) ^ 3 \\overset { \\mathbb { P } } { \\longrightarrow } 2 B ( f _ { 3 , 0 } ) _ T \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} u _ { t } - a \\Delta u = F \\left ( x , t ; u \\right ) \\quad Q _ { T } , \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} u _ R ( x , t ) \\le \\frac { \\| u _ 0 \\| _ \\infty } { \\left ( 1 - ( p - 1 ) \\| u _ 0 \\| _ \\infty ^ { p - 1 } \\ , t \\right ) ^ { \\frac { 1 } { p - 1 } } } \\Longrightarrow T _ R \\ge T : = \\frac { 1 } { ( p - 1 ) \\| u _ 0 \\| _ \\infty ^ { p - 1 } } \\ , . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} & f _ t ( x , v ) = \\mathcal { Z } _ t ^ { - 1 } \\exp \\bigg ( - \\frac { ( v - A ( t , x ) ) ^ 2 } { 2 B ( t , x ) } \\bigg ) \\ , , \\mathcal { Z } _ t : = \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } x \\ , \\sqrt { 2 \\pi B ( t , x ) } \\ , , \\\\ & \\mbox { a n d } \\ , \\ , B ( t , x ) > 0 \\ , , \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\lambda \\left ( t \\right ) = t - T - \\eta . \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} A ^ { \\lambda , \\delta , \\varepsilon } ( x ) : = - \\operatorname { d i v } ( a ^ { \\ast } \\phi ^ { \\lambda } ( a \\nabla x ) ) - \\varepsilon L ^ { a } x - \\frac { 1 } { 2 } J _ { \\delta } ^ { a } L ^ { b } J _ { \\delta } ^ { a } x , \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} g _ E = d r ^ 2 + 4 r ^ 2 ( \\omega _ 2 \\otimes \\omega _ 2 + \\omega _ 3 \\otimes \\omega _ 3 ) , \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ t \\binom { { _ { \\neq k } } \\Lambda _ { \\neq d } - 1 } { r } \\binom { { _ { \\neq d } } \\Lambda _ { \\neq k } - 3 r - 1 } { t - r } \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } ( | R _ n - R _ n ( r ) | > \\delta ) = 0 \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} N ! = \\sqrt { 2 \\pi } N ^ { N + 1 / 2 } e ^ { - N } \\cdot e ^ { r _ N } , \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} v _ { m _ i i } ( x ) s _ i ( x ) + v _ { ( m _ i - 1 ) i } ( x ) r _ i ( x ) + 2 \\ell _ i v _ { m _ i i } ' ( x ) r _ i ( x ) & = ( - 1 ) ^ { m _ i } r _ i ( x ) ( m _ i v _ { m _ i i } ' ( x ) ^ * - v _ { ( m _ i - 1 ) i } ( x ) ^ * ) \\\\ & + ( - 1 ) ^ { m _ i } s _ i ( x ) v _ { m _ i i } ( x ) ^ * \\\\ & + ( - 1 ) ^ { m _ i } m _ i r _ i ' ( x ) v _ { m _ i i } ( x ) ^ * . \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} \\sigma \\left ( \\frac { L ^ S ( r , \\pi \\boxtimes \\chi , \\varrho _ { 5 } ) } { ( 2 \\pi i ) ^ { 2 k + 3 r } G ( \\chi ) ^ 3 \\langle F , F \\rangle } \\right ) = \\frac { L ^ S ( r , { } ^ \\sigma \\pi \\boxtimes { } ^ \\sigma \\ ! \\chi , \\varrho _ { 5 } ) } { ( 2 \\pi i ) ^ { 2 k + 3 r } G ( { } ^ \\sigma \\ ! \\chi ) ^ 3 \\langle { } ^ \\sigma \\ ! F , { } ^ \\sigma \\ ! F \\rangle } . \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} | J _ 1 | \\leq P _ 1 ^ { 2 ^ * - 2 } \\bigl ( u _ 0 + \\sum _ { j = 2 } ^ k P _ j ) + J _ 3 , \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\theta _ { \\emptyset } ^ { ( 0 ) } ( e ) = \\left | \\left \\{ ( a , b ) \\in ( [ 0 , r _ e - 1 ] \\cap \\N ) ^ 2 : \\ a - b \\equiv 0 \\mod n \\right \\} \\right | = r _ e . \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\bar { d } \\le & \\big ( \\sqrt { 8 \\cdot 4 ^ { d _ 1 ^ 2 + 1 } } + 1 \\big ) ^ { 2 n ^ 2 } - \\big ( \\sqrt { 8 \\cdot 4 ^ { d _ 1 ^ 2 + 1 } } - 1 \\big ) ^ { 2 n ^ 2 } \\\\ \\le & \\big ( \\sqrt { 3 2 } \\cdot 2 ^ { 8 ^ { ( 2 n ^ 3 ) ^ { 1 4 9 n ^ 8 } } } + 1 \\big ) ^ { 2 n ^ 2 } - \\big ( \\sqrt { 3 2 } \\cdot 2 ^ { 8 ^ { ( 2 n ^ 3 ) ^ { 1 4 9 n ^ 8 } } } - 1 \\big ) ^ { 2 n ^ 2 } \\\\ \\le & \\big ( 2 \\sqrt { 3 2 } \\cdot 2 ^ { 8 ^ { ( 2 n ^ 3 ) ^ { 1 4 9 n ^ 8 } } } \\big ) ^ { 2 n ^ 2 } = 2 ^ { 1 0 n ^ 2 } 4 ^ { n ^ 2 8 ^ { ( 2 n ^ 3 ) ^ { 1 4 9 n ^ 8 } } } \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} | u ( t ) | \\leq K _ 0 = \\max \\{ R , | A | , | B | \\} . \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} \\alpha | _ { X _ { i j k l } } & = ( G _ { i j k } d F _ i \\wedge d F _ j + G _ { j k i } d F _ j \\wedge d F _ k + G _ { i k j } d F _ i \\wedge d F _ k ) | _ { X _ { i j k l } } \\\\ & = ( G _ { j k l } d F _ j \\wedge d F _ k + G _ { k l j } d F _ k \\wedge d F _ l + G _ { j l k } d F _ j \\wedge d F _ l ) | _ { X _ { i j k l } } . \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} 2 ^ { - \\nu ( N ) - 2 } \\zeta _ { \\Q } ( 2 ) ^ { - 1 } \\mu _ { N , M } c ( \\mathfrak d _ { 4 \\xi } ) \\chi ( \\mathfrak f _ { \\xi } ) \\mathfrak f _ { 4 \\xi } ^ { k - 1 / 2 } \\mathcal W _ { \\infty } \\times \\sum _ { \\substack { d \\mid ( b _ 1 , b _ 2 , b _ 3 ) , \\\\ ( d , N ) = 1 } } d ^ { 1 / 2 } \\prod _ { p \\nmid N } \\Psi _ p \\left ( \\frac { 4 \\xi } { d ^ 2 } ; \\alpha _ p \\right ) \\prod _ { p \\mid N } \\Psi _ p \\left ( \\xi ; \\alpha _ p \\right ) . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} \\partial _ t \\mathsf { G } _ { t _ 1 - \\tau , x - z } Q ^ T \\mathsf { G } ^ T _ { \\tau - t _ 2 , z - y } = \\frac { Q ^ T \\mathsf { G } ^ T _ { t _ 1 - t _ 2 , x - y } - \\mathsf { G } _ { t _ 1 - t _ 2 , x - y } Q ^ T } { 2 } \\ , . \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\lambda _ M ( G ) = 2 n - L - 1 . \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} t _ { s _ { 1 , 1 } } ^ n & = ( t _ { c _ 1 } ^ { - 1 } t _ { z _ 1 } ^ { - 1 } ) ^ n t _ { a _ 1 } ^ n t _ { a _ 1 } ^ n t _ { d _ { g - 1 } } ^ n t _ { d _ 1 } ^ n \\\\ & = { } _ { t _ { c _ 1 } ^ { - 1 } } ( t _ { z _ 1 } ^ { - 1 } ) { } _ { t _ { c _ 1 } ^ { - 2 } } ( t _ { z _ 1 } ^ { - 1 } ) \\cdots { } _ { t _ { c _ 1 } ^ { - n } } ( t _ { z _ 1 } ^ { - 1 } ) \\cdot t _ { c _ 1 } ^ { - n } t _ { a _ 1 } ^ n t _ { a _ 1 } ^ n t _ { d _ { g - 1 } } ^ n t _ { d _ 1 } ^ n . \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} f _ m ( r ) = - \\frac { 1 } { 2 r } - \\frac { 2 m } { ( e ^ { 4 m r } - 1 ) ^ 3 } \\left ( ( 8 m ^ 2 r ^ 2 - 4 m r - 1 ) e ^ { 8 m r } + ( 8 m ^ 2 r ^ 2 + 4 m r + 2 ) e ^ { 4 m r } - 1 \\right ) . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} = d ( f _ { \\alpha ^ { 1 } } ( ( f _ { \\alpha ( 1 ) } ( x _ { 1 } ) , . . . , f _ { \\alpha ( m ) } ( x _ { m } ) ) ) , f _ { \\alpha ^ { 1 } } ( ( f _ { \\alpha ( 1 ) } ( y _ { 1 } ) , . . . , f _ { \\alpha ( m ) } ( y _ { m } ) ) ) ) \\overset { ( 2 ) } { < } l - \\lambda _ { l } \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} & \\sum _ { j _ 1 + \\cdots + j _ m = n } \\binom { n } { j _ 1 , \\ldots , j _ m } B _ { j _ 1 } ( x _ 1 ) \\cdots B _ { j _ m } ( x _ m ) = B _ n ( m ; x ) \\\\ & = \\sum _ { k = 0 } ^ n \\dfrac { s ( m + k , m ) } { k ! \\binom { m + k } { m } } \\Delta ^ k I _ n ( x ) . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} [ L ^ { 2 } ( \\R ) ] ^ d = L _ { \\sigma } ^ { 2 } \\oplus G ^ { 2 } , \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} \\frac { \\partial w _ { \\beta } ^ { \\varepsilon } } { \\partial t } & + A \\Delta w _ { \\beta } ^ { \\varepsilon } - \\rho _ { \\beta } w _ { \\beta } ^ { \\varepsilon } \\\\ & = \\mathbf { P } _ { \\varepsilon } ^ { \\beta } w _ { \\beta } ^ { \\varepsilon } + e ^ { \\rho _ { \\beta } \\left ( t - T \\right ) } \\mathbf { Q } _ { \\varepsilon } ^ { \\beta } u + e ^ { \\rho _ { \\beta } \\left ( t - T \\right ) } \\left [ F \\left ( x , t ; u _ { \\beta } ^ { \\varepsilon } \\right ) - F \\left ( x , t ; u \\right ) \\right ] . \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} R _ { L _ { \\infty } / K } : = T _ { L _ { \\infty } / K } + C _ { L _ { \\infty } / K } + U ' _ { L _ { \\infty } / K } - M _ { L _ { \\infty } / K } \\in K _ 0 ( \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) ) . \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} h ^ 0 \\left ( N \\left ( - \\sum D _ j \\right ) \\right ) = h ^ 0 ( N ) - r l = 0 . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} e ( T ) & = e ( T [ W \\cup S \\cup \\{ u _ 1 , \\ldots , u _ { _ { 1 0 - w - s } } \\} ] ) + e _ T ( \\{ u _ { _ { 1 1 - w - s } } , \\ldots , u _ r \\} , W \\cup S \\cup \\{ u _ 1 , \\ldots , u _ { _ { 1 0 - w - s } } \\} ) \\\\ & \\le e ( \\mathcal { T } _ { 1 0 } ) + ( 2 n - 4 ) = 2 4 + ( 2 n - 4 ) , \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} \\tilde { H } _ 0 : \\vartheta = 0 _ { l _ 2 ( \\mathbb { N } ^ * ) } \\mathrm { a g a i n s t } \\tilde { H } _ 1 : \\vartheta \\in \\tilde { \\Theta } , \\ \\| \\vartheta \\| ^ 2 \\geq \\mu _ { \\varepsilon } ^ 2 , \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} A _ { k } : = \\{ | u | > 2 ^ { k } \\} , \\ ; k \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} G ( A ( 0 ) , z ) = G ( \\boldsymbol { a } , e ^ z - 1 ) . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} K \\oplus G _ \\bullet : = \\left ( \\left \\{ 0 \\right \\} \\subset K \\subset K \\oplus G _ 1 \\subset \\dots \\subset K \\oplus G _ { 2 n - 1 } \\subset K \\oplus W = V \\right ) . \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} G ( \\vec z ) & = \\frac { r - ( \\vec z , | \\vec v | ) } { \\| \\vec z \\| _ p } , \\\\ B _ \\infty & = \\{ \\vec z \\in \\mathbb { R } ^ n : \\| \\vec z \\| _ \\infty \\le 1 \\} , \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} h _ 2 = \\ , & a _ 1 ^ 2 + a _ 1 ^ 6 + b + a _ 1 ^ 4 b + a _ 1 ^ 2 b ^ 2 + a _ 1 ^ 6 b ^ 2 + b ^ 3 + a _ 1 ^ 4 b ^ 3 + a _ 1 ^ 2 b ^ 4 + a _ 1 ^ 4 b ^ 5 + a _ 1 ^ 2 b ^ 6 + a _ 1 ^ 4 b ^ 7 + b ^ 9 \\\\ & + b ^ { 1 1 } + ( a _ 1 ^ 4 + a _ 1 ^ 8 + a _ 1 ^ 4 b ^ 4 ) k + ( a _ 1 ^ 2 + a _ 1 ^ 6 + a _ 1 ^ 4 b + a _ 1 ^ 8 b + a _ 1 ^ 6 b ^ 4 + a _ 1 ^ 4 b ^ 5 + a _ 1 ^ 2 b ^ 8 ) k ^ 2 \\cr & + ( a _ 1 ^ 4 + a _ 1 ^ 8 + a _ 1 ^ 4 b ^ 2 + a _ 1 ^ 8 b ^ 2 + a _ 1 ^ 4 b ^ 4 + a _ 1 ^ 4 b ^ 6 ) k ^ 3 + ( a _ 1 ^ 6 + a _ 1 ^ 8 b ^ 3 + a _ 1 ^ 6 b ^ 4 ) k ^ 4 + ( a _ 1 ^ 8 + a _ 1 ^ 8 b ^ 2 ) k ^ 5 . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} M ( m ) = L ( Y _ { n , r } ) \\otimes L ( Y _ { n , r + 2 p } ) \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} S ( M ; X , Y , Z ) = \\sum _ { A \\subseteq E } ( X - 1 ) ^ { k ( M \\backslash A ^ c ) - k ( M ) } ( Y - 1 ) ^ { | A | - | V | + k _ o ( M \\backslash A ^ c ) } ( Z - 1 ) ^ { k ( M \\backslash A ^ c ) - k _ o ( M \\backslash A ^ c ) } , \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} D _ { 3 } = \\frac { N _ { 0 } 2 ^ { N _ { 0 } } M _ { N _ { 0 } } } { ( 1 + M ) ^ { v _ { 0 } N _ { 0 } } } \\left ( \\frac { D ( c _ { 0 } ) P _ { N _ { 0 } } } { M \\sqrt { v _ { 0 } } } \\right ) ^ { v _ { 0 } } , D _ { 4 } = \\frac { 2 P _ { N _ { 0 } } ^ { 2 } } { M ^ { 2 } ( 3 M + 1 ) ^ { 2 N _ 0 } } . \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} \\begin{array} { c c } & \\varphi \\left ( \\mathrm { C a p } _ t ( x _ 1 , \\dots , x _ { t - 1 } , x _ t ^ g , y _ 1 , \\dots , y _ { t + 1 } ) \\right ) = \\\\ & \\mathrm { C a p } _ t ( v _ 1 , \\dots , v _ { t - 1 } , v _ t , w _ 1 , \\dots , w _ { t + 1 } ) \\otimes b = 0 . \\end{array} \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} Y ^ \\delta _ t ( x ) & \\stackrel { d } { = } \\lambda \\widetilde { Y } ^ n _ t ( x ) + \\lambda _ 2 \\widehat { Y } ^ { n + 1 } _ t ( x ) , \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} \\bar { \\sigma } - \\frac { k ^ { 2 } } { d + 1 } = k \\left [ \\frac { k - 1 } { d } - \\frac { k } { d + 1 } \\right ] = - k \\left [ \\frac { d + 1 - k } { d ( d + 1 ) } \\right ] \\leq 0 . \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} \\rho _ { \\delta , a } ( t ) = \\begin{cases} 1 & \\ | t | \\in [ 0 , a ] , \\\\ 0 & \\ | t | \\in [ a + \\delta , \\infty ) . \\end{cases} \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} g = \\frac { \\sum _ { i \\in \\Lambda } f _ { i } } { \\norm { \\sum _ { i \\in \\Lambda } f _ { i } } } . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} F _ { T V } ( p ) = \\sum _ { i = 1 } ^ n \\lambda _ i ( p ) , \\ ; p \\in \\R ^ { n \\times n } . \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} A _ { \\rho , k } = \\left ( \\begin{array} { c c c c c c } A _ { 1 , 1 } & \\cdots & A _ { 1 , k - 1 } & A _ { 1 , k + 1 } & \\cdots & A _ { 1 , n } \\\\ \\vdots & & \\vdots & \\vdots & & \\vdots \\\\ A _ { n - 1 , 1 } & \\cdots & A _ { n - 1 , k - 1 } & A _ { n - 1 , k + 1 } & \\cdots & A _ { n - 1 , n } \\end{array} \\right ) . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n B _ { n , k } t ^ k = \\sum _ { k = 0 } ^ n C _ { n , k } ( 1 + t ) ^ k , \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} \\omega _ a ( x , y ) : = \\sum _ { n = 0 } ^ \\infty a _ n \\omega _ n ( x , y ) + \\sum _ { n = 1 } ^ \\infty a _ { - n } \\omega _ { - n } ( x , y ) \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} \\zeta _ i ^ { ( k ) } & = \\frac { 1 } { 2 } \\ell _ i ^ { ( k ) } \\\\ \\varphi _ i ^ { ( k ) } & = \\psi _ i ^ { ( k - 1 ) } + \\mbox { ` ` h i g h e r t e r m s . . . '' } \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} N ( t ) = 3 e ^ { - t } - 3 e ^ { - 2 t } + e ^ { - 4 t } \\ , , \\beta = 1 / 2 0 \\mbox { ( n o t e $ N ( 0 ) = 1 $ ) } \\ , . \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} & \\mathbb E \\left [ x ^ T ( t ) P ^ { - 1 } x ( t ) \\right ] \\\\ & \\leq \\mathbb E \\int _ 0 ^ t x ^ T ( s ) ( A ^ T P ^ { - 1 } + P ^ { - 1 } A + \\sum _ { k = 1 } ^ m N _ k ^ T P ^ { - 1 } N _ k + \\sum _ { i , j = 1 } ^ v H _ i ^ T P ^ { - 1 } H _ j k _ { i j } ) x ( s ) d s \\\\ & \\quad + \\mathbb E \\int _ 0 ^ t 2 x ^ T ( s ) P ^ { - 1 } B u ( s ) d s + \\int _ 0 ^ t \\mathbb E \\left [ x ^ T ( s ) P ^ { - 1 } x ( s ) \\right ] \\left \\| u ( s ) \\right \\| _ 2 ^ 2 d s . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} ( A _ { \\lambda } , \\Phi _ { \\lambda } ) = ( s _ { \\lambda } ^ * A , { \\lambda } ^ { - 1 } s _ { \\lambda } ^ * \\Phi ) \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} L _ { \\chi } ^ 2 ( \\Lambda ) = \\{ y \\ | \\ y \\ \\textbf { i s } \\ \\textbf { m e a s u r a b l e } \\ \\& \\ | | y | | _ { \\chi } < \\infty \\} , \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{gather*} h _ 4 ( \\tau ) = \\eta ( 3 \\tau ) ^ 8 = \\sum _ { k \\ge 0 , ~ m , n \\in \\Z \\atop { m \\equiv n \\mod 2 } } ( - 8 ) ^ k \\big ( ( 3 m + 1 ) ^ 3 - ( 3 m + 1 ) 9 n ^ 2 \\big ) q ^ { 4 ^ k ( ( 3 m + 1 ) ^ 2 + 3 n ^ 2 ) } . \\end{gather*}"}
-{"id": "9238.png", "formula": "\\begin{align*} \\mathbf F ( g ) = \\det ( g _ { \\infty } ) ^ { ( k + 1 ) / 2 } \\det ( C \\sqrt { - 1 } + D ) ^ { - k - 1 } F ( g _ { \\infty } \\sqrt { - 1 } ) \\underline { \\chi } ( k ) , \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} \\sup _ { x = ( z , y ) \\in B _ { \\frac { 1 } { 4 } } ( 0 ) \\times Y } \\sup _ { x ' \\in B ^ { g _ { z , t } } ( x , \\frac { 1 } { 8 } ) } \\frac { | { g } _ t ^ \\bullet ( x ) - \\mathbf { P } ^ { g _ { z , t } } _ { x ' x } ( { g } _ t ^ \\bullet ( x ' ) ) | _ { g _ { z , t } ( x ) } } { d ^ { g _ { z , t } } ( x , x ' ) ^ \\alpha } \\leq C _ \\alpha \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} \\Lambda _ { + } ( X ) = \\sum _ { i \\in \\mathcal { I } } \\left ( ( A _ i - X ) ^ T \\otimes ( \\Phi ^ { + } _ { i , T } ( X ) ) \\right ) + \\sum _ { i \\in \\mathcal { I } ^ - } \\left ( ( A _ i - \\overline { X } _ i ) ^ T \\otimes ( \\Phi ^ { + } _ { i , T } ( X ) ) \\right ) ( \\overline { M } ^ + - I _ 4 ) , \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} \\tau ^ - = \\{ i \\mid \\tau _ i = - \\} , \\tau ^ 0 = \\{ i \\mid \\tau _ i = 0 \\} , \\tau ^ + = \\{ i \\mid \\tau _ i = + \\} . \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} l _ 0 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) , & l _ 1 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } [ \\Delta \\varphi + \\langle b , \\nabla \\varphi \\rangle ] \\varrho \\ , d x = 0 \\forall \\varphi \\in C _ 0 ^ \\infty ( \\mathbb { R } ^ d ) . \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal { I } } \\left ( ( A _ i - X ) ^ T \\otimes ( \\partial _ { x _ j } \\Phi ^ { s } _ i ( X ) ) \\right ) + \\sum _ { i \\in \\mathcal { I } ^ { s ' } } \\left ( ( A _ i - \\overline { X } _ i ) ^ T \\otimes ( \\partial _ { x _ j } \\Phi ^ { s } _ i ( X ) ) \\right ) ( \\overline { M } ^ s - I _ 4 ) = I ^ j , \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\Delta _ { I ( f _ c ) } \\Delta _ { I ( f _ c ' ) } = \\Delta _ { I ( f _ s ) } { \\Delta _ { I ( f _ n ) } + \\Delta _ { I ( f _ e ) } \\Delta _ { I ( f _ w ) } } . \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} \\epsilon ( h , \\alpha _ n ) = ( x ( h ) x ( h \\alpha _ n ) , x ( \\alpha _ n ) x ( h \\alpha _ n ) ) _ p . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } g _ { 0 } \\ ! \\ ! \\ ! \\ ! & = \\overline g , \\\\ g ' _ t \\ ! \\ ! \\ ! \\ ! & = \\lambda _ t g _ t \\qquad a . e . \\ t \\in [ 0 , 1 ] , \\end{array} \\right . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} L = \\frac { \\sigma ^ 2 } { 2 } \\Delta - \\mu ( x \\circ \\nabla ) . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} c ( \\xi ) = c ( \\mathfrak d _ { \\xi } ) \\chi ( \\mathfrak f _ { \\xi } ) a ( \\mathfrak f _ { \\xi , N } ) \\sum _ { \\substack { d \\mid \\mathfrak f _ { \\xi } , \\\\ d > 0 } } \\mu ( d ) \\chi _ { - \\xi } ( d ) d ^ { k - 1 } a ( \\mathfrak f _ { \\xi , 0 } / d ) . \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} { \\rm { P } } \\left ( 1 \\right ) : \\qquad \\left ( { 1 + x } \\right ) ^ 1 _ q = \\left ( { 1 + x } \\right ) \\ge 1 + x = 1 + \\left [ 1 \\right ] _ q x \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} \\sup _ { s \\in [ 0 , T ] } E ^ { P ^ n } \\big [ \\ 1 _ K ( X _ 0 ) \\| X _ { s \\wedge \\tau _ \\infty } \\| ^ p \\big ] = \\sup _ { s \\in [ 0 , T ] } E ^ { P ^ n } \\big [ \\ 1 _ K ( X _ 0 ) \\| X _ s \\| ^ p \\big ] \\leq C _ { p , T } , \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { m - k - 1 } \\binom { m } { t + k + 1 } t ! S _ { a _ { 1 } , b _ { 1 } + a _ { 1 } m } ^ { a _ { 2 } , b _ { 2 } + a _ { 2 } m , p _ { 2 } } \\left ( p _ { 1 } , t \\right ) = \\sum _ { t = k + 1 } ^ { m } \\binom { t - 1 } { k } \\left ( b _ { 1 } + a _ { 1 } t \\right ) ^ { p _ { 1 } } \\left ( b _ { 2 } + a _ { 2 } t \\right ) ^ { p _ { 2 } } . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } \\left [ s , 1 \\right ] \\varphi ( x ) = \\int _ { \\Q _ p } \\varphi ( y ) \\psi ( 2 x y ) d y . \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} n a + ( n b + r ) - i = s + n t , \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} P ( z , d z , \\dots , d ^ k z ) = \\sum _ { | \\alpha | = m } c _ \\alpha ( z ) ( d ^ { 1 } z ) ^ { \\alpha _ 1 } ( d ^ { 2 } z ) ^ { \\alpha _ 2 } \\cdots ( d ^ { k } z ) ^ { \\alpha _ k } , \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\Phi \\colon [ 0 , T ] \\rightarrow \\mathcal { L } _ 2 ( U , L ^ 2 _ \\eta ( S ; \\R ) ) , \\Phi ( t ) u : = \\langle u , g ( \\cdot , t ) \\rangle , \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} S = \\ker W \\subseteq \\R ^ n \\tilde S = \\ker \\tilde W \\subseteq \\R ^ n . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\mathfrak s = \\{ X \\in \\mathfrak r \\rtimes \\mathfrak k \\ | \\ B ( X , Y ) = 0 \\mbox { f o r a l l } Y \\in \\mathfrak k \\} \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} \\begin{cases} c ' _ { k - 1 } = c _ k + c _ { k + 1 } - \\min \\{ c _ { k - 1 } , c _ { k + 1 } \\} , \\\\ c ' _ k = \\min \\{ c _ { k - 1 } , c _ { k + 1 } \\} , \\\\ c ' _ { k + 1 } = c _ { k - 1 } + c _ k - \\min \\{ c _ { k - 1 } , c _ { k + 1 } \\} , \\\\ c ' _ s = c _ s \\ \\ s \\ \\ s \\neq k , k \\pm 1 , \\end{cases} \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\tilde { J } _ \\beta = \\sum _ { 0 \\leq \\gamma \\leq \\beta } \\frac { 1 } { ( v ^ 2 ) _ { \\beta - \\gamma } } v ^ { ( \\gamma , \\gamma ) - 2 ( \\lambda + \\rho , \\gamma ) } c ^ { \\beta - \\gamma } v ^ { \\tau _ \\lambda ( \\beta - \\gamma , \\beta ) } \\cdot \\tilde { J } _ \\gamma , \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} \\Phi _ { \\breve { \\mathbf g } _ p } ( g ) : = \\frac { \\langle \\tau _ p ( g ) \\breve { \\mathbf g } _ p , \\breve { \\mathbf g } _ p \\rangle } { | | \\breve { \\mathbf g } _ p | | ^ 2 } , \\Phi _ { \\mathbf h _ p } ( g ) : = \\frac { \\langle \\tilde { \\pi } _ p ( g ) \\mathbf h _ p , \\mathbf h _ p \\rangle } { | | \\mathbf h _ p | | ^ 2 } , \\Phi _ { \\pmb { \\phi } _ p } ( g ) : = \\frac { \\langle \\omega _ { \\overline { \\psi } _ p } ( g ) \\pmb { \\phi } _ p , \\pmb { \\phi } _ p \\rangle } { | | \\pmb { \\phi } _ p | | ^ 2 } . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} X ^ m = L ^ \\infty ( 0 , T _ 0 ; H ^ m ) \\times \\big ( L ^ \\infty ( 0 , T _ 0 ; H ^ m ) \\cap L ^ 2 ( 0 , T _ 0 ; H ^ { m + 1 } ) \\big ) \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} d _ 1 d _ 2 d _ 3 \\neq 0 , \\ ; \\ ; d _ 1 c _ 2 = d _ 2 c _ 1 , \\ ; \\ ; d _ 1 b _ 3 = d _ 3 b _ 1 , \\ ; \\ ; d _ 2 a _ 3 = a _ 2 d _ 3 , \\ ; \\ ; a _ i , b _ i , c _ i \\neq 0 . \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} \\mathbb P \\Omega _ j = \\{ ( \\omega , [ v ] ) \\ , : \\ , v \\in \\mathcal V _ j ( \\omega ) \\setminus \\mathcal V _ { j - 1 } ( \\omega ) \\} . \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} \\rho ^ k _ { n , e } ( V ) \\ : & = \\ \\dim U ( n , e ) - k \\left ( k - \\chi ( C , V \\otimes E ) \\right ) \\\\ & = \\ n ^ 2 ( g - 1 ) + 1 - k \\left ( k - r e - n d + r n ( g - 1 ) \\right ) . \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & - 1 & 0 & \\cdots & 0 \\\\ - 1 & 2 & - 1 & \\ddots & \\vdots \\\\ 0 & \\ddots & \\ddots & - 1 & 0 \\\\ \\vdots & \\ddots & - 1 & 2 & - 1 \\\\ 0 & \\cdots & 0 & - 2 & 2 \\end{pmatrix} \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} q _ 2 ( t ) = 2 b _ { 2 , \\inf } ( t ) \\underbar r _ 2 + b _ { 1 , \\inf } ( t ) \\underbar r _ 1 + \\frac { \\chi _ 2 \\left ( k \\underbar r _ 1 + l \\underbar r _ 2 \\right ) } { 2 d _ 3 } , \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} [ U _ { \\nu } ( t ) ] _ { \\upsilon } = U _ { \\nu + \\upsilon } ( t ) = [ U _ { \\upsilon } ( t ) ] _ { \\nu } \\quad ( \\nu , \\upsilon \\in \\mathbf { R } , t \\in \\mathbf { N } , U _ { \\nu } ( t ) \\neq 0 , U _ { \\upsilon } ( t ) \\neq 0 , \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} q : = \\sum _ k q _ { z _ k } u _ k , \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} f ' _ j f _ i - f ' _ i f _ j = ( j - i ) f _ { i + j - 1 } , g ' _ j f _ i - f ' _ i g _ j = j g _ { i + j - 1 } g ' _ j g _ i = g ' _ i g _ j . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} t _ { i , n } ^ { ( 2 ) } = \\begin{cases} i / n , & n , \\\\ i / ( 2 n ) , & n , \\end{cases} \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\mathfrak { C M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) \\ , \\Bigl ( \\frac { \\Gamma ( 1 - \\tau ) } { \\pi \\ , \\Gamma ( \\tau ) } \\Bigr ) ^ { \\frac { q } { \\tau } } \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { C M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) \\Bigl ( \\frac { \\Gamma ( 1 - \\frac { 1 } { \\tau } ) } { \\pi \\ , \\Gamma ( 1 / \\tau ) } \\Bigr ) ^ { q } \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} 0 < \\mu _ A ( f ) = \\frac { 1 } { \\lambda ( A ) } \\int _ A f ( \\iota ( a ) ) \\ , d \\lambda ( a ) , \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} C ( 2 , i ) = \\sum _ { k } B _ { i - 1 , k } = \\sum _ { k } C _ { i - 1 , k } 2 ^ k , \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} V _ n ( \\lambda , \\ell ) - V _ n ( \\lambda - 1 , \\ell ) = V _ { n - 1 } ( \\lambda , \\ell ) - \\delta _ { n , 1 } , \\ \\ \\ n \\ge 1 \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} \\begin{cases} \\underbar s _ 1 ^ { k + 1 } - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 1 ^ { k + 1 } + \\epsilon \\cr \\underbar s _ 2 ^ { k + 1 } - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 2 ^ { k + 1 } + \\epsilon \\end{cases} \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} \\Pi = \\begin{pmatrix} P & P & P & \\cdots & P & 1 \\\\ P & L _ { 1 , 1 } & L _ { 1 , 2 } & \\cdots & L _ { 1 , b } & 1 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ P & L _ { a , 1 } & L _ { a , 2 } & \\cdots & L _ { a , b } & 1 \\\\ 1 & 1 & 1 & \\cdots & 1 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} C _ { n } = M _ { n } - \\sum _ { j = 1 } ^ { n - 1 } \\binom { n - 1 } { j - 1 } M _ j C _ { n - j } \\ , . \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} h _ d b ^ { w _ d } z \\equiv 0 \\ ; ( N ) & \\Leftrightarrow b ^ { w _ d } h _ d z = k b ^ m \\Leftrightarrow h _ d z = k b ^ { m - w _ d } \\Leftrightarrow h _ d z = k ' z b ^ { m - w _ d } \\\\ & \\Leftrightarrow h _ d = k ' b ^ { m - w _ d } \\Leftrightarrow b ^ { m - w _ d } \\mid h _ d , \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} t = 2 \\left ( \\left \\lfloor \\frac { m + 1 } { 2 } \\right \\rfloor - \\frac { d - 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\left \\Vert c \\right \\Vert _ { Y } : = \\sup _ { t \\in \\left [ 0 , T \\right ] } \\sum _ { j = 1 } ^ { n } \\left | c _ { j } \\left ( t \\right ) \\right | \\quad c = \\left ( c _ { j } \\right ) _ { 1 \\le j \\le n } . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\Theta ) ( e ) = - \\mathrm { d i v } ( \\check { \\Theta } ) ( \\check { e } ) \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} \\varphi ( u ) = \\frac { e ^ { - u ^ 2 / 2 } } { \\sqrt { 2 \\pi } } , \\qquad Q ( \\mu ) = \\int _ \\mu ^ { + \\infty } \\varphi ( u ) d u \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\ , \\longmapsto \\ , \\begin{pmatrix} P ( x , y ) \\\\ Q ( x , y ) \\end{pmatrix} \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & \\tilde { K } _ { 2 } ( s ) \\\\ & = ( 1 - p ( s ) \\sigma _ { z } ( s ) ) ^ { - 1 } \\left \\{ p ( s ) \\sigma _ { y } ( s ) + 2 \\left [ \\sigma _ { x } ( s ) + \\sigma _ { y } ( s ) p ( s ) + \\sigma _ { z } ( s ) K _ { 1 } ( s ) \\right ] \\right \\} \\tilde { P } ( s ) \\\\ & \\ \\ + ( 1 - p ( s ) \\sigma _ { z } ( s ) ) ^ { - 1 } \\left \\{ \\tilde { Q } ( s ) + p ( s ) ( 1 , p ( s ) , K _ { 1 } ( s ) ) D ^ { 2 } \\sigma ( s ) ( 1 , p ( s ) , K _ { 1 } ( s ) ) ^ { \\intercal } \\right \\} . \\end{array} \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\omega _ { 0 n } = B _ { 2 \\Lambda { n } } \\left ( M _ { 0 n } \\right ) \\cap { \\Omega _ { n } } , \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} H ( \\rho , \\tau ) = \\lim _ { p \\rightarrow 1 } \\frac { 1 - \\left \\| \\rho \\right \\| _ { L ^ p ( M , \\tau ) } } { p - 1 } = \\lim _ { p \\rightarrow 1 } \\frac { p } { 1 - p } \\log ( \\left \\| \\rho \\right \\| _ { L ^ p ( M , \\tau ) } ) = \\lim _ { p \\rightarrow 1 } h _ p ( \\rho , \\tau ) . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } ( | \\overline { C } ^ * ( p _ 1 + p _ 2 , f , \\pi _ n ) _ T - \\overline { C } ^ { * , r } ( p _ 1 + p _ 2 , f , \\pi _ n ) _ T | > \\delta ) = 0 \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\sin ( z ) & = \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n z ^ { 2 n + 1 } } { ( 2 n + 1 ) ! } \\\\ \\cos ( z ) & = \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n z ^ { 2 n } } { ( 2 n ) ! } \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} \\Delta p = - \\partial _ i \\partial _ j ( u ^ i u ^ j ) . \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } \\gamma _ { i , i + k } ( T ^ { p k } - T ^ { - p k } ) + \\gamma _ { i , i - k } ( T ^ { - p k } - T ^ { p k } ) . \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} \\left ( v \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) ^ { r } & = \\left ( \\prod _ { j = 0 } ^ { r - 1 } \\rho ^ { k _ { j } } v \\tau ^ { a _ { 2 } v _ { 2 } j } \\right ) \\left ( \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) ^ { r } \\\\ & = \\rho ^ { l _ { 1 } } v ^ { r } \\tau ^ { l _ { 2 } a _ { 2 } v _ { 2 } } \\left ( \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) ^ { r } , \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} P = \\varepsilon \\frac { \\rho ^ \\gamma } { \\gamma } , \\ \\ \\gamma > 1 . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\big \\{ \\P _ 0 ( \\psi _ n ( A ) = 1 ) + \\P _ 1 ( \\psi _ n ( A ) = 0 ) \\big \\} \\ge 1 / 3 \\ , . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} \\beta _ h u ^ h = \\beta . \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} W _ { \\tilde { \\varphi } _ p , \\xi } ( 1 ) = \\begin{cases} 2 p ^ { - e _ p } & \\chi _ { - \\xi } ( p ) + w _ p \\neq 0 , \\\\ 0 & \\chi _ { - \\xi } ( p ) + w _ p = 0 . \\end{cases} \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} \\mathbb { P } _ { a , b , r , p } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( z \\right ) = \\sum _ { i = 0 } ^ { r p + \\sigma } A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) z ^ { r p + \\sigma - i } , \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\Lambda , \\eta , \\beta , H } ^ { \\vec { h } } ( A ) = \\mathbb { P } _ { \\Lambda , w , \\beta , H } ^ { | \\vec { h } | } ( A | E ( \\Lambda , \\eta ) ) \\leq \\mathbb { P } _ { \\Lambda , w , \\beta , H } ^ { | \\vec { h } | } ( A ) . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\tau ( C , p ) = ( k _ p - 1 ) ^ 2 . \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\int _ M w _ i \\phi _ k \\ , d M = \\int _ M \\bar w _ i \\phi _ k \\ , d M = 0 , \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla \\varphi | ^ 2 = \\int _ \\Omega | U _ * + \\varphi | ^ { 2 ^ { * } - 2 } ( U _ * + \\varphi ) \\varphi - \\int _ \\Omega \\Bigl ( u _ 0 ^ { 2 ^ - 1 } - \\sum _ { j = 1 } ^ k U _ j ^ { 2 ^ * - 1 } \\Bigr ) \\varphi . \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\alpha _ 1 \\nu ' ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ) p ^ { - 1 / 2 } \\mathbf 1 _ { p ^ { - 1 } \\Z _ p } ( x ) \\int _ { \\Q _ p } \\psi ( - 2 x y p - \\gamma ^ { - 1 } y ^ 2 p ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) d y . \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} P _ \\ast ( T ) = S _ a . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} S ( X , X ' ) = \\sum _ { X < n \\leq X ' } \\exp \\Big ( 2 \\pi i \\big ( \\xi P ( n ) + m ( \\varphi _ 1 ( n ) - \\tau \\psi ( n ) ) \\big ) \\Big ) \\Lambda ( n ) . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} \\mathbf L _ { k j } f ( T ( 1 , \\boldsymbol { \\theta } ) ) = \\tilde { \\mathbf L } _ { k j } f ( T ( 1 , \\boldsymbol { \\theta } ) ) . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} ( v _ 1 , \\dots , v _ n ) . ( t _ 1 , \\dots , t _ n ) : = ( t _ 1 v _ 1 , t _ 2 v _ 2 , \\dots , t _ n v _ n ) . \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} C _ { e , k } = \\log \\bigg ( 1 + \\frac { p _ { s } | h _ { e , k } | ^ { 2 } } { \\frac { P _ { B S } } { N _ { T } - 1 } \\| \\mathbf { g } _ { e , k } \\mathbf { T } \\| ^ { 2 } + \\sigma _ { e } ^ { 2 } } \\bigg ) , \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } = & \\frac { \\tau ^ { 1 / \\tau } } { \\Gamma ( 1 - 1 / \\tau ) } \\ , \\beta ^ { - 1 } _ { 2 2 } ( \\tau , b _ 0 = \\tau , \\ , b _ 1 = 1 + \\tau \\lambda _ 1 , \\ , b _ 2 = 1 + \\tau \\lambda _ 2 ) \\times \\\\ & \\times \\beta _ { 1 , 0 } ^ { - 1 } ( \\tau , b _ 0 = \\tau ( \\lambda _ 1 + \\lambda _ 2 + 1 ) + 1 ) , \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Delta u = f & & \\mbox { i n } ~ ~ \\Omega ; \\\\ & u = g & & \\mbox { o n } ~ ~ \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} - \\frac { 2 P _ 3 ( 1 ) } { L } \\sum _ { \\substack { p \\leq y _ 2 \\\\ ( p , q ) = 1 } } \\frac { ( \\log p ) P _ 2 [ p ] } { p } \\frac { 1 } { 2 } \\sum _ { w \\mid q } \\varphi ( w ) \\mu ( q / w ) \\sum _ { n \\equiv \\pm 1 ( w ) } \\frac { 1 } { n ^ { \\frac { 1 } { 2 } } } F \\left ( \\frac { n } { p } \\right ) . \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} L _ 1 = \\int _ { \\mathbb R } \\zeta _ k \\nu _ { n _ k } [ \\mathcal T , \\varphi _ k ] \\mathcal H \\nu ' _ { n _ k } \\leqq \\| \\nu _ { n _ k } \\| \\| [ \\mathcal T , \\varphi _ k ] \\mathcal H \\nu ' _ { n _ k } \\| \\leqq C | \\varphi ' _ k | _ { \\infty } \\| \\nu _ { n _ k } \\| ^ 2 \\leqq C \\frac { | \\varphi ' | _ { \\infty } } { R _ k } \\| { \\bf { h } } _ { n _ k } \\| _ { 1 \\times 1 } ^ 2 = O ( \\epsilon ) . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} L + T = \\begin{pmatrix} L ^ \\perp & 0 \\\\ 0 & T \\end{pmatrix} = L ^ \\perp \\oplus T \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} J _ i \\ll \\vert s _ 0 \\vert ^ { O ( 1 ) } i = 2 , \\ 3 . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\displaystyle \\Delta u - \\frac { n - 2 } { 4 ( n - 1 ) } \\mathcal { R } ( x ) u + \\frac { n - 2 } { 4 ( n - 1 ) } K ( x ) u ^ { \\frac { n + 2 } { n - 2 } } = 0 , \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} x _ { 0 1 } + ( 2 p - 1 ) x _ { 0 p } + ( p - 1 ) x _ { 1 p } + \\cdots + ( p - 1 ) x _ { p - 1 p } = m . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\sum \\limits _ { i ' \\in [ I ] } \\overline { \\zeta } _ { j , i ' } ( z ^ { \\varphi , h } _ { \\iota } ( t ) ) / ( 1 - \\epsilon ^ h _ { j , \\iota } ) = 1 ; \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\int \\limits _ { \\beta _ { k n } } \\ , \\Delta f _ 0 ( M ) \\ , d m _ 3 + \\int \\limits _ { \\mathbb { R } ^ 3 } \\ , \\phi _ { k n } ( M ) \\ , d m _ 3 ( M ) = 0 . \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} \\varphi _ B ( x ) = \\frac { 2 } { \\pi } \\sum _ { k = 1 } ^ { \\infty } ( - 1 ) ^ k ( 2 k - 1 ) \\exp ( - ( 2 k - 1 ) \\beta ) J _ { 2 k - 1 } ( x ) . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} S _ x = 1 \\{ \\eta _ x \\neq \\eta ^ { \\prime } _ x \\} , x \\in \\partial _ { e x } \\Lambda . \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ( t ) = & \\ , \\alpha ( T - t ) ^ { - \\alpha - 1 } \\ , [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } } \\\\ & \\ , + \\frac { C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } \\ , ( T - t ) ^ { - \\alpha m } \\ , [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } } \\\\ \\leq & \\ , \\frac { 2 C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } \\ , ( T - t ) ^ { - \\alpha m } \\ , [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} A ^ T P ^ { - 1 } + P ^ { - 1 } A + \\sum _ { k = 1 } ^ m N ^ T _ k P ^ { - 1 } N _ k + \\sum _ { i , j = 1 } ^ v H _ i ^ T P ^ { - 1 } H _ j k _ { i j } \\leq - P ^ { - 1 } B B ^ T P ^ { - 1 } . \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\abs { b _ j } \\leq \\sum _ { \\ell \\mid j } \\Lambda ( \\ell ) = \\log ( j ) , \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} W _ { p , \\xi } ( t ( a ) ) = \\chi _ { \\psi } ( a ^ { - 1 } ) \\chi _ { \\delta } ( a ^ { - 1 } ) | a | _ p ^ { 1 / 2 } W _ { p , a ^ 2 \\xi } ( 1 ) = W _ { p , a ^ 2 \\xi } ( 1 ) = \\Psi _ p ( a ^ 2 \\xi ; \\alpha _ p ) = \\Psi _ p ( \\xi ; \\alpha _ p ) . \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} 0 \\otimes v \\cong v \\otimes 0 \\cong \\bigg ( v \\otimes \\bigvee _ { a \\in \\varnothing } a \\bigg ) = \\bigvee _ { a \\in \\varnothing } ( v \\otimes a ) = 0 . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\cal { T } = \\inf \\{ n > 0 : H _ n \\not \\in \\Z _ + \\} ~ = ~ \\inf \\{ n > 0 : H _ n \\in \\{ \\vartheta \\} \\cup \\Z _ - \\} , \\Z _ - = \\Z \\setminus \\Z _ + . \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} \\mathcal { M } ( t ) & \\\\ = \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } m _ { u _ \\iota } ( y ) j _ { u _ \\mu } ( x ) \\cdot \\nabla _ x \\psi ( x , y ) + m _ { u _ \\mu } ( x ) j _ { u _ \\iota } ( y ) \\cdot \\nabla _ y \\psi ( x , y ) \\ , d x \\ , d y , & \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} [ \\mathcal { X } _ { | n | + 2 } , \\mathcal { Y } _ { | n | + 2 } ] [ t _ { \\delta _ 1 } ^ { - 4 n } t _ { \\delta _ 2 } ^ { - 8 n } \\cdots t _ { \\delta _ { k - 1 } } ^ { - 4 ( k - 1 ) n } , r _ k ] = [ \\mathcal { X } _ { | n | + 2 } t _ { \\delta _ 1 } ^ { - 4 n } t _ { \\delta _ 2 } ^ { - 8 n } \\cdots t _ { \\delta _ { k - 1 } } ^ { - 4 ( k - 1 ) n } , \\mathcal { Y } _ { | n | + 2 } r _ k ] , \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} d x _ r & = [ A _ { 1 1 } x _ r + B _ 1 u + \\sum _ { k = 1 } ^ m N _ { k , 1 1 } x _ r u _ k ] d t + \\sum _ { i = 1 } ^ v H _ { i , 1 1 } x _ r d M _ i , \\\\ y _ r ( t ) & = C _ 1 x _ r ( t ) , \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} \\partial _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) } ( x _ { L _ { \\infty } / K } ) = - C _ { L _ { \\infty } / K } - U ' _ { L _ { \\infty } / K } + M _ { L _ { \\infty } / K } \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} T _ d : = \\begin{bmatrix} \\Delta & - 1 & * & \\dots & * \\\\ & \\ddots & \\ddots & \\ddots & \\vdots \\\\ & & \\ddots & \\ddots & * \\\\ & & & \\Delta & - 1 \\\\ & & & & 1 \\end{bmatrix} = \\begin{bmatrix} \\Delta I & \\\\ & 1 \\end{bmatrix} + \\left [ t _ { j k } \\right ] _ { j , k = 0 , \\dots , d } , \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} ( r ^ k - \\Vert \\vec s ^ k \\Vert _ 1 ) \\norm { \\vec x ^ { k + 1 } } \\le r ^ k \\norm { \\vec x ^ { k + 1 } } - ( \\vec x ^ { k + 1 } , \\vec s ^ k ) = L ( r ^ k , \\vec s ^ k ) < 0 , \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} [ g , \\sigma ( h ) ] ^ { n + 1 } = [ g , \\sigma ( h ) ^ 2 ] [ \\sigma ( h ) g \\sigma ( h ) ^ { - 1 } , \\sigma ( h ) ] . \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} f _ v ( Q _ n \\cdot ( k _ 1 h k _ 2 , 1 ) , s ) = f _ v ( Q _ n \\cdot ( h , 1 ) ( k _ 2 , k _ 1 ^ { - 1 } ) , s ) \\qquad k _ 1 , k _ 2 \\in K \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} C _ { i n v } h ^ { - 1 } \\| P u - I _ { S Z } u \\| \\leq C _ { i n v } h ^ { - 1 } \\Big ( \\| P u - u \\| + \\| u - I _ { S Z } u \\| \\Big ) \\leq C h ^ { - 1 } h \\| \\nabla u \\| = C \\| \\nabla u \\| . \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} C \\le C ( T ) = \\frac { 1 } { 2 } \\frac { ( [ 2 T ] + 1 ) ( [ 2 T ] + 2 ) } { [ 2 T ] + 1 - T } \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} & \\dot M _ 0 = 0 \\ , , \\\\ & \\dot M _ 1 = G ( M _ 1 ) - M _ 1 \\ , , \\\\ & \\dot M _ 2 = 2 [ G ( M _ 1 ) M _ 1 - M _ 2 + \\sigma ] \\ , . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} | J | ^ 2 = Q ^ 3 . \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} e \\left ( \\frac { n } { m \\ell _ 1 \\ell _ 3 v w } \\right ) = 1 + O \\left ( \\frac { N } { M L _ 1 L _ 3 V W } \\right ) . \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} v _ k = m ! \\sum _ { j = 0 } ^ k \\binom { - r } { j } \\beta ^ j \\dfrac { s ( m + k - j , m ) } { ( m + k - j ) ! } . \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} \\Delta ^ 2 _ x | x - y | = \\begin{cases} - \\frac { ( n - 1 ) ( n - 3 ) } { | x - y | ^ 3 } \\ \\ \\ \\ \\ \\ \\ \\ \\ d \\geq 4 , \\\\ \\\\ - 4 \\pi \\delta _ { x = y } \\ \\ \\ \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ \\ \\ d = 3 . \\end{cases} \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\mathcal { H } ^ u = \\bigcap _ { n \\geq 0 } V _ { N + 1 } ^ n \\mathcal { H } , \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( 1 - \\gamma _ 1 ) & + 1 5 & ( 1 - \\gamma _ 2 ) & + 1 6 4 & \\epsilon & < 1 , \\\\ 3 & ( 1 - \\gamma _ 1 ) & + 1 2 & ( 1 - \\gamma _ 2 ) & + 1 2 0 & \\epsilon & < 2 \\end{aligned} \\right . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} L & = \\exp \\bigl ( \\mathcal { N } ( 0 , 4 \\log 2 ) \\bigr ) \\ ; ( ) , \\\\ X _ 2 & = \\beta ^ { - 1 } _ { 2 , 2 } \\bigl ( a = ( 1 , 1 ) , \\ , b _ 0 = 2 , \\ , b _ 1 = b _ 2 = 1 / 2 \\bigr ) , \\\\ X _ 3 & = \\frac { 2 } { y ^ 3 } \\ , d y , \\ ; y > 1 \\ ; ( ) , \\\\ Y & = \\frac { 1 } { y ^ 2 } \\ , e ^ { - 1 / y } \\ , d y , \\ ; y > 0 \\ ; ( ) . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} \\textrm { H e s s } _ W \\mathcal { H } ( \\rho | \\rho ^ * ) ( V _ { \\Phi _ 1 } , V _ { \\Phi _ 2 } ) = & \\int _ { M } \\frac { \\Delta _ \\rho \\Phi _ 1 \\Delta _ \\rho \\Phi _ 2 } { \\rho } + \\big ( \\nabla \\nabla \\log { \\rho } \\nabla \\Phi _ 1 , \\nabla \\Phi _ 2 \\big ) \\rho + \\big ( \\nabla \\nabla h \\nabla \\Phi _ 1 , \\nabla \\Phi _ 2 \\big ) \\rho d x . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\sigma ^ B ( t ) = u _ \\delta ^ { B } ( t ) = \\sigma _ { 0 } ^ { B } + \\beta t , \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} \\delta M _ { s t } = - \\int _ s ^ t ( \\nabla _ x + \\nabla _ y ) ^ 2 M _ r d r - \\Gamma _ { s t } ^ { 1 , * } M _ s + \\Gamma _ { s t } ^ { 2 , * } M _ s + M _ { s t } ^ { \\natural } \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} e _ n = \\left \\{ \\begin{array} { c l } + 1 & ( f ( n ) , p ) = 1 , \\ r _ p ( f ( n ) ^ { - 1 } ) < \\frac { p } { 2 } \\\\ - 1 & ( f ( n ) , p ) = 1 , \\ r _ p ( f ( n ) ^ { - 1 } ) > \\frac { p } { 2 } p \\mid f ( n ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} V _ { \\mu } : = \\{ u \\in V \\mid t _ i . u = v _ i ^ { \\langle h _ i , \\mu \\rangle } u \\ i \\in I \\} . \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta \\varphi _ 1 ( x ) = \\frac { 1 } { 2 } \\varphi _ 1 ( x ) , \\ ; \\mbox { i n } \\ ; \\Omega , \\ ; \\ ; n \\geq 1 , \\\\ \\varphi _ 1 | _ { \\partial \\Omega } = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{gather*} E _ 2 ^ \\ast ( \\tau ) = 1 - \\frac { 3 } { \\pi \\Im ( \\tau ) } - 2 4 \\sum _ { n > 0 } n \\frac { q ^ { n } } { 1 - q ^ { n } } \\end{gather*}"}
-{"id": "7400.png", "formula": "\\begin{align*} y = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} p \\\\ v \\end{bmatrix} + \\begin{bmatrix} O _ { 3 , 2 } & I _ 2 \\end{bmatrix} w . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\gamma ( \\alpha ( g ) ) = \\alpha _ A \\circ \\gamma ( g ) \\circ \\alpha _ A ^ { - 1 } \\quad \\mbox { f o r a l l $ g \\in G $ . } \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} & [ \\gamma _ i , \\gamma _ j ] = 1 , ( 1 \\leq i , j \\leq n ) , \\\\ & ( \\gamma _ 0 \\gamma _ i ) ^ 2 = ( \\gamma _ i \\gamma _ 0 ) ^ 2 , ( 1 \\leq i \\leq n ) . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} ( a , b ) ^ { s _ 1 } = ( a , b + 1 ) , ( a , 0 ) ^ { s _ 2 } = ( a + 1 , 0 ) , \\quad \\mbox { a n d } ( a , 1 ) ^ { s _ 2 } = ( a + 2 , 1 ) . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} & F _ { i _ k } \\cdots F _ { i _ 2 } F _ { i _ 1 } ( w _ i ) = \\delta _ { i _ 1 , i } \\delta _ { i _ 2 , i _ 1 + 1 } \\cdots \\delta _ { i _ k , i _ { k - 1 } + 1 } w _ { i + k } , \\\\ & E _ { j _ k } \\cdots E _ { j _ 2 } E _ { j _ 1 } ( w _ j ) = \\delta _ { j _ 1 , j - 1 } \\delta _ { j _ 2 , j _ 1 - 1 } \\cdots \\delta _ { j _ k , j _ { k - 1 } - 1 } w _ { j - k } . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\langle \\rho , \\psi _ t \\rangle + \\langle \\rho u , \\psi _ x \\rangle = 0 , \\\\ \\langle \\rho u , \\psi _ t \\rangle + \\langle \\rho u ^ 2 , \\psi _ x \\rangle = - \\langle \\beta \\rho , \\psi \\rangle , \\end{array} \\right . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} U ( x ) = [ \\vec u _ { \\ell _ 1 1 } ( x ) \\ \\vec u _ { \\ell _ 2 2 } ( x ) \\ \\dots \\ \\vec u _ { \\ell _ N N } ( x ) ] ^ T . \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} r ^ k \\norm { \\vec x } - ( \\vec x , \\vec s ^ k ) \\ge r ^ k \\norm { \\vec x } - \\Vert \\vec s ^ k \\Vert _ 1 \\norm { \\vec x } = ( r ^ k - \\Vert \\vec s ^ k \\Vert _ 1 ) \\norm { \\vec x } , \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} M = \\begin{bmatrix} I & - E ^ { - 1 } U ^ T \\\\ 0 & I \\end{bmatrix} , \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { 1 } { \\mathbb { E } [ Y _ i ] } \\le & \\lim _ { m \\to \\infty } \\dfrac { 1 } { 2 ^ { m \\epsilon } \\frac { 1 } { m ( 1 + \\epsilon ) + 1 } \\left ( 1 - e ^ { - c _ 1 ( 1 + \\epsilon ) } \\right ) } = 0 \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} \\left \\langle \\mathbf { K } \\left [ 1 - e ^ { - h } \\right ] ( e _ n ) , e _ n \\right \\rangle & = \\sum _ { i = 1 } ^ { + \\infty } \\int _ { E \\times E } \\lambda _ i \\varphi _ i ( x ) \\varphi _ i ( y ) e _ n ( x ) e _ n ( y ) d x d y \\\\ & = \\sum _ { i = 1 } ^ { + \\infty } \\lambda _ i \\left ( \\int _ { E } \\varphi _ i ( x ) e _ n ( x ) d x \\right ) ^ 2 \\\\ & = \\sum _ { i = 1 } ^ { + \\infty } \\lambda _ i \\left \\langle \\varphi _ i , e _ n \\right \\rangle ^ 2 \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} \\tilde { S } = \\{ E \\in S : \\\\ G ( E + \\iota 0 ) K \\} \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} S \\left ( p _ { 1 } + p _ { 2 } , l \\right ) = \\sum _ { m = 0 } ^ { p _ { 2 } } \\sum _ { j = 0 } ^ { p _ { 1 } } \\binom { p _ { 1 } } { j } m ^ { p _ { 1 } - j } S \\left ( j , l - m \\right ) S \\left ( p _ { 2 } , m \\right ) . \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} F ( t ) [ 0 ] = \\int _ 0 ^ t \\nabla _ x b ( s , \\omega , X _ \\theta ( s ) ( \\omega ) ) F ( s ) [ 0 ] d s + \\int _ 0 ^ t \\nabla _ x \\sigma ( s , \\omega , X _ \\theta ( s ) ( \\omega ) ) F ( s ) [ 0 ] d W ( s ) , F ( 0 ) [ 0 ] = 0 \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} r ^ t ( \\mathbf { S } ^ * , \\delta ^ * ) & \\leq r ^ t ( \\mathbf { S } , \\delta ^ * ) \\forall \\ , \\mathbf { S } \\in \\mathbb { S } \\ ; , \\\\ r ^ r ( \\mathbf { S } ^ * , \\delta ^ * ) & \\leq r ^ r ( \\mathbf { S } ^ * , \\delta ) \\forall \\ , \\delta \\in \\Delta \\ ; . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } M _ 0 ^ n = \\dfrac { v \\ , u ^ T } { \\langle u , v \\rangle } \\ , , \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} p ( x \\star u , y \\star u ) = p ( x , y ) , \\forall x , y \\in E . \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} s _ { i } = | \\{ j > n + 2 - i : \\sigma _ j < \\sigma _ { n + 2 - i } \\} | + 1 . \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} P _ n = n ! | \\mathcal { S R P } _ n | . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} f ( \\alpha ( t _ 0 ) ) - f ( \\alpha ( a ) ) = f ( \\beta ( t _ 0 ) ) - f ( \\alpha ( a ) ) = t _ 0 - a , f ( \\beta ( b ) ) - f ( \\beta ( t _ 0 ) ) = b - t _ 0 , \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\widetilde { c } _ { n i } ( - 2 ( n - i ) ) & = - \\widetilde { c } _ { n i } ( - 2 ( n - i - 1 ) ) + \\widetilde { c } _ { n - 1 i } ( - 2 ( n - i ) ) + \\widetilde { c } _ { n - 1 i } ( - 2 ( n - i - 1 ) ) \\\\ & = 0 + 1 + 0 = 1 . \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} \\mu ( \\gamma ) = \\int ^ { + \\infty } _ { 0 } \\phi ( \\rho ) d \\rho \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} G _ S ( x , y ) ~ = ~ \\sum _ { n = 0 } ^ \\infty \\P _ x ( S ( n ) = y ) , \\ ; x , y \\in \\Z ^ d \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} ( a ; q ) _ n : = & \\prod _ { k = 0 } ^ { n - 1 } ( 1 - a q ^ k ) , \\\\ ( a ; q ) _ \\infty : = & \\prod _ { k = 0 } ^ \\infty ( 1 - a q ^ k ) . \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{gather*} \\xi ' ( \\mathfrak P _ v ) = \\xi _ v ( \\pi _ v ) \\end{gather*}"}
-{"id": "2597.png", "formula": "\\begin{align*} P V ( e ) \\leq f ( e ) = 1 . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} N _ { w , G } ( g ) = | \\{ ( g _ { 1 } , \\cdots , g _ { r } ) \\in G ^ { r } : \\ w ( g _ { 1 } , \\cdots , g _ { r } ) = g \\} | , \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\mu = \\dfrac { 1 } { r + 1 } \\Bigl ( \\sum _ { j = 1 } ^ { r } \\mu _ { j } + \\rho \\Bigr ) . \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} S _ 0 & : = ( x + L ) \\cap \\overline { B _ r ( x ) } , \\\\ \\Sigma _ x & : = \\Sigma \\cap \\overline { B _ r ( x ) } , \\\\ \\tau _ 0 & \\colon S _ 0 \\to S _ 0 ; \\ \\ z \\mapsto z , \\\\ \\delta _ 0 & < \\left ( 4 8 ( 3 C _ 1 ( m ) + 2 ) \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} f _ i ( x ) = a , f _ i ( x ) ^ q = a ^ q , f _ i ( x ) ^ { q ^ 2 } = a ^ { q ^ 2 } . \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} 4 S \\sum _ { \\substack { \\beta \\geq \\frac { 1 } { 2 } - \\frac { R } { \\log q } \\\\ 0 \\leq \\gamma \\leq \\frac { 2 S } { 3 \\log q } \\\\ L ( \\beta + i \\gamma , f ) = 0 } } \\cos \\left ( \\frac { \\pi \\ , \\gamma \\ , \\log q } { 2 S } \\right ) \\sinh \\left ( \\frac { \\pi \\big ( R + ( \\beta - 1 / 2 ) \\log q \\big ) } { 2 S } \\right ) \\ ; \\leq \\ ; I _ 1 ( f ) + I _ 2 ( f ) + I _ 3 ( f ) , \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\mathbf I ( \\vect { v } ) : = \\frac { 1 } { ( \\pi \\hbar ) ^ 2 } \\int \\beta _ { \\vect { z } } ( \\vect { u } , \\overline { \\vect { u } } ) \\mathrm { e x p } \\left ( - \\frac { 1 } { \\hbar } | \\vect u - \\vect v | ^ 2 \\right ) \\mathrm d \\vect { u } \\mathrm d \\overline { \\vect { u } } \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) : \\ , v \\in e } B ( v , e ) & = \\mu \\sum _ { e \\in E ( G ) : \\ , v \\in e } B _ 1 ( v , e ) + ( 1 - \\mu ) \\sum _ { e \\in E ( G ) : \\ , v \\in e } B _ 2 ( v , e ) \\\\ & = \\mu + ( 1 - \\mu ) = 1 . \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} \\int _ { \\mathcal L _ { 2 t } } ( c , d ) _ p \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h = - p ^ { 1 + 3 t } \\mathrm { v o l } ( \\mathcal L _ { 2 t } ) = - p ^ { 1 + t } ( 1 - p ^ { - 1 } ) ^ 2 . \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\begin{gathered} d X _ t = ( k - a X _ t ) d t + \\sigma \\sqrt { X _ t } \\circ d B _ t ^ H , t \\ge 0 , \\end{gathered} \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} x = \\frac { 4 \\mathfrak { S } _ { 0 } } { y ^ { 2 } } - \\frac { 2 \\omega } { K _ { 0 } } \\frac { 1 } { y } , \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} f ( x ) & = \\int _ { 0 } ^ { 1 } [ ( 1 - \\lambda ) + \\lambda x ^ { - 1 } ] ^ { - 1 } d \\mu ( \\lambda ) \\\\ & \\leq \\int _ { 0 } ^ { 1 } [ ( 1 - \\lambda ) + \\lambda x ] d \\mu ( \\lambda ) = 1 - f ' ( 1 ) + f ' ( 1 ) x \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ' ( v ) = 0 F _ j ( v ) - F _ k ( v ) = D _ v ( F _ j ' ( v ) - F _ k ' ( v ) ) \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} r _ { i , k } = s _ { i , k } e ^ { j \\theta _ { i , k } } + n _ { i , k } , \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{gather*} \\begin{aligned} [ \\alpha ] ^ 2 & = [ 1 ] \\ , , & [ \\alpha ] [ \\sigma _ g ] & = [ \\sigma _ g ] \\ , , & [ \\rho ] ^ 2 & = [ 1 ] + \\sum _ { g \\in G _ + } [ \\sigma _ g ] \\ , , \\\\ [ \\alpha ] [ \\rho ] & = [ \\alpha \\rho ] \\ , , & [ \\sigma _ g ] [ \\sigma _ { g } ] & = [ 1 ] + [ \\alpha ] + [ \\sigma _ { | 2 g | } ] \\ , , & [ \\sigma _ g ] [ \\sigma _ h ] & = [ \\sigma _ { | g + h | } ] + [ \\sigma _ { | g - h | } ] \\ , , \\end{aligned} \\end{gather*}"}
-{"id": "8014.png", "formula": "\\begin{align*} \\kappa = \\max \\{ \\alpha ( \\partial _ + ( \\{ 0 , \\overline { e } _ i \\} ) ) : i = 1 , . . . , d \\} \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} L ^ { 1 / \\tau } \\bigl ( \\frac { 1 } { \\tau } \\bigr ) & \\overset { { \\rm i n \\ , l a w } } { = } L ( \\tau ) , \\\\ X ^ { 1 / \\tau } _ i \\bigl ( \\frac { 1 } { \\tau } , \\tau \\lambda \\bigr ) & \\overset { { \\rm i n \\ , l a w } } { = } X _ i ( \\tau , \\lambda ) , \\ ; i = 1 , 2 , 3 , \\\\ M ^ { 1 / \\tau } _ { \\bigl ( a ( 1 / \\tau ) , \\ , x ( 1 / \\tau , \\tau \\lambda ) \\bigr ) } & \\overset { { \\rm i n \\ , l a w } } { = } M _ { \\bigl ( a ( \\tau ) , \\ , x ( \\tau , \\lambda ) \\bigr ) } . \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{gather*} L _ M ( w \\ , | \\ , a ) = - \\frac { 1 } { M ! } B _ { M , M } ( w \\ , | \\ , a ) \\ , \\log ( w ) + \\sum \\limits _ { k = 0 } ^ M \\frac { B _ { M , k } ( 0 \\ , | \\ , a ) ( - w ) ^ { M - k } } { k ! ( M - k ) ! } \\sum \\limits _ { l = 1 } ^ { M - k } \\frac { 1 } { l } + R _ M ( w \\ , | \\ , a ) , \\\\ R _ M ( w \\ , | \\ , a ) = O ( w ^ { - 1 } ) , \\ , | w | \\rightarrow \\infty , \\ , | \\arg ( w ) | < \\pi . \\end{gather*}"}
-{"id": "6500.png", "formula": "\\begin{align*} | P _ 1 ^ { 2 ^ * - 2 } \\bigl ( u _ 0 + \\sum _ { j = 2 } ^ k P _ j ) | \\le C U _ 1 ^ { 2 ^ * - 2 } \\le \\lambda ^ { - \\frac { N - 2 } 2 } \\frac { \\lambda ^ { \\frac { N + 2 } 2 } } { ( 1 + \\lambda | y - \\xi _ 1 | ) ^ { \\frac { N + 2 } 2 + \\tau } } , y \\in S . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} h _ { \\widetilde { \\nu } _ i } ( f ^ n ) = h _ { \\nu ^ i } ( \\sigma ) > h _ { \\omega ^ i } ( \\sigma ) - \\eta = h _ { \\widetilde { \\omega } _ i } ( f ^ n ) - \\eta . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} ( - \\Delta + q ) u = 0 \\Omega , \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & A _ 5 & B _ 7 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} M \\simeq \\mathbb { R } ^ { 4 } : \\sum \\limits _ { i = 1 } ^ { 3 } m _ { i } \\mathbf { a } _ { i } = 0 \\mathbf { a } _ { i } \\in \\mathbb { R } ^ { 2 } \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} \\left [ u _ 1 , \\dots , u _ { p } , u _ { p + 1 } , \\dots , u _ q \\right ] c = g . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} \\begin{aligned} \\rho _ 1 ( \\vect z ) & = z _ 1 z _ 3 + z _ 2 z _ 4 , & \\rho _ 2 ( \\vect z ) & = \\imath ( z _ 1 z _ 3 - z _ 2 z _ 4 ) \\\\ \\rho _ 3 ( \\vect z ) & = \\imath ( z _ 1 z _ 4 + z _ 2 z _ 3 ) , & \\rho _ 4 ( \\vect z ) & = z _ 1 z _ 4 - z _ 2 z _ 3 . \\end{aligned} \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} q \\left ( 0 _ { X } ^ { \\ast } \\right ) = \\inf _ { S \\in U , { { { \\lambda } \\in } } \\mathbb { R } _ { + } ^ { S } } \\left \\{ { { { \\sum \\limits _ { s \\in S } { { \\lambda _ { s } b _ { s } : } \\sum \\limits _ { s \\in S } { { \\lambda _ { s } a _ { s } ^ { \\ast } = } } } } - c ^ { \\ast } } } \\right \\} = - \\sup \\mathrm { ( O D P ) } _ { 0 _ { X } ^ { \\ast } } . \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} \\mathbf A _ k = \\mathbf M \\mathbf E _ k \\mathbf M ^ { - 1 } . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} T _ \\alpha \\big ( \\sum _ { g \\in G } ( \\delta _ g \\otimes x _ g ) \\big ) = \\sum _ { g \\in G } ( \\delta _ g \\otimes \\alpha _ g ^ { - 1 } ( x _ g ) ) . \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } a \\\\ & b \\end{array} \\right ) ^ { - 1 } \\left ( \\begin{array} { c c } x \\\\ & x ^ { - 1 } \\end{array} \\right ) \\left ( \\begin{array} { c c } a \\\\ & b \\end{array} \\right ) = \\left ( \\begin{array} { c c } x \\\\ & x ^ { - 1 } \\end{array} \\right ) \\neq g ^ { s } , \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} \\nu _ u = \\inf \\{ n \\geq 0 : H _ n \\not \\in E \\star u \\} \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} F ( \\gamma Z ) = j ( \\gamma , Z ) ^ k F ( Z ) \\qquad \\gamma \\in \\Gamma . \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} \\begin{aligned} U \\leq \\Lambda \\mbox { l o c a l l y i n $ \\R ^ n \\times \\left ( 0 , \\infty \\right ) $ } . \\end{aligned} \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ \\infty \\theta _ { B _ R ( 0 ) } ^ \\beta ( r _ i ) & \\geq \\frac 1 { 4 ^ \\beta } \\sum \\limits _ { i = 1 } ^ \\infty \\frac 1 { - \\log ( \\frac { r _ 1 } 4 ) + \\log ( C ^ { i - 1 } ) } \\\\ & \\geq \\frac 1 { 4 ^ \\beta } \\sum \\limits _ { i = 1 } ^ \\infty \\frac 1 { - \\log ( \\frac { r _ 1 } 4 ) + ( i - 1 ) \\log ( C ) } \\\\ & = \\infty . \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} P _ \\mu ( A ) \\geqslant \\sum _ { { \\sigma \\in S _ n } \\atop { \\sigma ( S ) = S } } \\prod _ { i = 1 } ^ { n } a _ { i \\sigma ( i ) } \\ , \\mu ^ { \\ell ( \\sigma ) } \\ ; . \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} | X | = \\frac { 1 } { n ! \\ , \\omega _ n } \\int _ { \\mathbb { R } ^ n } ( 1 + \\left \\| x \\right \\| ^ 2 ) ^ { p + 1 } \\bigl | \\widehat { h } ( x ) \\bigr | ^ 2 \\ , \\mathrm { d } x , \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} \\pi _ { 0 } ( F _ { \\alpha ^ { 1 } \\alpha ^ { 2 } . . . \\alpha ^ { k } } ( \\Lambda _ { 1 } , . . . , \\Lambda _ { m ^ { k } } ) ) = f _ { \\alpha ^ { 1 } \\alpha ^ { 2 } . . . \\alpha ^ { k } } ( \\pi _ { 0 } ( \\Lambda _ { 1 } ) , . . . , \\pi _ { 0 } ( \\Lambda _ { m ^ { k } } ) ) \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} g \\Bigl ( \\frac { x + z + 1 } { x + z } \\Bigr ) = \\frac { y + z + 1 } { y + z } . \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\varphi _ { 0 , B } ^ { \\rm { C F S K } } - \\varphi _ { 0 , A } ^ { \\rm { C F S K } } = 0 { \\rm { = } } 2 \\pi { k _ 0 } \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} x ( t ) = \\sigma _ { 0 } t + \\frac { 1 } { 2 } \\beta t ^ 2 . \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} h = T - U \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G _ 1 \\times G _ 2 ) = ( r - 1 ) ! \\lambda ^ { ( p ) } ( G _ 1 ) \\lambda ^ { ( p ) } ( G _ 2 ) . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} ( S _ I ^ * ) ^ { m _ I } = \\prod _ { i \\in I } ( S _ i ^ * ) ^ { m _ i } . \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} M ( u _ \\mu ) ( t ) = \\int _ { \\R ^ d } | u _ \\mu ( t ) | ^ 2 \\ , d x , \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { 1 \\leq a < b \\leq m } A _ { b , a } ( \\epsilon _ b - \\epsilon _ a ) & = \\nu ^ { ( 0 ) } = \\sum _ { i \\in X } ( \\epsilon _ { c _ i } - \\epsilon _ { a _ i } ) , \\\\ \\sum _ { m < a < b \\leq m + n } A _ { b , a } ( \\epsilon _ { b } - \\epsilon _ { a } ) & = \\nu ^ { ( 1 ) } = \\sum _ { i \\in X } ( \\epsilon _ { b _ i } - \\epsilon _ { d _ i } ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} P ^ { \\vec { \\mathcal { H } } } _ { \\Lambda , \\eta , \\beta } ( \\sigma ) = \\frac { 1 } { Z ^ { \\vec { \\mathcal { H } } } _ { \\Lambda , \\eta , \\beta } } \\exp { \\left [ \\beta \\sum _ { \\{ u , v \\} } \\sigma _ u \\sigma _ v + \\beta \\sum _ { \\{ u , v \\} : u \\in \\Lambda , v \\in \\partial _ { e x } \\Lambda } \\sigma _ u \\eta _ v + \\sum _ { u \\in \\Lambda } \\mathcal { H } _ u \\sigma _ u \\right ] } , \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} y = \\int ^ { \\psi ( t , x , y ) } _ 0 \\frac { d \\eta } { \\hat h _ 1 ( t , x , \\eta ) } . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\dim B _ n ( h , s , t ) = \\sum _ i 2 ^ { L - i } \\binom { L } { i } \\binom { N - 1 } { n - i } = \\sum _ k \\binom { L } { k } \\binom { N + k - 1 } { n } \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N c ^ d _ { i } \\leq C _ { \\mathrm { D L } } . \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} c ( x + \\beta ) y _ { n } ( x + 1 ) - ( x ( c + 1 ) + \\beta c ) y _ { n } ( x ) + x y _ { n } ( x - 1 ) = n ( c - 1 ) y _ { n } ( x ) . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} r ^ { 2 } R _ { 2 } - ( r ^ { 2 } + 2 r + 2 ) R = 0 . \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} B ( U , \\zeta ) = - C ( U , \\psi ) . \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} f _ n ( x ) = \\P _ x ( \\tau _ \\vartheta > n ) / \\P _ e ( \\tau _ \\vartheta > n ) \\geq \\P _ x ( \\tau _ \\vartheta > n + 1 ) / \\P _ e ( \\tau _ \\vartheta > n ) = P f _ n ( x ) \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} \\delta ( \\alpha ) = 1 + W \\left ( \\frac { 2 \\langle b \\rangle + \\upsilon } { \\langle a \\rangle } e ^ { - \\alpha - 1 } \\right ) , \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} F : \\R ^ { n \\times N } \\rightarrow \\R , \\ ; p \\mapsto \\sum _ { i = 1 } ^ n \\rho \\big ( \\lambda _ i ( p ) \\big ) \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\pi _ j = P \\left ( ( - 1 ) ^ M \\cdot ( L ( E _ M ) - \\mu _ M ) + \\frac { 2 } { 9 } \\in I _ j \\right ) , \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} w '' = \\Delta w - \\tfrac { p } { 2 } | w | ^ { p - 2 } \\ , w \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} \\mathcal { H } _ { [ N ] } = \\bigoplus _ { p \\in \\N ^ N } V _ { [ N ] } ^ p \\left ( \\mathcal { W } _ { [ N ] } \\right ) . \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} \\frac 1 { p } + \\frac { 1 } { q } = \\frac 1 2 , \\ \\ 2 < q \\leq \\infty . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} \\exp \\Bigl ( - \\bigl ( \\mathcal { S } _ N \\log \\Gamma _ M \\bigr ) ( q \\ , | \\ , a , b ) \\Bigr ) = & \\prod \\limits _ { n _ 1 , \\cdots , n _ M = 0 } ^ \\infty \\Bigl [ \\frac { q + b _ 0 + \\Omega } { \\prod \\limits _ { j _ 1 = 1 } ^ N q + b _ 0 + b _ { j _ 1 } + \\Omega } \\star \\\\ & \\star \\frac { \\prod \\limits _ { j _ 1 < j _ 2 } ^ N q + b _ 0 + b _ { j _ 1 } + b _ { j _ 2 } + \\Omega } { \\prod \\limits _ { j _ 1 < j _ 2 < j _ 3 } ^ N q + b _ 0 + b _ { j _ 1 } + b _ { j _ 2 } + b _ { j _ 3 } + \\Omega } \\cdots \\Bigr ] , \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} ( B ( q _ 2 ) - B ( q _ 1 ) ) _ l = \\langle q _ 2 - q _ 1 , e _ l r _ l ( q _ 2 ) \\rangle + \\langle q _ 1 , e _ l ( r _ l ( q _ 2 ) - r _ l ( q _ 1 ) ) \\rangle , \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\sigma ^ { \\alpha } & = \\rho ^ { b _ { 1 } } \\sigma ^ { a _ { 1 } } \\tau ^ { a _ { 3 } } \\\\ \\tau ^ { \\alpha } & = \\rho ^ { b _ { 2 } } \\sigma ^ { a _ { 2 } } \\tau ^ { a _ { 4 } } \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} c ( | D | ) a _ { \\chi } ( n ) = \\sum _ { \\substack { d \\mid n , \\\\ d > 0 } } \\left ( \\frac { D } { d } \\right ) \\chi ( d ) d ^ { k - 1 } c ( n ^ 2 | D | / d ^ 2 ) . \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} \\Psi _ 2 = \\Phi _ 2 + \\Theta _ h ^ \\Phi ; \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} T ( \\{ \\Gamma _ 2 \\} _ { D R } ) = [ [ \\alpha ^ { n - 2 , \\ , n } - \\beta ^ { n - 2 , \\ , n } ] _ { \\bar \\partial } ] _ { d _ 1 } \\in E _ 2 ^ { n - 2 , \\ , n } ( X ) . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} f _ i : = \\begin{cases} p _ i & 1 \\leq i \\leq m , \\\\ q _ { i - l + 1 } & m + 1 \\leq i \\leq m + l - 1 , \\\\ 0 & i \\leq 0 i \\geq m + l . \\end{cases} \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = N _ { k - 1 } } ^ { N _ k - 1 } \\abs { m _ { n + 1 } - m _ n } \\Big \\| _ { \\ell ^ 1 } \\leq \\big ( \\Psi _ { N _ { k - 1 } } ^ { - 1 } - \\Psi _ { N _ k } ^ { - 1 } \\big ) \\Psi _ { N _ { k - 1 } } + 2 \\Psi _ { N _ { k - 1 } } ^ { - 1 } \\big ( \\Psi _ { N _ { k - 1 } } - \\Psi _ { N _ k } \\big ) . \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\mu _ { p } ( \\Gamma _ { r } ) = \\sum _ { \\{ i \\in [ 1 , 2 ^ { p } ] \\ , ; \\ , \\lambda _ { i } \\in \\Gamma _ { r } \\} } \\mu _ { p } ( \\{ \\lambda _ { i } \\} ) . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} q _ { j } ( n ) = p \\left ( \\dfrac { n - j ( 2 j - 1 ) } { 2 } \\right ) , \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} t _ { a } t _ { b } t _ { c } t _ { b } t _ { a } ( b ) = t _ { a } t _ { b } t _ { c } ( a ) = b , \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} I _ k ( \\ell , r ) : = I ( U _ * + \\varphi ) . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\tilde P _ m : = \\tilde \\Sigma _ 2 ( \\tilde P _ { m - 1 } ) ^ { \\sigma } \\qquad \\qquad \\tilde P _ { m + 1 } : = \\tilde \\Sigma _ { 1 } \\tilde \\Sigma _ 2 \\tilde P _ { m - 1 } = \\tau ^ { - 1 } \\tilde P _ { m - 1 } , \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} | | c | | _ { p } ( T ) = \\left ( \\frac { \\sum _ { j = 1 } ^ { k ( T ) } | c _ { j } - \\overline { c } | ^ { p } l _ { j } } { \\sum _ { j = 1 } ^ { k ( T ) } l _ j } \\right ) ^ { { 1 } / { p } } , \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} J ( u , v ) ~ \\geq ~ \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\log \\delta ^ { \\kappa | v _ n - u _ n | } ~ = ~ \\kappa | u - v | \\log \\delta \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} \\mathcal C ( s _ 1 ) = \\left \\{ u \\in \\R ^ p : \\max \\left ( \\| u \\| , \\frac { \\| u \\| _ 1 } { 2 \\sqrt { s _ 1 } } \\right ) \\le \\frac { \\| \\Sigma ^ { 1 / 2 } u \\| } { \\theta _ 1 } \\right \\} . \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} I I \\leq C \\sum _ { k = 2 } ^ n r _ { k - 1 } ^ { 2 / d } \\epsilon _ 1 + C r _ n ^ { 2 / d } \\epsilon _ 1 \\leq C \\epsilon _ 1 . \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} H _ L : = \\bigoplus _ { \\sigma \\in \\Sigma ( L ) } \\Z _ p . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\delta ( \\imath _ 0 , r _ 0 ) = \\delta ( \\imath _ 0 - 2 , r _ 0 ) = 0 . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} f \\cdot g = \\sum _ \\sigma f \\cdot _ \\sigma g . \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} - a \\mu ( | D | ^ 2 + \\sigma ^ 2 ) \\nu = \\frac { \\gamma } { 1 - \\gamma } G _ 1 ( \\nu ) + \\frac { \\sqrt { \\mu } } { \\gamma } | D | \\mathcal T \\nu - \\frac { \\mu } { \\gamma ^ 2 } | D | ^ 2 \\mathcal T _ 1 v \\equiv G _ 2 ( \\nu ) \\in L ^ 2 ( \\mathbb R ) , \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} \\forall s > \\frac { 1 } { 2 } , \\lim _ { t \\to \\pm \\infty } \\| u ( t ) \\| _ { H ^ s } = + \\infty . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} B _ 2 ( m ) = p ^ { 3 m / 2 } ( - 1 ) ^ m \\chi _ { \\psi } ( p ^ m ) \\int _ { \\mathcal B _ 2 ( m ) } e ( h ) ( - d , p ^ m ) \\chi _ { \\psi } ( - d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h , \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} K ^ { \\ast } & = ( \\cos \\varphi ) \\theta ^ { \\prime } ( 1 + \\varphi ^ { \\prime 2 } ) + \\sin \\varphi ( \\varphi ^ { \\prime } \\theta ^ { \\prime \\prime } - \\theta ^ { \\prime } \\varphi ^ { \\prime \\prime } ) \\\\ & = \\frac { 1 } { v ^ { 3 } } \\left \\{ ( \\cos \\varphi ) \\dot { \\theta } ( v ^ { 2 } + \\dot { \\varphi } ^ { 2 } ) + \\sin \\varphi ( \\dot { \\varphi } \\ddot { \\theta } - \\dot { \\theta } \\ddot { \\varphi } ) \\right \\} \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { d , c } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\bar E _ { a , b } \\bar E _ { d , c } } E _ { d , c } E _ { a , b } , & M = N = 1 ; \\\\ E _ { d , c } ^ { ( N ) } E _ { a , b } ^ { ( M ) } , & . \\end{cases} \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} I _ { \\delta } = \\frac { 1 } { 2 \\pi i } \\int \\limits _ { 3 - i \\infty } ^ { 3 + i \\infty } \\Lambda ( 1 / 2 + \\delta + i t + s , f ) \\Lambda ( 1 / 2 + \\delta - i t + s , f ) G ( s + i t ) G ( s - i t ) \\frac { d s } { s - \\delta } , \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} d ( x , y ) = \\sum _ { i = 1 } ^ N d _ i ( x _ i , y _ i ) \\quad \\forall x , y \\in X , \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} \\int _ { \\R ^ d \\times \\R ^ d } \\nabla _ x \\zeta ( t , x ) \\cdot \\nabla _ y \\zeta ( t , y ) \\Delta ^ 2 _ x \\psi ( x , y ) \\ , d x d y & \\\\ = - ( \\nabla \\zeta ( t , \\cdot ) , ( - \\Delta ) ^ { \\frac { 3 - d } 2 } \\nabla \\zeta ( t , \\cdot ) ) \\leq 0 , \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} S \\left ( p _ { 1 } + p _ { 2 } , l \\right ) = \\sum _ { k = 1 } ^ { p _ { 2 } - 1 } \\left ( - 1 \\right ) ^ { p _ { 2 } + 1 + k } s \\left ( p _ { 2 } , k \\right ) S \\left ( p _ { 1 } + k , l \\right ) + \\sum _ { j = 0 } ^ { p _ { 1 } } \\binom { p _ { 1 } } { j } p _ { 2 } ^ { p _ { 1 } - j } S \\left ( j , l - p _ { 2 } \\right ) . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\pi ( g ) Z ( s , f _ k , w _ \\lambda ) = j ( g , I ) ^ { - k } Z ( s , f _ k , w _ \\lambda ) . \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} z _ j & = \\min \\left \\{ t \\ge 0 \\Big | \\ , ( a _ l \\ldots a _ 1 ) _ 3 < 3 ^ l - k , \\ , \\ , a _ i = \\vec 1 _ { x _ j ^ { ( i ) } > t } + 2 \\vec 1 _ { - x _ j ^ { ( i ) } > t } \\right \\} , \\\\ z _ { i j } ^ { ( a _ l \\ldots a _ 1 ) _ 3 } & = \\min \\left \\{ - ( - 1 ) ^ { a _ \\alpha } x _ { i ' } ^ { ( \\alpha ) } - | x _ { j ' } ^ { ( \\beta ) } | \\Big | a _ \\alpha > 0 , a _ \\beta = 0 , i ' , j ' \\in \\{ i , j \\} \\right \\} _ + , \\\\ z _ + & = \\max \\{ z , 0 \\} . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} 0 \\leqslant | u _ i ( z _ i ) | ^ 2 | \\Psi _ i | ^ 2 - | \\Psi | ^ 2 & = \\Big ( 1 - \\prod _ { j \\ne i } f _ { i j } ^ 2 ( z _ i - z _ j ) \\Big ) | u _ i ( z _ i ) | ^ 2 | \\Psi _ i | ^ 2 \\\\ & \\leqslant \\sum _ { j \\ne i } \\frac { C \\| u _ i \\| _ { L ^ \\infty } \\mathbf { 1 } ( | z _ i - z _ j | \\leqslant \\ell ) } { N | z _ i - z _ j | } | \\Psi _ i | ^ 2 . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\frac { ( u + v ) ( u + w ) } { u ^ 2 ( v + w ) v w } = \\frac { 1 } { ( v + w ) v w } + \\frac { 1 } { u v w } + \\frac { 1 } { u ^ 2 ( v + w ) } . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\dot { P } = A ^ * P + P A + C ^ * C - P B B ^ * P , P ( 0 ) = G ^ * G , \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , 1 \\right ) = \\left ( r ! \\binom { a + b } { r } \\right ) ^ { p } \\prod _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\binom { \\alpha _ { s } + \\beta _ { s } } { r _ { s } } \\right ) ^ { p _ { s } } - S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , 0 \\right ) , \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} { \\rm t r } ( - C ^ T C _ r X '^ T ) = { \\rm t r } ( ( - X A _ r '^ T + B B _ r '^ T ) ^ T Y ) . \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\left \\langle \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right \\rangle \\ \\ a _ { 1 } , a _ { 2 } , a _ { 3 } = 0 , . . . , p - 1 \\ \\ ( a _ { 1 } , a _ { 2 } , a _ { 3 } ) \\neq ( 0 , 0 , 0 ) , \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\sum _ { i j = k } S _ i T _ j , \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} \\delta u _ { s t } ^ n = \\int _ s ^ t \\Delta u _ r ^ n + \\mathrm { d i v } F ( u _ r ^ n ) d r + ( A _ { s t } ^ 1 ( n ) + A _ { s t } ^ 2 ( n ) ) u _ s ^ n + u _ { s t } ^ { n , \\natural } , \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} L ( f , s ) = \\prod _ p ( 1 - a _ p p ^ { - s } + e _ p p ^ { \\ell - 1 } p ^ { - 2 s } ) ^ { - 1 } , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} J _ { 5 } & = J _ { 3 } + 4 \\mathfrak { S } _ { 0 } J _ { 4 } = \\frac { 2 u _ { 1 } } { w _ { 0 } } - \\frac { 2 K _ { 1 } } { K _ { 0 } ^ { 2 } } J _ { 6 } = - \\frac { 2 J _ { 4 } } { K _ { 0 } } = \\frac { K _ { 1 } } { K _ { 0 } ^ { 2 } w _ { 0 } } \\\\ a & = 4 \\mathfrak { S } _ { 0 } J _ { 2 } \\ , \\ b = J _ { 5 } - \\frac { 2 J _ { 2 } } { K _ { 0 } } , c = J _ { 6 } d = 4 \\mathfrak { S } _ { 0 } e = - \\frac { 2 } { K _ { 0 } } \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\lambda ^ { n } \\right ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\lambda ^ { n } } { z ^ { n } } = \\frac { 1 } { 1 - \\frac { \\lambda } { z } } = \\frac { z } { z - \\lambda } , \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} \\limsup _ { t \\to T _ * } \\| u ( t ) \\| _ { \\dot B ^ { - 1 + d / p } _ { p , q } ( \\R ^ d ) } = \\infty . \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} | A | & \\leq \\sum _ { i = 0 } ^ { \\min \\{ k , N - 1 \\} } \\sum _ { z \\in B _ i ( x , y ) } | A ( z , i ) | \\\\ & \\leq \\sum _ { i = 0 } ^ { \\min \\{ k , N - 1 \\} } | B _ i ( x , y ) | \\tbinom { N - 1 + i } { N - 1 } \\\\ & \\leq \\sum _ { i = 0 } ^ { N - 1 } N ^ i \\tbinom { N - 1 + i } { N - 1 } , \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} \\mathbb { P } ^ * ( \\rho ) = \\frac { 1 } { K } e ^ { - \\frac { \\mathcal { F } ( \\rho ) } { \\beta } } , \\textrm { w h e r e $ K = \\int _ { \\mathcal { B } } \\Pi ( \\rho ) ^ { - \\frac { 1 } { 2 } } e ^ { - \\frac { \\mathcal { F } ( \\rho ) } { \\beta } } d \\textrm { v o l } $ } . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} | | c | | _ { p } ( T ) = \\left ( \\frac { \\int _ 0 ^ L | c ( X , T ) - \\bar { c } | ^ p \\ , \\mathrm { d } X } { \\int _ 0 ^ L \\mathrm { d } X } \\right ) ^ { 1 / p } , \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} T _ { 1 / c , v } = U _ { 1 / c } ^ { - 1 } V T _ { 1 / c } ^ v . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d \\Bigg ( \\sum _ { j , k = 1 } ^ d b '' _ { i j k } ( X ( s ) ) \\tilde { Y } _ { k n } ( r , s ) \\mathcal { Y } ^ { ( \\epsilon ) } _ { j m } ( s ) \\Bigg ) ^ 2 \\leq C _ 2 ^ 2 \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} f _ G ( p ) : = \\bigg ( \\frac { \\lambda ^ { ( p ) } ( G ) } { \\Delta } \\bigg ) ^ { p / ( p - r ) } , ~ g _ G ( p ) : = \\bigg ( \\frac { \\lambda ^ { ( p ) } ( G ) } { \\delta } \\bigg ) ^ { p / ( p - r ) } . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} c ^ { m - n } n ! ( 1 + \\beta ) _ { m - 1 } M _ { n } ( m ; \\beta , c ) = c ^ { n - m } m ! ( 1 + \\beta ) _ { n - 1 } M _ m ( n ; \\beta , c ) , \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} Q _ { 2 r } ( i ) = Q _ { 2 r , 1 } ( i ) + Q _ { 2 r , - 1 } ( i ) . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} a _ { j k } ^ * ( z ) = \\sum _ { \\ell = j } ^ d \\frac { p _ { j { \\ell } } ^ * ( z ) } { ( z ^ { - 1 } - 1 ) ^ { \\ell - j } } \\left ( ( z ^ { - 1 } + 1 ) \\widetilde b _ { { \\ell } k } ^ * ( z ) - \\sum _ { r = 0 } ^ { k - 1 } w _ { k , r + 1 } \\frac { \\widetilde b _ { { \\ell } r } ^ * ( z ) } { z ^ { - 1 } - 1 } \\right ) , j , k = 0 , \\dots , d . \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\underset { { \\tilde \\tau } _ 1 \\rightarrow \\tau _ 1 } \\lim \\nu _ { \\theta _ 1 } ( { \\tilde \\tau } _ 1 ) = \\underset { { \\tilde \\tau } _ 1 \\rightarrow \\tau _ 1 } \\lim \\int \\limits _ { \\Theta _ 2 } \\nu _ { \\theta } ( { \\tilde \\tau } _ 1 , \\tau _ 2 ) \\ , d \\tau _ 2 = \\int \\limits _ { \\Theta _ 2 } \\nu _ { \\theta } ( \\tau _ 1 , \\tau _ 2 ) \\ , d \\tau _ 2 = \\nu _ { \\theta _ 1 } ( \\tau _ 1 ) \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} & \\log \\prod _ p \\big ( 1 + p ^ { \\alpha } r ( p ) ^ 2 \\omega ( p ) \\big ) - \\log \\prod _ p \\big ( 1 + r ( p ) ^ 2 \\omega ( p ) \\big ) \\\\ & \\qquad = \\sum _ p \\log \\left ( 1 + \\frac { ( p ^ { \\alpha } - 1 ) r ( p ) ^ 2 \\omega ( p ) } { 1 + r ( p ) ^ 2 \\omega ( p ) } \\right ) \\\\ & \\qquad \\leqslant \\alpha \\sum _ p \\log p \\cdot r ( p ) ^ 2 \\omega ( p ) + O \\bigg ( \\alpha ^ 2 \\sum _ p \\log ^ 2 p \\cdot r ( p ) ^ 2 \\omega ( p ) \\bigg ) \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} [ h \\alpha _ n , \\epsilon ] = \\left [ \\left ( \\begin{array} { c c } c ^ { - 1 } p ^ { - n } & \\ast \\\\ 0 & c p ^ { n } \\end{array} \\right ) , e \\right ] \\left [ \\left ( \\begin{array} { c c } 0 & - 1 \\\\ 1 & d c ^ { - 1 } p ^ { - 2 n } \\end{array} \\right ) , 1 \\right ] , \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\lim _ { N \\to \\infty } \\langle u \\otimes v , P _ x \\otimes P _ y U _ R ( x - y ) P _ x \\otimes P _ y u \\otimes v \\rangle = 4 \\pi \\int _ { \\R ^ 3 } | u ( x ) | ^ 2 | v ( x ) | ^ 2 \\d x . \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\frac { \\partial x } { \\partial t } = \\alpha \\left ( \\frac { \\partial ^ 2 x } { \\partial z ^ 2 _ 1 } + \\frac { \\partial ^ 2 x } { \\partial z ^ 2 _ 2 } \\right ) , \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} m = 1 + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( b ^ y _ { \\ell } - 1 ) . \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\Omega _ z ( S _ 2 ^ { h + k } ( S _ 2 ^ * ) ^ k ) & = ( S _ 2 ^ { h + k } ( S _ 2 ^ * ) ^ k v , v ) = ( S _ 2 ^ { k } v , v ) - \\sum _ { i = 1 } ^ { 2 ^ k - 1 } ( S _ 2 ^ h U ^ i S _ 2 ^ { k } ( S _ 2 ^ * ) ^ k U ^ { - i } v , v ) \\\\ & = - \\sum _ { i = 1 } ^ { 2 ^ k - 1 } \\bar { z } ^ i ( U ^ { 2 ^ h i } S _ 2 ^ h S _ 2 ^ { k } ( S _ 2 ^ * ) ^ k v , v ) = - \\sum _ { i = 1 } ^ { 2 ^ k - 1 } \\bar { z } ^ i ( S _ 2 ^ h S _ 2 ^ { k } ( S _ 2 ^ * ) ^ k v , U ^ { - 2 ^ h i } v ) \\\\ & = - \\sum _ { i = 1 } ^ { 2 ^ k - 1 } z ^ { 2 ^ h i - i } ( S _ 2 ^ h S _ 2 ^ { k } ( S _ 2 ^ * ) ^ k v , v ) \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} ( 2 m + n ) u ' + m v ' & = 1 , \\\\ ( 2 n + m ) u ' + n v ' & = 1 , \\\\ ( 2 m - n ) u ' + m v ' & = 1 \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ^ q ( 1 , b _ 0 , b _ 1 , b _ 2 ) \\bigr ] = & \\frac { G ( b _ 0 ) } { G ( q + b _ 0 ) } \\frac { G ( q + b _ 0 + b _ 1 ) } { G ( b _ 0 + b _ 1 ) } \\frac { G ( q + b _ 0 + b _ 2 ) } { G ( b _ 0 + b _ 2 ) } \\times \\\\ & \\times \\frac { G ( b _ 0 + b _ 1 + b _ 2 ) } { G ( q + b _ 0 + b _ 1 + b _ 2 ) } . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} t _ a t _ b t _ c ( a ) & = t _ a t _ b ( a ) = b , \\\\ t _ a t _ b t _ c ( b ) & = t _ a ( c ) = c . \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\mathcal { A } _ v [ N _ - ] : = \\{ x \\in \\mathcal { U } _ { v } ^ - \\mid ( x , \\mathcal { U } _ { v , \\mathbb { Z } } ^ - ) _ L \\subset \\mathbb { Z } [ v ^ { \\pm 1 / 2 } ] \\} . \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} 2 \\rho _ { 0 } ^ { 3 } v _ { 0 } v _ { 1 } + 3 \\rho _ { 0 } ^ { 2 } v _ { 0 } ^ { 2 } \\rho _ { 1 } - 4 v _ { 0 } \\mathfrak { S } _ { 1 } + \\frac { 2 \\omega } { K _ { 0 } } [ v _ { 0 } \\rho _ { 1 } + \\rho _ { 0 } v _ { 1 } - \\frac { K _ { 1 } } { K _ { 0 } } \\rho _ { 0 } v _ { 0 } ^ { 2 } ] = 0 \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} z = \\frac { y ( a + b \\omega y + c \\omega ^ { 2 } y ^ { 2 } ) } { d + e \\omega y } , \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} \\frac { ( M _ { 1 } + \\cdots + M _ { \\ell } - 2 ) ! } { ( M _ { 1 } - 1 ) ! M _ { 2 } ! \\cdots M _ { i } ! \\cdots M _ { \\ell } ! } \\left ( M _ { 1 } - 1 + M _ { 2 } + \\cdots + M _ { \\ell } \\right ) = \\frac { ( M _ { 1 } + \\cdots + M _ { \\ell } - 1 ) ! } { ( M _ { 1 } - 1 ) ! M _ { 2 } ! \\cdots M _ { i } ! \\cdots M _ { \\ell } ! } . \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} \\overline { a } = \\overline { b } \\partial a = \\partial b = f _ 1 X _ n - U _ n . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} { \\bf Q } D _ { { \\bf x } , t } ^ + { \\bf u } + { \\bf Q } T _ G { \\bf D } p = { \\bf Q } T _ G ( { \\bf f } ) . \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} \\alpha \\geq \\frac 1 { m - 2 p + 1 } , \\beta = \\frac { 1 + \\alpha ( m - 1 ) } 2 \\ , , \\tau \\geq 1 \\ , . \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} p ( x , \\vartheta ) = 1 - \\sum _ { y \\in E } \\mu ( y - x ) \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} V _ { \\pi ( w ) g } ( \\pi ( w ) f ) ( z ) = e ^ { 2 \\pi i z \\cdot \\mathcal { I } w } V _ { g } f ( z ) \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k \\right ) = c _ { 1 } ^ { - p _ { 1 } } c _ { 2 } ^ { - p _ { 2 } } \\sum _ { j _ { 1 } = 0 } ^ { p _ { 1 } } \\sum _ { j _ { 2 } = 0 } ^ { p _ { 2 } } \\binom { p _ { 1 } } { j _ { 1 } } \\binom { p _ { 2 } } { j _ { 2 } } a _ { 1 } ^ { j _ { 1 } } a _ { 2 } ^ { j _ { 2 } } \\left ( b _ { 1 } c _ { 1 } - a _ { 1 } d _ { 1 } \\right ) ^ { p _ { 1 } - j _ { 1 } } \\left ( b _ { 2 } c _ { 2 } - a _ { 2 } d _ { 2 } \\right ) ^ { p _ { 2 } - j _ { 2 } } S _ { c _ { 1 } , d _ { 1 } } ^ { c _ { 2 } , d _ { 2 } , j _ { 2 } } \\left ( j _ { 1 } , k \\right ) . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta y = u & \\Omega , \\\\ y = 0 & \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\xi = \\xi ' / s ^ \\ell = \\sum _ { i \\ge \\ell } \\xi _ i t ^ i s ^ { i - \\ell } \\in F _ \\ell s ^ { - \\ell } = B _ \\ell . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} \\Lambda = C ^ * _ { \\Theta } \\subsetneq \\Theta ~ \\textrm { a n d } ~ t _ a ^ { \\Theta } > t _ a - \\eta . \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} X _ { t } ^ { m } = x _ { n } - \\int _ { 0 } ^ { t } A ^ { \\lambda , \\delta , \\varepsilon } ( X _ { s } ^ { m } ) \\ , d s + \\int _ { 0 } ^ { t } B ^ { \\delta , m } ( X _ { s } ^ { m } ) \\ , d W _ { s } , \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} h _ \\theta ( x \\star u ) ~ = ~ T _ u h _ \\theta ~ = ~ h _ \\theta ~ + ~ G A _ { u } h _ \\theta , u \\in E , \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} \\chi _ { \\psi } ( a ) = ( a , - 1 ) _ F \\frac { \\gamma ( \\psi ^ a ) } { \\gamma ( \\psi ) } , \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} j _ ! ( \\theta _ { \\tilde Z } ) = \\theta \\in N ^ p H ^ { 2 p + k + 1 } ( Y ) . \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} \\mathcal { S } _ N \\bigl ( B ^ { ( f ) } _ M ( w ) \\ , \\log ( w ) \\bigr ) ( q \\ , | \\ , b ) & = \\log ( q ) \\ , \\mathcal { S } _ N \\bigl ( B ^ { ( f ) } _ M ( w ) \\bigr ) ( q \\ , | \\ , b ) + O ( q ^ { - 1 } ) , \\ ; { \\rm i f } \\ ; M < N , \\\\ & = \\log ( q ) \\ , \\mathcal { S } _ N \\bigl ( B ^ { ( f ) } _ M ( w ) \\bigr ) ( q \\ , | \\ , b ) + O ( 1 ) , \\ ; { \\rm i f } \\ ; M = N . \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} ( g \\gamma _ { i } ) ^ { t _ { i } } = \\left ( 0 , \\frac { y } { e ^ { - z - s r _ i } - 1 } ( e ^ { - t _ i ( z + s r _ i ) } - 1 ) , t _ i ( z + s r _ i ) \\right ) , \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} P ^ k ( I d - T _ x P ) \\ 1 ( e ) ~ = ~ \\sum _ { y \\in E } \\P _ e ( X ( k ) = y ) \\left ( 1 - \\sum _ { z \\in E } p ( y \\star x , z ) \\right ) ~ \\geq ~ 0 , \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\| \\mathcal L _ { \\mu _ 2 } ( f , g ) ^ t \\| ^ 2 & = \\alpha \\int _ { \\mathbb R } | | \\xi | \\coth ( \\sqrt { \\mu _ 2 } | \\xi | ) - | \\xi | | ^ 2 | \\widehat { g } ( \\xi ) | ^ 2 d \\xi \\\\ \\\\ & \\leqq \\frac { \\alpha } { \\mu _ 2 } \\| g \\| ^ 2 \\to 0 \\end{array} \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} 2 n + 4 \\le e ( G ) & = e ( G [ W \\cup S ] ) + e _ G ( U , S ) \\le ( 3 ( w + s ) - 6 ) + ( 2 ( n - w ) - 4 ) \\\\ & = 2 n + w + 3 s - 1 0 = 2 n + ( 1 0 - 2 s ) + 3 s - 1 0 = 2 n + s , \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} a _ n = \\sum _ { k = 0 } ^ n s ( n , k ) A _ k ( 0 ) \\Leftrightarrow A _ n ( 0 ) = \\sum _ { k = 0 } ^ n S ( n , k ) a _ k . \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\vec { h } } _ { - } ( \\Lambda ) : = \\{ \\{ z , g ^ { - } \\} : z \\in \\Lambda h _ z < 0 \\} . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} P ^ * ( x ) = x + \\frac { x P ^ * ( x ) } { 1 - x \\widetilde { P } ' ( x ) } . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} D _ r ( n ) = \\sum _ { j = s } ^ n { j - 1 \\choose s - 1 } \\frac { n ! } { ( n - j ) ! } D _ { r - s } ( n - j ) . \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} \\pi _ 1 & = \\left ( \\psi _ 1 , \\psi _ 2 , \\psi _ 3 , \\psi _ 4 , \\psi _ 5 \\right ) , \\\\ \\pi _ 2 & = \\left ( \\psi _ { \\sigma ( 1 ) } , \\psi _ { \\sigma ( 2 ) } , \\psi _ { \\sigma ( 3 ) } , \\psi _ { \\sigma ( 4 ) } , \\psi _ { \\sigma ( 5 ) } \\right ) , \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i \\sum _ { z \\in X } a ( y _ 0 \\cdots y _ { i - 1 } \\bar { z } y _ i \\cdots y _ { n - 1 } ) = 0 . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} B _ { 2 , 2 } ( x \\ , | \\ , a ) = \\frac { x ^ 2 } { a _ 1 a _ 2 } - \\frac { x ( a _ 1 + a _ 2 ) } { a _ 1 a _ 2 } + \\frac { a _ 1 ^ 2 + 3 a _ 1 a _ 2 + a _ 2 ^ 2 } { 6 a _ 1 a _ 2 } , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} \\nabla ^ 0 _ 2 s & = s \\\\ \\nabla ^ { k } _ 2 s & = \\pi _ 2 ^ * d _ 2 \\ ! \\ ! \\left ( \\frac { s } { \\pi _ 2 ^ * \\sigma } \\right ) = _ { \\rm l o c } d _ 2 \\nabla ^ { k - 1 } _ 2 s - \\nabla ^ { k - 1 } _ 2 s \\frac { d \\pi _ 2 ^ * \\sigma } { \\pi _ 2 ^ * \\sigma } , \\ \\ \\ \\ k \\geqslant 0 . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} F ( z , z ' ) : = \\int _ L d \\mu ( \\ell ) F _ \\ell ( z , z ' ) , \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} \\left ( v \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) ^ { p } = 1 \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} s _ p \\left ( \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) \\right ) = \\begin{cases} ( c , d ) _ p & c d \\neq 0 \\mathrm { o r d } _ p ( c ) , \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} B ( \\alpha ) ^ { - 1 } \\Phi ( \\alpha , \\beta , Z ) = B ( \\beta ) ^ { - 1 } \\Phi ( \\beta , \\alpha , Z ^ t ) , \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} M _ X = \\begin{pmatrix} 0 & \\infty & 3 & \\infty \\\\ 2 & 0 & \\infty & 5 \\\\ \\infty & 3 & 0 & \\infty \\\\ \\infty & \\infty & 6 & 0 \\end{pmatrix} M _ X ^ 2 = \\begin{pmatrix} 0 & 6 & 3 & \\infty \\\\ 2 & 0 & 5 & 5 \\\\ 5 & 3 & 0 & 8 \\\\ \\infty & 9 & 6 & 0 \\end{pmatrix} M _ X ^ 3 = M _ X ^ 4 = \\begin{pmatrix} 0 & 6 & 3 & 1 1 \\\\ 2 & 0 & 5 & 5 \\\\ 5 & 3 & 0 & 8 \\\\ 1 1 & 9 & 6 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} F _ { 8 , 9 } ^ { ( 1 ) } = \\frac { 6 7 5 } { 1 6 } \\nu ( \\xi _ 3 ^ { ( 1 ) } ) \\chi _ 5 , F _ { 8 , 9 } ^ { ( 2 ) } = \\frac { 1 0 1 2 5 } { 4 } \\nu ( \\xi _ 3 ^ { ( 2 ) } ) \\chi _ 5 , \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} R i c ( \\omega ( t ) ) = - \\omega ( t ) + \\frac { n + 1 } { 2 n } \\chi + t \\hat \\omega . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} \\Lambda _ { \\sigma } f = \\left . \\sigma \\frac { \\partial u } { \\partial \\nu } \\right | _ { \\partial \\Omega } , \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} & \\lim _ { k \\to \\infty } \\lambda _ k = 0 , \\ \\ \\sum _ { k = 0 } ^ { \\infty } \\lambda _ k = + \\infty . \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} \\Lambda _ { n , s } ^ \\beta \\hat { f } = \\Theta _ { n , s } ^ \\beta \\hat { F } , \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} | f _ { \\omega , 1 } ^ { n + 1 } ( z , w ) | & \\le \\begin{cases} | f _ { \\omega , 1 } ^ n ( z , w ) | & \\textrm { i f } f _ n = f ; \\\\ | f _ { \\omega , 1 } ^ n ( z , w ) | \\left ( 1 + | f ^ n _ { \\omega , 2 } ( z , w ) | \\right ) & \\textrm { i f } f _ n = g ; \\end{cases} \\\\ | f _ { \\omega , 2 } ^ { n + 1 } ( z , w ) | & = \\begin{cases} \\frac { 1 } { 2 } | f _ { \\omega , 2 } ^ n ( z , w ) | & \\textrm { i f } f _ n = f ; \\\\ | f _ { \\omega , 2 } ^ n ( z , w ) | & \\textrm { i f } f _ n = g . \\end{cases} \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} P _ { U } ( n ) = \\max \\{ \\dim \\alpha _ { 1 } , \\dotsc , \\dim \\alpha _ { N } \\} , \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} \\| T ' \\| _ { S _ 1 } \\leq \\left [ \\prod _ { i = 1 } ^ N \\frac { q _ i + 1 } { q _ i - 1 } \\right ] \\| T \\| _ { S _ 1 } . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\phi ( ( T , p ) ) = ( \\sigma , ( T _ \\sigma , p _ \\sigma ) ) , \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} [ n ] = ( 0 < 1 < \\ldots < n ) \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J \\setminus J _ { K ( n ) } } } \\Phi ( \\frac { B _ { t _ { j } } + B _ { t _ { j + 1 } } } { 2 } , t _ { j } ) [ B _ { t _ { j + 1 } } - B _ { t _ { j } } ] = \\int _ { 0 } ^ { T } \\Phi ( B _ { t } , t ) \\circ d B _ { t } \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} \\psi _ 3 ( Z ( G _ 3 ) ) \\subseteq Z ( G _ 2 ) \\times Z ( G _ 2 ) \\times Z ( G _ 2 ) = G _ 2 ' \\times G _ 2 ' \\times G _ 2 ' . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\begin{aligned} \\xi _ \\gamma u ^ { \\gamma ' } p ^ { \\gamma '' } & \\equiv \\frac { a _ \\gamma } { q } u ^ { \\gamma ' } p ^ { \\gamma '' } + \\theta _ \\gamma n ^ { \\gamma ' } p ^ { \\gamma '' } \\pmod 1 \\\\ & \\equiv \\frac { a _ \\gamma } { q } ( r ' ) ^ { \\gamma ' } ( r '' ) ^ { \\gamma '' } + \\theta _ \\gamma u ^ { \\gamma ' } p ^ { \\gamma '' } \\pmod 1 , \\end{aligned} \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\overline { \\varphi } ( t ) - \\widetilde { \\varphi } ( t ) = ( t _ i - t ) \\partial _ t \\widetilde { \\varphi } ( t ) . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} & \\tau _ - ^ { ( l ) } ( s ) = \\sup \\{ t _ { i , n } ^ { ( l ) } \\leq s | i \\in \\mathbb { N } _ 0 \\} , \\\\ & \\tau _ + ^ { ( l ) } ( s ) = \\inf \\{ t _ { i , n } ^ { ( l ) } \\geq s | i \\in \\mathbb { N } _ 0 \\} \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} B = \\begin{bmatrix} a & u ^ T \\\\ u & B ' \\end{bmatrix} \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\iota ( \\gamma ( q ( x ) ) ( a ) ) = x \\iota ( a ) x ^ { - 1 } \\ ; \\ ; \\mbox { f o r a l l $ a \\in A $ a n d $ x \\in \\widehat { G } $ . } \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} E ( \\theta , \\lambda , P _ { B S } ) = \\lambda \\theta P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\rho ^ { 3 } v ^ { 2 } + \\frac { 2 \\omega } { K ^ { \\ast } } \\rho v = 4 \\mathfrak { S } \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} { \\rm { e } } _ q ^ x { \\rm { e } } _ q ^ y = { \\rm { e } } _ q ^ { x + y } . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} \\int _ 0 ^ t S _ { t - s } \\sigma ^ { n } ( X _ { s \\wedge \\tau _ m } ) \\dd W _ s = \\tfrac { \\sin ( \\pi \\theta ) } { \\pi } \\big ( G _ \\theta Y ^ { n , m , \\theta } \\big ) ( t ) , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 1 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; f _ t ( x _ t , x _ { t - 1 } , \\xi _ { t j } ) + \\mathcal { Q } _ { t + 1 } ( x _ { t } , 1 ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ { t j } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n a _ i ^ t = \\sum _ { j = 1 } ^ m b _ j ^ t \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} { } \\det ( z - \\mathcal { C } _ { [ 0 , n - 1 ] } ) - z \\det ( z - \\mathcal { E } _ { [ 1 , n - 1 ] } ) = \\frac { z \\det ( z - \\mathcal { E } _ { [ 1 , n - 1 ] } ) - \\det ( z - \\mathcal { E } _ { [ 0 , n - 1 ] } ) } { \\alpha _ { - 1 } } . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\sum _ i \\alpha _ i y _ i = \\sum _ i \\alpha _ i c _ i + \\boldsymbol \\epsilon ^ 3 \\sum _ i \\alpha _ i n _ i c _ i < u \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} \\dim \\mathcal { H } ^ 1 _ T ( M ) = \\dim \\mathcal { H } ^ 1 _ N ( M ) = 2 g + k - 1 . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\sup _ { 0 \\leq t \\leq T } | | \\P _ { s , x _ 0 } ( X _ { t + s } \\in \\cdot | \\tau _ { A } > s + T ) - \\P _ { x _ 0 } ( X _ t \\in \\cdot | \\tau _ { A _ \\infty } > T ) | | _ { T V } = 0 . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} \\mathfrak { M } ( i q \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\int \\limits _ \\mathbb { R } e ^ { i q x } \\ , f ( x ) \\ , d x . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} \\tau _ n = \\log ( n + \\rho ) - \\log W + o ( 1 ) = \\log n + O ( 1 ) . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} h ( r ) = \\begin{cases} 1 & r < R \\\\ \\sum _ { i = 0 } ^ p \\alpha _ i \\psi _ i ( r ) & r \\ge R , \\end{cases} \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\lambda p _ 1 + ( 1 - \\lambda ) p _ 2 = \\hat { \\lambda } p _ 3 + ( 1 - \\hat { \\lambda } ) p _ 2 \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} B ( v , e ) = \\frac { w ( e ) } { \\sum _ { f : \\ , v \\in f } w ( f ) } \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} \\chi ( M _ n ( \\pi / 2 ) ) = & 2 \\sum _ { s = 1 } ^ m ( - 1 ) ^ { m + s } \\left ( s - \\big \\lfloor s / 2 \\big \\rfloor \\right ) { n \\choose m - s } \\\\ = & 2 ( - 1 ) ^ m \\sum _ { s = 1 } ^ m ( - 1 ) ^ s \\big \\lfloor ( s + 1 ) / 2 \\big \\rfloor { 2 m + 1 \\choose m - s } \\\\ = & 2 ( - 1 ) ^ { m + 1 } \\sum _ { i = 0 } ^ { m + 1 } ( - 1 ) ^ i \\big \\lfloor i / 2 \\big \\rfloor { 2 m + 1 \\choose m + i } . \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} P _ r = \\begin{bmatrix} ( p _ i + p _ j ) ^ r \\end{bmatrix} . \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} d x ( t ) & = [ A x ( t ) + B u ( t ) + \\sum _ { k = 1 } ^ m N _ k x ( t ) u _ k ( t ) ] d t + \\sum _ { i = 1 } ^ v H _ i x ( t - ) d M _ i ( t ) , \\\\ y ( t ) & = { C } x ( t ) , \\ ; \\ ; \\ ; t \\geq 0 . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\Vert z _ i - a _ i \\Vert & \\le \\Big \\Vert z _ i - \\frac { y _ i } { 1 + 1 4 \\rho } \\Big \\Vert + \\Big \\Vert \\frac { y _ i } { 1 + 1 4 \\rho } - y _ i \\Big \\Vert + \\Vert y _ i - a _ i \\Vert \\\\ & < \\Bigl ( 1 - \\frac { 1 } { 1 + 1 4 \\rho } \\Bigr ) + \\Vert y _ i \\Vert \\Bigl ( 1 - \\frac { 1 } { 1 + 1 4 \\rho } \\Bigr ) + 8 6 \\rho \\\\ & < 2 8 \\rho + 8 6 \\rho = 1 1 4 \\rho < \\varepsilon . \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} \\zeta _ i = \\frac { 2 m - 2 \\Phi ( n ) + 2 i } { 2 m - 2 \\Phi ( n ) + 2 i + 1 } \\pi . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} \\tilde { f } ( x ) = L ( x ) f ( x ) \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\begin{pmatrix} a & - b & 0 \\\\ - a _ 1 & - b _ 2 & c \\end{pmatrix} . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{gather*} g ( \\tau ) = \\eta ( 4 \\tau ) ^ 6 , L \\big ( \\psi ^ 2 , s - 1 \\big ) = L \\big ( \\eta ( 4 \\tau ) ^ 6 , s \\big ) , L \\big ( \\eta ( 4 \\tau ) ^ 6 , 2 \\big ) = \\frac { 1 } { 6 4 } B ( 1 / 4 , 1 / 4 ) ^ 2 . \\end{gather*}"}
-{"id": "7655.png", "formula": "\\begin{align*} T & = \\frac { 1 } { 2 } t r ( \\dot { X } \\dot { X } ^ { t } ) = \\frac { 1 } { 2 } ( | \\mathbf { \\dot { x } } _ { 1 } | ^ { 2 } + | \\mathbf { \\dot { x } } _ { 2 } | ^ { 2 } ) \\\\ & = \\frac { 1 } { 2 } \\dot { \\rho } ^ { 2 } + \\frac { \\rho ^ { 2 } } { 8 } ( \\dot { \\varphi } ^ { 2 } + \\dot { \\theta } ^ { 2 } ) + \\frac { 1 } { 2 } \\rho ^ { 2 } \\dot { \\alpha } ^ { 2 } - \\frac { 1 } { 2 } \\rho ^ { 2 } ( \\cos \\varphi ) \\dot { \\alpha } \\dot { \\theta } , \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} x _ { 1 3 } x _ { 2 4 } \\ast _ g x _ { \\sigma ( 1 ) \\sigma ( 2 ) } x _ { \\sigma ( 3 ) \\sigma ( 4 ) } = x _ { i _ 1 i _ 2 i _ 3 i _ 4 } x _ { i _ 5 i _ 6 i _ 7 i _ 8 } . \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} C _ t \\triangleq \\{ y \\in \\mathbb { B } \\colon y = \\alpha ( t ) , \\alpha \\in \\Lambda ( R , \\xi , m ) \\} \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} c _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } \\overline { x } ( a ) + c _ 2 ~ ^ { A B R } D _ b ^ \\alpha \\overline { x } ( a ) = 0 , \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} f _ u \\times f _ { v v } = - a ^ 2 E \\lambda _ v ( \\nu \\times \\nu _ v ) \\times \\nu _ v = a ^ 2 E \\lambda _ v R \\nu \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} \\Sigma ^ s : = A ^ { - 1 } ( S _ { m , n } ^ s ) . \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} W : = w , \\alpha + 1 , \\omega ^ 1 , \\alpha + 2 , \\omega ^ 2 , \\dots , n , \\omega ^ \\tau . \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} \\mu ( \\mathbb { C } _ { \\mathbb { A } } ) = \\mu ^ \\Lambda ( \\mathbb { A } ) \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} \\gamma _ i \\mapsto ( i , \\xi ( i ) ) , & & & \\phi _ { \\mathcal { Q } } ( \\beta ) \\mapsto ( i , p - 2 ) \\ \\ \\Omega _ { \\xi } ( \\beta ) = ( i , p ) \\ \\ \\phi _ { \\mathcal { Q } } ( \\beta ) , \\beta \\in \\Delta _ + . \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} \\Gamma ^ { - 1 } _ M ( z \\ , | \\ , a ) = e ^ { P ( z \\ , | \\ , a ) } \\ , w \\prod \\limits _ { n _ 1 , \\cdots , n _ M = 0 } ^ \\infty { } ' \\Bigl ( 1 + \\frac { z } { \\Omega } \\Bigr ) \\exp \\Bigl ( \\sum _ { k = 1 } ^ M \\frac { ( - 1 ) ^ k } { k } \\frac { z ^ k } { \\Omega ^ k } \\Bigr ) , \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} & \\sigma _ { j - 1 } \\left ( 2 - \\frac { r } { r _ j } \\right ) + \\sigma _ { j } \\left ( 2 - { r } \\right ) + \\sum _ { i = 1 , i \\neq j - 1 , j } ^ { 2 q } 2 \\sigma _ i \\\\ & \\qquad = - 2 d _ { j - 1 } + \\frac { d } { r } + d _ { j + 1 } + 3 t - \\frac { e } { 2 q } \\left ( \\frac { 1 } { r _ j } + \\frac { 1 } { r _ { j + 1 } } \\right ) \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} Q _ 0 = \\begin{cases} x _ { r - 1 } ^ 2 + x _ { r - 1 } x _ r + \\alpha x _ r ^ 2 & \\\\ x _ { r - 1 } ^ 2 - \\beta x _ r ^ 2 & \\end{cases} \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} \\Sigma ( \\bar { x } , \\bar { \\Lambda } , \\bar { \\Lambda } ^ { ( 1 ) } , \\dots , \\bar { \\Lambda } ^ { ( n ) } ) = 0 \\mathrm { w h e n e v e r } \\Sigma ( x , \\Lambda , \\Lambda ^ { ( 1 ) } , \\dots , \\Lambda ^ { ( n ) } ) = 0 \\ ; . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\big | L ( 1 / 2 + \\delta + i t , f ) \\big | ^ 2 = \\left ( \\frac { q } { 4 \\pi ^ 2 } \\right ) ^ { - \\delta } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda _ f ( n ) \\eta _ { i t } ( n ) } { n ^ { 1 / 2 } } V _ { \\delta , t } \\left ( \\frac { 4 \\pi ^ 2 n } { q } \\right ) . \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} K ( z , z ' ) = ( z + z ' ) ^ { - 1 } . \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\sum _ { n = N _ { k - 1 } } ^ { N _ k - 1 } \\abs { m _ { n + 1 } ( x ) - m _ n ( x ) } \\leq 2 \\Psi _ { N _ { k - 1 } } ^ { - 1 } \\frac { \\log x } { \\psi ( x ) } . \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\frac { 1 } { | I | } \\sum _ { i \\in I } \\frac { r _ { i c _ { \\ell _ 0 } } } { r _ { 0 c _ { \\ell _ 0 } } } = \\frac { 1 } { c _ { \\ell _ 0 } - | I | } \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\mu _ M = \\frac { M } { 2 } + \\frac { 4 + r _ 2 ( M ) } { 1 8 } \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} \\gamma ^ n = ( 1 - \\varepsilon ) ^ n + \\gamma _ n ' ~ \\geq ~ ( 1 - \\varepsilon ) ^ n , \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} t r _ { N _ A } ( B A ) = t r _ { N _ A } ( A B ) \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{cases} & F ( u ) = \\frac { 3 } { 4 } \\ , u ^ 4 + \\frac { 5 \\beta } { 6 } \\ , u ^ 6 + F _ { f _ 5 } , \\\\ \\\\ & G ( u ) = u ^ 4 + \\frac { \\beta } { 6 } \\ , u + G _ { f _ 5 } , \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} \\Q _ v ^ { \\pm } ( \\pi _ v ) : = \\left \\lbrace D \\in \\Q _ v ^ { \\times } : \\left ( \\frac { D } { \\pi _ v } \\right ) = \\pm 1 \\right \\rbrace . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } ( { \\mathbb D } ^ { A } { } _ { B } { \\mathbb D } ^ { C } { } _ { D } + \\ , { \\mathbb D } ^ { C } { } _ { D } { \\mathbb D } { } ^ { A } { } _ { B } ) = { \\mathbb D } ^ { ( A } { } _ { ( B } { \\mathbb D } ^ { C ) } { } _ { D ) } + { \\mathbb D } ^ { [ A } { } _ { [ B } { \\mathbb D } ^ { C ] } { } _ { D ] } , \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} [ \\underline { W _ { k , r } ^ { ( i ) } } ] [ \\underline { W _ { k , r + 2 r _ i } ^ { ( i ) } } ] = t ^ { \\alpha / 2 } [ \\underline { W _ { k + 1 , r } ^ { ( i ) } } ] [ \\underline { W _ { k - 1 , r + 2 r _ i } ^ { ( i ) } } ] + t ^ { \\beta / 2 } [ \\underline { S _ { k , r } ^ { ( i ) } } ] . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} \\int _ { A } P _ { r _ { k k _ 0 } } ( x , y ) d y = C \\int _ { \\tilde { A } } \\left ( \\frac { 1 - | \\tilde { x } | ^ 2 } { | \\tilde { y } | ^ 2 - 1 } \\right ) ^ { s / 2 } \\cdot \\frac { 1 } { | \\tilde { x } - \\tilde { y } | ^ n } d \\tilde { y } \\geq \\mu , \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { j _ 1 , . . . , j _ N = 0 } ^ 1 \\sum _ { m _ 1 , . . . , m _ N = 0 } ^ \\infty \\eta _ { 2 m _ 1 + j _ 1 } ^ + ( x _ 1 ) \\otimes \\cdots \\otimes & \\eta _ { 2 m _ N + j _ N } ^ + ( x _ N ) \\\\ & \\otimes B e _ { ( 2 m _ 1 + j _ 1 , . . . , 2 m _ N + j _ N ) } . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\int _ { X } S ^ \\omega \\omega ^ n = \\int _ X n R i c ( \\omega ) \\wedge \\omega ^ { n - 1 } = - 2 \\pi n c _ 1 ( K _ X ) \\cdot \\alpha ^ { n - 1 } . \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} \\{ B ^ { ( N , g ) } _ { R r ^ { y , \\ell } _ k } ( p ^ { y , \\ell } _ k ) \\} ^ { J _ y } _ { \\ell = 1 } \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { n - \\delta + 1 } { n - r \\brack \\delta - 1 } C ' _ r = { n \\brack \\delta - 1 } ( c ^ { n - \\delta + 1 } - \\abs { Y } ) . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} S ^ 3 _ { r } ( C _ { p , q } ( K ) ) = \\begin{cases} S ^ 3 _ { p / q } ( K ) \\# L ( q , p ) , & ~ r = p q , \\\\ S ^ 3 _ { r / q ^ 2 } ( K ) , & ~ m = n p q \\pm 1 , \\\\ ( S ^ 3 \\backslash N ( K ) ) \\cup _ { T } S F S _ r , & o t h e r w i s e . \\\\ \\end{cases} \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\mathcal { G } _ t : = \\varprojlim _ { n \\ge 0 } G _ { n t } . \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } & \\displaystyle A ( x ) = | a ( x ) | ^ { p ( x ) - 1 } | \\nabla a ( x ) | \\\\ & \\displaystyle B ( x ) = | a ( x ) | ^ { p ( x ) } \\\\ & \\displaystyle C ( x ) = | a ( x ) | ^ { p ( x ) - 1 } | \\nabla p ( x ) | \\\\ & \\displaystyle D ( x ) = B ( x ) | \\nabla p ( x ) | . \\end{array} \\right . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} T ' = \\left [ \\prod _ { i = 1 } ^ N \\left ( 1 - \\frac { 1 } { q _ i } \\right ) ^ { - 1 } \\left ( I - \\frac { \\tau _ i } { q _ i } \\right ) \\right ] T \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} v _ { i p } = \\begin{cases} C , & ( i , j ) = ( 0 , p ) , \\\\ \\frac { C } { p - | I | } , & i \\in I , \\\\ 0 , & i \\in \\{ 1 , \\dots , p - 1 \\} \\setminus I , \\end{cases} \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} f ( \\sum _ { i = 1 } ^ n \\lambda _ i e _ i ) = \\sum _ { j = 1 } ^ 4 \\sum _ { i = 1 } ^ \\frac { k a } { 8 } \\lambda _ { \\frac { ( j - 1 ) k a } { 8 } + i } \\lambda _ { n - \\frac { j k a } { 8 } + i } + \\sum _ { i = \\frac { k a } { 2 } + 1 } ^ { \\frac { s } { 2 } } \\lambda _ { i } \\lambda _ { \\frac { s } { 2 } + i } . \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} - 2 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } \\Delta _ x m _ { u _ \\mu } ( t , x ) \\Delta _ y m _ { u _ \\iota } ( t , y ) \\Delta _ x \\psi ( x , y ) \\ , d x d y & \\\\ = - 2 \\int _ { \\R ^ d \\times \\R ^ d } \\sum _ { \\mu = 1 } ^ N \\Delta _ x m _ { u _ \\mu } ( t , x ) \\sum _ { \\iota = 1 } ^ N \\Delta _ y m _ { u _ \\iota } ( t , y ) \\Delta _ x \\psi ( x , y ) \\ , d x d y & \\\\ = - 2 \\int _ { \\R ^ d \\times \\R ^ d } \\Delta _ x \\zeta ( t , x ) \\Delta _ y \\zeta ( t , y ) \\Delta _ x \\psi ( x , y ) , \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} \\psi ^ { - 1 } \\ ( \\psi ( c ) + \\sum _ { i = 1 } ^ d g _ i v _ i \\ ) & = \\psi ^ { - 1 } \\ ( \\sum _ { j \\in J ( z ) } \\ ( \\log ( c _ j ) + \\sum _ { i = 1 } ^ d a _ { i j } g _ i \\ ) e _ j \\ ) = \\sum _ { j \\in J ( z ) } \\exp \\ ( \\log ( c _ j ) + \\sum _ { i = 1 } ^ d a _ { i j } g _ i \\ ) e _ j \\\\ & = \\sum _ { j \\in J ( z ) } \\ ( c _ j \\prod _ { i = 1 } ^ d \\exp ( g _ i a _ { i j } ) \\ ) e _ j = g ' \\cdot c \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\deg ( K _ 2 ( \\lambda ) ^ T \\ , y ( \\lambda ) ) = \\deg ( K _ 2 ( \\lambda ) ) + \\deg ( y ( \\lambda ) ) = \\ell + \\deg ( y ( \\lambda ) ) \\ , , \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} \\mathrm { W a l d } _ { \\psi } ( \\pi ) = \\left \\lbrace \\tilde { \\pi } ^ { \\epsilon } : \\prod _ { v \\in \\Sigma ( \\pi ) } \\epsilon _ v = \\epsilon ( \\pi , 1 / 2 ) \\right \\rbrace . \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} \\| \\psi \\| : = \\sqrt { \\psi ^ * \\psi } \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} A \\subseteq \\omega _ f ( x ) \\subseteq \\cup _ { l = 0 } ^ { \\infty } f ^ { - l } A . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} ( x ) _ n = \\sum _ { k = 0 } ^ n s ( n , k ) x ^ k , x ^ n = \\sum _ { k = 0 } ^ n S ( n , k ) ( x ) _ k , \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} K \\theta ( s ) \\ge \\theta ( 2 s ) = \\int _ { 0 } ^ { 2 s } \\theta _ { + } ^ { \\prime } ( r ) \\ , d r \\ge \\int _ { s } ^ { 2 s } \\theta _ { + } ^ { \\prime } ( r ) \\ , d r \\ge s \\theta _ { + } ^ { \\prime } ( s ) . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} \\dot { k } _ t ( \\varnothing ) = ( L ^ \\Delta _ { \\alpha _ 2 } k _ t ) ( \\varnothing ) = 0 . \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} \\abs { v _ 1 } \\ge \\abs { v _ 2 } \\ge \\cdots \\ge \\abs { v _ n } \\ge \\abs { v _ { n + 1 } } = 0 , \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta \\phi _ 0 ( x ) = 0 , \\ ; \\mbox { i n } \\ ; \\Omega , \\ ; \\ ; n \\geq 3 , \\\\ \\phi _ 0 | _ { \\partial \\Omega } = 0 , \\\\ | x | \\rightarrow \\infty , \\phi _ 0 ( x ) \\rightarrow 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} d _ 1 d _ 2 - 2 ( d _ 1 + d _ 2 ) = ( d _ 1 - 2 ) ( d _ 2 - 2 ) - 4 \\leq 0 . \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ j \\| \\nabla ^ j \\sigma \\| _ { L ^ \\infty ( B ^ { g _ P } ( p , \\rho ) ) } \\leq C _ k ( ( R - \\rho ) ^ { k + \\alpha } [ \\nabla ^ k \\sigma ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( p , R ) ) } + \\| \\sigma \\| _ { L ^ \\infty ( B ^ { g _ P } ( p , R ) ) } ) \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} \\mathcal { L } = \\Delta _ M + | \\sigma | ^ 2 + \\mathrm { R i c } ( \\eta ) \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q \\ , | \\ , b ) & = \\exp \\Bigl ( - \\bigl ( \\mathcal { S } _ N \\log \\Gamma _ M \\bigr ) ( 0 \\ , | \\ , b ) \\Bigr ) \\exp \\Bigl ( - \\frac { 1 } { M ! } \\mathcal { S } _ N \\bigl ( B ^ { ( f ) } _ M ( w ) \\ , \\log ( w ) \\bigr ) ( q \\ , | \\ , b ) + \\\\ & + \\sum \\limits _ { k = 0 } ^ M \\frac { B ^ { ( f ) } _ k ( 0 ) \\bigl ( \\mathcal { S } _ N ( - w ) ^ { M - k } \\bigr ) ( q \\ , | \\ , b ) } { k ! ( M - k ) ! } \\sum \\limits _ { l = 1 } ^ { M - k } \\frac { 1 } { l } + O ( q ^ { - 1 } ) \\Bigr ) . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{gather*} \\Lambda ( \\xi , s ) = W ( \\xi ) N _ { K / \\Q } ( \\mathfrak { d f } ) ^ { \\frac 1 2 - s } \\Lambda \\big ( \\xi ^ { - 1 } , 1 - s \\big ) , \\end{gather*}"}
-{"id": "2279.png", "formula": "\\begin{align*} \\tilde H ^ j ( F , \\C ) = F ^ 0 \\tilde H ^ j ( F , \\C ) \\supset F ^ 1 \\tilde H ^ j ( F , \\C ) \\supset \\ldots \\supset F ^ j \\tilde H ^ j ( F , \\C ) \\supset F ^ { j + 1 } \\tilde H ^ j ( F , \\C ) = 0 , \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} C _ j ( Q ) : = \\underbrace { \\left [ \\begin{array} { c c c c } Q _ q \\\\ Q _ { q - 1 } & Q _ { q } \\\\ \\vdots & Q _ { q - 1 } & \\ddots \\\\ Q _ 0 & \\vdots & \\ddots & Q _ { q } \\\\ & Q _ 0 & \\vdots & Q _ { q - 1 } \\\\ & & \\ddots & \\vdots \\\\ & & & Q _ 0 \\end{array} \\right ] } _ { \\displaystyle j + 1 \\ ; \\mbox { b l o c k c o l u m n s } } , \\mbox { f o r $ j = 0 , 1 , 2 , \\ldots $ . } \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} \\beta ^ { \\gamma } : = \\displaystyle \\prod _ { x \\in A } \\beta ( x ) ^ { \\gamma ( x ) } . \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} ( g _ t ^ \\natural - g _ { z _ 0 , t } ) | _ { ( z , y ) } = e ^ { - t } ( C ( z _ 0 , z , y ) \\circledast d z \\otimes d z + D ( z _ 0 , z , y ) \\circledast d z \\otimes d y + E ( z _ 0 , z , y ) \\circledast d y \\otimes d y ) , \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} & \\int _ { \\mathcal A _ 1 ^ + ( n ) } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h = \\sum _ { t > - n } \\int _ { \\mathcal L _ { 2 t } } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h = \\\\ & = - p \\sum _ { t > - n } \\mathrm { v o l } ( \\mathcal L _ { 2 t } ) = - p ( 1 - p ^ { - 1 } ) ^ 2 \\sum _ { t > - n } p ^ { - 2 t } = - p ^ { 2 n - 1 } \\frac { ( 1 - p ^ { - 1 } ) ^ 2 } { 1 - p ^ { - 2 } } = - p ^ { 2 n } \\frac { p - 1 } { p ( p + 1 ) } . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} \\mu ( P ) = t r ( P T ) \\ : , \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} E & = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 0 & 0 \\end{array} \\right ) , & H & = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) , & F & = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 1 & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} \\mathcal N = \\frac { \\mu } { \\gamma ^ 2 } \\Big [ \\coth ^ 2 ( \\sqrt { \\mu _ 2 } | D | ) - \\frac { 1 } { \\mu _ 2 | D | ^ 2 } - a \\gamma ^ 2 \\Big ] | D | ^ 2 + \\frac { \\mu } { \\mu _ 2 \\gamma ^ 2 } . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\begin{gathered} F _ \\varepsilon ( x ) = \\frac { k } { 4 \\varepsilon } x - \\frac { \\sigma } { 2 } C x ^ { H - \\delta } + \\varepsilon . \\end{gathered} \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} | ( f _ i \\pm \\varphi ) ( x ) & - ( f _ i \\pm \\varphi ) ( y ) | = | f _ i ( x ) \\pm \\varphi ( y ) | \\\\ & \\leq | f _ i ( x ) | + | \\varphi ( y ) | = | f _ i ( x ) - f _ i ( u ) | + | \\varphi ( y ) - \\varphi ( u ) | \\\\ & \\leq ( 1 - \\delta ) d ( x , u ) + ( 1 - \\delta ) d ( u , y ) \\leq d ( x , y ) . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\dim ( H _ { J _ { k + 1 } } ) = \\dim ( H _ { J _ { k } } ) + \\dim ( H _ { k + 1 } ) - \\dim ( A [ \\ell ] ( \\overline { K } ) ) = 2 g - k + 2 g - 1 - 2 g = 2 g - ( k + 1 ) = 2 g - \\# { J _ { k + 1 } } , \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } B _ { t } d B _ { t } = \\displaystyle \\frac { 1 } { 2 } B _ { T } ^ 2 - \\frac { 1 } { 2 } T = \\displaystyle \\lim _ { \\substack { n \\longrightarrow + \\infty } } \\sum _ { \\substack { j \\in J \\setminus J _ { K ( n ) } } } B _ { t _ { j } } ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) . \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} L _ p ( f , 0 , 0 , 0 ) = 1 p \\nmid r . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\sup _ { P \\in \\mathcal { P } _ { [ \\underline { x } , \\overline { x } ] } } \\sum _ { i = 0 } ^ { | P | - 1 } \\left \\vert f \\big ( x ^ { ( i + 1 ) } \\big ) - f \\big ( x ^ { ( i ) } \\big ) \\right \\vert < + \\infty , \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} | \\det d f ^ n _ \\omega ( z ) | & = \\prod _ { k = 1 } ^ n | \\det d f _ { T ^ { k - 1 } \\omega } ( f _ \\omega ^ { k - 1 } ( p ) ) | \\\\ & \\le \\prod _ { k = 1 } ^ n \\left ( | \\det d f _ { T ^ { k - 1 } \\omega } ( 0 ) | + \\varepsilon \\right ) . \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} Y _ n \\to Y _ { n - 1 } \\to \\cdots \\to Y _ 1 \\to Y _ 0 : = Y \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} ( X Y ) ^ n & = ( X Y X ^ { - 1 } ) ( X ^ 2 Y X ^ { - 2 } ) \\cdots ( X ^ n Y X ^ { - n } ) X ^ n , \\\\ ( X Y ) ^ n & = X ^ n ( X ^ { - n + 1 } Y X ^ { n - 1 } ) \\cdots ( X ^ { - 2 } Y X ^ 2 ) ( X ^ { - 1 } Y X ) Y . \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\widetilde { \\nu } : = \\big ( \\widetilde { \\nu } _ 1 , . . . , \\widetilde { \\nu } _ 6 \\big ) . \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} X _ 2 = \\beta _ { 2 , 2 } ^ { - 1 } \\bigl ( 1 , b _ 0 = 2 , \\ , b _ 1 = 1 / 2 , \\ , b _ 2 = 1 / 2 \\bigr ) , \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} [ \\beta ] = [ \\imath _ * \\imath ^ { ! * } \\beta ] + [ \\jmath _ { ! * } \\jmath ^ * \\beta ] . \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} X ' X ^ { - 1 } = \\Omega + F ( X ) . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { r ^ { ( 1 ) } _ { i } } \\frac { 1 } { 2 } \\sum _ { q = 1 } ^ { r ^ { ( 1 ) } _ { i - 1 } } \\mathrm { m i n } \\{ n _ { p , i } , n _ { q , { i - 1 } } \\} & = \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { k } r ^ { ( s ) } _ { i - 1 } r _ { i } ^ { ( s ) } , & \\\\ \\sum _ { p = 1 } ^ { r _ { i } ^ { ( 1 ) } } ( \\sum _ { p > p ' > 0 } \\mathrm { m i n } \\{ n _ { p , i } , n _ { p ' , i } \\} + \\frac { 1 } { 2 } n _ { p , i } ) & = \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { k } r ^ { ( s ) ^ { 2 } } _ { i } , & \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} f _ 1 X _ { n + 1 } ( M ) - U ^ a _ { n + 1 } ( M ) = \\partial f ^ a _ { n + 1 } ( M ) f _ 1 X _ { n + 1 } ( M ) - U ^ b _ { n + 1 } ( M ) = \\partial f ^ b _ { n + 1 } ( M ) . \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} K ( t , x , y ) \\\\ = \\nabla _ x m _ { u _ \\mu } ( t , x ) \\cdot \\nabla _ y m _ { u _ \\iota } ( t , y ) + \\kappa m _ { u _ \\mu } ( t , x ) m _ { u _ \\iota } ( t , y ) + 2 m _ { \\nabla u _ \\mu } ( t , x ) m _ { u _ \\iota } ( t , y ) . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} \\tilde { p } _ H \\bigl ( ( y \\star u , y ) , ( \\vartheta , \\vartheta ) \\bigr ) ~ = ~ 1 - \\sum _ { z \\in E } p _ H ( y \\star u , z ) \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\mathrm { ( a ) } \\ k > k ' & & & \\mathrm { ( b ) } \\ ( \\alpha _ { i _ k } , \\alpha _ { i _ { k ' } } ) < 0 & & & & & & \\mathrm { ( c ) } \\ \\{ t \\mid k ' < t < k , i _ t = i _ { k ' } \\ \\ i _ k \\} = \\emptyset . \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} P _ n ^ * = n \\sum \\limits _ { r \\geq 0 } \\sum \\limits _ { \\sum k _ i = n - 1 } { n - 1 \\choose k _ 0 , k _ 1 , \\hdots , k _ r } P _ { k _ { 0 } } ^ * \\widetilde { P } _ { k _ { 1 } } \\cdots \\widetilde { P } _ { k _ { r } } k _ 1 k _ 2 \\cdots k _ r , \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} B ( v , e ) & = \\begin{cases} B _ i ( v , e ) , & \\ v \\in V ( G _ i ) , e \\in E ( G _ i ) , \\\\ 0 , & , \\end{cases} \\\\ [ 1 m m ] w ( e ) & = \\frac { w _ i ( e ) } { C \\alpha _ i ^ { 1 / ( p - r ) } } , \\ \\ e \\in E ( G _ i ) . \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} r _ { i j } = \\begin{cases} 1 , & ( i , j ) = ( 0 , c _ { \\ell _ 0 } ) , \\\\ \\frac { 1 } { c _ { \\ell _ 0 } - | I | } , & i \\in I j = c _ { \\ell _ 0 } , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} ( T _ { \\rho } \\cdot X ) ( A , w ) : = R ( \\rho ) X ( R ( \\rho _ { w } ) ^ { - 1 } A , \\rho ( w ) ) R ( \\rho ) ^ { - 1 } \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\frac { ( \\vec x ^ * , \\vec v ) } { \\norm { \\vec x ^ * } } \\ge \\sum _ { i = 1 } ^ { m } | v _ i | , \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} B _ 1 ( m ) = \\begin{cases} p ^ { 1 - 3 m / 2 } ( - 1 ) ^ { m + 1 } \\chi _ { \\psi } ( p ^ m ) ( 1 - p ^ { - 1 } ) ( 1 + p - p ^ m ) & m > 0 , \\\\ p ^ { 3 m / 2 } ( - 1 ) ^ { m + 1 } \\chi _ { \\psi } ( p ^ m ) ( 1 - p ^ { - 1 } ) p ^ { - m } & m \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\| f \\| & \\leq \\sum _ { I \\subset [ N ] } \\left \\| f ^ I \\right \\| \\\\ & \\leq \\sum _ { I \\subset [ N ] } \\left \\| \\xi ^ I \\right \\| \\left \\| \\eta ^ I \\right \\| \\\\ & \\leq \\left ( \\sum _ { I \\subset [ N ] } \\left \\| \\xi ^ I \\right \\| ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\left ( \\sum _ { I \\subset [ N ] } \\left \\| \\eta ^ I \\right \\| ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\\\ & = \\| \\xi \\| \\| \\eta \\| \\\\ & = \\| f \\| . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} \\pi : ( X , \\circ ) \\rightarrow ( X _ A , \\diamond ) , \\pi ( a \\circ x ) = a \\diamond \\pi ( x ) . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} S ( \\lambda , \\mu ) \\coloneqq \\sup _ { u \\ne 0 } \\frac { \\sum _ { j = 1 } ^ p g _ j u _ j + \\lambda \\sum _ { j \\in T } ( | \\beta ^ * _ j | - | \\beta ^ * _ j + u _ j | ) - \\mu \\sum _ { j \\in T ^ c } | u _ j | } { \\| u \\| } . \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} \\overline X _ u \\cong _ { h t } L _ { m , n } ^ { t , m \\cdot n - N } \\vee \\bigvee _ { s _ 0 \\leq s \\leq t } \\bigvee _ { i = 1 } ^ { r ( s ) } S ^ { N - ( m - s + 1 ) ( n - s + 1 ) + 1 } ( L _ { m - s + 1 , n - s + 1 } ^ { t - s , 1 } ) \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\phi _ { \\breve { \\mathbf g } , p } \\left ( \\left ( \\begin{smallmatrix} x _ 1 & x _ 2 \\\\ x _ 3 & x _ 4 \\end{smallmatrix} \\right ) \\right ) = \\mathbf 1 _ { \\Z _ p } ( x _ 1 ) \\mathbf 1 _ { \\Z _ p } ( x _ 4 ) \\mathbf 1 _ { p \\Z _ p } ( x _ 3 ) \\left ( \\mathbf 1 _ { \\Z _ p } ( x _ 2 ) - p ^ { - 1 } \\mathbf 1 _ { p ^ { - 1 } \\Z _ p } ( x _ 2 ) \\right ) . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } _ r = - U ^ T A U x _ r + U ^ T B u , \\\\ y _ r = C U x \\end{cases} \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ N \\ , B ^ { ( f ) } _ n \\bigr ) ( q \\ , | \\ , b ) & = 0 , \\ ; n = 0 \\cdots N - 1 , \\\\ \\bigl ( \\mathcal { S } _ N \\ , B ^ { ( f ) } _ N \\bigr ) ( q \\ , | \\ , b ) & = f ( 0 ) \\ , N ! \\prod \\limits _ { j = 1 } ^ N b _ j , \\\\ \\bigl ( \\mathcal { S } _ N \\ , x ^ n \\bigr ) ( q \\ , | \\ , b ) & = 0 , \\ ; n = 0 \\cdots N - 1 , \\\\ \\bigl ( \\mathcal { S } _ N \\ , x ^ N \\bigr ) ( q \\ , | \\ , b ) & = ( - 1 ) ^ N N ! \\prod \\limits _ { j = 1 } ^ N b _ j . \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} \\abs { \\Q } = \\abs { Y } \\ , \\abs { Y ^ \\circ } = \\abs { Z } \\ , \\abs { Z ^ \\circ } = \\abs { \\S } . \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\prod _ { x = 1 } ^ M \\frac { 1 + e _ { ( j - 1 ) M + x } g _ x ^ { ( \\ell ) } } { 2 } = \\left \\{ \\begin{array} { c l } 1 & E _ { N } ^ { ( j , M ) } = G ^ { ( \\ell ) } , \\\\ 0 & E _ { N } ^ { ( j , M ) } \\neq G ^ { ( \\ell ) } , \\end{array} \\right . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} \\Delta _ { k - 2 m _ 0 } ^ { m _ 0 } ( E _ { k - 2 m _ 0 , N } ^ \\chi ( Z , 0 ; h ) ) = d E _ { k , N } ^ \\chi ( Z , - m _ 0 ; h ) , \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} v ( M / A ) = k ( M / A ) \\quad \\mbox { a n d } f ( M \\backslash A ^ c ) = k ( M \\backslash A ^ c ) . \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} \\nabla ^ { - \\nu } \\nabla { U ( t ) } = \\nabla ^ { 1 - \\nu } U ( t ) - { t + \\nu - 2 \\choose t - 1 } U ( 0 ) , \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} h \\varpi _ p = \\left ( \\begin{array} { c c } x p ^ { - 1 } & y \\\\ z p ^ { - 1 } & t \\end{array} \\right ) = \\left ( \\begin{array} { c c } - z ^ { - 1 } \\det ( h ) & x p ^ { - 1 } \\\\ 0 & z p ^ { - 1 } \\end{array} \\right ) \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & z ^ { - 1 } t p \\end{array} \\right ) . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\frac { \\varphi _ \\lambda ( u ) } { K ( u ) } - a ^ \\top ( u ) A ^ { - 1 } m ( \\varphi _ { \\lambda } ) = \\frac { f ( u ) } { G _ C ( u ) } , u \\in [ 0 , \\tau ] . \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ \\exp \\bigl ( q \\log \\beta _ { M , N } ( a , b ) \\bigr ) \\Bigr ] = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty ( e ^ { - t q } - 1 ) e ^ { - b _ 0 t } \\frac { \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } \\frac { d t } { t } \\Bigr ) . \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} F ( V _ N ) \\cdots F ( V _ 1 ) F ( V _ 0 ) = F _ { N + 1 } ( U _ { N + 1 } ) \\cdots F _ 2 ( U _ 2 ) F _ 1 ( U _ 1 ) . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} A R ^ { U } g = - g \\mbox { a . e . i n } U . \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} B ( i _ 1 , e ) x _ { i _ 1 } ^ p = B ( i _ 2 , e ) x _ { i _ 2 } ^ p = \\cdots = B ( i _ r , e ) x _ { i _ r } ^ p = \\frac { w ( e ) } { r } , \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} { \\rm e x p } ( b ) \\cdot m : = m - \\sum _ { i \\geq 0 } \\frac { { \\rm a d } _ b ^ { i } } { ( i + 1 ) ! } \\left ( { \\rm d } b + [ m , b ] \\right ) . \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} V _ N \\triangleq \\max \\Big \\{ V _ { \\varepsilon } ( x _ i ) + \\lambda _ 1 \\ , \\log ( x _ i ) + \\lambda _ 2 \\ , \\log ( 1 - x _ i ) , \\ , i = 1 \\cdots N \\Big \\} \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} \\nabla = d + \\Bigg ( p _ { - 1 } - \\frac { \\varphi } { h ^ \\vee } \\rho + \\sum _ { j \\in E } v _ j p _ j \\Bigg ) d z , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} N _ 7 = c _ 7 \\int _ { \\mathbb R } b _ k J ( | D | ) \\partial _ x ^ 2 ( \\zeta _ k \\nu _ { n _ k } + \\varphi _ k \\nu _ { n _ k } ) + \\zeta _ k \\nu _ { n _ k } J ( | D | ) \\partial _ x ^ 2 ( \\varphi _ k \\nu _ { n _ k } ) d x \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} Z _ { \\lambda _ 1 , \\lambda _ 2 , \\varepsilon } ( \\beta ) \\triangleq \\sum \\limits _ { i = 1 } ^ N x _ i ^ { \\beta \\lambda _ 1 } ( 1 - x _ i ) ^ { \\beta \\lambda _ 2 } e ^ { \\beta V _ \\varepsilon ( x _ i ) } . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\nabla = d + \\bar p _ { - 1 } d t + \\sum _ { i \\in \\bar E } \\bar v _ i ( t ) \\bar p _ i d t , \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} \\partial \\alpha ^ { n - 2 , \\ , n } = - \\bar \\partial \\Omega ^ { n - 1 , \\ , n - 1 } . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\displaystyle \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) ( \\psi w ) = \\psi \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) w + 2 \\nabla w \\nabla \\psi + w \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) \\psi . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} G _ \\alpha ( y , y ) = G ( y , y ) ~ \\leq ~ G _ S ( 0 , 0 ) \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\lim \\limits _ { A , B \\rightarrow 0 } \\int _ { x _ 1 ^ { A B } } ^ { x _ 2 ^ { A B } } \\rho _ * ^ { A B } u _ * ^ { A B } d x = ( \\sigma _ { 0 } + \\beta t ) \\sqrt { \\rho _ { + } \\rho _ { - } } ( u _ - - u _ + ) t . \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} A = B + \\sum _ { r = 1 } ^ t \\delta _ 1 ( \\mathbf { u } _ r , \\mathbf { v } _ r ) . \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} d y _ { n + 1 } = [ y ^ 2 , x _ n ] - \\sum _ { s + t = n } x _ s y x _ t - \\sum _ { \\substack { s + t = n \\\\ t \\geqslant 1 } } ( x _ s y _ t - ( - 1 ) ^ t y _ t x _ s ) , d x _ { n + 1 } = \\sum _ { s + t = n } ( - 1 ) ^ s x _ s x _ t . \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} \\widehat { \\sigma _ { 0 } ^ { B } } = \\sigma _ { 0 } ^ { B } = \\frac { u _ + + u _ - } { 2 } \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} S _ i ^ * S _ i = \\sum _ { i = 1 } A _ { i j } S _ j S _ j ^ * \\sum _ { i = 1 } ^ n S _ i S _ i ^ * = 1 . \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} { \\rm { V a r } } \\left ( { w _ 1 ^ i \\left ( { { t _ 0 } } \\right ) } \\right ) = \\int _ { - \\infty } ^ { + \\infty } { { S _ { w _ 1 ^ i } } \\left ( f \\right ) d f } = { { { N _ 0 } } \\mathord { \\left / { \\vphantom { { { N _ 0 } } 2 } } \\right . \\kern - \\nulldelimiterspace } 2 } \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} \\iota _ { 4 2 } = \\frac { 3 ^ { 1 0 } } { 2 ^ { 1 8 } \\cdot 5 ^ 5 } \\ , I _ 3 ^ 5 \\ I _ { 2 7 } \\ , . \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} \\left \\{ \\zeta _ i ^ { ( k ) } , \\zeta _ { q } ^ { ( p ) } \\right \\} _ t & = O ( e ^ { - \\delta t } ) . \\\\ \\left \\{ \\zeta _ i ^ { ( k ) } , \\varphi _ { q } ^ { ( p ) } \\right \\} _ t & = \\frac { 1 } { 4 } ( \\varepsilon ( k - p ) - 1 ) ( C - R ) + O ( e ^ { - \\delta t } ) . \\\\ \\left \\{ \\varphi _ i ^ { ( k ) } , \\varphi _ { q } ^ { ( p ) } \\right \\} _ t & = O ( e ^ { - \\delta t } ) . \\\\ \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\underset { p \\rightarrow + \\infty } { \\lim } \\sqrt { 1 - e ^ { - h _ p ( x ) } } K ( x , y ) & \\sqrt { 1 - e ^ { - h _ p ( y ) } } e _ n ( x ) e _ n ( y ) \\\\ & = \\sqrt { 1 - e ^ { - h ( x ) } } K ( x , y ) \\sqrt { 1 - e ^ { - h ( y ) } } e _ n ( x ) e _ n ( y ) \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\mathcal { O } _ { E _ { \\tau } } [ G _ n ] = \\mathcal { O } _ { E _ { \\tau } } \\otimes _ { \\Z _ p } \\Z _ p [ G _ n ] , \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} { \\bf E } [ \\beta _ { 1 , 0 } ( a , b ) ^ q ] = & \\frac { \\Gamma _ 1 ( q + b _ 0 \\ , | \\ , a ) } { \\Gamma _ 1 ( b _ 0 \\ , | \\ , a ) } , \\\\ = & a ^ { \\frac { q } { a } } \\frac { \\Gamma ( \\frac { q + b _ 0 } { a } ) } { \\Gamma ( \\frac { b _ 0 } { a } ) } , \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} \\Phi _ { m , n } ( \\chi ) = \\begin{cases} \\overline { \\chi ( m ) } + \\chi ( n ) \\chi ( m ) & s = e , \\\\ - 1 & s < e = 1 m = p ^ k k > 0 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\omega _ { \\infty } ( z ) = 0 \\ \\ \\textrm { o n } \\ \\ B ( 0 , 2 R _ 1 ) ^ c \\times ( - 5 / 4 , 0 ] . \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\beta _ k = + \\infty , \\ \\ \\sum _ { k = 0 } ^ { \\infty } \\beta ^ 2 _ k < + \\infty . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} \\Lambda ( t ) = \\int _ { - \\infty } ^ { \\infty } x u ^ 2 ( t , x ) d x \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\mathcal { A } ( x , k ) = \\bigcup _ { l = 0 } ^ { N - 1 } \\mathcal { A } ( x , k ) ^ { ( l ) } . \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} f ^ { \\star } _ 0 ( x ) = \\int \\limits _ { 0 } ^ { x } \\frac { \\omega ( t ) } { t } \\ , d t , x \\in [ 0 , \\ , 1 ] , \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} p ( \\rho ) = \\frac { g } { 2 } \\rho ^ 2 \\quad \\mu ( \\rho ) = 4 \\nu \\rho . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\lambda _ { s t } = ( \\varpi _ { i _ s } - w _ { \\leq s } \\varpi _ { i _ s } , \\varpi _ { i _ t } + w _ { \\leq t } \\varpi _ { i _ t } ) \\ \\ s < t . \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} ( g ) _ s : = \\{ h g h ^ { - 1 } \\in G : h \\in G _ { \\C } \\} ; \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} Q = \\bullet _ { r _ 1 , d _ 1 } \\longrightarrow \\bullet _ { r _ 2 , d _ 2 } \\longrightarrow \\cdots \\longrightarrow \\bullet _ { r _ n , d _ n } \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} \\mathrm { v o l } ( \\mathcal B _ 1 ^ + ( m ) ) = \\sum _ { j \\geq - m } \\mathrm { v o l } ( \\mathcal R _ { 2 j + 1 } ) = p ^ { - 1 } ( 1 - p ^ { - 1 } ) ^ 2 \\sum _ { j \\geq - m } p ^ { - 2 j } = p ^ { 2 m - 1 } ( 1 - p ^ { - 1 } ) ^ 2 \\frac { 1 } { 1 - p ^ { - 2 } } = p ^ { 2 m - 1 } \\frac { 1 - p ^ { - 1 } } { 1 + p ^ { - 1 } } . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{gather*} \\mathfrak { M } ( q \\ , | \\tau , 0 , 0 ) = \\frac { 1 } { \\Gamma ^ q \\bigl ( 1 - \\frac { 1 } { \\tau } \\bigr ) } \\Gamma \\bigl ( 1 - \\frac { q } { \\tau } \\bigr ) , \\\\ M _ { ( \\tau , 0 , 0 ) } = \\frac { \\tau ^ { 1 / \\tau } } { \\Gamma ( 1 - 1 / \\tau ) } \\beta _ { 1 , 0 } ^ { - 1 } ( \\tau , b _ 0 = \\tau ) . \\end{gather*}"}
-{"id": "3165.png", "formula": "\\begin{align*} & \\partial _ t g + v \\partial _ x g + G ( M ^ { n - 1 } ) \\partial _ v g - \\partial _ v ( { v g } ) - \\sigma \\partial _ { v v } g = \\left [ G ( M ^ { n - 2 } ) - G ( M ^ { n - 1 } ) \\right ] \\partial _ v f ^ { n - 1 } \\ , , \\\\ & g _ 0 = 0 \\ , , \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\tilde W = \\begin{pmatrix} 1 & 0 & - 1 \\end{pmatrix} W = \\begin{pmatrix} 1 & 1 & - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} P ^ t ( q ) = \\sum _ { ( i , j ) } x _ { i , j } , \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} S _ { N , N } ^ { ( r ) } & = \\sum _ { m = 1 } ^ N \\left ( S _ { N , m } ^ { ( r + 1 ) } - S _ { N , m - 1 } ^ { ( r + 1 ) } \\right ) \\log m \\\\ & = S _ { N , N } ^ { ( r + 1 ) } ( \\log N ) + \\sum _ { m = 1 } ^ { N - 1 } S _ { N , m } ^ { ( r + 1 ) } \\big ( \\log ( m ) - \\log ( m + 1 ) \\big ) . \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} g ( t ) \\triangleq f ( t ) e ^ { - q t } \\frac { d ^ r } { d t ^ r } \\bigl [ e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) \\bigr ] . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} g _ n ( X ) = \\sum _ { i \\in \\mathcal { I } ^ - } L ( A _ i ) \\nabla \\psi _ { i , T } ( X ) \\cdot \\bar { \\mathbf { n } } , ~ ~ ~ ~ g _ t ( X ) = \\sum _ { i \\in \\mathcal { I } ^ - } L ( A _ i ) \\nabla \\psi _ { i , T } ( X ) \\cdot \\bar { \\mathbf { t } } . \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} b _ { \\mu } b _ { \\lambda } & = \\psi _ { w _ { 0 } } y _ { \\mu } \\psi _ { w _ { 0 } } y _ { \\min } \\psi _ { w _ { 0 } } y _ { \\lambda } \\psi _ { w _ { 0 } } y _ { \\min } \\\\ & = ( - 1 ) ^ { \\tfrac { n ( n - 1 ) } { 2 } } \\psi _ { w _ { 0 } } ( y _ { \\mu } \\psi _ { w _ { 0 } } y _ { \\lambda } ) \\psi _ { w _ { 0 } } y _ { \\min } \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} \\lim _ { t \\to t _ 0 + } \\| u ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - u _ 0 ( \\cdot ) \\| _ { C ^ 0 ( \\bar \\Omega ) } + \\| v ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - v _ 0 ( \\cdot ) \\| _ { C ^ 0 ( \\bar \\Omega ) } = 0 . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} Y _ { u t } = { \\bf \\Sigma } _ { \\{ u \\} A } { \\bf \\Sigma } _ { A } ^ { - 1 } Y _ { A t } , \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} k _ { i j } ( x , \\sigma ) = | \\{ t \\le \\sigma : x _ { t - } = i , x _ t = j \\} | . \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} D _ q \\left ( { 1 + x } \\right ) _ q ^ \\alpha = \\left [ \\alpha \\right ] _ q \\left ( { 1 + q x } \\right ) _ q ^ { \\alpha - 1 } . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} 0 < \\alpha < \\frac 1 { m - 1 } \\ , , \\beta = \\frac { \\alpha ( m - 1 ) + 1 } 2 \\ , , \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\pi ( h ) \\ast _ g f & = \\frac { 1 } { n ! } \\sum _ { \\sigma , \\tau \\in \\Sigma _ n } g h _ { \\tau ( 1 ) } \\wedge g ^ c f _ { \\sigma ( 1 ) } \\otimes \\cdots \\otimes g h _ { \\tau ( n ) } \\wedge g ^ c f _ { \\sigma ( n ) } \\\\ \\pi ( h \\ast _ g f ) & = \\frac { 1 } { n ! } \\sum _ { \\sigma , \\tau \\in \\Sigma _ n } g h _ { \\tau ( 1 ) } \\wedge g ^ c f _ { \\sigma \\tau ( 1 ) } \\otimes \\cdots \\otimes g h _ { \\tau ( n ) } \\otimes g ^ c f _ { \\sigma \\tau ( n ) } . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} | \\alpha _ { f , i } ( p ) | & \\leq 1 , \\ i = 1 , 2 , \\\\ \\Re \\mu _ { f , i } & \\leq 0 , \\ i = 1 , 2 . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} S _ \\infty = \\Big \\{ ( \\sum _ { j = 1 } ^ p \\gamma _ { n _ j } ) + k : p \\ge 1 , \\ n _ 1 , n _ 2 , \\ldots , n _ p \\ge 1 , k \\ge 0 \\Big \\} . \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} k ( i , r ) : = \\# \\{ r ' \\in \\mathbb { Z } \\mid ( i , r ' ) \\in \\overline { I } _ { \\xi } ^ { \\mathrm { t w } , \\flat } , r ' - r \\in 2 r _ i \\mathbb { Z } _ { \\geq 0 } \\} . \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} \\dot { u } = C ( t ) u \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} ( ( u _ 1 , u _ 2 ) | ( v _ 1 , v _ 2 ) ) = K ( u _ 1 , v _ 1 ) - K ( u _ 2 , v _ 2 ) , \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} N _ { m } = ( 2 m + 1 ) ! . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\mathfrak v _ i \\mathfrak b _ i = \\mathfrak b _ i \\mathfrak v _ i ^ * . \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} K \\mathbf { c } _ j + L ( A _ j ) \\sum _ { i \\in \\mathcal { I } ^ - } \\hat { \\sigma } ( \\Psi _ { i , T } \\mathbf { c } _ i ) ( F ) \\bar { \\mathbf { n } } = K \\mathbf { v } _ j - L ( A _ j ) \\sum _ { i \\in \\mathcal { I } ^ + } \\hat { \\sigma } ( \\Psi _ { i , T } \\mathbf { v } _ i ) ( F ) \\bar { \\mathbf { n } } , ~ ~ j \\in \\mathcal { I } ^ - . \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} J _ 1 = J _ { 1 , 1 } + J _ { 1 , 2 } + J _ { 1 , 3 } \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} I I ^ { \\partial M } ( X , Y ) = \\langle A X , Y \\rangle , \\textrm { f o r } X , Y \\in T M , \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} L ( w ( R _ 1 ( \\epsilon ) , R _ 2 ( \\epsilon ) ) \\geq L ( S _ \\epsilon ) = \\epsilon . \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} E _ { \\mu _ 2 } ( { \\bf { h } } _ { n _ k } ) = E _ { \\mu _ 2 } ( { \\bf { h } } _ { k , 1 } ) + E _ { \\mu _ 2 } ( { \\bf { h } } _ { k , 2 } ) + O ( \\epsilon ) . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} \\{ \\tau _ { m - 1 } \\leq t \\} = \\{ M ^ * _ t \\geq m - 1 \\} \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + A x + B \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} v _ p ( S ( n , k ) ) = \\beta + l v _ p ( n - x _ 0 ) \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} \\Lambda ( f , s ) = \\varepsilon ( f ) N ^ { \\ell / 2 - s } \\Lambda ( f , \\ell - s ) , \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} \\exp _ G \\circ L ( \\alpha ) = \\alpha \\circ \\exp _ G \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\partial _ t ( \\rho X ) = - \\partial _ x ( \\rho u X ) - \\partial _ x p ( \\rho ) + \\rho f . \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} V _ 1 ( f ) = \\int _ { \\Omega } u _ 1 ( f ( \\omega ) , \\omega ) d \\mathbb { P } _ 1 \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n / 2 - 1 } \\gamma _ { i , i + k } ( T ^ { p k } - T ^ { - p k } ) + \\gamma _ { i , i - k } ( T ^ { - p k } - T ^ { p k } ) + \\gamma _ { i , i + n / 2 } ( T ^ { p n / 2 } - T ^ { - p n / 2 } ) . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} | V _ n | = | \\widetilde { \\Gamma } _ n | . \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} r - \\lambda = \\frac { b k ( d + 1 - k ) } { ( d + 1 ) d } , \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\widetilde T _ { S , d } : = \\begin{bmatrix} \\Delta & - 1 & \\dots & - 1 \\\\ & \\ddots & \\ddots & \\vdots \\\\ & & \\ddots & - 1 \\\\ & & & \\Delta \\end{bmatrix} , \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} r _ { 1 2 } ( \\lambda _ 1 , \\lambda _ 2 ) = { 1 \\over \\lambda _ 2 - \\lambda _ 1 } \\begin{pmatrix} 0 & 0 & 0 & 0 \\cr 0 & \\lambda _ 2 & - \\lambda _ 1 & 0 \\cr 0 & - \\lambda _ 2 & \\lambda _ 1 & 0 \\cr 0 & 0 & 0 & 0 \\end{pmatrix} \\ ; . \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} y _ i = x _ i \\land \\bigwedge _ { k \\in N ^ - ( i ) \\setminus \\{ i \\} } y _ k = 0 , \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} \\Tilde { e } _ { i } ^ { \\pm P } T = \\mathrm { P R } \\circ \\Tilde { e } _ { i } ^ { P } \\circ \\mathrm { D P R } ( T ) , \\\\ \\Tilde { f } _ { i } ^ { \\pm P } T = \\mathrm { P R } \\circ \\Tilde { f } _ { i } ^ { P } \\circ \\mathrm { D P R } ( T ) , \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} \\log \\mathfrak { M } ( i q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = i q \\ , \\mathfrak { m } ( \\tau ) - \\frac { 1 } { 2 } q ^ 2 \\sigma ^ 2 ( \\tau ) + \\int _ { \\mathbb { R } \\setminus \\{ 0 \\} } \\bigl ( e ^ { i q u } - 1 - i q u / ( 1 + u ^ 2 ) \\bigr ) d \\mathcal { M } _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ( u ) \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} \\psi \\colon & \\C \\to \\C \\\\ & c \\mapsto \\sum _ { i = 0 } ^ { \\ell - 1 } { \\sigma ^ i ( c ) } . \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\begin{cases} \\bar h _ 1 \\bar r _ 1 = \\bar h _ 2 + \\bar h _ 3 \\bar r _ 2 \\cr \\bar p _ 1 \\bar r _ 2 = \\bar p _ 2 + \\bar p _ 3 \\bar r _ 1 \\cr \\underbar h _ 1 \\underbar r _ 1 = \\underbar h _ 2 + \\underbar h _ 3 \\underbar r _ 2 \\cr \\underbar p _ 1 \\underbar r _ 2 = \\underbar p _ 2 + \\underbar p _ 3 \\underbar r _ 1 . \\end{cases} \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\tilde { E } ~ = ~ \\bigl \\{ ( y \\star u , y ) , y \\in E \\bigr \\} \\cup \\bigl \\{ ( y , \\vartheta ) , y \\in E \\bigr \\} \\cup \\bigl \\{ ( \\vartheta , \\vartheta ) \\bigr \\} \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} \\ell ' = b ' _ 1 c ' + b ' _ 2 c ' _ 2 = b ' _ 1 c ' _ 1 + b ' c ' _ 2 \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\Sigma \\cap B _ { \\frac r 3 } ( x ) & = x + \\left ( \\operatorname { g r a p h } ( f ) \\cap B _ { \\frac r 3 } ( 0 ) \\right ) \\\\ & = x + \\left ( \\operatorname { g r a p h } ( \\tilde f ) \\cap B _ { \\frac r 3 } ( 0 ) \\right ) . \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} D \\sim \\frac { 1 } { \\sqrt { 2 \\pi e } } \\frac { \\sum \\limits _ { i = 1 } ^ { n } b ^ { 2 } ( i ; n ) } { \\lambda _ n } . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} ( \\frac { e ^ { n + 1 } _ { u } - e ^ { n } _ { u } } { \\Delta t } , v _ { h } ) + \\nu ( \\nabla e ^ { n + 1 } _ { u } , \\nabla v _ { h } ) + b ( u ^ { n + 1 } - u ^ { n } , u ^ { n + 1 } , v _ { h } ) + b ( e ^ { n } _ { u } , u ^ { n + 1 } , v _ { h } ) \\\\ + b ( u ^ { n } _ { h } , e ^ { n + 1 } _ { u } , v _ { h } ) - ( e ^ { n + 1 } _ { p } , \\nabla \\cdot v _ { h } ) = ( \\frac { u ^ { n + 1 } - u ^ { n } } { \\Delta t } - u ^ { n + 1 } _ { t } , v _ { h } ) \\ ; \\ ; \\forall v _ { h } \\in X _ { h } . \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} ( f V ) _ t : = f ( \\cdot , t ) V _ t \\quad \\mbox { f o r } \\mathcal { L } ^ 1 \\mbox { - a . e . \\ } t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ { + } } \\beta \\left ( \\varepsilon \\right ) = 0 , \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} \\psi _ { n + 1 } ( ( a b ^ i ) ^ { k p ^ n } ) = \\big ( ( ( a ^ { \\alpha } b ) ^ { k i } u _ 1 ) ^ { p ^ { n - 1 } } , ( ( a ^ { \\alpha } b ) ^ { k i } u _ 2 ) ^ { p ^ { n - 1 } } , \\dots ( ( a ^ { \\alpha } b ) ^ { k i } u _ p ) ^ { p ^ { n - 1 } } \\big ) , \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} & \\underline { m } ^ * \\circ S _ { k , k } \\\\ = \\ ; \\ ; & \\underline { m } ^ * \\circ ( ( R i ) ^ { \\simeq } ) ^ { - 1 } \\circ \\mathsf { c o f } ^ { \\underline { 1 } } \\circ ( L i ) ^ { \\simeq } \\\\ = \\ ; \\ ; & \\underline { m } ^ * \\circ \\mathsf { c o f } ^ { \\underline { 1 } } \\circ ( L i ) ^ { \\simeq } \\\\ \\cong \\ ; \\ ; & C ^ { | \\underline { m } | } \\circ ( 1 _ { \\underline { m } ^ \\vee } ) ^ \\ast \\circ ( L i ) ^ \\simeq . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| x ^ k - y ^ { k } \\| = 0 \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} \\begin{array} { c c l } 0 & = & \\mathrm { C a p } _ t ( v _ 1 \\otimes 1 , \\dots , v _ { t - 1 } \\otimes 1 , v _ t \\otimes b _ j , w _ 1 \\otimes 1 , \\dots , w _ { t + 1 } \\otimes 1 ) \\\\ & = & \\mathrm { C a p } _ t ( v _ 1 , \\dots , v _ { t - 1 } , v _ t , w _ 1 , \\dots , w _ { t + 1 } ) \\otimes b _ j . \\end{array} \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} W _ { \\pi } ( u g ) = \\psi ( u ) \\mathcal { J } _ { \\pi } ( g ) u \\in U _ n ( F ) , g \\in F ^ \\times K _ n , \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } = \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { X } \\Delta ( y ) d y & = O \\left ( X \\right ) \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} D _ k = \\frac { \\max ( n \\psi ( m ) , m \\psi ( n ) ) } { e ^ k ( m , n ) } = \\frac { \\textup { l c m } ( m , n ) \\max \\left ( \\frac { \\psi ( m ) } { m } , \\frac { \\psi ( n ) } { n } \\right ) } { e ^ k } = \\Delta r t e ^ { - k } . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} P ( \\Pi ) & = \\prod _ { i = 1 } ^ d ( \\Pi - 1 ^ { \\otimes ( i - 1 ) } \\otimes \\pi \\otimes 1 ^ { \\otimes ( d - i ) } ) \\\\ & = \\sum _ { r = 0 } ^ d ( - 1 ) ^ r \\Pi ^ { d - r } \\sum _ { \\substack { ( i _ 1 , \\dots , i _ d ) \\in \\{ 0 , 1 \\} ^ d \\\\ | \\{ s | i _ s = 1 \\} | = r } } \\pi ^ { i _ 1 } \\otimes \\dots \\otimes \\pi ^ { i _ d } \\\\ & = \\sum _ { r = 0 } ^ d ( - 1 ) ^ r e _ { \\alpha _ r } \\Pi ^ { d - r } , \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} w ' ( e ) ^ { p ' - r } \\prod _ { v \\in e } B ' ( v , e ) = w ( e ) ^ { p ' - p } \\alpha & \\leq ( \\alpha \\Delta ^ r ) ^ { ( p ' - p ) / ( p - r ) } \\alpha \\\\ & = \\alpha ^ { ( p ' - r ) / ( p - r ) } \\Delta ^ { r ( p ' - p ) / ( p - r ) } \\\\ & = r ^ { p ' - r } \\bigg ( \\frac { \\Delta ^ { r ( p ' - p ) } } { ( \\lambda ^ { ( p ) } ( G ) ) ^ { p ( p ' - r ) } } \\bigg ) ^ { 1 / ( p - r ) } . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\sum _ i \\int _ { \\partial M } ( w _ i \\eta ( w _ i ) + I I ^ { \\partial W } ( N , N ) w _ i ^ 2 ) \\ , d s = 2 \\int _ { \\partial M } H ^ { \\partial W } \\| \\xi \\| ^ 2 \\ , d s \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} S _ x ( u - P _ { \\tilde { y } } ( \\cdot - \\tilde { y } ) ) ( z ) & = ( u - P _ { \\tilde { y } } ( \\cdot - \\tilde { y } ) ) ( \\Psi _ x ^ { - 1 } ( z ) ) \\\\ & = ( u - P _ { \\tilde { y } } ( \\cdot - \\tilde { y } ) ) ( \\Psi _ y ^ { - 1 } \\circ \\Psi _ y \\circ \\Psi _ x ^ { - 1 } ( z ) ) \\\\ & = S _ y ( u - P _ { \\tilde { y } } ( \\cdot - \\tilde { y } ) ) ( \\Psi _ y \\circ \\Psi _ x ^ { - 1 } ( z ) ) . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} t r _ N ( B A ) & = \\sum _ { u \\in N } \\langle u | B A u \\rangle = \\sum _ { u \\in N _ e } \\langle u | B u \\rangle s ( u ) = \\sum _ { u \\in N _ e } \\langle A u | B u \\rangle = \\sum _ { u \\in N } \\langle A u | B u \\rangle = \\\\ & = \\sum _ { u \\in N } \\langle u | A B u \\rangle = t r _ N ( A B ) \\ : , \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + \\Delta u = - n u , ( x , t ) \\in \\R ^ + \\times \\R ^ + , \\\\ n _ { t t } + ( 1 - \\Delta ) n = | u | ^ 2 , \\\\ u ( x , 0 ) = u _ 0 ( x ) \\in H ^ { s _ 0 } ( \\R ^ + ) \\\\ n ( x , 0 ) = n _ 0 ( x ) \\in H ^ { s _ 1 } ( \\R ^ + ) , n _ t ( x , 0 ) = n _ 1 ( x ) \\in { H } ^ { s _ 1 - 1 } ( \\R ^ + ) , \\\\ u ( 0 , t ) = g ( t ) \\in H ^ { \\frac { 2 s _ 0 + 1 } 4 } ( \\R ^ + ) , n ( 0 , t ) = h ( t ) \\in H ^ { s _ 1 } ( \\R ^ + ) , \\end{cases} \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\omega = 0 \\ \\ \\textrm { o n } \\ \\ \\R ^ d \\times ( t _ 1 + 2 \\sigma , t _ 1 + \\kappa ) . \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} \\sum _ { f : \\ , ( u , v ) \\in f } B ( ( u , v ) , f ) & = \\frac { 1 } { ( r - 1 ) ! } \\sum _ { f : \\ , ( u , v ) \\in f } B _ 1 ( u , \\pi _ 1 ( f ) ) B _ 2 ( v , \\pi _ 2 ( f ) ) \\\\ & = \\sum _ { e _ 1 : \\ , u \\in e _ 1 } \\sum _ { e _ 2 : \\ , v \\in e _ 2 } B _ 1 ( u , e _ 1 ) B _ 2 ( v , e _ 2 ) \\\\ & = 1 . \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\sum _ { \\begin{subarray} { l } \\widetilde { \\Sigma } < D \\\\ \\vert \\widetilde { \\Sigma } \\rvert = 2 \\end{subarray} } B ^ { \\widetilde { \\Sigma } } = B ^ \\Sigma + B ^ { \\Sigma ' } . \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} \\dim ( D ^ \\circ ) = \\frac { n ^ 2 } { 4 } + \\frac { 3 } { 4 } s ^ 2 - 1 . \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\varepsilon _ { \\delta , R } : = \\min \\left \\{ C _ { R } ^ { - 1 } R \\delta ^ 2 , \\varepsilon _ 0 \\right \\} . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} \\nabla _ { a } ^ { \\nu } U ( t ) = \\nabla ^ { n } \\nabla _ { a } ^ { - ( n - \\nu ) } U ( t ) = \\nabla ^ { n } \\sum _ { s = a } ^ { t } \\frac { [ t - \\varphi ( s ) ] ^ { \\overline { n - \\nu - 1 } } } { \\Gamma ( n - \\nu ) } U ( s ) , \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} P _ { \\Lambda , \\eta , \\beta , H } ( \\sigma ) = \\frac { 1 } { Z _ { \\Lambda , \\eta , \\beta , H } } \\exp { \\left [ 2 \\beta \\sum _ { \\{ u , v \\} } \\delta _ { \\sigma _ u , \\sigma _ v } + 2 \\beta \\sum _ { e = \\{ u , v \\} : u \\in \\Lambda , v \\in \\partial _ { e x } \\Lambda } \\delta _ { \\sigma _ u , \\eta _ v } + 2 H \\sum _ { u \\in \\Lambda } \\delta _ { \\sigma _ u , 1 } \\right ] } , \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} v ( r , \\theta ) = \\sum _ { k \\ge 0 } a _ { k } ( r ) \\tilde \\Phi _ i ^ k ( \\theta ) , r \\in ( 1 , R ) , \\ \\theta \\in \\mathbb S ^ { n - 1 } , \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} ( \\alpha - \\mu b | \\omega | - \\frac { 1 } { 4 \\eta } \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } ) \\| \\nu ' \\| ^ 2 = \\frac { 4 \\mu \\epsilon \\gamma ^ 4 \\beta _ 0 ^ 2 } { 1 + 4 \\epsilon \\gamma ^ 4 \\beta _ 0 } \\| \\nu ' \\| ^ 2 \\equiv \\theta \\| \\nu ' \\| ^ 2 . \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} & \\sum _ { k \\leq x } \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } \\tau ( \\gcd ( k , j ) ) \\log j \\\\ & = \\sum _ { n \\leq x } \\frac { \\sigma ( n ) } { n } \\log \\frac { n } { e } + \\frac { 1 } { 2 } \\sum _ { n \\leq x } \\frac { l ( n ) } { n } + \\log \\sqrt { 2 \\pi } \\sum _ { n \\leq x } \\frac { \\tau ( n ) } { n } + { \\Theta } \\sum _ { n \\leq x } \\frac { \\sigma _ { - 1 } ( n ) } { n } . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( n \\right ) = - z \\frac { d } { d z } \\frac { z } { z - 1 } = \\frac { z } { \\left ( z - 1 \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} P _ i ( k ) = \\frac { v _ i } { \\mu _ k } Q _ k ( i ) . \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} \\{ y \\in \\mathbb { B } \\colon y = S _ t x , x \\in K \\} , t \\in ( 0 , T ] , \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\mathbb { E } [ f ( \\Delta _ { i , n } ^ { ( 1 ) } X ^ { t o y , ( 1 ) } , \\Delta _ { j , n } ^ { ( 2 ) } X ^ { t o y , ( 2 ) } ) | \\mathcal { S } ] = \\sum _ { k = 1 } ^ N g _ k ( | \\mathcal { I } _ { i , n } ^ { ( 1 ) } | , | \\mathcal { I } _ { j , n } ^ { ( 2 ) } | , | \\mathcal { I } _ { j , n } ^ { ( 2 ) } \\cap \\mathcal { I } _ { j , n } ^ { ( 2 ) } | ) h _ k ( \\sigma ^ { ( 1 ) } , \\sigma ^ { ( 2 ) } , \\rho ) ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} { \\rm m u l t } _ \\lambda ( \\mathbf { m } ) = \\sum _ { w \\in W _ K } \\varepsilon ( w ) Q ( w ( \\mathbf { m } + \\rho _ c ) - \\lambda - \\rho _ n ) . \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\Psi _ { \\Upsilon , M } ( w ) = \\frac { M ^ { w } - M ^ { ( 1 - \\Upsilon ) w } } { \\Upsilon w ^ 2 \\log M } . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\widetilde { P } ( x ) = \\sum _ { n \\leq x } \\frac { ( \\mu * \\mu ) ( n ) } { n } \\Delta \\left ( \\frac { x } { n } \\right ) \\log \\frac { x } { e } + O \\left ( ( \\log x ) ^ 3 \\right ) . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} \\stackrel { \\ast } { \\bar { \\nabla } } _ { X } Y = \\bar { \\nabla } _ X Y + \\eta ( X ) \\phi Y + ( f _ 1 - f _ 3 ) \\eta ( Y ) \\phi X - ( f _ 1 - f _ 3 ) g ( \\phi X , Y ) \\xi . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} C = \\frac { m } { 2 p - 1 + \\frac { ( p - 1 ) | I | } { p - | I | } } . \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} { n \\brack k } = \\prod _ { i = 0 } ^ { k - 1 } \\frac { q ^ { 2 n } - q ^ { 2 i } } { q ^ { 2 k } - q ^ { 2 i } } \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} \\chi ( M _ k ) = m \\chi ( M ) + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( \\chi ( \\Sigma ^ y _ { \\ell } ) - b ^ y _ { \\ell } ) \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\alpha ( i , r ; j , s ) & = 2 ( - \\delta _ { r - s , ( \\alpha _ i , \\alpha _ j ) } + \\delta _ { r - s , - ( \\alpha _ i , \\alpha _ j ) } ) \\\\ & = \\begin{cases} 2 ( - \\delta _ { r - s , 2 r _ i } + \\delta _ { r - s , - 2 r _ i } ) & \\ i = j , \\\\ 2 \\sum _ { l = 0 } ^ { - c _ { i j } - 1 } ( - \\delta _ { r - s , - r _ i + c _ { i j } + 1 + 2 l } + \\delta _ { r - s , r _ i + c _ { i j } + 1 + 2 l } ) & \\ i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} \\sigma _ 2 \\circ \\tau ( n ) & = \\sigma _ 2 \\circ \\tau \\circ \\sigma _ 1 ^ { k - 1 } \\circ \\sigma _ 2 ( m ) = \\sigma _ 2 \\circ \\sigma _ 2 ^ { k - 1 } \\circ \\sigma _ 1 ( m ) = \\sigma _ 2 ^ k \\circ \\sigma _ 1 ( m ) \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} \\Psi _ 2 ^ { \\min } ( p ^ e , \\chi ) = 0 , \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{gather*} L ( f _ { 3 2 } , s ) = L \\ ( \\psi , s - \\frac 1 2 \\ ) = \\sum _ { \\mathfrak I \\in S _ 1 } \\tilde \\psi ( \\mathfrak I ) ( N \\mathfrak I ) ^ { - s } , \\end{gather*}"}
-{"id": "6984.png", "formula": "\\begin{align*} x _ { \\sigma ( \\iota ) } = y _ { \\sigma ( \\iota ) } ; \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Delta u + e ^ u = 0 \\\\ & \\int _ { \\mathbb { R } ^ 2 } e ^ u < \\infty . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} D _ x ^ { \\alpha = n } [ f ( x ) \\cdot g ( x ) ] = \\sum ^ { \\infty } _ { l = 0 } C ^ { l } _ { n } g ^ { ( l ) } ( x ) D _ x ^ { ( n - l ) } [ f ( x ) ] = \\sum ^ { n } _ { l = 0 } C ^ { l } _ { n } g ^ { ( l ) } ( x ) f ^ { ( n - l ) } ( x ) = ( f ( x ) \\cdot g ( x ) ) ^ { ( n ) } . \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} B _ { \\chi _ 1 } ^ { \\chi _ 2 } ( V z , s ) = \\chi _ 1 ( - 1 ) B _ { \\chi _ 1 } ^ { \\chi _ 2 } ( z , s ) . \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} e ^ { \\mathbf { C } T _ { 1 , 1 } } = \\frac { 1 + M _ 0 } { m ( 1 + M _ 0 ^ 3 ) } . \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\Phi _ { \\mathbf h _ p } ( \\nu \\beta _ 1 \\nu ' ) = \\frac { \\chi _ { \\psi } ( p ) \\underline { \\chi } _ p ( \\gamma ) p ^ { - 1 / 2 } } { p - 1 } ( p - G ( - \\gamma , p ) ) . \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} D ^ { ( l ) } f = \\sum _ { j = 0 } ^ 2 \\sum _ { k = 0 } ^ 3 \\frac { \\partial f _ k } { \\partial x _ 0 } e _ j e _ k , D ^ { ( r ) } f = \\sum _ { j = 0 } ^ 2 \\sum _ { k = 0 } ^ 3 \\frac { \\partial f _ k } { \\partial x _ 0 } e _ k e _ j , \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } V , N ) = \\breve { g } ( V , \\breve { J } N ) = \\frac { 1 } { p } \\breve { g } ( \\breve { J } V , \\breve { J } N ) . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\| ( Y _ 1 + h _ 1 ) ^ { \\ast i _ 1 + 1 } - Y _ 1 ^ { \\ast i _ 1 + 1 } \\| _ { \\mu } & = \\| Y _ 1 \\ast \\left ( ( Y _ 1 + h _ 1 ) ^ { \\ast i _ 1 } - Y _ 1 ^ { \\ast i _ 1 } \\right ) + h _ 1 \\ast ( Y _ 1 + h _ 1 ) ^ { \\ast i _ 1 } \\| _ { \\mu } \\\\ & \\leq i _ 1 \\| Y _ 1 \\| _ { \\mu } \\left ( \\| Y _ 1 \\| _ { \\mu } + \\| h _ 1 \\| _ { \\mu } \\right ) ^ { i _ 1 - 1 } \\| h _ 1 \\| _ { \\mu } + \\left ( \\| Y _ 1 \\| _ { \\mu } + \\| h _ 1 \\| _ { \\mu } \\right ) ^ { i _ 1 } \\| h _ 1 \\| _ { \\mu } , \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} \\partial _ t \\| u \\| _ { L ^ 2 _ x ( \\R ^ + ) } ^ 2 = 2 \\Re \\int _ 0 ^ \\infty u _ t \\overline { u } \\d x & = - 2 \\Im \\int _ 0 ^ \\infty u _ { x x } \\overline { u } \\d x \\\\ & = - 2 \\Im \\overline { u } ( 0 , \\cdot ) u _ x ( 0 , \\cdot ) = - 2 \\Im \\overline { g } ( \\cdot ) u _ x ( 0 , \\cdot ) . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\omega J _ b \\xi + \\mathcal L _ { + \\infty } \\nu = 2 r \\xi \\nu \\\\ - \\omega J _ b \\nu + ( 1 - \\gamma ) J _ c \\xi = r \\nu ^ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} a _ \\epsilon ( x + h ) = a _ \\epsilon ( x ) + \\bar { P } _ { x , \\epsilon } ( h ) + O ( \\abs { h } ^ { q - \\abs { \\epsilon } } ) , \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} \\xi _ k ^ n = n ^ { p / 2 - 1 } \\sum _ { i \\in L ( n , k , T ) } f ( \\Delta _ { i , n } C ( r ) ) , \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} \\begin{cases} a : = \\sinh ( \\log \\sqrt { X } ) \\sin ( ( 2 T ) ^ { - 1 } ) ; \\\\ b : = \\cosh ( \\log \\sqrt { X } ) \\cos ( ( 2 T ) ^ { - 1 } ) . \\end{cases} \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { H ^ { n - 1 } ( \\partial B ( x _ 0 , r _ k ) \\cap \\Omega ^ c ) } { r _ k ^ { n - 1 } } = + \\infty . \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { \\varepsilon \\in E } \\int _ 0 ^ 1 \\chi _ { P + \\varepsilon } ( g _ x ( t ) ) \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\partial _ { i n } \\Lambda : = \\{ z \\in \\mathbb { Z } ^ d : z \\in \\Lambda , z \\Lambda ^ C \\} , \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} \\alpha = \\frac { \\pi ^ 2 } { 4 \\gamma } - \\frac { \\pi ^ 2 } { 8 \\gamma ^ 2 } \\log ( 2 e \\gamma ) - \\frac { \\pi ^ 2 } { 1 6 \\gamma } \\approx 0 . 3 7 5 , \\beta = \\frac { \\pi } { 2 \\gamma } \\approx 0 . 5 8 6 , a _ { \\gamma } = 1 - \\frac { \\pi } { 2 \\gamma } . \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} \\mathcal { Z } _ { N , w _ j } & = \\begin{cases} \\{ z \\in \\{ 1 , 2 , \\ldots , b ^ { m - w _ j } - 1 \\} : \\gcd ( z , N ) = 1 \\} & \\mbox { i f } w _ j < m , \\\\ \\{ 1 \\} & \\mbox { i f } w _ j \\ge m , \\end{cases} \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{gather*} x \\cdot ( m \\otimes h ) = x ^ { 2 1 } m \\otimes x ^ { 2 2 } h S \\big ( x ^ 1 \\big ) . \\end{gather*}"}
-{"id": "2744.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } ( | R _ n - R _ n ( r ) | > \\delta ) = 0 \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} [ \\phi ( x ) ] ^ { \\beta [ \\frac { n } { x } ] } = i ( n , N _ \\phi | A _ \\phi ) \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\in \\mathbb { Z } ^ { d - 1 } \\\\ n \\neq 0 } } \\frac { 1 } { | n | \\prod _ { k = 1 } ^ { d - 1 } ( | L _ k ( n ) | + 1 ) \\left \\| n _ 1 \\alpha _ 1 + \\cdots + n _ { d - 1 } \\alpha _ { d - 1 } \\right \\| } < \\infty . \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} Q ' _ i ( k ) = P _ i ( k ) \\quad P ' _ k ( i ) = Q _ k ( i ) , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\nabla = d + p _ { - 1 } d \\sigma + \\sum _ { i = 1 } ^ \\ell u _ i ( \\sigma ) \\alpha _ i d \\sigma \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} N ( w ) = \\{ \\alpha _ 1 , \\alpha _ 1 + \\alpha _ 2 , \\alpha _ 1 + \\alpha _ 2 + \\alpha _ 3 , \\alpha _ 2 + \\alpha _ 3 , \\alpha _ 3 \\} . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\bar r _ 2 = \\frac { b _ { 0 , \\sup } - b _ { 1 , \\inf } \\underbar r _ 1 - k \\frac { \\chi _ 1 } { d _ 3 } \\underbar r _ 2 } { b _ { 2 , \\inf } - l \\frac { \\chi _ 2 } { d _ 3 } } , \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} P _ { a } ( x ) = \\sum _ { n \\leq x } \\frac { ( \\mu * \\mu ) ( n ) } { n } \\Delta _ { a } \\left ( \\frac { x } { n } \\right ) + O _ { a } \\left ( ( \\log x ) ^ 2 \\right ) , \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} \\Delta _ { y _ 1 , \\ldots , y _ m } ^ m p _ n ( x ) = 0 , m = n + 1 , n + 2 , \\ldots \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} H _ i ( F ' ) : = \\{ T ' \\in C ( F ' ) \\colon r ( T ' ) \\in V ( F _ i ) \\} \\enspace \\enspace H _ i ( F ^ { \\# } ) : = \\{ T ' \\in C ( F ^ { \\# } ) \\colon r ( T ' ) \\in V ( F _ i ) \\} . \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} F ( t ) = \\xi P ( j t ) + m \\big ( \\varphi _ 1 ( j t ) - \\tau \\psi ( j t ) \\big ) . \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { n - \\delta + 1 } { n - r \\brack \\delta - 1 } C ' _ r = c ^ { n - \\delta + 1 } \\sum _ { s = 0 } ^ n { n - s \\brack n - \\delta + 1 } B _ s \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} N = \\sum _ { v \\in V ( H ) } | \\R ^ v | \\le \\sum _ { v \\in V ( H ) } 2 | D ^ v | = 2 | D | \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} Q _ n = { 1 \\over W ^ { ( 1 ) } _ { n } } , P _ { n } = { W ^ { ( 2 ) } _ { n - 1 } \\over W ^ { ( 1 ) } _ { n - 1 } W ^ { ( 1 ) } _ { n } } \\ , , \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} \\bar A _ 1 = \\frac { a _ { 0 , \\sup } } { a _ { 1 , \\inf } - \\frac { k \\chi _ 1 } { d _ 3 } } , \\bar A _ 2 = \\frac { b _ { 0 , \\sup } } { b _ { 2 , \\inf } - \\frac { l \\chi _ 2 } { d _ 3 } } . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} f \\colon [ 0 , T ] \\times [ 0 , T ] \\to U , f ( s , r ) = \\ 1 _ { [ 0 , s ] } ( r ) B ^ \\ast T ^ \\ast ( s - r ) g ( s ) . \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} T _ { u } \\varphi ( x ) ~ = ~ \\varphi ( x \\star u ) , x \\in E , \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} h ^ { ( 1 ) } u _ 0 = 0 , h ^ { ( 2 ) } v _ 0 = 0 \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} 0 = x _ { n + 1 } - 2 x _ { n + 1 } x _ n - 2 x _ 1 x _ n ^ 2 + x _ 1 x _ { n } + \\frac 1 2 \\sum _ { i = 2 } ^ { n - 1 } x _ i ^ 2 . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} \\rho ( g h ) = \\frac { \\Delta _ H ( h ) } { \\Delta _ G ( h ) } \\rho ( g ) ; \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} \\mathcal { H } ^ s = \\bigoplus _ { n \\geq 0 } V _ { N + 1 } ^ n \\left ( V _ { N + 1 } ^ * \\right ) . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} X _ t ( x ) \\stackrel { d } { = } \\mathrm { e } ^ { - \\theta t } Y ^ \\delta \\biggl ( \\frac { \\sigma ^ 2 } { 4 \\theta } \\bigl ( \\mathrm { e } ^ { \\theta t } - 1 \\bigr ) , x \\biggr ) , t \\ge 0 , \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} u _ { \\infty } ( x , t ) = \\frac { M _ 0 } { M } \\mathcal { B } _ M ( x , t ) \\forall ( x , t ) \\in \\R ^ n \\times \\left [ 0 , \\infty \\right ) . \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} \\max _ { \\lambda } & ~ U _ { L } ( \\lambda ) = a \\bigg [ \\log ( 1 + d ( \\lambda C - 2 D ) ) - \\log ( 1 + \\rho _ { e } ^ { \\textrm { o p t } } ) \\bigg ] \\\\ & ~ ~ ~ ~ ~ ~ - \\lambda ^ { 2 } C + 2 \\lambda D , \\\\ s . t . & ~ \\lambda \\geq 0 , \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} \\bar { P } ^ { i n } _ { \\mathbb { Z } ^ d , \\vec { H } } ( T ^ { \\vec { H } } ( x ) = 1 ) = \\bar { E } ^ { i n } _ { \\mathbb { Z } ^ d , \\vec { H } } [ p _ x ( \\beta , H , H _ x ) ] \\leq P r o b ( | H _ x | < \\delta ) + 1 - a ^ 2 ( \\beta , H , \\delta ) . \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} \\rho = \\begin{cases} - 2 \\min ( f , g ) & 2 \\mid u _ 2 2 \\mid \\frac { c u _ 2 + 2 } { N / c } , \\\\ 1 - \\min ( f , g ) - \\min ( f + 1 , g - 1 ) & 2 \\mid u _ 2 2 \\nmid \\frac { c u _ 2 + 2 } { N / c } , \\\\ 1 - \\min ( f , g ) - \\min ( f - 1 , g + 1 ) & 2 \\nmid u _ 2 2 \\mid \\frac { c u _ 2 + 2 } { N / c } , \\\\ 2 - 2 \\min ( f , g ) & 2 \\nmid u _ 2 2 \\nmid \\frac { c u _ 2 + 2 } { N / c } . \\end{cases} \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\lambda g ( U , \\zeta + \\phi \\psi ) = 0 . \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} \\tilde \\alpha _ 0 ^ { [ n ] } = \\frac { \\det [ \\chi _ { i + j } ( R ) ] _ { i , j = 1 } ^ { p } } { \\det [ \\chi _ { i + j } ( R ) ] _ { i , j = 0 } ^ { p } } . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} \\nu _ { 1 1 } ( x , \\zeta ^ 2 ) = \\mu _ { 1 1 } ( x , \\zeta ) , \\nu _ { 1 2 } ( x , \\zeta ^ 2 ) = \\mu _ { 1 2 } ( x , \\zeta ) / \\zeta . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\phi _ { \\mathbf h , q } ( z ) = \\pmb { \\phi } _ q ( x _ 3 ) \\phi _ { \\breve { \\mathbf g } , q } ( X ) = \\mathbf 1 _ { \\Z _ q } ( x _ 1 ) \\mathbf 1 _ { \\Z _ q } ( x _ 2 ) \\mathbf 1 _ { \\Z _ q } ( x _ 3 ) \\mathbf 1 _ { \\Z _ q } ( x _ 4 ) \\mathbf 1 _ { \\Z _ q } ( x _ 5 ) . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} - \\alpha ^ { 2 } = \\frac { 2 m \\epsilon } { \\hbar ^ { 2 } } , \\quad \\beta = \\frac { 2 m b } { \\hbar ^ { 2 } } , \\quad \\gamma = \\frac { 2 m c } { \\hbar ^ { 2 } } , \\quad \\delta = \\frac { 2 m a } { \\hbar ^ { 2 } } + \\ell ( \\ell + 1 ) , \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\left ( V _ { j n } ^ { \\varepsilon } \\right ) _ { t } - ( A + \\rho _ { \\varepsilon } ) V _ { j n } ^ { \\varepsilon } = e ^ { \\rho _ { \\varepsilon } \\left ( t - T \\right ) } \\left \\langle F \\left ( \\cdot , t ; e ^ { \\rho _ { \\varepsilon } \\left ( T - t \\right ) } v _ { n } ^ { \\varepsilon } \\right ) , \\phi _ { j } \\right \\rangle + \\left \\langle \\mathbf { P } _ { \\varepsilon } ^ { \\beta } v _ { n } ^ { \\varepsilon } , \\phi _ { j } \\right \\rangle , \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} \\frac { 1 } { d _ j ( A ) } = \\underset { { \\mathcal { M } \\subset \\mathbb { C } ^ { 2 n } } \\atop { \\dim \\mathcal { M } = 2 n - j + 1 } } { \\min } \\ , \\ , \\ , \\underset { { x \\in \\mathcal { M } } \\atop { \\langle x , A x \\rangle = 1 } } { \\max } \\ , \\ , \\ , \\langle x , i J x \\rangle , \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\lim _ { \\sigma \\rightarrow 2 } ( 2 - \\sigma ) c ( n , \\sigma ) ( - \\Delta ) ^ { \\sigma / 2 } f ( x ) = - \\Delta f ( x ) . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\Delta u + e ^ u = 0 \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} k _ { u , v } = \\frac { u - 3 v } { 2 v } , \\lambda _ { i , j } = \\frac { i - 1 } { 2 } - \\frac { 1 + ( - 1 ) ^ { i + j } } { 4 } - \\frac { u } { 2 v } j , s _ { i , j } = \\frac { i } { 2 } - \\frac { u } { 2 v } j , q _ { i , j } = \\frac { ( u j - v i ) ^ 2 - 4 v ^ 2 } { 8 v ^ 2 } , \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{gather*} w _ { 0 1 } ^ 1 \\ = \\ l _ { 1 1 } + l _ { 1 2 } , w _ { 2 1 } ^ 1 \\ = \\ \\frac { l _ { 1 1 } + l _ { 1 2 } } { l _ { 2 1 } } , \\\\ w _ { 1 2 } ^ 2 \\ = \\ \\frac { w _ { 0 1 } ^ 2 + 1 } { l _ { 1 2 } } , w _ { 2 2 } ^ 2 \\ = \\ \\frac { w _ { 0 1 } ^ 2 + 1 } { l _ { 2 2 } } . \\end{gather*}"}
-{"id": "7364.png", "formula": "\\begin{align*} \\nu _ k ^ * \\omega _ { \\log } ( s _ 1 , \\ldots , s _ k ) & = \\tilde { \\omega } _ { \\log } ( s _ 1 , \\ldots , s _ k ) \\cdot F , \\\\ \\nu _ k ^ * \\omega ' _ { \\log } ( s _ 1 , \\ldots , s _ k ) & = \\tilde { \\omega } _ { \\log } ( s _ 1 , \\ldots , s _ k ) \\cdot F ' . \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} \\prod _ { t = 1 } ^ { \\lambda _ m + 1 } { \\lambda _ { m + i _ { t } } + \\lambda _ m + 1 - t \\brack 1 } _ q \\neq 0 . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} P _ i = \\sum _ { j = 1 } ^ k [ \\tau ^ i ( 1 - \\tau ) ( R - \\rho ) ] ^ { \\beta _ j } f _ j ( \\rho _ i ) , \\ ; \\ , Q _ i = \\sum _ { \\ell = 1 } ^ m A _ \\ell [ \\tau ^ i ( 1 - \\tau ) ( R - \\rho ) ] ^ { \\gamma _ \\ell } . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\mathcal { Q } _ { t } ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } ) = \\mathbb { E } _ { \\xi _ t , D _ t } \\Big [ \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } , \\xi _ t , D _ t ) | D _ { t - 1 } , \\xi _ { [ t - 1 ] } \\Big ] , \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} Z ( G , F _ 3 ) \\subset Z ( F _ 1 , F _ 3 ) = Z ( F _ 1 , F _ 2 , F _ 3 ) \\cup Z ( X _ 1 , Z _ 2 ) . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} d ( p , \\gamma _ i ( \\delta ) ) - d ( \\gamma _ i ( 0 ) , \\gamma _ i ( \\delta ) ) = - \\int _ 0 ^ { d _ i } h ' ( t ) d t = \\int _ 0 ^ { d _ i } g _ { w _ i ( t ) } ( w _ i ( t ) , \\dot \\sigma _ i ( t ) ) d t \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\| A \\| _ F ^ 2 = \\sum _ { i , j \\in [ d ] } A _ { i j } ^ 2 \\ , . \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} \\overline X _ u \\cong _ { h t } \\begin{cases} S ^ 2 \\vee \\bigvee _ { i \\in I } S ^ { d } & \\textnormal { i f } n < N < 2 n \\\\ \\left ( \\bigvee _ { i = 1 } ^ { e - 1 } S ^ 1 \\right ) \\vee \\left ( \\bigvee _ { i = 1 } ^ r S ^ { 1 } \\right ) & \\textnormal { i f } N = n \\\\ e ' \\textnormal { p o i n t s } & \\textnormal { i f } N = n - 1 , \\end{cases} \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\gamma ( i , r ; j , s ) = \\begin{cases} \\widetilde { c } _ { j i } ( - r _ j - r + s ) - \\widetilde { c } _ { j i } ( r _ j - r + s ) & \\ r > s , \\\\ 0 & \\ r = s . \\end{cases} \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} U _ { \\omega } ( \\eta ) : = \\upsilon | \\eta | + \\Phi _ { \\omega } ( \\eta ) \\ge 0 , \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\dot { z } = [ A ( t ) + J f ( t , y ( t ) ) ] z \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} C ' _ { \\max } ( \\alpha , \\beta ) \\sqrt { k } \\leq C _ { \\min } ( \\alpha , \\beta ) b _ k ^ 2 \\sqrt { \\sum _ { j = 1 } ^ k b _ j ^ { - 4 } } \\hbox { f o r a l l } \\ : k \\in \\mathbb { N } ^ * , \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } m ( s ) = & W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) h ( s ) , \\\\ n ( s ) = & \\left ( 1 - W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) \\sigma _ { z } \\left ( s \\right ) \\right ) ^ { - 1 } b _ { z } ( s ) ( W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) ) ^ { 2 } \\\\ & + g _ { z } ( s ) W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ) + W _ { x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ) \\sigma ( s ) h ( s ) , \\end{array} \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\mathcal { T S } ^ { o p } & = \\{ \\pi ^ * ( B _ 1 ) , \\ldots , \\pi ^ * ( B _ { k - 2 } ) , \\pi ^ * ( B _ { k - 1 } ) - E , E , \\pi ^ * ( B _ { k } ) - E , \\pi ^ * ( B _ { k + 1 } ) , \\ldots , \\pi ^ * ( B _ n ) \\} \\\\ & = \\{ \\pi ^ * ( A _ { n - 1 } ) , \\ldots , \\pi ^ * ( A _ { n - k + 2 } ) , \\pi ^ * ( A _ { n - k + 1 } ) - E , E , \\pi ^ * ( A _ { n - k } ) - E , \\pi ^ * ( A _ { n - k - 1 } ) , \\ldots , \\pi ^ * ( A _ 1 ) , \\pi ^ * ( A _ n ) \\} \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} & \\upsilon _ { 2 ^ { - n } } ( M _ 0 ) = - \\frac { 1 } { 4 \\pi } \\int \\limits _ { \\mathbb { R } ^ 3 \\setminus \\widetilde { \\Omega } _ { n - 2 } } \\ , \\frac { \\Delta f _ 0 ( M ) } { \\rho _ { M _ 0 } ( M ) } \\ , d m _ 3 ( M ) + \\\\ & + \\frac { 1 } { 4 \\pi } \\int \\limits _ { \\mathbb { R } ^ 3 } \\ , \\frac { \\Phi _ { n } ( M ) } { \\rho _ { M _ 0 } ( M ) } \\ , d m _ 3 ( M ) \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} | { u } _ { \\iota } x _ 0 - { u } _ 0 x _ 0 | & \\geqslant \\sum _ { l = 0 } ^ { \\iota - 1 } | { u } _ l x _ 0 - { u } _ { l + 1 } x _ 0 | - 4 8 \\iota \\delta \\\\ & \\geqslant \\iota \\ ; | { u } _ j x _ 0 - { u } _ { j + 1 } x _ 0 | - 4 8 \\iota \\delta . \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} F ^ { 0 , q } & = F ^ { 0 , q } ( x , a , b ) : = \\int _ { - 1 } ^ 1 f ^ { ( q ) } ( A + x + 2 a \\sqrt { x } s ) \\ , d s , \\\\ F ^ { p , q } & = F ^ { p , q } ( x , a , b ) \\\\ & : = \\int _ { - 1 } ^ 1 \\int _ { 0 } ^ { s _ 1 } \\dots \\int _ { 0 } ^ { s _ p } f ^ { ( q ) } ( A + x + 2 a \\sqrt { x } s _ { p + 1 } ) \\ , d s _ { p + 1 } \\ldots \\ , s _ 2 \\ , d s _ 2 \\ , s _ 1 \\ , d s _ 1 , \\\\ & p = 1 , \\dots , l , \\ , q = l + 1 , \\ldots , 2 l + 1 . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} p _ h ( x , y ) ~ = ~ p ( x , y ) h ( y ) / h ( x ) , \\ , x , y \\in E . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} H _ { 1 , \\boldsymbol { \\alpha } } ( \\tau ) = 2 \\int _ { \\R ^ 2 } \\left ( \\mathcal G _ { \\alpha _ 1 } ( w _ 1 ) \\mathcal G _ { \\alpha _ 2 } ( w _ 2 ) - \\mathcal F _ { \\alpha _ 1 } ( w _ 1 ) \\mathcal F _ { \\alpha _ 2 } ( w _ 2 ) \\right ) e ^ { 2 \\pi i \\tau Q ( \\boldsymbol { w } ) } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} R _ \\alpha \\circ \\pi ( g \\cdot ( t , c ) ) & = R _ \\alpha \\circ \\pi ( ( t , c ) ) , \\ , & Z \\circ \\pi ( g \\cdot ( t , c ) ) & = Z \\circ \\pi ( ( t , c ) ) \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\left ( \\mathbb { C } [ X ] \\otimes _ { G _ c } M \\right ) _ { G _ d } \\cong \\mathbb { C } [ G _ d \\backslash X ] \\otimes _ { G _ c } M = \\mathbb { C } \\otimes _ { G _ c } M \\cong M _ { G _ c } . \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\Phi _ 1 \\Phi _ 2 \\cdots \\Phi _ n = \\left ( ( - 1 ) ^ { a - 1 } P \\right ) \\mathrm { I d } _ { a \\times a } \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{gather*} \\zeta ( 2 n ) = { \\frac 1 2 } \\sum _ { m \\in \\Z \\setminus \\{ 0 \\} } \\frac 1 { m ^ { 2 n } } = ( - 1 ) ^ { n + 1 } \\frac { B _ { 2 n } \\cdot ( 2 \\pi ) ^ { 2 n } } { 2 ( 2 n ) ! } , \\end{gather*}"}
-{"id": "3596.png", "formula": "\\begin{align*} \\langle [ \\nabla _ r , A _ q ] _ - u ^ { } , w _ r | w | ^ { q - 2 } \\rangle = c \\left \\langle \\bigl [ ( \\nabla _ r \\mathsf { f } ^ i ) \\mathsf { f } ^ l + \\mathsf { f } ^ i \\nabla _ r \\mathsf { f } ^ l \\bigr ] w _ l , \\nabla _ i ( w _ r | w | ^ { q - 2 } ) \\right \\rangle = : S _ { 1 } + S _ { 2 } , \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} W ^ { s , p , q } ( \\mathbb { G } ) = \\{ u \\in L ^ { p } ( \\mathbb { G } ) : u , [ u ] _ { s , p , q } < + \\infty \\} , \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} ( \\phi + _ r \\psi ) ( K ) = \\phi ( K ) + _ r \\psi ( K ) \\forall K \\in \\Sigma X \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} F _ 2 = \\ , & b + a _ 1 b + a _ 1 ^ 3 b + a _ 1 ^ 4 b + b ^ 2 + a _ 1 b ^ 2 + a _ 1 ^ 3 b ^ 2 + a _ 1 ^ 4 b ^ 2 + b ^ 3 + a _ 1 ^ 2 b ^ 3 + a _ 1 ^ 3 b ^ 3 + b ^ 4 \\\\ & + a _ 1 ^ 2 b ^ 4 + a _ 1 ^ 3 b ^ 4 + b ^ 5 + a _ 1 b ^ 5 + a _ 1 ^ 2 b ^ 5 + b ^ 6 + a _ 1 b ^ 6 + a _ 1 ^ 2 b ^ 6 + b ^ 7 + b ^ 8 \\cr & + ( a _ 1 ^ 3 + a _ 1 ^ 4 + a _ 1 ^ 3 b + a _ 1 ^ 3 b ^ 2 + a _ 1 ^ 5 b ^ 2 + a _ 1 ^ 3 b ^ 3 + a _ 1 ^ 4 b ^ 3 ) k \\cr & + ( a _ 1 ^ 4 + a _ 1 ^ 4 b + a _ 1 ^ 4 b ^ 2 + a _ 1 ^ 6 b ^ 2 + a _ 1 ^ 4 b ^ 3 ) k ^ 2 + a _ 1 ^ 7 k ^ 3 + a _ 1 ^ 8 k ^ 4 , \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} \\delta ( u \\otimes v ) _ { s t } = \\int _ s ^ t \\big ( \\dot { \\mu } _ r \\otimes v _ r + u _ r \\otimes \\dot { \\nu } _ r \\big ) d r + \\Gamma _ { s t } ^ 1 ( u \\otimes v ) _ s + \\Gamma _ { s t } ^ 2 ( u \\otimes v ) _ s + ( u \\otimes v ) _ { s t } ^ { \\natural } \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} \\langle w , 1 \\rangle & = \\tfrac { 1 } { c _ { 5 } } \\Bigl ( \\omega _ { 5 } R ^ { 5 } - \\tfrac { 3 ( R ^ 3 + 9 R ^ 2 + 2 7 R + 2 4 ) \\sigma _ { 4 } R ^ { 2 } } { ( R + 3 ) } + \\tfrac { 6 ( R ^ 5 + 9 R ^ 4 + 3 1 R ^ 3 + 4 8 R ^ 2 + 3 6 R + 1 2 ) \\sigma _ { 4 } } { ( R + 3 ) } \\Bigr ) \\\\ & = \\frac { R ^ { 5 } } { 5 ! } + \\frac { 3 R ^ 5 + 2 7 R ^ 4 + 1 0 5 R ^ 3 + 2 1 6 R ^ 2 + 2 1 6 R + 7 2 } { 4 ! ( R + 3 ) } , \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\| \\Phi _ { i j } \\| _ { L ^ 1 } & = \\int _ { \\R ^ 3 } | \\nabla _ { z } f _ { i j } ( z ) | ^ 2 + \\frac { 1 } { 2 } V _ { i j } ( z ) | f _ { i j } ( z ) | ^ 2 \\d z \\\\ & = \\lambda _ { N , i j } \\int _ { \\R ^ 3 } | f _ { i j } ( z ) | ^ 2 \\mathbf { 1 } ( | z | \\leqslant \\ell ) \\d z \\\\ & \\leqslant \\Big ( \\frac { 3 a _ { i , j } } { N \\ell ^ 3 } + \\frac { C } { N ^ 2 \\ell ^ 4 } \\Big ) \\int _ { \\R ^ 3 } \\mathbf { 1 } ( | z | \\leqslant \\ell ) \\d z \\leqslant \\frac { 4 \\pi a } { N } + \\frac { C } { N ^ 2 \\ell } . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} p _ x ( \\beta , H , H _ x ) : = 1 - a ^ 2 ( \\beta , H , | H _ x | ) , x \\in \\Lambda . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} l _ 0 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) & l _ 1 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} & \\sum _ { j _ 1 + \\cdots + j _ m = n } \\binom { n } { j _ 1 , \\ldots , j _ m } E _ { j _ 1 } ( \\beta ; x _ 1 ) \\cdots E _ { j _ m } ( \\beta ; x _ m ) \\\\ & = E _ n ( m , \\beta ; x ) = \\sum _ { k = 0 } ^ n \\binom { - m } { k } \\beta ^ k \\Delta ^ k I _ n ( x ) . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{gather*} w ^ 0 ( x ) : = ( I - x H _ u ^ 2 ) ^ { - 1 } ( 1 ) , \\\\ w ^ 1 ( x ) : = ( I - x H _ u ^ 2 ) ^ { - 1 } ( u ) . \\end{gather*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\nu = K _ 2 \\ast G _ 3 ( \\nu ) \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} H _ { 1 } ( t _ { 1 } ) = \\frac { 1 } { 2 } \\sum _ { \\lambda \\lambda ^ { \\prime } \\mu \\mu ^ { \\prime } } \\int d ^ { 4 } x _ { 1 } d ^ { 4 } x _ { 1 } ^ { \\prime } \\hat { \\psi } _ { \\lambda } ^ { \\dagger } ( x _ { 1 } ) \\hat { \\psi } _ { \\mu } ^ { \\dagger } ( x _ { 1 } ^ { \\prime } ) U ( x _ { 1 } , x _ { 1 } ^ { \\prime } ) _ { \\lambda \\lambda ^ { \\prime } \\mu \\mu ^ { \\prime } } \\hat { \\psi } _ { \\mu ^ { \\prime } } ( x _ { 1 } ) \\hat { \\psi } _ { \\lambda ^ { \\prime } } ( x _ { 1 } ^ { \\prime } ) , \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} \\Delta ^ m f ( x ) = \\sum _ { k = 0 } ^ m \\binom { m } { k } ( - 1 ) ^ { m - k } f ( x + k ) . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} B _ { n , N _ { 0 } } = \\left \\{ \\min _ { 0 \\leqslant N \\leqslant N _ { 0 } } \\min _ { 1 \\leqslant j \\leqslant m _ { N } } \\mathbb { P } _ { n } \\left ( A _ { j } ^ { ( N ) } \\right ) > 0 \\right \\} , \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} K _ { \\widehat { X } } = f ^ * K _ Y - \\frac { n - 3 } { n - 1 } E . \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\lambda _ f ^ { \\ast } ( p ) = \\lambda _ f ( p ) + O ( p ^ { \\theta - 1 } ) , \\lambda _ { g } ^ { \\ast } ( p ) = \\lambda _ { g } ( p ) + O ( p ^ { \\theta - 1 } ) , \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} \\Omega _ + = \\Omega _ 1 \\cup \\Omega _ 4 \\Omega _ - = \\Omega _ 2 \\cup \\Omega _ 3 \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} \\mathcal I _ { \\infty } ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { L ( 1 , \\pi _ { \\infty } , \\mathrm { A d } ) L ( 1 , \\tau _ { \\infty } , \\mathrm { A d } ) } { L ( 1 / 2 , \\pi _ { \\infty } \\times \\mathrm { A d } ( g ) ) } \\alpha _ { \\infty } ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = 1 . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} h \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a ^ 2 & 2 a b & b ^ 2 \\\\ a c & a d + b c & b d \\\\ c ^ 2 & 2 c d & d ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} P ( x ) = 2 x C ( x ) + \\ln ( 1 - x C ( x ) ) . \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} \\gamma _ j = 0 . \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\sum _ { \\substack { e _ 1 , \\ldots , e _ n \\in \\Z \\\\ 0 \\leq e _ 1 \\leq \\ldots \\leq e _ n } } \\mathcal { S } ( T _ { e _ 1 , \\ldots , e _ n } ) Y ^ { e _ 1 + \\ldots + e _ n } = \\frac { 1 - Y } { 1 - q ^ n Y } \\prod _ { i = 1 } ^ n \\frac { 1 - q ^ { 2 i } Y ^ 2 } { ( 1 - X _ i q ^ n Y ) ( 1 - X _ i ^ { - 1 } q ^ n Y ) } . \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\beta ^ { \\kappa } _ { M , M - 1 } ( \\kappa \\ , a , \\kappa \\ , b , \\kappa \\ , \\bar { b } ) \\overset { { \\rm i n \\ , l a w } } { = } \\beta _ { M , M - 1 } ( a , \\ , b , \\bar { b } ) . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} \\langle g , f _ { j } \\rangle = \\begin{cases} \\frac { 1 } { \\sqrt { d + 2 } } & \\\\ - \\sqrt { \\frac { d + 2 } { d ^ { 2 } } } & \\end{cases} \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\langle P _ { k _ 1 } ( x _ 1 ) , Q _ { k _ 2 } ( x _ 2 ) \\rangle = \\left \\{ \\begin{array} { l l } 1 & \\exists j \\geq 0 , \\ k _ i = l _ i + j \\ ( i = 1 , 2 ) \\\\ 0 & \\end{array} \\right . , \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { n \\to \\infty } & \\frac { 1 } { n } \\log \\ ! \\Bigg [ \\binom { w n } { \\lambda n } S _ { d + 1 , k } \\big ( n - \\lambda n ( d + 1 ) , ( w - \\lambda ) n \\big ) \\Bigg ] \\\\ & = \\ ; w H \\Big ( \\frac { \\lambda } { w } \\Big ) + ( w - \\lambda ) \\log \\sum _ { i = d + 2 } ^ { k + 1 } \\rho _ { w , \\lambda } ^ { i - \\frac { 1 - \\lambda ( d + 1 ) } { w - \\lambda } } , \\end{aligned} \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} T _ l ( i , j ) & = T _ { l } ( i 1 , j 1 ) + C _ l ( i 1 , j ) + R _ { l } ( i , j 1 ) + S _ l ( i , j ) , \\\\ C _ l ( i , j ) & = C _ l ( i 1 , j ) + S _ l ( i , j ) , \\\\ R _ l ( i , j ) & = R _ l ( i , j 1 ) + S _ l ( i , j ) . \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} \\mathcal { F } ( x , y , z ) & = a ^ 2 + d ^ 2 + \\frac { ( a ^ 2 + d ^ 2 ) ( \\gamma + 1 ) } { ( 1 + x ^ 2 ) ( 1 + y ^ 2 ) ( 1 + z ^ 2 ) } { \\sigma ( x , y , z ) , } \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\overline { \\nu } = ( \\nu _ 0 , \\ , \\nu _ { - d } ) : v _ 0 \\oplus \\Sigma ^ { - d } v _ { - d } \\rightarrow f . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\omega _ { x _ i } ( k _ i ) = \\omega _ { y _ i } ( m _ i ) \\iff \\exists \\ , j _ i \\in \\N , \\ k _ i = k _ 0 + j _ i , \\ m _ i = m _ 0 + j _ i , \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\ P ^ { n , m } ( \\tau _ { m - 1 } \\leq t ) \\leq P ( \\tau _ { m - 1 } \\leq t ) . \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} \\mathrm { d i m } [ ( \\mathrm { I m } \\rho ) _ s + ( \\mathrm { I m } \\rho _ * ) _ s ] = \\mathrm { d i m } T _ s S \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} y = z C ( z ) , \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} h = ( h _ 1 , h _ 2 ) , v _ 1 = \\frac { h _ 2 } { h } , \\ v _ 2 = \\frac { h _ 1 } { h } . \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} | a | = 1 , \\bar { a } b + \\bar { b } = 0 , | c | ^ 2 e ^ { | b | ^ 2 } = 1 , \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} N _ { m + 1 } = \\sum _ { n = 0 } ^ { m + 1 } \\binom { m + 1 } { m - n + 1 } N _ { \\mathrm { d } \\ , m - n + 1 } N _ { \\mathrm { c } n } . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} x _ { \\mathbb A } : = \\frac { 1 } { \\lambda } \\begin{pmatrix} x _ 1 \\\\ \\vdots \\\\ x _ d \\end{pmatrix} \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ \\Omega ( \\sigma ( t , x ) - \\tau ( t , x ) ) : ( \\dot { e } ( t , x ) - \\dot { \\eta } ( t , x ) ) \\ , d x \\ , d t = \\frac { 1 } { 2 } \\langle \\mathbb { A } _ r ( \\sigma ( T ) - \\tau ( T ) ) , \\sigma ( T ) - \\tau ( T ) \\rangle . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} [ \\rho ] [ \\rho ] & = \\sum _ { g \\in G } [ g ] \\ , , & [ g ] [ \\rho ] & = [ \\rho ] [ g ] = [ \\rho ] \\ , , & [ g ] [ h ] & = [ g h ] \\ , , \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} s : = \\frac { 4 \\Lambda c _ 1 } { m ( 1 - \\delta ) c _ 2 } < r _ { \\delta } ( x ) . \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} j f _ 1 g _ { i + j - 1 } & = g ' _ j f _ 1 f _ i - f _ 1 f ' _ i g _ j = g ' _ j f _ 1 f _ i - ( ( i - 1 ) f _ i - f ' _ 1 f _ i ) g _ j \\\\ & = ( g ' _ j f _ 1 - g _ j f ' _ 1 ) f _ i - ( i - 1 ) f _ i = ( j - i + 1 ) f _ i g _ j \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\lambda \\phi _ i ( \\lambda ) = \\alpha _ i \\phi _ { i + 1 } ( \\lambda ) + \\beta \\phi _ i ( \\lambda ) + \\gamma \\phi _ { i - 1 } ( \\lambda ) , i = 1 , 2 , \\hdots , \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} U = U _ { j - 1 } + L _ { j - 1 } \\leq U _ { j - 1 } + M \\Rightarrow 2 ^ { j + 1 } U _ { j - 1 } \\geq U \\mbox { i f $ U _ j > 0 $ . } \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} | \\pi _ n | _ T = \\sup \\big \\{ t _ { i , n } ^ { ( l ) } \\wedge T - t _ { i - 1 , n } ^ { ( l ) } \\wedge T \\big | i \\geq 1 , ~ l = 1 , 2 \\big \\} \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} m * e = m , e * m = m , ( m * n ) * p = m * ( n * p ) \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} \\underbar r _ 1 = \\frac { a _ { 0 , \\inf } - a _ { 2 , \\sup } \\bar r _ 2 - k \\frac { \\chi _ 1 } { d _ 3 } \\bar r _ 1 } { a _ { 1 , \\sup } - k \\frac { \\chi _ 1 } { d _ 3 } } , \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} L ( \\lambda \\mid \\hat { x } _ n , \\hat { X } _ n ) M ( \\lambda ; \\hat { u } _ n , \\hat { U } _ n ) = M ( \\lambda ; \\hat { v } _ n , \\hat { V } _ n ) L ( \\lambda \\mid \\hat { y } _ n , \\hat { Y } _ n ) \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} \\log \\Gamma ( a + z ) = & \\log \\Gamma ( a ) + \\sum \\limits _ { p = 1 } ^ \\infty \\frac { ( - z ) ^ p } { p } \\zeta ( p , a ) , \\\\ \\sum \\limits _ { j = x } ^ y j ^ p = & \\frac { B _ { p + 1 } ( y + 1 ) - B _ { p + 1 } ( x ) } { p + 1 } . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} \\triangle ( u ( x + h ) - p _ { u , x } ( h ) - e _ { u , x } ( h ) ) = 0 B _ \\delta . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} j ^ * \\sigma ^ + _ \\lambda = \\alpha ; j ^ * \\sigma ^ + _ \\mu = \\beta ; j ^ * \\sigma ^ + _ \\nu = \\gamma . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta \\phi _ 0 ( x ) = 0 , \\ ; x > 0 , \\\\ \\phi _ 0 | _ { x = 0 } = 0 , \\\\ x \\rightarrow \\infty , \\phi _ 0 ( x ) \\rightarrow + \\infty , \\ ; \\ ; \\mbox { a n d } \\ ; \\phi _ 0 ( x ) \\ ; \\mbox { i n c r e a s e a t t h e r a t e o f l i n e a r f u n c t i o n $ x $ } . \\end{array} \\right . \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\psi _ 3 ( ( a v ) ^ 3 ) = ( [ a , b ] , [ a , b ] , [ a , b ] ) . \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} n ( t , s ) = T _ t n _ 0 ( s ) + \\int _ 0 ^ t T _ { t - \\tau } A [ n ( \\tau , \\cdot ) ] ( s ) d \\tau , t \\in [ 0 , T ) , \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\sum _ { v \\in V ( G ) } ( x _ v ^ * ) ^ p & = \\sum _ { e \\in E ( G ) } \\sum _ { u \\in e } B ( u , e ) ( x _ u ^ * ) ^ p \\\\ & = \\sum _ { e \\in E ( G ) } r B ( u , e ) ( x _ u ^ * ) ^ p \\\\ & = \\sum _ { e \\in E ( G ) } w ( e ) = 1 , \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} & v _ + \\left ( x , t \\right ) = 0 \\mbox { a . e . i n $ \\R ^ n \\times ( 0 , T ) $ } \\\\ & \\Rightarrow u _ 1 ( x , t ) \\leq u _ { 2 } ( x , t ) \\mbox { a . e . i n $ \\R ^ n \\times ( 0 , T ) $ } . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} \\ < [ w ] | T \\psi = \\sum c _ \\ell \\ < [ w ] | [ z _ \\ell ] \\ > = \\sum c _ \\ell K ( [ w ] , [ z _ \\ell ] ) = \\sum c _ \\ell K ( w , z _ \\ell ) = 0 . \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} C = \\theta ^ { - 1 } \\theta C ( C \\in \\sigma ( \\theta ) ) . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{gather*} P ^ t = l _ { 0 , 0 } , \\\\ \\mathcal { W } ^ t = \\min \\left ( \\{ l _ { 0 , 0 } - l _ { 1 , 1 } , l _ { a , b } \\} \\cup \\left ( \\bigcup _ { \\substack { 1 \\leq i \\leq a - 1 \\\\ 1 \\leq j \\leq b } } \\{ l _ { i , j } - l _ { i + 1 , j } \\} \\right ) \\cup \\left ( \\bigcup _ { \\substack { 1 \\leq i \\leq a \\\\ 1 \\leq j \\leq b - 1 } } \\{ l _ { i , j } - l _ { i , j + 1 } \\} \\right ) \\right ) . \\end{gather*}"}
-{"id": "7495.png", "formula": "\\begin{align*} f \\Big ( t , \\omega + \\varepsilon h , X ( t ) ( \\omega ) \\Big ) - f \\Big ( t , \\omega , X ( t ) ( \\omega ) \\Big ) = \\int _ 0 ^ \\varepsilon D ^ h f ( t , \\omega + r h , X ( t ) ( \\omega ) ) d r . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} & \\{ \\lambda ( 1 + \\sum _ { i \\in I } t _ i ) + \\sum _ { j \\in J } \\dfrac { a _ j } { t _ j } = 0 \\} , \\\\ & \\{ \\lambda ( \\sum _ { i \\in I } t _ i ) + 1 + \\sum _ { j \\in J } \\dfrac { a _ j } { t _ j } = 0 \\} \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} t _ 0 = \\gamma + u _ 0 , \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} ^ { C } L _ 1 x _ { \\lambda _ 2 } ( t ) = \\lambda _ 2 r ( t ) x _ { \\lambda _ 2 } ( t ) , \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} H = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 1 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} | \\zeta _ l | \\le \\Delta ( d ) V ( E _ l ^ h ( x ) ) / c _ d r ^ d = h ^ d \\frac { \\Delta ( d ) } { c _ d } \\left ( \\frac { h + 2 r } { h r } \\right ) ^ d = h ^ d g _ d ( h , r ) , \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\partial _ t \\| v \\| _ { L ^ 2 _ x ( \\R ^ + ) } ^ 2 & = \\Im \\int _ 0 ^ \\infty u _ 1 ( m ^ + + m ^ - ) \\overline { v } \\d x \\\\ \\partial _ t \\| m ^ \\pm \\| _ { L ^ 2 _ x ( \\R ^ + ) } ^ 2 & = \\mp 2 \\Im \\int _ 0 ^ \\infty D ^ { - 1 } ( u _ 1 \\overline { v } + \\overline { u _ 2 } v ) \\overline { m } ^ \\pm \\d x . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} H ( m ) . ( s \\otimes e ^ { \\alpha } ) : = ( H ( m ) + \\delta _ { m , 0 } ( H | \\alpha ) ) s \\otimes e ^ { \\alpha } \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} x _ { i j } ( x _ { i j } - 1 ) = 0 , \\ \\forall i , j . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} f ( x + \\gamma h ) = f ( x ) + \\sum _ { \\abs { \\sigma } < q } a _ \\sigma ( x ) ( \\gamma h ) ^ \\sigma + O ( \\abs { h } ^ q ) . \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} K _ i : = \\left \\{ x \\in \\bar \\Omega : p ( x ) = \\frac { \\alpha } { 2 } ( u _ i + u _ { i + 1 } ) \\right \\} . \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} ( \\Gamma \\boldsymbol { \\xi } ) _ n = A _ { n - 1 } \\xi _ { n - 1 } + d _ { y _ { n - 1 } } f _ { n - 1 } \\xi _ { n - 1 } , \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} & \\sim \\sum _ { r = 0 } ^ t \\frac { \\big ( { _ { \\neq k } } \\Lambda _ { \\neq d } \\big ) ^ r } { r ! } \\frac { \\big ( { _ { \\neq d } } \\Lambda _ { \\neq k } \\big ) ^ { t - r } } { ( t - r ) ! } \\ ; \\ ; \\sim \\frac { 2 ^ t } { t ! } \\big ( { _ { \\neq k } } \\Lambda _ { \\neq d } \\big ) ^ t \\\\ & \\sim \\frac { 2 ^ t } { t ! } \\big ( w ^ * ( 1 - \\rho ^ { d + 1 } ) ( 1 - \\rho ^ { k + 1 } ) \\big ) ^ t n ^ t . \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} k = 2 n - N . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k _ 1 , . . . , k _ N = 0 } ^ \\infty \\delta _ { \\omega _ { x _ 1 } ( k _ 1 ) } \\otimes \\cdots \\otimes \\delta _ { \\omega _ { x _ N } ( k _ N ) } \\otimes B e _ { ( k _ 1 , . . . , k _ N ) } \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} \\bar { U } = U - \\frac { \\omega ^ { 2 } } { 2 \\rho ^ { 2 } } \\bar { T } - \\bar { U } = T - U = h \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\big ( { \\mathbb D } ^ 2 \\big ) ^ { A } { } _ { B } = \\ , { \\mathbb D } ^ { A } { } _ { R } { \\mathbb D } ^ { R } { } _ { B } + \\ , { \\mathbb D } ^ { R } { } _ { B } { \\mathbb D } ^ { A } { } _ { R } , \\ , \\langle { \\mathbb D } , { \\mathbb D } \\rangle = \\ , { \\mathbb D } ^ { S } { } _ { R } { \\mathbb D } ^ { R } { } _ { S } , \\ , { \\mathbb D } ^ 2 _ o = \\ , \\mathrm { t f } \\big ( { \\mathbb D } ^ 2 \\big ) . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\{ e \\} = H _ 0 \\lhd H _ 1 \\lhd \\cdots \\lhd H _ m = G \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} Z _ { t } = Z _ { 0 } + \\int _ { 0 } ^ { t } G _ { s } \\ , d s + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } P ^ { \\ast } L ^ { b } P Z _ { s } \\ , d s + \\int _ { 0 } ^ { t } B P Z _ { s } \\ , d W _ { s } \\quad \\forall t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} a _ { i j } = \\sum _ { k = 1 } ^ { n } \\delta _ k \\Re ( u _ { i k } v _ { j k } ) + \\sum _ { k = 1 } ^ { n } \\sigma _ k \\Re ( u _ { i k } \\overline { v } _ { j k } ) . \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} d _ { 2 } \\le 4 ^ { ( 2 n ^ 3 ) ^ { 1 4 9 n ^ 8 } } 3 n ^ 2 ( 2 n ^ 3 ) ^ { 1 4 8 . 5 n ^ 6 } 1 8 ^ { n ^ 2 } ( 2 n ^ 3 ) ^ { 1 4 8 . 5 n ^ 8 } \\\\ = 3 n ^ 2 1 8 ^ { n ^ 2 } 4 ^ { ( 2 n ^ 3 ) ^ { 1 4 9 n ^ 8 } } ( 2 n ^ 3 ) ^ { ( 1 4 8 . 5 n ^ 8 + 1 4 8 . 5 n ^ 6 ) } , \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} ( \\mathrm { d i v } F ( u ) , \\phi ) = - ( F ( u ) , \\nabla \\phi ) . \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} ( u _ 1 , \\theta , h _ 1 ) | _ { t = 0 } ~ = ~ ( u _ { 1 , 0 } , \\theta _ 0 , h _ { 1 , 0 } ) ( x , y ) , ( x , y ) \\in \\mathbb { T } \\times \\mathbb { R } _ + , \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} F S ( e ) = \\frac { p ^ { 3 e } - r _ e ^ 3 } { n } + 1 + { { 3 r _ e - n - 1 } \\choose { 2 } } - 3 { { 2 r _ e - n - 1 } \\choose { 2 } } + { { 3 r _ e - 2 n - 1 } \\choose { 2 } } , \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\begin{bmatrix} \\lambda P _ d + P _ { d - 1 } & P _ { d - 2 } & \\cdots & P _ 0 \\end{bmatrix} & ( \\Lambda _ { d - 1 } ( \\lambda ) \\otimes I _ n ) = \\\\ & ( \\Lambda _ { d - 1 } ( \\lambda ) ^ T \\otimes I _ m ) \\begin{bmatrix} \\lambda P _ d + P _ { d - 1 } \\\\ P _ { d - 2 } \\\\ \\vdots \\\\ P _ 0 \\end{bmatrix} = P ( \\lambda ) , \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\Psi & = \\mathsf { G } ( - \\mathsf { V } + \\Xi ) \\ , . \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\frac { d } { d s } = \\frac { 1 } { v } \\frac { d } { d t } \\frac { d ^ { 2 } } { d s ^ { 2 } } = \\frac { 1 } { v ^ { 2 } } \\frac { d ^ { 2 } } { d t ^ { 2 } } - \\frac { \\dot { v } } { v ^ { 3 } } \\frac { d } { d t } \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} A _ m = \\sum \\limits _ { k = 0 } ^ m ( - 1 ) ^ { m - k } \\frac { ( k + 1 ) } { ( m - k ) ! } . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} \\sum _ { y \\in \\eta \\setminus x } \\beta ( x - y ) \\le s g _ d ( h , r ) \\sum _ { l = 1 } ^ { \\infty } \\beta _ l h ^ d \\le s g _ d ( h , r ) ( \\langle b \\rangle + \\varepsilon ) \\le s \\delta . \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} Z ( G _ 3 ) \\cap \\{ [ b , g ] \\mid g \\in G _ 3 \\} = 1 . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} F _ \\psi ^ \\infty ( p ) : = \\lim _ { t \\rightarrow \\infty } \\frac { F _ \\psi ( t p ) } { t } , \\ ; p \\in \\R ^ { n \\times n } , \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} f ( x + \\gamma h ) = f ( x + h ) + \\sum _ { \\abs { \\sigma } < q } a _ \\sigma ( x + h ) ( ( \\gamma - 1 ) h ) ^ \\sigma + O ( \\abs { h } ^ q ) . \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} \\frac { ( M _ { 1 } - 1 + M _ { 2 } + \\cdots + M _ { \\ell } ) ! } { ( M _ { 1 } - 1 ) ! \\cdots M _ { \\ell } ! } + \\cdots + \\frac { ( M _ { 1 } + \\cdots + M _ { \\ell - 1 } + M _ { \\ell } - 1 ) ! } { M _ { 1 } ! \\cdots ( M _ { \\ell } - 1 ) ! } = & \\frac { ( M _ { 1 } + \\cdots + M _ { \\ell } - 1 ) ! } { M _ { 1 } ! \\cdots M _ { \\ell } ! } \\left ( M _ { 1 } + \\cdots + M _ { \\ell } \\right ) \\\\ = & \\frac { ( M _ { 1 } + M _ { 2 } + \\cdots + M _ { \\ell } ) ! } { M _ { 1 } ! M _ { 2 } ! \\cdots M _ { \\ell } ! } \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} H _ { 1 , ( 0 , \\alpha _ 2 ) } ( \\tau ) = - \\int _ { \\R ^ 2 } \\mathcal F _ 0 ( w _ 1 ) \\mathcal F _ { \\alpha _ 2 } ( w _ 2 ) e ^ { 2 \\pi i \\tau \\left ( 3 w _ 1 ^ 2 + w _ 2 ^ 2 \\right ) } \\sum _ { \\pm } \\pm e ^ { \\pm 6 \\pi i \\tau w _ 1 w _ 2 } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} E _ { 1 , n } & : 0 = ( n + 2 ) ( n + 1 ) \\rho _ { 0 } ^ { 2 } \\rho _ { n + 2 } + . . . . . \\\\ E _ { 2 , n } & : 0 = ( n + 2 ) ( n + 1 ) \\rho _ { 0 } ^ { 3 } \\varphi _ { n + 2 } + . . . . . \\\\ E _ { 3 , n } & : 0 = ( n + 2 ) ( n + 1 ) g _ { 0 } \\rho _ { 0 } ^ { 3 } \\theta _ { n + 2 } + . . . . \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\big \\| S ^ { ( s ) } _ { N , N } f - ( \\log N ) S ^ { ( s + 1 ) } _ { N , N } f \\big \\| _ { L ^ p } & \\leq \\sum _ { m = 2 } ^ { N - 1 } \\big \\| S ^ { ( s + 1 ) } _ { N , m } f \\big \\| _ { L ^ p } m ^ { - 1 } \\\\ & \\lesssim N ^ { k - 1 } ( \\log N ) ^ { - s } \\sum _ { m = 2 } ^ { N - 1 } \\| f \\| _ { L ^ p } ( \\log m ) ^ { - 1 } \\\\ & \\lesssim N ^ k ( \\log N ) ^ { - s - 1 } \\| f \\| _ { L ^ p } , \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\varepsilon _ { n , r } = \\begin{cases} ( - 1 ) ^ { \\lfloor n / 2 \\rfloor } & \\textrm { i f $ r > n - 2 $ } \\\\ ( - 1 ) ^ { n - k + 1 } & \\textrm { i f $ 2 k < r < 2 k + 1 \\le n - 2 , $ $ k \\ge 0 $ i s a n i n t e g e r } \\\\ ( - 1 ) ^ { k + 1 } & \\textrm { i f $ 2 k + 1 < r < 2 k + 2 \\le n - 2 , $ $ k \\ge 0 $ i s a n i n t e g e r } . \\end{cases} \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\sum _ n \\frac { \\overline { \\chi } ( n ) } { n ^ { \\frac { 1 } { 2 } } } V \\left ( \\frac { p n } { q } \\right ) & = L \\left ( \\frac { 1 } { 2 } , \\overline { \\chi } \\right ) - \\sum _ n \\frac { \\chi ( n ) } { n ^ { \\frac { 1 } { 2 } } } F \\left ( \\frac { n } { p } \\right ) . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\frac { m _ 0 + \\gamma } { \\alpha + \\beta } = \\frac { m _ 0 } { \\alpha } , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\alpha > 0 . \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} \\dd Y _ t = b ( Y _ { t - } ) \\dd t + \\sigma ( Y _ { t - } ) \\dd W _ t + \\int v ( x , Y _ { t - } ) ( p - q ) ( \\dd x , \\dd t ) , \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\bar { B _ 2 } = \\frac { b _ { 0 , \\sup } ( a _ { 1 , \\inf } - \\frac { k \\chi _ 1 } { d _ 3 } ) + \\frac { k \\chi _ 2 } { d _ 3 } a _ { 0 , \\sup } } { ( a _ { 1 , \\inf } - \\frac { k \\chi _ 1 } { d _ 3 } ) ( b _ { 2 , \\inf } - \\frac { l \\chi _ 2 } { d _ 3 } ) - \\frac { l k \\chi _ 1 \\chi _ 2 } { d _ 3 ^ 2 } } . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} g _ { \\infty } = n ( X ) m ( A , 1 ) = \\left ( \\begin{array} { c c } \\mathbf 1 _ 2 & X \\\\ 0 & \\mathbf 1 _ 2 \\end{array} \\right ) \\left ( \\begin{array} { c c } A & 0 \\\\ 0 & { } ^ t A ^ { - 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} L _ b ^ { * } \\mu = 0 \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} w ( x , t ) \\leq \\frac { M _ 2 } { M } \\mathcal { B } _ { M } \\left ( 0 , t \\right ) = c ^ { \\ast } M _ 2 M ^ { a _ 3 - 1 } t ^ { - a _ 1 } , \\forall t > 0 \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} \\left ( K _ X + \\Delta \\right ) ^ 3 = \\left ( \\lambda \\pi ^ * H + \\alpha D + \\beta E \\right ) ^ 3 \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} H ( z ) & = e ^ { \\bf f } H ( z ) e ^ { - { \\bf f } } - 2 f ( z ) F ( z ) , \\\\ e ^ { - { \\bf f } } H \\otimes x ( \\zeta ) e ^ { \\bf f } & = H \\otimes x ( \\zeta ) - 2 F \\otimes f ( \\zeta ) x ( \\zeta ) . \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } \\| x ^ { k } - z \\| = 0 . \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\langle ( \\xi _ 1 , \\eta _ 1 ) , ( \\xi _ 2 , \\eta _ 2 ) \\rangle _ { ( A _ r , B _ r ) } = { \\rm t r } ( A _ r ^ { - 1 } \\xi _ 1 A _ r ^ { - 1 } \\xi _ 2 ) + { \\rm t r } ( \\eta _ 1 ^ T \\eta _ 2 ) , \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} \\left [ \\alpha \\right ] _ q = \\frac { { q ^ \\alpha - 1 } } { { q - 1 } } , \\alpha \\in \\mathbb { C } . \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} i \\hbar \\frac { \\partial \\psi } { \\partial t } = \\frac { 1 } { 2 m } ( - i \\hbar \\nabla + A ( x ) ) ^ 2 \\psi + P ( x ) \\psi - \\rho ( x , | \\psi | ) \\psi , \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\lambda _ n ( f ) = \\frac { 1 } { | T _ n | } \\sum _ { \\gamma \\in T _ n } f ( \\gamma ) , \\textrm { f o r $ f \\in \\ell ^ \\infty ( \\Gamma ) $ } . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mathrm { m i n i m i z e } & F ( x ) : = f ( x ) + \\psi ( x ) \\\\ \\mathrm { s u b j e c t ~ t o ~ } & x = ( x ^ 1 , \\dots , x ^ n ) \\in \\R ^ n , \\end{array} \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} \\mathcal { D } ^ \\dagger = \\left \\{ R \\in \\mathcal { R } : \\int _ { \\Gamma _ 0 } \\Psi ( \\eta ) | R ( \\eta ) | \\lambda ( d \\eta ) < \\infty \\right \\} . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} x _ i = \\left ( 1 - \\frac { i } { p } \\right ) y _ i + \\frac { i } { p } y _ { i + 1 } + \\frac { r } { p ( p + 1 ) } . \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} ( \\psi - \\psi _ V ) ( \\phi ) = \\ < \\phi , \\psi - \\psi _ V \\ > = \\ < \\phi , \\psi \\ > - \\ < \\phi , \\psi _ V \\ > \\to 0 . \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} s ' & = ( 1 + b + a \\circ ( 1 + b ) ) \\circ ( 1 + c ) - ( 1 + a ) \\circ ( 1 + c + b \\circ ( 1 + c ) ) \\\\ & = ( 1 + c + b \\circ ( 1 + c ) + ( a \\circ ( 1 + b ) ) \\circ ( 1 + c ) ) \\\\ & - ( 1 + c + b \\circ ( 1 + c ) + a \\circ ( 1 + c + b \\circ ( 1 + c ) ) ) \\\\ & = ( a \\circ ( 1 + b ) ) \\circ ( 1 + c ) - a \\circ ( 1 + c + b \\circ ( 1 + c ) ) . \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} \\eta ( t ) W _ 0 ^ t \\big ( u _ 0 ^ e , g \\big ) + i \\eta ( t ) W _ 0 ^ t \\big ( 0 , q \\big ) = \\eta ( t ) e ^ { i t \\Delta } u _ 0 ^ e + \\eta ( t ) W _ 0 ^ t \\big ( 0 , g - p + i q \\big ) . \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} h ( R ) = 1 ; h ' ( R ) = 0 ; h '' ( R ) = 0 ; \\dots ; h ^ { ( p ) } ( R ) = 0 . \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k c _ i ^ d = 1 . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} L + \\frac { B } { \\rho _ { - } ^ { \\alpha } } = \\rho _ { + } ( u _ + - \\widehat { \\sigma _ { 0 } ^ { B } } ) ^ { 2 } . \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\langle w \\rangle _ f : = \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } w \\ , f ( w ) \\ , w \\ , . \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} | ( \\nabla ^ { k , \\hat { g } _ t } \\hat { g } ^ \\bullet _ t ) ( \\hat { x } _ t ) | _ { \\hat { g } _ t ( \\hat { x } _ t ) } = 1 . \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} \\| y ^ { k + 1 } - z \\| ^ 2 & \\leq \\| w ^ k - z \\| ^ 2 - \\| w ^ k - y ^ { k + 1 } \\| ^ 2 \\\\ & = \\| x ^ k - \\lambda _ k g ( y ^ k ) - z \\| ^ 2 - \\| x ^ k - \\lambda _ k g ( y ^ k ) - y ^ { k + 1 } \\| ^ 2 \\\\ & = \\| x ^ k - z \\| ^ 2 - \\| x ^ k - y ^ { k + 1 } \\| ^ 2 + 2 \\lambda _ k \\langle z - y ^ { k + 1 } , g ( y ^ k ) \\rangle \\\\ & = \\| x ^ k - z \\| ^ 2 - \\| x ^ k - y ^ { k + 1 } \\| ^ 2 + 2 \\lambda _ k \\langle g ( y ^ k ) , y ^ k - y ^ { k + 1 } \\rangle - 2 \\lambda _ k \\langle g ( y ^ k ) , y ^ k - z \\rangle . \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} w _ t ( \\eta ) = ( - 1 ) ^ { | \\eta | } q _ t ( \\eta ) . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} H _ n = \\mathrm { I n d } _ { H _ n ^ k } ^ { H _ n } H _ n ^ k , \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) ^ { \\mathrm { a l g } } : = \\frac { \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) } { \\langle g , g \\rangle ^ 2 c ^ + ( f ) } \\in \\Q ( f , g ) , \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} & \\rho _ { 1 3 4 } ( S _ 1 ) e _ { 2 k } = e _ { 4 k + 1 } \\\\ & \\rho _ { 1 3 4 } ( S _ 1 ) e _ { 2 k + 1 } = e _ { 4 k + 2 } \\\\ & \\rho _ { 1 3 4 } ( S _ 2 ) e _ { 2 k } = e _ { 4 k + 3 } \\\\ & \\rho _ { 1 3 4 } ( S _ 2 ) e _ { 2 k + 1 } = e _ { 4 k } \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\big ( \\exp _ \\star ( { \\bf a } ) , \\tau \\big ) = \\sum _ { n = 0 } ^ { | \\tau | } \\frac { 1 } { n ! } \\sum a ^ { \\tau _ 1 } \\cdots a ^ { \\tau _ n } , \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} e ( { R } / { ( g _ 0 , \\dots , g _ { n - 2 } ) : I ^ \\infty } ) = e ( { R } / { ( g _ 0 , \\dots , g _ { n - 2 } ) : I } ) . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} \\Vert u _ { \\rho } - { \\check { u } } _ { n } \\Vert _ { { H ^ { 1 } } ( { { \\mathbb { R } } } ) } + \\Vert { \\mathcal { L } } u _ { \\rho } - { \\check { v } } _ { n } \\Vert _ { { H ^ { 1 } } ( { { \\mathbb { R } } } ) } < \\rho \\ ; \\ ; J ( u _ { \\rho } ) < c _ { n } = J ( { \\check { u } } _ { n } ) \\ ; . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} L ^ p = \\mathbb { F } \\oplus \\mathbb { F } ( - 1 ) \\oplus \\dots \\oplus \\mathbb { F } ( 1 - p ) \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} u ( t ) = S ( t ) u _ { 0 } + \\ \\int _ { 0 } ^ { t } S ( t - s ) B ( u ( s ) ) \\ d s \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\sum _ { u \\in N } \\langle u | Q T Q u \\rangle = \\lim _ { n \\to + \\infty } \\sum _ { u \\in N } \\langle u | Q _ n T Q _ n u \\rangle \\ : . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} \\mathbf g ( \\gamma \\gamma _ { \\infty } k _ 0 ) = g ( \\gamma _ { \\infty } ( i ) ) ( c i + d ) ^ { - \\ell } ( \\det \\gamma _ { \\infty } ) ^ { \\ell / 2 } \\underline { \\chi } ( k _ 0 ) \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} \\pi _ { \\mathfrak { k } ^ * } = \\sum _ { 1 \\leq i \\leq k < n } \\sum _ { 1 \\leq j \\leq i } c _ { i , j } ^ { ( k ) } \\frac { \\partial } { \\partial \\ell _ i ^ { ( k ) } } \\wedge \\frac { \\partial } { \\partial \\psi _ j ^ { ( k ) } } , \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} D _ { i , i } = \\tbinom { N + i - 1 } { N - 1 } ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} \\mathcal C _ { U _ 1 , \\dots U _ n } : = \\left \\{ \\omega \\in \\Omega \\ , | \\ , M _ \\omega \\in U _ 1 , \\dots , M _ { T ^ { n - 1 } \\omega } \\in U _ n \\right \\} . \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} A = \\frac { 1 } { 2 } \\left ( U \\Delta V ^ T + U \\Sigma V ^ { \\ast } + \\overline { U } \\Sigma V ^ T + \\overline { U } \\Delta V ^ { \\ast } \\right ) . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} d _ { \\Gamma ( G ) } ( \\Psi ( x ) , \\Psi ( y ) ) = 2 d ( x , y ) , \\quad \\forall x , y \\in X . \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\lbrace Q \\stackrel { \\otimes } { , } Q \\rbrace + [ Q \\otimes I + I \\otimes Q , r ] = 0 , \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} \\tilde \\Sigma = \\Sigma + \\partial \\Omega _ 1 + \\ldots + \\partial \\Omega _ N \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} { S } = \\bigcup _ { j \\in \\mathbb { N } } { S } _ j , \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} \\Delta _ { \\mathbf { m } } ( \\underline { a } _ 0 , \\underline { a } _ 1 , \\underline { a } _ 2 , \\ldots , \\underline { a } _ { r } ) = \\det ( p _ j ( \\underline { a _ i } ) ) _ { 0 \\leq i , j \\leq r } . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} K ( u ) & = \\displaystyle \\int _ { \\Omega } \\widetilde { b } ( x ) \\left | u \\right | ^ { q ( x ) } d x \\geq \\displaystyle \\int _ { \\Omega _ { u } } \\widetilde { b } ( x ) \\left | u \\right | ^ { q ( x ) } d x \\\\ & \\geq \\epsilon _ { 1 } \\left \\| u \\right \\| ^ { q ^ { - } } m \\left ( \\Omega _ { u } \\right ) \\geq \\epsilon _ { 1 } ^ { 2 } \\left \\| u \\right \\| ^ { q ^ { - } } . \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\gamma _ j ^ { \\frac { 1 } { 2 } } \\ , b ^ { w _ j } & = \\sum _ { j = 1 } ^ { \\infty } j ^ { - \\frac { a } { 2 } } \\ , b ^ { \\lfloor c \\log _ b j \\rfloor } \\asymp \\sum _ { j = 1 } ^ { \\infty } j ^ { - \\frac { a } { 2 } } \\ , b ^ { c \\log _ b j } = \\sum _ { j = 1 } ^ { \\infty } j ^ { c - \\frac { a } { 2 } } , \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\nu \\beta _ 1 \\nu ' ) \\mathbf h _ p ( x ) = C \\int _ { \\Z _ p ^ { \\times } } \\psi \\left ( \\frac { - \\gamma ^ { - 1 } y ^ 2 } { p } \\right ) \\underline { \\chi } _ p ( y ) \\left ( \\int _ { \\Z _ p } \\psi \\left ( \\frac { 2 x z - 2 z y - \\gamma z ^ 2 } { p } \\right ) d z \\right ) d y . \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} \\arg { c } = - \\pi / 2 + \\gamma , 0 < \\gamma \\ll T ^ { - 1 } . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} & { \\rm H e s s } \\ , J ( A _ r , B _ r , C _ r ) [ ( A ' _ r , B ' _ r , C ' _ r ) ] \\\\ = & ( - 2 A _ r { \\rm s y m } ( Q ' P + Q P ' + Y '^ T X + Y ^ T X ' ) A _ r \\\\ & \\ , - 2 { \\rm s y m } ( A ' _ r { \\rm s y m } ( Q P + Y ^ T X ) A _ r ) , \\\\ & \\ , \\ , 2 ( Q ' B _ r + Q B ' _ r + Y '^ T B ) , 2 ( C ' _ r P + C _ r P ' - C X ' ) ) , \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\alpha > \\frac 1 { m - 1 } \\ , , \\ \\beta > 0 \\alpha = \\frac 1 { m - 1 } \\ , , \\ 0 < \\beta \\leq 1 \\ , . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} ( v + v ' ) \\cdot ( z + z ' ) = v \\cdot z + v ' \\cdot z ' \\ ; , \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\epsilon ( h s , \\alpha _ m ) = ( x ( h s ) x ( h \\beta _ m ) , x ( \\alpha _ m ) x ( h \\beta _ m ) ) _ p \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} B _ \\lambda ( s ) & = \\alpha _ n i ^ { n k } \\ , 2 ^ { - n ( 2 n + 1 ) ( s + \\frac 1 2 ) + n } \\int \\limits _ T \\bigg ( \\prod _ { 1 \\leq i < j \\leq n } ( t _ i ^ 2 - t _ j ^ 2 ) \\bigg ) \\Big ( \\prod _ { j = 1 } ^ n t _ j \\Big ) ^ { - ( 2 n + 1 ) s + \\frac 1 2 - n - k } \\ , d \\mathbf { t } . \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} X _ { \\xi } ( t ) - X _ { \\theta } ( t ) = \\xi - \\theta & + \\int _ 0 ^ t \\Big [ b ( s , \\omega , X _ { \\xi } ( s ) ) - b ( s , \\omega , X _ { \\theta } ( s ) ) \\Big ] d s \\\\ & + \\int _ 0 ^ t \\Big [ \\sigma ( s , \\omega , X _ { \\xi } ( s ) ) - \\sigma ( s , \\omega , X _ { \\theta } ( s ) ) \\Big ] d W ( s ) . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} t = { \\gamma _ 2 \\beta _ 2 \\over \\alpha _ 2 \\delta _ 2 } , s = { \\alpha _ 1 \\alpha _ 2 \\over \\gamma _ 1 \\gamma _ 2 } \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} \\tau { ( \\underline { a } ) } : = ( \\ell + 1 - ( a _ { n } + n ) , \\ell + 1 - ( a _ { n - 1 } + n - 1 ) , \\cdots , \\ell + 1 - ( a _ 1 + 1 ) ) \\in \\P _ { \\ell , n } . \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} \\mathcal U : = \\mathcal C _ { U _ 1 , \\dots U _ N } \\cap T ^ { - N } ( \\Omega _ { K _ 0 } ) = U _ 1 \\times \\dots \\times U _ N \\times \\Omega _ { K _ 0 } . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} V _ 1 \\big ( V _ 2 f \\big ) ( x ) = ( D _ x f ) \\big ( ( V _ 1 V _ 2 ) ( x ) \\big ) + ( D _ x ^ 2 f ) \\big ( V _ 1 ( x ) , V _ 2 ( x ) \\big ) . \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} X _ i X _ j 1 _ { \\lambda } = X _ { j } X _ i 1 _ { \\lambda } & & \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { E ^ { \\operatorname { G P } } _ { N } } { N } = e _ { \\operatorname { G P } } . \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\langle \\rho , \\psi _ t \\rangle + \\langle \\rho u , \\psi _ x \\rangle = 0 , \\\\ \\langle \\rho u , \\psi _ t \\rangle + \\langle \\rho u ^ 2 - \\frac { B } { \\rho ^ { \\alpha } } , \\psi _ x \\rangle = - \\langle \\beta \\rho , \\psi \\rangle , \\end{array} \\right . \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} h \\beta ^ { n + 1 } ( U ) = \\beta ^ m ( h _ { j _ m } ) \\cdots \\beta ^ n ( h _ { j _ n } ) \\beta ^ { n + 1 } ( U ) \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} e ^ { - t ( | y _ j - y _ k | ^ 2 + \\lambda ) } = \\frac { e ^ { - \\lambda t } } { 4 \\pi t } \\int _ { \\R ^ 2 } e ^ { i p ( y _ j - y _ k ) } e ^ { - p ^ 2 / ( 4 t ) } d p , \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} 1 = f ^ * ( F ) f ^ * \\left ( \\frac { 1 } { F } \\right ) \\in \\mathfrak { q } B _ \\mathfrak { q } , \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ( v ) = \\beta _ v f ' ( v ) . \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} S _ 1 ( z ) \\psi = \\mathcal S _ 1 ( z ) \\psi - h ^ \\vee \\bigg ( \\Delta ' ( z ) - \\frac { \\varphi ( z ) } { h ^ \\vee } \\Delta ( z ) \\bigg ) \\psi . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} d ( a , b ) = b ^ { \\binom n 2 } q ^ { \\binom n 3 } \\prod _ { j = 1 } ^ n ( q ^ { j - n } a / b , q ^ { n - j } a b ) _ { j - 1 } \\prod _ { 1 \\leq j < k \\leq n } z _ k ^ { - 1 } \\theta ( z _ k z _ j ^ \\pm ) . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} L _ i = \\left ( \\dfrac { 4 \\pi d _ i f } { c } \\right ) ^ 2 \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\begin{pmatrix} \\lambda _ { 1 1 } & \\lambda _ { 1 2 } & \\lambda _ { 1 3 } \\\\ \\lambda _ { 2 1 } & \\lambda _ { 2 2 } & \\lambda _ { 2 3 } \\\\ \\lambda _ { 3 1 } & \\lambda _ { 3 2 } & \\lambda _ { 3 3 } \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} A u ( x ) = L u ( x ) \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} V ^ * T V = V ^ * U V V ^ * | T | V , \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} G ( a , p ) = \\sum _ { j = 0 } ^ { p - 1 } \\zeta ^ { a x ^ 2 } , \\zeta = e ^ { 2 \\pi \\sqrt { - 1 } / p } . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} \\frac { d \\mathcal { Q } } { d \\mathcal { P } } = e ^ { q ^ 2 \\beta ^ 2 \\log \\varepsilon } e ^ { - q \\beta V _ \\varepsilon ( \\phi ) } , \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} e ( M _ { 1 } , M _ { 1 } , 1 ) & = \\widetilde { e } ( M _ { 1 } , M _ { 1 } , 1 ) = 1 \\ \\\\ e ( G , M _ { 1 } , 1 ) & = \\widetilde { e } ( G , M _ { 1 } , 1 ) = 0 \\ \\ G \\neq M _ { 1 } . \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} \\langle s , t \\rangle _ { \\mathcal { E } ^ 0 } ( g ) & = \\mu ( g ) ^ { - 1 / 2 } \\sum _ { k } \\langle \\sqrt { \\rho _ k } s , \\sqrt { \\rho _ k } g t \\rangle _ { L ^ 2 \\left ( E | _ { U _ k } \\right ) } \\\\ & = \\sum _ { k } \\langle \\sqrt { \\rho _ k } s , \\sqrt { \\rho _ k } t \\rangle _ { \\mathcal { E } ^ 0 _ { U _ k } } ( g ) , \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to \\infty } \\vert \\Phi \\vert = m . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} \\hat { \\bar { \\nabla } } _ { X } Y = \\bar { \\nabla } _ X Y + ( f _ 1 - f _ 3 ) \\eta ( Y ) \\phi X - ( f _ 1 - f _ 3 ) g ( \\phi X , Y ) \\xi . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} ( i _ { Y _ { j _ 0 } } ( \\alpha ) \\wedge d \\omega ) | _ { X _ { j _ 0 l _ 0 } } + ( i _ { Y _ { j _ 0 } } ( \\omega ) \\wedge d \\alpha ) | _ { X _ { j _ 0 l _ 0 } } = 0 . \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} f ( \\gamma ( t ) ) - f ( \\gamma ( s ) ) = t - s \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} \\int _ { \\Omega } F _ n \\varphi c ^ { - 2 } d x = 0 , \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} H = ( \\mathfrak { d } _ 1 \\dot { \\phi } ( i + j ) ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} \\left ( \\rho \\ , \\sigma _ i \\right ) _ k = \\begin{cases} \\tau _ k & k \\neq i , i + 1 \\\\ \\prescript { \\tau _ i } { } \\tau _ { i + 1 } & k = i \\\\ \\tau _ i & k = i + 1 \\end{cases} \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\Delta _ { m + 1 } ^ { * } & = \\delta _ { m + 1 } ^ { * } \\Delta _ { 2 , m + 1 } ^ { * } \\\\ & = \\lambda _ { m + 1 } ^ { - 1 } \\Delta _ { 2 , m + 1 } ^ { - 1 } \\\\ & = \\left ( \\Delta _ { 2 , m + 1 } \\lambda _ { m + 1 } \\right ) ^ { - 1 } \\\\ & = \\Delta _ { m + 1 } ^ { - 1 } \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\sigma _ 1 ^ { A B } ( \\rho _ * ^ { A B } - \\rho _ - ) = \\rho _ { * } ^ { A B } ( v _ { * } ^ { A B } + \\beta t ) - \\rho _ - ( u _ - + \\beta t ) , \\\\ \\sigma _ 2 ^ { A B } ( \\rho _ + - \\rho _ { * } ^ { A B } ) = \\rho _ + ( u _ + + \\beta t ) - \\rho _ { * } ^ { A B } ( v _ { * } ^ { A B } + \\beta t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} Q _ n \\cdot ( h , 1 ) Q _ \\tau ^ { - 1 } = p \\gamma \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } \\Phi _ { t } \\circ d W _ { t } = \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { i = 0 } ^ { n - 1 } \\frac { \\Phi _ { t _ { i + 1 } } + \\Phi _ { t _ { i } } } { 2 } ( B _ { t _ { i + 1 } } - B _ { t _ { i } } ) \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} v _ { k + 1 } ( x ) = \\int _ { B ^ c _ { r _ { k k _ 0 } } } P _ { r _ { k k _ 0 } } ( x , y ) \\tilde { g } ( y ) d y + \\int _ { B _ { r _ { k k _ 0 } } } G ( x , y ) | f | ( y ) d y , \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} \\bigg | \\int \\frac { G _ 1 ( z ) F ( z ) \\overline { F _ 1 ( z ) } } { z - w } d \\nu ( z ) \\bigg | + \\bigg | \\int \\frac { G _ 1 ( z ) H ( z ) \\overline { F _ 1 ( z ) } } { z - w } d \\nu ( z ) \\bigg | = O \\Big ( \\frac { 1 } { | w | } \\Big ) , \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\int _ { \\mathbb { R } ^ n } w ' ( t , x ) \\varphi ' ( t , x ) \\ , d x \\ , d t = & \\ , \\int _ 0 ^ { \\infty } \\int _ { \\mathbb { R } ^ n } \\nabla w ( t , x ) \\cdot \\nabla \\varphi ( t , x ) \\ , d x \\ , d t + \\\\ [ . 2 c m ] & \\ , + \\int _ 0 ^ { \\infty } \\int _ { \\mathbb { R } ^ n } \\tfrac { p } { 2 } | w ( t , x ) | ^ { p - 2 } \\ , w ( t , x ) \\varphi ( t , x ) \\ , d x \\ , d t \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\frac { m - 1 } { l } = \\frac { d } { 2 } . \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} ( 1 - \\gamma ) \\xi = \\omega J ^ { - 1 } _ c J _ b \\nu + r J ^ { - 1 } _ c ( \\nu ^ 2 ) \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} k = k ^ { * } \\pm c \\sqrt { \\frac { n } { A _ n } } \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { a _ 1 } \\left | u ( x , t ) - \\frac { M _ 0 } { M } \\mathcal { B } _ M ( x , t ) \\right | = 0 \\mbox { u n i f o r m l y i n $ \\R ^ n $ } \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} 0 = r ^ k \\| \\vec x ^ k \\| _ \\infty - I ( \\vec x ^ k ) = & r ^ k \\| \\vec x ^ k \\| _ \\infty - ( \\vec x ^ k , \\vec s ^ k ) \\\\ \\ge & r ^ k \\| \\vec x ^ { k + 1 } \\| _ \\infty - ( \\vec x ^ { k + 1 } , \\vec s ^ k ) \\\\ \\ge & r ^ k \\| \\vec x ^ { k + 1 } \\| _ \\infty - I ( \\vec x ^ { k + 1 } ) \\\\ = & \\| \\vec x ^ { k + 1 } \\| _ \\infty ( r ^ k - r ^ { k + 1 } ) . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\{ E _ i , F _ i , H _ i \\} _ { i = 1 , \\ldots , n - 1 } \\cup \\{ X _ n , Y _ { n } , Z _ n \\} \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} \\hat f \\circ \\phi ( z ) & = f _ 0 + ( f _ + \\circ \\phi ) ( 0 ) + \\sum _ { n > 0 } ( f _ + \\circ \\phi ) _ n z ^ n + \\overline { ( f _ - \\circ \\phi ) ( 0 ) } + \\sum _ { n < 0 } \\overline { ( f _ - \\circ \\phi ) _ { - n } } z ^ { n } \\\\ & = f ( \\phi ( 0 ) ) + \\sum _ { n > 0 } ( f _ + \\circ \\phi ) _ n z ^ n + \\sum _ { n < 0 } \\overline { ( f _ - \\circ \\phi ) _ { - n } } z ^ { n } . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} \\tfrac { \\binom { M d _ j } { M d _ 1 } } { \\binom { d _ j } { M d _ 1 } } = \\tfrac { ( M d _ j ) ( M d _ j - 1 ) \\cdots ( M d _ j - M d _ 1 + 1 ) } { ( d _ j ) ( d _ j - 1 ) \\cdots ( d _ j - M d _ 1 + 1 ) } = \\tfrac { M ^ { M d _ 1 } ( d _ j ) ( d _ j - \\tfrac { 1 } { M } ) \\cdots ( d _ j - \\tfrac { M d _ 1 } { M } + \\tfrac { 1 } { M } ) } { ( d _ j ) ( d _ j - 1 ) \\cdots ( d _ j - M d _ 1 + 1 ) } . \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} \\sigma \\ ( f _ \\lambda \\circ \\theta : \\lambda \\in \\Lambda \\ ) = \\theta ^ { - 1 } \\sigma \\ ( f _ \\lambda : \\lambda \\in \\Lambda \\ ) , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} l _ { 1 } & = l _ { 1 } ( r ) \\stackrel { \\mathrm { d e f } } { = } \\sum _ { j = 1 } ^ { r - 1 } k _ { j } + \\frac { a _ { 2 } v _ { 2 } ^ { 2 } } { 2 } \\sum _ { j = 1 } ^ { r - 2 } j \\left ( j + 1 \\right ) \\ \\\\ l _ { 2 } & = l _ { 2 } ( r ) \\stackrel { \\mathrm { d e f } } { = } \\sum _ { j = 1 } ^ { r - 1 } j . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} I _ { k , s , e } ( \\lambda ) & = \\left \\{ ( A , B ) \\in I _ { k , s , e } \\mid \\exists \\ , \\Sigma \\in X _ { \\lambda } , A \\subset \\Sigma \\subset B \\right \\} , \\\\ I _ { k , s , e } ^ + ( \\lambda ) & = \\left \\{ ( A , B ) \\in I _ { k , s , e } \\mid \\exists \\ , \\Sigma ^ + \\in X _ { \\lambda } ^ + , A \\subset \\Sigma ^ + \\subset B \\right \\} . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} P _ k ( x ) = \\sum _ { l = 0 } ^ { N - 1 } f _ k ^ { ( l ) } ( x ) \\quad Q _ k ( x ) = \\sum _ { l = 0 } ^ { N - 1 } ( - 1 ) ^ { l } f _ k ^ { ( l ) } ( x ) . \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} h _ 1 ( \\partial _ x u _ 1 + \\partial _ y u _ 2 ) = - \\partial _ \\tau \\hat h _ 1 + \\hat h _ 1 \\partial _ \\xi \\hat u _ 1 - \\hat u _ 1 \\partial _ \\xi \\hat h _ 1 + \\nu \\hat h _ 1 \\big ( \\partial _ \\eta ( \\hat h _ 1 \\partial _ \\eta \\hat h _ 1 ) - ( \\partial _ \\eta \\hat h _ 1 ) ^ 2 \\big ) . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} \\dfrac { d ^ 2 f } { d X _ u d Y _ u } = \\dfrac { 8 ( X _ u - x _ i ) ( Y _ u - y _ i ) } { ( ( X _ u - x _ i ) ^ 2 + ( Y _ u - y _ i ) ^ 2 + z ^ 2 _ { m i n } ) ^ 3 } \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{gather*} \\mathcal G _ { 2 , ( 1 ; 3 ) } ^ \\ast ( \\tau ) = \\frac { \\pi ^ 2 } { 3 } \\ ( E _ 2 ^ \\ast ( 3 \\tau ) - \\phi ^ 2 ( \\tau ) \\ ) , \\end{gather*}"}
-{"id": "3302.png", "formula": "\\begin{align*} U ( s , t ) ~ U ( t , s ) = U ( s , s ) = I , \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} [ f _ * I _ Y ] = \\sum _ S [ { \\jmath _ S } _ { ! * } \\jmath _ S ^ * \\imath _ S ^ { ! * } f _ * I _ Y ] . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} A _ 2 ( n ) = \\begin{cases} p ^ { - 3 n / 2 } ( - 1 ) ^ n \\chi _ { \\psi } ( p ^ n ) ( 1 - p ^ { - 1 } ) p ^ { n } & n > 0 , \\\\ p ^ { 3 n / 2 } ( - 1 ) ^ n \\chi _ { \\psi } ( p ^ n ) ( 1 - p ^ { - 1 } ) ( 1 + p - p ^ { 1 - n } ) & n \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\delta ( t ) : = \\frac { \\zeta ( t ) } { m - 1 } \\frac { \\eta ' ( t ) } { \\eta ( t ) } + \\frac { C ^ { m - 1 } m } { a ^ 2 ( m - 1 ) ^ 2 } \\zeta ^ m ( t ) \\eta ^ 2 ( t ) \\ , , \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} a _ 0 ( r ) = 0 \\ \\hbox { f o r a n y } \\ r \\in ( R , 1 ) . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b , \\bar { b } ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl ( ( e ^ { - t q } - 1 ) e ^ { - b _ 0 t } + ( e ^ { t q } - 1 ) e ^ { - \\bar { b } _ 0 t } \\Bigr ) \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } \\Bigr ) . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} Q ^ { ( m ) } _ { 2 r } ( 1 ) & = \\mu ^ { ( m ) } _ { 2 r , 1 } + \\mu ^ { ( m ) } _ { 2 r , - 1 } - q ^ m \\mu ^ { ( m - 1 ) } _ { 2 r - 1 } , \\\\ Q ^ { ( m ) } _ { 2 r + 1 } ( 1 ) & = \\mu ^ { ( m ) } _ { 2 r + 1 } - q ^ m \\big [ \\mu ^ { ( m - 1 ) } _ { 2 r , 1 } + \\mu ^ { ( m - 1 ) } _ { 2 r , - 1 } \\big ] . \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\widetilde { c } _ { n i } ( - 2 ) = - \\widetilde { c } _ { n i } ( 0 ) + \\widetilde { c } _ { n - 1 i } ( - 2 ) + \\widetilde { c } _ { n - 1 i } ( 0 ) = \\delta _ { n - 1 i } . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} ( G f ) ( x ) = \\beta f ( x ) , \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} r = \\| \\vec v \\| _ 1 , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; p = \\infty . \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\begin{aligned} v ^ m _ { r r } ( r , t ) & = - \\frac { C ^ m m } { a ( m - 1 ) } \\eta ( t ) \\zeta ^ m ( t ) G ^ { \\frac 1 { m - 1 } } + \\frac { C ^ m m } { a ^ 2 ( m - 1 ) ^ 2 } \\zeta ^ m ( t ) \\eta ^ 2 ( t ) r ^ 2 G ^ { \\frac 1 { m - 1 } - 1 } \\\\ & \\geq - \\frac { C ^ m m } { a ( m - 1 ) } \\eta ( t ) \\zeta ^ m ( t ) G ^ { \\frac 1 { m - 1 } } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} V _ k \\cdots V _ 2 V _ 1 & = t _ { c _ k } ^ { - 1 } t _ { b _ k } ^ { \\epsilon _ k } \\cdots t _ { b _ 2 } ^ { \\epsilon _ 2 } t _ { b _ 1 } ^ { \\epsilon _ 1 } t _ { c _ 0 } = t _ { f ( a ) } ^ { - 1 } f t _ a \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial x _ i } ( x ) = \\sum _ { k ; k \\neq i } \\frac { 1 } { x _ i - x _ k } = : u _ { i 1 } ( x ) , & & \\frac { \\partial ^ 2 F } { \\partial x _ i ^ 2 } ( x ) = - \\sum _ { k ; k \\neq i } \\frac { 1 } { ( x _ i - x _ k ) ^ 2 } = : - u _ { i 2 } ( x ) , \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\partial _ x ^ 2 \\mathbb { E } f \\bigl ( Y ^ n _ t ( x ) \\bigr ) & = \\partial _ x \\mathbb { E } f ' \\bigl ( Y ^ { n } _ { t } ( x ) \\bigr ) + \\frac { 1 } { 2 } \\sqrt { t } \\ , \\mathbb { E } \\bigl ( \\xi \\partial _ x g _ 1 \\bigl ( x , \\xi \\sqrt t , Y ^ { n - 1 } _ { t } \\bigr ) \\bigr ) \\\\ & = : P _ 2 ( t , x ) + R _ 2 ( t , x ) . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} H = ( \\mathfrak { d } _ 2 \\dot { \\phi } ( i + j ) ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} z = y ( 1 - y ) . \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\mathcal J _ 1 ( m ) = \\mathrm { v o l } ( \\mathcal B _ 1 ^ + ( m ) ) + \\int _ { \\mathcal B _ 1 ^ - ( m ) } ( c , d ) _ p \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h . \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} \\dim ( X ) \\ & = \\ n + m + 2 - ( r + 1 ) - \\delta ( X ) \\\\ & \\ge \\ \\frac { 2 \\delta ( X ) + 2 \\alpha - m } { 2 } + m + 2 - \\delta ( X ) \\\\ & = \\ \\alpha + \\frac { m } { 2 } + 2 . \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} R [ I t ] : = R \\oplus I t \\oplus I ^ 2 t ^ 2 \\oplus \\cdots \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\Big ( \\big ( S _ t - X _ t , S _ t \\big ) : t \\geq 0 \\Big ) \\stackrel { \\mathcal { L } } { = } \\Big ( \\big ( | X _ t | , L _ t ^ 0 ( X ) \\big ) : t \\geq 0 \\Big ) \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} - d S _ 1 T ^ { d - 1 } + \\dots + ( - 1 ) ^ d d S _ d & = d F \\\\ & = - F \\sum _ { 1 \\le i \\le d } \\frac { d { \\it p r } ^ \\ast _ i t } { T - { \\it p r } ^ \\ast _ i t } \\\\ & = F \\sum _ { 1 \\le i \\le d } \\frac { 1 } { { \\it p r } ^ \\ast _ i t } \\frac { d { \\it p r } ^ \\ast _ i t } { 1 - \\frac { T } { { \\it p r } ^ \\ast _ i t } } \\\\ & = F \\sum _ { r \\ge 0 } \\omega _ { r + 1 } T ^ r . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} S \\cap \\theta ^ { - k } ( S ) = \\emptyset . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { n - \\delta } { n - r \\brack \\delta } ( q ^ \\delta A ' _ r + B ' _ r ) = q ^ { ( m + 1 ) ( n - \\delta ) } q ^ \\delta \\left ( { n \\brack \\delta } ( A _ 0 + q ^ \\delta C _ 0 ) + A _ \\delta + q ^ \\delta C _ \\delta \\right ) . \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} 0 \\ge \\bar { p } _ { j _ 2 } \\ge \\bar { p } _ { i _ 0 } \\ge \\bar { p } _ { j _ 1 } \\ge 0 \\ ; \\ ; \\ ; \\Rightarrow \\ ; \\ ; \\ ; \\bar { p } _ { i _ 0 } = 0 \\ ; \\ ; \\ ; \\Rightarrow \\ ; \\ ; \\ ; p _ { i _ 0 } = q _ { i _ 0 } = 0 \\ ; \\ ; \\ ; \\Rightarrow \\ ; \\ ; \\ ; s _ { i _ 0 } = 0 , \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\nu : = \\big ( \\nu _ 1 , . . . , \\nu _ 5 \\big ) . \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} \\omega ( y , z ) - \\omega ( x + y , z ) + \\omega ( x , y + z ) - \\omega ( x , y ) = 0 \\ ; \\ , \\mbox { f o r a l l $ x , y , z \\in A $ } \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\tau _ { s } ( n g ) = f ( n g \\iota ( n g ) ^ { - 1 } ) = f ( n ) + f ( g \\iota ( g ) ^ { - 1 } ) = f ( n ) + \\tau _ { s } ( g ) \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\langle I _ { \\lambda , p } ' ( \\tilde { u } _ { 0 } , \\tilde { v } _ { 0 } ) , ( \\varphi , \\psi ) \\rangle = \\langle I _ { \\lambda , p } ' ( u _ { n } , v _ { n } ) , ( \\varphi _ { n } , \\psi _ { n } ) \\rangle + o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} Q ^ - ( u ) = \\rho , \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\widehat U _ { L _ { \\infty } / K } ' : = \\varprojlim _ n \\widehat U _ { L _ n / K } ' \\in K _ 0 ( \\Lambda ( \\mathcal { G } ) , \\Lambda ^ { \\mathcal { O } _ F } ( \\mathcal { G } ) ) \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} \\mathcal { Y } ^ m : = \\big \\{ g \\in X ^ { m } = L ^ 2 ( [ 0 , T ] \\times \\mathbb { T } ; \\mathcal { H } ^ { 1 , m } _ v ) \\ ; \\colon \\partial _ t g + v \\partial _ x g \\in ( X ^ { m } ) ^ * \\big \\} \\ , . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} p ( { \\bf M ^ { ( 2 ) } } ) = \\Sigma \\ C _ { a b c d e f g j k } \\lambda ^ a \\mu ^ b v ^ c z _ 1 ^ d \\bar { z } _ 1 ^ e z _ 2 ^ f \\bar { z } _ 2 ^ g z _ 3 ^ j \\bar { z } _ 3 ^ k , \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} c = c _ { p , q } = 1 - \\frac { 6 ( p - q ) ^ { 2 } } { p q } \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\mathcal { A } ( x , k ) = \\bigcup _ { y \\in A ( x , k ) } \\{ P \\in \\mathcal { A } ( x , k ) \\ : \\ y \\in P \\} . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\Big ( \\frac { 1 } { \\gamma } - | \\omega | - \\eta \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } \\Big ) \\| \\nu \\| ^ 2 = ( f ( x _ 0 ) - \\epsilon _ 0 ) \\| \\nu \\| ^ 2 , \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\ < z | : = | z \\ > ^ * \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} p _ t ( x , y ) = \\sum _ { i = 0 } ^ { \\infty } e ^ { - \\lambda _ i t } u _ i ( x ) u _ i ( y ) , \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\left \\langle \\eta _ k ^ + ( x ) , \\eta _ { n - k } ^ - ( y ) \\right \\rangle = \\left \\{ \\begin{array} { l l } 1 & d ( x , y ) \\leq n \\\\ 0 & . \\end{array} \\right . \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} { \\rm g r a d } \\ , J ( A _ r , B _ r , C _ r ) = & ( - 2 A _ r { \\rm s y m } ( Q P + Y ^ T X ) A _ r , \\\\ & 2 ( Q B _ r + Y ^ T B ) , 2 ( C _ r P - C X ) ) . \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\big \\langle \\widetilde \\Lambda _ v \\widetilde P _ { v _ 1 , \\rm R } \\widetilde F ( v _ 1 ) , \\widetilde P _ { v _ 1 , \\rm R } \\widetilde F ( v _ 1 ) \\big \\rangle & = C _ { v _ 1 } \\sum _ { j = 1 } ^ { d + 1 } \\big | \\widetilde F _ j ( v ) \\big | ^ 2 = C _ { v _ 1 } \\sum _ { j = 1 } ^ { d } \\big | F _ j ( v ) \\big | ^ 2 \\\\ & = \\big \\langle \\Lambda _ { v _ 1 } P _ { v _ 1 , \\rm R } F ( v _ 1 ) , P _ { v _ 1 , \\rm R } F ( v _ 1 ) \\big \\rangle , \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} r _ { \\max } = \\max \\limits _ { \\vec x \\in \\mathbb { R } ^ n \\setminus \\{ \\vec 0 \\} } F ( \\vec x ) . \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\Phi _ { i , T } ( X ) \\left ( \\mathbf { r } ^ T \\otimes I _ 2 \\right ) = \\mathbf { r } ^ T \\otimes \\Phi _ { i , T } ( X ) . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\alpha ^ { * } \\alpha + \\gamma ^ { * } \\gamma = I , & \\alpha \\alpha ^ { * } + q ^ { 2 } \\gamma ^ { * } \\gamma = I , & \\gamma ^ { * } \\gamma = \\gamma \\gamma ^ { * } , \\\\ \\alpha \\gamma = q \\gamma \\alpha , & \\alpha \\gamma ^ { * } = q \\gamma ^ { * } \\alpha . \\end{array} \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{gather*} \\phi ( \\tau _ 0 ) = \\frac { \\sqrt { 3 } } { 2 \\pi } \\frac 1 { 2 ^ { 1 / 3 } } B ( 1 / 3 , 1 / 3 ) \\phi _ 1 ( \\tau _ 0 ) = \\frac 1 { 2 ^ { 2 / 3 } \\cdot 3 ^ { 1 / 2 } } \\frac { 1 } { 2 \\pi } B ( 1 / 3 , 1 / 3 ) . \\end{gather*}"}
-{"id": "2139.png", "formula": "\\begin{align*} p ^ 2 - 1 = 0 \\mod 5 . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} t r _ N ( T ) : = \\sum _ { x \\in N } \\langle x | T x \\rangle \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ { r _ { k k _ 0 } } } G ( x , y ) | f | ( y ) d y \\leq C _ 1 M r _ { k k _ 0 } ^ { s - n / p } \\leq C _ 1 M r _ { k k _ 0 } ^ { \\beta } \\leq \\hat { C } M r _ { ( k + 1 ) k _ 0 } ^ { \\beta } \\cdot \\frac { C _ 1 } { \\hat { C } \\tau _ 1 ^ { \\beta k _ 0 } } . \\end{aligned} \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } ( 1 , \\ , \\tau ) = ( 1 , \\ , \\frac { 1 } { \\tau } ) \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} V _ N = 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\log X + \\log Y + \\log Y ' + o ( 1 ) . \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} U _ N ^ * \\ ; \\Phi = \\sum _ { j = 0 } ^ { N _ 1 } \\sum _ { k = 0 } ^ { N _ 2 } \\frac { 1 } { \\sqrt { ( N _ 1 - j ) ! ( N _ 2 - k ) ! } } \\Big ( ( a ^ * _ 0 ) ^ { N _ 1 - j } \\ , ( b _ 0 ^ * ) ^ { N _ 2 - k } \\Phi \\Big ) _ { N _ 1 N _ 2 } . \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\binom { n } { k ^ * } = \\binom { n } { c n } \\sim & \\frac { ( n / e ) ^ n } { ( c n / e ) ^ { c n } ( ( 1 - c ) n / e ) ^ { ( 1 - c ) n } } \\frac { \\sqrt { 2 \\pi n } } { \\sqrt { 2 \\pi n c } \\sqrt { 2 \\pi n ( 1 - c ) } } \\\\ = & \\frac { 1 } { c ^ { n c } ( 1 - c ) ^ { ( 1 - c ) n } } \\frac { 1 } { \\sqrt { 2 \\pi n c ( 1 - c ) } } . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\rho _ { \\mathbf { m } } , \\mathbf { m } = ( \\ell _ 1 + 1 + m ) ( e _ 1 + \\ldots + e _ n ) , \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} \\mathrm { e v } _ { t = 1 } ( E _ { t } ( \\widetilde { m } ) ) & = \\chi _ q \\left ( M ( m ) \\right ) & \\mathrm { e v } _ { t = 1 } ( \\mathcal { K } _ { t } ) & = \\chi _ q ( K ( \\mathcal { C } _ { \\bullet } ) ) \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} \\begin{pmatrix} \\mathcal A & \\mathcal B \\\\ \\mathcal B ^ t & \\mathcal D \\\\ \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} \\mathbf I _ \\ell + \\mathcal { B D } ^ { - 1 } \\mathcal B ^ t & - \\mathcal { B D } ^ { - 1 } \\\\ - \\mathcal D ^ { - 1 } \\mathcal B ^ t & \\mathcal D ^ { - 1 } \\\\ \\end{pmatrix} ^ { - 1 } . \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } \\hat { X } ( r ) : = & X ^ { s , x ^ { \\prime } ; \\bar { u } } ( r ) - \\bar { X } ^ { t , x ; \\bar { u } } ( r ) , \\ \\hat { Y } ( r ) : = Y ^ { s , x ^ { \\prime } ; \\bar { u } } ( r ) - \\bar { Y } ^ { t , x ; \\bar { u } } ( r ) , \\\\ \\hat { Z } ( r ) : = & Z ^ { s , x ^ { \\prime } ; \\bar { u } } ( r ) - \\bar { Z } ^ { t , x ; \\bar { u } } ( r ) , \\ \\hat { \\Theta } ( r ) : = \\left ( \\hat { X } ( r ) , \\hat { Y } ( r ) , \\hat { Z } ( r ) \\right ) . \\end{array} \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} ( \\mathrm { d e g } _ G a _ i ) \\cdot ( \\mathrm { d e g } _ G b _ i ) = g , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ \\infty e ^ { - s t } d K ^ { ( f ) } _ { M , N } ( t \\ , | \\ , b ) = \\int \\limits _ 0 ^ \\infty e ^ { - s t } e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) f ( t ) \\ , d t / t ^ M < \\infty , \\ ; s > 0 . \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} X _ { C _ 1 } \\cdot X _ { C _ 2 } = \\sum _ { j = 1 } ^ r \\lambda _ j X _ { D _ j } \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} C _ { k , l } ( \\gamma ) = \\frac { k ^ \\perp \\cdot l } { | k | ^ \\gamma } . \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} | N | \\cdot N _ { w , G } ( g ) = | N | \\cdot N _ { w , G } ( h ) , \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} & R _ n = \\overline { C } ( p , f , \\pi _ n ) _ T , \\\\ & R = \\int _ 0 ^ T m _ { c _ s } ( f ) d { G _ p } ( s ) , \\\\ & R _ n ( r ) = n ^ { p / 2 - 1 } \\sum _ { i : t _ { i , n } \\leq T } f \\big ( \\Delta _ { i , n } C ( r ) \\big ) , \\\\ & R ( r ) = \\int _ 0 ^ T m _ { c _ s ( r ) } ( f ) d { G _ p } ( s ) , \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\int _ { \\rho ^ { - 1 } ( 0 , r ] } \\lvert _ A \\Phi \\rvert ^ 2 = \\frac { 1 } { 2 } \\int _ { \\rho ^ { - 1 } ( r ) } \\ast \\lvert \\Phi \\rvert ^ 2 . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { d , c } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\bar E _ { a , b } \\bar E _ { d , c } } E _ { d , c } E _ { a , b } + K _ { c , d } E _ { a , c } , & M = N = 1 ; \\\\ \\displaystyle \\sum _ { t = 0 } ^ { \\min ( M , N ) } q _ { b } ^ { - t ( N - t ) } E _ { d , c } ^ { ( N - t ) } K _ { c , d } ^ { t } E _ { a , b } ^ { ( M - t ) } E _ { a , c } ^ { ( t ) } , & . \\end{cases} \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} G ( x , y ) & \\leq \\exp ( - \\langle \\alpha , y - x \\rangle ) G _ \\alpha ( x , y ) \\\\ & \\leq ~ \\exp ( - \\langle \\alpha , y - x \\rangle ) \\P _ x ( X _ \\alpha ( t ) = y \\ ; \\ ; t \\geq 0 ) G _ S ( 0 , 0 ) \\\\ & \\leq ~ \\exp ( - \\langle \\alpha , y - x \\rangle ) G _ S ( 0 , 0 ) , \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} L \\big ( \\eta ( 3 \\tau ) ^ 8 , 3 \\big ) & = L \\big ( \\chi ^ 3 , 3 / 2 \\big ) = \\frac 4 3 L ( G _ 3 ( q ) , 3 ) = \\frac 4 3 \\frac { - { \\rm i } } { 3 ^ { 3 / 2 } } \\mathcal G _ { 3 , ( 1 ; 3 ) } ^ \\ast ( \\tau _ 0 ) \\\\ & = \\frac 4 3 \\frac 1 2 \\ ( \\frac { 2 \\pi } { \\sqrt { 3 } } \\ ) ^ 3 \\phi _ 1 ( \\tau _ 0 ) ^ 3 = \\frac 8 3 L ( f _ { 3 6 } , 1 ) ^ 3 . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} & \\partial _ t u + ( u \\cdot \\nabla ) u - \\nu \\Delta u = - \\nabla p \\\\ & \\nabla \\cdot u = 0 \\\\ & u ( x , 0 ) = u _ 0 ( x ) \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} \\left ( W ( \\rho ^ 0 , \\rho ^ 1 ) \\right ) ^ 2 : = \\inf _ { \\rho ( t ) , v ( t ) } \\Big \\{ \\int _ 0 ^ 1 ( v ( t ) , v ( t ) ) _ { \\rho ( t ) } d t ~ : ~ \\frac { d \\rho } { d t } + \\mathrm { d i v } ( \\rho v ) = 0 , ~ \\rho ( 0 ) = \\rho ^ 0 , ~ \\rho ( 1 ) = \\rho ^ 1 \\Big \\} . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} M _ n ( a ) : = A _ n ( a ) / S O ( 3 ) . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} [ g _ 1 , \\epsilon _ 1 ] [ g _ 2 , \\epsilon _ 2 ] = [ g _ 1 g _ 2 , \\epsilon ( g _ 1 , g _ 2 ) \\epsilon _ 1 \\epsilon _ 2 ] , \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} x _ i = x _ 0 - \\sum _ { j = 1 } ^ { i - 1 } \\frac { \\tilde \\gamma _ { i , j } } { \\tilde \\gamma _ { i , i } } ( x _ j - x _ 0 ) - \\sum _ { j = 0 } ^ { i - 1 } \\frac { \\tilde \\beta _ { i , j } } { \\tilde \\gamma _ { i , i } } f ' ( x _ j ) , i = 1 , \\hdots , N , \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\varphi : C ( N ) : = ( 1 , \\infty ) _ r \\times N \\to X \\setminus K \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} \\Sigma = \\frac { 1 } { 2 } ( \\Gamma + \\Gamma ^ { - 1 } ) , \\Delta = \\frac { 1 } { 2 } ( \\Gamma - \\Gamma ^ { - 1 } ) \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} \\langle M u , M v \\rangle _ { \\mathcal G } = \\langle u , v \\rangle _ { \\mathcal G } . \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} - \\Delta f + \\frac { 1 } { 2 } N ^ 2 V ( N . ) f = \\lambda _ N f \\mathbf { 1 } _ { B ( 0 , \\ell ) } \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} M _ k ( t ) = & \\int _ 0 ^ t \\sigma ( u , X ( u ) ) e _ k ( u ) d u + \\int _ 0 ^ t \\Big ( \\int _ 0 ^ r U ( u , r ) e _ k ( u ) d u \\Big ) d r + \\int _ 0 ^ t \\Big ( \\int _ 0 ^ r V ( u , r ) e _ k ( u ) d u \\Big ) d W ( r ) \\\\ & + \\int _ 0 ^ t \\nabla _ x b ( r , X ( r ) ) M _ k ( r ) d r + \\int _ 0 ^ t \\nabla _ x \\sigma ( r , X ( r ) ) M _ k ( r ) d W ( r ) . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} \\dim _ k \\mathcal { F } ( f _ i , f _ j ) = \\begin{cases} 1 & 0 \\leq j - i \\leq l - 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} a ( H ^ { i j } ) = ( \\lambda _ i , \\lambda _ j ) / \\kappa , a ( H ^ i _ j ) = - ( \\lambda _ i , \\alpha _ { c ( j ) } ) / \\kappa , a ( H _ { i j } ) = ( \\alpha _ { c ( i ) } , \\alpha _ { c ( j ) } ) / \\kappa , \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} _ a D _ { t } ^ { - \\nu } f ( t ) = [ f ( t ) ] _ { - \\nu } = \\frac { 1 } { \\Gamma ( \\nu ) } \\int _ a ^ t \\frac { f ( s ) } { ( t - s ) ^ { 1 - \\nu } } d s \\quad ( t > a , \\nu > 0 ) . \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} P ^ n A _ u V ( x ) & = ~ \\varphi _ { u , n } ( x ) + \\sum _ { k = 1 } ^ { n } B _ k P ^ { n - k } A _ u V ( x ) \\\\ & = \\varphi _ { u , n } ( x ) + \\sum _ { k = 1 } ^ { n } \\sum _ { y \\in E } b _ k ( x , y ) P ^ { n - k } A _ u V ( y ) . \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\bar { \\bf C } _ { A S } ^ \\bullet ( M ) : = \\left \\{ C ^ \\bullet _ \\wedge ( M ) / \\bar { C } _ { \\wedge , 0 } ^ \\bullet ( M ) , \\delta \\right \\} \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} w ^ B ( t ) = ( \\rho _ - u _ - - \\rho _ + u _ + ) t , \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N B _ n ( k \\lambda ) = \\sum _ { m = 0 } ^ n \\binom { N + 1 } { m + 1 } m ! S _ Y ( n , m ) = \\sum _ { k = 0 } ^ n c _ { n , N } ( k ) B _ n ( k \\lambda ) . \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\varphi ^ { 2 } U = p \\varphi U + q ( U ) - u ( U ) \\xi , \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} L _ { 1 } x ( t ) = \\lambda r ( t ) x ( t ) , \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} m _ V - m _ H = | V | - | H | . \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} \\mathcal { G } \\left ( V ^ { \\varepsilon } \\right ) \\left ( t \\right ) = H ^ { \\varepsilon } - \\int _ { t } ^ { T } K ^ { \\varepsilon } \\left ( s , V ^ { \\varepsilon } \\right ) d s , \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\alpha + \\beta & = ( \\vec z ^ * , | \\vec v | ) - r > 0 , \\\\ m _ 0 + \\gamma & = \\| \\vec z ^ * \\| _ p ^ p > 0 . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { C ^ { k } \\left ( \\left [ 0 , T \\right ] ; X \\right ) } = \\sum _ { n = 0 } ^ { k } \\sup _ { t \\in \\left [ 0 , T \\right ] } \\left \\Vert u ^ { \\left ( n \\right ) } \\left ( t \\right ) \\right \\Vert _ { X } < \\infty . \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} J ( { \\underline \\gamma } ^ * ) = \\inf _ { { { \\underline \\gamma } } \\in { { \\bf \\Gamma } } } J ( { { \\underline \\gamma } } ) = : J ^ * . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\Phi _ k ( s , x ) : = \\frac { \\Gamma ( k - s / 2 ) \\Gamma ( 1 - k - s / 2 ) } { \\Gamma ( 1 / 2 ) } { } _ 2 F _ 1 \\left ( k - \\frac { s } { 2 } , 1 - k - \\frac { s } { 2 } , \\frac { 1 } { 2 } ; \\frac { 1 } { 4 } \\right ) , \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} H = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 2 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} \\sum _ { x = 0 } ^ \\infty \\binom { x + \\beta - 1 } { x } c ^ x \\ , M _ { n } ( x ; \\beta , c ) M _ m ( x ; \\beta , c ) = 0 , m \\ne n , \\ m , n = 0 , 1 , \\dots \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\abs { Y } = q ^ { m ( m - 2 \\delta + 1 ) / 2 } . \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} f ( x ) ~ = ~ \\lim _ { k \\to \\infty } f _ { n _ k } ( x ) , x \\in E , \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} u _ { i } : = \\begin{cases} u & i \\leqslant N _ 1 , \\\\ v & i > N _ 1 . \\end{cases} \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} p ^ * \\left ( \\prod _ { f \\in I } X _ f \\right ) = \\prod _ { ( f , g ) \\in I \\times \\tilde { I } } A _ { g } ^ { \\tilde { \\varepsilon } _ { g f } } = \\prod _ { g \\in \\tilde { I } } A _ { g } ^ { \\sum \\limits _ { f \\in I } \\tilde { \\varepsilon } _ { g f } } . \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} \\langle P ( x ) , Q ( y ) \\rangle = \\sum _ { \\substack { k _ 1 , . . . , k _ N = 0 \\\\ m _ 1 , . . . , m _ N = 0 } } ^ { \\infty } \\left ( \\prod _ { i = 1 } ^ N \\left \\langle \\delta _ { \\omega _ { x _ i } ( k _ i ) } , \\delta _ { \\omega _ { y _ i } ( m _ i ) } \\right \\rangle \\right ) \\left \\langle A ^ * B e _ { ( k _ 1 , . . . , k _ N ) } , e _ { ( m _ 1 , . . . , m _ N ) } \\right \\rangle . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} x _ i = x _ 0 - \\sum _ { j = 0 } ^ { i - 1 } h _ { i , j } f ' ( x _ j ) . \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\max _ { 0 \\leqslant N \\leqslant N _ { 0 } } \\left \\Vert \\alpha _ { n } ^ { ( N ) } \\right \\Vert _ { \\mathcal { F } } \\geqslant \\lambda \\right ) \\leqslant \\sum _ { N = 1 } ^ { N _ { 0 } } \\mathbb { P } \\left ( \\left \\Vert \\alpha _ { n } ^ { ( N ) } \\right \\Vert _ { \\mathcal { F } } \\geqslant \\lambda \\right ) \\leqslant N _ { 0 } \\mathbb { P } \\left ( \\left \\Vert \\alpha _ { n } ^ { ( N _ { 0 } ) } \\right \\Vert _ { \\mathcal { F } } \\geqslant \\lambda \\right ) . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\begin{aligned} d X _ { t } & = \\div ( a ^ { \\ast } \\phi ( a \\nabla X _ { t } ) ) \\ , d t + \\langle b \\nabla X _ { t } , \\circ \\ , d W _ { t } \\rangle , t \\in ( 0 , T ] , \\\\ X _ { 0 } & = x . \\end{aligned} \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} \\chi _ { j k } \\ ; & \\equiv \\ ; \\chi _ { j k } ( x _ 1 , \\dots , x _ j ; y _ 1 , \\dots , y _ k ) \\\\ u _ 0 ^ { \\otimes ( N _ 1 - j ) } \\ ; & \\equiv \\ ; u _ 0 ( x _ { j + 1 } ) \\cdots u _ 0 ( x _ { N _ 1 } ) \\\\ v _ 0 ^ { \\otimes ( N _ 2 - k ) } \\ ; & \\equiv \\ ; v _ 0 ( y _ { k + 1 } ) \\dots v _ 0 ( y _ { N _ 2 } ) \\ , . \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} i \\Bigg ( \\sum _ k n _ k | A _ k , N | A \\Bigg ) = \\cup _ { k } ( i ( n _ k , N | A ) \\cap A _ k ) \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} J ( \\gamma ) = \\int _ { \\gamma } T d t = \\frac { 1 } { \\sqrt { 2 } } \\int _ { \\gamma } d s _ { h } \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} h _ { d _ i } ^ { ( i ) } ( q _ i ) = \\sum _ { j = 0 } ^ { n _ i / d _ i - 1 } q _ i ^ { d _ i j } . \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\delta _ { \\bar g } \\delta _ { \\bar g } ^ * X ^ { \\flat } = - \\delta _ { \\bar g } ( e ^ { - 2 u } ( \\mathring { Q } _ m ( g , s ) + \\rho \\bar g ) . \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} S _ { d , k } ^ { ( d ) } ( n , W , \\ell ) = \\binom { W } { \\ell } S _ { d + 1 , k } \\big ( n - \\ell ( d + 1 ) , W - \\ell \\big ) \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} f ^ { \\prime \\prime } ( x ) \\ , - \\ , \\left [ \\left ( \\pi ( x ) - { 1 \\over 2 } \\varphi ^ { \\prime } ( x ) \\right ) ^ 2 + \\left ( \\pi ( x ) - { 1 \\over 2 } \\varphi ^ { \\prime } ( x ) \\right ) ^ { \\prime } \\right ] f ( x ) \\ , = \\ , 0 \\ ; . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} \\theta ( \\beta _ c ( d ) + \\epsilon _ 1 , \\tilde { H } _ 0 ) = 0 \\epsilon _ 1 , \\tilde { H } _ 0 > 0 , \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} u _ \\delta ^ B ( t ) = \\sigma _ { 0 } ^ B + \\beta t , \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} Z _ n ( X ) = \\bigoplus _ { u \\in X ^ + } Z _ n ( X ; u ) \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} \\max _ { \\theta } & ~ U _ { L } ( \\theta , \\lambda ^ { \\textrm { o p t } } ) = \\mu ( 1 - \\theta ) \\bigg [ \\log \\bigg ( 1 + t _ { 1 } \\frac { \\theta } { 1 - \\theta } \\bigg ) \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - \\log \\bigg ( 1 + t _ { 2 } \\frac { \\theta } { 1 - \\theta } \\bigg ) \\bigg ] - \\theta t _ { 3 } , \\\\ s . t . & ~ 0 < \\theta < 1 , \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\mathcal { B } & : = \\ ! \\left \\{ ( p + 1 , p - 1 ) , ( 1 , - 1 ) , ( p + 2 , p ) , ( 2 , 0 ) , ( 1 , 1 ) , ( p + 1 , p + 1 ) \\right \\} , \\ \\varepsilon _ 2 ( \\boldsymbol { B } ) : = \\varepsilon \\left ( \\frac { B _ 2 - 3 B _ 1 } { 2 p } , \\frac { B _ 1 } { p } \\right ) . \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} F \\left ( u \\right ) = \\begin{pmatrix} \\left ( b _ { 1 0 } - b _ { 1 1 } u _ { 1 } - b _ { 1 2 } u _ { 2 } \\right ) u _ { 1 } \\\\ \\left ( b _ { 2 0 } - b _ { 2 1 } u _ { 1 } - b _ { 2 2 } u _ { 2 } \\right ) u _ { 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{gather*} S \\big ( h ^ 1 \\big ) \\alpha h ^ 2 = \\varepsilon ( h ) \\alpha , h ^ 1 \\beta S \\big ( h ^ 2 \\big ) = \\epsilon ( h ) \\beta . \\end{gather*}"}
-{"id": "6456.png", "formula": "\\begin{align*} x _ i ^ * ( y _ i ) & = \\frac { x _ i ^ * ( u _ i + x _ i - z _ i ) } { 1 + \\varepsilon } \\\\ & = \\frac { P x _ i ^ * ( u _ i ) + ( x _ i ^ * - P x _ i ^ * ) ( x _ i ) + P x _ i ^ * ( x _ i - z _ i ) } { 1 + \\varepsilon } \\\\ & > \\frac { ( \\| P x _ i ^ * \\| - \\varepsilon ) ( 1 + \\varepsilon ) + ( \\| x _ i ^ * - P x _ i ^ * \\| - \\varepsilon ) ( 1 + \\varepsilon ) - \\varepsilon } { 1 + \\varepsilon } \\\\ & > \\| x _ i ^ * \\| - 3 \\varepsilon = 1 - 3 \\varepsilon . \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} f _ u = \\dfrac { 1 } { \\hat { H } } \\Big ( \\lambda \\nu _ u + F \\nu \\times \\nu _ u - E \\nu \\times \\nu _ v \\Big ) , f _ v = \\dfrac { 1 } { \\hat { H } } \\Big ( \\lambda \\nu _ v + G \\nu \\times \\nu _ u - F \\nu \\times \\nu _ v \\Big ) , \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\ < u ( \\omega _ N ) \\cdot \\nabla \\omega _ N , \\phi \\ > = - \\int \\ ! \\ ! \\int \\omega _ N ( x ) \\omega _ N ( y ) H _ \\phi ( x , y ) \\ , \\d x \\d y = - \\ < \\omega _ N \\otimes \\omega _ N , H _ \\phi \\ > . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} \\sum _ { \\{ i \\in [ 1 , 2 ^ { p + 1 } ] \\ , ; \\ , i \\equiv r \\ ! \\ ! \\ ! \\ ! \\mod 2 ^ { q } \\} } \\mu _ { p + 1 } ( \\{ \\lambda _ { i } \\} ) & = \\sum _ { \\{ i \\in [ 1 , 2 ^ { p } ] \\ , ; \\ , i \\equiv r \\ ! \\ ! \\ ! \\ ! \\mod 2 ^ { q } \\} } \\left ( \\mu _ { p + 1 } ( \\{ \\lambda _ { i } \\} ) + \\mu _ { p + 1 } ( \\{ \\lambda _ { 2 ^ p + i } \\} ) \\right ) \\\\ & = \\sum _ { \\{ i \\in [ 1 , 2 ^ { p } ] \\ , ; \\ , i \\equiv r \\ ! \\ ! \\ ! \\ ! \\mod 2 ^ { q } \\} } \\mu _ { p } ( \\{ \\lambda _ { i } \\} ) \\le ( 1 - \\varepsilon ) ^ { q } \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\frac { a _ { t _ 2 } ^ { i _ { t _ 2 } } q ^ { \\binom { i _ { t _ 2 } } { 2 } } } { ( q ; q ) _ { i _ { t _ 2 } } } . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} \\mathcal { Y } _ t = \\bigoplus _ { m : \\ \\mathcal { Y } } \\mathbb { Z } [ t ^ { \\pm 1 / 2 } ] \\underline { m } \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( 1 - \\gamma _ 1 ) & + 1 5 & ( 1 - \\gamma _ 2 ) & < 1 , \\\\ 3 & ( 1 - \\gamma _ 1 ) & + 1 2 & ( 1 - \\gamma _ 2 ) & < 2 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} g e n u s ( M _ k ) \\geq \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } g e n u s ( M _ k \\cap B ^ { ( N , g ) } _ { R r ^ { y , \\ell } _ k } ( p ^ { y , \\ell } _ k ) ) , \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} \\sum _ { j > 0 } \\int _ { \\mathcal L _ { 2 j } } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h = - p ( 1 - p ^ { - 1 } ) ^ 2 \\sum _ { j > 0 } p ^ { - 2 j } = - p ^ { - 1 } \\frac { ( 1 - p ^ { - 1 } ) ^ 2 } { ( 1 - p ^ { - 2 } ) } = \\frac { 1 - p } { p ( p + 1 ) } . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{gather*} | \\varepsilon \\boldsymbol { f } _ \\varepsilon ' | _ { L ^ 2 ( 0 , T ; \\boldsymbol { H } ) } = | \\boldsymbol { f } - \\boldsymbol { f } _ \\varepsilon | _ { L ^ 2 ( 0 , T ; \\boldsymbol { H } ) } \\le M _ { 1 0 } , \\\\ \\varepsilon ^ { 1 / 2 } | \\boldsymbol { f } _ \\varepsilon | _ { L ^ \\infty ( 0 , T ; \\boldsymbol { H } ) } \\le M _ { 1 0 } , \\end{gather*}"}
-{"id": "3611.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , 1 , \\xi _ t , 1 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + \\mathcal { Q } _ { t + 1 } ( x _ { t } , \\xi _ { [ t - 1 ] } , \\xi _ t , 1 ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} \\kappa = 3 \\theta _ 2 P _ 3 ( 1 ) \\widetilde { P _ 2 } ( 1 ) - 2 \\theta _ 2 \\int _ 0 ^ 1 P _ 2 ( x ) P _ 3 ( x ) d x \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} \\Phi _ N : = U _ N \\psi _ N ^ { \\operatorname { M F } } \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} f _ 2 \\circ u _ 1 & = u _ 4 \\circ f _ 1 \\circ e _ 1 \\\\ g _ 2 \\circ u _ 1 & = u _ 4 \\circ g _ 1 \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} z ( \\lambda ) : = \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} h ( \\lambda ) \\ , \\in \\mathcal { N } _ r ( \\mathcal { L } ) \\ , . \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} \\tilde { H } _ { i , j } = \\langle H \\delta _ j , S D \\delta _ i \\rangle = \\left ( \\frac { 1 + i } { 2 + i } \\right ) ^ { \\alpha } \\langle H \\delta _ j , \\delta _ { i + 1 } \\rangle = ( 1 + i ) ^ \\alpha ( 1 + j ) ^ \\alpha a _ { i + j + 1 } . \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\Omega = \\omega _ 1 + \\omega _ 2 \\ ; . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\sum _ { j \\in \\tau } ( - 1 ) ^ { j } x _ { j i } \\ , X _ { \\tau \\setminus \\{ j \\} } = 0 \\mbox { f o r a l l } i \\in [ d ] , \\tau \\in \\binom { [ n ] } { d + 1 } \\enspace , \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} D _ { n + 2 , t } ^ 0 D _ { n + 1 } ^ 0 - D _ { n + 1 , t } ^ 0 D _ { n + 2 } ^ 0 = 2 D _ { n + 2 } ^ { - 1 } D _ { n + 1 } ^ 1 , \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} X _ y = \\left ( A - \\mathbf Y ( y ) \\right ) ^ { - 1 } ( M _ { m , n } ^ t ) = q \\left ( p ^ { - 1 } ( \\{ y \\} ) \\right ) \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\sphericalangle \\big ( P ( y , r _ { x , k } ) , P ( y , r _ { x , k + i } ) \\big ) & \\leq \\sum \\limits _ { l = 0 } ^ { i - 1 } \\sphericalangle \\big ( P ( y , r _ { x , k + l } ) , P ( y , r _ { x , k + l + 1 } ) \\big ) \\\\ & \\leq \\tilde C C _ 1 ( m ) \\sum \\limits _ { l = 0 } ^ { i - 1 } ( \\delta _ { x , k + l + 1 } + 2 C \\delta _ { x , k + l } ) \\\\ & \\xrightarrow [ k \\to \\infty ] { } 0 , \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} \\limsup _ { t \\to T _ * } \\| u ( t ) \\| _ { \\dot B ^ { s _ p } _ { p , q } ( \\R ^ d ) } = \\infty . \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\Omega ^ { n + 1 } ( A , B , f ) & = \\Omega ( \\Omega ^ n A , M \\otimes P ^ A _ { n - 1 } , M \\otimes i ^ A _ n ) \\oplus ( 0 , \\Omega ^ { n + 1 } B , 0 ) . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} P _ { \\epsilon } ( ( x _ { \\epsilon , \\eta } ) _ { * } ) & = u _ { \\epsilon } ( x _ { \\epsilon , \\eta } ) - \\frac { 1 } { \\epsilon } | ( x _ { \\epsilon , \\eta } ) _ { * } - x _ { \\epsilon , \\eta } | ^ { 2 } = u ( x _ { \\epsilon , \\eta } ) _ { * } ) , \\\\ \\nabla P _ { \\epsilon } ( ( x _ { \\epsilon , \\eta } ) _ { * } ) & = \\nabla u _ { \\epsilon } ( x _ { \\epsilon , \\eta } ) , \\\\ \\nabla ^ { 2 } P _ { \\epsilon } ( ( x _ { \\epsilon , \\eta } ) _ { * } ) & = \\nabla ^ { 2 } u _ { \\epsilon } ( x _ { \\epsilon , \\eta } ) . \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} c _ v ( s ) = \\frac { L ( ( 2 n + 1 ) s + 1 / 2 , \\pi _ v \\boxtimes \\chi _ v , \\varrho _ { 2 n + 1 } ) } { L ( ( 2 n + 1 ) ( s + 1 / 2 ) , \\chi _ v ) \\prod \\limits _ { i = 1 } ^ n L ( ( 2 n + 1 ) ( 2 s + 1 ) - 2 i , \\chi _ v ^ 2 ) } . \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} \\sigma = \\begin{pmatrix} \\sigma ^ { ( 1 ) } & 0 \\\\ \\rho \\sigma ^ { ( 2 ) } & \\sqrt { 1 - \\rho ^ 2 } \\sigma ^ { ( 2 ) } \\end{pmatrix} \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} \\tilde { L } _ 0 \\varphi : = - \\Delta \\varphi - ( 2 ^ * - 1 ) u _ 0 ^ { 2 ^ * - 2 } \\varphi , \\varphi \\in H ^ 1 _ 0 ( \\Omega ) , \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } 1 - ( f ' ( 0 ) + \\varepsilon ( x , 0 ) / x ) ^ a = 0 . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} X _ t ( x _ { t - 1 } , \\xi _ t ) = \\{ x _ t \\in \\mathbb { R } ^ n : x _ t \\in \\mathcal { X } _ t , \\ ; g _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) \\leq 0 , \\ ; \\ ; \\displaystyle A _ { t } x _ { t } + B _ { t } x _ { t - 1 } = b _ t \\} , \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} \\mathcal B _ \\delta ^ { ( 1 ) } : = \\left \\{ \\omega \\in \\mathcal A _ r \\ , \\Big | \\ , \\begin{array} { c } \\forall z , w \\in \\mathbb B _ r \\textrm { w i t h } \\Vert z - w \\Vert < \\delta : \\\\ \\sup _ n \\Vert f ^ n _ \\omega ( z ) - f ^ n _ \\omega ( w ) \\Vert < \\varepsilon _ 1 \\end{array} \\right \\} , \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} F ^ { ( m + 1 ) } _ { r } ( s ) = q ^ { 2 r } F ^ { ( m - 1 ) } _ { r } ( s - 1 ) - q ^ { 2 r - 2 } F ^ { ( m - 1 ) } _ { r - 1 } ( s - 1 ) , \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} C _ L ^ { \\bullet } : = R \\Gamma ( L , \\Z _ p ( 1 ) ) [ 1 ] \\in \\mathcal { D } ( \\Z _ p [ G ] ) \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( u + \\delta ) \\le \\mu ( u + \\delta ) \\le \\mu ( u ) = \\tilde { \\varphi } ( u ) \\le 1 \\ , , \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\hat { \\omega } _ p ( t ( a ) r , 1 ) \\hat { \\phi } _ { \\mathbf h , p } ( \\beta ; 0 , 1 ) = \\omega _ p ( ( t ( a ) r , 1 ) ) \\phi _ { 1 , p } ( \\beta ) \\phi _ { 2 , p } ( ( 0 , 1 ) t ( a ) r ) , \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} s _ n ^ * { \\rm { = } } \\mathop { { \\rm { a r g m a x } } } \\limits _ { { s _ n } = 0 , 1 } \\Pr \\left ( { \\left . { { s _ n } } \\right | { { \\bf { R } } _ q } } \\right ) \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} p _ { N } = \\min _ { 1 \\leqslant j \\leqslant m _ { N } } P ( A _ { j } ^ { ( N ) } ) > 0 , N \\in \\mathbb { N } _ { \\ast } , \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} G _ { \\omega , n } ^ E : = ( J _ 0 ^ n ( \\omega ) - E ) ^ { - 1 } G _ { \\omega , n } ^ E ( j , k ) : = \\langle \\delta _ j , G _ { \\omega , n } ^ E \\delta _ k \\rangle , \\ j , k \\in [ 1 , n ] . \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} { M ^ g } ' ( t ) = & \\int _ 0 ^ t \\sigma ( s , X ( s ) ) g ( s ) d s + \\int _ 0 ^ t \\Big ( \\int _ 0 ^ r U ( s , r ) g ( s ) d s \\Big ) d r + \\int _ 0 ^ t \\Big ( \\int _ 0 ^ r V ( s , r ) g ( s ) d s \\Big ) d W ( r ) \\\\ & + \\int _ 0 ^ t \\nabla _ x b ( r , X ( r ) ) { M ^ g } ' ( r ) d r + \\int _ 0 ^ t \\nabla _ x \\sigma ( r , X ( r ) ) { M ^ g } ' ( r ) d W ( r ) . \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\widetilde b _ { j k } ^ * ( z ) = ( z ^ { - 1 } - 1 ) ^ j h _ { j k } ^ * ( z ) , j = 0 , \\dots , d - 1 , k = 0 , \\dots , d , \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} \\frac { \\partial U _ { B S } } { \\partial P _ { B S } } & = \\lambda \\theta \\| \\mathbf { h } \\| ^ { 2 } - 2 \\theta A P _ { B S } - \\theta B = 0 , \\\\ & ~ ~ ~ ~ \\Rightarrow P _ { B S } = \\frac { \\lambda \\| \\mathbf { h } \\| ^ { 2 } - B } { 2 A } . \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} \\mathcal { I I } = \\sum _ { \\substack { \\mu , \\iota = 1 } } ^ N A ^ { \\mu \\iota } , & \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} \\mathbb W ^ s _ \\omega ( z ) : = \\{ w \\in U _ \\omega \\ , | \\ , \\Vert f ^ n _ \\omega ( z ) - f ^ n _ \\omega ( w ) \\Vert \\to 0 \\} . \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\begin{gathered} d X _ t = ( k - a X _ t ) d t + \\sigma \\sqrt { X _ t } \\circ d B _ t ^ H , \\end{gathered} \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} d : M \\times M \\to [ 0 , \\infty ) , d ( p , q ) : = \\inf _ { \\gamma \\in \\Omega _ { p , q } } { \\cal L } _ \\gamma \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\frac { w ( e _ 1 ) } { B ( v , e _ 1 ) } = \\frac { w ( e _ 2 ) } { B ( v , e _ 2 ) } = \\cdots = \\frac { w ( e _ d ) } { B ( v , e _ d ) } = r x _ v ^ p . \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{gather*} 2 L ( \\psi , 1 / 2 ) ^ 2 = 2 L ( f _ { 3 2 } , 1 ) ^ 2 = \\frac { 1 } { 6 4 } B ( 1 / 4 , 1 / 4 ) ^ 2 = L \\big ( \\psi ^ 2 , 1 \\big ) , \\end{gather*}"}
-{"id": "924.png", "formula": "\\begin{align*} \\sup \\limits _ { t \\in [ 0 , T ] } \\left \\| \\Phi ( t ) - \\sum _ { j = 1 } ^ { m } e _ j \\otimes \\Phi ( t ) e _ j \\right \\| _ { } \\le \\frac { \\epsilon } { 2 } . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} a ' = a ' _ 1 + a ' _ 2 , b ' = b ' _ 1 + b ' _ 2 , c ' = c ' _ 1 + c ' _ 2 . \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} g ( q ) = \\sum _ { n \\geq 0 } a ( g , n ) q ^ n . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\phi ( x , \\gamma \\gamma ' ) = \\phi ( x , \\gamma ) \\phi ( x \\gamma , \\gamma ' ) \\quad \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\left ( u _ { \\infty } \\right ) _ t = \\nabla \\cdot \\left ( m \\ , \\mathcal { B } _ { M } ^ { m - 1 } \\nabla u _ { \\infty } \\right ) \\mbox { a n d } u _ { \\infty } \\leq \\mathcal { B } _ { M } \\forall ( x , t ) \\in \\R ^ n \\times [ 0 , \\infty ) . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} f _ k ( Q _ n \\cdot ( a n , 1 ) ) & = i ^ { n k } \\bigg ( \\prod _ { j = 1 } ^ n \\frac 1 { a _ j + a _ j ^ { - 1 } } \\bigg ) ^ { ( 2 n + 1 ) ( s + \\frac 1 2 ) } \\ ! f _ k ( \\left [ \\begin{smallmatrix} C & & & S \\\\ & C & S \\\\ & - S & C \\\\ - S & & & C \\end{smallmatrix} \\right ] \\ ! \\ ! \\left [ \\begin{smallmatrix} U \\\\ & I _ n \\\\ & & ^ t U ^ { - 1 } \\\\ & & & I _ n \\end{smallmatrix} \\right ] \\ ! \\ ! \\left [ \\begin{smallmatrix} I _ n & & - X \\\\ & I _ n \\\\ & & I _ n \\\\ & & & I _ n \\end{smallmatrix} \\right ] ) . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} b ( u , v , w ) & = ( u \\cdot \\nabla v , w ) + \\frac { 1 } { 2 } ( ( \\nabla \\cdot u ) v , w ) , \\\\ b ( u , v , w ) & \\leq C _ { 1 } \\| \\nabla u \\| \\| \\nabla v \\| \\| \\nabla w \\| , \\\\ b ( u , v , w ) & \\leq C _ { 2 } \\sqrt { \\| u \\| \\| \\nabla u \\| } \\| \\nabla v \\| \\| \\nabla w \\| . \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} b = \\frac { \\lambda \\binom { v } { 2 } } { \\binom { k } { 2 } } \\quad r ( k - 1 ) = \\lambda ( v - 1 ) . \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} \\langle r ^ { - } _ { \\psi } ( \\nu _ c \\alpha _ n ) \\mathbf h _ p , \\mathbf h _ p \\rangle = \\chi _ { \\psi } ( p ^ n ) p ^ { - n / 2 } \\int _ { \\Z _ p ^ { \\times } } \\left ( \\int _ { \\Q _ p } \\psi ( - 2 x p ^ n z - c p ^ { 2 n + 1 } z ^ 2 ) \\mathfrak G ( 2 z , \\underline { \\chi } _ p ^ { - 1 } ) d z \\right ) \\underline { \\chi } _ p ( x ) d x . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { s - 1 } \\left ( 1 + \\gamma _ j ^ { \\frac { 1 } { \\alpha - 2 \\delta } } 2 \\zeta \\left ( \\frac { \\alpha } { \\alpha - 2 \\delta } \\right ) \\right ) = \\exp \\left ( \\log \\left ( \\prod _ { j = 1 } ^ { s - 1 } \\left ( 1 + \\gamma _ j ^ { \\frac { 1 } { \\alpha - 2 \\delta } } 2 \\zeta \\left ( \\frac { \\alpha } { \\alpha - 2 \\delta } \\right ) \\right ) \\right ) \\right ) , \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} S ( m , n ) = \\prod _ { i = 1 } ^ N S _ { i , m _ i , n _ i } \\in \\mathcal { B } ( \\ell _ 2 ( \\N ^ N ) ) , \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\lambda _ i ( t ) & = | t | - \\frac { x _ i ^ k } { \\| \\vec x ^ k \\| _ \\infty } t \\ge 0 , \\ , \\ , \\ , \\forall \\ , i \\in \\{ 1 , \\ldots , n \\} , \\\\ \\| \\vec s \\| _ 1 - r ^ k & = \\| \\vec s \\| _ 1 - \\frac { ( \\vec x ^ k , \\vec s ) } { \\| \\vec x ^ k \\| _ \\infty } = \\sum _ { i = 1 } ^ n \\lambda _ i ( s _ i ) , \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 ^ { n - 2 } } = \\sum _ { k = 0 } ^ { k _ { 0 } - 2 } + \\sum _ { k = k _ { 0 } - 1 } ^ { k _ { 0 } + 2 } + \\sum _ { k = k _ { 0 } + 3 } ^ { 2 ^ { n - 2 } } \\stackrel { \\rm d e f } = \\Sigma _ 1 + \\Sigma _ 2 + \\Sigma _ 3 . \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { \\mathbb { G } _ { \\alpha } ^ { p , q } ( \\mathbb { T } ^ { d } ) } : = \\left ( \\sum _ { k \\in \\mathbb { Z } ^ { d } } \\left ( 2 \\alpha \\left \\Vert k \\right \\Vert ^ { q } _ { p } \\right ) \\hat { u } _ { k } \\right ) ^ { 1 / 2 } < \\infty , \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} 1 - \\varepsilon & < \\| w \\| = N ( \\| u \\| , \\| v \\| ) \\\\ & \\leq N ( a - \\delta , a - \\delta ) \\\\ & \\leq N ( ( 1 - \\varepsilon ) a , ( 1 - \\varepsilon ) a ) \\\\ & = ( 1 - \\varepsilon ) N ( a , a ) = 1 - \\varepsilon , \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} \\mathcal { Q } _ t ( \\cdot , 1 ) \\geq \\mathcal { Q } _ t ^ { k - 1 } ( \\cdot , 1 ) , t = 2 , \\ldots , T _ { \\max } + 1 , \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} P _ n = n ! | \\mathcal { S R P } _ n | = n ! \\cdot ( n - 1 ) ! \\cdot C _ { n - 1 } = ( 2 n - 2 ) ! . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} { \\theta _ { 1 , A } } \\left ( t \\right ) = 2 \\pi \\left ( { f ' - f '' } \\right ) t + { \\varphi ' _ n } - { \\varphi '' _ n } \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} \\mathbf { D } \\mathbf { A } = \\mathbf { I } _ m , \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\beta = 0 & \\gamma \\in \\{ 2 m \\mid m = 0 , 1 , \\ldots , 6 5 , 6 8 , 7 1 , 7 9 \\} , \\\\ \\beta = 1 & \\gamma \\in \\{ 2 m \\mid m = 8 , 9 , \\ldots , 5 8 , 6 3 \\} , \\\\ \\beta = 2 & \\gamma \\in \\{ 2 m \\mid m = 0 , 4 , 6 , \\ldots , 5 5 \\} \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} L ( f \\otimes g \\otimes g , s ) = L ( f , s - k ) L ( f \\otimes \\mathrm { A d } ( g ) , s ) . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow + \\infty } \\langle I _ { \\lambda _ { n } } ' ( u _ { n } ) - I ' _ { \\lambda _ { n } } ( u _ { 0 } ) , u _ { n } - u _ { 0 } \\rangle = 0 . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\Q ^ { \\times } = \\bigsqcup _ { \\epsilon } \\Q ^ { \\epsilon } ( \\pi ) . \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} z _ t = z _ s + \\sum _ { I ; \\vert I \\vert \\leq [ p ] } X ^ I _ { t s } ( V _ I ) ( z _ s ) + O \\big ( \\vert t - s \\vert ^ a \\big ) , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} \\hat { W } _ n ( k ) = \\begin{cases} 2 ^ { - n + 1 } ( k - 2 ^ { n - 1 } ) , & k \\in [ 2 ^ { n - 1 } , 2 ^ n ] \\\\ 2 ^ { - n } ( 2 ^ { n + 1 } - k ) , & k \\in [ 2 ^ { n } , 2 ^ { n + 1 } ] \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} j : L \\otimes _ { K } H \\longrightarrow \\mathrm { E n d } _ { K } ( L ) \\ \\ j ( x \\otimes y ) ( z ) = x y ( z ) \\ \\ x , z \\in L , y \\in H \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\left ( - 2 L _ { - 2 } + \\frac { \\kappa } { 2 } L _ { - 1 } ^ { 2 } + \\frac { \\tau } { 2 } \\sum _ { r = 1 } ^ { 3 } X _ { r } ( - 1 ) ^ { 2 } \\right ) v _ { \\pm \\Lambda } = 0 \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} p _ I \\ = \\ \\max _ { \\sigma \\in S _ d } \\left ( \\sum _ { \\ell = 1 } ^ { d } A _ { i _ { \\ell } , \\sigma ( \\ell ) } \\right ) \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} f '' = { { \\left ( { { f _ { 1 , B } } - { f _ { 1 , A } } } \\right ) } \\mathord { \\left / { \\vphantom { { \\left ( { { f _ { 1 , B } } - { f _ { 1 , A } } } \\right ) } 2 } } \\right . \\kern - \\nulldelimiterspace } 2 } \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) } w ( e ) = \\frac { 1 } { C } \\sum _ { i = 1 } ^ k \\frac { \\sum _ { e \\in E ( G _ i ) } w _ i ( e ) } { \\alpha _ i ^ { 1 / ( p - r ) } } = \\frac { 1 } { C } \\sum _ { i = 1 } ^ k \\frac { 1 } { \\alpha _ i ^ { 1 / ( p - r ) } } = 1 . \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\delta _ 1 ^ { \\mathfrak { K } } ( t , x + w , y ; z ) - \\delta _ 1 ^ { \\mathfrak { K } } ( t , x , y ; z ) = \\int _ 0 ^ 1 \\left < w , \\nabla _ x \\delta _ 1 ^ { \\mathfrak { K } } ( t , x + \\theta w , y ; z ) \\right > d \\theta \\ , . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} B ( X ) = \\ , & ( 1 + a + b ) X ^ 3 + ( 1 + b + z + a z + b z ) X ^ 2 + ( b + k + a k + b k + z + a z + b z ) X \\\\ & + b + a k + a z + b z + k z + a k z + b k z . \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} E ( A ) : = \\Bigl \\{ \\sum n _ { a , b } & \\bigl ( a + b + ( - 1 ) ( a \\dot + b ) \\bigr ) \\\\ & \\Bigl | \\ , a , b \\in A , n _ { a , b } \\in \\mathbb Z , n _ { a , b } = 0 \\ , \\ , \\Bigr \\} . \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} ( \\bar R ^ \\ast \\otimes 1 _ \\rho ) ( 1 _ \\rho \\otimes R ) & = 1 _ \\rho \\ , ; & ( R ^ \\ast \\otimes 1 _ { \\bar \\rho } ) ( 1 _ { \\bar \\rho } \\otimes \\bar R ) & = 1 _ { \\bar \\rho } \\ , . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} C _ { r - i } = \\frac { - 1 } { B _ q } ( A _ { p - i } + \\sum _ { j = 0 } ^ { i - 1 } C _ { r - j } B _ { m - i + j } ) \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} w : = \\min \\{ f ( x ) : h ( x ) = 0 , g ( x ) \\leq 0 \\} , \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) & = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\dot { \\phi } \\left ( | m | + | n | + 2 | I | \\right ) \\\\ & = \\sum _ { k = 0 } ^ N \\tbinom { N } { k } ( - 1 ) ^ { k } \\dot { \\phi } \\left ( | m | + | n | + 2 k \\right ) \\\\ & = \\mathfrak { d } _ 2 ^ N \\dot { \\phi } \\left ( | m | + | n | \\right ) . \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} \\partial _ t w & = c _ \\mu \\rho ^ { \\alpha - 1 } \\partial _ x ^ 2 w - ( u + c _ \\mu \\rho ^ { \\alpha - 2 } \\partial _ x \\rho ) \\partial _ x w + \\frac { c _ p } { c _ \\mu } \\left ( \\gamma - 2 ( \\alpha + 1 ) \\right ) \\rho ^ { \\gamma - \\alpha } w \\\\ & - \\frac { 1 } { c _ \\mu } ( \\alpha + 1 ) \\rho ^ { - \\alpha } w ^ 2 + \\frac { c ^ 2 _ p } { c _ \\mu } \\left ( \\gamma - ( \\alpha + 1 ) \\right ) \\rho ^ { 2 \\gamma - \\alpha } . \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} \\sum _ { l \\in \\mathbb { Z } , \\ , \\ , l \\geq k } d _ { l } = a _ { k } \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\Theta ( x ) : = \\lim _ { r \\downarrow 0 } \\mu ( B _ r ( x ) ) , \\quad \\forall x \\in X . \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } , \\xi _ t , D _ { t } ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; D _ { t } f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + ( D _ { t - 1 } - D _ t ) { \\overline { f } } _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + \\mathcal { Q } _ { t + 1 } ( x _ { t } , \\xi _ { [ t ] } , D _ t ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\nabla _ { k , l } : & M ^ { k , l } \\rightarrow \\C = M ^ { k + l } ~ ; ~ \\sum _ { g \\in G } k _ g g \\rightarrow \\sum _ { g \\in G } k _ g & \\displaybreak [ 1 ] \\\\ \\Delta _ { k , l } : & \\C = M ^ { k + l } \\rightarrow M ^ { k , l } ~ ; ~ 1 \\rightarrow \\sum _ { g \\in G } g = \\frac { 1 } { k ! l ! } \\sum _ { g \\in S _ { k + l } } S _ { k , l } g \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} \\left ( - 2 L _ { - 2 } + \\frac { \\kappa } { 2 } L _ { - 1 } ^ { 2 } + \\frac { \\tau } { 2 } \\sum _ { r = 1 } ^ { 3 } X _ { r } ( - 1 ) ^ { 2 } \\right ) e ^ { - \\Lambda } = 0 \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} & 1 + d \\bigg ( \\frac { - ( 1 - 3 d D ) - \\sqrt { ( 1 - d D ) ^ { 2 } + 2 a d ^ { 2 } C } } { 2 d } - 2 D \\bigg ) \\\\ & = 1 + d \\bigg ( \\frac { - ( 1 - d D ) - \\sqrt { ( 1 - d D ) ^ { 2 } + 2 a d ^ { 2 } C } } { 2 d } \\bigg ) \\\\ & < 1 + d \\bigg ( \\frac { - ( 1 - d D ) - | 1 - d D | } { 2 d } \\bigg ) \\leq 1 . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} & E _ 2 ^ 2 D _ 1 ^ 2 + E _ 1 E _ 2 ^ 2 D _ 1 + F _ 3 ^ 2 + E _ 1 E _ 2 F _ 3 + E _ 1 ^ 2 F _ 4 = 0 , \\\\ & E _ 1 D _ 1 ^ 2 + E _ 1 ^ 2 D _ 1 ^ 2 + E _ 0 F _ 3 + E _ 2 F _ 1 + E _ 1 F _ 2 = 0 , \\\\ & E _ 0 ^ 2 D _ 1 ^ 2 + E _ 0 ^ 2 E _ 1 D _ 1 + F _ 1 ^ 2 + E _ 0 E _ 1 F _ 1 + E _ 1 ^ 2 F _ 0 = 0 . \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} \\widetilde { m } _ { i j } = \\frac { 1 } { 2 } \\left ( a _ { i j } ^ 2 + b _ { i j } ^ 2 + c _ { i j } ^ 2 + g _ { i j } ^ 2 \\right ) . \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} ( \\alpha , i _ ! ( \\beta ) ) _ { X } = ( i ^ \\ast ( \\alpha ) , \\beta ) _ { Z } \\neq 0 \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{gather*} \\tilde \\chi ( \\mathfrak I ) = \\tilde \\chi \\big ( a + b \\sqrt { - 3 } \\big ) = a - b \\sqrt { - 3 } . \\end{gather*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\chi ^ I _ i = \\begin{cases} 1 & i \\in I \\\\ 0 & i \\notin I . \\end{cases} \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\omega _ j ( g _ 1 , g _ 2 ) = \\iota ^ { - 1 } ( \\sigma _ j ( g _ 1 ) \\sigma _ j ( g _ 2 ) \\sigma _ j ( g _ 1 g _ 2 ) ^ { - 1 } ) \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} g ^ { + } _ i ( u ) & : = \\min \\{ g _ i ( x ) + d ( x , u ) \\colon x \\in N \\} , \\\\ g ^ { + } _ i ( v ) & : = \\max \\{ g ^ { + } _ i ( x ) - d ( x , v ) \\colon x \\in N \\cup \\{ u \\} \\} . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } \\left . \\frac { \\partial V } { \\partial x } ( s , x , W ( s , x ) , W _ { x } ( s , x ) , \\bar { u } ( s ) ) \\right \\vert _ { x = \\bar { X } ^ { t , x ; \\bar { u } } ( s ) } & = K _ { 1 } ( s ) , \\\\ \\left . \\frac { \\partial ^ { 2 } V } { \\partial x ^ { 2 } } ( s , x , W ( s , x ) , W _ { x } ( s , x ) , \\bar { u } ( s ) ) \\right \\vert _ { x = \\bar { X } ^ { t , x ; \\bar { u } } ( s ) } & = \\tilde { K } _ { 2 } ( s ) , \\end{array} \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} \\Omega _ 1 ( N , q ) = \\begin{cases} 1 & 0 \\leq e _ 2 \\leq 1 s _ 2 = e _ 2 , \\\\ \\frac { 3 } { 2 } & e _ 2 = 2 s _ 2 < e _ 2 , \\\\ 2 & e _ 2 \\geq 3 s _ 2 < e _ 2 , \\end{cases} \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} \\mathfrak D = \\partial _ x ^ 2 D _ 2 ( x ) + \\partial _ x D _ 1 ( x ) + D _ 0 ( x ) \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} \\nu \\beta _ 1 \\nu ' = ( - 1 ) s \\left ( \\begin{array} { c c } 1 & - \\gamma p \\\\ 0 & 1 \\end{array} \\right ) s s \\alpha _ 1 \\nu ' = s \\left ( \\begin{array} { c c } 1 & - \\gamma p \\\\ 0 & 1 \\end{array} \\right ) \\alpha _ 1 \\nu ' \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} s _ 2 s _ 3 s _ 1 s ^ { h } s _ 1 s _ 3 s _ 2 & = ( s _ 2 s _ 3 s _ 1 ) ( s _ 1 s _ 2 s _ 3 s _ 1 ) ( s _ 1 s _ 3 s _ 2 ) = s _ 2 s _ 3 , \\\\ s _ 2 s _ 3 s _ 1 s ^ { h v } s _ 1 s _ 3 s _ 2 & = ( s _ 2 s _ 3 s _ 1 ) ( s _ 1 s _ 2 s _ 1 s _ 2 s _ 3 s _ 1 ) ( s _ 1 s _ 3 s _ 2 ) = ( s _ 2 s _ 3 ) ( s _ 2 s _ 1 ) . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} \\psi ' _ \\infty : = \\lim _ { t \\rightarrow \\infty } \\psi ' ( t ) , \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} I & \\leq t ^ { 1 - \\eta - \\theta } \\left [ h ^ { - 1 } ( 1 / t ) \\right ] ^ { \\gamma + \\beta } \\int _ 0 ^ 1 s ^ { \\gamma / 2 - \\eta } ( 1 - s ) ^ { \\beta / 2 - \\theta } \\ , d s \\\\ & = B ( \\beta / 2 + 1 - \\theta , \\gamma / 2 + 1 - \\eta ) t ^ { 1 - \\eta - \\theta } \\left [ h ^ { - 1 } ( 1 / t ) \\right ] ^ { \\gamma + \\beta } \\ , . \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\Sigma _ B ( s ) = \\frac { 1 } { \\pi ^ 2 } \\sum _ { k = 1 } ^ { \\infty } ( 2 k - 1 ) \\exp ( - ( 2 k - 1 ) \\beta ) \\Sigma _ B ( k , s ) , \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} { \\varphi '' _ n } = { { \\left ( { \\varphi _ { n , B } ^ { { \\rm { C F S K } } } + { \\varphi _ B } - \\varphi _ { n , A } ^ { { \\rm { C F S K } } } - { \\varphi _ A } } \\right ) } \\mathord { \\left / { \\vphantom { { \\left ( { \\varphi _ { n , B } ^ { { \\rm { C F S K } } } + { \\varphi _ B } - \\varphi _ { n , A } ^ { { \\rm { C F S K } } } - { \\varphi _ A } } \\right ) } 2 } } \\right . \\kern - \\nulldelimiterspace } 2 } \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\langle w , g \\rangle & : = \\bigl \\langle \\tfrac { 1 } { c _ n } ( I - \\Delta ) ^ m \\phi , g \\bigr \\rangle = \\frac { 1 } { c _ n } \\biggl \\langle \\sum _ { k = 0 } ^ m ( - 1 ) ^ m \\binom { m } { k } \\Delta ^ k \\phi , g \\biggr \\rangle \\\\ & : = \\frac { 1 } { c _ n } \\sum _ { k = 0 } ^ m ( - 1 ) ^ m \\binom { m } { k } \\langle \\phi , \\Delta ^ k g \\rangle \\\\ & : = \\frac { 1 } { c _ n } \\sum _ { k = 0 } ^ m ( - 1 ) ^ m \\binom { m } { k } \\int _ { \\R ^ n } \\phi \\Delta ^ k g . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} ( \\omega \\cdot f ) ( \\gamma _ 1 , \\ldots , \\gamma _ { p + q } ) = \\omega ( \\gamma _ 1 , \\ldots , \\gamma _ p ) f ( \\gamma _ { p + 1 } , \\ldots , \\gamma _ { p + q } ) \\ ; . \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} \\begin{array} { r c l c r } - \\nabla \\cdot ( e ^ { u ( x , \\omega ) } \\nabla q ( x , \\omega ) ) = 1 , & & x \\in G = ( - 0 . 5 , 0 . 5 ) ^ d , & & \\omega \\in \\Omega , \\\\ q ( x , \\omega ) = 0 , & & x \\in \\partial G , & & \\omega \\in \\Omega , \\end{array} \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\Delta ( v _ 1 \\cdots v _ n ) = \\sum _ { S \\subseteq [ n ] } v _ S \\otimes v _ { [ n ] \\setminus S } , \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} u _ 1 = & \\ \\frac { 1 } { 2 } ( d _ { 1 1 2 } , d _ { 2 1 2 } ) , & u _ 2 = & \\ u _ 1 \\ + \\ \\left ( \\frac { 1 } { 2 } , 0 \\right ) , \\\\ u _ 3 = & \\ \\frac { 1 } { 2 } ( - 1 , 0 ) , & u _ 4 = & \\ u _ 3 \\ + \\ ( 1 , 0 ) , \\\\ u _ 5 = & \\ \\frac { 1 } { 2 } ( - d _ { 1 1 2 } - 1 , - d _ { 2 1 2 } ) , & u _ 6 = & \\ u _ 5 \\ + \\ \\left ( \\frac { 1 } { 2 } , 0 \\right ) . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} \\Phi _ { p _ k } ( \\nu ^ { m / p _ k } ) = \\dfrac { \\nu ^ { m } - 1 } { \\nu ^ { m / p _ k } - 1 } = \\dfrac { \\frac { \\nu ^ { m } - 1 } { \\nu - 1 } } { \\frac { \\nu ^ { m / p _ k } - 1 } { \\nu - 1 } } = \\dfrac { [ m ] _ \\nu } { [ m / p _ k ] _ \\nu } . \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} e ( { R } / { ( g _ 0 , \\dots , g _ { n - 2 } ) : I ^ \\infty } ) = \\sum _ { k = 1 } ^ { m } ( - 1 ) ^ { m - k } \\cdot k \\cdot f _ { m - k } + ( n - m ) . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} p _ n = \\sum _ { k \\geq \\log n } \\frac { 1 } { 2 ^ k } \\sim \\frac { 2 } { n } . \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} \\tilde { J } _ { 1 } = \\frac { p } { 2 } I + \\left ( \\frac { 2 \\sigma _ { p , q } - p } { 2 } \\right ) \\tilde { F } , \\tilde { J } _ { 2 } = \\frac { p } { 2 } I - \\left ( \\frac { 2 \\sigma _ { p , q } - p } { 2 } \\right ) \\tilde { F } . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\prod \\limits _ { j = 0 } ^ { p - 1 } { ( \\delta _ j ( X + b Y ) - \\gamma _ j Y ) } = \\prod \\limits _ { j = 0 } ^ { p - 1 } { ( \\delta _ j X - ( \\gamma _ j - b \\delta _ j ) Y ) } = \\prod \\limits _ { j = 0 } ^ { p - 1 } { ( \\delta _ j X - \\gamma _ j Y ) } . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\rho _ c = \\frac 1 2 \\sum _ { j = 1 } ^ n ( n + 1 - 2 j ) e _ j , \\qquad \\rho _ n = \\frac { n + 1 } 2 \\sum _ { j = 1 } ^ n e _ j . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\displaystyle \\frac { | \\nabla h | } { ( \\beta - h ) } ( x , t ) = \\Big ( \\frac { | \\nabla u | } { u } \\frac { 1 } { \\beta - \\ln u / D } \\Big ) ( x . t ) . \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} [ \\sigma ] _ { C ^ { \\alpha } ( B ^ g ( p , R ) ) } = \\sup \\left \\{ \\frac { | \\sigma ( x ) - \\mathbf { P } ^ g _ { x ' x } ( \\sigma ( x ' ) ) | _ { h ( x ) } } { d ^ g ( x , x ' ) ^ \\alpha } : x , x ' \\in B ^ g ( p , R ) , \\ ; x \\neq x ' , \\ ; \\mathbf { P } ^ g _ { x x ' } \\ ; \\right \\} \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} { f _ { 1 , B } } - { f _ { 1 , A } } = { f _ { 2 , B } } - { f _ { 2 , A } } = f _ B ^ { { \\rm { R F } } } - f _ A ^ { { \\rm { R F } } } \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} \\mathcal B _ 2 ^ - ( m ) : = \\{ h \\in \\mathcal B _ 2 ( m ) : \\mathrm { o r d } _ p ( d ) \\} , \\mathcal B _ 2 ^ + ( m ) : = \\{ h \\in \\mathcal B _ 2 ( m ) : \\mathrm { o r d } _ p ( d ) \\} . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} A y = u & \\Omega , \\\\ y = 0 & \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} F _ { i , j } ^ t + F _ { i , j + 1 } ^ { t + 1 } = \\max \\left [ F _ { i + 1 , j + 1 } ^ t + F _ { i - 1 , j } ^ { t + 1 } , F _ { i , j + 1 } ^ t + F _ { i , j } ^ { t + 1 } \\right ] . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\mathcal { F } ( x ) = A x ^ { 2 } + B x \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} & \\sum _ { g ^ 2 | n } \\lambda _ f \\left ( \\frac { n } { g ^ 2 } \\right ) \\sum _ { \\alpha \\beta = n / g ^ 2 } \\mu ( \\beta g ) ^ 2 F _ { \\Upsilon , M } ( \\beta g ) \\sum _ { c d = g } \\mu ( \\beta d ) = \\lambda _ f ( n ) \\sum _ { \\alpha \\beta = n } \\mu ( \\beta ) F _ { \\Upsilon , M } ( \\beta ) , \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} \\emph { V e c } ( \\nabla \\mathbf { u } ^ + ( \\widetilde { X } ) ) = M ^ - ( \\widetilde { X } ) \\emph { V e c } ( \\nabla \\mathbf { u } ^ - ( \\widetilde { X } ) ) , ~ ~ ~ ~ ~ ~ \\emph { V e c } ( \\nabla \\mathbf { u } ^ - ( \\widetilde { X } ) ) = M ^ + ( \\widetilde { X } ) \\emph { V e c } ( \\nabla \\mathbf { u } ^ + ( \\widetilde { X } ) ) . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} p _ G ^ 1 ( M ) = | G | ^ { n ^ * - g } \\sum _ { h _ 1 , \\dots , h _ g \\in G } \\bold { 1 } \\left ( \\prod _ { i = 1 } ^ { g } h _ i ^ 2 \\right ) , \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } w ( x , t ) \\ , d x = M _ 0 \\forall t \\geq 0 . \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\textrm { D e t } ( N ^ s ( \\widetilde { X } ) ) = \\textrm { D e t } ( \\overline { N } ^ s ) = \\mu ^ s ( \\lambda ^ s + 2 \\mu ^ s ) , ~ ~ s = \\pm . \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} L ( 1 / 2 , \\pi _ p \\otimes \\tau _ p \\otimes \\tau _ p ) = \\frac { 1 } { ( 1 + w _ p p ^ { - 2 } ) ( 1 + w _ p p ^ { - 1 } ) ^ 2 } = \\frac { p ^ 4 } { ( p ^ 2 + w _ p ) ( p + w _ p ) ^ 2 } , \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\sum _ { \\substack { A \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | A | = k } } \\prod _ { i \\in A } b ( i ; n ) \\leq \\frac { \\lambda _ n ^ k } { k ! } . \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } g _ M \\Phi _ N = 0 , \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} H _ n = \\displaystyle \\sum _ { l = n - N } ^ n ( l + 1 ) ^ { N - 1 } \\tbinom { N } { n - l } ( - 1 ) ^ { n - l } D ^ { ( l ) } . \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} h ( \\cdot ) = \\sqrt n \\lambda \\| \\cdot \\| _ 1 , \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\Lambda = | \\Lambda ( H ) | , \\Lambda ( H ) = \\frac { d S ( \\rho ( t ) ) } { d t } \\bigg | _ { t = 0 } . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} ( D _ { { \\bf x } , t } ^ + { \\bf u } - F _ { \\Gamma } D _ { { \\bf x } , t } ^ + { \\bf u } ) + T _ G { \\bf D } p = T _ G ( { \\bf f } ) . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} C _ r ( n ) = \\sum _ { j = r } ^ { n } { j \\choose r } ( - 1 ) ^ { n - j } ( n - r ) _ { j - r } \\equiv ( - 1 ) ^ n \\sum _ { j = r } ^ { r + d - 1 } { j \\choose r } ( - 1 ) ^ j ( n - r ) _ { j - r } \\pmod { d } . \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} \\| P _ N U ( q _ 1 ) - P _ N U ( q _ 2 ) \\| _ { \\ell ^ 2 } & \\leq C \\ , e ^ { C N ^ { \\frac { s } { d } } } \\sqrt { \\sum _ { l = 1 } ^ N \\| ( \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } ) \\psi _ 1 ^ l \\| _ { H ^ { - 1 / 2 } ( \\partial \\Omega ) } ^ 2 } \\\\ & = C \\ , e ^ { C N ^ { \\frac { 1 } { 2 } + \\alpha } } \\norm { \\left ( ( \\Lambda _ { q _ 1 } - \\Lambda _ { q _ 2 } ) \\psi _ 1 ^ l \\right ) _ { l = 1 } ^ N } _ { H ^ { - 1 / 2 } ( \\partial \\Omega ) ^ N } , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} { \\theta _ { 1 , B } } \\left ( t \\right ) = 2 \\pi \\left ( { f ' + f '' } \\right ) t + { \\varphi ' _ n } + { \\varphi '' _ n } \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} K : = \\left \\{ \\overline { x y } \\subset \\Omega ; \\ x , y \\in \\partial \\Omega , x \\neq y , | u _ \\rho ( x ) - u _ \\rho ( y ) | = | x - y | \\right \\} , \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\Big | \\sum _ { n _ 1 = M _ 0 } ^ { M _ 1 } A _ { n _ 1 , \\tilde { n } } \\big ( S _ { n _ 1 , \\tilde { n } } ^ { ( r ) } - S _ { n _ 1 - 1 , \\tilde { n } } ^ { ( r ) } \\big ) \\Big | \\lesssim Q ^ { - 1 } N ( \\log N ) ^ { - \\alpha } . \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} & \\min \\xi ( x ) \\\\ \\hbox { s u b j e c t t o } & g ( y ) \\leq 0 , \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | m _ { n + 1 } - m _ n \\big | \\Big \\| _ { \\ell ^ 1 } \\leq \\bigg ( \\frac { 1 } { \\vartheta _ B ( N _ 1 ) } - \\frac { 1 } { \\vartheta _ B ( N _ 2 ) } \\bigg ) \\vartheta _ B ( N _ 1 ) + \\frac { 2 } { \\vartheta _ B ( N _ 1 ) } \\big ( \\vartheta _ B ( N _ 2 ) - \\vartheta _ B ( N _ 1 ) \\big ) , \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} \\mathcal { T S } ^ { o p } & = \\{ \\pi ^ * ( B _ 1 ) , \\ldots , \\pi ^ * ( B _ { k - 1 } ) - E , E , \\pi ^ * ( B _ { k } ) - E , \\ldots , \\pi ^ * ( B _ n ) \\} \\\\ & = \\{ \\pi ^ * ( A _ { n - 1 } ) , \\ldots , \\pi ^ * ( A _ { n - k + 1 } ) - E , E , \\pi ^ * ( A _ { n - k } ) - E , \\ldots , \\pi ^ * ( A _ 1 ) , \\pi ^ * ( A _ n ) \\} \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\begin{aligned} \\left | a \\nabla J _ { \\delta } ^ { 0 } u \\right | = & \\left | a \\nabla \\int _ { 0 } ^ { \\infty } e ^ { - t } P _ { \\delta t } ^ { 0 } u \\ , d t \\right | = \\left | a \\nabla \\int _ { 0 } ^ { \\infty } e ^ { - t + 2 K \\delta t } P _ { \\delta t } ^ { a } u \\ , d t \\right | \\\\ \\le & \\int _ { 0 } ^ { \\infty } e ^ { - t + 2 K \\delta t } \\left | a \\nabla P _ { \\delta t } ^ { a } u \\right | \\ , d t \\le \\int _ { 0 } ^ { \\infty } e ^ { - t } P _ { \\delta t } ^ { a } \\left | a \\nabla u \\right | \\ , d t = J _ { \\delta } ^ { a } | a \\nabla u | , \\end{aligned} \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\theta _ i ' = \\omega _ i + \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\sum _ { p = 1 } ^ \\infty a _ p \\sin ( p ( \\theta _ j - \\theta _ i ) ) . \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} \\widetilde { P _ 2 } ( x ) = \\int _ 0 ^ x P _ 2 ( u ) d u . \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ( v ) = 0 F _ j ' ( v ) - F _ k ' ( v ) = C _ v ( F _ j ( v ) - F _ k ( v ) ) \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} p _ n ( x ) = \\begin{cases} \\frac { - i } { 2 \\pi } L ^ { \\frac { n + 1 } { 2 } } ( f _ 2 - f _ 1 ) ( x ) & \\textnormal { i f } n \\textnormal { i s o d d } \\\\ 0 & \\textnormal { i f } n \\textnormal { i s e v e n . } \\end{cases} \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} \\varepsilon ^ { 2 s } ( - \\Delta ) ^ s u + V ( x ) u = f ( x , u ) , \\ \\ \\ \\ x \\in \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{gather*} \\theta _ 3 ^ 4 = \\theta _ 2 ^ 4 + \\theta _ 4 ^ 4 . \\end{gather*}"}
-{"id": "6252.png", "formula": "\\begin{align*} 2 \\cdot ( n - 1 ) + 1 = 2 n - 2 + 1 = 2 n - 1 \\geq n + d - 1 \\enspace . \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} h ( R ) & = 1 & h ' ( R ) & = 0 & \\Delta h ( R ) & = 0 ; & ( \\Delta h ) ' ( R ) & = 0 , \\\\ \\Delta ^ 2 h ( R ) & = 0 , & ( \\Delta ^ 2 h ) ' ( R ) & = 0 , & \\cdots \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} p _ { s } ^ { \\textrm { o p t } } = \\frac { \\theta \\xi P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } } { 1 - \\theta } , \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} T ^ { { A } _ { ( L ) } } _ { i } = \\max \\{ T ^ { u ' } _ { i } , T ^ { d ' } _ { i } , T ^ { a ' } _ { i } \\} , \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} [ h \\alpha _ n , \\epsilon ] = \\left [ \\left ( \\begin{array} { c c } d ^ { - 1 } p ^ n & \\ast \\\\ 0 & d p ^ { - n } \\end{array} \\right ) , e \\right ] \\left [ \\left ( \\begin{array} { c c } 1 & 0 \\\\ c d ^ { - 1 } p ^ { 2 n } & 1 \\end{array} \\right ) , 1 \\right ] , \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} \\frac { \\beta ( \\ell _ j , s ) } { \\beta ( k - j , s ) } = \\prod _ { i = 0 } ^ { m - 1 } \\frac { t - \\frac { \\ell _ j } 2 - m + i } { t + \\frac { \\ell _ j } 2 + m - 1 - i } . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } | \\bar { u } _ t | _ { L ^ 2 ( U ) } ^ 2 + \\int _ 0 ^ T | \\nabla \\bar { u } _ r | _ { L ^ 2 ( U ) } ^ 2 d r & = \\sup _ { t \\in [ 0 , T ] } | u _ t | _ { L ^ 2 ( U ) } ^ 2 + \\int _ 0 ^ T | \\nabla u _ r | ^ 2 _ { L ^ 2 ( U ) } d r \\\\ & \\leq \\sup _ { t \\in [ 0 , T ] } | u _ t | ^ 2 _ { L ^ 2 ( \\R ^ d ) } + \\int _ 0 ^ T | \\nabla u _ r | ^ 2 _ { L ^ 2 ( \\R ^ d ) } d r , \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} r _ { f _ 1 , f _ 2 , \\frac { d } { c } } ( \\tau ) & : = \\int _ { \\frac { d } { c } } ^ { i \\infty } \\int _ { w _ 1 } ^ { \\frac { d } { c } } \\frac { f _ 1 ( w _ 1 ) f _ 2 ( w _ 2 ) } { ( - i ( w _ 1 + \\tau ) ) ^ { 2 - \\kappa _ 1 } ( - i ( w _ 2 + \\tau ) ) ^ { 2 - \\kappa _ 2 } } d w _ 2 d w _ 1 , \\\\ r _ { f _ 1 , \\frac d c } ( \\tau ) & : = \\int _ { \\frac d c } ^ { i \\infty } \\frac { f _ 1 ( w ) } { \\left ( - i ( w + \\tau ) \\right ) ^ { 2 - \\kappa _ 1 } } d w . \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} \\lim \\limits _ { h \\rightarrow + \\infty } | R ^ { \\bar { \\varphi } ( o ) , h } - \\max \\limits _ { \\phi \\in \\Phi } R ^ { \\phi , h } | = 0 . \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} \\mathcal F & \\ll ( 1 + | t | ) ^ B q ^ { - 1 / 2 + \\varepsilon } . \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} \\big \\langle \\Lambda _ v P _ { v , \\rm R } F ( v ) , P _ { v , \\rm R } F ( v ) \\big \\rangle & = \\frac { 1 } { D _ v } \\bigg ( \\sum _ { j = 1 } ^ { \\deg ( v ) } | F _ j ( v ) | ^ 2 - \\frac { 1 } { \\deg ( v ) } \\bigg | \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ( v ) \\bigg | ^ 2 \\bigg ) . \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} \\norm { ( 1 - \\lambda ) a + \\lambda b } ^ 2 = ( 1 - \\lambda ) \\norm { a } ^ 2 + \\lambda \\norm { b } ^ 2 - \\lambda ( 1 - \\lambda ) \\norm { a - b } ^ 2 \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} & \\mathcal { C } _ { 2 1 } ' : = \\left [ \\frac { 2 A + 1 } { \\log q } - i q , \\alpha _ 0 - i q \\right ] \\cup \\left [ \\alpha _ 0 - i q , \\alpha _ 0 + i q \\right ] \\cup \\left [ \\alpha _ 0 + i q , \\frac { 2 A + 1 } { \\log q } + i q \\right ] \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} r ( s , u ) = p ^ { | s - u | } , 0 \\leq s , u \\leq 1 , \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} 0 + _ r 1 = \\begin{cases} 0 & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k } \\binom { r p + \\sigma - i } { k - i } A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) = \\sum _ { j = 0 } ^ { k } \\left ( - 1 \\right ) ^ { j } \\binom { k } { j } \\binom { \\left ( k - j \\right ) a + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\left ( k - j \\right ) \\alpha _ { s } + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} \\Psi _ { k } ^ { ( N ) } ( f ) = \\Phi _ { k } ^ { ( N ) } ( f ) - 2 \\mathbb { E } \\left [ f \\mid \\mathcal { A } ^ { ( k ) } \\right ] + 2 \\sum _ { k < l \\leqslant N } \\mathbf { P } _ { \\mathcal { A } ^ { ( l ) } | \\mathcal { A } ^ { ( k ) } } \\cdot \\Phi _ { l } ^ { ( N ) } ( f ) = - \\Phi _ { k } ^ { ( N ) } ( f ) . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} | B _ \\delta | ( 1 + \\delta ) \\leq \\int _ { B _ \\delta } \\frac { | \\sigma | _ r } { \\alpha _ 0 } d x d t \\leq \\liminf _ { N \\rightarrow \\infty } \\int _ { B _ \\delta } \\frac { | \\sigma ^ N | _ r } { \\alpha _ 0 } d x d t \\leq \\liminf _ { N \\rightarrow \\infty } \\Big ( \\frac { C } { \\alpha _ 0 } \\Big ) ^ { \\frac { 1 } { N } } | B _ \\delta | ^ { \\frac { N - 1 } { N } } = | B _ \\delta | . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} c _ \\delta ( x \\cup \\chi ) = - ( c _ \\delta x ) \\cup \\chi \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} M _ s ( t ) ( \\omega ) = & \\sigma ( s , \\omega , X ( s ) ( \\omega ) ) + \\int _ s ^ t U ( s , r , \\omega ) d r + \\int _ s ^ t V ( s , r , \\omega ) d W ( r ) \\\\ & + \\int _ s ^ t \\nabla _ x b ( r , \\omega , X ( r ) ( \\omega ) ) M _ s ( r ) ( \\omega ) d r + \\int _ s ^ t \\nabla _ x \\sigma ( r , \\omega , X ( r ) ( \\omega ) ) M _ s ( r ) ( \\omega ) d W ( r ) , \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} \\psi ( g ^ p ) = \\big ( ( a ^ { \\alpha } b ) ^ { k i } w _ 1 , ( a ^ { \\alpha } b ) ^ { k i } w _ 2 , \\dots ( a ^ { \\alpha } b ) ^ { k i } w _ p \\big ) , \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ L ^ { q / \\tau } \\bigl ( \\frac { 1 } { \\tau } \\bigr ) \\Bigr ] & = { \\bf E } \\Bigl [ e ^ { ( q / \\tau ) \\mathcal { N } ( 0 , \\ , 4 \\tau \\log 2 ) } \\Bigr ] , \\\\ & = e ^ { 4 q ^ 2 \\log 2 / 2 \\tau } \\equiv { \\bf E } \\bigl [ L ^ { q } ( \\tau ) \\bigr ] . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\sum _ n \\frac { \\overline { \\chi } ( n ) } { n ^ { 1 / 2 } } V \\left ( \\frac { T n } { q } \\right ) & = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 ) } \\frac { \\Gamma ^ 2 \\left ( \\frac { s } { 2 } + \\frac { 1 } { 4 } \\right ) } { \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } \\frac { G ( s ) } { s } \\left ( \\frac { q } { \\pi } \\right ) ^ s T ^ { - s } L \\left ( \\frac { 1 } { 2 } + s , \\overline { \\chi } \\right ) d s . \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} B ( u , v ) = \\frac { 1 } { 2 } [ Q ( v + u ) - Q ( u ) - Q ( v ) ] , \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} A _ x T _ u + T _ x A _ u & = ~ T _ x P T _ u - P T _ x T _ u + T _ x T _ u P - T _ x P T _ u ~ = ~ T _ x T _ u P - P T _ x T _ u \\\\ & = ~ T _ { x \\star u } P - P T _ { x \\star u } ~ = ~ A _ { x \\star u } \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} { { \\rm { E } } _ q ^ x } = \\left ( { 1 + \\left ( 1 - q \\right ) x } \\right ) _ q ^ \\infty . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} ( h \\cdot f ) ( x ) : = f ( S ^ { - 1 } ( h ) x ) , \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} E ( t ) : = \\begin{bmatrix} V & 0 \\\\ 0 & I \\end{bmatrix} e ^ { - t J } \\begin{bmatrix} e & 0 \\\\ 0 & 0 \\end{bmatrix} e ^ { t J } \\begin{bmatrix} V ^ { - 1 } & 0 \\\\ 0 & I \\end{bmatrix} , t \\in [ 0 , \\frac \\pi 2 ] , \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} M ' = \\begin{bmatrix} x & y ^ 2 \\\\ y & z ^ 2 \\\\ z & x ^ 2 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} & E _ 1 ( x , \\zeta ) = e ^ { i ( x _ 1 \\zeta _ 1 + x _ 2 \\zeta _ 2 - x _ 0 ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) ) } = e ^ { i ( x _ 1 \\zeta _ 1 + x _ 2 \\zeta _ 2 ) } e ^ { - i x _ 0 ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) } . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} c _ 1 \\textbf { E } ^ 1 _ { \\overline { \\alpha , 1 } , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x ( a ) + c _ 2 ~ ^ { A B R } \\nabla _ b ^ \\alpha x ( a ) = 0 , \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} & D ^ u _ { i } + D ^ d _ { i } + \\sum _ { j = 1 } ^ M ( T ^ { a c } _ { i j } + T ^ { c } _ { i j } ) x _ { i j } \\leq t _ i , \\ \\forall i , \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} T _ 1 ( s , f ) = \\sum _ { M ^ { 1 - \\Upsilon } < n \\leq X ( \\log q ) ^ 2 } \\frac { \\lambda _ f ( n ) c ( n ) } { n ^ s } e ^ { - n / X } + \\mathcal { O } \\Big ( q ^ { - B } \\Big ) . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + { \\rm c o n s t } ) } \\ , { \\bf E } \\bigl [ M ^ q _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\bigr ] , \\ ; N \\rightarrow \\infty . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} E _ { 3 } & : y ^ 2 + w x y + v y = x ^ 3 , \\\\ E ' _ { 3 } & : y ^ 2 + w x y + v y = x ^ 3 - 5 w v x - v ( w ^ 3 + 7 v ) . \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\tilde { w } _ t - r _ t = \\int _ 0 ^ t Q _ { \\alpha _ 2 \\alpha ' } ( t - s ; B _ { 2 , \\upsilon } ^ \\Delta ) D _ { \\alpha ' \\alpha } r _ s d s , \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} \\begin{array} { l l } l _ { k } & = \\Bigl [ | U _ * | ^ { 2 ^ { * } - 2 } U _ * - u _ 0 ^ { 2 ^ * - 1 } + \\sum _ { j = 1 } ^ { k } P _ j ^ { 2 ^ { * } - 1 } \\Bigr ] + \\sum _ { j = 1 } ^ { k } \\bigl ( U _ j ^ { 2 ^ * - 1 } - P _ j ^ { 2 ^ { * } - 1 } \\bigr ) \\\\ & = : J _ 1 + J _ 2 . \\end{array} \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\ddot \\rho - L ( \\dot \\rho ) L ( \\rho ) ^ { \\dagger } \\dot \\rho + \\frac { 1 } { 2 } L ( \\rho ) ( \\nabla _ G L ( \\rho ) ^ { \\dagger } \\dot \\rho \\circ \\nabla _ G L ( \\rho ) ^ { \\dagger } \\dot \\rho ) = - d _ \\rho \\mathcal { F } ( \\rho ) . \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} \\left ( B _ n ( m ; \\cdot ) \\times E _ n ( r , \\beta ; \\cdot ) \\right ) ( x ) = \\sum _ { k = 0 } ^ n \\dfrac { ( a ( r ) \\times b ( m ) ) _ k } { k ! } \\Delta ^ k I _ n ( x ) . \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} \\bigg | \\frac { f ( z ) } { \\phi _ t ' ( z ) } \\bigg | \\le M \\begin{cases} \\exp { - \\rho | \\phi _ t ( z ) | } \\forall z \\in \\psi _ t \\big ( ( - \\infty , 0 ) \\big ) \\\\ \\exp { - \\mu | \\phi _ t ( z ) | } \\forall z \\in \\psi _ t \\big ( [ 0 , - \\infty ) \\big ) \\end{cases} . \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} P _ { + } ^ { k } = \\{ \\Lambda \\in P _ { + } | ( \\theta | \\Lambda ) \\le k \\} . \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} G _ 2 ( v , c ) : = a [ - \\frac { A } { \\alpha } + \\frac { \\beta } { \\alpha } \\psi ( c ) + \\int _ 0 ^ 1 \\psi ( \\int _ 0 ^ \\tau v ( s ) \\ , d s + c ) \\ , d \\tau ) ] + \\psi ( \\int _ 0 ^ 1 v ( s ) \\ , d s + c ) \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} G ( z ) \\triangleq G ( z \\ , | \\ , \\tau = 1 ) . \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} S _ r = \\begin{bmatrix} ( q _ i + p _ j ) ^ r \\end{bmatrix} . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} \\left ( \\sum _ { i \\geq 0 } \\alpha _ i t ^ { i + 1 } \\right ) \\circ \\left ( \\sum _ { j \\geq 0 } \\beta _ j t ^ { j + 1 } \\right ) : = \\sum _ { i \\geq 0 } \\alpha _ i \\left ( \\sum _ { j \\geq 0 } \\beta _ j t ^ { j + 1 } \\right ) ^ { i + 1 } . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} S ( T \\acute { N } ) = \\mu _ { 0 } \\bot \\{ \\tilde { J } ( R a d T \\acute { N } ) \\oplus \\tilde { J } ( l t r ( T \\acute { N } ) ) \\} , \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\tilde { J } U = \\varphi U + u ( U ) N , \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha \\left ( \\tau + t \\right ) ^ { - \\alpha - 1 } \\left [ \\log \\left ( t + \\tau \\right ) \\right ] ^ { \\frac { \\beta } { m - 1 } } \\ge C ^ { p - 1 } \\left ( t + \\tau \\right ) ^ { - \\alpha p } \\left [ \\log ( \\tau + t ) \\right ] ^ { \\frac { \\beta p } { m - 1 } } t > 0 \\ , , \\end{aligned} \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} S _ 1 = \\{ ( 0 1 2 3 4 5 6 ) , ( 0 2 4 ) ( 1 6 5 3 ) \\} \\mbox { a n d } S _ 2 = \\{ ( 0 1 2 3 4 ) ( 5 6 ) , ( 0 2 4 5 3 1 6 ) \\} . \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} _ { \\mathcal F } ( E _ H ) \\ , : = \\ , ( \\widehat { \\mathrm { d } } p ) ^ { - 1 } ( { \\mathcal F } ) \\ , \\subset \\ , ( E _ H ) \\ , . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\mathcal { D } ^ { \\Delta , \\sigma } _ { \\alpha _ 2 , u } = \\{ u \\in \\mathcal { U } _ { \\sigma , \\alpha _ 2 } : \\Psi _ \\upsilon u \\in \\mathcal { U } _ { \\sigma , \\alpha _ 2 } \\} \\subset \\mathcal { D } ^ \\Delta _ { \\alpha _ 2 } , \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} \\lvert B ( \\pi ) \\rvert = \\lvert A ( \\pi ) \\setminus A ( e ) \\rvert = w _ B ( \\pi ) . \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { \\abs { \\sigma } \\leq k } a _ \\sigma x ^ \\sigma + O ( \\abs { x } ^ { k + 1 } ) , \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} [ X \\xleftarrow { p _ 1 } V _ 1 \\xrightarrow { s _ 1 } Y ; E _ 1 ] + [ X \\xleftarrow { p _ 2 } V _ 2 & \\xrightarrow { s _ 2 } Y ; E _ 2 ] \\\\ & : = [ X \\xleftarrow { p _ 1 + p _ 2 } V _ 1 \\sqcup V _ 2 \\xrightarrow { s _ 1 + s _ 2 } Y ; E _ 1 + E _ 2 ] , \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} & [ X \\xleftarrow { f } M \\xrightarrow { g } Y ] \\circ \\Bigl ( [ Y \\xleftarrow { h _ 1 } N _ 1 \\xrightarrow { k _ 1 } Z ] + [ Y \\xleftarrow { h _ 2 } N _ 2 \\xrightarrow { k _ 2 } Z ] \\Bigr ) \\\\ & = [ X \\xleftarrow { f } M \\xrightarrow { g } Y ] \\circ [ Y \\xleftarrow { h _ 1 } N _ 1 \\xrightarrow { k _ 1 } Z ] + [ X \\xleftarrow { f } M \\xrightarrow { g } Y ] \\circ [ Y \\xleftarrow { h _ 2 } N _ 2 \\xrightarrow { k _ 2 } Z ] . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} \\phi _ \\pm ( x ) = n _ { 0 } ^ e ( x ) \\mp i D ^ { - 1 } n _ { 1 } ^ e ( x ) \\in H ^ { s _ 1 } ( \\R ) . \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ { 1 } = \\frac { - ( 1 - 3 d D ) + \\sqrt { ( 1 - d D ) ^ { 2 } + 2 a d ^ { 2 } C } } { 2 d C } , \\\\ \\lambda _ { 2 } = \\frac { - ( 1 - 3 d D ) - \\sqrt { ( 1 - d D ) ^ { 2 } + 2 a d ^ { 2 } C } } { 2 d C } . \\end{cases} \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} \\| { \\bf { w } } _ { k } \\| _ { 1 \\times 1 } = O ( \\epsilon ) . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\norm { x _ i - x _ j } ^ 2 & = \\norm { x _ 0 - \\sum _ { k = 0 } ^ { i - 1 } h _ { i , k } g _ k + v _ i - x _ 0 + \\sum _ { k = 0 } ^ { j - 1 } h _ { j , k } g _ k - v _ j } ^ 2 \\\\ & = \\norm { \\hat x _ i - \\hat x _ j } ^ 2 + \\norm { v _ i - v _ j } ^ 2 \\\\ & \\geq \\norm { \\hat x _ i - \\hat x _ j } ^ 2 , { i , j = 0 , \\hdots , N } , \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} \\Lambda _ 1 = \\frac { \\theta ^ { n - 1 } } { 1 + ( 1 - 2 R ) q ^ { n - 1 } } \\frac { 1 + q ^ { n - 1 } } { 1 - q ^ { n - 1 } } , \\Lambda _ 2 = \\frac { q ^ { n - 1 } } { 1 + ( 1 - 2 R ) q ^ { n - 1 } } . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} P ( \\lambda ) = \\sum _ { i = 0 } ^ d P _ i \\lambda ^ i , \\mbox { w i t h } P _ 0 , \\hdots , P _ d \\in \\mathbb { F } ^ { m \\times n } , \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} ( [ a ] \\cup [ b ] ) \\cup [ c ] = [ a ] \\cup ( [ b ] \\cup [ c ] ) + ( - 1 ) ^ { q r } [ a ] \\cup ( [ c ] \\cup [ b ] ) . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} \\begin{array} { l l } \\psi ( t , x ) = - 2 \\mu ^ { - 1 / 2 } ~ \\partial _ x \\log u ( t , x ) , \\end{array} \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} A ^ T Q + Q A + \\sum _ { k = 1 } ^ m N _ k ^ T Q N _ k + \\sum _ { i , j = 1 } ^ v H _ i ^ T Q H _ j k _ { i j } = - C ^ T C . \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} & \\Bigl ( [ X \\xleftarrow { f _ 1 } M _ 1 \\xrightarrow { g _ 1 } Y ] + [ X \\xleftarrow { f _ 2 } M _ 2 \\xrightarrow { g _ 2 } Y ] \\Bigr ) \\circ [ Y \\xleftarrow h N \\xrightarrow k Z ] \\\\ & = [ X \\xleftarrow { f _ 1 } M _ 1 \\xrightarrow { g _ 1 } Y ] \\circ [ Y \\xleftarrow h N \\xrightarrow k Z ] + [ X \\xleftarrow { f _ 2 } M _ 2 \\xrightarrow { g _ 2 } Y ] \\circ [ Y \\xleftarrow h N \\xrightarrow k Z ] , \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} J ( p ) : = p p ^ T \\ ; \\big ( ( p ^ T ) _ { i j } = p _ { j i } \\big ) , \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} \\operatorname { T r } ( r _ { ( \\alpha , \\alpha - 1 , - 1 ) } r _ { ( - \\alpha , \\alpha - 1 , 1 ) } ) & = \\operatorname { T r } ( r _ { ( \\alpha , \\alpha - 1 , - 1 , 0 ) } r _ { ( - \\alpha , \\alpha - 1 , 1 , 0 ) } ) \\\\ & = \\alpha - 2 . \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} A ' _ { i , j } = \\Delta _ { I ' ( i , j ) } = C _ a ^ * ( A _ { i , j } ) . \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} r _ i ( x ) = \\Re \\left [ a _ { t _ i i } ( x ) ^ { \\left ( \\frac { 2 \\ell _ i n _ i } { m _ i } - n _ i + \\frac { 2 n _ i ( m _ i + t _ i ) - 2 t _ i } { m _ i } \\right ) } h _ { k _ i i } ( x ) ^ { - 2 } \\exp \\int \\frac { 2 n _ i } { m _ i } \\frac { a _ { ( t _ i - 1 ) i } ( x ) } { a _ { t _ i i } ( x ) } d x \\right ] \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\frac { e ^ { - z - s r _ i } - 1 + y / \\tilde { y } } { e ^ { - z - s r _ j } - 1 + y / \\tilde { y } } = e ^ { s \\tilde { r _ { i j } } } \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} ( N | A ) ^ \\sqsubset : = W | D \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} [ { M _ \\epsilon } ^ { - 1 } \\Gamma _ { 0 } { M _ \\epsilon } , { M _ { \\epsilon ' } } ^ { - 1 } \\Gamma _ { 0 } { M _ { \\epsilon ' } } ] = 1 . \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} \\frac { N _ { \\mathrm { c } m } } { ( 2 m ) ! ! } = \\frac { 1 } { 2 ^ { n + 1 } } \\sum _ { i = 0 } ^ { m } ( - 1 ) ^ { i } \\sum _ { a _ { 1 } , \\cdots , a _ { i + 1 } = 1 } ^ { \\infty } \\delta _ { a _ { 1 } + \\cdots + a _ { i + 1 } , m + 1 } \\prod _ { j = 1 } ^ { i + 1 } \\frac { ( 2 a _ { j } ) ! } { a _ { j } ! } , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} \\underbrace { f \\circ \\pi _ J ( a ) } _ { = \\rho ( a ) } = f \\circ \\pi _ { J , k } ( x ) \\xrightarrow [ x \\to \\pi _ k ( c ) ] { } f \\circ \\pi _ { J , k } \\big ( \\pi _ k ( c ) \\big ) = \\underbrace { f \\circ \\pi _ J ( c ) } _ { = \\rho ( c ) = c } . \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} R _ { e } = \\max _ { k \\in [ 1 , K ] } R _ { e , k } = \\log ( 1 + \\rho _ { e } ) , \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{gather*} L \\left ( \\psi _ v , s - \\frac 1 2 \\right ) = \\frac { 1 } { 1 - ( - 1 ) N v ^ { 1 / 2 - s } } = \\frac { 1 } { 1 - ( - p ) p ^ { - 2 s } } . \\end{gather*}"}
-{"id": "2252.png", "formula": "\\begin{align*} \\mathcal { C } _ { 2 1 } : = \\left [ \\frac { 2 A + 1 } { \\log q } - i q , - \\frac { 1 } { 2 } - i q \\right ] \\cup \\left [ - \\frac { 1 } { 2 } - i q , - \\frac { 1 } { 2 } + i q \\right ] \\cup \\left [ - \\frac { 1 } { 2 } + i q , \\frac { 2 A + 1 } { \\log q } + i q \\right ] , \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} M : = { \\rm S y m } _ + ( r ) \\times { \\bf R } ^ { r \\times m } \\times { \\bf R } ^ { p \\times r } . \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} \\left ( { 1 + x } \\right ) _ q ^ n = \\prod \\limits _ { j = 0 } ^ { \\infty } { \\left ( { 1 + q ^ j x } \\right ) } . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} d \\breve { s } ^ { 2 } & = 2 T ^ { h } d t ^ { 2 } = 2 ( T - T ^ { \\omega } ) d t ^ { 2 } = 2 ( T - \\frac { \\omega ^ { 2 } } { 2 \\rho ^ { 2 } } ) d t ^ { 2 } \\\\ & = d \\bar { s } ^ { 2 } = 2 \\bar { T } d t ^ { 2 } = d \\rho ^ { 2 } + \\frac { \\rho ^ { 2 } } { 4 } ( d \\varphi ^ { 2 } + \\sin ^ { 2 } \\varphi d \\theta ^ { 2 } ) \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} I ( \\vec y ) = ( \\vec y , \\vec s ) , \\ , \\ , \\ , \\forall \\ , \\vec s \\in \\partial I ( \\vec y ) , \\ , \\ , \\ , \\forall \\ , \\vec y \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} \\hat { \\Lambda } _ { f , c } ( x , k ) = \\hat { u } ( x , k ) , \\ \\ \\ \\ ( x , k ) \\in \\R ^ 3 \\times ( 0 , \\epsilon ) . \\\\ \\\\ \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} \\left [ y ( y ^ { \\alpha } x ) ^ { - 1 } \\right ] x = y ( y ^ { \\alpha } x ) ^ { - 1 } x \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} \\mathcal { D } & = \\big \\{ g ( \\langle \\cdot , y ^ * \\rangle ) \\colon g \\in C ^ 2 _ c ( \\mathbb { R } ) , y ^ * \\in D ( A ^ * ) \\big \\} , \\\\ \\mathcal { B } & = \\big \\{ g ( \\langle \\cdot , y ^ * \\rangle ) \\colon g \\in C ^ 2 _ b ( \\mathbb { R } ) , y ^ * \\in D ( A ^ * ) \\big \\} . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} \\begin{bmatrix} I _ { r } & 0 \\\\ 0 & B \\end{bmatrix} \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\int _ A e ^ { - 2 \\pi i \\langle n , x \\rangle } \\ , \\textrm { d } x = \\sum _ { \\substack { ( F _ { d - 1 } , F _ { d - 2 } , \\dots , F _ k ) \\\\ \\textrm { r e l e v a n t f l a g s } } } C _ n ( F _ { d - 1 } , F _ { d - 2 } , \\dots , F _ k ) \\int _ { \\pi _ k ( F _ k ) } e ^ { - 2 \\pi i \\langle \\pi _ k ( n ) , x \\rangle } \\ , \\textrm { d } x \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} \\gamma ( W _ 1 \\otimes \\cdots \\otimes W _ 2 ) = \\gamma _ 1 ( W _ 1 ) \\cdots \\gamma _ N ( W _ N ) . \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} \\lbrace { \\tilde z _ 1 } , { \\tilde z _ 3 } \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace { \\tilde z _ 2 } , { \\tilde z _ 4 } \\rbrace = 1 . \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} \\Phi _ x ( \\alpha ; Z ) : = \\int _ { \\Delta ^ { ( x ) } } \\varphi _ x ( \\alpha ; Z ; u ) _ x u = \\Phi ( ( \\alpha _ { e } ) _ { \\underline { e } = x } , ( \\alpha _ e ) _ { \\overline { e } = x } ; ( Z _ { e , e ' } ) _ { \\overline { e } = x = \\underline { e } ' } ) , \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} ( a , e _ A ) \\circledcirc ( c , x ) = ( a \\circ c , x ) , \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\mathbb { P } [ \\mathbf { D } \\mathbf { x } = \\mathbf { e } _ i | w t ( \\mathbf { x } ) = k ] & = \\mathbb { P } [ \\sum _ { j = 1 } ^ k d ^ { e _ j } _ { i } = 1 ] \\prod _ { \\substack { l ' = 1 \\\\ l ' \\neq i } } ^ m \\mathbb { P } [ \\sum _ { j = 1 } ^ k d ^ { e _ j } _ { l ' } = 0 ] . \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\lbrace z _ 1 , z _ 3 \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace z _ 2 , z _ 4 \\rbrace = 1 . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} \\partial ^ \\sigma _ \\xi u ( \\xi ) = \\sum _ { 2 j + \\frac { k } { \\beta } < q } \\rho ^ { 2 j + \\frac { k } { \\beta } } \\left ( a _ { j , k , \\sigma } ( \\xi ) \\cos k \\theta + b _ { j , k , \\sigma } ( \\xi ) \\sin k \\theta \\right ) + O ( \\rho ^ q ) \\rho < 1 / 2 . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\chi ( x , y ) = ( - 1 ) ^ { d ( x , y ) } \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\ell ^ * = \\ell ^ * ( \\hat { T } ) = \\pi \\sqrt { \\frac { \\hat { T } } { 2 P e } } , \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} C ^ { - 1 } & = \\frac { 1 } { 3 } \\begin{pmatrix} 2 & 1 \\\\ 1 & 2 \\end{pmatrix} & C ( z ) & = \\begin{pmatrix} z + z ^ { - 1 } & - 1 \\\\ - 1 & z + z ^ { - 1 } \\end{pmatrix} & \\widetilde { C } ( z ) & = \\frac { 1 } { z ^ { 2 } + 1 + z ^ { - 2 } } \\begin{pmatrix} z + z ^ { - 1 } & 1 \\\\ 1 & z + z ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} \\mathrm { G S p } _ 2 ^ + ( \\R ) : = \\{ g \\in \\mathrm M _ { 4 } ( \\R ) : g J _ 2 { } ^ t g = \\nu ( g ) J _ 2 , \\ , \\nu ( g ) > 0 \\} , J _ 2 = \\begin{pmatrix} 0 & \\mathrm { I d } _ 2 \\\\ - \\mathrm { I d } _ 2 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} W ^ I = \\{ x \\in W : \\ \\ell ( x w ) = \\ell ( x ) + \\ell ( w ) \\ w \\in W _ I \\} = \\{ x \\in W : \\ x ( \\Phi _ I ^ + ) \\subset \\Phi ^ + \\} . \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} e ( M _ { 1 } , M _ { 1 } ) & = ( 2 p ^ { 3 } - 3 p + 1 ) p ^ { 2 } , \\\\ e ( C _ { p } ^ { 3 } , M _ { 1 } ) & = ( p ^ { 3 } - 1 ) ( p ^ { 2 } + p - 1 ) p ^ { 2 } , \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ( t ) = & \\ , \\alpha ( T - t ) ^ { - \\alpha - 1 } + \\frac { C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } \\ , ( T - t ) ^ { - \\alpha m + \\beta } - \\frac { \\beta } { m - 1 } \\ , ( T - t ) ^ { - \\alpha - 1 } \\\\ \\leq & \\ , \\frac { 2 C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } \\ , ( T - t ) ^ { - \\alpha m + \\beta } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} ( ( f \\circ g ) ( T _ { \\lambda , \\mu } ) ) ^ i = \\{ k + d | k \\in ( g ( T _ { \\mu } ) ) ^ { i - n } \\} \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} k ! _ { \\underline { S } } = \\operatorname * { l c m } \\limits _ { \\vert \\mathbf { i } \\vert = k } \\mathbf { i } ! _ { \\underline { S } } . \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} \\left [ \\sum _ k \\binom { L } { k } \\binom { N + k - 1 } { n } \\right ] \\left [ \\sum _ k \\binom { L } { k } \\binom { N + k - 1 } { n + 1 } \\right ] , \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\Lambda _ z ( x ) = \\int _ z ^ x e ^ { 2 \\int _ 0 ^ y V ( \\xi ) d \\xi } d y , ~ ~ ~ ~ \\forall x \\geq z \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} | u y _ 0 - y _ 0 | & \\leqslant | u y _ 0 - u x _ 0 | + | u x _ 0 - x _ 0 | + | x _ 0 - y _ 0 | \\\\ & = 2 | x _ 0 - y _ 0 | + | u x _ 0 - x _ 0 | \\leqslant | u x _ 0 - x _ 0 | + 2 0 0 0 d \\delta . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\begin{aligned} W ^ { - } ( n , \\phi ) = \\textnormal { s u p } C ^ { - } ( \\phi ^ { n } ) \\\\ W ^ { + } ( n , \\phi ) = \\textnormal { i n f } C ^ { + } ( \\phi ^ { n } ) . \\end{aligned} \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} A - A ^ { - 1 } = B ^ { - 1 } - B . \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{gather*} ( Q m _ { \\langle 0 \\rangle } ) _ { \\langle 0 \\rangle } \\otimes ( Q m _ { \\langle 0 \\rangle } ) _ { \\langle 1 \\rangle } S ^ 2 ( P ) \\otimes R m _ { \\langle 1 \\rangle } \\\\ \\qquad { } = P ( R m ) _ { \\langle 0 \\rangle } \\otimes Q ( ( R m ) _ { \\langle 1 \\rangle } ) ^ 1 S ^ 2 ( P ) \\otimes R ( ( R m ) _ { \\langle 1 \\rangle } ) ^ 2 S ^ 2 ( Q ) . \\end{gather*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\mathbb { P } [ \\exists \\mathbf { x } \\in \\mathcal { F } ^ l ~ \\textrm { s . t . } ~ \\mathbf { D } \\mathbf { x } = \\mathbf { e } _ i ] = \\mathbb { P } [ Y _ i > 0 ] \\ge \\frac { \\mathbb { E } ^ 2 [ Y _ i ] } { \\mathbb { E } [ Y _ i ^ 2 ] } \\ge \\frac { 1 } { 1 + \\frac { 1 } { \\mathbb { E } [ Y _ i ] } } \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} ( a \\rhd b ) \\rhd ( ( a \\lhd b ) \\circ ( \\overline { a \\lhd b } \\cdot _ x c ) ) = a \\circ ( \\bar { a } \\cdot _ { b \\circ x } ( b \\rhd c ) ) . \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\nabla \\tilde w _ i = \\nabla f ( | w _ i | ^ 2 ) \\otimes w _ i + f ( | w _ i | ^ 2 ) \\nabla w _ i , \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} [ X , Y ] [ Y , Z ] & = X Y X ^ { - 1 } Y ^ { - 1 } \\cdot Y Z Y ^ { - 1 } Z ^ { - 1 } = X Y X ^ { - 1 } Z Y ^ { - 1 } Z ^ { - 1 } , \\\\ [ X Z ^ { - 1 } , Z Y Z ^ { - 1 } ] & = ( X Z ^ { - 1 } ) ( Z Y Z ^ { - 1 } ) ( Z X ^ { - 1 } ) ( Z Y ^ { - 1 } Z ^ { - 1 } ) = X Y X ^ { - 1 } Z Y ^ { - 1 } Z ^ { - 1 } . \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) = \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) \\Bigl ( \\frac { \\tau ^ { \\frac { 1 } { \\tau } } } { \\Gamma ( 1 - \\frac { 1 } { \\tau } ) } \\Bigr ) ^ { - q } \\frac { \\Gamma _ 2 ( 1 - q \\ , | \\tau ) \\Gamma _ 2 ( \\tau \\ , | \\tau ) } { \\Gamma _ 2 ( 1 \\ , | \\tau ) \\Gamma _ 2 ( \\tau - q \\ , | \\tau ) } . \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} a \\ell & = b m , \\\\ a _ 1 \\ell + b _ 2 m & = c n . \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} \\| f + g \\| _ m \\leq \\| f \\| _ m + \\| g \\| _ m , \\| f g \\| _ { m + k } \\leq \\| f \\| _ m \\| g \\| _ k \\left \\| \\frac { \\d f } { \\d x _ j } \\right \\| _ { m + 1 } \\leq e ( m + 1 ) r ^ { n - 1 } \\| f \\| _ m , j = 1 , \\dots , n . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\left ( \\sum _ { m \\geq 0 } \\alpha _ m \\right ) \\circ \\left ( \\sum _ { n \\geq 0 } \\beta _ n \\right ) : = \\sum _ { m \\geq 0 } \\alpha _ m \\left ( \\sum _ { n \\geq 0 } \\beta _ n \\right ) ^ { m + 1 } = \\sum _ { k \\geq 0 } \\gamma _ k \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} \\begin{pmatrix} Q _ 1 & Q _ 2 & Q _ 3 \\\\ Q _ 4 & Q _ 5 & Q _ 6 \\\\ Q _ 7 & Q _ 8 & Q _ 9 \\end{pmatrix} , \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} G ( B ( m ; x ) , z ) = G ( ( \\stackrel { \\stackrel { m } { \\smile } } { B \\times \\cdots \\times B } ) ( x ) , z ) , \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} v _ 2 ( k ! S ( 2 ^ n , k ) ) = k - 1 \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} ( y z / x ) ( b \\backslash y ^ { \\alpha } x ) = y ( b \\backslash \\left [ ( b z / x ) ( b \\backslash y ^ { \\alpha } x ) \\right ] ) \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} B ( U , \\zeta + \\phi \\psi ) = 0 . \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} ( A _ { m } , \\Phi _ { m } ) = ( s _ m ^ * A , m ^ { - 1 } s _ m ^ * \\Phi ) , \\ \\ g _ m = m ^ 2 s _ { m } ^ * g . \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} \\langle \\Lambda _ 0 , \\check \\Lambda _ i \\rangle = 0 \\qquad \\langle \\alpha _ i , \\check \\Lambda _ j \\rangle = \\delta _ { i , j } , i , j \\in I . \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} Q ( x _ 1 , \\tilde { x } , x _ 1 ' , \\tilde { x } ' ) = P ( x _ 1 ; \\tilde { x } ) - P ( x _ 1 ' ; \\tilde { x } ) - P ( x _ 1 ; \\tilde { x } ' ) + P ( x _ 1 ' ; \\tilde { x } ' ) . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} \\| P _ m - f \\| _ { S _ n ^ p } & = | P _ m ( 0 ) - f ( 0 ) | + | P ' _ m ( 0 ) - f ' ( 0 ) | + \\cdots \\\\ & \\phantom { = } + | P ^ { ( n - 1 ) } _ m ( 0 ) - f ^ { ( n - 1 ) } ( 0 ) | + \\| P ^ { ( n ) } _ m - f ^ { ( n ) } \\| _ { H ^ p } \\\\ & = \\| p _ m - f ^ { ( n ) } \\| _ { H ^ p } \\ , . \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\norm { V - V _ n } _ t \\leq \\lim _ { n \\to \\infty } \\lim _ { m \\to \\infty } \\sum _ { i = n } ^ { m } \\norm { V _ { i + 1 } - V _ i } _ t \\stackrel { \\eqref { e q : b o u n d V } } = 0 , \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\gcd ( \\ell , m ) = 1 , \\quad \\ell < m , \\quad ( \\ell - 2 ) m < n . \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\nu \\left ( \\sum _ { \\ell = w } ^ \\infty t _ \\ell x ^ { - \\ell } \\right ) = \\sum _ { \\ell = \\max ( 1 , w ) } ^ m t _ { \\ell } b ^ { - \\ell } . \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} e _ k ( x ) = \\begin{cases} \\frac { \\sqrt 2 } 2 ( \\tilde e _ k ( x ) + \\tilde e _ { - k } ( x ) ) , & k \\in \\Z ^ 2 _ + , \\\\ \\frac { \\sqrt 2 } { 2 i } ( \\tilde e _ k ( x ) - \\tilde e _ { - k } ( x ) ) , & k \\in \\Z ^ 2 _ - . \\end{cases} \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\begin{array} { l } { \\inf } \\ ; \\mathbb { E } _ { \\xi _ 2 , \\ldots , \\xi _ { T _ { \\max } } , D _ 2 , \\ldots , D _ { T _ { \\max } } } \\Big [ \\displaystyle { \\sum _ { t = 1 } ^ { T _ { \\max } } } \\ ; D _ { t } f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + ( D _ { t - 1 } - D _ t ) { \\overline { f } } _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) \\Big ] \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) \\ ; \\mbox { a . s . } , x _ { t } \\ ; { \\overline { \\mathcal { F } } } _ t \\mbox { - m e a s u r a b l e , } t = 1 , \\ldots , T _ { \\max } , \\end{array} \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} \\sigma _ { \\phi _ { ( j , N ) } f } ^ { 2 } = P ( f _ { ( j , N ) } ^ { 2 } ) \\leqslant \\sigma _ { \\phi _ { ( N ) } f } ^ { 2 } = P ( f _ { ( N ) } ^ { 2 } ) = \\sum _ { j = 1 } ^ { m _ { N } } \\sigma _ { \\phi _ { ( j , N ) } f } ^ { 2 } \\leqslant \\sigma _ { f } ^ { 2 } . \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} \\partial _ r f ( x ) = \\sum _ { j = 1 } ^ d \\frac { x _ j } { | x | } \\ , \\partial _ j f ( x ) , x \\in B \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { 1 } { 3 \\cdot 2 ^ d } ( 1 - \\gamma _ 1 ) + \\bigg ( 1 + \\frac { 1 } { 6 ( 2 ^ d - 1 ) } \\bigg ) & ( 1 - \\gamma _ 2 ) & < \\frac { 1 } { 3 \\cdot 2 ^ d } , \\\\ & ( 1 - \\gamma _ 2 ) & < \\frac { 1 } { 4 \\cdot 2 ^ d } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} \\mathrm { v o l } ( \\Gamma _ 0 \\alpha _ n \\Gamma _ 0 ) = \\begin{cases} p ^ { 2 n - 1 } ( 1 - p ^ { - 1 } ) & n > 0 , \\\\ p ^ { - 2 n - 1 } ( 1 - p ^ { - 1 } ) & n < 0 . \\end{cases} \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\mathbb { E } [ w t ( \\mathbf { d } _ i ) ] = \\frac { 1 } { 2 } \\lim _ { m \\to \\infty } m ( 1 - \\sigma ) = c _ 1 > 0 . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} \\mathbf F _ { \\chi } ( g ) = \\det ( g _ { \\infty } ) ^ { ( k + 1 ) / 2 } \\det ( C \\sqrt { - 1 } + D ) ^ { - k - 1 } F _ { \\chi } ( g _ { \\infty } \\sqrt { - 1 } ) \\underline { \\chi } ( k ) , \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} d X _ s = \\sigma _ i ( X _ s ) d B ^ i _ s + \\beta _ j ( X _ s ) \\dot { Z } _ s ^ j d s , \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\sum _ { j = p + 1 } ^ \\kappa \\psi _ j ( u ) \\overline { z } ^ { j - p - 1 } = \\sum _ { j = 0 } ^ \\kappa \\left ( \\dfrac { \\overline { \\psi _ j ( z ) } - \\overline { Q _ { j , p } ( z ) } } { \\overline { z } ^ { p + 1 } } \\right ) u ^ j , \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} V f = \\psi _ 0 f + \\psi _ n f ^ { ( n ) } , f \\in ( V ) , \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} \\displaystyle \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) u ( x , t ) + a u ( x , t ) \\ln u ( x , t ) = 0 , \\ \\ \\ a \\in \\mathbb { R } \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} \\left ( v \\alpha _ { 1 } \\right ) u \\left ( v \\alpha _ { 1 } \\right ) ^ { - 1 } = v \\left ( \\alpha _ { 1 } \\cdot u \\right ) v ^ { - 1 } u ^ { - 1 } = \\rho ^ { u _ { 2 } + u _ { 2 } v _ { 3 } - u _ { 3 } v _ { 2 } } \\in \\left \\langle u \\right \\rangle , \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} C ( n , k , w ) = \\sum _ { j = 0 } ^ k ( - 1 ) ^ j \\binom { k } { j } \\binom { n - j w - 1 } { k - 1 } . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} a + b = \\sum _ { j = 1 } ^ { \\deg ( v _ 0 ) } F _ j ( v _ 0 ) \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} F ' ( \\tilde { x } ) & = \\frac { k } { 4 \\varepsilon } - \\frac { \\sigma ( H - \\delta ) } { 2 } C \\tilde { x } ^ { H - \\delta - 1 } = 0 \\\\ \\implies \\tilde { x } & = \\biggl ( \\frac { k } { 2 \\sigma \\varepsilon C ( H - \\delta ) } \\biggr ) ^ { 1 / ( H - \\delta - 1 ) } \\\\ & = \\biggl ( \\frac { 2 \\sigma C ( H - \\delta ) } { k } \\biggr ) ^ { 1 / ( 1 + \\delta - H ) } \\varepsilon ^ { 1 / ( 1 + \\delta - H ) } . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{gather*} y = \\big ( 1 - x ^ 2 \\big ) ^ { 1 / 4 } \\end{gather*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\| P ( x ) \\| ^ 2 = \\sum _ { n \\in \\N ^ N } ^ \\infty \\| B e _ { n } \\| ^ 2 = \\| B \\| _ { S _ 2 } ^ 2 , \\quad \\forall x \\in X . \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} & \\frac 1 4 \\left ( \\langle D \\bar F _ 2 - D F _ 2 , D \\bar F _ 1 + D F _ 1 \\rangle - \\langle D \\bar F _ 2 + D F _ 2 , D \\bar F _ 1 - D F _ 1 \\rangle \\right ) \\\\ & = \\frac 1 4 \\left ( ( 1 - \\mu _ 2 ) ( 1 + \\mu _ 1 ) - ( 1 + \\mu _ 2 ) ( 1 - \\mu _ 1 ) \\right ) \\langle D \\bar F _ 2 , D \\bar F _ 1 \\rangle \\\\ & = \\frac { \\mu _ 1 - \\mu _ 2 } { 2 } \\langle D \\bar F _ 2 , D \\bar F _ 1 \\rangle \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} d s ^ 2 \\ , = \\ , d r ^ 2 + \\psi ( r ) ^ 2 \\ , d \\theta ^ 2 , \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} \\mathcal Q ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = C _ 1 ^ { - 2 } \\mathcal P _ { \\chi } ( \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) \\otimes \\underline { \\chi } , \\mathbf Y _ M \\mathbf G ) , \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} t _ i = - \\frac { 1 } { z + s r _ i } \\ln \\left ( \\frac { \\tilde { y } } { y } ( e ^ { - z - s r _ i } - 1 ) + 1 \\right ) ; \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} V ( x ) & = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 ) } \\frac { \\Gamma ^ 2 \\left ( \\frac { s } { 2 } + \\frac { 1 } { 4 } \\right ) } { \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } \\frac { G ( s ) } { s } \\pi ^ { - s } x ^ { - s } d s . \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} - \\frac { \\chi _ 1 } { 2 } \\int _ { \\Omega } \\psi _ + ^ 2 \\Delta w = \\frac { \\chi _ 1 } { 2 d _ 3 } \\int _ { \\Omega } \\psi _ + ^ 2 ( k u + l v - \\lambda w ) , \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} \\sigma ( g | _ { \\partial D } , g | _ { \\partial D } ) = - \\delta \\tau ( g , h ) \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\dots , n \\} , ~ f ( P _ { \\sigma ( i ) } ) = f ( P _ i ) & \\Leftrightarrow \\forall i \\in \\{ 1 , \\dots , n \\} , ~ f \\circ \\sigma ( P _ i ) = f ( P _ i ) \\\\ & \\Leftrightarrow \\forall i \\in \\{ 1 , \\dots , n \\} , ~ ( f \\circ \\sigma - f ) ( P _ i ) = 0 . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} \\leq \\sum _ { l \\in \\mathbb { Z } , \\ , \\ , a _ { l - 1 } \\neq 0 } \\sum _ { i \\in \\mathbb { Z } , \\ , \\ , i \\leq l - 1 } 2 ^ { i p } a ^ { - s p / Q } _ { l - 1 } d _ { l } = \\sum _ { l \\in \\mathbb { Z } , \\ , a _ { l - 1 } \\neq 0 } \\sum ^ { + \\infty } _ { k = 0 } 2 ^ { p ( l - 1 - k ) } a _ { l - 1 } ^ { - s p / Q } d _ { l } \\leq S . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\widehat { h _ \\theta } ( y ) = \\frac { s i n h ( y ) } { y c o s h ( y ) + \\theta s i n h ( y ) } . \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} - 4 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } \\nabla _ x m _ { \\nabla u _ \\mu } ( t , x ) \\cdot \\nabla _ y m _ { u _ \\iota } ( t , y ) \\Delta _ x \\psi ( x , y ) \\ , d x d y \\leq 0 . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} T _ { m , n } = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) , \\quad \\forall m , n \\in \\N ^ N \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} Y _ n \\to Y _ { n - 1 } \\to \\cdots \\to Y _ 1 \\to Y _ 0 : = Y \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} I _ { 1 , 2 } = \\frac 1 2 \\sum _ { d l \\leq x } \\frac { \\Lambda ( l ) } { l } = - \\frac { \\zeta ' ( 2 ) } { 2 \\zeta ( 2 ) } x + O ( \\log x ) , \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\mathbf I ( \\vect v ) = \\left . \\sum _ { \\ell = 0 } ^ { \\infty } \\hbar ^ \\ell \\big ( \\partial _ { \\vect u \\overline { \\vect u } } \\big ) ^ \\ell \\beta _ \\vect z \\right | _ { \\vect u = \\vect v } , \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} N _ { w , G } ( g ) > 0 = N _ { w , G } ( g ^ { \\prime } ) . \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\bar { \\Omega } ( x ) \\partial \\Lambda ( x ) = \\Omega _ 0 ( x ) - \\Omega _ 1 ( x ) \\Lambda ( x ) + \\Omega _ 2 ( x ) \\Lambda ( x ) ^ 2 \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} E q 1 : \\rho _ { 0 } ^ { 3 } v _ { 0 } ^ { 2 } + 2 \\omega \\frac { v _ { 0 } \\rho _ { 0 } } { K _ { 0 } } = 4 \\mathfrak { S } _ { 0 } , \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} \\ell f _ j ( \\rho ) & \\leq C f _ { j - 1 } ( \\rho + \\ell ) + \\begin{cases} C \\ell ^ { 1 + \\alpha } [ \\nabla ^ { j } \\sigma ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( p , \\rho + \\ell ) ) } & { \\rm f o r \\ ; a l l } \\ ; j , \\\\ C \\ell ^ 2 f _ { j + 1 } ( \\rho + \\ell ) & { \\rm f o r \\ ; a l l } \\ ; j < k . \\end{cases} \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} \\{ 0 , 1 , \\dots , 1 0 ^ 4 \\} = \\{ 0 , 1 , \\dots , 1 0 \\} \\cup \\{ 1 1 \\} \\cup \\{ 1 2 \\} \\cup \\{ 1 3 \\} \\cup \\{ 1 4 \\} \\cup \\{ 1 5 \\} \\cup \\{ 1 6 , 1 7 , \\dots , 1 0 ^ 4 \\} . \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\tilde W = \\begin{pmatrix} 1 & 0 & - 1 \\\\ 0 & 1 & - 1 \\end{pmatrix} W = \\begin{pmatrix} 1 & 0 & - 1 \\\\ 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 1 } ^ { t } ( N - k ) \\leq b _ B ( N , t ) \\leq \\prod \\limits _ { k = 0 } ^ { t } ( N - k ) . \\\\ \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} V _ N = \\bigoplus _ { \\pi \\in \\Pi _ N ( \\mathbf { k } ) } V _ N ( \\pi ) . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\lim \\limits _ { A , B \\rightarrow 0 } \\int _ { x _ 1 ^ { A B } ( t ) } ^ { x _ 2 ^ { A B } ( t ) } \\rho _ * ^ { A B } d x = = \\lim \\limits _ { A , B \\rightarrow 0 } \\int _ { 0 } ^ { t } \\rho _ * ^ { A B } ( \\sigma _ 2 ^ { A B } - \\sigma _ 1 ^ { A B } ) d t = \\sqrt { \\rho _ { + } \\rho _ { - } } ( u _ - - u _ + ) t . \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} t _ { s _ h } ^ n & = ( t _ x t _ y ) ^ n t _ z ^ n t _ a ^ { - n } t _ b ^ { - n } t _ c ^ { - n } \\\\ & = { } _ { t _ x } ( t _ y ) { } _ { t _ x ^ 2 } ( t _ y ) \\cdots { } _ { t _ x ^ n } ( t _ y ) t _ x ^ n t _ z ^ n t _ a ^ { - n } t _ b ^ { - n } t _ c ^ { - n } . \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} [ L _ { m } , L _ { n } ] = ( m - n ) L _ { m + n } + \\frac { c } { 1 2 } ( m ^ { 3 } - m ) \\delta _ { m + n , 0 } , \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} S ^ { o , h } _ C ( i _ { \\kappa } , \\kappa , j ) = f ^ { o , \\kappa , h } _ A ( \\rho ^ h _ { \\kappa } , j ) \\leq \\min \\{ C _ j ^ 0 - \\frac { R ^ { h } _ { \\kappa } ( j ) } { h } , C _ j ^ 0 - \\frac { R ^ h _ { \\kappa } ( j ) } { h } - \\Delta ^ { \\kappa } _ C ( h , j ) \\} . \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} d X _ t = d W _ t - V ( X _ t ) d t , \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\big | h \\big | = \\big | h \\big | h \\in H . \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} ( \\mathfrak { d } _ 1 ^ { m } a ) _ n = \\sum _ { j \\geq 0 } ( \\mathfrak { d } _ 1 ^ { m } a ) _ { n + j } - ( \\mathfrak { d } _ 1 ^ { m } a ) _ { n + j + 1 } , \\quad \\forall n \\in \\N . \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} a _ f ( n ) = \\mu ^ 2 ( n ) \\lambda _ f ( n ) , \\omega ( n ) = \\omega ' ( n ) = \\mu ^ 2 ( n ) | \\lambda _ f ( n ) | ^ 2 \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} \\begin{aligned} | \\underline { \\vec z _ { k + d } } | & = 3 \\sqrt 2 | \\underline { \\vec z _ { k + d - 1 } } | ^ { 1 + \\beta } + O \\big ( | \\underline { \\vec z _ { k + d - 1 } } | \\big ) , \\\\ | \\underline { \\vec z _ { k + d - 1 } } \\wedge \\underline { \\vec z _ { k + d } } | & = 3 | \\underline { \\vec z _ { k + d - 1 } } | ^ { 2 + \\beta } + O \\big ( | \\underline { \\vec z _ { k + d - 1 } } | ^ 2 \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} [ \\beta ] = [ \\imath _ * \\imath ^ { ! * } \\beta ] + [ \\jmath _ { ! * } \\jmath ^ * \\beta ] \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} F _ { 8 , 5 } = \\frac { 1 3 5 } { 8 } \\nu ( \\xi _ 1 ) \\chi _ 5 , F _ { 8 , 7 } = - \\frac { 4 0 5 } { 4 } \\nu ( \\xi _ 2 ) \\chi _ 5 , \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\begin{cases} \\overline u _ t \\geq \\overline u ( t ) ( a _ 0 ( t ) - a _ 1 ( t ) \\overline u ( t ) - a _ 2 ( t ) \\underline v ( t ) ) \\\\ \\underline v _ t \\leq \\underline v ( t ) ( b _ 0 ( t ) - b _ 1 ( t ) \\overline u ( t ) - b _ 2 ( t ) \\underline v ( t ) ) . \\end{cases} \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} & u _ { 2 } = 0 , \\ v _ { 2 } = - u _ { 3 } v _ { 2 } , \\ w _ { 2 } = u _ { 3 } - u _ { 3 } w _ { 2 } , \\ u _ { 3 } = - w _ { 2 } , \\\\ & u _ { 1 } - v _ { 2 } ( w _ { 2 } - 1 ) - \\frac { 1 } { 2 } w _ { 2 } ( w _ { 2 } - 1 ) = v _ { 3 } + w _ { 3 } v _ { 2 } - v _ { 3 } w _ { 2 } . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} \\Omega _ { \\Z [ H _ 1 \\oplus H _ 2 ] | \\Z } & = \\Omega _ { \\Z [ H _ 1 ] \\otimes _ \\Z \\Z [ H _ 2 ] | \\Z } \\\\ & = \\Omega _ { \\Z [ H _ 1 ] | \\Z } \\otimes _ \\Z \\Z [ H _ 2 ] \\oplus \\Z [ H _ 1 ] \\otimes _ \\Z \\Omega _ { \\Z [ H _ 2 ] | \\Z } \\\\ & = H _ 1 \\otimes _ \\Z \\Z [ H _ 1 ] \\otimes _ \\Z \\Z [ H _ 2 ] \\oplus \\Z [ H _ 1 ] \\otimes _ \\Z H _ 2 \\otimes _ \\Z \\Z [ H _ 2 ] \\\\ & = ( H _ 1 \\oplus H _ 2 ) \\otimes _ \\Z ( \\Z [ H _ 1 ] \\otimes _ \\Z \\Z [ H _ 2 ] ) \\\\ & = ( H _ 1 \\oplus H _ 2 ) \\otimes _ \\Z \\Z [ H _ 1 \\oplus H _ 2 ] . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\begin{bmatrix} M ( \\lambda ) & K _ 2 ( \\lambda ) ^ T \\\\ K _ 1 ( \\lambda ) & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} h ' ( a ) = \\frac { d } { a } - M R ( a M ) + \\sum _ { i = 1 } ^ { d + 1 } \\gamma _ i R ( a \\gamma _ i ) + \\sum _ { i = 1 } ^ { d + 1 } \\gamma _ i \\log \\left ( \\frac { \\gamma _ i / M } { x _ i } \\right ) , \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} F = F _ f \\colon P \\to \\mathrm { S u b } ( A ) \\colon p \\mapsto A _ { \\leq p } . \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} X _ 3 = \\sum _ { j = 0 } ^ 6 \\frac { ( v _ j - t \\cdot \\pi _ j ) ^ 2 } { t \\cdot \\pi _ j } , \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} C _ { 2 } \\left ( \\ell ^ { \\varepsilon } \\right ) : = \\frac { \\underline { M } + 1 } { 4 } + \\frac { L _ { F } ^ { 2 } \\left ( \\ell ^ { \\varepsilon } \\right ) } { \\underline { M } } . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\zeta _ 1 ^ { ( 2 ) } \\circ { \\rm g w } _ t = \\lim _ { t \\to \\infty } \\frac { 1 } { 2 t } \\ln \\left ( e ^ { t \\lambda ^ { ( 2 ) } _ 1 } + e ^ { t \\lambda ^ { ( 2 ) } _ 2 } - e ^ { t ( \\lambda ^ { ( 2 ) } _ 1 + \\lambda ^ { ( 2 ) } _ 2 - \\lambda ^ { ( 1 ) } _ 1 ) } - e ^ { t \\lambda ^ { ( 1 ) } _ 1 } \\right ) = \\frac { 1 } { 2 } \\lambda ^ { ( 2 ) } _ 1 . \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} T _ { \\phi , \\mathbb { C } } ^ { s } = ( T _ { \\phi , \\mathbb { C } } ^ { u } ) ^ { * } . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} S ^ { ( s ) } _ { N , N } f & = \\sum _ { m = 2 } ^ { N } \\left ( S ^ { ( s + 1 ) } _ { N , m } f - S ^ { ( s + 1 ) } _ { N , m - 1 } f \\right ) \\log m \\\\ & = ( \\log N ) S ^ { ( s + 1 ) } _ { N , N } f + \\sum _ { m = 2 } ^ { N - 1 } \\big ( \\log m - \\log ( m + 1 ) \\big ) S ^ { ( s + 1 ) } _ { N , m } f . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} \\Sigma _ N & = \\sum _ { d \\mid v } \\mu ( d ) \\sum _ { ( a , w ) = 1 } e \\left ( \\frac { a d \\overline { m \\ell _ 1 \\ell _ 3 v } } { w } \\right ) \\sum _ { n \\equiv a ( w ) } G \\left ( \\frac { d n } { N } \\right ) . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\Big | & \\frac { f ( t , \\omega + \\varepsilon h , X ( t ) ( \\omega ) ) - f ( t , \\omega , X ( t ) ( \\omega ) ) } { \\varepsilon } - D ^ h f ( t , \\omega , X ( t ) ( \\omega ) ) \\Big | ^ m d t \\\\ & = \\int _ 0 ^ T \\Big | \\frac { 1 } { \\varepsilon } \\int _ 0 ^ \\varepsilon D ^ h f \\big ( t , \\omega + r h , X ( t ) ( \\omega ) \\big ) d r - D ^ h f ( t , \\omega , X ( t ) ( \\omega ) ) \\Big | ^ m d t . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} K ( x ) = - \\frac { 2 \\ell } { \\sqrt { 2 \\pi } } \\int _ 0 ^ { + \\infty } \\frac { y e ^ { - | x | y } } { ( c - y ^ 2 ) ^ 2 + \\ell ^ 2 y ^ 2 } d y + \\frac { 2 \\sqrt { 2 \\pi } } { \\sqrt { 4 c - \\ell ^ 2 } } e ^ { - \\frac { \\sqrt { 4 c - \\ell ^ 2 } } { 2 } | x | } c o s ( \\ell x / 2 ) . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\partial _ t m _ t = - \\Delta m _ t + \\mathrm { d i v } ( \\beta _ j m _ t ) \\dot { Z } _ t ^ j , m _ T = 1 , \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\min \\limits _ { \\{ \\mathbf { x } _ { i j } \\} } & \\sum _ { i = 1 } ^ N \\bigg [ \\sum _ { j = 1 } ^ M ( E ^ l _ { i j } x ^ l _ { i j } + E ^ A _ { i j } x ^ a _ { i j } + E ^ C _ { i j } x ^ c _ { i j } ) \\\\ & \\ \\ + \\rho _ i \\max \\{ T ^ { L } _ { i } , T ^ { { A } _ { ( U ) } } _ { i } , T ^ { { C } _ { ( U ) } } _ { i } \\} \\bigg ] \\allowdisplaybreaks \\\\ & \\eqref { e q _ p l a c e m e n t } \\ \\eqref { x y _ n e w _ C A P } . \\allowdisplaybreaks \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} d _ { i } b _ { i j } = - d _ j b _ { j i } \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} J ( A _ r , B _ r , C _ r ) & = { \\rm t r } ( C \\Sigma _ c C ^ T + C _ r P C _ r ^ T - 2 C _ r X ^ T C ^ T ) \\\\ & = { \\rm t r } ( B ^ T \\Sigma _ o B + B _ r ^ T Q B _ r + 2 B ^ T Y B _ r ) , \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} K _ L ^ { \\bullet } : = R \\Gamma ( L , \\Z _ p ( 1 ) ) [ 1 ] \\oplus H _ L [ - 1 ] = C _ L ^ { \\bullet } \\oplus H _ L [ - 1 ] \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} \\phi ( \\lambda ) = b \\lambda + \\int _ { ( 0 , \\infty ) } ( 1 - e ^ { - \\lambda x } ) \\mu ( d x ) \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} f \\colon [ 0 , T ] \\times [ 0 , T ] \\to U , f ( s , r ) = \\ 1 _ { [ 0 , s ] } ( r ) B ^ \\ast T ^ \\ast ( s - r ) v . \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} \\Psi & = \\mathsf { G } _ { [ + \\infty , t _ 0 ] } ( - \\mathsf { V } + \\Xi ) + e ^ { - \\tilde { \\mathsf { D } } ( t - t _ 0 ) } \\Psi _ 0 , & \\mathsf { G } & = \\begin{pmatrix} G & 0 & 0 \\\\ 0 & \\bar { G } & 0 \\\\ 0 & 0 & P \\end{pmatrix} \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} m ' = Y _ { n - ( p + 1 ) / 2 , r + p } . \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} ( W ^ * , f ^ * ) . t : = ( W ^ * , t f ^ * ) . \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\begin{cases} C _ { p } ^ { q } = C ^ { q } ( G ; A ) , & \\mbox { i f $ p \\leq 0 $ } , \\\\ C _ { p } ^ { q } = 0 , & \\mbox { i f $ q < p $ } \\end{cases} \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} \\mathcal G _ 0 ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * g ^ * : \\mathcal G _ 0 ( Y ) \\to \\mathcal G _ 0 ( X ) , \\\\ \\mathcal H ^ { T o d d } _ * ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * \\bigl ( t d ( T _ g ) \\cap g ^ * \\bigr ) : \\mathcal H ^ { T o d d } _ * ( Y ) \\to \\mathcal H ^ { T o d d } _ * ( X ) . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} ( \\frac { u ^ { n + 1 } _ { h } - u ^ { n } _ { h } } { \\Delta t } , v _ { h } ) + b ( u ^ { n } _ { h } , u ^ { n + 1 } _ { h } , v _ { h } ) + \\nu ( u ^ { n + 1 } _ { h } , v _ { h } ) - ( p ^ { n + 1 } _ { h } , \\nabla \\cdot v _ { h } ) = ( f ^ { n + 1 } , v _ { h } ) \\ ; \\ ; \\forall v _ { h } \\in X _ { h } , \\\\ ( \\nabla \\cdot u ^ { n + 1 } _ { h } , q _ { h } ) \\ ; \\ ; \\forall q _ { h } \\in Q _ { h } . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} ( ( r s ) t ) = ( r ( s t ) ) + ( r ( t s ) ) , \\forall ~ r , s , t \\in R . \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\Gamma _ { \\mathbf { m } , k } ( \\underline { S } ) = \\operatorname * { l c m } \\limits _ { \\substack { \\mathbf { 0 } \\leq \\mathbf { i } \\leq \\mathbf { m } \\\\ \\vert \\mathbf { i } \\vert \\leq k } } \\mathbf { i } ! _ { \\underline { S } } . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} E _ { k , N } ^ { \\chi } ( Z , s ; h ) = \\left ( E _ { k , N } ^ { \\chi } \\big | _ { k } h _ 0 \\right ) ( Z , s ) : = j ( h _ 0 , Z ) ^ { - k } E _ { k , N } ^ { \\chi } ( h _ 0 Z , s ) , \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} g ( \\varphi U , V ) = g ( U , \\varphi V ) + u ( V ) \\theta ( U ) - u ( U ) \\theta ( V ) , \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} M _ { } & = 2 P _ 1 ( 1 ) P _ 3 ( 1 ) \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\epsilon ( \\chi ) \\sum _ n \\frac { \\overline { \\chi } ( n ) } { n ^ { 1 / 2 } } V \\left ( \\frac { n } { q } \\right ) . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} \\Gamma ( q ^ k z ) = ( z ) _ k \\ , \\Gamma ( z ) , \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\Big | \\sum _ { \\atop { K < k \\leq K ' } { X < j k \\leq X ' } } e ^ { 2 \\pi i F ( j k ) } B _ k \\Big | ^ 2 & = \\sum _ { \\abs { r } \\leq K } \\sum _ { \\atop { K < k , k + r \\leq K ' } { X < j k , j ( k + r ) \\leq X ' } } \\exp \\Big ( 2 \\pi i \\big ( F ( j k ) - F ( j ( k + r ) ) \\big ) \\Big ) B _ k \\overline { B _ { k + r } } . \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} u ( \\xi ) = \\sum _ { 2 j + \\frac { k } { \\beta } < q } \\rho ^ { 2 j + \\frac { k } { \\beta } } \\left ( a _ { j , k } ( \\xi ) \\cos k \\theta + b _ { j , k } ( \\xi ) \\sin k \\theta \\right ) + O ( \\rho ^ q ) \\rho < 1 / 2 . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} m m p ( 0 , m , 0 , 0 ) ( \\sigma ) = | \\{ i : | \\{ j : j < i , \\sigma _ j > \\sigma _ i \\} | \\geq m \\} | , \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} \\begin{array} { l } P = [ \\ g _ 0 \\ g _ 1 \\ \\hdots \\ g _ N \\ | \\ x _ 1 - x _ 0 \\ \\hdots \\ x _ N - x _ 0 \\ | \\ x _ * - x _ 0 \\ ] , \\\\ F = [ \\ f _ 0 \\ f _ 1 \\ \\hdots \\ f _ N \\ f _ * \\ ] ^ \\top . \\end{array} \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\pm \\infty } \\| u ( t , \\cdot ) \\| _ { L ^ q } = 0 , \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( N ) } ( f ) - P ( f ) = \\frac { 1 } { \\sqrt { n } } \\mathbb { G } _ { n } ^ { ( N ) } ( f ) + \\frac { 1 } { \\sqrt { n } } \\mathbb { R } _ { n } ^ { ( N ) } ( f ) , \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} e ^ { n } _ { u } & = u ^ { n } - u ^ { n } _ { h } , \\ ; e ^ { n } _ { p } = p ^ { n } - p ^ { n } _ { h } . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} I ( u ) & = \\displaystyle \\int _ { \\Omega } \\frac { B ( x ) } { p ( x ) } | \\nabla u ( x ) | ^ { p ( x ) } d x + \\displaystyle \\int _ { \\Omega } \\frac { A ( x ) } { p ( x ) } | u ( x ) | ^ { p ( x ) } d x + \\displaystyle \\int _ { \\Omega } \\frac { C ( x ) } { p ( x ) - 1 } | u ( x ) | ^ { p ( x ) - 1 } d x \\\\ & + \\displaystyle \\int _ { \\Omega } \\frac { D ( x ) } { p ( x ) + 1 } | u ( x ) | ^ { p ( x ) + 1 } d x - \\displaystyle \\int _ { \\Omega } b ( x ) \\frac { | u ( x ) | ^ { q ( x ) } } { q ( x ) } d x . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} T _ w T _ s = \\left \\{ \\begin{aligned} & T _ { w s } , & \\mbox { i f } \\ell ( w s ) > \\ell ( w ) ; \\\\ & ( q ^ 2 - 1 ) T _ w + q ^ 2 T _ { w s } , & \\mbox { i f } \\ell ( w s ) < \\ell ( w ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} | F ( x , t , | y ' | ) y ' - F ( x , t , | y | ) y | & \\le ( 1 + \\alpha _ N ) ( N + 1 ) \\max _ { i = 0 , \\ldots , N } a _ i | y ' - y | \\Big ( 1 + \\int _ 0 ^ 1 | \\gamma ( \\tau ) | ^ { \\alpha _ N } d \\tau \\Big ) \\\\ & \\le ( 1 + \\alpha _ N ) ( N + 1 ) \\max _ { i = 0 , \\ldots , N } a _ i | y ' - y | \\Big ( 1 + \\int _ 0 ^ 1 ( | y ' | + | y | ) ^ { \\alpha _ N } d \\tau \\Big ) \\\\ & \\le 2 ^ { \\alpha _ N } ( 1 + \\alpha _ N ) ( N + 1 ) \\max _ { i = 0 , \\ldots , N } a _ i \\Big ( 1 + | y ' | ^ { \\alpha _ N } + | y | ^ { \\alpha _ N } \\Big ) | y ' - y | , \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{gather*} \\log \\mathfrak { M } ( q \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ \\frac { 1 } { ( e ^ t - 1 ) ( e ^ { t \\tau } - 1 ) } \\Bigl ( e ^ { t ( q - \\lambda _ 1 \\tau ) } + e ^ { t ( q - \\lambda _ 2 \\tau ) } + e ^ { t ( q + 1 ) } + \\\\ + e ^ { t ( q - 1 - \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) ) } - e ^ { t ( 2 q - 1 - \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) ) } \\Bigr ) + A ( t ) + B ( t ) \\ , q + C ( t ) \\ , q ^ 2 \\Bigr ] \\end{gather*}"}
-{"id": "2683.png", "formula": "\\begin{align*} \\big | \\omega ^ N _ t \\big | _ { H _ N } = | \\xi | _ { H _ N } . \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} \\alpha \\cdot & v = \\rho ^ { \\widetilde { v } _ { 1 } } \\sigma ^ { b _ { 1 } v _ { 2 } + b _ { 2 } v _ { 3 } } \\tau ^ { b _ { 3 } v _ { 2 } + b _ { 4 } v _ { 3 } } , \\ \\\\ \\widetilde { v } _ { 1 } & \\stackrel { \\mathrm { d e f } } { = } \\mathrm { d e t } ( \\beta ) v _ { 1 } + \\frac { 1 } { 2 } \\left ( b _ { 3 } b _ { 1 } v _ { 2 } ^ { 2 } + b _ { 4 } b _ { 2 } v _ { 3 } ^ { 2 } \\right ) + b _ { 2 } b _ { 3 } v _ { 2 } v _ { 3 } + r _ { 1 } \\left ( b _ { 1 } v _ { 2 } + b _ { 2 } v _ { 3 } \\right ) + r _ { 3 } \\left ( b _ { 3 } v _ { 2 } + b _ { 4 } v _ { 3 } \\right ) \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} H ( \\lambda , u ) = I d - K ( \\lambda N ( u ) ) . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} E & = \\begin{pmatrix} & & E _ e \\\\ & E _ { n - 2 k - 2 l _ 2 } \\\\ E _ e \\end{pmatrix} , \\\\ F & = \\begin{pmatrix} & & F _ d \\\\ & E _ { k - 2 l _ 1 } \\\\ F _ d \\end{pmatrix} , \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} U ( \\lambda ) = \\begin{bmatrix} K ( \\lambda ) \\\\ \\widehat { K } ( \\lambda ) \\end{bmatrix} \\mbox { a n d } U ( \\lambda ) ^ { - 1 } = \\begin{bmatrix} \\widehat { N } ( \\lambda ) ^ T & N ( \\lambda ) ^ T \\end{bmatrix} . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} H _ t : = \\left ( 1 , h _ t - h _ 0 , \\int _ 0 ^ t ( h _ { u _ 1 } - h _ 0 ) \\ , d u _ 1 , \\dots , \\int _ { 0 \\leq u _ { [ p ] } \\leq \\cdots \\leq u _ 1 \\leq t } d h _ { u _ { [ p ] } } \\cdots d h _ { u _ 1 } \\right ) ; \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} \\tau _ K ( \\gamma ) ( f \\lambda _ g ) = f ( \\gamma ^ { - 1 } \\cdot ) \\lambda _ g \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} & - c b _ { 1 } ^ { - 1 } b _ { 3 } + r _ { 3 } b _ { 1 } ^ { - 2 } b _ { 4 } + \\frac { 1 } { 2 } b _ { 1 } ^ { - 1 } b _ { 4 } \\left ( b _ { 1 } ^ { - 1 } - 1 \\right ) \\left ( c + 1 \\right ) \\\\ & = - c b _ { 1 } ^ { - 1 } b _ { 3 } + r _ { 3 } + \\frac { 1 } { 2 } b _ { 1 } \\left ( b _ { 1 } ^ { - 1 } - 1 \\right ) \\left ( c + 1 \\right ) = 0 , \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} \\psi \\leq s _ 2 \\log 2 , 0 \\leq \\left ( \\psi \\right ) ' \\leq \\frac { 2 ^ { s _ 2 + 1 } } { \\omega } \\mbox { a n d } \\psi '' = \\left ( \\psi ' \\right ) ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} X _ \\alpha ( t . u _ i ) & = \\sum \\limits _ { \\chi \\in A _ Q } t ^ { d ( i , \\chi ) } X _ { ( \\alpha , \\chi ) } ( u _ { ( i , \\chi ) } ) = \\sum \\limits _ { \\chi \\in A _ Q } t ^ { d ( i , \\chi ) } u _ { ( j , \\chi + e _ \\alpha ) } = t ^ { - c _ \\alpha } \\sum \\limits _ { \\chi \\in A _ Q } t . u _ { ( j , \\chi + e _ \\alpha ) } = t ^ { - c _ \\alpha } t . u _ j . \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} & \\sum _ { j _ 1 + \\cdots + j _ { m + r } = n } \\binom { n } { j _ 1 , \\cdots , j _ { m + r } } B _ { j _ 1 } ( x _ 1 ) \\cdots B _ { j _ m } ( x _ m ) E _ { j _ { m + 1 } } ( \\beta ; x _ { m + 1 } ) \\cdots E _ { j _ { m + r } } ( \\beta ; x _ { m + r } ) \\\\ & = \\left ( B _ n ( m , \\cdot ) \\times E ( r , \\beta ; \\cdot ) \\right ) ( x ) = \\sum _ { k = 0 } ^ n v _ k \\Delta ^ k I _ n ( x ) , \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} f ( X , Y ) = \\frac { R ( X ^ \\ell , Y ^ \\ell ) } { { \\Big ( \\big ( \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { \\delta _ j } \\big ) X ^ \\ell - ( - 1 ) ^ { \\ell - 1 } \\big ( \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { \\gamma _ j } \\big ) Y ^ \\ell \\Big ) } ^ t } \\cdot \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} a _ { n , k } = 2 ^ { n - 2 k - 1 } \\binom { n - 1 } { 2 k } C _ k \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} \\delta _ \\Phi ( \\psi _ 1 , & \\ldots , \\psi _ { n } ) = \\\\ & ( \\psi _ { 2 } \\phi _ 1 - \\phi _ { 1 } \\psi _ 1 , \\mbox { } \\psi _ { 3 } \\phi _ 2 - \\phi _ { 2 } \\psi _ 2 , \\ldots , \\psi _ { n } \\phi _ { n - 1 } - \\phi _ { n - 1 } \\psi _ { n - 1 } ) \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} & ( f _ 1 ( U ^ 2 ) + f _ 3 ( U ^ 2 ) U ) e _ 4 = U ^ 2 e _ 4 \\\\ & ( f _ 2 ( U ^ 2 ) U ^ * + f _ 4 ( U ^ 2 ) ) e _ 7 = U ^ { - 2 } e _ 7 \\ ; . \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} \\epsilon ( g _ 1 , g _ 2 ) = ( x ( g _ 1 ) x ( h \\alpha _ n ) , x ( g _ 2 ) x ( h \\alpha _ n ) ) _ p = ( c ^ 2 p ^ { 2 n } , c p ^ n ) _ p = 1 , \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} 3 n - 6 = e ( T ) & = e ( T [ W \\cup S \\cup \\{ u _ 1 , u _ 2 \\} ] ) + e _ T ( \\{ u _ 3 , \\ldots , u _ r \\} , W \\cup S \\cup \\{ u _ 1 , u _ 2 \\} ) \\\\ & \\le e ( \\mathcal { T } _ { 1 1 } ) + ( 2 n - 4 ) = 2 7 + ( 2 n - 4 ) , \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\left \\{ f , g \\right \\} _ { \\mathfrak { k } ^ * } ( \\xi ) = \\langle \\xi , [ d f _ { \\xi } , d g _ { \\xi } ] \\rangle \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} \\Sigma _ { P } ^ { ( \\epsilon , \\eta ) } ( \\lambda ) : = \\begin{bmatrix} B _ k ( \\lambda ) & B _ { k - 1 } ( \\lambda ) & \\cdots & B _ { \\eta + 1 } ( \\lambda ) \\\\ 0 & \\cdots & 0 & \\vdots \\\\ \\vdots & \\ddots & \\vdots & B _ 2 ( \\lambda ) \\\\ 0 & \\cdots & 0 & B _ 1 ( \\lambda ) \\end{bmatrix} , \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\omega ( \\gamma _ { \\tau , n } ( 1 ) ) = \\gamma _ { \\tau ' , n } ( 1 ) \\phi _ n ^ { z _ { \\tau } ( \\omega ) } , \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} A ^ { W } : = \\sum _ { k = 0 } ^ { 2 } A ^ { W } _ { k } \\omega _ { k } \\in M _ { j } ( \\mathbb { C } ) \\otimes \\Gamma . \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} = \\begin{cases} ( i , b ) & , \\\\ ( a , n - i ) & , \\\\ ( 0 , 0 ) & . \\end{cases} \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} \\eta ( \\boldsymbol { \\alpha } ) & : = \\begin{cases} 1 & \\quad \\boldsymbol { \\alpha } \\in \\left \\{ \\left ( 1 - \\frac 1 p , \\frac 2 p \\right ) , \\left ( 0 , 1 - \\frac 1 p \\right ) , \\left ( \\frac 1 p , 1 - \\frac 1 p \\right ) \\right \\} , \\\\ - 1 & \\quad \\end{cases} \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} s _ { \\sigma _ { h - 1 , 1 } + \\sum _ { j = 1 } ^ { k - 1 } m _ { h , j } + x } \\cdot \\Pi _ t = \\Pi _ t \\cdot s _ { \\sigma _ { h - 1 , t + 1 } + \\sum _ { j = t + 1 } ^ { k - 1 } m _ { h , j } + x } . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{gather*} I ( f _ c ) = J \\cup \\{ i , k \\} , I ( f _ s ) = J \\cup \\{ k , l \\} , I ( f _ w ) = J \\cup \\{ i , l \\} , \\\\ I ( f _ e ) = J \\cup \\{ i , j \\} , I ( f _ n ) = J \\cup \\{ j , k \\} , \\end{gather*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\frac { S _ 2 ( w + 1 - k \\ , | \\ , 1 , \\tau ) } { S _ 2 ( w + 1 \\ , | \\ , 1 , \\tau ) } = & \\prod \\limits _ { j = 0 } ^ { k - 1 } S _ 1 \\bigl ( w - j \\ , | \\ , 1 , \\tau \\bigr ) , \\\\ = & \\prod \\limits _ { j = 0 } ^ { k - 1 } 2 \\sin \\pi \\bigl ( \\frac { w } { \\tau } - \\frac { j } { \\tau } \\bigr ) . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} G ^ { 2 } = \\{ \\nabla q , q \\in W ^ { 1 , 2 } ( \\R ) \\} . \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} w _ 1 ^ K & = \\left ( \\rho _ 1 e ^ { i \\varphi _ 1 } \\frac { \\| u _ 1 ^ H \\| _ { L ^ 2 } ^ 2 } { \\rho _ 1 ^ 2 - \\sigma _ 1 ^ 2 } - \\rho _ 2 e ^ { i \\varphi _ 2 } \\frac { \\| u _ 2 ^ H \\| _ { L ^ 2 } ^ 2 } { \\sigma _ 1 ^ 2 - \\rho _ 2 ^ 2 } \\right ) u _ 1 ^ K \\\\ & = \\left ( \\rho _ 1 e ^ { i \\varphi _ 1 } \\frac { \\rho _ 1 ^ 2 - \\sigma _ 2 ^ 2 } { \\rho _ 1 ^ 2 - \\rho _ 2 ^ 2 } - \\rho _ 2 e ^ { i \\varphi _ 2 } \\frac { \\rho _ 2 ^ 2 - \\sigma _ 2 ^ 2 } { \\rho _ 1 ^ 2 - \\rho _ 2 ^ 2 } \\right ) u _ 1 ^ K . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\phi _ { \\breve { \\mathbf g } , p } = p ^ { - 1 } \\omega _ p ( \\varpi _ p , h _ p ) \\phi _ { \\mathbf g ^ { \\sharp } , p } = p ^ { - 1 } \\omega _ p ( \\varpi _ p , h _ p ) \\phi _ { \\mathbf g , p } \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} z _ i ^ * = \\min \\{ 1 , { a _ i } ^ { \\frac { 1 } { p - 1 } } \\} , \\ , \\ , \\ , \\ , i = 1 , 2 , \\ldots , n , \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} U ^ { \\ast } ( \\varphi , \\theta ) = \\bar { m } ^ { 5 / 2 } \\sum _ { i = 1 } ^ { 3 } \\frac { ( \\mu _ { j } \\mu _ { k } ) ^ { 3 / 2 } ( \\mu _ { i } ^ { \\ast } ) ^ { - 1 / 2 } } { \\sqrt { ( 1 - \\sin \\varphi \\cos ( \\theta - \\theta _ { i } ) } } \\mu _ { i } ^ { \\ast } = \\frac { 1 } { 2 } ( 1 - \\mu _ { i } ) \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} H ( a ) = \\sum _ w \\gamma _ { a , w } ( t ) ~ w , a \\in Q _ 1 . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} \\mathcal O = \\sigma \\ ( A X , R _ \\alpha ( \\theta ) , \\ , Z ( \\theta ) ; \\alpha \\in \\ker A \\ ) \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} \\overline { V } ( 2 , f _ { ( 1 , 1 ) } , \\pi _ n ) \\overset { \\mathbb { P } } { \\longrightarrow } \\int _ 0 ^ T \\rho _ s \\sigma ^ { ( 1 ) } _ s \\sigma ^ { ( 2 ) } _ s d { H _ { 0 , 0 , 2 } } ( s ) = \\int _ 0 ^ T \\rho _ s \\sigma ^ { ( 1 ) } _ s \\sigma ^ { ( 2 ) } _ s d s \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\alpha = 0 \\beta = & - 6 4 0 , - 6 5 0 , - 6 5 6 , - 6 6 0 , - 6 6 2 , - 6 6 4 , - 6 6 8 , - 6 7 2 , - 6 7 6 , \\\\ & - 6 7 8 , - 6 8 0 , - 6 8 4 , - 6 8 6 , - 6 8 8 , - 6 9 0 , - 6 9 2 , - 6 9 4 , - 6 9 6 , \\\\ & - 6 9 8 , \\\\ \\alpha = 1 \\beta = & - 6 5 0 , - 6 6 8 , - 6 8 0 , - 6 8 2 , - 6 8 6 , - 6 8 8 , - 6 9 2 , - 6 9 4 , - 6 9 6 , \\\\ & - 6 9 8 , - 7 0 0 , - 7 0 2 , - 7 0 4 , - 7 1 2 , - 7 2 2 , - 7 3 8 , - 7 4 8 , \\\\ \\alpha = 2 \\beta = & - 6 7 2 , - 7 2 0 , - 7 3 2 , - 7 3 4 \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} \\gamma _ 1 & = \\begin{pmatrix} 0 , & 1 \\\\ 1 , & 0 \\end{pmatrix} & \\gamma _ 2 & = \\begin{pmatrix} 0 , & - i \\\\ i , & 0 \\end{pmatrix} & \\gamma _ 3 & = \\begin{pmatrix} 1 , & 0 \\\\ 0 , & - 1 \\end{pmatrix} \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} \\Delta ( \\hat { u } _ 2 ( x , k ) - \\hat { u } _ 1 ( x , k ) ) + \\frac { k ^ 2 } { c _ 2 ^ 2 ( x ) } \\hat { u } _ 2 ( x , k ) - \\frac { k ^ 2 } { c _ 1 ^ 2 ( x ) } \\hat { u } _ 1 ( x , k ) = - \\frac { i k } { 2 \\pi } \\left ( \\frac { f _ 2 ( x ) } { c _ 2 ^ { 2 } ( x ) } - \\frac { f _ 1 ( x ) } { c _ 1 ^ { 2 } ( x ) } \\right ) , \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} \\gamma _ j \\wedge d F _ i \\wedge d F _ j \\wedge d F _ k = 0 , \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} e = \\begin{cases} ( c d , d ) _ p ( d , p ^ n ) _ p & c \\neq 0 \\mathrm { o r d } _ p ( c ) \\\\ ( d , p ^ n ) _ p & \\end{cases} \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} F _ 1 ^ L ( \\vec x ) = \\int _ 0 ^ { \\norm { \\vec x } } \\vert \\partial V _ t ^ + ( \\vec x ) \\vert + \\vert \\partial V _ t ^ - ( \\vec x ) \\vert d t . \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} M = P D _ L P ^ T , \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} { \\left | { \\hat h _ { \\min } ^ { { \\rm { r o u g h } } } } \\right | ^ { \\rm { 2 } } } = { \\sum \\limits _ { { { \\bf { r } } _ n } \\in { I _ 1 } } ^ { } { \\min \\left ( { { { \\left | { r _ { n , 1 } ^ { } } \\right | } ^ { \\rm { 2 } } } , { { \\left | { r _ { n , 2 } ^ { } } \\right | } ^ { \\rm { 2 } } } } \\right ) } } / { \\left | { { I _ 1 } } \\right | } - { N _ 0 } \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} { r } _ { \\delta } ( x ) : = \\sup \\left \\{ r \\in [ 0 , \\infty ) : \\sup _ { B _ r ( x ) } \\lvert \\Phi \\rvert < m \\delta \\right \\} \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} \\frac { \\tilde { S } _ { k , n } } { \\sum \\limits _ { \\substack { I ^ { c } _ { k + l } ( n ) } } \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k + l } ; n ) } = 1 + \\frac { \\sum \\limits _ { \\substack { I _ { k + l } ( n ) } } \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k + l } ; n ) } { \\sum \\limits _ { \\substack { I ^ { c } _ { k + l } ( n ) } } \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k + l } ; n ) } , \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\breve { \\nabla } _ { W } \\breve { J } U = \\breve { J } \\breve { \\nabla } _ { W } U \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} R i c ( \\omega _ { i _ 0 , \\epsilon } ( t ) ) = - \\omega _ { i _ 0 , \\epsilon } ( t ) + t \\omega _ { i _ 0 , \\epsilon } . \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ N I _ n ( x + k ) = \\sum _ { k = 0 } ^ N \\sum _ { m = 0 } ^ k \\binom { k } { m } m ! S ( n , m ; x ) \\\\ & = \\sum _ { m = 0 } ^ N m ! S ( n , m ; x ) \\sum _ { k = m } ^ N \\binom { k } { m } = \\sum _ { m = 0 } ^ { n \\wedge N } \\binom { N + 1 } { m + 1 } m ! S ( n , m ; x ) , \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\int _ { \\mathcal L _ { 2 j } } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h = - p \\int _ { \\mathcal L _ { 2 j } } 1 d h = - p \\mathrm { v o l } ( \\mathcal L _ { 2 j } ) = - p ^ { 1 - 2 j } ( 1 - p ^ { - 1 } ) ^ 2 , \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\varepsilon ^ { 2 s } M ( [ u ] _ { s , A _ \\varepsilon } ^ 2 ) ( - \\Delta ) _ { A _ \\varepsilon } ^ s u + V ( x ) u = | u | ^ { 2 _ s ^ \\ast - 2 } u + h ( x , | u | ^ 2 ) u , \\ x \\in \\mathbb { R } ^ N , \\smallskip \\smallskip \\\\ u ( x ) \\rightarrow 0 , \\mbox { a s } \\ | x | \\rightarrow \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} r ^ { ( \\jmath ) } : = \\begin{cases} - 2 | \\jmath - \\imath | & \\ 1 \\leq \\jmath \\leq n - 1 , \\\\ - 2 ( n - \\imath ) + 1 & \\ \\jmath = n , \\\\ - 2 ( \\jmath - \\imath ) + 2 & \\ n + 1 \\leq \\jmath \\leq 2 n - 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} G ( z \\ , | \\ , \\tau ) \\triangleq \\exp \\left ( a ( \\tau ) \\frac { z } { \\tau } + b ( \\tau ) \\frac { z ^ 2 } { 2 \\tau ^ 2 } \\right ) \\frac { z } { \\tau } \\prod _ { m , \\ , n = 0 } ^ \\infty { } ' \\left ( 1 + \\frac { z } { \\Omega } \\right ) \\exp \\left ( - \\frac { z } { \\Omega } + \\frac { z ^ 2 } { 2 \\Omega ^ 2 } \\right ) , \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} F ^ \\infty ( P ) = \\lim _ { t \\to \\infty } \\frac { 1 } { t } \\ , F ( t P ) , \\ P \\in \\R ^ { N \\times n } , \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} \\frac { w ( e _ 1 ) } { B ( v , e _ 1 ) } = \\frac { w ( e _ 2 ) } { B ( v , e _ 2 ) } = \\cdots = \\frac { w ( e _ d ) } { B ( v , e _ d ) } . \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} \\mathbb { G } ^ { ( N ) } ( f ) = \\mathbb { G } ( f ) - \\sum _ { k = 1 } ^ { N } \\Phi _ { k } ^ { ( N ) } ( f ) ^ { t } \\cdot \\mathbb { G } \\left [ \\mathcal { A } ^ { ( k ) } \\right ] . \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} \\Delta \\coloneqq \\begin{pmatrix} \\Omega _ 0 ( 0 , \\mathcal { I } _ m ) & \\Omega _ 1 ( 0 , \\mathcal { I } _ m ) & \\bar { \\Omega } ( 0 , \\mathcal { I } _ m ) & \\Omega _ 2 ( 0 , \\mathcal { I } _ m ) \\\\ \\Omega _ 0 ( 0 , \\mathcal { I } _ { \\bar { m } } ) & \\Omega _ 1 ( 0 , \\mathcal { I } _ { \\bar { m } } ) & \\bar { \\Omega } ( 0 , \\mathcal { I } _ { \\bar { m } } ) & \\Omega _ 2 ( 0 , \\mathcal { I } _ { \\bar { m } } ) \\end{pmatrix} \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } 1 _ { A _ { j } ^ { ( N ) } } ( X ) f _ { k } ( X ) = 1 _ { A _ { j } ^ { ( N ) } } ( X ) f ( X ) \\quad P ( 1 _ { A _ { j } ^ { ( N ) } } | f _ { k } | ) \\leqslant P \\left ( | f _ { k } | \\right ) \\leqslant M , \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} w = | \\nabla \\ln ( \\beta - h ) | ^ 2 = \\frac { | \\nabla h | ^ 2 } { ( \\beta - h ) ^ 2 } \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} ( f \\cdot _ \\sigma b ) \\ast _ g a = \\sum _ { i = 1 } ^ r h _ i \\cdot _ \\sigma ( f \\ast _ { g } p _ i ) . \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} \\bar { M } = C ( M ^ { \\ast } ) : d \\bar { s } ^ { 2 } = d \\rho ^ { 2 } + \\frac { \\rho ^ { 2 } } { 4 } ( d \\varphi ^ { 2 } + \\sin ^ { 2 } ( \\varphi ) d \\theta ^ { 2 } ) \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} \\| v _ n \\| _ { L ^ { \\infty } ( - 2 , 0 ; \\dot B ^ { s _ p } _ { p , p } ) } = \\| u \\| _ { L ^ { \\infty } ( t _ n , T _ * ; \\dot B ^ { s _ p } _ { p , p } ) } & \\leq \\widetilde { M } , \\\\ \\| v _ { 0 , n } \\| _ { \\dot B ^ { s _ p } _ { p , p } } = \\| u ( t _ n ) \\| _ { \\dot B ^ { s _ p } _ { p , p } } & \\leq \\widetilde { M } . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} P _ { B S } ^ { \\textrm { o p t } } \\ ! = \\ ! \\left [ \\frac { - ( 2 \\theta A \\ ! + \\ ! d B \\theta ) \\ ! + \\ ! \\sqrt { ( 2 \\theta A \\ ! + \\ ! d B \\theta ) ^ { 2 } \\ ! - \\ ! 8 \\theta A d ( B \\theta \\ ! - \\ ! a d ) } } { 4 \\theta A d } \\right ] ^ { + } \\ ! \\ ! . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\| C _ \\phi ( f ) \\| _ { 2 , \\omega } ^ 2 & = | f ( \\phi ( 0 ) ) | ^ 2 + \\| f _ + \\circ \\phi \\| _ { 2 , \\omega } ^ 2 + \\| f _ - \\circ \\phi \\| _ { 2 , \\omega } ^ 2 \\\\ & \\le | f ( \\phi ( 0 ) ) | ^ 2 + \\Big ( \\frac { 1 + r } { 1 - r } \\Big ) ^ { 2 \\Lambda } \\Big ( \\| f _ + \\| _ { 2 , \\omega } ^ 2 + \\| f _ - \\| _ { 2 , \\omega } ^ 2 \\Big ) . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N - 1 } \\int _ { w _ n } ^ { w _ { n + 1 } } \\frac { f ( z ) - f ( z _ 0 ) } { z - z _ 0 } \\ , d z \\ = \\ \\int _ T \\frac { f ( z ) - f ( z _ 0 ) } { z - z _ 0 } \\ , d z . \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} \\mathcal P _ { \\chi } ( \\mathfrak R _ M \\mathbf F _ { \\chi } , \\mathbf Y _ M \\mathbf G ) = C _ 3 ^ 2 M ^ { 3 - k } | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 . \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} \\chi _ { k } ( b d ) = \\epsilon ( b ) \\chi _ { k } ( d ) + \\chi _ { k } ( b ) f _ { k } ( d ) . \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} \\Pi ( V ^ { r , s } ) = \\bigcup _ { j = 0 } ^ { \\min \\{ r , s \\} } \\{ \\lambda \\in \\Z ^ n | \\sum _ { i } | \\lambda _ i | = r + s - 2 j , \\sum _ i \\lambda _ i = r - s \\} \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\| v \\otimes u \\| _ { L ^ 2 ( 0 , T ; L ^ 2 ) } \\leq C \\| v \\| _ { L ^ r ( 0 , T ; L ^ q ) } \\| u \\| ^ { 1 - \\theta } _ { L ^ { \\infty } ( 0 , T ; L ^ 2 ) } \\| u \\| ^ { \\theta } _ { L ^ { 2 } ( 0 , T ; \\dot H ^ 1 ) } , \\ \\ \\ \\theta = d / q . \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\tau & = { \\pi _ 0 ^ r ( C _ { 1 0 } - C _ { 0 0 } ) \\over \\pi _ 1 ^ r ( C _ { 0 1 } - C _ { 1 1 } ) } \\ , , \\\\ k _ 0 & = \\pi _ 0 ^ t \\sqrt { \\pi _ 1 ^ r \\over \\pi _ 0 ^ r } \\sqrt { ( C _ { 1 0 } - C _ { 0 0 } ) ( C _ { 0 1 } - C _ { 1 1 } ) } \\ , , \\\\ k _ 1 & = \\pi _ 1 ^ t \\sqrt { \\pi _ 0 ^ r \\over \\pi _ 1 ^ r } \\sqrt { ( C _ { 1 0 } - C _ { 0 0 } ) ( C _ { 0 1 } - C _ { 1 1 } ) } \\ , . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} | \\tilde \\rho | ^ q _ q = t ^ { N - q } | \\rho | _ q ^ q , \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} \\widetilde { e } ( M _ { 1 } , M _ { 1 } ) & = 1 + 2 ( p - 1 ) + ( 2 p - 3 ) p + 4 = 2 p ^ { 2 } - p + 3 , \\\\ \\widetilde { e } ( C _ { p } ^ { 3 } , M _ { 1 } ) & = 2 + 2 p - 1 = 2 p + 1 , \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} \\frac { a _ { t _ 1 } ^ { i _ { t _ 1 } } q ^ { \\binom { i _ { t _ 1 } } { 2 } + i _ { t _ 1 } } } { ( q ; q ) _ { i _ { t _ 1 } } } . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\partial _ t ( u ^ 2 _ t , m _ t ) & = ( u _ t ^ 2 , \\Delta m _ t ) - 2 ( | \\nabla u _ t | ^ 2 , m _ t ) - 2 ( F ( u _ t ) , \\nabla ( u _ t m _ t ) ) \\\\ & - ( u _ t ^ 2 , \\mathrm { d i v } ( \\beta _ j m _ t ) ) \\dot { Z } _ t ^ j + ( u _ t ^ 2 , \\partial _ t m _ t ) \\\\ & = ( u _ t ^ 2 , \\Delta m _ t + \\partial _ t m _ t - \\mathrm { d i v } ( \\beta _ j m _ t ) \\dot { Z } _ t ^ j ) - 2 ( | \\nabla u _ t | ^ 2 , m _ t ) - 2 ( F ( u _ t ) , \\nabla ( u _ t m _ t ) ) . \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\overline { \\Lambda } : = \\Lambda \\cup \\partial _ { e x } \\Lambda . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} \\lim _ { \\sigma \\to 0 ^ + } \\langle \\ ! \\langle G , Q ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) k _ 0 \\rangle \\ ! \\rangle = \\langle \\ ! \\langle G , k _ t \\rangle \\ ! \\rangle . \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} | \\Q | = \\left [ \\left ( { n \\over k } \\right ) ! \\right ] ^ k . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} M _ X = \\begin{array} { c | c c c c } \\nearrow & A & B & C & D \\\\ \\hline A & 0 & \\ ? & \\ ? & \\ ? \\\\ B & 2 & 0 & \\infty & \\ ? \\\\ C & \\ ? & \\ ? & \\ ? & \\ ? \\\\ D & \\ ? & \\ ? & \\ ? & \\ ? \\end{array} \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} v _ k ( x ) & = i s \\ , u _ k ( x ) - f _ k ( x ) , & x \\in ( - 1 , 0 ) , \\\\ u _ k '' ( x ) & = ( k ^ 2 \\pi ^ 2 - s ^ 2 ) u _ k ( x ) - i s \\ , f _ k ( x ) - g _ k ( s ) , & x \\in ( - 1 , 0 ) , \\\\ w _ k '' ( x ) & = ( k ^ 2 \\pi ^ 2 + i s ) w _ k ( x ) - h _ k ( x ) , & x \\in ( 0 , 1 ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\omega ( n ) = | a _ f ( n ) | ^ 2 , \\quad \\omega ' ( n ) = \\overline { a _ f ( n ) } \\lambda _ f ( n ) \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} & ( i ) x y ^ { 2 } + \\frac { 2 \\omega } { K _ { 0 } } y = 4 \\mathfrak { S } _ { 0 } \\\\ & ( i i ) \\frac { z } { y } - \\omega \\lbrack J _ { 3 } \\frac { 1 } { x y } + J _ { 4 } y \\ ] = J _ { 2 } \\\\ & ( i i i ) x ( z ^ { 2 } - 2 h ) + \\frac { \\omega ^ { 2 } } { x } - \\frac { \\omega } { 2 K _ { 0 } } y = J _ { 1 } \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} V ' _ I \\Lambda _ k = \\sum _ { j = 0 } ^ n \\binom { k + j - 1 } { k - 1 } V ' _ { I [ n - j ] , k + j } . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} p ( t ) = \\frac { 2 p t _ p } { ( 2 - p ) t + p t _ p } \\ , . \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\tilde { f } ( t , x ) = \\Bigg \\{ \\begin{array} { c l } f ( t , x ) , & x \\in X t \\in T , \\\\ - \\phi ( x ) , & x \\in X t = \\omega _ 1 , \\\\ \\delta _ { K } ( x ) , & x \\in X t = \\omega _ 2 . \\end{array} \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\rho _ { e } ^ { \\textrm { o p t } } = \\frac { \\theta \\xi \\| \\mathbf { h } \\| ^ { 2 } ( N _ { T } - 1 ) \\{ [ 1 - ( 1 - \\varepsilon ) ^ { \\frac { 1 } { K } } ] ^ { \\frac { 1 } { 1 - N _ { T } } } - 1 \\} \\delta _ { e } ^ { 2 } } { ( 1 - \\theta ) \\gamma _ { e } ^ { 2 } } . \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} \\begin{pmatrix} a & - b & 0 \\\\ a & 0 & c \\end{pmatrix} , \\begin{pmatrix} a & - b & 0 \\\\ 0 & - b & c \\end{pmatrix} , \\begin{pmatrix} a & 0 & - c \\\\ 0 & b & - c \\end{pmatrix} . \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{gather*} | \\pi _ v | _ v = \\frac { 1 } { N v } { \\rm a n d } | u _ v | _ v = 1 { \\rm f o r } \\ \\ u _ v \\in \\mathcal U _ v \\end{gather*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\frac { \\partial \\varphi _ 0 } { \\partial F } ( F , t ) = \\sigma - \\frac { p - 2 + m } { m - 1 } \\ , \\gamma \\ , F ^ { \\frac { p - 1 } { m - 1 } } \\ , ; \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} \\mathcal W = 1 + g ( D ) , \\mathcal Z = \\frac { 1 } { \\gamma } \\Big ( 1 + \\frac { \\beta - 1 } { \\gamma } \\sqrt { \\mu } | D | \\coth ( \\sqrt { \\mu _ 2 } | D | ) \\Big ) , \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} \\left ( P ( B ) \\right ) ^ { N + 1 } ( k ^ d \\lambda ^ k ) & = \\sum _ { r = 0 } ^ d P ( \\lambda ) ^ { N + r - d } A _ { d , N , r } \\sum _ { s = 0 } ^ r Q _ { r , s } ( \\lambda ) \\left ( k ^ s \\lambda ^ k \\right ) \\\\ & = \\sum _ { s = 0 } ^ d \\sum _ { r = s } ^ d P ( \\lambda ) ^ { N + r - d } A _ { d , N , r } Q _ { r , s } ( \\lambda ) \\left ( k ^ s \\lambda ^ k \\right ) . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} \\prod _ { d | m , d > 1 } \\Phi _ d ( \\nu ) & = [ m ] _ { \\nu } , & \\prod _ { p _ k | d , d | m } \\Phi _ d ( \\nu ) & = \\frac { [ m ] _ { \\nu } } { [ m / p _ k ] _ { \\nu } } . \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} \\left \\{ \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} h _ 1 ( \\lambda ) , \\ldots , \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} h _ p ( \\lambda ) \\right \\} , \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} E _ 0 ( f ) = \\lim \\limits _ { n \\to \\infty } \\inf \\frac 1 n \\log f ( n ) . \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) ( 2 \\pi ) ^ { - \\frac { q } { \\tau } } \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) ( 2 \\pi ) ^ { - q } \\times \\\\ & \\times \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\left \\langle \\mathbf { K } \\left [ 1 - e ^ { - h } \\right ] ( e _ n ) , e _ n \\right \\rangle & = \\int _ { E \\times E } \\sqrt { 1 - e ^ { - h ( x ) } } K ( x , y ) \\sqrt { 1 - e ^ { - h ( y ) } } e _ n ( x ) e _ n ( y ) d x d y \\\\ & = \\int _ { E \\times E } \\sum _ { i = 1 } ^ { + \\infty } \\lambda _ i \\varphi _ i ( x ) \\varphi _ i ( y ) e _ n ( x ) e _ n ( y ) d x d y \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} ( U _ { j - 1 } ^ * U ''' _ j L ) = - d _ { j - 1 } + ( U ''' _ j ) + t < - d _ { j - 1 } + \\frac { d } { r } + t \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\psi ( \\gamma ^ 0 \\phi ) = \\psi ( \\varphi ^ 0 \\alpha \\phi ) . \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} Q ( \\sum _ { i = 1 } ^ n \\lambda _ i e _ i ) = \\sum _ { i = 1 } ^ { \\frac { k b } { 2 } } \\lambda _ i \\lambda _ { i + \\frac { k b } { 2 } } + Q ' ( \\sum _ { i = k b + 1 } ^ n \\lambda _ i e _ i ) \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q V _ N } ] = \\lim _ { \\beta \\rightarrow \\infty } \\Bigl [ { \\bf E } \\bigl [ X ^ { \\frac { q } { \\beta } } \\bigr ] \\Gamma ( 1 - \\frac { q } { \\beta } ) \\Bigr ] . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} \\delta ( \\wp ) : = \\begin{cases} 1 & \\ \\wp \\ , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} \\frac { 1 } { \\beta _ { v _ 1 } } | a | ^ 2 = r | a | ^ 2 = r | a + b - b | ^ 2 \\leq p | a + b | ^ 2 + q | b | ^ 2 = \\frac { 1 } { \\beta _ { v _ 0 } } | a + b | ^ 2 - \\frac { 1 } { \\beta _ { v _ 2 } } | b | ^ 2 . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} \\dot { \\phi } ( n ) = \\begin{cases} - \\displaystyle \\sum _ { j = 0 } ^ { k - 1 } ( 2 j + 1 ) ^ { - N - \\frac { 1 } { 2 } } , & n = 2 k \\\\ - \\displaystyle \\sum _ { j = 1 } ^ { k } ( 2 j ) ^ { - N - \\frac { 1 } { 2 } } , & n = 2 k + 1 . \\end{cases} \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} F _ \\varepsilon ( u ) : = \\int _ 0 ^ { \\infty } \\int _ { \\mathbb { R } ^ n } e ^ { - t / \\varepsilon } \\ , \\left ( | u '' ( t , x ) | ^ 2 + \\tfrac { 1 } { \\varepsilon ^ 2 } | \\nabla u ( t , x ) | ^ 2 + \\tfrac { 1 } { \\varepsilon ^ 2 } | u ( t , x ) | ^ { 2 k } \\right ) \\ , d x \\ , d t \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} H _ K ( \\beta ) \\begin{cases} \\infty , & \\beta \\in [ 0 , \\beta _ P ( d ) ] , \\\\ \\in ( 0 , \\infty ) , & \\beta \\in ( \\beta _ P ( d ) , \\beta _ P ( d ) + \\epsilon _ d ] , \\\\ \\in [ 0 , \\infty ) , & \\beta \\in ( \\beta _ P ( d ) + \\epsilon _ d , \\beta _ c ( d ) ) , \\\\ 0 , & \\beta \\geq \\beta _ c ( d ) . \\end{cases} \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\alpha ' = r ^ { p ' - r } \\bigg ( \\frac { \\Delta ^ { r ( p ' - p ) } } { ( \\lambda ^ { ( p ) } ( G ) ) ^ { p ( p ' - r ) } } \\bigg ) ^ { 1 / ( p - r ) } . \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} F _ { \\eta } = \\prod _ { k = 0 } ^ { \\eta - 1 } 7 \\sqrt { \\frac { 2 } { \\pi c } } \\frac { ( \\pi c ) ^ k } { k ! } e ^ { - \\frac { \\pi } { 2 } c } = e ^ { - \\frac { \\pi } { 2 } c \\eta } \\left ( 7 \\sqrt { \\frac { 2 } { \\pi c } } \\right ) ^ { \\eta } A _ { \\eta } , A _ { \\eta } = \\prod _ { k = 0 } ^ { \\eta - 1 } \\frac { ( \\pi c ) ^ k } { k ! } . \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} \\cup _ { \\psi \\in \\Gamma } [ \\neg \\phi ] ^ { \\beta } \\cup \\cup _ { \\psi \\in \\Delta } [ \\psi ] ^ { \\beta } \\cup \\cap _ { n \\in L ^ 0 ( \\mathbb { N } ) } [ \\phi ] ^ { \\beta [ \\frac { n } { x } ] } = \\Omega \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} f ( t ) = t ^ M \\prod \\limits _ { j = 1 } ^ M ( 1 - e ^ { - a _ j t } ) ^ { - 1 } \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} \\forall x \\in \\gamma \\sum _ { \\eta \\Subset \\gamma } \\prod _ { z \\in \\eta } g ( z ) = ( 1 + g ( x ) ) \\sum _ { \\eta \\Subset \\gamma \\setminus x } \\prod _ { z \\in \\eta } g ( z ) , \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\overline { \\omega } \\circ \\begin{pmatrix} \\gamma _ 0 & 0 \\\\ 0 & \\gamma _ { - d } \\end{pmatrix} = ( \\omega _ 0 \\circ \\gamma _ 0 , \\ , \\omega _ { - d } \\circ \\gamma _ { - d } ) = \\nu \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} \\widetilde { c } _ { j i } ( - 2 k ) = - \\widetilde { c } _ { j i } ( - 2 ( k - 2 ) ) + \\widetilde { c } _ { j + 1 i } ( - 2 ( k - 1 ) ) + \\widetilde { c } _ { j - 1 i } ( - 2 ( k - 1 ) ) , \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} W = T W + Q W , \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} P _ { \\epsilon , h } [ f ] ( y ) = \\frac { 1 } { h ^ \\epsilon } \\sum _ { 0 \\leq \\gamma \\leq \\epsilon } ( - 1 ) ^ { \\abs { \\gamma } + 1 } C ^ \\gamma _ \\epsilon f ( y + \\gamma h ) . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} L ( t ) u : = \\sum \\limits _ { k = 1 } ^ { \\infty } \\langle e _ k , u \\rangle \\sigma _ k \\ell _ k ( t ) t \\in [ 0 , T ] , \\ , u \\in U , \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} x _ i = x _ 0 - \\sum _ { j = 1 } ^ { i - 1 } \\frac { \\tilde \\gamma _ { i , j } } { \\tilde \\gamma _ { i , i } } ( x _ j - x _ 0 ) - \\sum _ { j = 0 } ^ { i - 1 } \\frac { \\tilde \\beta _ { i , j } } { \\tilde \\gamma _ { i , i } } f ' ( x _ j ) , i = 1 , \\hdots , N , \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} A _ 2 ( z ) \\sum _ { t _ n \\in T _ 2 } \\frac { G _ 2 ( t _ n ) \\bar c _ n } { A _ 2 ' ( t _ n ) ( \\mu ^ { ( 2 ) } _ n ) ^ { 1 / 2 } ( z - t _ n ) } = G _ 2 ( z ) S _ 2 ( z ) \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} b _ s ^ { } = \\begin{cases} ( - 1 ) ^ { 2 n + 1 - m - s } q \\sigma _ { p } ' & \\\\ ( - 1 ) ^ { 2 n + 1 - m - s } q ( \\sigma _ { p } ' + \\tau _ { 1 ^ { p - 2 n - 2 + 2 m } } ) & \\end{cases} \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} \\Phi _ { \\mathbf h _ p } ( \\nu \\beta _ 1 \\nu ' ) = \\frac { \\chi _ { \\psi } ( p ) \\underline { \\chi } _ p ( \\gamma ) p ^ { - 1 / 2 } } { p - 1 } ( p - G ( - \\gamma , p ) ) . \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} \\Psi _ k ( s , x ) : = x ^ k \\frac { \\Gamma ( k - s / 2 ) \\Gamma ( k + 1 / 2 - s / 2 ) } { \\Gamma ( 2 k ) } { } _ 2 F _ 1 \\left ( k - \\frac { s } { 2 } , k + \\frac { 1 - s } { 2 } , 2 k ; x \\right ) . \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\overline { e } _ { t } ( \\zeta ) = 0 , \\ \\ \\overline { h } _ { t } ( \\zeta ) = - \\frac { \\tau } { 2 } \\rho _ { t } ( \\zeta ) ^ { - 2 } , \\ \\ \\overline { f } _ { t } ( \\zeta ) = 0 . \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} \\partial \\int D = \\left ( \\int \\partial + 1 \\right ) D = c _ p \\otimes \\partial x + w _ p \\otimes x . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} V _ N = 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\log Y + \\log Y ' + o ( 1 ) , \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} \\varphi ( b ' a - a ' b ) = \\varphi ( b ) ' \\varphi ( a ) - \\varphi ( a ) ' \\varphi ( b ) \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} L \\left ( \\left ( \\prod _ { s = 0 } ^ 3 Y _ { 1 , - 4 s } ^ { u _ { 1 , - 4 s } } \\right ) \\cdot Y _ { 1 , - 6 } ^ { u _ { 1 , - 6 } } \\cdot \\left ( \\prod _ { s = 0 } ^ 2 Y _ { 2 , - 2 - 4 s } ^ { u _ { 2 , - 2 - 4 s } } \\right ) \\cdot \\left ( \\prod _ { s = 0 } ^ 1 Y _ { 2 , - 4 - 4 s } ^ { u _ { 2 , - 4 - 4 s } } \\right ) \\cdot \\left ( \\prod _ { s = 0 } ^ 4 Y _ { 3 , - 1 - 2 s } ^ { u _ { 3 , - 1 - 2 s } } \\right ) \\right ) \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} H = ( \\mathfrak { d } _ 2 \\dot { \\phi } ( i + j ) ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} ( b / x ) ( b \\backslash ( z x ) ^ { \\alpha } x ) = b ( ( z x ) \\backslash \\left [ z ( b \\backslash ( z x ) ^ { \\alpha } x ) \\right ] ) \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ K | \\mathcal { G } _ i | = \\sum _ { i = 0 } ^ K \\gamma _ i = 2 ^ M . \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} b a c & \\equiv _ 2 b c a \\\\ a b c & \\equiv _ 2 a c b . \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\int _ { 0 } ^ T \\int _ { \\R ^ n } m \\ , U ^ { m - 1 } \\left | \\nabla v _ + \\right | ^ 2 \\ , d x d t + \\frac { 1 } { 2 } \\int _ { \\R ^ n } v _ + ^ 2 \\left ( x , T \\right ) \\ , d x \\ , \\leq \\ , \\frac { 1 } { 2 } \\int _ { \\R ^ n } v _ + ^ 2 \\left ( x , 0 \\right ) \\ , d x . \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\displaystyle \\tilde { t } = t , \\tilde { x } = x , \\tilde { y } = \\frac { y } { \\sqrt { \\varepsilon } } , \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} | z ^ * - z | = \\sqrt { ( a - \\alpha ) ^ 2 + ( b - \\beta ) ^ 2 } \\leq | a - \\alpha | + | b - \\beta | \\leq \\epsilon + \\epsilon ' = : \\hat \\epsilon , \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} 0 = \\frac { c _ { \\ell _ 0 } } { n r _ { 0 c _ { \\ell _ 0 } } } \\sum _ { i \\in I } \\left ( \\left ( - n + \\frac { n } { c _ { \\ell _ 0 } } \\right ) r _ { i c _ { \\ell _ 0 } } + \\sum _ { k \\neq i } \\frac { n } { c _ { \\ell _ 0 } } r _ { k c _ { \\ell _ 0 } } \\right ) = ( - c _ { \\ell _ 0 } + | I | ) \\sum _ { i \\in I } \\frac { r _ { i c _ { \\ell _ 0 } } } { r _ { 0 c _ { \\ell _ 0 } } } + | I | . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} g ( ( - t _ i + ( q ^ 2 - 1 ) 1 _ \\omega ) v ) = g ( t _ i ^ \\sharp v ) = \\tau ( t _ i ) g ( v ) = t _ { m + n - r + i } g ( v ) . \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} d _ { n + 1 } = d _ { n + 2 } \\ge d _ { n + 3 } = d _ { n + 4 } \\ge \\cdots \\ge d _ { l - 1 } = d _ l . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\textnormal { l o g } \\mathcal { P } _ { X _ { A } } ( n ) } { n } = h _ { t o p } ( \\sigma _ { A } ) . \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} & \\iota _ l ( x ) = ( x , 0 ) \\\\ & p _ l ( x , y ) = y \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} B ( v , e ) & = \\begin{dcases} \\frac { \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G [ S ] ) x _ v ^ p } , & ~ v \\in e ~ ~ v \\in S , \\ , e \\in E ( G [ S ] ) , \\\\ 0 , & , \\end{dcases} \\\\ [ 2 m m ] w ( e ) & = \\frac { r \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G [ S ] ) } . \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} D _ s X ( t ) ( \\omega ) = & \\sigma ( s , \\omega , X ( s ) ( \\omega ) ) + \\int _ s ^ t U ( s , r , \\omega ) d r + \\int _ s ^ t V ( s , r , \\omega ) d W ( r ) \\\\ & + \\int _ s ^ t \\nabla _ x b ( r , \\omega , X ( r ) ( \\omega ) ) D _ s X ( r ) ( \\omega ) d r + \\int _ s ^ t \\nabla _ x \\sigma ( r , \\omega , X ( r ) ( \\omega ) ) D _ s X ( r ) ( \\omega ) d W ( r ) , \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} ( { \\mathcal R } { f } ) ( r ) = \\left \\{ \\begin{array} { c l } { f } ( r ) , & r > 0 , \\\\ 0 , & r \\leq 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ N \\frac { \\psi ( n ) } { \\log n } = N \\frac { \\psi ( N ) } { \\log N } - 2 \\frac { \\psi ( 2 ) } { \\log 2 } + \\sum _ { n = 2 } ^ { N - 1 } n \\bigg ( \\frac { \\psi ( n ) } { \\log n } - \\frac { \\psi ( n + 1 ) } { \\log ( n + 1 ) } \\bigg ) . \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { c , d } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\overline { E } _ { a , b } \\overline { E } _ { c , d } } q _ b E _ { c , d } E _ { a , b } , & M = N = 1 , b = d ; \\\\ ( - 1 ) ^ { \\overline { E } _ { a , b } \\overline { E } _ { c , d } } q _ a E _ { c , d } E _ { a , b } , & M = N = 1 , a = c ; \\\\ q _ b ^ { M N } E _ { c , d } ^ { ( N ) } E _ { a , b } ^ { ( M ) } , & . \\end{cases} \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} f ( x , y ) : = \\begin{cases} \\langle A x , y - x \\rangle , & \\ x , y \\in C , \\\\ + \\infty , & . \\end{cases} \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} A _ y = A - \\mathbf Y ( y ) \\colon U \\to Y . \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} C = \\frac { \\theta \\| \\mathbf { h } \\| ^ { 4 } } { 2 A } , ~ D = \\frac { \\theta B \\| \\mathbf { h } \\| ^ { 2 } } { 4 A } , ~ a = \\mu ( 1 - \\theta ) , ~ d = \\frac { \\xi | h _ { s } | ^ { 2 } } { ( 1 - \\theta ) \\sigma _ { s } ^ { 2 } } . \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} T = \\frac { 1 } { 2 } | \\dot { \\gamma } ^ { h } | ^ { 2 } + \\frac { 1 } { 2 } | \\dot { \\gamma } ^ { \\omega } | ^ { 2 } = T ^ { h } + T ^ { \\omega } , \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} & ( g , \\alpha \\wedge \\beta ) ( x ) \\\\ = ~ & ( d l _ { g ^ { - 1 } } ) ^ * ~ ( \\alpha \\wedge \\beta ) ( g ^ { - 1 } x ) \\\\ = ~ & ( d l _ { g ^ { - 1 } } ) ^ * ~ ( \\alpha ( g ^ { - 1 } x ) \\wedge \\beta ( g ^ { - 1 } x ) ) \\\\ = ~ & ( d l _ { g ^ { - 1 } } ) ^ * \\alpha ( g ^ { - 1 } x ) \\wedge ( d l _ { g ^ { - 1 } } ) ^ * \\beta ( g ^ { - 1 } x ) = \\big ( ( g , \\alpha ) \\wedge ( g , \\beta ) \\big ) ( x ) . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} \\theta _ i ' = \\omega _ i + \\sum _ { j = 1 } ^ n \\gamma _ { i j } H ( \\theta _ j - \\theta _ i ) \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\tilde { D } _ k = \\{ \\alpha \\in \\R ^ d : R _ k ( \\alpha + \\alpha _ k ) \\leq 1 \\} = \\{ \\alpha \\in \\R ^ d : ~ R _ k ( \\alpha ) \\leq 1 \\} - \\alpha _ k = D _ k - \\alpha _ k . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} \\sum \\limits _ { r = 1 } ^ \\infty \\frac { \\zeta ( r + 1 ) } { r + 1 } / \\tau ^ { r + 1 } = \\log \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) + \\frac { \\psi ( 1 ) } { \\tau } , \\ , | \\tau | > 1 . \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} ( d \\cdot \\nabla \\psi ) \\psi + 2 \\nabla \\psi \\nabla w = \\Big ( d + 2 \\frac { \\nabla \\psi } { \\psi } \\Big ) \\nabla ( \\psi w ) - ( d \\cdot \\nabla \\psi ) w - 2 \\frac { | \\nabla \\psi | ^ 2 } { \\psi } w \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} ( x , y , z ) = ( t ^ 5 , t ^ { 1 7 } + s t ^ { 1 8 } , t ^ { 2 8 } ) \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\widetilde { c } _ { n i } ( - 2 k ) = - \\widetilde { c } _ { n i } ( - 2 ( k - 1 ) ) + \\widetilde { c } _ { n - 1 i } ( - 2 k ) + \\widetilde { c } _ { n - 1 i } ( - 2 ( k - 1 ) ) . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{gather*} L \\left ( \\psi _ v , s - \\frac 1 2 \\right ) = \\frac { 1 } { 1 - ( - 1 ) ^ { b / 2 } ( a - b { \\rm i } ) p ^ { - s } } . \\end{gather*}"}
-{"id": "6829.png", "formula": "\\begin{align*} X _ { t } ( x ) = x + \\int _ { 0 } ^ { t } u ( r , X _ { r } ( x ) ) \\ d r + B _ { t } , \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\sum _ { t _ j \\le T } X ^ { i t _ j } = \\int _ { - 1 } ^ 1 \\hat { g } ( \\xi ) \\left ( \\sum _ { j } ( X \\exp ( - 2 \\pi \\xi ) ) ^ { i t _ j } \\exp ( - t _ j / T ) \\right ) d \\xi + O ( T \\log T ) . \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} [ M ( m ) ] = [ L ( m ) ] + [ L ( m ' ) ] \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} p ^ * ( L _ { a , b } ) = p ^ * ( X _ { a , b } ) = \\lambda _ n \\frac { \\Delta _ { \\{ 2 , \\dots , a , n \\} } } { \\Delta _ { \\{ 1 , 2 , \\dots , a \\} } } = \\lambda _ n \\frac { A _ { a - 1 , b - 1 } } { A _ { a , b } } . \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} ( x , y , z ) = ( t ^ 4 , t ^ 7 , t ^ 9 + s t ^ { 1 0 } ) . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} | u | _ \\vartheta \\leq \\| u \\| _ { \\sigma , \\alpha } \\int _ { \\Gamma _ 0 } \\exp \\left ( ( \\alpha + \\vartheta ) | \\eta | \\right ) e ( \\phi _ \\sigma , \\eta ) \\lambda ( d \\eta ) = \\| u \\| _ { \\sigma , \\alpha } \\exp \\left ( ( \\alpha + \\vartheta ) \\langle \\phi \\rangle \\right ) . \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} \\mathcal B ( \\Theta _ z ) = \\sigma ( A \\iota _ z , R _ \\alpha \\iota _ z ; \\alpha \\in \\ker A \\iota _ z ) . \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} \\partial _ t B = 2 ( \\sigma - B ) \\Rightarrow B _ t = e ^ { - 2 t } B _ 0 + \\sigma ( 1 - \\mathrm { e } ^ { - 2 t } ) \\stackrel { t \\rightarrow \\infty } { \\longrightarrow } \\sigma \\ , . \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\iota \\left ( D _ 1 \\otimes \\cdots \\otimes D _ n \\right ) = \\iota \\left ( M _ 1 \\otimes \\cdots \\otimes M _ n \\right ) = \\sum _ { i = 1 } ^ n \\iota ( D _ i ) + \\sum _ { 1 \\leq j < k \\leq n } m \\left ( h _ k , \\partial h _ j \\right ) . \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} i n d e x ( M _ k \\cap B ^ { ( N , g ) } _ { R r ^ { y , \\ell } _ k } ( p ^ { y , \\ell } _ k ) ) = i n d e x ( \\tilde { M } ^ { y , \\ell } _ k \\cap B ^ { ( \\tilde { N } , \\tilde { g } ) } _ R ( 0 ) ) \\geq i n d e x ( \\Sigma ^ y _ \\ell \\cap B _ R ( 0 ) ) = i n d e x ( \\Sigma ^ y _ \\ell ) \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} { \\mathbb Y } _ { a } { } ^ A { } _ { B } : = Y _ a { } ^ A Y _ B , & { \\mathbb Z } _ a { } ^ { b } { } ^ A { } _ { B } : = Y _ a { } ^ A X ^ b { } _ B , \\\\ { \\mathbb W } ^ { A } { } _ { B } : = X ^ A Y _ B , & { \\mathbb X } ^ { b } { } ^ A { } _ { B } : = X ^ A X ^ b { } _ B , \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} t _ { \\beta } t _ { \\gamma } \\cdot t _ b t _ c t _ a t _ b \\cdot t _ { \\alpha } t _ { \\beta } ( a ) & = t _ { \\beta } t _ { \\gamma } \\cdot t _ b t _ c t _ a t _ b ( a ) = t _ { \\beta } t _ { \\gamma } ( c ) = c , \\\\ t _ { \\beta } t _ { \\gamma } \\cdot t _ b t _ c t _ a t _ b \\cdot t _ { \\alpha } t _ { \\beta } ( \\alpha ) & = t _ { \\beta } t _ { \\gamma } \\cdot t _ b t _ c t _ a t _ b ( \\beta ) = t _ { \\beta } t _ { \\gamma } ( \\beta ) = \\gamma , \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} \\lim _ { | s | \\to \\infty } \\frac { f ' ( s ) } { | s | ^ { 4 } } = 0 , \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} \\lambda _ * = \\tfrac { 2 n - 2 p - 1 + 2 m _ - + m _ * } { p + 2 + m _ * } . \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\breve { J } ^ { 2 } = p \\breve { J } + q I , \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} p _ H ( x , \\vartheta ) ~ = ~ 1 - \\sum _ { y \\in \\Z _ + } p _ H ( x , y ) , x \\in \\Z _ + . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\frac { 4 ^ n } { \\sqrt { \\pi ( n + 1 / 2 ) } } \\leq \\binom { 2 n } { n } \\leq \\frac { 4 ^ n } { \\sqrt { \\pi n } } . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} V ( \\alpha ) & \\triangleq \\left \\{ t > 0 \\colon \\tau _ { t } ( \\alpha ) < \\tau _ { t + } ( \\alpha ) \\right \\} , \\\\ V ' ( \\alpha ) & \\triangleq \\left \\{ t > 0 \\colon \\alpha ( \\tau _ t ( \\alpha ) ) \\not = \\alpha ( \\tau _ t ( \\alpha ) - ) \\| \\alpha ( \\tau _ t ( \\alpha ) - ) \\| = t \\right \\} . \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} a ( \\mathfrak f _ { \\xi , N } ) = \\prod _ { p \\mid \\mathfrak f _ { \\xi , N } } a ( p ^ { e _ p } ) = \\mathfrak f _ { \\xi , N } ^ { k - 1 / 2 } \\prod _ { p \\mid \\mathfrak f _ { \\xi , N } } \\alpha _ p ^ { e _ p } = 2 ^ { - \\nu ( \\mathfrak f _ { \\xi , N } ) } \\mathfrak f _ { \\xi , N } ^ { k - 1 / 2 } \\prod _ { p \\mid \\mathfrak f _ { \\xi , N } } \\Psi _ p ( \\xi ; \\alpha _ p ) . \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} f _ n = O ( f ) \\ \\ \\iff \\ \\ \\lim _ { n \\to \\infty } \\frac { f _ n } { f } = C < \\infty \\textrm { a n d } f _ n = o ( f ) \\ \\ \\iff \\ \\ \\lim _ { n \\to \\infty } \\frac { f _ n } { f } = 0 . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\lim _ { | n | \\to \\infty } \\tilde { \\psi } ( n ) = 0 , \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} \\eta = \\frac { \\sigma } { \\hat { \\sigma } } , \\quad \\quad \\hat { \\sigma } ^ 2 = \\dfrac { \\Gamma ( \\nu ) \\ \\nu ^ { d / 2 } } { \\Gamma ( \\nu + d / 2 ) } \\left ( \\frac { 2 } { \\pi } \\right ) ^ { d / 2 } \\lambda ^ { - d } , \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} h _ \\beta ' ( t ) = - ( t + 1 - q ) ^ { - 2 } ( 1 - t ) ^ { 2 \\beta - 1 } \\Bigg \\{ 2 \\beta t ^ 2 + ( 2 \\beta + 1 ) ( 1 - q ) t - ( 1 - q ) \\Bigg \\} . \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} \\rho : = \\rho _ { n } , \\eta : = \\rho _ { n - 1 } , u : = u _ { n } , v : = u _ { n - 1 } . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} & \\| M _ { k n } M _ { k _ 1 n _ 1 } \\| \\leq \\| M _ { k n } N \\| + \\| N N _ 1 \\| + \\| N _ 1 M _ { k _ 1 n _ 1 } \\| \\leq \\\\ & \\leq 4 \\Lambda _ { n } + 1 8 \\frac { 1 } { 8 } \\Lambda _ { n } + 4 \\Lambda _ { n _ 1 } \\leq \\left ( 4 + 1 8 \\frac { 1 } { 8 } + 4 \\cdot 8 \\right ) < 5 5 \\Lambda _ { n } \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\left ( w \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) u \\left ( w \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) ^ { - 1 } & = w \\left ( \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\cdot u \\right ) w ^ { - 1 } u ^ { - 1 } = \\\\ & \\rho ^ { \\frac { 1 } { 2 } a _ { 2 } u _ { 2 } \\left ( u _ { 2 } - 1 \\right ) + a _ { 3 } u _ { 3 } + u _ { 2 } w _ { 3 } - u _ { 3 } w _ { 2 } - a _ { 2 } u _ { 2 } w _ { 2 } - a _ { 2 } u _ { 2 } ^ { 2 } } \\tau ^ { a _ { 2 } u _ { 2 } } \\in \\left \\langle u \\right \\rangle , \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} & \\left | \\frac { 1 } { \\rho _ { M _ 0 } ( M ) } - \\frac { 1 } { \\rho _ { M _ 0 } ( M _ { k , n - 2 } ) } \\right | = \\left | \\frac { 1 } { \\| M _ 0 M \\| } - \\frac { 1 } { \\| M _ 0 M _ { k , n - 2 } \\| } \\right | \\leq \\\\ & \\leq \\frac { C \\Lambda _ { n - 2 } } { { \\| M _ 0 M _ { k , n - 2 } \\| } ^ 2 } \\leq \\frac { C \\Lambda _ { n - 2 } } { { | k - k _ 0 | } ^ 2 \\Lambda ^ { 2 } _ { n - 2 } } \\leq \\frac { C } { \\Lambda _ n { | k - k _ 0 | } ^ 2 } . \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} \\mathcal { O } _ { \\mathbb { P } ( E ) } ( 2 k m ) \\otimes \\pi ^ { * } A ^ { - 1 } & = K _ { \\mathbb { P } ( E ) } \\otimes \\mathcal { O } _ { \\mathbb { P } ( E ) } ( 2 k m ) \\otimes \\pi ^ { * } A ^ { - 2 m } \\otimes \\mathcal { O } _ { \\mathbb { P } ( E ) } ( r ) \\otimes \\pi ^ { * } ( A ^ { - 1 } \\otimes K _ { X } ^ { - 1 } \\otimes \\det E ^ { * } \\otimes A ^ m ) \\otimes \\pi ^ { * } A ^ { m } \\\\ & = K _ { \\mathbb { P } ( E ) } \\otimes \\widetilde { L } \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} { L } ( r , \\vec v ) : = \\min _ { \\| \\vec x \\| _ p = 1 } \\{ r \\norm { \\vec x } - ( \\vec x , \\vec v ) \\} , \\ , \\ , \\ , r \\in \\mathbb { R } , \\ , \\ , \\vec v \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\partial _ { x _ j x _ k } I _ { h , T } \\mathbf { u } ( X ) = \\sum _ { i \\in \\mathcal { I } ^ { s ' } } \\partial _ { x _ j x _ k } \\Phi _ { i , T } ( X ) ( \\mathbf { E } _ i ^ s ( X ) + \\mathbf { F } _ i ^ s ( X ) ) + \\sum _ { i \\in \\mathcal { I } } \\partial _ { x _ j x _ k } \\Phi _ { i , T } ( X ) \\mathbf { R } _ i ^ s ( X ) , ~ s = \\pm , \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} \\left \\langle f , g \\right \\rangle = ( 1 - c ) ^ { \\beta } \\sum _ { x = 0 } ^ \\infty f ( x ) g ( x ) \\frac { c ^ { x } \\Gamma ( \\beta + x ) } { \\Gamma ( \\beta ) \\Gamma ( x + 1 ) } + \\lambda f ( 0 ) g ( 0 ) , \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} ( 1 - 8 y _ { * * } ^ 2 ) ( 2 y _ { * * } ^ 4 + 2 y _ { * * } ^ 2 ) = ( 4 y _ { * * } ^ 4 - 4 ) ( 8 y _ { * * } ^ 4 + 4 y _ { * * } ^ 2 - 1 ) . \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} | \\mathcal { I } _ { i , n } ^ { ( 1 ) } | | \\mathcal { I } _ { j , n } ^ { ( 2 ) } | = ( | \\mathcal { I } _ { i , n } ^ { ( 1 ) } \\cap \\mathcal { I } _ { j , n } ^ { ( 2 ) } | + | \\mathcal { I } _ { i , n } ^ { ( 1 ) } \\setminus \\mathcal { I } _ { j , n } ^ { ( 2 ) } | ) ( | \\mathcal { I } _ { i , n } ^ { ( 1 ) } \\cap \\mathcal { I } _ { j , n } ^ { ( 2 ) } | + | \\mathcal { I } _ { j , n } ^ { ( 2 ) } \\setminus \\mathcal { I } _ { i , n } ^ { ( 1 ) } | ) . \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\deg \\operatorname { I m } ( \\eta \\circ a _ 1 ) + \\dim R = e \\left ( \\frac { R } { ( g _ 0 , \\dots , g _ { n - 2 } ) : I } \\right ) = e \\left ( \\frac { R } { ( g _ 0 , \\dots , g _ { n - 2 } ) : I ^ \\infty } \\right ) . \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{gather*} \\frac { 3 } { 2 } L \\big ( \\eta ( 6 \\tau ) ^ 4 , 1 \\big ) ^ 2 = L \\big ( \\eta ( 2 \\tau ) ^ 3 \\eta ( 6 \\tau ) ^ 3 , 2 \\big ) \\frac 8 3 L \\big ( \\eta ( 6 \\tau ) ^ 4 , 1 \\big ) ^ 3 = L \\big ( \\eta ( 3 \\tau ) ^ 8 , 3 \\big ) . \\end{gather*}"}
-{"id": "6127.png", "formula": "\\begin{align*} ( - 1 ) ^ { m + 1 } \\sum _ { | \\alpha | = 2 m } \\gamma _ \\alpha ( x ) \\xi ^ \\alpha \\geq C | \\xi | ^ { 2 m } \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} \\begin{tabular} { | r c l | } \\hline & N o n - z e r o v a l u e s o f $ \\chi _ { k } $ & \\\\ $ \\chi _ { 0 } ( \\gamma ^ { * } ) = - q ^ { - 1 } $ & $ \\chi _ { 1 } ( \\alpha ) = 1 $ & $ \\chi _ { 1 } ( \\alpha ^ { * } ) = - q ^ { 2 } $ \\\\ $ \\chi _ { 2 } ( \\gamma ) = - q $ & & \\\\ & A l l o t h e r v a l u e s o f $ \\chi _ { k } $ o n t h e g e n e r a t o r s a r e z e r o & \\\\ \\hline \\end{tabular} \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\alpha \\left ( v \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) \\alpha ^ { - 1 } = \\left ( \\alpha \\cdot v \\right ) \\alpha _ { 1 } ^ { a _ { 1 } b _ { 4 } - a _ { 3 } b _ { 3 } + r _ { 3 } a _ { 2 } b _ { 1 } ^ { - 1 } b _ { 4 } + \\frac { 1 } { 2 } a _ { 2 } b _ { 4 } \\left ( b _ { 1 } ^ { - 1 } - 1 \\right ) } \\alpha _ { 2 } ^ { a _ { 2 } b _ { 1 } ^ { - 1 } b _ { 4 } } \\alpha _ { 3 } ^ { a _ { 3 } b _ { 1 } } , \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} c _ { l } = \\frac { 1 } { \\lambda ^ { n _ { l } } } \\big ( o ( \\| \\varphi \\| _ { \\alpha , * } ) + O ( \\| h \\| _ { \\alpha , * * } ) \\big ) . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} N \\ge \\sum _ { i = 1 } ^ k ( | M _ i | + | R _ i | ) \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\widetilde { U } ( \\lambda ) L ( \\lambda ) \\widetilde { V } ( \\lambda ) = \\begin{bmatrix} Z ( \\lambda ) & X ( \\lambda ) & I _ { t } \\\\ Y ( \\lambda ) & P ( \\lambda ) & 0 \\\\ I _ { s } & 0 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - c \\mathcal D \\xi + \\mathcal B \\nu = \\frac { \\epsilon } { \\gamma } \\xi \\nu , \\\\ \\xi = \\frac { 1 } { 1 - \\gamma } \\Big ( c \\nu + \\frac { \\epsilon } { 2 \\gamma } \\nu ^ 2 \\Big ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} X _ 0 ( t ) = X _ N ( t ) = 0 , \\ , \\ , t \\ge 0 , \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\mathcal I _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { L ( 1 , \\pi _ p , \\mathrm { a d } ) L ( 1 , \\tau _ p , \\mathrm { a d } ) } { L ( 1 / 2 , \\pi _ p \\times \\mathrm { a d } \\tau _ p ) } \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) . \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} \\langle r ^ { - } _ { \\psi } ( \\beta _ m ) \\mathbf h _ p , \\mathbf h _ p \\rangle = \\chi _ { \\psi } ( p ^ m ) p ^ { m / 2 } \\int _ { \\Z _ p ^ { \\times } } \\mathfrak G ( 2 x p ^ { - m } , \\underline { \\chi } _ p ^ { - 1 } ) \\underline { \\chi } _ p ( x ) d x . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} J ' + C _ { \\gamma ' } \\circ J = 0 , \\ \\ J ( 0 ) = I . \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( \\frac { f _ 2 ( y ) } { c _ 2 ^ { 2 } ( y ) } - \\frac { f _ 1 ( y ) } { c _ 1 ^ { 2 } ( y ) } \\right ) \\varphi ( y ) d y = 0 , \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\rm B I } = b ^ 2 \\left ( 1 - \\sqrt { 1 - \\frac { | \\bf E | ^ 2 } { b ^ 2 } } \\right ) , \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} & f ( 1 ) > f ( y _ { j _ 1 } ) \\geq f ( p ) , \\ j _ { i _ 1 } \\in I _ 0 = \\{ 1 , \\dots , n _ 1 \\} \\\\ & f ( p ) > f ( y _ { j _ { 2 } } ) \\geq 2 f ( p ) - f ( 1 ) , \\ j _ { i _ 2 } \\in I _ 1 = \\{ n _ 1 + 1 , \\dots , n _ 2 \\} \\\\ & \\dots \\\\ & ( k - 1 ) f ( p ) - ( k - 2 ) f ( 1 ) > f ( y _ { i _ k } ) > { f ( 0 ) \\geq } k f ( p ) - ( k - 1 ) f ( 1 ) , \\ j _ { i _ k } \\in I _ { k - 1 } = \\{ n _ k + 1 , \\dots , n _ { k + 1 } = N \\} . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} T = \\left ( \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) \\right ) _ { m , n \\in \\N ^ N } \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( - \\Delta ) ^ { s } u + u = ( | u | ^ { 2 p } + b | u | ^ { p - 1 } | v | ^ { p + 1 } ) u , & x \\in \\mathbb { R } ^ { N } , \\\\ ( - \\Delta ) ^ { s } v + \\omega ^ { 2 s } v = ( | v | ^ { 2 p } + b | v | ^ { p - 1 } | u | ^ { p + 1 } ) v , & x \\in \\mathbb { R } ^ { N } , \\end{array} \\right . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} E _ m = \\begin{pmatrix} J \\\\ & J \\\\ & & \\ddots \\\\ & & & J \\end{pmatrix} \\in M _ { m , m } ( q ) \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\Omega _ { n o r } = \\{ \\omega \\in \\Omega \\ , | \\ , \\mathcal F _ \\omega \\textrm { i s a n o r m a l f a m i l y n e a r t h e o r i g i n } \\} . \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} x M _ { n } ( x ; \\beta , c ) = M _ { n + 1 } ( x ; \\beta , c ) + \\frac { ( c + 1 ) n + \\beta c } { 1 - c } M _ { n } ( x ; \\beta , c ) + \\gamma _ n M _ { n - 1 } ( x ; \\beta , c ) , \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{gather*} \\theta _ 2 ( \\tau ) = 2 \\frac { \\eta ( 2 \\tau ) ^ 2 } { \\eta ( \\tau ) } , \\theta _ 3 ( \\tau ) = \\frac { \\eta ( \\tau ) ^ 5 } { \\eta ( \\tau / 2 ) ^ 2 \\eta ( 2 \\tau ) ^ 2 } , \\theta _ 4 ( \\tau ) = \\frac { \\eta ( \\tau / 2 ) ^ 2 } { \\eta ( \\tau ) } . \\end{gather*}"}
-{"id": "9111.png", "formula": "\\begin{align*} N _ 1 = \\Big | \\| \\varphi _ k \\xi _ { n _ k } \\| ^ 2 _ 1 + \\| \\varphi _ k \\nu _ { n _ k } \\| ^ 2 _ 1 - \\int _ { \\mathbb R } \\rho _ { k , 2 } ( x ) d x \\Big | \\leqq \\epsilon . \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\delta u _ { s t } = \\int _ s ^ t \\big [ \\Delta u _ r + \\mathrm { d i v } F ( u _ r ) \\big ] d r + A _ { s t } ^ 1 u _ s + A _ { s t } ^ 2 u _ s + u _ { s t } ^ { \\natural } \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\left ( \\frac { d ^ n } { d z ^ n } M _ z f \\right ) ( z ) & = \\frac { d ^ n } { d z ^ n } ( z f ( z ) ) = \\binom { n } { 0 } z f ^ { ( n ) } ( z ) + \\binom { n } { 1 } f ^ { ( n - 1 ) } ( z ) \\\\ & = ( M _ z f ^ { ( n ) } ) ( z ) + ( n T _ { z } f ^ { ( n ) } ) ( z ) \\\\ & = \\left ( T \\frac { d ^ n } { d z ^ n } f \\right ) ( z ) \\ , . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} 4 ! h _ 4 & = p _ { 1 1 1 1 } + 6 p _ { 2 1 1 } + 3 p _ { 2 2 } + 8 p _ { 3 1 } + 6 p _ 4 , \\\\ 4 ! S _ 4 & = \\Psi ^ { 1 1 1 1 } + 3 \\Psi ^ { 1 1 2 } + 2 \\Psi ^ { 1 2 1 } + 6 \\Psi ^ { 1 3 } + \\Psi ^ { 2 1 1 } + 3 \\Psi ^ { 2 2 } + 2 \\Psi ^ { 3 1 } + 6 \\Psi _ 4 , \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} { I _ 0 } & = \\left \\{ { { \\bf { r } } _ n } : { { \\bf { r } } _ n } \\right \\} \\\\ { I _ 1 } & = \\left \\{ { { \\bf { r } } _ n } : { { \\bf { r } } _ n } \\right \\} \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} \\gamma _ { i } ^ { q + 1 } = - w _ { i } . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ \\infty \\mathsf { E } _ X \\big \\| P _ k ( W , t ) \\big \\| < \\infty . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} S : = \\sum _ { \\l \\in \\mathbb { Z } , \\ , \\ , a _ { l - 1 } \\neq 0 } 2 ^ { l p } a _ { l - 1 } ^ { - s p / Q } d _ { l } . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} h _ 1 ^ 2 \\cdots h _ g ^ 2 h _ { g + 1 } h _ { g + 1 } ^ { - 1 } \\cdots h _ { n ^ * } h _ { n ^ * } ^ { - 1 } = 1 \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} B ^ { k , l } ( T ) = c _ { 1 , T ^ { 2 } } c _ { 2 , T ^ { 1 } } T = B _ { k , l } ( T ) \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\frac { d } { d t } u _ k ^ K = \\frac { d } { d t } f ( K _ u ^ 2 ) u & = [ B _ u , f ( K _ u ^ 2 ) ] u + f ( K _ u ^ 2 ) ( B _ u u - i J H _ u u ) \\\\ & = B _ u f ( K _ u ^ 2 ) u - i J f ( K _ u ^ 2 ) H _ u u \\\\ & = B _ u u _ k ^ K - i J w _ k ^ K , \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} \\left | B \\right | = \\frac { R ^ { n } } { n ! } + \\frac { R ^ { n - 1 } } { ( n - 1 ) ! } \\sum _ { k = \\lfloor m / 2 \\rfloor + 1 } ^ { m } ( - 1 ) ^ { k } \\binom { m } { k } ( \\Delta ^ { k - 1 } h ) ' ( R ) . \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} S _ p ( X ) = - p \\ , \\partial _ s I _ { p + s , p } ( X ) | _ { s = 0 } \\ , , \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} 1 6 y _ { * * } ^ 6 - 1 1 y _ { * * } ^ 2 + 2 = 0 , \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} \\mathrm { L H S } & = ( 1 + u ( u + w ) ) w ^ 2 = w ^ 2 + u ^ 2 w ^ 2 + u w ^ m , \\\\ \\mathrm { R H S } & = ( ( u + w ) w - 1 ) ( 1 + u w ) = w ^ 2 - 1 + u ^ 2 w ^ 2 + u w ^ m . \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} P _ { B S } ( \\lambda ) = \\bigg [ \\frac { \\mu ( 1 - \\theta ) } { \\lambda \\theta \\| \\mathbf { h } \\| ^ { 2 } } - \\frac { ( 1 - \\theta ) \\sigma _ { s } ^ { 2 } } { \\xi \\theta \\| \\mathbf { h } \\| ^ { 2 } | h _ { s } | ^ { 2 } } \\bigg ] ^ { + } . \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq \\gamma _ i \\leq \\epsilon _ i } ( - 1 ) ^ { \\gamma _ i } C ^ { \\gamma _ i } _ { \\epsilon _ i } F ( \\gamma _ i ; j ) = 0 . \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} ( a ) \\dfrac { d ^ 2 f } { d X ^ 2 _ u } \\leq 0 , ~ ~ ~ ~ ~ \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\\\ ( b ) \\dfrac { d ^ 2 f } { d Y ^ 2 _ u } \\leq 0 , ~ ~ ~ ~ ~ \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\\\ ( c ) \\dfrac { d ^ 2 f } { d X ^ 2 _ u } \\dfrac { d ^ 2 f } { d Y ^ 2 _ u } - ( \\dfrac { d ^ 2 f } { d X d Y _ u } ) ^ 2 \\geq 0 , ~ ~ ~ ~ ~ \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\\\ \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} ( + \\infty ) + c & = c + ( + \\infty ) = + \\infty \\\\ c & < + \\infty c \\in { \\mathbb Q } . \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} P ^ * ( x ) = x + x ^ 2 \\widetilde { P } ' ( x ) . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} \\pi _ \\varkappa ( \\Gamma ^ { \\Lambda , n } ) = \\frac { \\left ( \\varkappa | \\Lambda | \\right ) ^ n } { n ! } \\exp \\left ( - \\varkappa | \\Lambda | \\right ) , \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} \\Sigma ( s ) \\ll \\frac { ( 1 + | r | ) ^ A X ^ { \\theta } } { X ^ { 1 / 4 } } \\sum _ { n = 1 } ^ { \\infty } f \\left ( \\frac { n } { 2 | c | } \\right ) , \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} 0 < \\inf _ { x \\in G } \\ ; \\lvert \\gamma ( x ) \\rvert \\ ; \\frac { ( \\omega _ 2 \\circ \\phi ) ( x ) } { \\omega _ 1 ( x ) } \\sup _ { x \\in G } \\lvert \\gamma ( x ) \\rvert \\ ; \\frac { ( \\omega _ 2 \\circ \\phi ) ( x ) } { \\omega _ 1 ( x ) } < \\infty . \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\mathcal W \\partial _ t \\zeta + \\mathcal Z \\partial _ x v - \\frac { \\epsilon } { \\gamma } \\partial _ x ( \\zeta v ) = 0 \\\\ \\partial _ t v + ( 1 - \\gamma ) \\partial _ x \\zeta - \\frac { \\epsilon } { 2 \\gamma } \\partial _ x ( v ^ 2 ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} C _ { L / K } : = \\chi _ { \\Z _ p [ G ] , \\Q _ p ^ c } ( K _ L ^ { \\bullet } , \\phi _ L ^ { - 1 } ) \\in K _ 0 ( \\Z _ p [ G ] , \\Q _ p ^ c ) . \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} b _ n ( x , y ) = P ^ { n - 1 } A _ x \\ 1 _ y ( e ) \\geq 0 , \\forall x , y \\in E , \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} \\mathcal B ( \\mathbf R _ { > 0 } ^ n ) & = \\varphi ^ { - 1 } \\mathcal B \\ ( ( \\mathrm { I m } F \\iota ) \\times \\mathbf R _ { > 0 } ^ m \\ ) = \\varphi ^ { - 1 } \\sigma \\ ( p , q _ 1 , \\dots , q _ m \\ ) = \\sigma \\ ( p \\varphi , q _ 1 \\varphi , \\dots , q _ m \\varphi \\ ) \\\\ & = \\sigma \\ ( F \\iota , \\ , R _ { \\alpha _ 1 } ( x ) , \\dots , R _ { \\alpha _ m } \\ ) \\subset \\sigma ( F \\iota , R _ \\alpha ; \\alpha \\in \\ker F ) \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} k \\geq K ^ * : = \\left \\lceil \\frac { 1 } { e } \\sqrt { C _ F + \\frac { C _ d } { \\mu ( x _ 0 ) } } - a \\right \\rceil \\ ; . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{gather*} L \\big ( \\rho ^ n , n / 2 \\big ) = C _ { \\rho , n } L ( \\rho , 1 / 2 ) ^ n . \\end{gather*}"}
-{"id": "1978.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( 1 - | \\lambda _ n | ) = \\infty . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} \\gamma _ x : A \\times A \\rightarrow A , ( a , c , x ) \\mapsto \\gamma _ x ( a ) ( c ) = a \\cdot _ x c \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mu _ t ( F ^ \\theta ) = ( L ^ * \\mu _ t ^ { \\Lambda _ \\theta } ) ( F ^ \\theta ) = \\langle \\ ! \\langle e ( \\theta , \\cdot ) , L ^ \\Delta _ { \\alpha _ T } k _ t \\rangle \\ ! \\rangle , \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J \\setminus J _ { K } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) = \\int _ 0 ^ T \\Phi _ { t } d B _ { t } . \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} P j _ { l _ 1 , k } = j _ { l _ 2 , n - 2 k } P , \\\\ j _ { l _ 1 , k } R = R j _ { l _ 2 , n - 2 k } , \\\\ j _ { l _ 1 , k } Q + Q j _ { l _ 1 , k } = R P . \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} l _ 0 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) & l _ 1 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} U ( s , s ) = I , \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} Y _ { 1 } & = 0 . 1 3 7 ; ( \\rho _ { 0 } , \\rho _ { 1 } , v _ { 0 } ) = ( 2 7 9 . 8 \\cdot \\omega ^ { 2 } , 0 . 0 4 9 \\cdot \\omega ^ { - 1 } , 0 . 0 0 0 4 9 \\cdot \\omega ^ { - 3 } ) \\\\ Y _ { 2 } & = 0 . 3 9 0 ; ( \\rho _ { 0 } , \\rho _ { 1 } , v _ { 0 } ) = ( 0 . 5 1 0 \\cdot \\omega ^ { 2 } , - 0 . 1 9 8 \\cdot \\omega ^ { - 1 } , 0 . 7 7 6 \\cdot \\omega ^ { - 3 } ) \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} { \\rm R H S } = \\alpha _ n 2 ^ n \\int \\limits _ T \\bigg ( \\prod _ { 1 \\leq i < j \\leq n } ( t _ i ^ 2 - t _ j ^ 2 ) \\bigg ) \\Big ( \\prod _ { j = 1 } ^ n t _ j \\Big ) ^ { 1 - 2 k } \\ , d \\mathbf { t } , \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} E ( z ) = \\sum _ { n \\geq 0 } E _ n \\frac { z ^ n } { n ! } = \\sec z + \\tan z \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\left ( v _ { n } ^ { \\varepsilon } \\right ) _ { t } & + A \\Delta v _ { n } ^ { \\varepsilon } - \\rho _ { \\varepsilon } v _ { n } ^ { \\varepsilon } = e ^ { \\rho _ { \\varepsilon } \\left ( t - T \\right ) } F \\left ( x , t ; e ^ { \\rho _ { \\varepsilon } \\left ( T - t \\right ) } v _ { n } ^ { \\varepsilon } \\right ) + \\mathbf { P } _ { \\varepsilon } ^ { \\beta } v _ { n } ^ { \\varepsilon } \\in \\left ( H ^ { 1 } \\left ( \\Omega \\right ) \\right ) ' , \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} \\bar { \\nabla } _ X Y = \\nabla _ X Y + h ( X , Y ) , \\ \\bar { \\nabla } _ X V = - A _ V X + \\nabla _ X ^ { \\perp } V \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\left ( - R i c ( \\hat \\omega ) + \\frac { n + 1 } { 2 n } \\chi + \\sqrt { - 1 } \\partial \\bar \\partial u \\right ) ^ n = e ^ { u } \\hat \\omega ^ n . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} L _ t ( Y _ { i , r } ) = [ \\underline { L ( Y _ { i , r } ) } ] ( = F _ t ( Y _ { i , r } ) ) \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} \\beta ^ { \\kappa } _ { M , N } ( \\kappa \\ , a , \\kappa \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } \\beta _ { M , N } ( a , \\ , b ) . \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} a _ { 0 , 0 , \\sigma } ( \\xi ) = \\partial _ \\xi ^ \\sigma a _ { 0 , 0 } ( \\xi ) . \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} \\deg ( L \\vert _ { \\partial B _ { 1 0 m _ i ^ { - 1 / 2 } } ( x _ j ) } ) = 1 , \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} \\frac 2 { 1 - 2 \\delta _ { x , i } } \\left ( \\delta _ { x , i } + \\frac { r _ { \\tilde x , k } } { r _ { x , i } } \\delta _ { \\tilde x , k } \\right ) & \\leq 4 ( \\delta _ { x , i } + 2 C _ x \\delta _ { \\tilde x , k } ) \\\\ & \\leq 4 \\left ( \\frac \\varepsilon { 2 4 C _ 1 } + 2 C _ x \\frac \\varepsilon { 4 8 C _ 1 C _ x } \\right ) \\\\ & = \\frac \\varepsilon { 3 C _ 1 } \\\\ & < \\frac 1 { \\sqrt 2 } . \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} X ^ { 1 / \\tau } ( 1 / \\tau , \\tau \\lambda ) & \\overset { { \\rm i n \\ , l a w } } { = } \\beta _ { 2 , 2 } ^ { - 1 / \\tau } \\bigl ( a / \\tau , b / \\tau \\bigr ) , \\\\ & \\overset { { \\rm i n \\ , l a w } } { = } \\beta _ { 2 , 2 } ^ { - 1 } \\bigl ( a , b \\bigr ) , \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} \\tilde { L } ^ { W , n } | i _ 1 . . . i _ n \\rangle \\otimes | j _ 1 . . . j _ n \\rangle & = \\left [ \\sum _ { k = 1 } ^ n ( - 1 ) ^ { i _ k } \\sum _ { k = 1 } ^ n ( - 1 ) ^ { j _ k } - \\frac { 1 } { 2 } \\sum _ { k , l = 1 } ^ n ( - 1 ) ^ { i _ k + i _ l } - \\frac { 1 } { 2 } \\sum _ { k , l = 1 } ^ n ( - 1 ) ^ { j _ k + j _ l } \\right ] | i _ 1 . . . i _ n \\rangle \\otimes | j _ 1 . . . j _ n \\rangle \\\\ & = - 2 ( | \\mathbf { i } | - | \\mathbf { j } | ) ^ 2 | i _ 1 . . . i _ n \\rangle \\otimes | j _ 1 . . . j _ n \\rangle , \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} \\displaystyle M A E = \\displaystyle \\frac { 1 } { N } \\sum _ { i = 0 } ^ { N } | [ \\sum _ { \\substack { j \\in J \\setminus J _ { K ( n ) } } } B _ { t _ { j } } ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) ] - [ \\frac { 1 } { 2 } B _ { T } ^ 2 - \\frac { 1 } { 2 } T ] | . \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} f ( x ) \\geq f ( x ) + \\tilde \\alpha ^ \\top h ( x ) + \\tilde \\beta ^ \\top g ( x ) = L ( x , \\tilde \\alpha , \\tilde \\beta ) \\geq \\tilde \\omega , \\forall x : M ^ \\top h ( x ) = 0 , g ( x ) \\leq 0 , \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\begin{cases} \\alpha ^ b = ( p + m - 1 - n ) , \\alpha ^ t = ( n + 2 - m , n + 1 - m , \\dots , 2 ) & \\\\ \\alpha ^ b = \\emptyset , \\alpha ^ t = ( n + 1 - m , n - m , \\dots , 1 ) & . \\end{cases} \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} \\ell ' = b _ 1 c + b _ 2 c _ 2 \\geq b ' c ' + b _ 2 c _ 2 > b ' c ' \\geq b ' d ' = \\ell ' . \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} \\sum _ { \\{ i , i _ 2 , \\ldots , i _ r \\} \\in E ( G ) } x _ { i _ 2 } \\cdots x _ { i _ r } = \\lambda ^ { ( p ) } ( G ) x _ i ^ { p - 1 } ~ ~ ~ x _ i \\neq 0 . \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} \\lbrace z _ 1 , z _ 3 \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace z _ 2 , z _ 4 \\rbrace = 1 . \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} p _ { 1 } ( x ) = & m 2 ^ { 1 - s } f ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) ^ s [ f ( b ) - m f ( a ) ] , \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} - \\sum _ { i = 1 } ^ N \\frac { ( \\lambda _ i | \\alpha _ { c ( j ) } ) } { w _ j - z _ i } + \\sum _ { \\substack { i = 1 \\\\ i \\neq j } } ^ m \\frac { ( \\alpha _ { c ( i ) } | \\alpha _ { c ( j ) } ) } { w _ j - w _ i } = 0 , j = 1 , \\ldots , m . \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\mathcal { G } _ \\alpha : = L ^ 1 ( \\Gamma _ 0 , e ^ { \\alpha | \\cdot | } d \\lambda ) , \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} ( - 1 ) ^ { d ( x , x _ 0 ) + d ( y , x _ 0 ) } = ( - 1 ) ^ { d ( x , y ) } = \\chi ( x , y ) . \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{gather*} \\frac { { \\rm d } s } { { \\rm d } \\tau } = 2 \\pi { \\rm i } \\cdot s \\big ( 1 - s ^ 3 \\big ) a ^ 2 / 3 . \\end{gather*}"}
-{"id": "3447.png", "formula": "\\begin{align*} G ( B ( t ; x ) , z ) = \\left ( \\dfrac { z } { e ^ z - 1 } \\right ) ^ t e ^ { x z } . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} T _ u g ( x ) ~ = ~ \\tilde { g } ( x ) + G ( \\ 1 + A _ u g ) ( x ) , \\forall x \\in E , \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\dot { x } = A ( t ) x + f ( t , x ) \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\int \\partial D = A _ P ^ * \\partial D = A _ P ^ * \\left ( w _ p \\otimes \\partial x \\right ) = c _ p \\otimes \\partial x . \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\langle \\pi _ t , \\psi \\rangle = \\langle \\pi _ 0 , e ^ { t \\mathcal { A } } \\psi \\rangle - \\int _ 0 ^ t \\ ! \\mathrm { d } s \\ , \\left \\langle G \\left ( \\frac { { \\boldsymbol { \\tilde { \\varphi } } } ^ { \\pi _ s } } { { \\boldsymbol { { \\varphi } } } ^ { \\pi _ s } } \\right ) \\pi _ s , \\partial _ v \\left ( e ^ { ( t - s ) \\mathcal { A } } \\psi \\right ) \\right \\rangle \\ , . \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\lVert \\delta _ { A } ^ { W ^ { - } ( n , \\phi ^ { - 1 } ) } \\rVert _ { 1 } = \\alpha ^ { - } ( \\phi ^ { - 1 } ) \\log \\rho ( \\delta _ { A } ) = \\alpha ^ { - } ( \\phi ^ { - 1 } ) h _ { t o p } ( \\sigma _ { A } ) \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} 4 S \\sum _ { \\substack { \\beta \\geq \\frac { 1 } { 2 } - \\frac { R } { \\log q } \\\\ 0 \\leq \\gamma \\leq \\frac { 2 S } { 3 \\log q } \\\\ L ( \\beta + i \\gamma , f ) = 0 } } \\sinh \\left ( \\frac { \\pi \\big ( R + ( \\beta - 1 / 2 ) \\log q \\big ) } { 2 S } \\right ) \\ ; \\leq \\ ; I _ 1 ( f ) + I _ 2 ( f ) + I _ 3 ( f ) . \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} v ( x , t ) \\equiv v ( r ( x ) , t ) : = C \\zeta ( t ) \\left [ 1 - \\frac { \\eta ( t ) } { 2 a } ( r ^ 2 + 1 ) \\right ] _ + ^ { \\frac 1 { m - 1 } } , ( x , t ) \\in B _ 1 \\times [ 0 , T ) \\ , . \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\begin{aligned} & = \\alpha a _ { \\omega } \\cdot \\frac { a _ { \\omega } ^ { - 1 } \\log N \\log \\log N } { \\log \\log N + \\log \\log \\log N + O _ { \\omega } ( 1 ) } + O _ { \\omega } \\left ( \\alpha \\frac { \\log N } { \\log \\log N } \\right ) \\\\ & = \\alpha \\left ( \\log N - \\frac { \\log N \\log \\log \\log N } { \\log \\log N } + O _ { \\omega } \\left ( \\frac { \\log N } { \\log \\log N } \\right ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} \\begin{array} { c } \\tilde { J } ( R a d T \\acute { N } ) = R a d T \\acute { N } , \\\\ \\tilde { J } ( l t r ( T \\acute { N } ) ) = l t r ( T \\acute { N } ) . \\end{array} \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} L ( f , s ) = \\sum _ { n > 0 } a _ n n ^ { - s } \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\rho _ 2 : = & ( 1 , 2 ) , ( 3 , 6 ) , ( 1 , 3 ) , ( 4 , 5 ) , ( 4 , 6 ) , ( 2 , 4 ) , ( 2 , 3 ) , ( 1 , 5 ) , \\\\ & ( 1 , 4 ) , ( 5 , 6 ) , ( 3 , 4 ) , ( 2 , 5 ) , ( 3 , 5 ) , ( 1 , 6 ) , ( 2 , 6 ) \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} & \\int _ G \\int _ G \\mu ^ { - 1 / 2 } ( t ) \\mu ^ { - 1 / 2 } ( s ) \\langle T ^ { 1 / 2 } s ( e ) , T ^ { 1 / 2 } ( t ( e ) ) \\rangle _ { H ^ i ( E ) } \\cdot \\langle s ( h ) , t ( h ) \\rangle _ H \\ , d s \\ , d t \\\\ & = \\int _ G \\bigg \\langle e , \\left ( \\int _ G s ( T ) \\ , d s \\right ) ( e ) \\bigg \\rangle _ { \\mathcal { E } ^ i } ( t ) \\cdot \\langle h , t ( h ) \\rangle _ H \\ , d t . \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} W _ 0 \\subset \\cdots \\subset W _ n = W \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} | d \\overline { f } ( \\gamma _ 1 , \\gamma _ 2 ) | = | \\overline { \\gamma _ 1 f ( \\gamma _ 2 ) + f ( \\gamma _ 1 ) - f ( \\gamma _ 1 \\gamma _ 2 ) } | = | \\overline { ( \\gamma _ 1 f ( \\gamma _ 2 ) - 1 ) ( f ( \\gamma _ 1 ) - 1 ) } | \\leq \\tfrac { \\| \\overline { f } \\| } { r c } . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} I I I \\leq C \\epsilon _ 1 + \\sum _ { k = 4 } ^ n r _ { k - 1 } ^ { \\gamma _ { d , \\alpha } } \\epsilon _ 1 + C r _ n ^ { \\gamma _ { d , \\alpha } } \\epsilon _ 1 \\leq C \\epsilon _ 1 . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} L _ M ( w \\ , | \\ , a ) = \\int \\limits _ 0 ^ \\infty \\frac { d t } { t ^ { M + 1 } } \\Bigl ( e ^ { - w t } \\ , f ( t ) - \\sum \\limits _ { k = 0 } ^ { M - 1 } \\frac { t ^ k } { k ! } \\ , B _ { M , k } ( w \\ , | \\ , a ) - \\frac { t ^ M \\ , e ^ { - t } } { M ! } \\ , B _ { M , M } ( w \\ , | \\ , a ) \\Bigr ) . \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { D _ \\varepsilon } \\theta _ k ^ 2 > C \\varepsilon ^ 2 \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } , \\forall \\varepsilon \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} \\Sigma ' _ q = \\left ( \\begin{matrix} 1 - \\frac { r } { r _ 2 } & 1 & 1 & \\cdots & 1 \\\\ 1 & 1 - r & 1 & & \\vdots \\\\ 1 & 1 & 1 - \\frac { r } { r _ 4 } & & \\\\ \\vdots & & & \\ddots & \\\\ 1 & 1 & 1 & \\cdots & 1 - r \\end{matrix} \\right ) \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} \\frac { \\partial \\ell _ i ^ { ( k ) } } { \\partial \\zeta _ q ^ { ( p ) } } = 2 \\delta _ { k , p } \\delta _ { i , q } e ^ { t ( 2 \\zeta _ q ^ { ( p ) } - \\ell _ i ^ { ( k ) } ) } + O ( e ^ { - t \\delta } ) , \\ , \\frac { \\partial \\ell _ i ^ { ( k ) } } { \\partial \\varphi _ q ^ { ( p ) } } = O ( e ^ { - t \\delta } ) , \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} \\varphi _ { n , B } ^ { \\rm { C F S K } } - \\varphi _ { n , A } ^ { \\rm { C F S K } } = 4 \\pi \\Delta f T \\sum \\limits _ { i = 0 } ^ { n - 1 } { \\left ( { { s _ { i , B } } - { s _ { i , A } } } \\right ) } \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ 4 \\log \\frac { 1 - | h ( \\omega _ k ) | ^ 2 } { | h ' ( \\omega _ k ) | } + \\sum _ { k = 1 } ^ 4 \\sum _ { { \\substack { l = 1 \\\\ l \\neq k } } } ^ 4 \\delta _ k \\delta _ l \\log \\left | \\frac { 1 - \\overline { h ( \\omega _ k ) } h ( \\omega _ l ) } { h ( \\omega _ k ) - h ( \\omega _ l ) } \\right | \\le 2 \\log \\frac { 4 \\rho ^ 2 } { 1 - \\rho ^ 2 } . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} f ( p g , s ) = \\chi ( p ) \\delta _ { P _ { 4 n } } ( p ) ^ { s + \\frac 1 2 } f ( g , s ) \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} a \\wedge \\left ( a ^ { \\prime } \\vee b \\right ) \\leq a \\wedge \\left ( a ^ { \\prime } \\vee b ^ { \\prime } \\right ) = a ^ { \\prime } \\vee \\left ( a \\wedge b ^ { \\prime } \\right ) < a \\wedge \\left ( a ^ { \\prime } \\vee b \\right ) \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} x ^ 2 + x = k + 1 + \\frac { ( c _ 1 ^ 2 + c _ 0 c _ 2 ) ( c _ 2 ^ 2 + c _ 1 c _ 3 ) } { ( c _ 1 c _ 2 + c _ 0 c _ 3 ) ^ 2 } . \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\begin{aligned} r ( X ) & \\leq H ( X ) , \\forall X \\subsetneq V , \\\\ r ( V ) & = H ( V ) . \\end{aligned} \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} \\Delta Q _ 2 = - { \\rm d i v } ( U \\cdot \\nabla U + U \\cdot \\nabla V + V \\cdot \\nabla U ) . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} d X _ s = V _ j ( X _ s ) d \\tilde { \\Z } _ s ^ j \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\left | \\nabla u _ { \\lambda } \\right | \\leq C _ { \\mathcal { K } } \\forall \\lambda \\geq \\lambda _ 3 , \\ , \\ , ( x , t ) \\in \\mathcal { K } . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} \\sum _ { \\substack { b \\leq y _ 2 } } \\frac { \\Lambda ( b ) P _ 2 [ b ] } { b } & = ( \\log y _ 2 ) \\int _ 0 ^ 1 P _ 2 ( u ) d u + O ( 1 ) . \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\Omega _ { a } : = \\left \\{ \\omega \\in \\Omega \\ , \\Big | \\ , \\forall n \\in \\mathbb N \\textrm { a n d } \\forall \\alpha \\in \\Omega , \\ , \\exists k _ j \\ , : \\ , M ^ n _ { T ^ { k _ j } \\omega } \\to M ^ n _ \\alpha \\right \\} \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} D _ s X ( t ) = X ( s ) + \\int _ 0 ^ s g ( r ) d W ( r ) + g ( s ) [ W ( t ) - W ( s ) ] + \\int _ s ^ t D _ s X ( r ) d W ( r ) . \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} S _ { k + 1 / 2 } ^ { + , n e w } ( 4 N M , \\chi ; f \\otimes \\chi ) : = \\{ h \\in S _ { k + 1 / 2 } ^ + ( 4 N M , \\chi ) : h | T _ { p ^ 2 } = a _ { \\chi } ( p ) h p \\nmid 2 N \\} \\subset S _ { k + 1 / 2 } ^ { + , n e w } ( 4 N M , \\chi ) , \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} \\kappa _ H = 1 - \\frac { o } { p _ H } \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} \\widetilde h [ f ] - h [ f ] & = \\frac { 1 } { \\beta _ { v _ 0 } } | a + b | ^ 2 - \\frac { 1 } { \\beta _ { v _ 1 } } | a | ^ 2 - \\frac { 1 } { \\beta _ { v _ 2 } } | b | ^ 2 . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} Q _ i ^ { \\epsilon } & : = ( V _ k \\cdots V _ { i + 2 } V _ { i + 1 } ) P _ i ^ { \\epsilon _ i } ( V _ k \\cdots V _ { i + 2 } V _ { i + 1 } ) ^ { - 1 } \\\\ & = ( V _ k \\cdots V _ { i + 2 } V _ { i + 1 } V _ i ) t _ { b _ i } ^ { - \\epsilon _ i } ( V _ k \\cdots V _ { i + 2 } V _ { i + 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ N \\left ( \\dfrac { p } { q } \\right ) ^ k L i _ { - n } ^ { \\star k } ( q ) = \\sum _ { m = 0 } ^ n \\binom { N + 1 } { m + 1 } m ! \\ , S _ { Y + 1 } ( n , m ) \\\\ & = \\sum _ { k = 0 } ^ n c _ { n , N } ( k ) \\left ( \\dfrac { p } { q } \\right ) ^ k L i _ { - n } ^ { \\star k } ( q ) . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} x ^ 3 + \\frac { c _ 2 } { c _ 3 } x ^ 2 + \\frac { c _ 1 } { c _ 3 } x + \\frac { c _ 0 } { c _ 3 } = 0 , \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} q _ G ( M ) = \\sum _ { A \\subseteq E } ( - 1 ) ^ { | A ^ c | } | G | ^ { | A | - | V | + k _ { \\mathrm { o } } ( M \\backslash A ^ c ) } | L _ { 1 } | ^ { k ( M \\backslash A ^ c ) - k _ { \\mathrm { o } } ( M \\backslash A ^ c ) } . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} L ^ { \\Delta , \\sigma } _ { \\alpha , u } u = L _ { \\vartheta } ^ { \\Delta , \\sigma } u , u \\in \\mathcal { D } ^ { \\Delta , \\sigma } _ { \\alpha , u } . \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} \\Omega ^ { n + 1 } ( A , B , f ) & = ( \\Omega ^ { n + 1 } A , M \\otimes P ^ A _ { n } , M \\otimes i ^ A _ { n + 1 } ) \\oplus ( 0 , \\Omega ^ { n + 1 } B , 0 ) . \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} D _ 2 = \\frac { E _ 2 D _ 1 + F _ 3 } { E _ 1 } , D _ 0 = \\frac { E _ 0 D _ 1 + F _ 1 } { E _ 1 } . \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} & D _ 2 ^ 2 + E _ 2 D _ 2 = F _ 4 , \\\\ & E _ 1 D _ 2 + E _ 2 D _ 1 = F _ 2 , \\\\ & E _ 0 D _ 2 + D _ 1 ^ 2 + E _ 1 D _ 1 + E _ 2 D _ 0 = F _ 2 , \\\\ & E _ 0 D _ 1 + E _ 1 D _ 0 = F _ 1 , \\\\ & D _ 0 ^ 2 + E _ 0 D _ 0 = F _ 0 . \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G ) : = \\max _ { | x _ 1 | ^ p + \\cdots + | x _ n | ^ p = 1 } r \\sum _ { \\{ i _ 1 , \\ldots , i _ r \\} \\in E ( G ) } x _ { i _ 1 } \\cdots x _ { i _ r } . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} d \\tilde { N } _ s ( t ) = \\nabla _ x b ( t , X ( t ) ) \\tilde { N } _ s ( t ) d t + \\nabla _ x \\sigma ( t , X ( t ) ) \\tilde { N } _ s ( t ) d W ( t ) , \\tilde { N } _ s ( s ) = 0 . \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\alpha ( t s ) = t ^ { y } s ^ { p - 1 } = s ^ { p - 1 } t ^ { ( p + 1 ) ^ { p - 1 } y } \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} f = \\sum _ { n \\geq 1 } a ( n ) q ^ n , f \\otimes \\chi = \\sum _ { n \\geq 1 } a _ { \\chi } ( n ) q ^ n \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { \\sigma } { 2 } C _ \\omega \\bigl ( \\tau ^ { ( k _ n ) } - \\tau ^ { ( k _ n ) } _ \\varepsilon \\bigr ) ^ { H - \\delta } > \\frac { k _ n } { 4 \\varepsilon } \\bigl ( \\tau ^ { ( k _ n ) } - \\tau ^ { ( k _ n ) } _ \\varepsilon \\bigr ) . \\end{gathered} \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} L _ t ( m ^ { ( i ) } _ { k , r } ) = [ \\underline { W ^ { ( i ) } _ { k , r } } ] . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} x _ 4 = x _ 1 x _ 3 + x _ 1 x _ 2 ^ 2 , \\\\ x _ 4 ^ 2 = x _ 1 x _ 3 + x _ 1 ^ 2 x _ 2 , \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} \\lim _ { \\| h \\| _ V \\to 0 } \\frac { \\| f ( x + h ) - f ( x ) - A h \\| _ W } { \\| h \\| _ V } = 0 . \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} \\sum _ { \\substack { B \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | B | = k } } \\prod _ { i \\in B } b ( i ; n ) \\leq \\frac { \\lambda _ n } { k } \\ , \\sum _ { \\substack { A \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | A | = k - 1 } } \\prod _ { i \\in A } b ( i ; n ) , \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } \\left ( V ( s ) ^ \\gamma - V ( r _ k / 1 6 ) ^ \\gamma \\right ) V ( s ) \\mathcal { P } _ 1 ( s ) \\leq c _ 8 \\lim _ { s \\rightarrow \\infty } \\frac { V ( s ) ^ \\gamma - V ( r _ k / 1 6 ) ^ \\gamma } { V ( s ) } = 0 , \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\tau _ g ( m _ { v , v } ) = 0 . \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} l ( t ) = - L _ { 1 } - \\int _ { t } ^ { T } F ( l ( r ) ) d r , \\ ; t \\in \\lbrack 0 , T ] . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} u _ 2 ( t , x , y ) = - \\frac { ( \\partial _ t \\psi + u _ 1 \\partial _ x \\psi - \\nu \\partial _ { y } ^ 2 \\psi ) } { h _ 1 } ( t , x , y ) , h _ 2 ( t , x , y ) = - \\partial _ x \\psi ( t , x , y ) . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} u ( t ) - u ^ { n } ( t ) = u _ { 0 } - u _ { 0 } ^ { n } + \\ \\int _ { 0 } ^ { t } S ( t - s ) \\big ( B ( u ( s ) ) - B ( u ^ { n } ( s ) ) \\big ) \\ d s . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} e ( N ( 1 ) ) & = ( - 1 ) ^ { m - 1 } ( 1 ) f _ { m - 1 } + ( - 1 ) ^ { m - 2 } ( 2 ) f _ { m - 2 } + \\cdots + ( m ) f _ 0 + n - m \\\\ & = \\sum _ { k = 1 } ^ { m } ( - 1 ) ^ { m - k } \\cdot k \\cdot f _ { m - k } + n - m . \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} B ' ( v , e \\cup \\{ v _ e \\} ) & = \\begin{cases} B ( v , e ) , & \\ v \\in e , \\\\ 1 , & \\ v = v _ e , \\\\ 0 , & , \\end{cases} \\\\ w ' ( e \\cup \\{ v _ e \\} ) & = w ( e ) . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} S _ * = \\Big \\{ ( \\sum _ { j = 1 } ^ p \\gamma _ { n _ j } ) + k : & p \\ge 1 , \\ 1 \\le n _ 1 , n _ 2 , \\ldots , n _ p \\le N _ 0 , \\\\ & k \\ge 0 \\Big \\} . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} B ( x _ 0 , p ^ { - m } ) & = \\{ x \\in \\Z _ p : | x - x _ 0 | _ p \\leq p ^ { - m } \\} = \\{ x \\in \\Z _ p : v _ p ( x - x _ 0 ) \\geq m \\} \\\\ & = \\{ x \\in \\Z _ p : x \\equiv x _ 0 \\pmod { p ^ m } \\} , \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\epsilon _ c = \\begin{bmatrix} 0 & \\sum _ { l = 1 } ^ L ( n _ l { r _ { a _ l } } ) \\lambda _ e \\\\ 0 & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} A _ m ( v ) = \\frac { m v ^ { 2 m - 1 } } { \\sqrt { 1 + v ^ { 2 m } } } \\ , . \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\lambda = \\begin{pmatrix} \\lambda _ { 1 1 } & \\lambda _ { 1 2 } & \\lambda _ { 1 3 } \\\\ & \\lambda _ { 1 1 } & \\\\ & \\lambda _ { 3 2 } & \\lambda _ { 3 3 } \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\omega _ i ( z ) = n _ 1 z ^ { w _ 1 } + n _ 2 z ^ { w _ 2 } + n _ 3 z ^ { w _ 3 } + n _ 4 z ^ { w _ 4 } + n _ 5 z ^ { w _ 5 } + n _ 6 z ^ { w _ 6 } , \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} & \\int _ 0 ^ t \\left \\| u \\right \\| _ { L ^ 2 _ s } ^ 2 \\left \\| u ( s ) \\right \\| _ 2 ^ 2 \\exp \\left ( \\int _ s ^ t \\left \\| u ( w ) \\right \\| _ 2 ^ 2 d w \\right ) d s \\leq \\left \\| u \\right \\| _ { L ^ 2 _ t } ^ 2 \\left [ - \\exp \\left ( \\int _ s ^ t \\left \\| u ( w ) \\right \\| _ 2 ^ 2 d w \\right ) \\right ] _ { s = 0 } ^ t \\\\ & = \\left \\| u \\right \\| _ { L ^ 2 _ t } ^ 2 \\left ( \\exp \\left ( \\int _ 0 ^ t \\left \\| u ( s ) \\right \\| _ 2 ^ 2 d s \\right ) - 1 \\right ) . \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} \\displaystyle \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) u ( x , t ) + q ( x , t ) u ^ \\alpha ( x , t ) = 0 , \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} v _ { 1 } = \\frac { 2 } { \\rho _ { 0 } ^ { 3 } } ( - \\rho _ { 0 } ^ { 2 } v _ { 0 } \\rho _ { 1 } + 2 u _ { 1 } ) \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} \\gamma _ i \\gamma _ j - \\gamma _ j \\gamma _ i = 2 \\omega _ { i j } , \\gamma _ { [ i } \\gamma _ { j ] } = \\omega _ { i j } , \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} \\begin{cases} \\frac { a _ 2 } { b _ 2 } < \\frac { a _ 0 } { b _ 0 } < \\frac { a _ 1 } { b _ 1 } , \\cr \\big ( a _ 1 - 2 k \\frac { \\chi _ 1 } { d _ 3 } \\big ) \\big ( b _ 2 - 2 l \\frac { \\chi _ 2 } { d _ 3 } \\big ) > a _ 2 b _ 1 . \\end{cases} \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { 1 } { 3 \\cdot 2 ^ d } ( 1 - \\gamma _ 1 ) + \\bigg ( 1 + \\frac { 1 } { 6 ( 2 ^ d - 1 ) } \\bigg ) & ( 1 - \\gamma _ 2 ) + 6 \\epsilon & < \\frac { 1 } { 3 \\cdot 2 ^ d } , \\\\ & ( 1 - \\gamma _ 2 ) + 4 \\epsilon & < \\frac { 1 } { 4 \\cdot 2 ^ d } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} F = Q ^ 2 + \\pi ^ s Q H + \\pi ^ { 2 s } G . \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow + \\infty } \\int _ { \\Omega } B ( x ) ( | \\nabla u _ { n } ( x ) | ^ { p ( x ) - 2 } \\nabla u _ { n } ( x ) - | \\nabla u _ { 0 } ( x ) | ^ { p ( x ) - 2 } \\nabla u _ { 0 } ( x ) ) ( \\nabla u _ { n } ( x ) - \\nabla u _ { 0 } ( x ) ) d x = 0 , \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} \\mu ^ { - 1 } ( a ) = M _ n ( a ) . \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} a ' _ k = 0 \\quad . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\rho ( 0 , y , z ) & = \\frac { y ^ 2 z ^ 2 } { ( 1 + y ^ 2 ) ^ 2 ( 1 + z ^ 2 ) ^ 2 } \\le \\frac { 1 } { 1 6 } < \\frac { 1 } { 1 2 } . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} \\left \\| \\alpha _ 1 n _ 1 + \\cdots + \\alpha _ { d - 1 } n _ { d - 1 } \\right \\| \\cdot \\prod _ { k = 1 } ^ { d - 1 } \\left ( | L _ k ( n ) | + 1 \\right ) < | n | ^ { - \\varepsilon } \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} \\deg ( z ( \\lambda ) ) = \\deg ( N _ 1 ( \\lambda ) ^ T \\ , h ( \\lambda ) ) = \\deg ( N _ 1 ( \\lambda ) ) + \\deg ( h ( \\lambda ) ) . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} \\rho _ { 1 4 2 } ( S _ 2 ) ^ 2 & = ( S _ 2 ^ 2 S _ 1 ^ * + S _ 1 ^ 2 S _ 2 ^ * ) ( S _ 2 ^ 2 S _ 1 ^ * + S _ 1 ^ 2 S _ 2 ^ * ) \\\\ & = S _ 2 ( S _ 2 S _ 1 ) S _ 2 ^ * + S _ 1 ( S _ 1 S _ 2 ) S _ 1 ^ * \\ ; . \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } u _ { \\lambda } \\left ( x , t \\right ) \\ , d x = \\int _ { \\R ^ n } \\lambda ^ { a _ 1 } u \\left ( \\lambda ^ { a _ 2 } x , \\lambda t \\right ) \\ , d x = \\int _ { \\R ^ n } u \\left ( y , \\lambda t \\right ) \\ , d y = M _ 0 < \\infty \\forall \\lambda > 0 , \\ , \\ , t \\geq 0 . \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} M ^ t = \\big ( M _ 1 ^ t , M _ 2 ^ t , \\dots , M _ n ^ t \\big ) = ( \\delta _ 1 , \\delta _ 2 - \\delta _ 1 , \\dots , \\delta _ n - \\delta _ { n - 1 } ) . \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} S _ 1 = - \\tau \\ , q ^ { m - s } \\ , Q ^ { ( m - 1 ) } _ { k - 1 } ( 2 s - 1 ) , \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} L _ f ( x ) = \\sigma \\sqrt { 2 \\log ( x ) - 5 \\log \\log ( x ) - \\log ( 4 \\pi ) + 2 f ( x ) } \\\\ f 0 \\le 2 f ( x ) \\le \\log ( 4 \\pi ) + 5 \\log \\log ( x ) . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} a _ j = \\sum _ { \\substack { I \\subset [ N ] \\\\ N \\notin I } } ( - 1 ) ^ { | I | } \\tilde { \\phi } \\left ( \\vec { d } ( x , y ) + 2 ( j _ 1 , . . . , j _ { N - 1 } , j ) + 2 \\chi ^ I \\right ) , \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} c _ { 1 , S } c _ { 2 , R } T = c _ { 1 , S \\cup Q } c _ { 2 , R \\cup Q } T \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} f _ i ( x ) = x \\cdot \\prod _ { j = 1 5 ( i - 1 ) + 1 } ^ { 1 5 i } ( x ^ 2 + j ^ 2 ) , i = 1 , \\dots , 2 0 . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{gather*} \\eta \\ ( - \\frac { 1 } { \\tau } \\ ) = \\sqrt { - { \\rm i } \\tau } \\eta ( \\tau ) . \\end{gather*}"}
-{"id": "6910.png", "formula": "\\begin{align*} \\frac 1 { p - 1 } < \\alpha < \\frac 1 { m - 1 } , \\beta = 1 - \\alpha ( m - 1 ) \\ , . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} 0 \\subset F _ 1 \\subset \\dots \\subset F _ r = U \\subset \\pi ^ { - 1 } ( F ^ \\prime _ 1 ) \\subset \\dots \\subset \\pi ^ { - 1 } ( F ^ \\prime _ s ) = W , \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} & \\Bigl | \\Bigl ( \\frac { V ( \\xi _ 1 + \\lambda ^ { - 1 } y ) } { \\lambda ^ { \\frac { N - 2 } 2 } } \\Bigr ) ^ { 2 ^ * - 1 } - \\Bigl ( \\frac { P _ 1 ( \\xi _ 1 + \\lambda ^ { - 1 } y ) } { \\lambda ^ { \\frac { N - 2 } 2 } } \\Bigr ) ^ { 2 ^ * - 1 } \\Bigr | \\\\ \\le & C \\Bigl ( \\frac 1 { ( 1 + | y | ) ^ 4 } \\frac { | \\ln k | ^ { \\sigma _ N } } { \\lambda ^ { \\frac { N - 2 } 2 } } + \\frac { | \\ln k | ^ { ( 2 ^ * - 1 ) \\sigma _ N } } { \\lambda ^ { \\frac { N + 2 } 2 } } \\Bigr ) . \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} & \\langle f ^ i _ k , \\bar u _ k ^ i - \\bar u _ k ^ { i - 1 } - \\bar w _ k ^ i + \\bar w _ k ^ { i - 1 } \\rangle = \\langle \\bar \\varrho _ k ^ i , E \\bar u _ k ^ i - E \\bar u _ k ^ { i - 1 } - E \\bar w _ k ^ i + E \\bar w _ k ^ { i - 1 } \\rangle \\\\ & = \\langle \\bar \\varrho _ k ^ i , e _ k ^ i - e _ k ^ { i - 1 } \\rangle + \\langle \\bar \\varrho _ k ^ i , p _ k ^ i - p _ k ^ { i - 1 } \\rangle - \\langle \\bar \\varrho _ k ^ i , E \\bar w _ k ^ i - E \\bar w _ k ^ { i - 1 } \\rangle \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} \\sum _ { | I | = i } \\prod _ { j \\in I } \\lambda _ j ( b b ^ * ) = \\sum _ { | I | = i } \\Delta _ { I , I } ( b b ^ * ) = \\sum _ { | I | = | J | = i } | \\Delta _ { I , J } ( b ) | ^ 2 , \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} A ( t ) u _ s : = \\mathop { \\rm w l i m } \\limits _ { h \\to 0 } h ^ { - 1 } ( U ( t + h , t ) - I ) u _ s \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} w ( 0 ) = w ( x , 0 ) = w _ 0 ( x ) \\ , . \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { d , c } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\bar E _ { a , b } \\bar E _ { d , c } } E _ { d , c } E _ { a , b } - ( q _ b - q _ b ^ { - 1 } ) ^ { - 1 } K _ { c , b } E _ { a , c } E _ { d , b } , M = N = 1 ; \\\\ \\displaystyle \\sum _ { t = 0 } ^ { \\min ( M , N ) } ( - 1 ) ^ t q _ { b } ^ { \\frac { - t ( 2 N - 3 t - 1 ) } { 2 } } ( q _ b - q _ b ^ { - 1 } ) ^ { t } [ t ] _ q ! E _ { d , c } ^ { ( N - t ) } E _ { d , b } ^ { ( t ) } K _ { c , b } ^ { t } E _ { a , b } ^ { ( M - t ) } E _ { a , c } ^ { ( t ) } , . \\end{cases} \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} \\| b _ { N , R , S } \\| _ 2 = \\left ( \\sum _ n \\sum _ r \\sum _ s | b _ { n , r , s } | ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} \\mathfrak { M } ( l \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\frac { \\Gamma \\bigl ( 1 - l / \\tau \\bigr ) } { \\Gamma ^ l \\bigl ( 1 - 1 / \\tau \\bigr ) } \\ , \\prod \\limits _ { j = 0 } ^ { l - 1 } \\frac { \\Gamma \\bigl ( 1 - j / \\tau \\bigr ) \\Gamma \\bigl ( 1 + \\lambda _ 1 - j / \\tau \\bigr ) \\Gamma \\bigl ( 1 + \\lambda _ 2 - j / \\tau \\bigr ) } { \\Gamma \\bigl ( 2 + \\lambda _ 1 + \\lambda _ 2 - ( l + j - 1 ) / \\tau \\bigr ) } . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} f ( \\psi ) : = \\lim _ { \\ell \\to \\infty } \\phi _ \\ell ^ * \\psi \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} \\begin{array} { c c c } m ( I \\otimes S ) \\Delta ( b ) = \\epsilon ( b ) I , & m ( S \\otimes I ) \\Delta ( b ) = \\epsilon ( b ) I , & \\forall b \\in S U _ { q } ^ { 0 } ( 2 ) . \\end{array} \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & d W _ { x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) \\\\ & = \\left \\{ W _ { s x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) + W _ { x x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) b ( s ) \\right . \\\\ & \\left . + \\frac { 1 } { 2 } W _ { x x x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) ( \\sigma ( s ) ) ^ { 2 } \\right \\} d s + W _ { x x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) \\sigma ( s ) d B ( s ) . \\end{array} \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} P ^ n ( X _ t \\not \\in C ( t , \\epsilon ) ) & = P ^ n ( X _ { t \\wedge \\tau _ { m ^ o - 1 } } \\not \\in C ( t , \\epsilon ) , \\tau _ { m ^ o - 1 } > t ) + P ^ n ( X _ t \\not \\in C ( t , \\epsilon ) , \\tau _ { m ^ o - 1 } \\leq t ) \\\\ & \\leq P ^ n ( X _ { t \\wedge \\tau _ { m ^ o - 1 } } \\not \\in C ( t , \\epsilon ) ) + P ^ { n , m ^ o } ( \\tau _ { m ^ o - 1 } \\leq t ) . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} \\frac { \\partial c } { \\partial T } = D \\frac { \\partial ^ 2 c } { \\partial X ^ 2 } , \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n = \\zeta . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} R D _ f & = \\{ \\omega \\in \\Omega : \\left ( \\phi ( f ( \\omega ) ^ + , \\omega ) - \\phi ( f ( \\omega ) , \\omega ) \\right ) > 0 \\} \\\\ L D _ f & = \\{ \\omega \\in \\Omega : \\left ( \\phi ( f ( \\omega ) , \\omega ) - \\phi ( f ( \\omega ) ^ - , \\omega ) \\right ) > 0 \\} \\\\ D _ f & = \\{ \\omega \\in \\Omega : \\left ( \\phi ( f ( \\omega ) ^ + , \\omega ) - \\phi ( f ( \\omega ) ^ - , \\omega ) \\right ) > 0 \\} \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} \\langle \\pmb { \\phi } _ 2 ^ { ( 2 ) } , \\pmb { \\phi } _ 2 ^ { ( 2 ) } \\rangle = \\int _ { \\Q _ p } \\mathbf 1 _ { \\frac { 1 } { 2 } \\Z _ 2 } ( x ) \\overline { \\mathbf 1 _ { \\frac { 1 } { 2 } \\Z _ 2 } ( x ) } d x = \\mathrm { v o l } ( 2 ^ { - 1 } \\Z _ 2 ) = 2 = 2 \\mathrm { v o l } ( \\Z _ 2 ) = 2 \\langle \\pmb { \\phi } _ 2 , \\pmb { \\phi } _ 2 \\rangle . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{gather*} \\lambda = \\left ( \\frac { \\theta _ 2 } { \\theta _ 3 } \\right ) ^ 4 = \\ ( \\frac { \\sqrt { 2 } \\eta ( \\tau / 2 ) { \\eta ( 2 \\tau ) ^ 2 } } { \\eta ( \\tau ) ^ 3 } \\ ) ^ 8 = 1 6 \\big ( q ^ { 1 / 2 } - 8 q + 4 4 q ^ { 3 / 2 } - 1 9 2 q ^ 2 + 7 1 8 q ^ { 5 / 2 } + \\cdots \\big ) , \\end{gather*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } u _ n / n = u , \\lim _ { n \\to \\infty } v _ n / n = v , \\lim _ { n \\to \\infty } \\hat { u } _ n / n = \\hat { u } , \\lim _ { n \\to \\infty } \\hat { v } _ n / n = \\hat { v } , \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} \\ell = b + 2 , m = 2 a + 1 , n = a b + b + 1 . \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} \\alpha = \\begin{pmatrix} A _ 1 & A _ 2 & A _ 3 & A _ 4 & B _ 1 & B _ 2 & B _ 3 & B _ 4 & C _ 1 \\\\ & & A _ 1 & A _ 2 & B _ 3 & B _ 4 \\\\ & & A _ 5 & A _ 6 & B _ 3 & B _ 4 & & & C _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} 4 f ' _ k f _ i g ' _ j = 2 j f ' _ k g _ { i + j - 1 } = - j ( i + j - 1 ) g _ { i + j + k - 2 } \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\max _ { P _ { B S } } U _ { B S } & = \\lambda \\theta P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } - \\theta ( A P _ { B S } ^ { 2 } + B P _ { B S } ) , \\\\ s . t . & ~ P _ { B S } \\geq 0 . \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\binom { a n + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\alpha _ { s } n + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } \\right ) = \\frac { 1 } { \\left ( r ! \\right ) ^ { p } \\prod _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\right ) ^ { p _ { s } } } \\sum _ { k = 0 } ^ { r p + \\sigma } k ! S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) \\frac { z } { \\left ( z - 1 \\right ) ^ { k + 1 } } , \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} V _ \\lambda = \\{ v \\in V \\ , | \\ , K _ a . v = q _ a ^ { \\lambda _ a } v , \\ { K _ a ; c \\brack t } . v = { \\lambda _ a + c \\brack t } _ q v , \\ \\forall \\ , t \\in \\mathbb N , c \\in \\mathbb Z \\} . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} P ( \\lambda ) = \\sum _ { i = 0 } ^ d P _ i \\phi _ i ( \\lambda ) \\in \\mathbb { F } [ \\lambda ] ^ { n \\times n } , \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} A _ { \\sigma , { \\tau } } A _ { \\kappa , { \\tau } } = I + S , \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} \\underbrace { \\begin{bmatrix} N _ { n } \\\\ \\vdots & N _ { n } \\\\ N _ { 0 } & \\vdots & \\ddots \\\\ & N _ { 0 } & \\vdots & N _ { n } \\\\ & & \\ddots & \\vdots \\\\ & & & N _ { 0 } \\end{bmatrix} } _ { b + 1 \\mbox { b l o c k c o l u m n s } } \\begin{bmatrix} B _ { b } \\\\ B _ { b - 1 } \\\\ \\vdots \\\\ B _ 0 \\end{bmatrix} = \\begin{bmatrix} Q _ { n + b } \\\\ Q _ { n + b - 1 } \\\\ \\vdots \\\\ Q _ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} t r _ { C _ { p ^ { k - 1 } } } ^ { C _ { p ^ { k } } } \\circ r e s _ { C _ { p ^ { k - 1 } } } ^ { C _ { p ^ { k } } } = p . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\frac { d } { d t } g _ { t } ( z ) = \\frac { 2 } { g _ { t } ( z ) - \\sqrt { \\kappa } B _ { t } } \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} g _ { 1 } ( M ) = \\frac { 1 } { \\left | B _ { \\star } ( M ) \\right | } \\int \\limits _ { B _ { \\star } ( M ) } g \\left ( M _ 1 \\right ) d { m _ 3 } , \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } \\Phi ( \\frac { B _ { t _ { j } } + B _ { t _ { j + 1 } } } { 2 } , t _ { j } ) [ B _ { t _ { j + 1 } } - B _ { t _ { j } } ] | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K \\} \\\\ = C _ { n } ^ { K } ( 2 ^ { K } - 1 ) . \\end{array} \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} [ \\beta ] = \\left ( [ \\imath _ * \\imath ^ { ! * } \\beta ] + [ \\imath _ * \\imath ^ { ! * } \\beta ' ] \\right ) + [ \\jmath _ { ! * } \\jmath ^ * \\beta ' ] . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\dot { z } = g _ 2 ( t , \\theta _ { t } ) z , \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} \\mathrm { d e t } \\left ( \\mathcal { M } \\right ) = b ( x _ 1 - x _ 0 ) \\frac { \\mathcal { R } ( x _ 0 , x _ 1 ) } { \\mathcal { S } ( x _ 0 , x _ 1 ) } \\ ; . \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} T _ x P V = P T _ x V + A _ x V \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} \\bar { c } = \\bar { c } ( r , N ) = \\frac { \\sum _ { j = 1 } ^ { N } c _ { j } l _ { j } } { \\sum _ { j = 1 } ^ { N } l _ { j } } ( T = 0 ) \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} \\mathcal { C } ^ { \\delta } = L ( A \\cup B ) = L ( A ) \\cup L ( B ) . \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} { \\rm H e s s } \\ , f ( S ) [ \\xi ] = S { \\rm s y m } ( { \\rm D } \\nabla \\bar { f } ( S ) [ \\xi ] ) S + { \\rm s y m } ( \\xi { \\rm s y m } ( \\nabla \\bar { f } ( S ) ) S ) . \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} ( z ^ 2 - 1 ) ^ N \\sum _ { n \\geq 0 } \\dot { \\phi } ( n ) z ^ n & = \\sum _ { k = 0 } ^ N \\sum _ { n \\geq 0 } \\tbinom { N } { k } ( - 1 ) ^ k \\dot { \\phi } ( n ) z ^ { n + 2 N - 2 k } \\\\ & = \\sum _ { k = 0 } ^ N \\sum _ { n \\geq 0 } \\tbinom { N } { k } ( - 1 ) ^ k \\dot { \\phi } ( n - 2 N + 2 k ) z ^ { n } \\\\ & = \\sum _ { n \\geq 0 } \\mathfrak { d } _ 2 ^ N \\dot { \\phi } ( n - 2 N ) z ^ { n } . \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} \\langle T ^ { - 1 } y , x \\rangle _ { \\mathcal { M } } = \\langle T ^ { - 1 } y , T ^ * ( T ^ * ) ^ { - 1 } x \\rangle _ { \\mathcal { M } } = \\langle y , ( T ^ * ) ^ { - 1 } x \\rangle _ { \\mathcal { N } } . \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} 0 \\neq \\xi ^ { d + 1 } = \\Sigma ^ d ( \\psi ^ 0 ) \\circ \\gamma ^ { d + 1 } = \\Sigma ^ d ( \\varphi \\circ \\nu ) \\circ \\gamma ^ { d + 1 } = \\Sigma ^ d ( \\varphi ) \\circ \\Sigma ^ d ( \\nu ) \\circ \\gamma ^ { d + 1 } , \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} V ( J _ G ( A , d , \\epsilon ) ) = A E ( J _ G ( A , d , \\epsilon ) ) = \\{ a a ' \\colon a \\in A , a ' \\in A , d _ { G } ( a , a ' ) \\neq ( d ^ 2 \\pm 3 \\epsilon ) | B | \\} . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} s _ n = \\sum _ { i = 1 } ^ n f _ i \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} d \\Big ( \\sum _ { n = 0 } ^ \\infty a _ n x ^ n \\Big ) & = d x \\sum _ { n = 0 } ^ \\infty a _ { n + 1 } x ^ n + x d x \\sum _ { n = 0 } ^ \\infty a _ { n + 2 } x ^ n + \\ldots \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} F _ 1 \\cong _ { h t } L _ 1 \\vee \\bigvee _ { s _ 0 \\leq s \\leq t } \\bigvee _ { i = 1 } ^ { r _ 1 ( s ) } S ^ { N - ( m - s + 1 ) ( n - s + 1 ) + 1 } ( L _ { m - s + 1 , n - s + 1 } ^ { t - s , 1 } ) \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} ( | a | _ { \\C } | b | _ { \\C } - a \\cdot b ) \\int _ { \\Omega } \\delta ( x ) e ^ { i x \\cdot k } d x = - \\frac { \\| k \\| ^ 2 } { 2 } \\int _ { \\Omega } \\delta ( x ) e ^ { i x \\cdot k } d x = 0 , \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k + 2 } | \\nabla ^ k \\hat { g } ^ \\bullet _ t \\{ j \\} | \\delta _ t ^ { - j } , \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} h ( a \\tau - b ) = - e ^ { - 2 \\pi i a \\left ( b + \\frac 1 2 \\right ) } q ^ { \\frac { a ^ 2 } { 2 } } \\int _ 0 ^ { i \\infty } \\frac { g _ { a + \\frac 1 2 , b + \\frac 1 2 } ( w ) } { \\sqrt { - i ( w + \\tau ) } } d w , \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} h ^ { ( \\epsilon ) , a } _ { m n } ( r , s ) & = \\langle \\Xi ^ { ( \\epsilon ) } _ { \\bullet m n } ( r , s ) , a ^ { ( \\epsilon ) , \\prime } ( X ( s ) ) \\Xi ^ { ( \\epsilon ) } _ { \\bullet m n } ( r , s ) \\rangle , \\\\ h ^ { ( \\epsilon ) , b } _ { m n } ( r , s ) & = \\langle \\Xi ^ { ( \\epsilon ) } _ { \\bullet m n } ( r , s ) , b ' ( X ( s ) ) \\Xi ^ { ( \\epsilon ) } _ { \\bullet m n } ( r , s ) \\rangle . \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} \\dot { \\bar { e } } _ { x } = - \\beta D D ^ { \\top } \\bar { e } _ { x } + \\Pi \\Omega , \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} H _ i ( L _ { \\infty } , \\Delta ( G _ K ) ) = H _ i ( L _ { \\infty } , \\Lambda ( G _ K ) ) = 0 \\mbox { f o r a l l } i \\geq 1 . \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\alpha ( X ( p _ m ) , H _ { t _ m } ^ { - 1 } ( X ( p _ m ) ) ) = 0 \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} \\gamma _ { j } = 0 \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} a _ 1 = \\frac { n } { ( m - 1 ) n + 2 } , a _ 2 = \\frac { a _ 1 } { n } , k = \\frac { a _ 1 ( m - 1 ) } { 2 m n } . \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{gather*} \\begin{aligned} [ \\rho _ \\pm ] ^ 2 & = \\sum _ { g \\in G } [ g ] \\ , , & [ \\rho _ \\pm ] [ \\rho _ \\mp ] & = \\sum _ { g \\in G } [ g \\tau ] \\ , , \\\\ [ g ] [ \\rho _ \\pm ] & = [ \\rho _ \\pm ] [ g ] = [ \\rho _ \\pm ] \\ , , & [ \\tau ] [ \\rho _ \\pm ] & = [ \\rho _ \\pm ] [ \\tau ] = [ \\rho _ \\mp ] \\ , . \\end{aligned} \\end{gather*}"}
-{"id": "6480.png", "formula": "\\begin{align*} - \\Delta U = U ^ { N + 2 \\over N - 2 } \\ \\hbox { i n } \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} & \\left \\langle \\rho , u \\alpha _ { 1 } , v \\alpha _ { 3 } \\right \\rangle \\cong M _ { 1 } \\\\ & \\ A = \\begin{pmatrix} u _ { 2 } & v _ { 2 } \\\\ u _ { 3 } & v _ { 3 } \\end{pmatrix} \\in \\mathrm { G L } _ { 2 } ( \\mathbb { F } _ { p } ) \\ \\ v _ { 2 } - u _ { 3 } - \\mathrm { d e t } ( A ) \\not \\equiv 0 \\ \\mathrm { m o d } \\ p . \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} ^ { C } L _ 1 x _ { \\lambda _ 1 } ( t ) = \\lambda _ 1 r ( t ) x _ { \\lambda _ 1 } ( t ) , \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} W ^ { ( 1 ) } - W ^ { ( 3 ) } = & b _ 0 y + ( 1 2 0 b _ 0 + b _ 1 ) y ^ 5 + ( 5 5 5 5 b _ 0 + 7 4 b _ 1 + b _ 2 ) y ^ 9 \\\\ & + ( 8 7 4 4 0 b _ 0 + 1 3 8 2 b _ 1 + 2 8 b _ 2 + b _ 3 ) y ^ { 1 3 } \\\\ & + ( - 2 6 6 6 6 1 0 b _ 0 - 3 8 6 7 0 b _ 1 - 6 7 5 b _ 2 - 1 8 b _ 3 ) y ^ { 1 7 } + \\cdots . \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\textrm { r a n k } ( M _ e ) = \\textrm { r a n k } ( M ^ \\ell _ e ) \\quad M ^ { \\ell - 1 } _ e = ( M ^ { \\ell , \\ell - 1 } _ e ) ^ T ( M ^ \\ell _ e ) ^ { - 1 } M ^ { \\ell , \\ell - 1 } _ e . \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} a = a _ 1 + a _ 2 , b = b _ 1 + b _ 2 , c = c _ 1 + c _ 2 \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} M ^ H : = \\{ m \\in M | h m = \\epsilon ( h ) m , ~ \\forall h \\in H \\} . \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} p : = \\{ n \\in \\mathbb { Z } ^ { > 0 } \\mid n \\leq j + l - 1 , \\ , f _ p \\in \\mathcal { W } \\} . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} T _ v = \\{ T ^ { 1 } _ v , \\ldots , T ^ { n } _ v \\} T ^ { j } _ v = \\{ k \\in \\underline { d } | i _ k = j \\} . \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\mathrm { d e t } ( p _ j ( \\underline { a } _ i ) ) = \\mathrm { d e t } ( M ) \\mathrm { d e t } ( F _ { \\mathbf { m } , j } ( \\underline { a } _ i ) ) . \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} Q _ r = \\left [ \\begin{smallmatrix} I _ n & 0 & 0 & 0 \\\\ 0 & I ' _ { n - r } & 0 & \\widetilde { I _ r } \\\\ 0 & 0 & I _ n & \\widetilde { I _ r } \\\\ \\widetilde { I _ r } & - \\widetilde { I _ r } & 0 & I ' _ { n - r } \\end{smallmatrix} \\right ] , \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} T _ { \\kappa ' \\kappa } ^ { ( n ) } ( t , t , t _ 2 , \\dots , t _ n ) = B _ 2 T _ { \\kappa ' \\kappa } ^ { ( n - 1 ) } ( t , t _ 2 , \\dots , t _ n ) . \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} \\mathcal M = \\mathcal M _ 0 ( V ) + O \\Bigl ( \\frac { ( | s _ 0 | \\log q ) ^ { C _ 4 } } { V ^ { 1 / 2 } } \\Bigr ) , \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} { } _ 0 ^ C D _ x ^ { \\varrho ( x ) } : = { } _ 0 I _ x ^ { n - q ( x , r ) } \\circ \\frac { { d ^ n } } { { d x ^ n } } , \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} a _ j = \\sum _ { \\atop { d > u } { d \\ell = j } } \\mu ( d ) , b _ j = \\sum _ { \\atop { d , \\ell \\leq u } { d \\ell = j } } \\mu ( d ) \\Lambda ( \\ell ) , \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} M ( u ) & = \\sum _ { 1 \\leq k < \\infty } \\sigma _ k ^ 2 \\cdot \\dim F _ u ( \\sigma _ k ) , \\\\ Q ( u ) ^ 2 & = \\sum _ { 1 \\leq k \\leq \\infty } \\ell _ k , \\\\ | J ( u ) | ^ 2 & = \\sum _ { 1 \\leq k \\leq \\infty } ( Q + \\sigma _ k ^ 2 ) \\ell _ k . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} L ( r ^ k , \\vec s ^ k ) & \\le r ^ k \\| \\vec x ^ * \\| _ \\infty - ( \\vec x ^ * , \\vec s ^ k ) \\\\ & = r ^ k \\| \\frac { \\vec y } { \\| \\vec y \\| _ p } \\| _ \\infty - ( \\frac { \\vec y } { \\| \\vec y \\| _ p } , \\vec s ^ k ) \\\\ & = \\frac { 1 } { \\| \\vec y \\| _ p } ( r ^ k - \\| \\vec s ^ k \\| _ 1 ) < 0 . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} V _ r ( a _ j : j \\in A ) = \\sup _ { \\atop { k _ 0 < \\ldots < k _ J } { k _ j \\in A } } \\bigg ( \\sum _ { j = 1 } ^ J \\abs { a _ { k _ j } - a _ { k _ { j - 1 } } } ^ r \\bigg ) ^ { \\frac { 1 } { r } } . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} C _ { k - 1 } ( N ) \\begin{bmatrix} B _ { k - 1 } \\\\ B _ { k - 2 } \\\\ \\vdots \\\\ B _ 0 \\end{bmatrix} = \\begin{bmatrix} Q _ { n + k - 1 } \\\\ Q _ { n + k - 2 } \\\\ \\vdots \\\\ Q _ 0 \\end{bmatrix} - A _ { 2 1 } ( N ) \\begin{bmatrix} B _ { b } \\\\ B _ { b - 1 } \\\\ \\vdots \\\\ B _ k \\end{bmatrix} , \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} [ ( g , \\alpha ) , ( g , \\beta ) ] = ~ & \\mathcal { L } _ { \\Pi ^ \\sharp ( g , \\alpha ) } ( g , \\beta ) - i _ { \\Pi ^ \\sharp ( g , \\beta ) } d ( g , \\alpha ) \\\\ = ~ & \\mathcal { L } _ { ( g , \\Pi ^ \\sharp \\alpha ) } ( g , \\beta ) - i _ { ( g , \\Pi ^ \\sharp \\beta ) } d ( g , \\alpha ) = ( g , [ \\alpha , \\beta ] ) . \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\varepsilon ^ { 2 s } ( - \\Delta ) ^ s u + u = Q ( x ) | u | ^ { 2 _ s ^ \\ast - 2 } u + P ( x ) | u | ^ { p - 2 } u , \\ \\ \\ \\ x \\in \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} M ( \\chi ) = \\begin{cases} p ^ { e - 1 } ( p + 1 ) & s = e , \\\\ \\frac { p - \\chi ( - 1 ) } { \\gcd ( 2 , p - 1 , e ) } \\varphi ( p ^ { e - 1 } ) & s < e \\le 2 , \\\\ \\frac { p ^ 2 - 1 } { \\gcd ( 2 , p - 1 , e ) } \\varphi ( p ^ { e - 2 } ) & e > \\max \\{ s , 2 \\} . \\end{cases} \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} \\begin{pmatrix} A & 0 \\\\ 0 & B \\end{pmatrix} \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} P ( H _ \\lambda - E ) \\psi = P \\phi + P ( H - E - i \\epsilon ) \\psi _ \\epsilon + \\lambda G ( E + i 0 ) \\phi + o ( 1 ) . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\omega _ { n } ^ { ( N ) } ( A ) = \\frac { 1 } { n } { \\displaystyle \\prod \\limits _ { k = 1 } ^ { N } } \\frac { P ( A _ { j _ { k } } ^ { ( k ) } ) } { \\mathbb { P } _ { n } ^ { ( k - 1 ) } ( A _ { j _ { k } } ^ { ( k ) } ) } . \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} { \\bf K } f ( x ) = \\int _ { \\real ^ d } K ( x , y ) f ( y ) \\ d y \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} | R - R ( r ) | & = \\Big | \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ 2 } ( f ( \\sigma _ { s - } x ) - f ( \\sigma _ s ( r ) x ) ) \\Phi _ { 0 , I _ 2 } ( d x ) d { G _ p } ( s ) \\Big | \\\\ & \\leq \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ 2 } | ( f \\psi _ A ) ( \\sigma _ { s - } x ) - ( f \\psi _ A ) ( \\sigma _ s ( r ) x ) | \\Phi _ { 0 , I _ 2 } ( d x ) d { G _ p } ( s ) \\\\ & ~ ~ ~ + \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ 2 } | ( f \\psi _ A ' ) ( \\sigma _ { s - } x ) - ( f \\psi _ A ' ) ( \\sigma _ s ( r ) x ) | \\Phi _ { 0 , I _ 2 } ( d x ) d { G _ p } ( s ) . \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} \\partial _ t \\Psi + \\tilde { \\mathsf { D } } \\Psi + \\mathsf { V } ( \\Psi ) - \\Xi = 0 \\ , , \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} V ( k ) = \\{ i \\ , \\ , \\big | \\ , \\ , \\ , z _ i ^ { k + 1 } \\ne z _ i ^ k \\} . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} v ^ n ( y , x ) \\triangleq \\frac { 1 } { \\epsilon _ n ^ n } \\int _ { \\mathbb { R } ^ n } \\phi ^ n \\Big ( \\frac { z - \\theta _ n x } { \\epsilon _ n } \\Big ) v \\Big ( y , \\sum _ { i = 1 } ^ n \\frac { z ^ i e _ i } { \\lambda _ i } \\Big ) \\dd z , y \\in E , x \\in \\mathbb { B } . \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} r _ f ( \\tau ) : = \\int _ 0 ^ { i \\infty } \\frac { f ( w ) } { ( - i ( w + \\tau ) ) ^ { 2 - k } } d w . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} \\tau _ V + \\sum _ { x } \\pm u _ e = 0 \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\varphi = \\mathcal { A } ( \\varphi ) = : L _ { k } ( N ( \\varphi ) ) + L _ { k } ( l _ { k } ) , \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | u _ n ( x ) - u _ n ( y ) | ^ 2 \\phi _ \\rho ( y ) } { | x - y | ^ { N + 2 s } } d y d x = \\int _ { \\mathbb { R } ^ N } \\phi _ \\rho d \\eta . \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} & x ^ k = \\dfrac { ( \\varphi - 1 ) y ^ k + x ^ { k - 1 } } { \\varphi } . \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} \\mathbb { \\mu } _ { n } = \\mathbb { E } \\left \\Vert \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } f ( X _ { i } ) \\right \\Vert _ { \\mathcal { F } } \\leqslant M \\int _ { 0 } ^ { 1 } \\sqrt { 1 + \\log N _ { \\left [ \\ \\right ] } ( \\mathcal { F } , M \\varepsilon , d _ { P } ) } d \\varepsilon . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( a _ { n + k } \\right ) = z ^ { k } \\left ( \\mathcal { A } \\left ( z \\right ) - \\sum _ { j = 0 } ^ { k - 1 } \\frac { a _ { j } } { z ^ { j } } \\right ) . \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} \\breve { J } W = K W + L W . \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} 2 P _ 1 ( 1 ) P _ 3 ( 1 ) \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\epsilon ( \\chi ) L \\left ( \\frac { 1 } { 2 } , \\overline { \\chi } \\right ) = ( 1 + o ( 1 ) ) 2 P _ 1 ( 1 ) P _ 3 ( 1 ) \\varphi ^ + ( q ) . \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ \\infty \\Big ( \\frac { n - 1 } { n } \\Big ) ^ k = n - 1 \\quad \\sum \\limits _ { k = 1 } ^ \\infty 2 k \\Big ( \\frac { n - 1 } { n } \\Big ) ^ k = 2 n ( n - 1 ) , \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} \\delta _ \\Phi ( \\psi _ 1 , & \\ldots , \\psi _ { n - q } ) = \\\\ & ( \\psi _ { 2 } \\phi _ 1 - \\phi _ { 1 + q } \\psi _ 1 , \\mbox { } \\psi _ { 3 } \\phi _ 2 - \\phi _ { 2 + q } \\psi _ 2 , \\ldots , \\psi _ { n - q } \\phi _ { n - ( q + 1 ) } - \\phi _ { n - 1 } \\psi _ { n - ( q + 1 ) } ) \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} R _ { G , o p t } ( N , t ) & \\geq 1 - \\frac { \\log \\left [ \\prod \\limits _ { k = 0 } ^ { \\min \\{ 8 t , N - 1 \\} } ( N - k ) \\right ] } { \\log N ! } \\\\ & > 1 - \\frac { ( 8 t + 1 ) \\log N } { ( N + \\frac { 1 } { 2 } ) \\log N - ( \\log e ) N } \\\\ & > 1 - \\frac { ( 8 t + 1 ) \\log N } { N \\log N - ( \\log e ) N } \\\\ & > 1 - \\frac { 8 t + 1 } { N } \\left ( 1 + \\frac { 2 \\log e } { \\log N } \\right ) , \\\\ \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} R _ { \\ell k } = \\frac { 1 } { 4 } \\left \\langle U ^ t \\mathcal A _ \\ell E U E , \\mathcal A _ k \\right \\rangle _ \\mathcal V \\ ; . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} \\mathbb { E } [ Y _ i ] & \\ge 2 ^ { - m } \\binom { l } { \\frac { l } { 2 } } ( 1 - \\sigma ^ { \\frac { l } { 2 } } ) ( 1 + \\sigma ^ { \\frac { l } { 2 } } ) ^ { m - 1 } \\\\ & \\ge \\frac { 2 ^ { l - m } } { l + 1 } ( 1 - \\sigma ^ { \\frac { l } { 2 } } ) ( 1 + \\sigma ^ { \\frac { l } { 2 } } ) ^ { m - 1 } \\ge \\frac { 2 ^ { l - m } } { l + 1 } ( 1 - \\sigma ^ { \\frac { l } { 2 } } ) . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} Z _ { \\lambda , \\varepsilon } ( \\beta ) = \\sum \\limits _ { j = - N / 2 } ^ { N / 2 } | 1 + e ^ { 2 \\pi i \\psi _ j } | ^ { 2 \\lambda \\ , \\beta } e ^ { \\beta V _ \\varepsilon ( \\psi _ j ) } . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} \\pi _ 1 & = \\left ( \\psi _ 1 , \\psi _ 2 , \\cdots , \\psi _ { d + 1 } \\right ) , \\\\ \\pi _ 2 & = \\left ( \\psi _ { \\sigma ( 1 ) } , \\psi _ { \\sigma ( 2 ) } , \\cdots , \\psi _ { \\sigma ( d + 1 ) } \\right ) , \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} \\begin{aligned} \\| X _ { t } - Z _ { t } \\| _ { H } ^ { 2 } = & \\| x - Z _ { 0 } \\| _ { H } ^ { 2 } + 2 \\int _ { 0 } ^ { t } ( - \\eta _ { s } - G _ { s } , X _ { s } - Z _ { s } ) _ { H } \\ , d s \\\\ & + \\int _ { 0 } ^ { t } ( L ^ { b } X _ { s } - P ^ { \\ast } L ^ { b } P Z _ { s } , X _ { s } - Z _ { s } ) _ { H } \\ , d s \\\\ & + 2 \\int _ { 0 } ^ { t } ( X _ { s } - Z _ { s } , B ( X _ { s } - P Z _ { s } ) \\ , d W _ { s } ) _ { H } \\\\ & + \\int _ { 0 } ^ { t } \\| B ( X _ { s } - P Z _ { s } ) \\| _ { L _ { 2 } ( U , H ) } ^ { 2 } . \\end{aligned} \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c c } 0 & a & b \\\\ - a & 0 & c \\\\ - b & - c & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} \\Gamma _ M ( \\kappa w \\ , | \\ , \\kappa a ) = \\kappa ^ { - B _ { M , M } ( w \\ , | \\ , a ) / M ! } \\ , \\Gamma _ M ( w \\ , | \\ , a ) . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} [ { \\mathbb D } , { \\mathbb D } ] ^ B { } _ { C } = { \\mathbb D } ^ { B } { } _ { R } { \\mathbb D } ^ { R } { } _ { C } - \\ , { \\mathbb D } ^ { R } { } _ { C } { \\mathbb D } ^ { B } { } _ { R } = 3 { \\mathbb D } ^ { B } { } _ { C } . \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} d _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x _ { \\lambda _ 1 } ( b ) + d _ 2 ~ ^ { A B R } D _ b ^ \\alpha x _ { \\lambda _ 1 } ( b ) = 0 , \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} \\partial _ t u ^ n _ t = \\Delta u _ t ^ n + \\mathrm { d i v } F ( u _ t ^ n ) + \\beta _ j \\nabla u _ t ^ n \\dot { Z } ^ j _ t ( n ) . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} L _ M ( w \\ , | \\ , a ) \\triangleq \\partial _ s \\zeta _ M ( s , \\ , w \\ , | \\ , a ) | _ { s = 0 } , \\ , \\ , \\Re ( w ) > 0 . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} E \\big \\langle y \\mid \\partial ( y ) = \\sum _ { i } s _ { i } t _ { i } ^ { u _ { i } - 1 } x _ { i } \\big \\rangle \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} h ' ( x _ i ) : = \\max \\{ h ' ( x ) \\colon x \\in H _ i ^ { h ( x _ i ) - 1 } , c ( x _ i x ) = 2 \\} . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\gamma _ j ^ { \\frac { 1 } { 2 ( 1 - \\delta ) } } \\ , b ^ { w _ j } & \\le \\sum _ { j = 1 } ^ { \\infty } \\gamma _ j ^ { \\frac { 1 } { 2 } } \\ , b ^ { w _ j } < \\infty . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\begin{pmatrix} j _ { l _ 1 , k } \\\\ & j _ { l _ 2 , n - 2 k } \\\\ & & j _ { l _ 1 , k } \\end{pmatrix} , \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\mathbf { u } ^ s ( A _ i ) = & \\mathbf { u } ^ s ( X ) + \\nabla \\mathbf { u } ^ s ( X ) ( A _ i - X ) + \\mathbf { R } ^ s _ { i } ( X ) , ~ i \\in \\mathcal { I } ^ s , ~ \\forall X \\in T _ { \\ast } ^ s , s = \\pm . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} G ( A , B ) = A \\# B = A ^ { 1 / 2 } ( A ^ { - 1 / 2 } B A ^ { - 1 / 2 } ) ^ { 1 / 2 } A ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} ( v , K u ) & = - 4 \\pi \\int _ 0 ^ \\infty ( \\overline { { \\mathcal F } ^ \\ast N _ v ) ( t ) } ( { \\mathcal F } ^ \\ast N _ u ) ( t ) d t \\\\ & = - 4 \\pi \\big ( { \\mathcal F } ^ \\ast N _ v , { \\mathcal R } { \\mathcal F } ^ \\ast N _ u \\big ) = - 4 \\pi \\big ( N _ v , { \\mathcal F } { \\mathcal R } { \\mathcal F } ^ \\ast N _ u \\big ) . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} \\left ( \\sum _ { l = 0 } ^ { k } \\underline { M _ { l } } \\right ) \\underline { m _ { k , r + 2 r _ i } ^ { ( i ) } } = t ^ { \\alpha / 2 } \\underline { m _ { k + 1 , r } ^ { ( i ) } } \\left ( \\sum _ { l = 0 } ^ { k - 1 } \\underline { M ' _ { l } } \\right ) + t ^ { \\beta ' / 2 } \\underline { m _ { k , r } ^ { ( i ) } m _ { k , r + 2 r _ i } ^ { ( i ) } \\prod _ { s = 0 } ^ { k - 1 } A _ { i , r + ( 2 k - 1 - 2 s ) r _ i } ^ { - 1 } } , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} K K ^ { T } = ( r - \\lambda ) I + \\lambda J \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\| P _ k ( x ) \\| ^ 2 = \\sum _ { l = 0 } ^ { N - 1 } \\| f _ k ^ { ( l ) } ( x ) \\| ^ 2 = \\sum _ { l = 0 } ^ { N - 1 } | \\mathcal { A } ( x , k ) ^ { ( l ) } | = | \\mathcal { A } ( x , k ) | . \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} e ( M _ { 1 } , M _ { 1 } ) & = 1 + ( p ^ { 3 } - p ^ { 2 } - 1 ) ( p + 1 ) + ( p ^ { 4 } - p ^ { 3 } - 2 p ^ { 2 } + 2 p + 1 ) p + ( p ^ { 2 } - 1 ) p ^ { 3 } \\\\ & = ( 2 p ^ { 3 } - 3 p + 1 ) p ^ { 2 } , \\\\ e ( C _ { p } ^ { 3 } , M _ { 1 } ) & = ( p ^ { 3 } - 1 ) ( p + 1 ) p ^ { 2 } + ( p ^ { 3 } - 1 ) ( p ^ { 2 } - 2 ) p ^ { 2 } = ( p ^ { 3 } - 1 ) ( p ^ { 2 } + p - 1 ) p ^ { 2 } . \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} A _ { i , j } : = \\Delta _ { I ( i , j ) } , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} B _ i = \\begin{cases} A _ { n - i } & \\\\ A _ n & \\end{cases} \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\Gamma ( q z ) = \\theta ( z ) \\Gamma ( z ) \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\ell ( g ) = \\sqrt { ( \\log | a _ 1 | ) ^ 2 + \\ldots + ( \\log | a _ d | ) ^ 2 } = \\| j ( g ) \\| _ 2 . \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} = \\int _ { \\mathbb { G } } \\int _ { \\mathbb { G } } \\frac { | u ( x ) - u ( y ) | ^ { p } } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d x d y = [ u ] ^ { p } _ { s , p , q } . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{gather*} \\lim _ { n \\uparrow \\infty } u ( x _ n ) - f ( x _ n ) = \\sup _ x u ( x ) - f ( x ) , \\\\ \\lim _ { n \\uparrow \\infty } u ( x _ n ) - \\lambda H _ \\dagger f ( x _ n ) - h ( x _ n ) \\leq 0 . \\end{gather*}"}
-{"id": "2218.png", "formula": "\\begin{align*} f ^ { k + 1 } ( x _ 0 ) = f ( f ^ k ( x _ 0 ) ) \\stackrel { ( \\star ) } { \\geq } f ( g ^ k ( x _ 0 ) ) \\stackrel { ( \\star \\star ) } { \\geq } g ( g ^ k ( x _ 0 ) ) = g ^ { k + 1 } ( x _ 0 ) , \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} p _ x = 1 - a ^ 2 ( \\beta , H , 1 ) < p _ c ^ s ( d ) / 2 x \\in \\mathbb { Z } ^ d , \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} [ L _ { n } , Y ( A , w ) ] = \\sum _ { m \\ge - 1 } \\binom { n + 1 } { m + 1 } Y ( L _ { m } A , w ) w ^ { n - m } . \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\begin{cases} y ^ 2 + G = 0 , & \\\\ \\pi ^ s y - Q = 0 . & \\end{cases} \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} A _ { i , j } = \\Phi \\left ( \\frac { \\partial r _ i } { \\partial x _ j } \\right ) , \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} 2 ^ { n ( d - k ) } \\left | \\Delta f _ n ^ { ( k ) } ( \\gamma ) - \\frac 1 { 2 ^ n } f _ n ^ { ( k + 1 ) } ( \\gamma ) + \\sum _ { \\ell = 2 } ^ { d - k } t _ { k , k + \\ell } \\frac 1 { 2 ^ { n \\ell } } f _ n ^ { ( k + \\ell ) } ( \\gamma ) \\right | \\le \\varepsilon _ n , \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} u ( x ) = \\left ( \\begin{array} { c c } 1 & x \\\\ 0 & 1 \\end{array} \\right ) , \\ , n ( x ) = \\left ( \\begin{array} { c c } 1 & 0 \\\\ x & 1 \\end{array} \\right ) , \\ , t ( \\alpha ) = \\left ( \\begin{array} { c c } \\alpha & 0 \\\\ 0 & \\alpha ^ { - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} E ( \\mathbf { g } , s , f ) = \\sum \\limits _ { \\gamma \\in P _ { 4 n } ( F ) \\backslash G _ { 4 n } ( F ) } f ( \\gamma \\mathbf { g } , s ) , \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} M ( a , s ) = \\frac { \\Psi ( a ) } { ( a + s ) ^ 2 } s ^ 2 M ( 0 , s ) . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} t _ { \\alpha , \\varepsilon } = C _ { \\alpha , 1 } \\ , \\varepsilon ^ 2 \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } \\mathrm { a n d } s _ { \\alpha , \\varepsilon } = C _ { \\alpha , 2 } \\ , \\varepsilon ^ 2 \\sqrt { D _ \\varepsilon } \\forall \\varepsilon \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { 0 } ^ { + \\infty } n ( t , s ) \\d s = 0 . \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} T _ x P ^ { k + 1 } ~ = ~ P T _ x P ^ { k } + A _ x P ^ { k } , \\forall k \\in \\N . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} \\tau _ { K } ( \\chi \\otimes \\rho ) = \\rho ( ( \\mathfrak { f } ( \\chi ) \\mathfrak { D } _ { K } ^ { \\chi ( 1 ) } ) ^ { - 1 } ) \\tau _ { K } ( \\chi ) = \\rho ( c _ { \\chi } ^ { - 1 } ) \\tau _ K ( \\chi ) , \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} V _ \\varepsilon ( \\psi ) - q \\beta { \\bf C o v } ( V _ \\varepsilon ( \\psi ) , V _ \\varepsilon ( \\phi ) ) \\big | _ \\mathcal { P } = V _ \\varepsilon ( \\psi ) \\big | _ \\mathcal { Q } . \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( n ) = \\begin{cases} \\dot { \\phi } \\left ( \\tfrac { 1 } { 2 } | n | \\right ) , & n \\in ( 2 \\N ) ^ N \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} V ( G _ 1 * G _ 2 ) = V ( G _ 1 ) \\cup V ( G _ 2 ) , ~ E ( G _ 1 * G _ 2 ) = \\{ e \\cup f \\ , | \\ , e \\in E ( G _ 1 ) , f \\in E ( G _ 2 ) \\} . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} \\widetilde u ( z ) \\coloneqq - \\sum _ { i = 1 } ^ N \\frac { \\dot { \\lambda } _ i } { z - z _ i } d z + \\sum _ { i = 1 } ^ \\ell \\Bigg ( \\sum _ { j = 1 } ^ { m _ i } \\frac { 1 } { z - w ^ i _ j } - \\sum _ { j = 1 } ^ { m _ 0 } \\frac { 1 } { z - w ^ 0 _ j } \\Bigg ) \\alpha _ i d z . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} g ^ { ( n ) } ( 0 ) & = \\sum \\limits _ { m = 0 } ^ n \\binom { n } { m } B ^ { ( f ) } _ { n - m } ( q ) \\frac { d ^ { m + r } } { d t ^ { m + r } } \\Big \\vert _ { t = 0 } \\bigl [ e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) \\bigr ] , \\\\ & = ( - 1 ) ^ r \\sum \\limits _ { p = 0 } ^ { M - 1 } ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ { M - 1 } \\bigl ( b _ 0 + \\sum b _ { k _ j } \\bigr ) ^ r \\ , B ^ { ( f ) } _ n \\bigl ( q + b _ 0 + \\sum b _ { k _ j } \\bigr ) . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} \\begin{cases} \\underbar s _ 1 ^ n - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 1 ^ n + \\epsilon \\cr \\underbar s _ 2 ^ n - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 2 ^ n + \\epsilon , \\end{cases} \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\mathfrak { M } ( q \\ , | \\tau , 0 , 0 ) = \\frac { \\Gamma \\bigl ( 1 - \\frac { q } { \\tau } \\bigr ) } { \\Gamma ^ q \\bigl ( 1 - \\frac { 1 } { \\tau } \\bigr ) } , \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} a \\rhd b = \\lambda _ a ( b ) . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { d \\leq x } \\frac { \\mu ( d ) \\log d } { d } \\sum _ { l \\leq x / d } 1 & = \\frac { x } { 2 } \\sum _ { d \\leq x } \\frac { \\mu ( d ) } { d ^ 2 } \\log d + O \\left ( \\sum _ { d \\leq x } \\frac { \\log d } { d } \\right ) \\\\ & = \\frac { \\zeta ' ( 2 ) } { 2 \\zeta ^ { 2 } ( 2 ) } x + O \\left ( \\log ^ { 2 } x \\right ) , \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} \\dot { w } _ t = \\left ( A _ 1 ^ { \\Delta } - A _ 2 ^ { \\Delta , \\sigma } - B _ 1 ^ { \\Delta } + B _ 2 ^ { \\Delta , \\sigma } \\right ) w _ t . \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} ~ ~ ^ { A B C } _ { a } D ^ \\alpha ( p ( t ) ~ ^ { A B R } D ^ \\alpha _ b x ( t ) ) + q ( t ) x ( t ) = \\lambda r ( t ) x ( t ) , ~ ~ t \\in ( a , b ) , \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} \\Theta ( f ) ( M _ k ) = \\Theta ( f ) ( M _ { \\overline { \\pi _ { i j } } } ) = \\int _ G f ( r ) ( \\rho _ r M _ { \\overline { \\pi _ { i j } } } \\rho _ r ^ * ) d r = M _ g \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\begin{cases} 2 a m + 2 a n = c _ 0 m + c _ 0 n \\\\ 2 b m = m c _ 0 \\\\ 2 n = n c _ 0 \\end{cases} , \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} f = \\begin{pmatrix} f _ { 1 1 } & f _ { 1 2 } & f _ { 1 3 } & \\dots & f _ { 1 n } \\\\ f _ { 2 1 } & f _ { 2 2 } & f _ { 2 3 } & \\dots & f _ { 2 n } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ f _ { m 1 } & f _ { m 2 } & f _ { m 3 } & \\dots & f _ { m n } \\end{pmatrix} \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} { \\bf u } - F _ { \\Gamma } { \\bf u } + T _ G { \\bf Q } T _ G { \\bf D } p = T _ G { \\bf Q } T _ G ( { \\bf f } ) . \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} [ h \\beta _ m , \\epsilon ] = \\left [ \\left ( \\begin{array} { c c } - d ^ { - 1 } p ^ { - m } & \\ast \\\\ 0 & - d p ^ m \\end{array} \\right ) , e \\right ] \\left [ \\left ( \\begin{array} { c c } 0 & - 1 \\\\ 1 & \\frac { c } { d } p ^ { - 2 m } \\end{array} \\right ) , 1 \\right ] , \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} d _ { k - \\deg ( F ) } : = \\sum _ { i = 1 } ^ s d _ { k - \\deg ( f _ i ) } , \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} p ( x , \\vartheta ) = 1 - \\sum _ { y \\in E } p ( x , y ) , x \\in E . \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} & x _ 0 ^ T Q x _ 0 \\int _ 0 ^ t \\left \\| u ( s ) \\right \\| _ 2 ^ 2 \\exp \\left ( \\int _ s ^ t \\left \\| u ( w ) \\right \\| _ 2 ^ 2 d w \\right ) d s = x _ 0 ^ T Q x _ 0 \\left [ - \\exp \\left ( \\int _ s ^ t \\left \\| u ( w ) \\right \\| _ 2 ^ 2 d w \\right ) \\right ] _ { s = 0 } ^ t \\\\ & = x _ 0 ^ T Q x _ 0 \\left ( \\exp \\left ( \\int _ 0 ^ t \\left \\| u ( s ) \\right \\| _ 2 ^ 2 d s \\right ) - 1 \\right ) . \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} f = \\displaystyle \\sum _ { I \\subset [ N ] } f ^ I \\quad \\| f \\| = \\displaystyle \\sum _ { I \\subset [ N ] } \\left \\| f ^ I \\right \\| . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\frac { 2 p } { 2 - p } \\log ( \\frac { \\left \\| f \\right \\| _ { L ^ p ( O _ N ^ + ) } } { \\left \\| \\widehat { f } \\right \\| _ { \\ell ^ { p ' } ( \\widehat { O _ N ^ + } ) } } ) & \\geq \\frac { 2 p } { 2 - p } \\frac { p - 2 } { p } \\log ( \\sup _ { k \\geq t } \\frac { C ( 1 + k ) } { n _ k ^ { \\frac { 1 } { 2 } } } ) \\\\ & = - 2 \\log ( \\sup _ { k \\geq t } \\frac { C ( 1 + k ) } { n _ { k } ^ { \\frac { 1 } { 2 } } } ) = 2 \\inf _ { k \\geq t } \\log ( \\frac { n _ { k } ^ { \\frac { 1 } { 2 } } } { C ( 1 + k ) } ) . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{gather*} \\mathcal G _ { 1 , ( 1 : 3 ) } ^ \\ast ( s , \\tau ) = \\Im ( \\tau ) ^ s \\sum _ { m , n \\in \\Z } \\frac { 1 } { \\ ( ( 3 m + 1 ) \\tau + n \\ ) } \\frac { 1 } { | ( 3 m + 1 ) \\tau + n | ^ { 2 s } } \\end{gather*}"}
-{"id": "8535.png", "formula": "\\begin{align*} Q ( K + \\sum _ { j \\in \\mathcal { I } ^ - } L ( A _ j ) \\overline { \\Psi } _ j ) Q ^ T = P ^ - + \\left [ \\begin{array} { c c } ( \\hat { \\lambda } + 2 \\hat { \\mu } ) g _ n ( F ) & \\hat { \\lambda } g _ t ( F ) \\\\ \\hat { \\mu } g _ t ( F ) & \\hat { \\mu } g _ n ( F ) \\end{array} \\right ] \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} f ( \\sigma _ G ( k ) g ) & = f ( k g ) \\\\ & = ( F k g , k g \\cdot t ( F k g ) ^ { - 1 } ) = ( F g , k g \\cdot t ( F g ) ^ { - 1 } ) \\\\ & = ( { \\mathrm { i d } } _ { F \\backslash G } \\times \\sigma _ F ( k ) ) ( f ( g ) ) , \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} L _ k ( \\lambda ) : = \\begin{bmatrix} - 1 & \\lambda \\\\ & - 1 & \\lambda \\\\ & & \\ddots & \\ddots \\\\ & & & - 1 & \\lambda \\\\ \\end{bmatrix} \\in \\mathbb { F } [ \\lambda ] ^ { k \\times ( k + 1 ) } , \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} N = \\left ( \\begin{array} { c c c c } 0 & 1 & 0 & \\\\ 0 & 0 & 1 & \\\\ 0 & 0 & 0 & \\\\ & & & \\ddots \\end{array} \\right ) . \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\partial _ t f _ t = - G ( \\langle w \\rangle _ { f _ t } ) \\partial _ v f _ t + \\partial _ v ( v f _ t ) + \\sigma \\ , \\partial _ { v v } f _ t ( v ) \\ , , \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{gather*} \\mbox { T h e r e e x i s t s s o m e $ N $ s u c h t h a t $ j > N \\Rightarrow W _ { i , j } = W _ { i , j + 1 } $ f o r a l l $ i $ } . \\\\ \\mbox { T h e r e e x i s t s s o m e $ d $ s u c h t h a t $ i > d \\Rightarrow W _ { i , j } = 0 $ f o r a l l $ j $ } . \\\\ \\textstyle \\sum _ { p \\geq 0 } W _ { i + p , j + p } \\geq \\sum _ { p \\geq 0 } W _ { i + 1 + p , j + p } . \\end{gather*}"}
-{"id": "2991.png", "formula": "\\begin{align*} \\partial _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) } ( x _ { L _ { \\infty } / K } ) = - C _ { L _ { \\infty } / K } - U ' _ { L _ { \\infty } / K } + M _ { L _ { \\infty } / K } . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\mathcal W _ { \\mathfrak R _ M \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) = \\mathcal W _ { \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) = A _ { \\chi } ( B ) \\det ( Y ) ^ { ( k + 1 ) / 2 } e ^ { 2 \\pi \\sqrt { - 1 } \\mathrm { T r } ( B Z ) } , \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} R _ Y A ( x ) = ( A \\times R _ Y I ) ( x ) . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\nu _ k ^ * \\omega ' _ { \\log , I _ 1 , \\ldots , I _ k } ( a _ { I _ 1 } , \\ldots , a _ { I _ k } ) = F ' \\cdot \\widetilde { \\omega } _ { \\log , I _ 1 , \\ldots , I _ k } ( a _ { I _ 1 } , \\ldots , a _ { I _ k } ) . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} [ x ^ U ] _ \\beta ^ T \\alpha = \\alpha x ^ T . \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} \\mathcal I _ 2 ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { \\mathcal I _ 2 ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) } { \\langle \\mathbf h _ 2 , \\mathbf h _ 2 \\rangle \\langle \\breve { \\mathbf g } _ 2 , \\breve { \\mathbf g } _ 2 \\rangle \\langle \\pmb { \\phi } _ 2 , \\pmb { \\phi } _ 2 \\rangle } , \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} j ^ * \\sigma ^ + _ \\lambda = \\alpha ; j ^ * \\sigma ^ + _ \\mu = \\beta ; j ^ * \\sigma ^ + _ \\nu = \\gamma . \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} 1 + a _ 1 ^ 2 + a _ 1 b + a _ 1 b ^ 2 + b ^ 4 = 0 . \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} \\ast F _ A = _ A \\Phi , \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} \\left | X \\right | = \\frac { 1 } { n ! \\ , \\omega _ n } \\int _ { \\R ^ n } h ( x ) \\ , \\mathrm { d } x . \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} M _ k = \\begin{bmatrix} 1 & a \\\\ 0 & 1 \\end{bmatrix} M _ { k ' } = \\begin{bmatrix} 1 & b \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\Delta _ x | x - y | = \\frac { ( n - 1 ) } { | x - y | } , \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} K _ L ^ { \\bullet } : = R \\Gamma ( L , \\Z _ p ( 1 ) ) [ 1 ] \\oplus H _ L [ - 1 ] \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} y ( t ) = z ( t ) + \\int _ 0 ^ t \\lambda ( s ) y ( s ) \\ , \\d s \\quad \\mbox { f o r e v e r y } t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\mathbb { V } \\left ( \\mathbb { G } ^ { ( N ) } ( f ) \\right ) - \\mathbb { V } \\left ( \\mathbb { G } ( f ) \\right ) = \\sum _ { k = 1 } ^ { N } \\Phi _ { k } ^ { ( N ) } ( f ) ^ { t } \\cdot \\mathbb { V } \\left ( \\mathbb { G } [ \\mathcal { A } ^ { ( k ) } ] \\right ) \\cdot \\Psi _ { k } ^ { ( N ) } ( f ) , \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} \\gamma ^ { \\ast } ( t ) = ( \\varphi ( t ) , \\theta ( t ) ) \\bar { \\gamma } ( t ) = ( \\rho ( t ) , \\gamma ^ { \\ast } ( t ) ) \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} A ( X ) = \\ , & X ^ 3 ( 1 + a _ 1 + b + a _ 1 z ) + X ^ 2 ( a _ 1 + a _ 1 k + z + a _ 1 z + b z ) \\cr & + X ( 1 + a _ 1 + k + b k + z + a _ 1 z + b z + a _ 1 k z ) \\cr & + a _ 1 + k + a _ 1 k + b k + a _ 1 k ^ 2 + a _ 1 z + b z + k z + b k z \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} { \\rm R H S } = \\alpha _ n 2 ^ n \\prod _ { m = 1 } ^ n ( m - 1 ) ! \\prod _ { j = 1 } ^ m \\frac { 1 } { 2 k - n - m - 2 + 2 j } . \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} \\beta _ { i _ 0 j _ 0 } = A _ { i _ 0 j _ 0 k _ 0 } ^ { i _ 0 j _ 0 } d F _ { i _ 0 } \\wedge d F _ { j _ 0 } + A _ { i _ 0 k _ 0 j _ 0 } ^ { i _ 0 j _ 0 } d F _ { i _ 0 } \\wedge d F _ { k _ 0 } + A _ { j _ 0 k _ 0 i _ 0 } ^ { i _ 0 j _ 0 } d F _ { j _ 0 } \\wedge d F _ { k _ 0 } . \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} ( 1 - \\gamma ) \\mathcal L _ { + \\infty } \\nu = G ( \\nu ) \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } c _ { k } ( j ) \\log j = \\frac { 1 } { 2 } \\sum _ { d | k } \\frac { \\mu ( d ) } { d } \\log d + \\log \\sqrt { 2 \\pi } \\sum _ { d | k } \\frac { \\mu ( d ) } { d } + \\frac { \\vartheta } { 1 2 } \\sum _ { d | k } \\frac { \\mu ( d ) } { d ^ 2 } . \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} { \\bf E } [ M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ^ q ] = \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) , \\ ; \\Re ( q ) < \\tau . \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} C ' _ { j } = \\{ e ^ { 2 i \\pi n p _ { j } ^ { - l } } \\ , ; \\ , n , l \\ge 0 \\} \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} \\alpha ( t s ) = \\alpha ( t ) \\alpha ( s ) = t ^ { n } s ^ { p - 2 } = s ^ { p - 2 } t ^ { k ^ { p - 2 } n } \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} \\frac { ( m + 1 ) ! } { a _ { 1 } ! \\cdots a _ { i } ! ( m + 1 - s - r ) ! s ! } = \\binom { m + 1 } { m + 1 - a _ { 1 } } \\binom { m + 1 - a _ { 1 } } { m + 1 - a _ { 1 } - a _ { 2 } } \\cdots \\binom { m + 1 - a _ { 1 } - \\cdots - a _ { i } } { m + 1 - a _ { 1 } - \\cdots - a _ { i } - a _ { i + 1 } } \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} \\mbox { $ u _ { \\infty } $ i s a w e a k s o l u t i o n o f $ u _ t - \\nabla \\left ( m \\ , \\mathcal { B } _ M ^ { m - 1 } \\nabla u \\right ) = 0 $ } \\mbox { i n $ \\R ^ n \\times ( 0 , \\infty ) $ } . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} N _ { \\mathrm { c } m } = \\sum _ { n = 1 } ^ { m } \\frac { n ! } { 2 } \\mathcal { C } _ { n } ^ { m } \\left [ \\frac { N _ { \\mathrm { d } \\ , n + 1 } } { ( n + 1 ) ! } - N _ { \\mathrm { d } 1 } \\frac { N _ { \\mathrm { d } n } } { n ! } \\right ] . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} { \\mathcal O } \\left ( ( 2 ^ { L } ( L - 2 ) + ( J - L ) 2 ^ { L } L \\right ) = { \\mathcal O } \\left ( m \\log m \\log \\frac { 2 N } { m } \\right ) , \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{gather*} G _ { 4 k } ( \\tau ) : = { \\frac 1 2 } \\sum _ { ( m , n ) \\in \\Z ^ 2 \\setminus \\{ ( 0 , 0 ) \\} } \\frac { 1 } { ( m + n \\tau ) ^ { 4 k } } = \\zeta ( 4 k ) E _ { 4 k } , E _ { 4 k } ( \\tau ) = 1 - \\frac { 8 k } { B _ { 4 k } } \\sum _ { n = 1 } ^ \\infty \\frac { n ^ { 4 k - 1 } q ^ n } { 1 - q ^ n } , \\end{gather*}"}
-{"id": "8596.png", "formula": "\\begin{align*} F ( T ) = y ^ 3 + A ^ 2 T ^ 2 + ( A B + A ) T - 1 \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\iota ^ { ( i ) } \\big ( D _ a ( w ) \\big ) = \\sum _ { \\{ s | b _ s = a \\} } \\iota ^ { ( i ) } \\big ( b _ { s + 1 } \\ldots b _ r b _ 1 \\ldots b _ { s - 1 } \\big ) = \\iota ^ { ( i ) } ( w ) _ { a } . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} H ^ { \\bullet _ a } _ { t s } : = h ^ a _ t - h ^ a _ s , H ^ { [ \\tau _ 1 \\dots \\tau _ n ] _ a } _ { t s } = \\int _ s ^ t \\prod _ { i = 1 } ^ n H ^ { \\tau _ i } _ { u s } \\ , d h ^ a _ u ; \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} \\varphi ( ( u ^ { \\beta } _ { s , t } ) ^ * u ^ { \\alpha } _ { i , j } ) = \\frac { \\delta _ { \\alpha , \\beta } \\delta _ { i , s } \\delta _ { j , t } ( Q _ { \\alpha } ) _ { i , i } ^ { - 1 } } { \\mathrm { t r } ( Q _ { \\alpha } ) } \\mathrm { ~ a n d ~ } \\varphi ( u ^ { \\beta } _ { s , t } ( u ^ { \\alpha } _ { i , j } ) ^ * ) = \\frac { \\delta _ { \\alpha , \\beta } \\delta _ { i , s } \\delta _ { j , t } ( Q _ { \\alpha } ) _ { j , j } } { \\mathrm { t r } ( Q _ { \\alpha } ) } \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} \\tilde { L } ^ { ( \\mu ) } : = ( L ^ { a } + 2 K - 1 ) ^ { ( \\mu ) } = \\frac { 1 } { \\mu } ( ( 1 - \\mu ( L ^ { a } + 2 K - 1 ) ) ^ { - 1 } - 1 ) . \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} K ( C , D , N , R , S ) ^ 2 = C S ( R S + N ) ( C + R D ) + C ^ 2 D S \\sqrt { ( R S + N ) R } + D ^ 2 N R S ^ { - 1 } . \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} \\frac { d } { d t } X _ i \\cdot X _ i ^ { - 1 } = \\Omega _ i + \\frac 1 2 \\sum _ { j = 1 } ^ n \\gamma _ { i j } ( g ( B X _ j X _ i ^ { - 1 } A ^ { - 1 } ) - g ( B A ^ { - 1 } ) ) . \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} H _ K ( \\beta ) : = \\sup \\{ H \\geq 0 : \\theta ( \\beta , H ) = 0 \\} . \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { 1 - p ^ { - 1 } } { p ^ 2 + w _ p } ( p ^ { - 1 } ( p ^ 2 - w _ p ) + 1 - w _ p ) = \\frac { p - 1 } { p ^ 2 } \\cdot \\frac { p ^ 2 + p ( 1 - w _ p ) - w _ p } { p ^ 2 + w _ p } = \\frac { p - w _ p } { p ^ 2 + w _ p } \\zeta _ p ( 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} \\forall x \\in X _ 1 : ( x ^ { S _ 1 } ) ^ \\theta = ( x ^ { \\theta } ) ^ { S _ 2 } , \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} M _ 4 : = \\sum _ { m , n , p , q \\geqslant 1 } \\Big [ & \\frac { 1 } { 2 N } V ^ { ( 1 ) } _ { m n p q } a ^ * _ m a ^ * _ n a _ p a _ q + \\frac { 1 } { 2 N } V ^ { ( 2 ) } _ { m n p q } b ^ * _ m b ^ * _ n b _ p b _ q \\\\ & + \\frac { 1 } { N } V ^ { ( 1 2 ) } _ { m n p q } a ^ * _ m a _ p b ^ * _ n b _ q \\Big ] . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} \\begin{dcases} \\sum _ { e \\in E ( G _ i ) } w _ i ( e ) = 1 , \\\\ \\sum _ { e \\in E ( G _ i ) : \\ , v \\in e } B _ i ( v , e ) = 1 , \\ \\ v \\in V ( G _ i ) , \\\\ w _ i ( e ) ^ { p - r _ i } \\prod _ { v \\in e } B _ i ( v , e ) = \\alpha _ i , \\ \\ e \\in E ( G _ i ) . \\end{dcases} \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} M _ { i j k l } ^ { ( 2 ) } = { \\bf { \\lambda } } \\delta _ { i j } \\delta _ { k l } + 2 { \\bf { \\mu } } \\delta _ { i \\hat { k } } \\delta _ { j \\hat { l } } + v ( \\delta _ { i \\hat { k } } \\epsilon _ { j \\hat { l } } + \\delta _ { j \\hat { k } } \\epsilon _ { i \\hat { l } } ) + \\delta _ { i j } D _ { k l } ^ { 1 } + \\delta _ { k l } D _ { i j } ^ { 2 } + D _ { i j k l } , \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} 3 & ( 1 - \\gamma _ 1 ) & + 6 2 & ( 1 - \\gamma _ 2 ) & < 3 , \\\\ 4 & ( 1 - \\gamma _ 1 ) & + 3 2 & ( 1 - \\gamma _ 2 ) & < 3 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\tau _ g ( f ) : = \\int _ { G / Z _ G ( g ) } f ( h g h ^ { - 1 } ) \\ , d ( h Z _ G ( g ) ) , \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} f = \\dfrac { 1 } { ( X _ u - x _ i ) ^ 2 + ( Y _ u - y _ i ) ^ 2 + z ^ 2 _ { m i n } } , \\forall i \\in I \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\tilde { R } _ k ( \\alpha ) = \\sum _ { x \\in \\Z ^ d } \\exp ( \\langle \\alpha + \\alpha _ k , x \\rangle ) \\mu _ k ( x ) ~ = ~ R _ k ( \\alpha + \\alpha _ k ) , \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} A \\cap B \\subseteq ( \\overline B \\cup C ) \\cap B = ( \\overline B \\cap B ) \\cup ( C \\cap B ) = C \\cap B \\subseteq C . \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} \\eta _ n ( \\xi ) = \\eta \\Big ( 2 ^ { - \\chi \\sqrt { \\log N _ n } } N _ n ^ A \\xi \\Big ) . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} Y ( A , w ) = R ( \\rho ) Y ( R ( \\rho _ { w } ) ^ { - 1 } A , \\rho ( w ) ) R ( \\rho ) ^ { - 1 } . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} C _ { 2 , 1 } & : = ( t _ { a _ 1 } t _ { b _ 1 } ) ^ 6 t _ { s _ { 1 , 1 } } ^ { - 1 } . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\langle P _ { k _ 1 } ( x ) , Q _ { k _ 2 } ( y ) \\rangle = \\left \\{ \\begin{array} { l l } 1 & \\exists j \\geq 0 , \\ k _ i = l _ i + j \\ ( i = 1 , 2 ) \\\\ 0 & \\end{array} \\right . , \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} \\pi _ { a \\circ x } ( b ) = \\lambda _ a \\left ( \\pi _ x ( \\bar { a } \\circ b ) \\cdot \\pi _ x ( \\bar { a } ) ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} I _ { h , T } \\mathbf { u } ( X ) - \\mathbf { u } ( X ) = \\sum _ { i \\in \\mathcal { I } } \\Phi _ { i , T } ( X ) \\widetilde { \\mathbf { R } } _ i ( X ) , \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} f ( A , B ) = f ^ L ( \\vec 1 _ { A , B } ) = f ^ L ( - \\vec 1 _ { A , B } ) = f ^ L ( \\vec 1 _ { B , A } ) = f ( B , A ) , \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} F ( \\omega , z ) = ( T \\omega , f _ \\omega ( z ) ) , F ^ n ( \\omega , z ) = ( T ^ n \\omega , f ^ n _ \\omega ( z ) ) . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\begin{cases} & - \\mu \\bigl ( \\log \\varepsilon - \\kappa + \\log | u - v | - \\log | u | - \\log | v | \\bigr ) + O ( \\varepsilon ) , \\ ; , \\\\ & - 2 \\mu \\bigl ( \\log \\varepsilon - \\kappa - \\log | u | \\bigr ) + O ( \\varepsilon ) , \\ ; \\end{cases} \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} \\Theta ( x ) = \\lim _ { j \\to \\infty } \\mu ( B _ { r _ j } ( x ) ) = 4 \\pi k _ x , \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} 1 & = \\norm { \\frac { x \\lambda ^ { - 1 } } { \\norm { x \\lambda ^ { - 1 } } } } = \\norm { \\frac { x \\lambda ^ { - 1 } } { \\norm { x } | \\lambda ^ { - 1 } | } } = \\norm { \\frac { x \\lambda ^ { - 1 } } { | A | | \\lambda | | \\lambda ^ { - 1 } | } } = \\norm { \\frac { x \\lambda ^ { - 1 } } { | A | } } \\\\ \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} c ( | D | ) \\left ( \\frac { D } { n } \\right ) \\chi ( n ) ^ { - 1 } n ^ { 1 - k } a _ { \\chi } ( n ) = \\sum _ { 0 < t \\mid n } \\left ( \\frac { D } { t } \\right ) \\chi ( t ) ^ { - 1 } t ^ { 1 - k } c ( t ^ 2 | D | ) . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} \\partial _ t u = - i \\left ( \\int _ \\mathbb { T } | u | ^ 2 u \\right ) \\Pi ( | u | ^ 2 ) - i \\bar J \\left ( \\int _ \\mathbb { T } | u | ^ 2 \\bar { u } \\right ) u ^ 2 - i J \\left ( \\int _ \\mathbb { T } | u | ^ 2 u \\right ) \\Pi ( | u | ^ 2 ) , \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} U _ 1 \\delta _ { ( x _ 1 , . . . , x _ N ) } = \\left ( U _ 1 \\delta _ { x _ 1 } \\right ) \\otimes \\delta _ { x _ 2 } \\otimes \\cdots \\otimes \\delta _ N , \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} ( u _ 1 , \\theta , h _ 1 ) = ( u _ 1 , \\theta , h _ 1 ) ( t , x , y ) : = ( \\hat u _ 1 , \\hat \\theta , \\hat h _ 1 ) \\big ( t , x , \\psi ( t , x , y ) \\big ) . \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { m i n } \\ ; \\ ; & \\sum _ { i = 1 } ^ n \\frac { 1 } { s _ i - s _ { i - 1 } } \\\\ \\mathrm { s . t . } \\ ; \\ ; & s _ i \\leq ( i + 1 ) ^ 2 , \\ ; \\ ; \\ ; s _ { i - 1 } \\leq s _ { i } , \\ ; \\ ; \\ ; i = 1 , \\dots , n \\end{aligned} \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\mu _ A ( f ) = \\frac { 1 } { \\lambda ( A ) } \\int _ A f ( \\iota ( \\gamma ) ) \\ , d \\lambda ( \\gamma ) , \\textrm { f o r $ f \\in C ( b \\Gamma ) $ } . \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} { n \\brack \\delta } ( q ^ \\delta A ' _ 0 + B ' _ 0 ) = q ^ { ( m + 1 ) ( n - \\delta ) } q ^ \\delta \\left ( { n \\brack \\delta } ( A _ 0 + q ^ \\delta C _ 0 ) + A _ \\delta + q ^ \\delta C _ \\delta \\right ) . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} s _ { i } \\left ( h \\right ) & = h - \\langle h , \\alpha _ { i } \\rangle h _ { i } & s _ { i } \\left ( \\mu \\right ) & = \\mu - \\langle h _ { i } , \\mu \\rangle \\alpha _ { i } \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} \\Delta f ( x ) : = f ( x + 1 ) - f ( x ) , \\nabla f ( x ) : = f ( x ) - f ( x - 1 ) . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} P ( \\tilde { V } _ { n } = k ) = \\sum \\limits ^ { n } _ { \\substack { l = 0 } } ( - 1 ) ^ { l } \\binom { k + l } { k } \\tilde { S } _ { k + l , n } . \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} | G | ^ { f ( M ) - e ( M ) } p ^ 1 _ G ( M ) = | G | ^ { f ( M ' ) - e ( M ' ) } p ^ 1 _ G ( M ' ) . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} H _ * ( \\Z [ N _ 4 ] \\otimes _ { \\Z [ N _ 3 ] } \\Z [ X ] ) & = H _ * ( \\Z [ N _ 3 \\oplus _ { N _ 1 } N _ 2 ] \\otimes _ { \\Z [ N _ 3 ] } \\Z [ X ] ) \\\\ & = H _ * ( \\Z [ N _ 3 ] \\otimes _ { \\Z [ N _ 1 ] } \\Z [ N _ 2 ] \\otimes _ { \\Z [ N _ 3 ] } \\Z [ X ] ) \\\\ & = H _ * ( \\Z [ N _ 2 ] \\otimes _ { \\Z [ N _ 1 ] } \\Z [ X ] ) \\overset { ( 2 ) } { = } \\Z [ N _ 2 ] \\otimes _ { \\Z [ N _ 1 ] } \\Z [ P ] \\\\ & = \\Z [ N _ 4 ] \\otimes _ { \\Z [ N _ 3 ] } \\Z [ P ] \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} \\sup _ { \\hat { x } = ( \\hat { z } , \\hat { y } ) \\in B ^ { \\hat { g } _ t } ( \\hat { x } _ t , R ) } \\Biggl ( \\sup _ { \\hat { x } ' \\in B ^ { \\hat { g } _ { t } } ( \\hat { x } _ t , R ) } \\frac { | \\hat \\eta _ t ( \\hat { x } ) - \\mathbf { P } ^ { \\hat { g } _ { \\hat { z } , t } } _ { \\hat { x } ' \\hat { x } } ( \\hat \\eta _ t ( \\hat { x } ' ) ) | _ { \\hat { g } _ { t } ( \\hat { x } ) } } { d ^ { \\hat { g } _ { t } } ( \\hat { x } , \\hat { x } ' ) ^ \\alpha } \\Biggr ) \\leq C . \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} \\langle - b \\cdot w , \\phi \\rangle & = \\langle - \\Delta u , | w | ^ { q - 2 } ( - b \\cdot w ) \\rangle + ( q - 2 ) \\langle | w | ^ { q - 3 } w \\cdot \\nabla | w | , - b \\cdot w \\rangle \\\\ & = : F _ 1 + F _ 2 . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z } \\Omega _ p ( \\alpha _ n ) \\mathrm { v o l } ( \\Gamma _ 0 \\alpha _ n \\Gamma _ 0 ) = p ^ { - 1 } ( 1 - p ^ { - 1 } ) \\left ( 1 + \\sum _ { n > 0 } ( - w _ p p ^ { - 2 } ) ^ n + \\sum _ { n < 0 } ( - w _ p p ^ { - 2 } ) ^ { - n } \\right ) . \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} u ( x , t ) \\equiv u ( r ( x ) , t ) : = C \\zeta ( t ) \\left [ 1 - \\frac r a \\eta ( t ) \\right ] _ + ^ { \\frac 1 { m - 1 } } \\ , . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N p _ n ( x + k ) = \\sum _ { m = 0 } ^ { n \\wedge N } \\binom { N + 1 } { m + 1 } \\Delta ^ m p _ n ( x ) = \\sum _ { k = 0 } ^ { n \\wedge N } c _ { n , N } ( k ) p _ n ( x + k ) , \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} \\mu _ { Z _ { 1 } } ( E _ 1 ) \\times \\mu _ { Z _ { 2 } } ( E _ 2 ) = 0 \\mu _ { Z _ 1 Z _ 2 } ( E _ 1 \\times E _ 2 ) \\neq 0 . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\sigma _ j \\nabla u _ j ) = 0 , \\ ; \\Omega , u _ j = f _ j , \\ ; \\partial \\Omega , \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} \\delta f ( g , h ) = f ( h ) - f ( g h ) + f ( g ) = 0 \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\overset { { \\rm i n \\ , l a w } } { = } \\int _ 0 ^ 1 \\ , s ^ { \\lambda _ 1 } ( 1 - s ) ^ { \\lambda _ 2 } \\ , M _ \\beta ( d s ) , \\ ; \\tau = 1 / \\beta ^ 2 > 1 . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} ( x / y ) ( z ^ { \\alpha } \\backslash x ^ { \\alpha } ) = x ( z ^ { \\alpha } y \\backslash x ^ { \\alpha } ) . \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\eta = \\frac { \\gamma ^ 2 } { \\sqrt { \\mu } } \\frac { 1 } { 4 \\gamma ^ 4 \\beta _ 0 } + \\epsilon _ 0 \\frac { \\gamma ^ 2 } { \\sqrt { \\mu } } \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\mu ( n , d ) = 2 ^ n + \\mu ( 2 ^ { n + 1 } , d - 1 ) . \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} \\left . \\mathbb E \\int _ 0 ^ T \\left \\| y ( s , x _ 0 , u ) \\right \\| _ 2 ^ 2 d s \\right \\vert _ { B = 0 } \\leq x _ 0 ^ T Q x _ 0 \\exp \\left ( \\int _ 0 ^ T \\left \\| u ( s ) \\right \\| _ 2 ^ 2 d s \\right ) . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} S ^ { ( J ) } = I _ { \\mu ^ { ( J ) } , 2 ^ { J } - 1 - \\mu ^ { ( J ) } } \\cup I _ { \\eta ^ { ( J ) } , 2 ^ { J } - 1 - \\eta ^ { ( J ) } } \\mu ^ { ( J ) } < 2 ^ { J - 1 } \\leq \\eta ^ { ( J ) } , \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\varphi ( x ) = \\frac { \\sinh \\beta } { \\pi } x \\exp ( i x \\cosh \\beta ) \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} \\frac { D _ 2 T _ { \\mathrm { m a x } , 2 } } { L _ 2 ^ 2 } = \\frac { D _ 1 T _ { \\mathrm { m a x } , 1 } } { L _ { 1 } ^ { 2 } } . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{gather*} L ( f , n + 1 ) = \\frac { ( 2 \\pi ) ^ { n + 1 } } { n ! } \\int _ 0 ^ \\infty f ( { \\rm i } t ) t ^ n { \\rm d } t , 0 \\le n \\le k - 2 . \\end{gather*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\bar { b } _ 0 = | a | - \\sum _ { j = 0 } ^ { N } b _ j > 0 . \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\Psi _ { \\Gamma } ( X ) = X + O \\left ( X ^ { 2 / 3 + \\theta / 6 } \\log ^ 3 X \\right ) . \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} \\lambda _ i ^ { ( k ) } ( A ) = \\ln \\left ( \\lambda _ i ^ { ( k ) } ( \\gamma ( A ) ) \\right ) , \\ , \\forall 1 \\leq i \\leq k \\leq n . \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} p < \\left ( \\beta + 1 - \\sqrt { ( \\beta + 1 ) ^ 2 - 1 } = \\gamma ( \\beta ) \\right ) ^ { 1 / \\beta } . \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} P _ H \\ 1 ( x ) & = ~ \\lim _ { n \\to \\infty } \\left ( \\sum _ { k = 0 } ^ n P ^ k T _ x P \\ 1 ( e ) - \\sum _ { k = 1 } ^ { n + 1 } P ^ { k } \\ 1 ( e ) \\right ) \\\\ & = ~ T _ x P \\ 1 ( e ) - \\lim _ { n \\to \\infty } \\left ( \\sum _ { k = 1 } ^ n P ^ k ( I d - T _ x P ) \\ 1 ( e ) + P ^ { n + 1 } \\ 1 ( e ) \\right ) . \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} r ^ { 2 } R _ { 2 } ( r ) - \\Big [ ( \\alpha ^ { 2 } + \\gamma ) r ^ { 2 } - \\beta { r } + \\delta \\Big ] R ( r ) = 0 , \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} F = u e _ 0 + u _ 2 e _ 1 - u _ 1 e _ 2 - u _ 0 e _ 3 , \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} A _ 1 = L + S N g R _ 2 S . \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} ( \\rho _ { p , q } \\otimes I d _ r ) \\circ \\rho _ { p + q , r } = ( I d _ p \\otimes \\rho _ { q , r } + ( - 1 ) ^ { r q } \\circ \\tau _ { r , q } \\circ \\rho _ { r , q } ) \\circ \\rho _ { p , q + r } \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} \\phi _ y ( t + h , x ) - \\phi _ y ( t , x ) & = \\int _ 0 ^ { t - \\varepsilon } \\left ( \\phi _ y ( t + h , x , s ) - \\phi _ y ( t , x , s ) \\right ) d s \\\\ & + \\int _ { t - \\varepsilon } ^ { t + h } \\phi _ y ( t + h , x , s ) \\ , d s - \\int _ { t - \\varepsilon } ^ t \\phi _ y ( t , x , s ) \\ , d s \\ , . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } c _ { k } ( j ) \\log j & = \\Lambda ( k ) + \\sum _ { d | k } \\frac { \\mu ( d ) } { d } \\log d ! . \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} \\sum ^ N _ { \\mu = 1 } \\| ( - \\Delta ) ^ { \\frac { 5 - d } { 4 } } | u _ \\mu ( t , x ) | ^ 2 \\| ^ 2 _ { L ^ 2 ( ( T _ 1 , T _ 2 ) ; L _ x ^ 2 ) } \\lesssim \\sup _ { t \\in [ T _ 1 , T _ 2 ] } | \\mathcal { M } ( t ) | . \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\ 1 \\leq \\imath \\leq n - 1 , & & & \\ \\imath = n , & & & \\ n + 1 \\leq \\imath \\leq 2 n - 1 . \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} \\mathrm { d e t } ( p _ j ( \\underline { x } _ i ) ) = \\mathrm { d e t } ( M ) \\mathrm { d e t } ( F _ { \\mathbf { m } , j } ( \\underline { x } _ i ) ) . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} \\begin{aligned} & \\Big | \\sum _ { n = 1 } ^ N \\exp \\Big ( 2 \\pi i \\big ( \\xi P ( n ) - m \\varphi _ 1 ( n ) \\big ) \\Big ) \\phi _ m ( n ) \\Lambda ( n ) \\Big | \\\\ & \\qquad \\qquad \\lesssim ( \\log N ) \\max _ { \\atop { X < X ' \\leq 2 X } { X ' \\leq N } } \\Big | \\sum _ { X < n \\leq X ' } \\exp \\Big ( 2 \\pi i \\big ( \\xi P ( n ) - m \\varphi _ 1 ( n ) \\big ) \\Big ) \\phi _ m ( n ) \\Lambda ( n ) \\Big | . \\end{aligned} \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} C = \\frac { m } { 2 p - 1 + \\frac { ( p - 1 ) | I | } { p - | I | } } , \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} \\exp \\Bigl ( \\bigl ( \\mathcal { S } _ { M - 1 } \\log S _ M \\bigr ) ( 0 \\ , | a , \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } \\log S _ M \\bigr ) ( q \\ , | a , \\ , b ) \\Bigr ) = & \\eta _ { M , M - 1 } ( q \\ , | \\ , a , b ) \\eta _ { M , M - 1 } ( - q \\ , | \\ , a , \\bar { b } ) , \\\\ = & \\eta _ { M , M - 1 } ( q | a , b , \\bar { b } ) , \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} \\Delta _ { K , \\rho } = \\displaystyle \\frac { \\det A _ { \\rho , k } } { \\det \\Phi ( x _ k - 1 ) } , \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} { } _ 0 ^ C \\mathfrak { D } _ x ^ { \\varrho ( x ) } u ( x ) = \\int _ 0 ^ x { \\frac { 1 } { { \\Gamma ( n - \\varrho ( r ) ) } } \\frac { { u ^ { ( n ) } ( r ) d r } } { { ( x - s ) ^ { \\varrho ( r ) - n + 1 } } } } . \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{gather*} L ( G _ k ( q ) , s ) = \\sum _ { m , n \\in \\Z } \\big ( 3 m + 1 - n \\sqrt { - 3 } \\big ) ^ k \\big ( ( 3 m + 1 ) ^ 2 + 3 n ^ 2 \\big ) ^ { - s } . \\end{gather*}"}
-{"id": "8275.png", "formula": "\\begin{align*} d _ 1 = ( a + 1 ) ( b + 2 ) , d _ 2 = ( 2 a + 1 ) ( b + 1 ) , d _ 3 = 2 a b + 2 b + 2 \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} \\aligned b _ N ( \\xi ) & = - 4 \\pi ^ 2 \\sum _ { k , l \\in \\Lambda _ N } \\textbf { 1 } _ { \\Lambda _ N } ( k + l ) \\xi _ k \\xi _ l \\frac { k \\cdot l ^ \\perp } { | l | ^ 2 } \\tilde e _ { k + l } \\\\ & = - 4 \\pi ^ 2 \\sum _ { j \\in \\Lambda _ N } \\bigg [ \\sum _ { l \\in \\Lambda _ N } \\textbf { 1 } _ { \\Lambda _ N } ( j - l ) \\xi _ l \\xi _ { j - l } \\frac { j \\cdot l ^ \\perp } { | l | ^ 2 } \\bigg ] \\tilde e _ j , \\endaligned \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} \\rho _ { p , q } ( G / H ) ( x _ 1 \\ldots x _ { p + q } ) = \\sum _ \\sigma \\mbox { s g n } ( \\sigma ) ( x _ 1 x _ { \\sigma ( 2 ) } \\ldots x _ { \\sigma ( p + q ) } ) , \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} \\Vert M ^ n _ { T ^ { k _ j } \\omega _ 0 } \\Vert = \\Vert M ^ { n + k _ j } _ { \\omega _ 0 } \\cdot M ^ { - k _ j } _ { \\omega _ 0 } \\Vert < C ^ 2 . \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} \\sum _ { k \\in I , r \\leq - 1 } C ( z ) _ { j k } \\widetilde { c } _ { k i } ( r ) z ^ { r } = \\delta _ { j i } . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} 0 = a ^ { \\prime } \\wedge b \\theta 1 \\wedge b = b = 0 \\vee b \\theta a \\vee b = 1 \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} x \\wedge y = ( x , y ) \\cdot ( e _ 1 \\wedge e _ 2 \\wedge e _ 3 \\wedge e _ 4 ) , x , y \\in \\tilde V . \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} G ( \\boldsymbol { u } \\times \\boldsymbol { v } , z ) = G ( \\boldsymbol { u } , z ) G ( \\boldsymbol { v } , z ) . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} K ( v ) = \\hat { K } ( v , c _ 1 ( v ) , c _ 2 ( v ) ) . \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} B _ { i + 1 } ( x , y ) = \\bigcup _ { z \\in B _ 1 ( x , y ) } B _ i ( x , z ) . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\sum a _ k K ( z , z _ k ) = \\sum a _ k \\ < z | z _ k \\ > = \\ < z | \\phi = \\sum b _ k \\ < z | z _ k \\ > = \\sum b _ k K ( z , z _ k ) . \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} \\mathcal I _ 1 ( n ) = \\int _ { \\mathcal A _ 1 ^ + ( n ) } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h + \\int _ { \\mathcal A _ 1 ^ - ( n ) } \\mathbf h _ p ( h ) d h . \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} Q _ c '' - c Q _ c + f ( Q _ c ) = 0 , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; Q _ c \\in H ^ 1 ( \\R ) , \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{gather*} \\partial _ t u _ \\Gamma - \\Delta _ \\Gamma \\mu _ \\Gamma = - \\partial _ { \\boldsymbol { \\nu } } \\mu \\mbox { o n } \\Sigma : = \\Gamma \\times ( 0 , T ) , \\\\ \\mu _ \\Gamma = - \\Delta _ \\Gamma u _ \\Gamma + { \\mathcal W } _ \\Gamma ' ( u _ \\Gamma ) - f _ \\Gamma + \\partial _ { \\boldsymbol { \\nu } } u \\mbox { o n } \\Sigma , \\end{gather*}"}
-{"id": "1350.png", "formula": "\\begin{align*} \\Vert z _ 1 - y _ 1 \\Vert = 1 - \\Vert y _ 1 \\Vert < \\frac { \\varepsilon } { 6 } < \\varepsilon . \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} | \\pi ( w ) - f | = | \\pi ( w ) - \\pi ( f ) | \\leq | w - f | \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} S _ { \\nu } ( N ; y ) & = \\sum _ { d \\in \\Phi ( N , y ) } \\sum _ { m \\in \\Phi ( N / d , y ) } \\tau _ { \\nu - 1 } ( m ) = \\sum _ { d \\in \\Phi ( N , y ) } S _ { \\nu - 1 } ( N / d ; y ) \\\\ & \\le \\sum _ { d \\in \\Phi ( N / y , y ) } S _ { \\nu - 1 } ( N / d ; y ) + S _ { \\nu - 1 } ( N ; y ) . \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} | K ( y , z ) - K ( y , z ' ) | = \\Big | \\ < y | \\Big ( | z \\ > - | z ' \\ > \\Big ) \\Big | \\le n ( y ) d ( z , z ' ) , \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} \\mathfrak { M } ( q \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\int \\limits _ 0 ^ \\infty y ^ q \\ , f _ { ( \\mu , \\lambda _ 1 , \\lambda _ 2 ) } ( \\log y ) \\ , \\frac { d y } { y } = { \\bf E } \\bigl [ M _ { ( \\mu , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G _ 1 * G _ 2 ) & = \\frac { ( r _ 1 + r _ 2 ) ^ { 1 - ( r _ 1 + r _ 2 ) / p } } { ( \\alpha _ 1 \\alpha _ 2 ) ^ { 1 / p } } \\\\ & = \\frac { ( r _ 1 + r _ 2 ) ^ { 1 - ( r _ 1 + r _ 2 ) / p } } { r _ 1 ^ { 1 - r _ 1 / p } r _ 2 ^ { 1 - r _ 2 / p } } \\lambda ^ { ( p ) } ( G _ 1 ) \\lambda ^ { ( p ) } ( G _ 2 ) . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\mu ( g ) = e ^ { t ( g ) \\cdot N } : L ^ p \\to L ^ p \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} \\frac { 1 } { F ( z ) } = \\frac { 1 } { F ( 0 ) } + \\sum _ n \\frac { 1 } { F ' ( t _ n ) } \\bigg ( \\frac { 1 } { z - t _ n } + \\frac { 1 } { t _ n } \\bigg ) , \\sum _ n \\frac { 1 } { | t _ n | ^ 2 | F ' ( t _ n ) | } < \\infty . \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} C _ 3 \\ , & = 1 + a + b , \\\\ C _ 2 ( Y ) \\ , & = 1 + b + ( 1 + a + b ) Y , \\\\ C _ 1 ( Y ) \\ , & = b + ( 1 + a + b ) k + ( 1 + a + b ) Y , \\\\ C _ 0 ( Y ) \\ , & = b + a k + ( a + b + k + a k + b k ) Y . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\ell ' ( A ) = d ( p v ) - a , \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\mathbf { \\Omega } = \\omega \\mathbf { k } \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\Psi ( s ; W _ \\pi , W _ { \\pi ^ \\prime } , \\phi ) = \\int _ { U _ n ( F ) \\backslash U _ n ( F ) H _ n ^ 1 } \\ , d g = 1 . \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} Q : = A , \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} \\phi _ t & = d _ 2 \\Delta \\phi - \\chi _ 2 \\nabla \\cdot ( \\phi \\nabla w ) - \\chi _ 2 \\nabla \\cdot ( v ^ { * * } \\nabla ( w - w ^ { * * } ) ) \\\\ & + \\psi \\Big ( b _ 0 ( t , x ) - b _ 1 ( t , x ) u - b _ 2 ( t , x ) ( v + v ^ { * * } ) \\Big ) - b _ 1 ( t , x ) v ^ { * * } \\psi . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} \\displaystyle \\Delta u + q ( x ) u ^ \\alpha + b ( x ) u = 0 \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} s _ l ( w _ { 2 , 1 } \\cdots w _ { h , 1 } ) = ( w _ { 2 , 1 } \\cdots w _ { h , 1 } ) s _ l . \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\ , \\ , \\displaystyle { u _ { t t } - \\Delta u - \\Delta u _ t = | u | ^ p } & \\displaystyle { t > 0 , x \\in { \\Omega } , } \\\\ { } \\\\ \\displaystyle { u ( 0 , x ) = \\varepsilon u _ 0 ( x ) , \\ ; \\ ; u _ t ( 0 , x ) = \\varepsilon u _ 1 ( x ) \\qquad \\qquad } & \\displaystyle { x \\in { \\Omega } , } \\\\ { } \\\\ \\displaystyle { u = 0 , \\qquad \\qquad } & \\displaystyle { t \\geq 0 , \\ ; x \\in { \\partial \\Omega } , } \\end{array} \\right . \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\nu = \\frac { 1 } { 1 - \\gamma } \\mathcal L ^ { - 1 } _ { + \\infty } [ G ( \\nu ) ] = \\frac { 1 } { \\beta _ 1 ( 1 - \\gamma ) } K \\ast G ( \\nu ) \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} x = \\frac { e ^ { a _ j } } { 1 + e ^ { 2 a _ j } } , y = \\frac 1 { 1 + e ^ { 2 a _ j } } , e ^ { i \\theta } = \\frac { 1 - i e ^ { a _ j } } { ( 1 + e ^ { 2 a _ j } ) ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} N ( m ) = \\frac { N _ { \\mathrm { c } m } } { ( 2 m ) ! ! } , \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\frac { | \\hat \\omega _ \\infty ^ \\bullet ( \\hat { x } _ \\infty ) - \\mathbf { P } ^ { g _ P } _ { \\hat { \\gamma } _ \\infty } ( \\hat \\omega _ \\infty ^ \\bullet ( \\hat { x } ' _ \\infty ) ) | _ { g _ P ( \\hat { x } _ \\infty ) } } { d ^ { g _ P } ( \\hat { x } _ \\infty , \\hat { x } ' ) ^ \\alpha } = 1 , \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} H ^ { \\partial W } = \\frac 1 2 \\textrm { t r } I I ^ { \\partial W } , \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} Y ( B , w ) = e ^ { \\bf a } Y ( e ^ { - { \\bf a } _ { w } } B , w ) e ^ { - { \\bf a } } \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} T _ x P = P T _ x + A _ x . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} \\mathcal { F C } _ P : = \\big \\{ F ( \\omega ) = f ( \\ < \\omega , e _ l \\ > ; l \\in \\Lambda ) \\mbox { f o r s o m e } \\Lambda \\Subset \\Z ^ 2 _ 0 \\mbox { a n d } f \\in C _ P ^ \\infty \\big ( \\R ^ \\Lambda \\big ) \\big \\} , \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} \\begin{pmatrix} a & - b _ 1 & - c _ 2 \\\\ - a _ 2 & b & - c _ 1 \\\\ - a _ 1 & - b _ 2 & c \\end{pmatrix} . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} & \\dim ( S _ 1 ) = \\dim ( S _ 2 ) = \\ldots = \\dim ( S _ { k - 1 } ) = \\dim ( S _ { k ' } ) = 1 \\\\ & \\dim ( S _ k ) = \\dim ( S _ { k + 1 } ) = \\ldots = \\dim ( S _ n ) = 0 \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} \\Gamma ( 1 - \\frac { 1 } { \\tau } ) \\sin \\pi / \\tau = \\frac { \\pi } { \\Gamma ( 1 / \\tau ) } . \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\Omega _ { k + 1 } = \\Omega _ k + 2 ( - 1 ) ^ { k + 1 } { n \\choose k } . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} { { { \\sum \\limits _ { s \\in S } { { \\lambda _ { s } } } } } } \\left ( { { { { { { a _ { s } ^ { \\ast } , } b _ { s } } } } } } \\right ) { = - } \\left ( c ^ { \\ast } , \\inf \\mathrm { ( R P ) } _ { 0 _ { X } ^ { \\ast } } \\right ) . \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} & Y ^ { n , m } _ t ( x , v ) : = f ^ n _ t ( x , v ) \\sqrt { 1 + v ^ { 2 m } } \\ , , \\\\ & Z ^ { n , m } _ t ( x , v ) : = ( D f ^ n _ t ( x , v ) ) ^ \\top \\sqrt { 1 + v ^ { 2 m } } \\ , , \\\\ & H ^ { n , m } _ t ( x , v ) : = D ^ 2 f ^ n _ t ( x , v ) \\sqrt { 1 + v ^ { 2 m } } \\ , . \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} \\det \\gamma = e ^ { - d ( M + 1 ) t } \\prod _ { i = 1 } ^ d ( e ^ { ( M + 1 ) t } - \\lambda _ i ) = \\prod _ { i = 1 } ^ d ( 1 - e ^ { - ( M + 1 ) t } \\lambda _ i ) . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} & \\abs { S _ x ( u - P _ { \\tilde { x } } ( \\cdot - \\tilde { x } ) ) ( z _ 1 ) - S _ x ( u - P _ { \\tilde { x } } ( \\cdot - \\tilde { x } ) ) ( z _ 2 ) } \\\\ = & \\abs { u ( \\Psi _ x ^ { - 1 } ( z _ 1 ) ) - u ( \\Psi _ x ^ { - 1 } ( z _ 2 ) ) } \\\\ \\leq & \\norm { u } _ { C ^ \\alpha ( \\hat { B } _ 2 ) } \\left ( \\rho ( x ) d ( z _ 1 , z _ 2 ) \\right ) ^ \\alpha . \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} \\begin{pmatrix} & & & & J \\\\ & & & J \\\\ & & J ' \\\\ & J \\\\ J \\end{pmatrix} \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} 4 \\pi a _ \\alpha = \\inf \\Big \\{ \\int _ { \\R ^ 3 } \\Big [ | \\nabla f ( x ) | ^ 2 + \\frac { 1 } { 2 } V ^ \\alpha ( x ) | f ( x ) | ^ 2 \\Big ] \\d x : \\ , \\ , \\ , f : \\R ^ 3 \\to \\R , \\lim _ { | x | \\to \\infty } f ( x ) = 1 \\Big \\} . \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { S _ { k , n } } { \\sum \\limits _ { \\substack { I _ { k } ( n ) ^ { c } } } b ( i _ { 1 } ; n ) b ( i _ { 2 } ; n ) \\cdots b ( i _ { k } ; n ) } = 1 , \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} F ( z ) & = e ^ { \\bf f } F ( z ) e ^ { - { \\bf f } } , \\\\ e ^ { - { \\bf f } } F \\otimes x ( \\zeta ) e ^ { \\bf f } & = F \\otimes x ( \\zeta ) . \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} - 2 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } \\Delta _ x \\psi ( x , y ) \\Delta _ x m _ { u _ \\mu } ( t , x ) \\Delta _ y m _ { u _ \\iota } ( t , y ) \\ , d x d y \\\\ - 4 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } \\Delta _ x \\psi ( x , y ) \\nabla _ x m _ { \\nabla u _ \\mu } ( t , x ) \\nabla _ y m _ { u _ \\iota } ( t , y ) \\ , d x d y , \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} 1 = \\sum _ { s = 0 } ^ { n - 1 } \\Xi _ { n , s } ^ { \\beta } + \\Big ( 1 - \\sum _ { s = 0 } ^ { n - 1 } \\Xi _ { n , s } ^ { \\beta } \\Big ) , \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} \\alpha _ r = \\left [ \\begin{smallmatrix} I _ n & 0 & 0 & 0 \\\\ 0 & I _ n & 0 & 0 \\\\ 0 & \\widetilde { I _ r } & I _ n & 0 \\\\ \\widetilde { I _ r } & 0 & 0 & I _ n \\end{smallmatrix} \\right ] , \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} & \\bigl \\| \\sqrt { n } \\bigl ( \\hat { M } ^ { - 1 } \\hat { \\varSigma } _ { \\beta } \\hat { M } ^ { - 1 } \\bigr ) ^ { - 1 / 2 } \\bigl ( \\hat { \\beta } ^ { ( 2 ) } _ n - \\beta _ 0 \\bigr ) \\bigr \\| ^ 2 \\\\ & \\quad = n \\bigl ( \\hat { \\beta } ^ { ( 2 ) } _ n - \\beta _ 0 \\bigr ) ^ \\top \\bigl ( \\hat { M } ^ { - 1 } \\hat { \\varSigma } _ { \\beta } \\hat { M } ^ { - 1 } \\bigr ) ^ { - 1 } \\bigl ( \\hat { \\beta } ^ { ( 2 ) } _ n - \\beta _ 0 \\bigr ) \\xrightarrow { } \\chi ^ 2 _ m . \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} - \\sqrt { 5 } = 2 e ^ { 4 \\pi { \\bf i } / 1 5 } + e ^ { 6 \\pi { \\bf i } / 1 5 } + 2 e ^ { 1 4 \\pi { \\bf i } / 1 5 } + 2 e ^ { 1 6 \\pi { \\bf i } / 1 5 } + e ^ { 2 4 \\pi { \\bf i } / 1 5 } + 2 e ^ { 2 6 \\pi { \\bf i } / 1 5 } \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} ( v _ 1 , \\vartheta , \\partial _ \\eta w ) | _ { \\eta = 0 } = \\mathbf { 0 } , \\lim _ { \\eta \\rightarrow + \\infty } ( v _ 1 , \\vartheta , w ) ( \\tau , \\xi , \\eta ) = \\mathbf { 0 } . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} \\mathcal { T } & : \\begin{pmatrix} x _ 1 , x _ 2 \\end{pmatrix} \\mapsto \\begin{pmatrix} - x _ 1 , x _ 2 \\end{pmatrix} , & \\mathcal { P } & : \\begin{pmatrix} x _ 1 , x _ 2 \\end{pmatrix} \\mapsto \\begin{pmatrix} x _ 1 , - x _ 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} \\pi _ t : = ( \\Delta _ t ) _ * ( t \\pi _ { K ^ * } ) \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} \\begin{pmatrix} A _ 5 & A _ 6 & A _ 7 & B _ 5 & B _ 6 & B _ 7 \\end{pmatrix} , \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} \\eqref { e q - t o } \\times ( v / m j ) & < ( m j u ^ 4 v - m ^ 2 j u ^ m v ^ 2 + m j u ^ 2 v ^ m - m u ^ 2 v + 2 u v ^ 2 ) a ^ 2 \\\\ & + ( m j u ^ 2 v - m ^ 2 j u v ^ 2 + m j v ^ m - 2 u + m v ) u ^ 2 a ^ 2 \\\\ & - 2 ( m j u ^ m v - m ^ 2 j u ^ 2 v ^ 2 + m j u v ^ m - u ^ 2 + v ^ 2 ) u a ^ 2 = 0 . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\sigma ^ c _ t ( f ) ( \\gamma , y ) = N ( \\gamma ) ^ { i t } f ( \\gamma , y ) , f \\in C _ c ( \\Gamma \\boxtimes Y ) \\ \\ ( \\gamma , y ) \\in \\Gamma \\boxtimes Y . \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} K ^ { \\ast } = 4 \\frac { U _ { \\mathbf { \\nu } } ^ { \\ast } } { \\rho ^ { 3 } v ^ { 2 } } - \\frac { 2 \\omega } { \\rho ^ { 2 } v } \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{gather*} f _ { 2 7 } \\ ( \\tau / 3 \\ ) = \\eta ( \\tau ) ^ 2 \\eta ( 3 \\tau ) ^ 2 = \\frac 1 3 b ( \\tau ) c ( \\tau ) = \\frac 1 3 s ( \\tau ) t ( \\tau ) a ( \\tau ) ^ 2 . \\end{gather*}"}
-{"id": "8545.png", "formula": "\\begin{align*} V ( X ) = \\left \\{ \\begin{array} { c c } ( X - \\overline { X } ) ^ T \\otimes I _ 2 & \\ ; X \\in \\overline { T } ^ + , \\\\ \\left ( ( X - \\overline { X } ) ^ T \\otimes I _ 2 \\right ) \\overline { M } ^ + & \\ ; X \\in \\overline { T } ^ - . \\end{array} \\right . \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} H _ 1 & = ( ( 1 + i ) ^ \\alpha ( 1 + j ) ^ \\beta a _ { i + j } ) _ { i , j \\in \\N } , \\\\ H _ 2 & = ( ( 1 + i + j ) ^ { \\alpha + \\beta } a _ { i + j } ) _ { i , j \\in \\N } . \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { \\sigma } } e ^ u = \\sum _ { e = \\{ x , y \\} \\in E , u ( x ) < \\sigma \\leq u ( y ) } w _ { x y } ( u ( y ) - u ( x ) ) . \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd R } | B _ R ^ n | \\stackrel { ? } { = } \\frac { R ^ { n - 1 } } { ( n - 1 ) ! } \\left [ h ^ { ( p + 1 ) } ( R ) \\right ] ^ 2 . \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} I ( x ) : = \\left \\{ \\begin{array} { l l } \\left \\{ u \\in U : F _ { u } ( x , 0 _ { u } ) = p ( x ) \\right \\} , & \\mathrm { i f } p ( x ) \\in \\mathbb { R } , \\\\ \\emptyset , & \\mathrm { i f } p ( x ) \\not \\in \\mathbb { R } . \\end{array} \\right . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} | \\tilde { \\psi } ( x ) - \\psi ( x _ 0 ) - \\nabla \\psi ( x _ 0 ) \\cdot ( x - x _ 0 ) | = | \\tilde { \\psi } ( x ) - \\tilde { \\psi } ( x _ 0 ) - \\nabla \\tilde { \\psi } \\cdot ( x - x _ 0 ) | \\le \\tilde { C } | x - x _ 0 | ^ { 2 } \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} u _ q : = \\mathcal { F } ^ { - 1 } ( \\varphi _ q ) * u q > - 1 u _ { - 1 } : = \\mathcal { F } ^ { - 1 } ( \\chi ) * u , \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} b \\varphi ( a ^ 0 , a ^ 1 , \\ldots , a ^ { q + 1 } ) & : = \\sum _ { j = 0 } ^ q ( - 1 ) ^ j ( a ^ 0 , \\ldots , a ^ j a ^ { j + 1 } , \\ldots , a ^ { q + 1 } ) \\\\ & + ( - 1 ) ^ { q + 1 } ( a ^ { q + 1 } a ^ 0 , a ^ 1 , \\ldots , a ^ { q } ) , \\\\ B \\varphi ( a _ 0 , \\ldots , a _ { q - 1 } ) & : = \\sum _ { j = 0 } ^ { n - 1 } ( - 1 ) ^ { ( n - 1 ) j } ( 1 , a _ j , \\ldots , a _ { q - 1 } , a _ 0 , \\ldots , a _ { j - 1 } ) . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q \\ , | \\ , b ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty ( e ^ { - q t } - 1 ) e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) f ( t ) \\ , d t / t ^ { M + 1 } \\Bigr ) . \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\zeta _ k = \\ ; \\theta ( n ) ^ c \\zeta _ { k + 1 } = \\ ; \\theta ( n ) . \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} W _ { f , \\varphi } ^ * K _ p ( q ) = \\overline { W _ { f , \\varphi } K _ q ( p ) } . \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} \\langle a , b \\rangle & = ( \\vert b \\vert + 1 ) b a - ( \\vert a \\vert + 1 ) a b , \\\\ \\langle a _ { i } ; b , c \\rangle & = i ( i + 1 ) a _ i [ c , b ] \\\\ \\langle a _ i , a _ j ; b , c \\rangle & = i ( i + 1 ) ( i + 2 j + 1 ) a _ i a _ j [ c , b ] - i ( i + 1 ) ( j + 1 ) a _ j a _ i [ c , b ] . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} ( a ) _ n : = ( a ) ( a + 1 ) \\cdots ( a + n - 1 ) , n \\in \\mathbb N _ 0 , \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} F S ( e ) = \\frac { p ^ { 3 e } - 1 } { 3 } + 1 \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { | \\sigma _ k | ^ { \\alpha } } { \\lambda _ k } < \\infty . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} E _ f & = \\dfrac { X } { \\hat { H } ^ 2 } E , & F _ f & = \\dfrac { X } { \\hat { H } ^ 2 } F , & G _ f & = \\dfrac { X } { \\hat { H } ^ 2 } G , \\\\ L _ f & = - \\dfrac { 1 } { \\hat { H } } \\left ( \\lambda P + E D \\right ) , & M _ f & = - \\dfrac { 1 } { \\hat { H } } \\left ( \\lambda Q + F D \\right ) , & N _ f & = - \\dfrac { 1 } { \\hat { H } } \\left ( \\lambda R + G D \\right ) . \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} M ( \\lambda ) N _ 1 ( \\lambda ) ^ T h ( \\lambda ) = K _ 2 ( \\lambda ) ^ T X ( \\lambda ) h ( \\lambda ) . \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\lambda _ 1 = u _ 1 - \\sqrt { \\frac { 2 R \\theta q } { P + ( 1 - 2 a ) q } } , \\lambda _ 2 = u _ 1 , \\quad \\lambda _ 3 = u _ 1 + \\sqrt { \\frac { 2 R \\theta q } { P + ( 1 - 2 a ) q } } , \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\begin{array} { l l } A ( t ) u _ s ( x ) = ( I + \\kappa U ( s , t ) ) ~ \\partial _ { t } { \\rm L o g } ~ ( U ( t , s ) + \\kappa I ) u _ s ( x ) \\end{array} \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} \\theta ( S ) = 2 \\theta _ 0 ( S ' ) \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} \\theta ^ { \\textrm { o p t } } = \\arg \\max _ { \\theta \\in ( 0 , 1 ) } U _ { L } ( \\theta , \\lambda ^ { \\textrm { o p t } } ) . \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} & f ( A , B ) + f ( C , D ) = p ( X _ I , X _ O ) + p ( Y _ I , Y _ O ) \\\\ \\ge \\ ; & p ( X _ I \\cap Y _ I , X _ O \\cap Y _ O \\setminus Z ) + p ( ( X _ I \\cup Y _ I ) \\setminus Z , ( X _ O \\cup Y _ O ) \\setminus Z ) \\\\ = \\ ; & p ( A \\cap C , ( A \\cap C ) \\cup ( B \\cap D ) ) \\\\ + & p ( ( A \\cup C ) \\setminus ( B \\cup D ) , ( ( A \\cup C ) \\setminus ( B \\cup D ) ) \\cup ( ( B \\cup D ) \\setminus ( A \\cup C ) ) ) \\\\ = \\ ; & f ( A \\cap C , B \\cap D ) + f ( ( A \\cup C ) \\setminus ( B \\cup D ) , ( B \\cup D ) \\setminus ( A \\cup C ) ) , \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{gather*} g _ { i } ( x _ { i } ^ { * } ) \\leq 0 , \\ , \\ , \\ , \\ , \\ , \\ , \\lambda _ { i } ^ { * } g _ { i } ( x _ { i } ^ { * } ) = 0 , \\ , \\ , \\ , \\ , \\ , i = 1 , \\ldots , N , \\\\ \\sum _ { i } ^ { N } \\frac { \\partial f _ { i } ( x _ { i } ^ { * } ) } { \\partial x _ { i } } + \\lambda _ { i } ^ { * } \\frac { \\partial g _ { i } ( x _ { i } ^ { * } ) } { \\partial x _ { i } } = 0 , \\ , \\ , \\ , \\ , i = 1 , \\ldots , N . \\end{gather*}"}
-{"id": "3572.png", "formula": "\\begin{gather*} t \\ ( \\tau _ 0 \\ ) = 3 \\big ( 2 ^ { 1 / 3 } - 1 \\big ) , \\end{gather*}"}
-{"id": "329.png", "formula": "\\begin{align*} \\varphi _ 0 = \\frac { - \\cosh \\beta } { 2 \\pi ^ 2 \\sinh ^ 2 \\beta } , \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} e ( G _ { A } ) = f \\cup e ( G / N ' ) = - d _ { 2 } f = [ - \\delta \\tau ] . \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} Z = \\lbrace x _ 1 , \\ldots , x _ k \\rbrace . \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} \\begin{array} { l l } N _ 3 = & \\int _ { \\mathbb R } ( \\zeta _ k + \\varphi _ k ) [ ( 2 c _ 1 a _ k - \\omega b _ k ) \\xi _ { n _ k } + ( c _ 2 b _ k - \\omega a _ k ) \\nu _ { n _ k } ] \\\\ \\\\ & + ( ( \\zeta _ k \\xi _ { n _ k } ) ' + ( \\varphi _ k \\xi _ { n _ k } ) ' ) ( 2 c _ 3 a ' _ k - \\omega c _ 4 b ' _ k ) + ( ( \\zeta _ k \\nu _ { n _ k } ) ' + ( \\varphi _ k \\nu _ { n _ k } ) ' ) ( 2 c _ 5 b ' _ k - \\omega c _ 4 a ' _ k ) \\end{array} \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} \\min \\ , f ( x ) = \\sum _ { i = 1 } ^ { N } f _ { i } ( x ) \\\\ \\ , g _ { i } ( x ) \\leq 0 , \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} \\tau = \\frac { 1 } { \\beta ^ 2 } > 1 . \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} Q _ 1 ^ { \\epsilon _ 1 } Q _ 2 ^ { \\epsilon _ 2 } \\cdots Q _ k ^ { \\epsilon _ k } = V _ k \\cdots V _ 2 V _ 1 t _ { b _ 1 } ^ { - \\epsilon _ 1 } t _ { b _ 2 } ^ { - \\epsilon _ 2 } \\cdots t _ { b _ k } ^ { - \\epsilon _ k } = t _ { f ( a ) } ^ { - 1 } f t _ a f ^ { - 1 } . \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} h \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } \\varpi _ p = \\left ( \\begin{array} { c c } t ^ { - 1 } p ^ { - 1 } \\det ( h ) & - y \\\\ 0 & - t \\end{array} \\right ) \\left ( \\begin{array} { c c } \\delta & 1 \\\\ 1 - \\delta \\eta ( h ) & - \\eta ( h ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\omega ( g _ 1 , g _ 2 ) = \\sigma ( g _ 1 ) \\sigma ( g _ 2 ) \\sigma ( g _ 1 g _ 2 ) ^ { - 1 } \\ ; \\ ; \\mbox { f o r a l l $ g _ 1 , g _ 2 \\in G $ } \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\Omega _ p ( 1 ) \\mathrm { v o l } ( \\Gamma _ { 0 0 } ) + \\Omega _ p ( r _ 1 ) \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ 1 \\Gamma _ { 0 0 } ) + \\Omega _ p ( r _ u ) \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ u \\Gamma _ { 0 0 } ) , \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\overline { A } \\setminus \\overline { W } _ 1 = \\big ( W _ 1 \\cup W _ 2 \\cup \\overline { B } _ 1 \\cup \\overline { B } _ 2 \\big ) \\setminus ( W _ 1 \\cup \\overline { B } _ 1 \\big ) = W _ 2 \\cup \\big ( \\overline { B } _ 2 \\setminus \\overline { B } _ 1 \\big ) . \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} \\sum _ { i : t _ { i , n } \\leq T } | \\mathcal { I } _ { i , n } | f ( | \\mathcal { I } _ { i , n } | ^ { - 1 / 2 } \\Delta _ { i , n } X ^ { t o y } ) = \\sum _ { i : t _ { i , n } \\leq T } | \\mathcal { I } _ { i , n } | f ( \\sigma Z _ i ^ n ) \\overset { \\mathbb { P } } { \\longrightarrow } T \\mathbb { E } [ f ( \\sigma Z ) ] \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} [ x _ t ] & = [ x _ 0 : x _ 1 : \\ldots : x _ n : t : 0 : \\ldots ] \\\\ [ y _ t ] & = [ t x _ 0 : t x _ 1 : \\ldots : t x _ n : 1 : 0 : \\ldots ] \\\\ [ z _ t ] & = [ t : 0 : \\ldots : 0 : 1 : 0 : \\ldots ] \\\\ [ w _ t ] & = [ 1 : 0 : \\ldots : 0 : t : 0 : \\ldots ] \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} T ^ i _ a = \\{ t \\in [ n ] | \\ , t + n ( i - 1 ) \\in T _ a \\} , T _ 0 = ( T _ 1 \\cap T _ 2 ) ^ c , a = 0 , 1 , 2 . \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\mathcal { A } : = v \\partial _ x - v \\partial _ v + \\sigma \\partial _ { v v } \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} g _ { a , r } : = - G ( g _ { a , 1 } + \\ldots + g _ { a , r - 1 } ) [ r ] , ~ ~ ~ r \\geq 2 . \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} W _ { k , r } ^ 2 & \\ll N x _ k ^ { - 1 } \\ ( y _ k N x _ k ^ { - 1 } + y _ k ^ 2 p ^ { 1 / 2 } \\log p \\ ) \\\\ & \\ll N ^ 2 x _ k ^ { - 1 } + N p ^ { 1 / 2 } x _ k \\log p . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\mathrm { v o l } ( \\Gamma _ 0 \\beta _ m \\Gamma _ 0 ) = \\begin{cases} p ^ { 2 m - 2 } ( 1 - p ^ { - 1 } ) & m > 0 , \\\\ p ^ { - 2 m } ( 1 - p ^ { - 1 } ) & m < 0 . \\end{cases} \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} g \\left ( { b } \\right ) - g \\left ( a \\right ) = D _ q g \\left ( \\eta \\right ) \\left ( { b - a } \\right ) \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\lambda ^ { ( 1 ) } ( G ) = 2 ^ { - 1 } \\cdot \\max \\Big \\{ \\frac { 4 } { 3 } , 1 \\Big \\} = \\frac { 2 } { 3 } . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} Q _ 1 = ( ( 1 _ { \\underline { m } } ) ^ * \\times \\underline { m } ^ * ) \\circ E ' \\qquad Q _ 2 = ( \\underline { m } ^ * \\times ( 1 _ { \\underline { m } } ) ^ * ) \\circ E ' \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} ( \\frac { \\phi ^ { n + 1 } _ { h } - \\phi ^ { n } _ { h } } { \\Delta t } , v _ { h } ) = ( \\frac { \\eta ^ { n + 1 } - \\eta ^ { n } } { \\Delta t } , v _ { h } ) + \\nu ( \\nabla e ^ { n + 1 } _ { u } , \\nabla v _ { h } ) + b ( u ^ { n + 1 } - u ^ { n } , u ^ { n + 1 } , v _ { h } ) + b ( e ^ { n } _ { u } , u ^ { n + 1 } , v _ { h } ) \\\\ + b ( u ^ { n } _ { h } , e ^ { n + 1 } _ { u } , v _ { h } ) - ( p ^ { n + 1 } - q ^ { n + 1 } _ { h } , \\nabla \\cdot v _ { h } ) - ( \\frac { u ^ { n + 1 } - u ^ { n } } { \\Delta t } - u ^ { n + 1 } _ { t } , v _ { h } ) \\ ; \\ ; \\forall v _ { h } \\in V _ { h } . \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} { \\Theta } \\sum _ { d \\leq x } \\frac { \\mu ( d ) } { d ^ 2 } \\sum _ { l \\leq x / d } 1 & = \\frac { \\Theta } { \\zeta ( 3 ) } x + O ( 1 ) . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 1 } ^ { t } ( N - k ) \\leq b _ { G } ( N , t ) \\leq \\prod \\limits _ { k = 0 } ^ { 4 t } ( N - k ) . \\\\ \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} u ^ { \\flat } _ { s t } = \\delta \\mu _ { s t } + A ^ 2 _ { s t } u _ s + u _ { s t } ^ { \\natural } , \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} A y ^ { k + 1 } - u ^ { k + 1 } & = 0 , \\\\ A ^ * p ^ { k + 1 } + y ^ { k + 1 } - z & = 0 , \\\\ - p ^ { k + 1 } + \\gamma u ^ { k + 1 } + \\alpha \\lambda ^ { k + 1 } & = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} \\lim _ { \\lvert n \\rvert \\to \\infty } \\lVert x _ n - y _ n \\rVert = 0 . \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\mathcal { F } } ( \\overline { \\pi _ { 0 } ( \\Omega ) } , . . . , \\overline { \\pi _ { 0 } ( \\Omega ) } ) = \\overline { \\pi _ { 0 } ( \\Omega ) } \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } u _ { i 1 } ( x ) \\mu _ i ( x ) = \\sum _ { i = 1 } ^ { d } \\mu _ i ( x ) \\sum _ { k \\neq i } \\frac { 1 } { x _ i - x _ k } = \\sum _ { k , l ; k > l } \\frac { \\mu _ k ( x ) - \\mu _ k ( x ) } { x _ k - x _ l } , \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} F _ \\beta ( z , z ' ) : = e ^ { \\beta F ( z , z ' ) } \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| w ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - w ^ { * * } ( \\cdot , t ) \\| _ { C ^ 0 ( \\bar \\Omega ) } = 0 . \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\Delta _ \\tau : = \\{ X \\in T M : \\phi _ \\tau ( X , \\ , \\cdot \\ , ) = 0 \\} , \\ \\ \\ \\ \\nu _ \\tau : = \\dim \\Delta _ \\tau , \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} \\left [ \\prod _ { i = 1 } ^ N \\frac { d _ i - 2 } { d _ i } \\right ] \\| T \\| _ { S _ 1 } + | c _ + | + | c _ - | \\leq \\| \\phi \\| _ { c b } \\leq \\| T \\| _ { S _ 1 } + | c _ + | + | c _ - | , \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} \\alpha = \\overline { \\omega } \\circ \\gamma = ( \\omega _ 0 , \\ , \\omega _ { - d } ) \\circ \\begin{pmatrix} \\gamma _ 0 \\\\ 0 \\end{pmatrix} = \\omega _ 0 \\circ \\gamma _ 0 . \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle { \\inf } \\ ; \\mathbb { E } _ { \\xi _ 2 , \\ldots , \\xi _ { T _ { \\max } , D _ 2 , \\ldots , D _ { T _ { \\max } } } } \\Big [ \\displaystyle { \\sum _ { t = 1 } ^ { T _ { \\max } } } \\ ; D _ { t - 1 } f _ { t } ( x _ { t } , x _ { t - 1 } , \\xi _ t ) \\Big ] \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) \\ ; \\mbox { a . s . } , \\ ; x _ { t } \\ ; { \\overline { \\mathcal { F } } } _ t \\mbox { - m e a s u r a b l e , } t = 1 , \\ldots , T _ { \\max } , \\end{array} \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} \\mu ( L ) = \\frac { \\sum _ { J } | P ^ { J } ( L ) | ^ 2 \\delta _ { J } } { \\sum _ { J } | P ^ { J } ( L ) | ^ 2 } , \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} \\| \\rho \\| ^ 2 _ { L ^ 2 ( K ) } = \\sum _ { \\sigma \\in \\widehat { K } } d _ \\sigma \\| \\sigma ( \\mu ) \\| _ { \\textrm { H S } } ^ 2 , \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} F _ { 8 , 1 8 } ^ { ( 2 ) } = - \\frac { 6 0 7 5 } { 1 6 } \\nu ( \\xi _ 6 ^ { ( 2 ) } ) \\chi _ 5 ^ 3 , F _ { 8 , 2 0 } = \\frac { 1 5 1 8 7 5 } { 3 2 } \\nu ( \\xi _ 7 ) \\chi _ 5 ^ 3 , F _ { 8 , 2 2 } = - \\frac { 1 3 6 6 8 7 5 } { 1 6 } \\nu ( \\xi _ 8 ) \\chi _ 5 ^ 3 \\ , . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} & \\max \\ , - ( \\tfrac { n } 6 + \\tfrac 1 4 + \\tfrac 1 4 m _ * ) ( 1 - \\lambda _ * ) ^ 2 m _ * , \\\\ & \\ , \\ , \\textrm { s . t . } \\ , \\ , \\ , \\ , \\lambda _ * \\in [ 0 , 1 ] , \\ , \\ , m _ * \\in \\{ 0 , \\ldots , p \\} , \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} t = 2 \\left ( \\left \\lfloor \\frac { m + 1 } { 2 } \\right \\rfloor - \\frac { d - 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} L + \\frac { B } { \\rho _ { + } ^ { \\alpha } } = \\rho _ { - } ( u _ - - \\widehat { \\sigma _ { 0 } ^ { B } } ) ^ { 2 } , \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{gather*} e ^ { \\beta ^ 2 \\log \\varepsilon } \\int _ \\phi ^ \\psi e ^ { \\beta V _ { \\varepsilon } ( \\theta ) } \\ , d \\theta \\longrightarrow M _ { \\beta } ( \\phi , \\psi ) , \\\\ { \\bf { E } } [ M _ { \\beta } ( \\phi , \\psi ) ] = | \\psi - \\phi | . \\end{gather*}"}
-{"id": "4519.png", "formula": "\\begin{align*} f ( T _ { i _ 1 i _ 2 \\ldots i _ m } ) = f ( Q _ { i _ 1 j _ 1 } Q _ { i _ 2 j _ 2 } \\ldots Q _ { i _ m j _ m } T _ { j _ 1 j _ 2 \\ldots j _ m } ) . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } [ P ( t ) ] = \\lim _ { t \\rightarrow \\infty } \\left [ \\begin{pmatrix} x \\\\ y \\\\ 1 / t \\end{pmatrix} \\right ] = \\left [ \\begin{pmatrix} x \\\\ y \\\\ 0 \\end{pmatrix} \\right ] . \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} \\beta _ { 2 , 2 } ( \\delta ) \\triangleq \\beta _ { 2 , 2 } \\bigl ( a = ( 1 , 1 ) , \\ , b _ 0 = \\delta , \\ , b _ 1 = b _ 2 = 1 / 2 \\bigr ) . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} ( q ; q ) _ { \\infty } & = 1 + \\sum _ { k = 1 } ^ { \\infty } ( - 1 ) ^ { k } q ^ { ( 3 k ^ { 2 } - k ) / 2 } ( 1 + q ^ { k } ) \\\\ & = \\sum _ { k = - \\infty } ^ { \\infty } ( - 1 ) ^ { k } q ^ { ( 3 k ^ { 2 } - k ) / 2 } . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} \\gamma \\left ( a , b \\right ) = \\left ( a \\vee b \\right ) \\wedge \\left ( a \\vee b { ^ { \\sim } } \\right ) \\wedge \\left ( a { ^ { \\sim } } \\vee b \\right ) \\wedge \\left ( a { ^ { \\sim } } \\vee b { ^ { \\sim } } \\right ) \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\lambda _ i \\cdot v _ i \\simeq 0 . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\partial _ t \\tilde g + v \\partial _ x \\tilde g + G ( M ^ { n - 1 } ( t ) ) \\partial _ v \\tilde g - \\partial _ v ( v \\tilde g ) + \\tilde g - \\sigma \\partial _ { v v } \\tilde g = R ^ n \\ , , \\tilde g _ 0 = 0 \\ , , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { c , d } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\overline { E } _ { a , b } \\overline { E } _ { c , d } } E _ { c , d } E _ { a , b } , & M = N = 1 ; \\\\ E _ { c , d } ^ { ( N ) } E _ { a , b } ^ { ( M ) } , & . \\end{cases} \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} D _ { n + 2 } ^ m D _ n ^ { m + 2 } = D _ { n + 1 } ^ { m + 2 } D _ { n + 1 } ^ m - ( D _ { n + 1 } ^ { m + 1 } ) ^ 2 , \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\min \\limits _ { x \\in K _ i } | \\nabla p ( x ) | > 0 \\qquad i = 1 , \\dots , d - 1 , \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } \\left . \\frac { \\partial V } { \\partial u } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) , W ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , W _ x ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , u ) \\right \\vert _ { u = \\bar { u } ( s ) } \\\\ = \\left ( 1 - W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) \\sigma _ { z } \\left ( s \\right ) \\right ) ^ { - 1 } W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) \\sigma _ { u } ( s ) . \\end{array} \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} G ( z \\ , | \\ , \\tau ) = ( 2 \\pi ) ^ { z / 2 } \\tau ^ { - \\bigl ( 1 + { } _ 2 S _ 0 ( z , | \\ , 1 , \\tau ) \\bigr ) } \\ , \\Gamma _ 2 ^ { - 1 } ( z \\ , | \\ , 1 , \\tau ) , \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} w _ \\infty : = \\lim _ { i \\to \\infty } \\dot \\gamma _ i ( 0 ) \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} A ( m _ 1 - 1 ) < r = A ( m _ 1 ) = A ( m _ 1 + 1 ) = \\cdots = A ( m _ 0 - 1 ) < A ( m _ 0 ) , \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} f _ { m , i } ^ + ( x ) = 1 - ( m d ( x , \\overline { B } _ i ) \\wedge 1 ) \\ ; , \\ ; f _ { m , i } ^ - ( x ) = m d ( x , ( B _ i ^ \\circ ) ^ c ) \\wedge 1 \\ , , \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} & \\tilde \\alpha ^ \\top h ( x ) = u ^ \\top M ^ \\top h ( x ) = 0 , \\\\ & \\tilde \\beta ^ \\top g ( x ) \\leq 0 , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} X ' _ { \\Pi _ { \\imath } } : = \\begin{cases} v ^ { \\frac { 1 } { 2 } \\lambda _ { ( n , 2 \\Xi _ n ) , \\Pi _ n } } X _ { ( n , 2 \\Xi _ n ) } ^ { - 1 } X _ { \\Pi _ { n } } & \\ \\flat = > \\ \\imath = n , \\\\ X _ { \\Pi _ { \\imath } } & \\end{cases} \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} x _ 1 & \\mapsto q x _ 1 \\\\ x _ 2 & \\mapsto r ^ 2 ( x _ 2 - 2 \\sum \\limits _ { j = 3 } ^ { n } s _ j x _ j + t ) \\\\ x _ j & \\mapsto r ( x _ j + s _ j ) , \\ ; j = 3 , \\ldots , n \\\\ x _ { n + 1 } & \\mapsto q r ^ 2 ( x _ { n + 1 } + x _ 1 \\sum \\limits _ { j = 3 } ^ { n } s _ j ^ 2 + t x _ 1 ) , \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\dot y _ u = V \\big ( { \\bf \\Lambda } _ { t s } \\big ) ( y _ u ) . \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} G ( A , B ) = A ^ { 1 / 2 } U B ^ { 1 / 2 } = B ^ { 1 / 2 } U ^ T A ^ { 1 / 2 } , \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} v = k \\ , x ^ { \\tilde y } , \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } g e n u s ( \\Sigma ^ y _ { \\ell } ) \\leq \\liminf _ { k \\to \\infty } g e n u s ( M _ k ) . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} g _ { \\sigma ( i ) } = \\begin{cases} \\beta _ { i } & \\mbox { i f $ 1 \\leq i \\leq p $ } , \\\\ \\beta _ { \\sigma ( i ) + p - i } ^ { - 1 } \\cdots \\beta _ { 1 } ^ { - 1 } \\alpha _ { i - p } \\beta _ { 1 } \\cdots \\beta _ { \\sigma ( i ) + p - i } , & \\mbox { i f $ p + 1 \\leq i \\leq p + q $ } \\end{cases} \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} T = \\min \\Big \\{ 1 \\leq t \\leq T _ { \\max } : D _ t = 0 \\Big \\} . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} R _ 0 = \\{ ( ( n - 2 , r + 2 ) , ( n - 2 , r + 4 s + 2 ) ) , ( ( n , r + 1 ) , ( n , r + 4 s + 1 ) ) \\} . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} s ^ { F _ { n } \\times F _ { n - 1 } } & = s _ 1 ( s _ 3 s _ 2 s _ 1 ) ^ { ( n - 3 ) / 2 } s _ 2 s _ 3 ( s _ 1 s _ 2 s _ 3 ) ^ { ( n - 3 ) / 2 } s _ 1 , \\ \\\\ s ^ { E _ { n } \\times E _ { n - 1 } } & = s _ 1 ( s _ 3 s _ 2 s _ 1 ) ^ { ( n - 3 ) / 2 } s _ 2 s _ 1 s _ 2 s _ 3 ( s _ 1 s _ 2 s _ 3 ) ^ { ( n - 3 ) / 2 } s _ 1 . \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} F ( z , z ' ) : = F _ 1 ( z , z ' ) F _ 2 ( z , z ' ) \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} S _ Y ( n , m ) = S _ Y ( n , m ; 0 ) , m \\leq n . \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} L M ( s , f ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda _ f ( n ) } { n ^ { s } } c ( n ) , \\textrm { w h e r e } c ( n ) = \\sum _ { d | n } \\mu ( d ) F _ { \\Upsilon , M } ( d ) . \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\sum _ { d = 1 } ^ { \\infty } \\gamma _ d ^ { \\frac { 1 } { \\alpha - 2 \\delta } } \\ , b ^ { w _ d } < \\infty . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} T N = \\mathring { D } \\oplus \\mathring { D } ^ { \\prime } . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} d B _ { t } ^ { ( 0 ) } \\cdot d B _ { t } ^ { ( 0 ) } = \\kappa d t , \\ \\ d B _ { t } ^ { ( r ) } \\cdot d B _ { t } ^ { ( r ) } = \\tau d t , \\ \\ r = 1 , 2 , 3 . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} \\Sigma ' _ q = \\left ( \\begin{matrix} - \\frac { r } { r _ 2 } & 0 & 0 & \\cdots & 0 \\\\ 0 & - r & 0 & & \\vdots \\\\ 0 & 0 & - \\frac { r } { r _ 4 } & & \\\\ \\vdots & & & \\ddots & \\\\ 0 & 0 & 0 & \\cdots & - r \\end{matrix} \\right ) + \\left ( \\begin{matrix} 1 \\\\ 1 \\\\ \\vdots \\\\ 1 \\\\ \\end{matrix} \\right ) \\left ( \\begin{matrix} 1 & 1 & \\cdots & 1 \\\\ \\end{matrix} \\right ) \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} \\sum _ { i \\in I } d _ i = d _ { i _ 1 } + \\dots + d _ { i _ k } < \\sum _ { l \\notin I } d _ l . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} { \\mathcal { E } } _ { 1 } ( u , w ) : = J ( u ) + \\frac { \\gamma } { 2 ( 1 + \\hat { \\delta } ) } \\| w \\| ^ { 2 } . \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\left ( s ^ { \\alpha } t ^ { \\beta } \\right ) ^ { - 1 } s ^ { x } \\left ( s ^ { \\alpha } t ^ { \\beta } \\right ) = s ^ { x } t ^ { \\beta - k ^ { x } \\beta } , \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} D _ { r + 1 } ( n ) = \\frac { n - r } { r + 1 } D _ r ( n ) + \\frac { n } { r + 1 } D _ r ( n - 1 ) . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} \\varphi ( x ) \\geq \\epsilon > 0 \\ , , \\varphi ( x ) = \\varphi ( - x ) \\ , , \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } x \\ , \\varphi ( x ) = 1 \\ , . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} \\lim _ { \\mu \\rightarrow \\infty } ( \\mu - 1 ) \\Phi _ \\mu \\big ( | p | \\big ) = | p | , \\ ; p \\in \\R ^ { n \\times N } . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\Phi _ d ( \\lambda ) ^ T \\otimes I _ n : = \\begin{bmatrix} \\phi _ { d - 1 } ( \\lambda ) & \\cdots & \\phi _ 1 ( \\lambda ) & \\phi _ 0 ( \\lambda ) \\end{bmatrix} \\otimes I _ n . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} Z ( u ) = \\sum _ { m < j \\le N } E _ { j - m , j } + \\sum _ { 1 \\le j \\le m } u E _ { N + j - m , j } = ( Z ^ { \\rho } ( u ) ) ^ m \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} g ( u ) \\ , D ^ w \\big ( ( Z ^ \\rho ( \\tfrac { \\eta } { u } ) ^ w ) \\big ) ^ k K ( \\pm u ) ( Z ^ \\rho ( \\eta u ) ) ^ k D = \\begin{cases} K ' ( u ) & \\\\ K '' ( u ) & \\end{cases} \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} ( r ^ { - } _ { \\psi } ( \\alpha _ n ) \\mathbf h _ p ) ( x ) = | p | _ p ^ { n / 2 } \\chi _ { \\psi } ( p ^ n ) \\mathbf h _ p ( p ^ n x ) = | p | _ p ^ { n / 2 } \\chi _ { \\psi } ( p ^ n ) \\mathbf 1 _ { p ^ { - n } \\Z _ p ^ { \\times } } ( x ) \\underline { \\chi } _ p ^ { - 1 } ( p ^ n x ) . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\varphi _ { n , B } ^ { \\rm { C F S K } } - \\varphi _ { n , A } ^ { \\rm { C F S K } } = 2 \\pi \\sum \\limits _ { i = 0 } ^ { n - 1 } { \\left ( { { s _ { i , B } } - { s _ { i , A } } } \\right ) } { \\rm { = } } 2 \\pi { k _ n } \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } \\frac { 2 ^ n } { \\| M _ 1 M _ 2 \\| } \\int \\limits _ { B _ { 3 \\cdot 2 ^ { - n } \\| M _ 1 M _ 2 \\| } ( M _ 2 ) } \\frac { \\omega ( d ( M ) ) } { d ^ 2 ( M ) } \\ , d m _ 3 ( M ) \\leq \\\\ & C \\sum _ { n = 0 } ^ { \\infty } \\frac { 2 ^ n } { \\| M _ 1 M _ 2 \\| } \\cdot 2 ^ { - n } \\cdot \\| M _ 1 M _ 2 \\| \\omega ( 2 ^ { - n } \\cdot \\| M _ 1 M _ 2 \\| ) = \\\\ & = C \\sum _ { n = 0 } ^ { \\infty } \\omega ( 2 ^ { - n } \\cdot \\| M _ 1 M _ 2 \\| ) \\leq C \\omega ( \\| M _ 1 M _ 2 \\| ) , \\\\ \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} I ( f _ c ' ) = J \\cup \\{ j , l \\} , \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} \\mathcal H _ 2 = \\{ Z \\in \\mathrm { M } _ 2 ( \\C ) : Z = { } ^ t Z , \\ , \\mathrm { I m } ( Z ) \\} \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{gather*} j = \\frac { ( u + a ) ^ 3 } { A u } \\ \\ a , \\ A , \\end{gather*}"}
-{"id": "6949.png", "formula": "\\begin{align*} N _ { m } = \\sum _ { n = 0 } ^ { m } \\binom { m } { m - n } N _ { \\mathrm { d } \\ , m - n } N _ { \\mathrm { c } n } \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} f _ j ^ { ( k ) } ( \\ell , \\zeta , \\varphi ) = 0 , \\ , 1 \\leq j \\leq k \\leq n . \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} \\tilde { P } ( u ) & = P ( \\psi ( u _ 1 ) , . . . , \\psi ( u _ N ) ) \\otimes \\delta _ { J ( u ) } , \\\\ \\tilde { Q } ( u ) & = Q ( \\psi ( u _ 1 ) , . . . , \\psi ( u _ N ) ) \\otimes \\delta _ { J ( u ) } . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} \\partial _ x ( \\rho ^ { \\gamma - \\alpha } u ) & = ( \\sqrt { \\rho } \\partial _ x u ) \\rho ^ { \\gamma - \\alpha - \\frac { 1 } { 2 } } + ( \\gamma - \\alpha ) \\rho ^ { \\gamma - 2 \\alpha } ( \\rho ^ { \\alpha - \\frac { 3 } { 2 } } \\partial _ x \\rho ) ( \\sqrt { \\rho } u ) \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} { \\rm a u g m } _ { p , 1 } ( A ' ) = & \\{ E , A _ 1 - E , A _ 2 , \\ldots , A _ { n - 1 } , A _ n - E \\} ; \\\\ { \\rm a u g m } _ { p , m } ( A ' ) = & \\{ A _ 1 , \\ldots , A _ { m - 2 } , A _ { m - 1 } - E , E , A _ { m } - E , A _ { m + 1 } , \\ldots , A _ n \\} \\quad 2 \\le m \\le n ; \\\\ { \\rm a u g m } _ { p , n + 1 } ( A ' ) = & \\{ A _ 1 - E , A _ 2 , \\ldots , A _ { n - 1 } , A _ n - E , E \\} . \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} & h _ { k _ i i } ( x ) v _ { ( m _ i - 1 ) i } ( x ) + h _ { ( k _ i - 1 ) i } ( x ) v _ { m _ i i } ( x ) + m _ i h ' _ { k _ i i } ( x ) v _ { m _ i i } ( x ) \\\\ = & a _ { t _ i i } ( x ) ^ { n _ i } h _ { ( k _ i - 1 ) i } ( x ) + ( n _ i a _ { t _ i i } ( x ) ^ { n _ i - 1 } a _ { ( t _ i - 1 ) i } ( x ) + ( t _ i n _ i - t _ i + m _ i n _ i ) a _ { t _ i i } ( x ) ^ { n _ i - 1 } a _ { t _ i i } ' ( x ) ) h _ { k _ i i } ( x ) . \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} x ^ { 2 } - p x - q = 0 , \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} \\partial _ t u _ t = \\partial _ t \\mu _ t + \\beta _ j \\nabla u _ t \\dot { Z } _ t ^ j , \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} 0 < \\int _ { t _ 0 } ^ t ( t - \\sigma ) \\Big ( \\lambda u ( \\sigma ) f \\big ( \\sigma , u ( \\sigma ) , \\dot { u } ( \\sigma ) \\big ) + \\dot { u } ( \\sigma ) \\varphi \\big ( \\dot { u } ( \\sigma ) \\big ) \\Big ) \\ , d \\sigma = \\int _ { t _ 0 } ^ t u ( \\sigma ) \\varphi ( \\dot u ( \\sigma ) ) \\ , d \\sigma \\leq 0 , \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} W _ { \\theta } ( x , t ) = \\frac { 1 } { \\theta ^ { a _ 1 } } W \\left ( \\frac { x } { \\theta ^ { a _ 2 } } , \\frac { t } { \\theta } \\right ) \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} s ^ 3 e _ 0 \\leq c \\varepsilon + c s ^ { 5 } e _ 0 ^ { 5 / 3 } = c \\varepsilon + s ^ { 3 } e _ 0 c ( s ^ 2 e _ 0 ^ { 2 / 3 } ) , \\quad \\forall s \\leq s _ 0 . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} \\mathbf { u } ^ { s ' } ( A _ i ) = & \\mathbf { u } ^ { s } ( X ) + ( ( A _ i - X ) ^ T \\otimes I _ 2 ) \\emph { V e c } \\left ( \\nabla \\mathbf { u } ^ { s } ( X ) \\right ) \\\\ & + ( ( A _ i - \\widetilde { Y } _ i ) ^ T \\otimes I _ 2 ) ( M ^ s - I _ 4 ) \\emph { V e c } \\left ( \\nabla \\mathbf { u } ^ { s } ( X ) \\right ) + \\mathbf { R } ^ s _ { i } ( X ) , ~ i \\in \\mathcal { I } ^ { s ' } , ~ \\forall X \\in T _ { \\ast } ^ s , ~ s = \\pm , \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} \\mathfrak R _ p F ( Z ) : = \\frac { 1 } { p } \\sum _ { j \\in \\Z / p \\Z } F ( u _ j Z ) = \\frac { 1 } { p } \\sum _ { j \\in \\Z / p \\Z } F | _ { k + 1 } [ u _ j ] ( Z ) . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} \\pi _ { 0 } ( F _ { \\alpha ^ { 1 } } ( \\Lambda _ { 1 } , . . . , \\Lambda _ { m } ) ) = f _ { \\alpha ^ { 1 } } ( \\pi _ { 0 } ( \\Lambda _ { 1 } ) , . . . , \\pi _ { 0 } ( \\Lambda _ { m } ) ) \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} A ^ - & : = \\{ x ^ * \\in X ^ \\ast \\mid \\langle x ^ * , x \\rangle \\leq 0 , \\forall x \\in A \\} . \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} E ^ { P ^ n } \\left [ \\left | k M ^ { f ^ n } _ { t \\wedge \\tau _ { \\lambda _ { m } } } - k M ^ { f } _ { t \\wedge \\tau _ { \\lambda _ { m } } } \\right | \\right ] & \\leq 2 c \\sup _ { m \\in \\mathbb { N } } P ^ m ( K ^ c ) + c \\sup _ { \\alpha \\in K } \\big | M ^ { f ^ n } _ { t \\wedge \\tau _ { \\lambda _ { m } } ( \\alpha ) } ( \\alpha ) - M ^ { f } _ { t \\wedge \\lambda _ { m } ( \\alpha ) } ( \\alpha ) \\big | \\\\ & \\leq \\frac { \\varepsilon } { 2 } + \\frac { \\varepsilon } { 2 } = \\varepsilon . \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} { \\rm H e s s } \\ , f ( S ) [ \\xi ] = { \\rm D } { \\rm g r a d } \\ , f ( S ) [ \\xi ] - { \\rm s y m } ( { \\rm g r a d } \\ , f ( S ) S ^ { - 1 } \\xi ) . \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\alpha _ n = p ^ { - n } \\rho _ { 2 n } , \\rho _ { 2 n } = \\left ( \\begin{array} { c c } p ^ { 2 n } & 0 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} \\mathcal { C } ( t , x ) = \\frac { C ( t x ) } { 1 - x C ( t x ) } , \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} a * ( b , \\bar { b } \\circ x ) = ( a \\circ ( \\bar { a } \\cdot _ x b ) , - ) . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} e _ n = \\left \\{ \\begin{array} { c l } \\left ( \\frac { f ( n G ) } { p } \\right ) & ( f ( n G ) , p ) = 1 , \\\\ + 1 & \\end{array} \\right . \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} \\Omega _ p ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) = \\overline { \\Phi _ { \\mathbf h _ p } ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) } \\Phi _ { \\breve { \\mathbf g } _ p } ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) \\Phi _ { \\pmb { \\phi } _ p } ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) = \\frac { p - \\overline { G ( - \\gamma , p ) } } { p ( p - 1 ) } = \\frac { p - G ( \\gamma , p ) } { p ( p - 1 ) } . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{gather*} \\tilde \\chi ^ k ( \\mathfrak I ) = \\big ( a - b \\sqrt { - 3 } \\big ) ^ k . \\end{gather*}"}
-{"id": "3705.png", "formula": "\\begin{align*} \\sigma _ { d , \\infty } ( w ) = ( 1 - w d ) H \\Big ( \\frac { w } { 1 - w d } \\Big ) , \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\beta _ c ( d ) : = \\sup \\{ \\beta \\geq 0 : \\theta ( \\beta , 0 ) = 0 \\} . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} H _ { i , j } ' = \\begin{cases} 0 , & i = 0 , \\\\ ( 1 + i ) ^ { \\alpha } ( 1 + j ) ^ { \\alpha } a _ { i + j - 1 } , & i > 0 . \\end{cases} \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} W ( c \\circ x ^ { \\tilde W } ) = y . \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} \\begin{aligned} v _ { x _ n } & = \\frac { 1 } { h _ z } , v _ { x _ i } = - \\frac { h _ { x ' } } { h _ z } , v _ t = - \\frac { h _ { t } } { h _ z } \\\\ v _ { x _ n x _ n } = - \\frac { h _ { z z } } { h _ z ^ 3 } , & v _ { x _ i x _ i } = - \\frac { 1 } { h _ z } \\left ( \\frac { h _ { x _ i } ^ 2 } { h _ z ^ 2 } h _ { z z } - \\frac { 2 h _ { x _ i } } { h _ z } h _ { x _ i z } + h _ { x _ i x _ i } \\right ) \\forall i = 1 , \\cdots , n - 1 . \\end{aligned} \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{gather*} \\l ' ( \\tau ) = \\frac { { \\rm d } \\l ( \\tau ) } { { \\rm d } \\tau } = 2 \\pi { \\rm i } \\cdot 8 \\cdot \\frac { \\eta ( \\tau / 2 ) ^ { 1 6 } \\eta ( 2 \\tau ) ^ { 1 6 } } { \\eta ( \\tau ) ^ { 2 8 } } = \\pi { \\rm i } \\cdot \\l ( \\tau ) \\cdot \\theta _ 4 ^ 4 ( \\tau ) . \\end{gather*}"}
-{"id": "6134.png", "formula": "\\begin{align*} U _ t \\triangleq \\big \\{ \\alpha \\in \\Omega \\colon \\alpha ( t \\wedge \\tau _ { \\lambda _ { m } } ( \\alpha ) ) \\not = \\alpha ( ( t \\wedge \\tau _ { \\lambda _ { m } } ( \\alpha ) ) - ) \\big \\} , \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} | \\eta _ r ( x , y ) - \\eta _ r ( y , x ) | = p ^ { - n _ 0 } \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\mathcal { C } ( x , y ) = \\dfrac { \\sigma ^ 2 } { 2 ^ { \\nu - 1 } \\Gamma ( \\nu ) } ( \\kappa r ) ^ \\nu \\mathcal { K } _ \\nu ( \\kappa r ) , \\ \\ r = \\Vert x - y \\Vert _ 2 , \\ \\ \\kappa = \\frac { \\sqrt { 8 \\nu } } { \\lambda } , \\ \\ x , y \\in D , \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial f ^ { ( k ) } } { \\partial \\zeta ^ { ( i ) } } \\right ) = - 2 t \\left ( \\begin{array} { c c c } e ^ { 2 t \\zeta _ 1 ^ { ( k ) } } & & \\\\ & \\ddots & \\\\ & & e ^ { 2 t \\zeta _ k ^ { ( k ) } } \\\\ \\end{array} \\right ) B ^ { ( k , i ) } , \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} S _ { m , n } ^ s = M _ { m , n } ^ s \\setminus M _ { m , n } ^ { s - 1 } \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} A ( N , \\dot { \\phi } ) & = \\left ( ( 1 + i + j ) ^ { N - 1 } \\mathfrak { d } _ 2 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\\\ B ( N , \\dot { \\phi } ) & = \\left ( ( 1 + i + j ) ^ { N - 1 } \\mathfrak { d } _ 1 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\\\ C ( N , \\dot { \\phi } ) & = \\left ( ( 1 + i + j ) ^ { N - 1 } \\mathfrak { d } _ 2 \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } , \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{gather*} L \\big ( \\eta ( 6 \\tau ) ^ 4 , 1 \\big ) = \\frac 1 { 3 \\cdot 2 ^ { 4 / 3 } } B ( 1 / 3 , 1 / 3 ) = \\frac { 1 } { 3 ^ { 5 / 4 } \\cdot 2 ^ { 5 / 6 } } b _ { \\Q ( \\sqrt { - 3 } ) } . \\end{gather*}"}
-{"id": "7945.png", "formula": "\\begin{align*} H _ K ( \\beta _ c ( d ) ) = 0 . \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} g _ 0 = l _ 0 , g _ 1 = A _ 0 y ^ 2 + A _ 1 y z + A _ 2 z ^ 2 , g _ 2 = l _ 2 , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\mathbf { c } = \\Big [ \\mathbf { c } _ { j _ k } \\Big ] _ { k = 1 } ^ { \\abs { \\mathcal { I } ^ - } } \\in \\mathbb { R } ^ { 2 \\abs { \\mathcal { I } ^ - } } , ~ ~ \\mathbf { v } ^ - = \\Big [ \\mathbf { v } _ { j _ k } \\Big ] _ { k = 1 } ^ { \\abs { \\mathcal { I } ^ - } } \\in \\mathbb { R } ^ { 2 \\abs { \\mathcal { I } ^ - } } , ~ ~ \\mathbf { v } ^ + = \\Big [ \\mathbf { v } _ { j _ k } \\Big ] _ { k = \\abs { \\mathcal { I } ^ - } + 1 } ^ { \\abs { \\mathcal { I } } } \\in \\mathbb { R } ^ { 2 \\abs { \\mathcal { I } ^ + } } . \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} [ c , d ] & : = [ ( 1 + m u v ) a - m v ^ 2 b , m u ^ 2 a + ( 1 - m u v ) b ] \\in \\mathcal P ^ + , \\\\ [ e , f ] & : = [ ( 1 - m v w ) a + m w ^ 2 b , - m v ^ 2 a + ( 1 + m v w ) b ] \\in \\mathcal P ^ + . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\begin{cases} x ^ \\prime = x ^ p \\ , , \\\\ x ( 0 ) = \\| u _ 0 \\| _ \\infty \\ , , \\end{cases} \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} [ \\beta ] = [ \\overline { \\alpha } ] + [ \\overline { \\gamma } ] \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} & P ( A _ { j } ^ { ( k ) } ) ( 1 - P ( A _ { j } ^ { ( k ) } ) ) \\mathbb { E } ( f \\mid A _ { j } ^ { ( k ) } ) - \\sum _ { j \\neq i \\leqslant m _ { k } } P ( A _ { i } ^ { ( k ) } ) P ( A _ { j } ^ { ( k ) } ) \\mathbb { E } ( f \\mid A _ { i } ^ { ( k ) } ) \\\\ & = \\mathbb { E } ( 1 _ { A _ { j } ^ { ( k ) } } f ) - P ( A _ { j } ^ { ( k ) } ) \\sum _ { 1 \\leqslant i \\leqslant m _ { k } } \\mathbb { E } ( 1 _ { A _ { i } ^ { ( k ) } } f ) \\\\ & = \\mathrm { C o v } \\left ( \\mathbb { G } ( A _ { j } ^ { ( k ) } ) , \\mathbb { G } ( f ) \\right ) . \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} V \\delta _ n = \\delta _ { 2 n } . \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\Psi ( s ; W _ { \\pi } , W _ { \\pi ^ \\prime } ) = \\int _ { U _ m ( F ) \\cap H ^ 1 _ m \\backslash H ^ 1 _ m } \\ , d h = 1 . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\hat { \\phi } _ i \\left ( e _ i \\right ) : = \\begin{cases} e _ { i - 1 } & , \\\\ ( - 1 ) ^ { a - 1 } e _ n & . \\end{cases} \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} \\phi ( b ) = \\left [ \\begin{array} { c c c c } \\epsilon ( b ) & \\chi _ { 0 } ( b ) & \\chi _ { 1 } ( b ) & \\chi _ { 2 } ( b ) \\\\ 0 & f _ { 0 } ( b ) & 0 & 0 \\\\ 0 & 0 & f _ { 1 } ( b ) & 0 \\\\ 0 & 0 & 0 & f _ { 2 } ( b ) \\\\ \\end{array} \\right ] \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\alpha _ 2 + \\alpha _ 3 + \\beta _ 6 + \\beta _ 7 = v ^ * ( x _ 0 - u _ 0 ) \\le \\Vert x _ 0 - u _ 0 \\Vert < \\varepsilon . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} s ( t ) : = \\frac { 1 } { q } - \\frac { 1 } { p ( t ) } \\ , , \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N \\langle k \\rangle _ n = \\sum _ { m = 0 } ^ n \\binom { N + 1 } { m + 1 } ( n ) _ m \\langle m \\rangle _ { n - m } = \\sum _ { k = 0 } ^ n c _ { n , N } ( k ) \\langle k \\rangle _ n . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} I _ { G } B = ( 1 - \\rho ) B \\subseteq \\bigl ( ( 1 + \\sigma ) \\rho \\bigr ) B + ( 1 + \\sigma \\rho ) B = \\bigl ( 1 + \\sigma \\bigr ) B + ( 1 + \\sigma \\rho ) B = N _ \\Sigma B + N _ { \\Sigma ' } B \\subseteq B ^ { \\Sigma } + B ^ { \\Sigma ' } . \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} C _ { b } ( N ) = \\begin{bmatrix} A _ { 1 1 } ( N ) & 0 \\\\ A _ { 2 1 } ( N ) & C _ { k - 1 } ( N ) \\end{bmatrix} , \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} u _ { t } + C u u _ { x } = D u _ { x x } + B u \\left ( 1 - u \\right ) , \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} A & = A ' + \\delta _ 1 ( ( i , j ' ) , ( i ' , j ) ) \\\\ & = B + \\sum _ { r = 1 } ^ t \\delta ( \\mathbf { u } _ r , \\mathbf { v } _ r ) + \\delta _ 1 ( ( i , j ' ) , ( i ' , j ) ) . \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} N _ 0 = \\Big | \\| \\zeta _ k \\xi _ { n _ k } \\| ^ 2 _ 1 + | \\zeta _ k \\nu _ { n _ k } \\| ^ 2 _ 1 - \\int _ { \\mathbb R } \\rho _ { k , 1 } ( x ) d x \\Big | \\leqq \\epsilon , \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} & ( A ! _ \\lambda B ) ^ { - 1 } - ( A \\nabla _ \\lambda B ) ^ { - 1 } = \\lambda ( 1 - \\lambda ) ( B - A ) ( A \\sharp B ) ^ { - 2 } ( A \\nabla _ \\lambda B ) ^ { - 1 } ( B - A ) \\\\ & ( A ! _ \\lambda B ) ^ { - 1 } ( A \\nabla _ \\lambda B ) - I = \\lambda ( 1 - \\lambda ) ( A - B ) A ^ { - 1 } B ^ { - 1 } ( A - B ) . \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} \\varphi ( z ) = \\sum _ { i = 1 } ^ N \\frac { k _ i } { z - z _ i } . \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{gather*} N ( k ) = N ( 1 ) ^ k \\cdot ( ) . \\end{gather*}"}
-{"id": "1886.png", "formula": "\\begin{align*} \\big ( P _ n ^ N ( a ) \\big ) ' _ t = P _ n ^ N ( a ' _ t ) a . e . \\ t . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} \\| f ( \\tau ) \\| _ { \\mathcal { H } ^ k ( \\Omega ) } : = \\left ( \\sum _ { | \\alpha | \\leq k } \\| \\partial ^ { \\alpha } f ( \\tau , \\cdot ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} \\tilde { H } _ 1 ( \\beta ) \\begin{cases} \\infty , & \\beta \\in [ 0 , \\beta _ P ( d ) ] , \\\\ \\in ( 0 , \\infty ) , & \\beta \\in ( \\beta _ P ( d ) , \\beta _ P ( d ) + \\epsilon _ d ] , \\\\ \\in [ 0 , \\infty ) , & \\beta \\in ( \\beta _ P ( d ) + \\epsilon _ d , \\beta _ c ( d ) ) , \\\\ \\end{cases} \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} x _ { i _ 1 i _ 2 } x _ { i _ 3 i _ 4 } \\ast _ g x _ { j _ 1 j _ 2 } x _ { j _ 3 j _ 4 } = \\tfrac { 1 } { 2 } ( x _ { g ( i _ 1 ) g ( i _ 2 ) j _ 1 j _ 2 } x _ { g ( i _ 3 ) g ( i _ 4 ) j _ 3 j _ 4 } + x _ { g ( i _ 1 ) g ( i _ 2 ) j _ 3 j _ 4 } x _ { g ( i _ 3 ) g ( i _ 4 ) j _ 1 j _ 2 } ) . \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} \\Gamma _ 0 ( z ) = & \\frac { 1 } { z } , \\\\ \\Gamma _ 1 ( z \\ , | \\ , \\tau ) = & \\frac { \\tau ^ { z / \\tau - 1 / 2 } } { \\sqrt { 2 \\pi } } \\ , \\Gamma \\bigl ( \\frac { z } { \\tau } \\bigr ) , \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} ( b _ N ( \\xi ) ) _ j = - 4 \\pi ^ 2 \\sum _ { l \\in \\Lambda _ N } \\textbf { 1 } _ { \\Lambda _ N } ( j - l ) \\xi _ l \\xi _ { j - l } \\frac { j \\cdot l ^ \\perp } { | l | ^ 2 } . \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{gather*} a ^ 1 \\otimes a ^ { 2 1 } \\otimes a ^ { 2 2 } = X a ^ { 1 1 } P \\otimes Y a ^ { 1 2 } Q \\otimes Z a ^ 2 R . \\end{gather*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\partial _ \\xi ^ 2 u ( x ) = \\partial _ \\xi ^ 2 ( u - P _ { \\tilde { x } } ( \\cdot - \\tilde { x } ) ) ( x ) + \\partial _ \\xi ^ 2 P _ { \\tilde { x } } ( x - \\tilde { x } ) . \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} ( u _ { t } , v ) + b ( u _ , u , v ) + \\nu ( \\nabla u , \\nabla v ) - ( p , \\nabla \\cdot v ) & = ( f , v ) \\ ; \\ ; \\forall v \\in X , \\\\ ( q , \\nabla \\cdot u ) & = 0 \\ ; \\ ; \\forall q \\in Q . \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ i } \\int \\limits _ E f ( x , \\omega ) \\mu ( d \\omega ) = \\int \\limits _ E \\frac { \\partial } { \\partial x _ i } f ( x , \\omega ) \\mu ( d \\omega ) \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} \\begin{aligned} & y _ { i } = x _ { i - 1 } - \\frac 1 L f ' ( x _ { i - 1 } ) \\\\ & x _ { i } = y _ { i } + \\zeta _ i ( y _ i - y _ { i - 1 } ) + \\eta _ i ( y _ i - x _ { i - 1 } ) \\end{aligned} \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} R ^ { \\nu , x , y } : = R ^ \\nu ( \\bullet | X _ 0 = x \\mbox { a n d } X _ 1 = y ) . \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} \\theta _ { B _ R ( 0 ) } ^ \\beta ( r _ i ) & \\geq \\frac 1 { 4 ^ \\beta } \\cdot \\frac { - 1 } { \\log ( \\frac { r _ i } 4 ) } \\\\ & \\geq \\frac 1 { 4 ^ \\beta } \\cdot \\frac { - 1 } { \\log ( \\frac { r _ 1 } { 4 C ^ { i - 1 } } ) } \\\\ & = \\frac 1 { 4 ^ \\beta } \\cdot \\frac { - 1 } { \\log ( \\frac { r _ 1 } 4 ) - \\log ( C ^ { i - 1 } ) ) } . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} f ^ { L } ( \\vec x ) & = \\int _ 0 ^ { \\| x \\| _ \\infty } f ( V _ t ^ + ( \\vec x ) , V _ t ^ - ( \\vec x ) ) d t \\\\ & = \\int _ 0 ^ { \\| x \\| _ \\infty } f ( V _ t ^ - ( \\vec x ) , V _ t ^ + ( \\vec x ) ) d t \\\\ & = \\int _ 0 ^ { \\| x \\| _ \\infty } f ( V _ t ^ + ( - \\vec x ) , V _ t ^ - ( - \\vec x ) ) d t = f ^ { L } ( - \\vec x ) , \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\sup _ { | y | \\leq R } \\sup _ { | \\xi | \\leq R ^ { - 1 } } | q ( y , \\xi ) | = 0 , \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} \\psi ^ { ( d + 1 ) } ( x ) = o \\bigg ( \\frac { \\varphi _ 1 ( x ) \\sigma _ 1 ( x ) } { x ^ { d + 1 } } \\bigg ) . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta ^ q _ { 2 , 2 } ( \\delta ) \\bigr ] & = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty ( e ^ { - t q } - 1 ) e ^ { - \\delta t } \\frac { ( 1 - e ^ { - t / 2 } ) ^ 2 } { ( 1 - e ^ { - t } ) ^ 2 } \\frac { d t } { t } \\Bigr ) , \\\\ & = \\exp \\Bigl ( \\frac { 1 } { 4 } \\int \\limits _ 0 ^ \\infty ( e ^ { - t q } - 1 ) e ^ { - ( \\delta - 1 / 2 ) t } \\frac { 1 } { \\cosh ^ 2 ( t / 4 ) } \\frac { d t } { t } \\Bigr ) . \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} r _ { A , 1 } & \\ ; : = l _ { A ^ \\bot } : ( 1 _ y , A ^ \\bot ) \\to ( A ^ \\bot ) , \\\\ r _ { A , 2 } & \\ ; : = e _ { A ^ \\bot } : ( A ^ \\bot , A ) \\to ( b ) , \\\\ r _ { A , 3 } & \\ ; : = r _ { A } : ( A , 1 _ y ) \\to ( A ) , \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} u ( t _ n ) \\rightharpoonup u ^ * : = u ( T _ * ) \\ \\ \\ \\textrm { w e a k l y i n } \\ \\ \\ \\dot B ^ { s _ p } _ { p , p } . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} B _ { n , k } = \\sum _ { s = k } ^ n \\binom { s } { k } C _ { n , s } . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\mathcal { P } } q ^ { | \\lambda | } x ^ { \\# \\mathcal { H } _ { 2 } ( \\lambda ) } b ^ { B G ( \\lambda ) } \\prod _ { h \\in \\mathcal { H } _ { 2 } ( \\lambda ) } \\dfrac { 1 } { h ^ { 2 } } = \\exp \\left ( \\dfrac { x q ^ { 2 } } { 2 } \\right ) \\sum _ { j = - \\infty } ^ { + \\infty } b ^ { j } q ^ { j ( 2 j - 1 ) } , \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\begin{cases} d u ^ { n } ( t ) + A u ^ { n } ( t ) \\ d t = B ( u ^ { n } ( t ) ) \\ d t , & t > 0 \\\\ u ( 0 ) = u _ { 0 } ^ { n } . \\end{cases} \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} ( A , B ; C , D ) = ( M \\cdot A , M \\cdot B ; M \\cdot C ; M \\cdot D ) . \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} Q \\bigg ( \\sum _ { i = 1 } ^ m x _ i \\alpha _ i \\bigg ) = \\sum _ { i , j = 1 } ^ m A _ { i j } x _ i x _ j \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} \\mathcal I & : = \\prod _ v \\mathcal I _ v ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) ^ { - 1 } = \\prod _ { p \\mid 2 N } \\mathcal I _ v ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) ^ { - 1 } = N \\prod _ { p \\mid M } \\frac { ( p + 1 ) } { 2 } = 2 ^ { - \\nu ( M ) } N \\prod _ { p \\mid M } ( p + 1 ) . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\dot { \\phi } ( n ) = \\frac { ( - 1 ) ^ n } { ( n + 1 ) ^ { \\alpha + 1 } } , \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\phi _ { M + D } = \\phi _ M - 2 Q \\phi _ { M _ u } + Q ^ 2 \\phi _ { M _ { u v } } . \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} p _ 0 \\leq \\bigwedge \\{ g ( q ) \\in P \\mid p _ 0 \\leq g ( q ) \\} \\cong g \\left ( \\bigwedge \\{ q \\in Q \\mid p _ 0 \\leq g ( q ) \\} \\right ) = g ( f ( p _ 0 ) ) , \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\| \\partial _ x f ^ n _ t - \\partial _ x f ^ { n - 1 } _ t \\| _ { L ^ 1 } & \\le \\int _ 0 ^ t \\ ! \\mathrm { d } s \\ , \\| \\left [ G ( M ^ { n - 1 } ( s ) - G ( M ^ { n - 2 } ( s ) \\right ] \\partial _ { x v } f ^ { n - 1 } _ s \\| _ { L ^ 1 } \\\\ & + \\int _ 0 ^ t \\ ! \\mathrm { d } s \\ , \\| \\partial _ x G ( M ^ { n - 1 } ( s ) ) \\partial _ v f ^ n _ s - \\partial _ x G ( M ^ { n - 2 } ( s ) ) \\partial _ v f ^ { n - 1 } _ s \\| _ { L ^ 1 } \\ , . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} M _ 0 = \\{ m _ { ( s + 1 ) / 2 , r + 1 } ^ { ( n - 1 ) } , m _ { ( s - 1 ) / 2 , r + 3 } ^ { ( n - 1 ) } \\} , \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} \\div _ { \\mu _ N } ( b _ N ) = \\div _ { \\mu _ N } \\big ( G _ N ^ k \\big ) = 0 , \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} C = \\frac { 1 } { 1 - \\alpha } > 1 , \\lambda = - \\frac { \\log ( 1 - \\alpha ) } { t _ 0 } , \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} f ( 1 1 ) = a , f ( 1 2 ) = a , f ( 1 3 ) = b , f ( 2 1 ) = c , f ( 2 2 ) = d , f ( 2 3 ) = d \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} ( v _ t ^ 2 , m _ t ) + \\int _ 0 ^ t 2 ( | \\partial _ x v _ r | ^ 2 , m _ r ) d r = ( v _ 0 ^ 2 , m _ 0 ) - 2 \\int _ 0 ^ t ( u _ r ^ { ( 1 ) } \\partial _ x u ^ { ( 1 ) } _ r - u _ r ^ { ( 2 ) } \\partial _ x u ^ { ( 2 ) } _ r , v _ r m _ r ) d r , \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} 1 = c _ 1 < c _ 2 < \\cdots < c _ d = n . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} I _ { 1 , 4 } = { \\Theta } \\frac { \\zeta ( 3 ) } { \\zeta ( 2 ) } x + O \\left ( \\log x \\right ) . \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} \\mathcal { A } ^ { ( N _ { 0 } - k ) } = \\mathcal { A } ^ { ( N _ { 1 } - k ) } , 0 \\leqslant k < N _ { 0 } . \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} \\overline { F _ { x } } = \\widehat { \\mu } ^ { - 1 } ( x ) \\cong \\cup _ { \\sigma \\in \\mathfrak { S } ( x ) } F _ { \\sigma } , \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\zeta _ 1 ^ { k , t } = - i ( \\pi k + t \\xi ) + \\sqrt { t ^ 2 + \\pi ^ 2 | k | ^ 2 } \\eta , \\\\ \\zeta _ 2 ^ { k , t } = - i ( \\pi k - t \\xi ) - \\sqrt { t ^ 2 + \\pi ^ 2 | k | ^ 2 } \\eta , \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{gather*} 2 L \\big ( \\eta ( 4 \\tau ) ^ 2 \\eta ( 8 \\tau ) ^ 2 , 1 \\big ) ^ 2 = L \\big ( \\eta ( 4 \\tau ) ^ 6 , 2 \\big ) , \\end{gather*}"}
-{"id": "2030.png", "formula": "\\begin{align*} [ a _ m , a _ n ] = 0 = [ a ^ * _ m , a ^ * _ n ] , [ a _ m , a ^ * _ n ] = \\delta _ { m , n } \\ 1 _ { \\mathcal { F } _ + ^ { ( 1 ) } } , \\\\ [ b _ m , b _ n ] = 0 = [ b ^ * _ m , b ^ * _ n ] \\ , , [ b _ m , b ^ * _ n ] = \\delta _ { m , n } \\ 1 _ { \\mathcal { F } _ + ^ { ( 2 ) } } . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} \\langle u '' ( t ) , u ' ( t ) \\rangle = \\frac { 1 } { 2 } \\frac { d } { d t } | u ' ( t ) | ^ 2 \\ \\mbox { f o r a l m o s t a l l } \\ t \\in T . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} S _ 1 & : = C _ R ( a + b ) \\\\ & = Z \\cup ( Z + ( a + b ) ) \\cup ( Z + 2 ( a + b ) ) \\cup ( Z + 3 ( a + b ) ) , \\\\ S _ 2 & : = C _ R ( a + 2 b ) = Z \\cup ( Z + ( a + 2 b ) ) , \\\\ S _ 3 & : = C _ R ( a ) = Z \\cup ( Z + a ) \\\\ S _ 4 & : = C _ R ( b ) = Z \\cup ( Z + b ) \\cup ( Z + 2 b ) \\cup ( Z + 3 b ) . \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} d g _ t = - \\frac 1 2 \\ , \\sum _ { k = 1 } ^ m \\ , L _ k ^ 2 \\ , g _ t \\ , d t + \\sum _ { k = 1 } ^ m \\ , L _ k \\ , g _ t \\ , d B _ t ^ k \\ , , \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ p \\lambda _ i f ( V _ i ^ + , V _ i ^ - ) & = \\sum _ { i = 0 } ^ p \\int _ { t _ { i - 1 } } ^ { t _ i } f ( V _ t ^ + ( \\vec x ) , V _ t ^ - ( \\vec x ) ) d t \\\\ & = \\int _ { 0 } ^ { t _ p } f ( V _ t ^ + ( \\vec x ) , V _ t ^ - ( \\vec x ) ) d t = f ^ L ( \\vec x ) . \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} B ( U , V ) = \\lambda g ( U , V ) . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\sigma ( L ( r , \\pi _ p \\boxtimes \\chi _ p , \\varrho _ { 5 } ) ) = L ( r , { } ^ \\sigma \\pi _ p \\boxtimes { } ^ \\sigma \\ ! \\chi _ p , \\varrho _ { 5 } ) . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} d _ 1 d _ 2 - 2 ( d _ 1 + d _ 2 ) + 1 = ( d _ 1 - 2 ) ( d _ 2 - 2 ) - 3 \\leq 0 . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\theta _ u ^ { { \\rm { R F } } } \\left ( t \\right ) = 2 \\pi \\int _ { \\rm { 0 } } ^ t { f _ u ^ { { \\rm { R F } } } \\left ( \\tau \\right ) d \\tau } + \\varphi _ u ^ { { \\rm { R F } } } + \\varepsilon _ u ^ { { \\rm { R F } } } ( t ) \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} ( a w ) ^ { i } = a ^ { i } w ^ { ( i ) } ( v + w ) ^ { ( h ) } = \\sum _ { i + j = h } v ^ { ( i ) } w ^ { ( j ) } \\ , . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 0 } ^ L \\eta _ { M - 1 , N } ( q \\ , | \\ , \\hat { a } _ i , b _ 0 + k a _ i ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } ( e ^ { - q t } - 1 ) \\frac { e ^ { - b _ 0 t } \\prod _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) } { \\prod _ { j \\neq i } ^ M ( 1 - e ^ { - a _ j t } ) } \\frac { 1 - e ^ { - a _ i t ( L + 1 ) } } { 1 - e ^ { - a _ i t } } \\Bigr ) . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} P ( \\omega ) = \\mathcal { P } [ x , u ( x , \\omega ) ] ( \\omega ) . \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} \\int _ { Q ( 1 ) } | q _ n | ^ { \\frac { 3 } { 2 } } d x d t \\leq C , \\ \\ \\ \\lim _ { n \\to \\infty } \\int _ { Q ( 1 ) } | v _ n | ^ 3 d x d t = 0 . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} & \\overline { E ( x , \\zeta ) } = e ^ { i ( x _ 1 \\zeta _ 1 + x _ 2 \\zeta _ 2 ) - x _ 0 | \\zeta | _ { \\C } } \\left ( 1 - \\frac { i \\zeta } { | \\zeta | _ { \\C } } \\right ) , \\\\ & \\left ( 1 - \\frac { i \\zeta } { | \\zeta | _ { \\C } } \\right ) ^ { - 1 } = \\frac 1 2 \\left ( 1 + \\frac { i \\zeta } { | \\zeta | _ { \\C } } \\right ) , \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } U _ N \\psi _ N ^ { \\operatorname { M F } } = \\Phi ^ { \\operatorname { g s } } . \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} { \\mu _ k } _ i ( R ) \\not = 0 \\ \\hbox { f o r a n y $ k \\ge 1 $ a n d $ i \\ge 1 $ } , \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { \\mathcal G } = \\langle H u , H v \\rangle _ E . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{gather*} \\beta _ \\lambda ( r ) : = \\frac { 1 } { \\lambda } \\bigl ( r - J _ \\lambda ( r ) \\bigr ) : = \\frac { 1 } { \\lambda } \\bigl ( r - ( I + \\lambda \\beta ) ^ { - 1 } ( r ) \\bigr ) , \\\\ \\beta _ { \\Gamma , \\lambda } ( r ) : = \\frac { 1 } { \\lambda \\varrho } \\bigl ( r - J _ { \\Gamma , \\lambda } ( r ) \\bigr ) : = \\frac { 1 } { \\lambda \\varrho } \\bigl ( r - ( I + \\lambda \\varrho \\beta _ \\Gamma ) ^ { - 1 } ( r ) \\bigr ) \\end{gather*}"}
-{"id": "8498.png", "formula": "\\begin{align*} \\| ( I - T ) a \\| ^ 2 & = \\| \\sum _ { i = 1 } ^ { \\infty } a _ i ( e _ i - x _ i ) \\| ^ 2 \\\\ & \\le \\sum _ { i = 1 } ^ { \\infty } | a _ i | \\| e _ i - x _ i \\| \\\\ & \\le \\left ( \\sum _ { i = 1 } ^ { \\infty } | a _ i | ^ 2 \\right ) \\left ( \\sum _ { i = 1 } ^ { \\infty } \\| x _ i - e _ i \\| ^ 2 \\right ) \\\\ & \\le ( 1 - \\epsilon ) \\| a \\| ^ 2 . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} D = \\frac { L ^ 2 } { P e \\ , T _ \\mathrm { m a x } } . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { n - \\delta } { n - r \\brack \\delta } ( q ^ \\delta A ' _ r + B ' _ r ) = q ^ { ( m + 1 ) ( n - \\delta ) } q ^ \\delta \\sum _ { s = 0 } ^ n { n - s \\brack n - \\delta } ( A _ s + q ^ \\delta C _ s ) \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} I _ { j k } & = \\Gamma ( t _ 1 t _ 2 q ^ { n - 1 } , t _ 1 t _ 3 q ^ { j + k - 2 } , t _ 1 t _ 4 q ^ { n + j - k - 1 } , t _ 1 t _ 5 q ^ { j - 1 } , t _ 1 t _ 6 q ^ { j - 1 } ) \\\\ & \\quad \\times \\Gamma ( t _ 2 t _ 3 q ^ { n - j + k - 1 } , t _ 2 t _ 4 q ^ { 2 n - j - k } , t _ 2 t _ 5 q ^ { n - j } , t _ 2 t _ 6 q ^ { n - j } , t _ 3 t _ 4 q ^ { n - 1 } ) \\\\ & \\quad \\times \\Gamma ( t _ 3 t _ 5 q ^ { k - 1 } , t _ 3 t _ 6 q ^ { k - 1 } , t _ 4 t _ 5 q ^ { n - k } , t _ 4 t _ 6 q ^ { n - k } , t _ 5 t _ 6 ) . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} t _ { s _ h } ^ { 2 m + 1 } = \\prod _ { i = 1 } ^ m { } _ { t _ x ^ { 2 i - 1 } } ( t _ y t _ a ^ { - 1 } t _ { t _ x ( y ) } t _ a ^ { - 1 } ) \\cdot t _ { t _ x ^ { 2 m + 1 } ( y ) } t _ a ^ { - 1 } \\cdot t _ x ^ { 2 m + 1 } t _ b ^ { - 2 m - 1 } t _ z ^ { 2 m + 1 } t _ c ^ { - 2 m - 1 } . \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} \\dot { F } _ t = L F _ t , F _ t | _ { t = 0 } = F _ 0 , \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u '' ( t ) + A ( t , u ' ( t ) ) + \\epsilon K _ r ( u ' ( t ) ) + B u ( t ) \\in F ( t , u ( t ) , u ' ( t ) ) \\ \\mbox { f o r a . a . } \\ t \\in T , \\\\ u ( 0 ) = u _ 0 , \\ u ' ( 0 ) = u _ 1 . \\end{array} \\right \\} \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} \\Big ( \\int _ 0 ^ \\cdot S _ { \\cdot - s } \\sigma ^ n ( X _ s ) \\dd W _ s \\Big ) ( t \\wedge \\tau _ m ) = \\tfrac { \\sin ( \\pi \\theta ) } { \\pi } \\big ( G _ \\theta Y ^ { n , m , \\theta } \\big ) ( t \\wedge \\tau _ m ) . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} t _ { s _ { 1 , 1 } } ^ { 5 n } t _ { s _ { 1 , 2 } } ^ { 5 n } \\cdots t _ { s _ { 1 , g - 1 } } ^ { 5 n } = [ \\mathcal { V } ' _ 1 , \\mathcal { W } ' _ 1 ] [ \\mathcal { V } ' _ 2 , \\mathcal { W } ' _ 2 ] \\cdots [ \\mathcal { V } ' _ { \\left [ \\frac { | 5 n | } { 2 } \\right ] + 1 } , \\mathcal { W } ' _ { \\left [ \\frac { | 5 n | } { 2 } \\right ] + 1 } ] \\cdot t _ { d _ 1 } ^ { 1 0 n } t _ { d _ 2 } ^ { 1 0 n } \\cdots t _ { d _ { g - 1 } } ^ { 1 0 n } . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } . v = 0 1 \\leq a < b \\leq m + n , a \\neq h , M \\in \\mathbb Z _ { > 0 } . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\| \\vec x \\| _ p = ( | x _ 1 | ^ p + | x _ 2 | ^ p + \\cdots + | x _ n | ^ p ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} \\begin{array} { l l } N _ 5 = & \\int _ { \\mathbb R } ( \\zeta _ k + \\varphi _ k ) [ ( 2 c _ 1 a _ k - \\omega b _ k ) \\xi _ { n _ k } + ( c _ 2 b _ k - \\omega a _ k ) \\nu _ { n _ k } ] \\\\ \\\\ & + ( ( \\zeta _ k \\xi _ { n _ k } ) ' + ( \\varphi _ k \\xi _ { n _ k } ) ' ) ( 2 c _ 3 a ' _ k - \\omega c _ 4 b ' _ k ) - \\omega c _ 4 a ' _ k ( ( \\zeta _ k \\nu _ { n _ k } ) ' + ( \\varphi _ k \\nu _ { n _ k } ) ' ) , \\end{array} \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} M _ { L _ { \\infty } / K } : = \\partial _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ( \\mathcal { G } ) } ( m _ { L _ { \\infty } / K } ) \\in K _ 0 ( \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ( \\mathcal { G } ) ) \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\begin{array} { c c c } A ^ { U } _ { 0 } : = ( \\iota \\otimes \\chi _ { 0 } ) U , & A ^ { U } _ { 1 } : = ( \\iota \\otimes \\chi _ { 1 } ) U , & A ^ { U } _ { 2 } : = ( \\iota \\otimes \\chi _ { 2 } ) U . \\\\ \\end{array} \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} a _ k = \\frac { \\theta _ { - 1 } ^ 2 } { \\theta _ { k - 1 } ^ 2 } a _ 0 - \\sum _ { l = 1 } ^ k \\frac { \\sigma ^ K _ l } { \\mu } \\frac { \\theta _ { - 1 } ^ 2 } { \\theta _ { l - 1 } ^ 2 } b _ l . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} D \\mapsto \\left . \\frac { d } { d x } \\right | _ { x = 0 } ( A + x \\ , D ) ^ { - s / 2 } \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} H _ 1 : = \\{ q _ x ( p ) = 0 \\} & H _ 2 : = \\{ q _ y ( p ) = 0 \\} \\\\ H _ 3 : = \\{ q _ z ( p ) = 0 \\} & H _ 4 : = \\{ f ( p ) = 0 \\} \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } , \\xi _ t , D _ { t } ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; D _ { t - 1 } f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + \\mathcal { Q } _ { t + 1 } ( x _ { t } , \\xi _ { [ t ] } , D _ t ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} C ^ { - 1 } g _ { \\C ^ { m + n } } \\leq \\tilde g _ t ^ \\bullet \\leq C g _ { \\C ^ { m + n } } , \\\\ \\frac { | \\tilde \\omega _ t ^ \\bullet ( 0 ) - \\mathbf { P } ^ { \\tilde { g } _ t } _ { \\tilde { x } ' _ t 0 } ( \\tilde \\omega ^ \\bullet _ t ( \\tilde { x } ' _ t ) ) | _ { \\tilde { g } _ t ( 0 ) } } { d ^ { \\tilde { g } _ t } ( 0 , \\tilde { x } ' _ t ) ^ \\alpha } = 1 + o ( 1 ) , \\\\ d ^ { \\tilde { g } _ t } ( 0 , \\tilde { x } _ t ' ) \\leq C . \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\delta u _ { s t } = \\delta \\mu _ { s t } + A _ { s t } ^ 1 u _ s + A _ { s t } ^ 2 u _ s + u _ { s t } ^ { \\natural } \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} \\mathcal { B } _ M ( x , t ) = t ^ { - a _ 1 } \\left ( \\mathcal { C } _ M - \\frac { k | x | ^ 2 } { t ^ { 2 a _ 2 } } \\right ) _ + ^ { \\frac { 1 } { m - 1 } } \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} b = ( 1 - \\rho ^ j ) b + \\rho ^ j b \\in I _ G B + ( B ^ { \\Sigma } + B ^ { \\Sigma ' } ) = B ^ { \\Sigma } + B ^ { \\Sigma ' } \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} Q _ { \\kappa ' \\kappa } ( t ) : = \\sum _ { n = 0 } ^ \\infty \\int _ 0 ^ t \\int _ 0 ^ { t _ 1 } \\cdots \\int _ 0 ^ { t _ { n - 1 } } T ^ { ( n ) } _ { \\kappa ' \\kappa } ( t , t _ 1 , t _ 2 , \\dots , t _ n ) d t _ n d t _ { n - 1 } \\cdots d t _ 1 . \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} Q ( x , - y , - y , - x ) ^ t & = ( A x - ( B + C ) y , - ( B + C ) x - A y , - ( B + C ) x - A y , - A x + ( B + C ) y ) ^ t \\\\ & = \\mu ( x , - y , - y , - x ) ^ t , \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} & e ( t + 1 ) = ( A + K C ) e ( t ) + ( B + K D ) w ( t ) , & e ( 0 ) = 0 \\\\ & r ( t ) = C e ( t ) + D w ( t ) , \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{gather*} h \\cdot m = S ^ 2 ( h ) m , m \\in M ^ \\# . \\end{gather*}"}
-{"id": "1054.png", "formula": "\\begin{align*} \\lambda _ { n } = \\sum _ { i = 1 } ^ { n } b ( i ; n ) , \\lambda = \\lim _ { n \\to \\infty } \\lambda _ n , \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} i _ { H } ^ { \\ast } N _ { H } ^ { G } ( r ) = \\prod _ { g \\in W _ { G } ( H ) } g r . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} \\Psi _ 4 ( p ^ { e _ p } , p ^ { s _ p } ) = p ^ { e _ p - s _ p } \\max \\{ 2 s _ p - e _ p - 1 , 0 \\} . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} y = \\begin{pmatrix} J _ { \\frac { k b } { 8 } , 4 } \\\\ & J _ { \\frac { k b } { 8 } , 4 } ^ { - T } \\\\ & & I _ { s } \\end{pmatrix} . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} \\delta _ \\Gamma ( \\vec x ^ { k + 1 } , \\vec x ^ k ) & = \\delta _ \\Gamma ( \\vec z ^ { k + 1 } , \\vec z ^ k ) = \\sum _ { i = 1 } ^ n \\delta _ \\Gamma ( \\vec z ^ { k , i } , \\vec z ^ { k , i - 1 } ) \\\\ & = \\sum _ { i \\in V ( k ) } \\delta _ \\Gamma ( \\vec z ^ { k , i } , \\vec z ^ { k , i - 1 } ) , \\ ; \\ ; \\ ; \\Gamma \\in \\{ F , \\vec q , \\vec p \\} \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} p \\log \\lambda ^ { ( p ) } ( G ) & \\leq p \\log ( r ^ { 1 - r / p } \\alpha ^ { - 1 / p } ) \\\\ & = ( p - r ) \\log r - \\mu \\log \\alpha _ 1 - ( 1 - \\mu ) \\log \\alpha _ 2 \\\\ & = \\mu [ ( p _ 1 - r ) \\log r - \\log \\alpha _ 1 ] + ( 1 - \\mu ) [ ( p _ 2 - r ) \\log r - \\log \\alpha _ 2 ] \\\\ & = \\mu p _ 1 \\log \\lambda ^ { ( p _ 1 ) } ( G ) + ( 1 - \\mu ) p _ 2 \\log \\lambda ^ { ( p _ 2 ) } ( G ) . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\int | \\nabla u _ 1 | ^ 2 = \\int | \\nabla | u _ 1 | | ^ 2 , \\int | \\nabla v _ 1 | ^ 2 = \\int | \\nabla | v _ 1 | | ^ 2 . \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} [ \\mathcal { P } _ 0 , \\alpha _ { \\infty } , H _ { L _ { \\infty } } ] - [ \\mathcal { P } _ 1 , \\alpha _ { \\infty } , H _ { L _ { \\infty } } ] = - \\partial _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ( \\mathcal { G } ) } \\left ( ^ { \\ast } ( q _ K e _ I ) \\right ) . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) : \\ , v \\in e } B ' ( v , e ) = \\sum _ { e \\in E ( G ) : \\ , v \\in e } B ( v , e ) = 1 . \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} \\dot C _ n ( t ) = - n C _ n ( t ) \\forall \\ , n \\ge 3 \\ , . \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} \\chi ( g , h ) = s ( g ) s ( h ) s ( g h ) ^ { - 1 } \\in A . \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} x _ { m + 1 } = A _ m x _ m m \\in \\Z , \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} G ( \\lambda \\vec z ) - G ( \\vec z ) = ( \\frac { 1 } { \\lambda } - 1 ) \\frac { r } { \\| \\vec z \\| _ p } < 0 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\lambda > 1 . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\mathbb { P } [ \\mathbf { D } \\mathbf { x } = \\mathbf { e } _ i ~ | ~ w t ( \\mathbf { x } ) = k ] = 2 ^ { - m } \\left ( 1 - \\sigma ^ k \\right ) \\left ( 1 + \\sigma ^ k \\right ) ^ { m - 1 } \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} H ( z ) & = e ^ { \\bf e } H ( z ) e ^ { - { \\bf e } } + 2 e ( z ) E ( z ) , \\\\ e ^ { - { \\bf e } } H \\otimes x ( \\zeta ) e ^ { \\bf e } & = H \\otimes x ( \\zeta ) + 2 E \\otimes e ( \\zeta ) x ( \\zeta ) . \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} \\min _ { 1 \\leq i \\leq k } \\{ h _ { \\omega _ i } \\} > t _ a - \\eta . \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} D ^ { \\varepsilon } ( x ) = \\mathrm { p r o j } _ { X ^ { \\ast } } ( \\partial ^ { \\varepsilon } F ) ( x , { 0 _ { Y } } ) , \\ \\forall ( \\varepsilon , x ) \\in \\mathbb { R } _ { + } \\times X . \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} E q 3 : \\rho _ { 0 } ( \\rho _ { 1 } ^ { 2 } - 2 h ) + \\frac { \\omega ^ { 2 } } { \\rho _ { 0 } } - \\frac { \\omega } { 2 K _ { 0 } } \\rho _ { 0 } v _ { 0 } = 2 u _ { 0 } - \\mathfrak { S } _ { 0 } \\ \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} M _ 0 = \\begin{pmatrix} z ^ { c _ 1 } & x ^ { a _ 1 } & y ^ { b _ 1 } \\\\ y ^ { b _ 2 } & z ^ { c _ 2 } & x ^ { a _ 2 } \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N - 1 } \\int _ { w _ n } ^ { w _ { n + 1 } } \\frac { f ( z ) - f ( z _ 0 ) } { z - z _ 0 } \\ , d z \\ = \\ 0 . \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\nu ( ( \\alpha , \\beta ) , ( s , t ) , Z ) = \\frac { 1 } { r } \\log \\Phi ( \\alpha + s , \\beta + t , Z ) + ( 1 - \\frac { 1 } { r } ) \\log \\Phi ( \\alpha + \\frac { 1 } { 1 - r } s , \\beta + \\frac { 1 } { 1 - r } t , Z ) - \\log \\Phi ( \\alpha , \\beta , Z ) , \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} \\sigma ^ c _ { \\check F } ( f ) ( g ) = F ( c ( g ) ) f ( g ) \\ \\ \\ \\ f \\in C _ c ( G ) \\ \\ g \\in G . \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} \\delta f _ { s t } : = f _ t - f _ s . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\chi ( U _ { a / c } ) = \\chi _ { \\alpha ( N , c ) } ( - 1 ) , \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} F _ 1 = f \\circ h _ 1 , \\ , \\ , F _ 2 ( \\prod _ { i = 1 } ^ l ( T ^ i _ 0 , T ^ i _ 1 , T ^ i _ 2 ) ) = F _ 1 ( ( A _ 0 , A _ 1 , \\ldots , A _ { 3 ^ l - 1 } ) ) , \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} I ( x , y , z ) = I ( x , y ) \\cap I ( y , z ) \\cap I ( z , x ) , \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} M _ d = A _ d ( I ) \\sqcup B _ d ( I ) \\sqcup C _ d ( I ) \\sqcup D _ d ( I ) . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{gather*} e ^ { \\beta ^ 2 \\log \\varepsilon } \\int _ a ^ b e ^ { \\beta V _ { \\varepsilon } ( u ) } \\ , d u \\longrightarrow M _ { \\beta } ( a , b ) , \\\\ { \\bf { E } } [ M _ { \\beta } ( a , b ) ] = | b - a | . \\end{gather*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } c _ { i _ p , j _ p } ^ { ' } z _ { i _ p , j _ p } \\prod \\limits _ { \\varepsilon _ { r p } = - 1 } z _ { i _ r , j _ r } = c _ { i _ p , j _ p } \\prod \\limits _ { \\varepsilon _ { r p } = 1 } z _ { i _ r , j _ r } , \\ ; p \\in V , \\ ; V \\subset \\{ s + 1 , \\ldots , l \\} , \\\\ z _ { i _ { p } , j _ { p } } = 0 \\ ; \\ ; p \\in \\{ s + 1 , \\ldots l \\} \\setminus V , \\\\ z _ { i , j } = 0 , \\ ; ( i , j ) \\notin \\{ ( i _ 1 , j _ 1 ) , \\ldots , ( i _ l , j _ l ) \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} P ( x ) = \\frac { a } { q } x ^ d + \\ldots + \\xi _ 1 x , \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} h ( x , y ) = \\sum _ { i = 1 } ^ { \\infty } f _ i ( x ) g _ i ( y ) , \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} \\Big ( \\bigcup _ { g \\in G } { \\langle x \\rangle } ^ g \\Big ) \\bigcap \\Big ( \\bigcup _ { g \\in G } { \\langle x t \\rangle } ^ g \\Big ) = 1 . \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} D ( f g ) = D ( g f ) = ( D f ) g + f ( D g ) . \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} \\Theta ( a , b , c ) : = \\{ \\theta _ q \\ , | \\ , q \\in Q ( a , b , c ) \\} , \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\kappa _ n ( 1 / 2 ) = \\frac { 3 2 ^ { n / 2 } } { 8 } \\Gamma \\bigl ( \\frac { n } { 2 } \\bigr ) \\frac { ( 2 ^ { 2 - n } - 1 ) } { 1 - n / 2 } \\bigl ( \\frac { 2 } { \\pi } \\bigr ) ^ { 1 - n / 2 } \\ , \\xi ( n - 1 ) . \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} V ^ { r , s } : = \\textstyle \\bigotimes ^ { r } \\C ^ n \\otimes \\textstyle \\bigotimes ^ { s } \\C ^ { * n } \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow 1 } S _ q ( \\rho ) = S ( \\rho ) . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dfrac { k n \\gamma } { \\ell } \\\\ \\dfrac { m d } { \\ell } - \\frac { k m \\gamma } { \\ell } - m \\\\ k \\gamma \\\\ d - \\ell - k \\gamma \\end{bmatrix} \\mbox { w i t h } k = 0 , 1 , \\hdots , \\left \\lfloor ( d - \\ell ) / \\gamma \\right \\rfloor , \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} \\sum _ { k = i _ r } ^ p ( - 1 ) ^ k \\binom { k - 1 } { i _ r - 1 } \\binom { p - 1 } { k - 1 } = ( - 1 ) ^ { i _ r } \\binom { p - 1 } { i _ r - 1 } \\sum _ { l = 0 } ^ { p - i _ r } ( - 1 ) ^ l \\binom { p - i _ r } { l } = ( - 1 ) ^ { i _ r } \\delta _ { p , { i _ r } } . \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} u _ { 0 } = U ^ { \\ast } ( p ) , \\tau _ { 0 } = u _ { 1 } = U _ { \\mathbf { \\tau } } ^ { \\ast } ( p ) , w _ { 0 } = U _ { \\mathbf { \\nu } } ^ { \\ast } ( p ) \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} V \\tilde \\pi _ 1 ( U ^ * ) V ^ * e _ { \\sigma _ 2 ^ k ( m _ { j '' } ) } & = V \\tilde \\pi _ 1 ( U ^ * ) e _ { \\sigma _ 2 ^ k ( m _ { j ' } ) } = V e _ { \\sigma _ 1 ^ { k } ( n _ { j } ) } = e _ { \\sigma _ 1 ^ { k } ( n _ { j } ) } = \\tilde \\pi _ 2 ( U ^ * ) e _ { \\sigma _ 2 ^ k ( m _ { j '' } ) } \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} \\langle I _ { 0 } ^ { \\prime } ( U _ { 0 } , V _ { 0 } ) , ( \\varphi , \\psi ) \\rangle = \\langle I _ { 0 } ^ { \\prime } ( u _ { \\lambda _ { n } } , v _ { \\lambda _ { n } } ) , ( \\varphi , \\psi ) \\rangle - ( u _ { \\lambda _ { n } } - U _ { 0 } , \\varphi ) _ { E _ { 1 } } - ( v _ { \\lambda _ { n } } - V _ { 0 } , \\psi ) _ { E _ { 2 } } \\\\ - \\int _ { \\mathbb { R } ^ { N } } ( f _ { 1 } ( u _ { n } ) - f _ { 1 } ( U _ { 0 } ) ) \\varphi \\ , \\mathrm { d } x - \\int _ { \\mathbb { R } ^ { N } } ( f _ { 2 } ( v _ { n } ) - f _ { 2 } ( V _ { 0 } ) ) \\psi \\ , \\mathrm { d } x = o _ { n } ( 1 ) , \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} N ( m ) = \\frac { 1 } { 2 ^ { m + 1 } } \\sum _ { i = 0 } ^ { m } ( - 1 ) ^ { i } \\sum _ { a _ { 1 } , \\cdots , a _ { i + 1 } = 1 } ^ { \\infty } \\delta _ { a _ { 1 } + \\cdots + a _ { i + 1 } , m + 1 } \\prod _ { j = 1 } ^ { i + 1 } \\frac { ( 2 a _ { j } ) ! } { a _ { j } ! } , \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} [ \\jmath _ { ! * } \\jmath ^ * \\beta ] = [ \\beta ' ] = [ \\imath _ * \\imath ^ { ! * } \\beta ' ] + [ \\jmath _ { ! * } \\jmath ^ * \\beta ' ] \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} F ( T _ i \\vec x ^ k ) - F ( \\vec x ^ k ) = - | \\bar p _ i ^ k | , i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } f _ M \\Phi ^ { } _ N = \\Phi ^ { \\operatorname { g s } } \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} \\Lambda _ { - } ( X ) = \\sum _ { i \\in \\mathcal { I } } \\left ( ( A _ i - X ) ^ T \\otimes ( \\Phi ^ { - } _ { i , T } ( X ) ) \\right ) + \\sum _ { i \\in \\mathcal { I } ^ + } \\left ( ( A _ i - \\overline { X } _ i ) ^ T \\otimes ( \\Phi ^ { - } _ { i , T } ( X ) ) \\right ) ( \\overline { M } ^ - - I _ 4 ) , \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} N = \\binom { n + 1 } { 2 } + r _ 1 + \\sum _ { i = 1 } ^ { n - 1 } ( n - i ) ( r _ i + 1 ) . \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( \\mathbf { H } \\right ) = \\left \\langle \\mathcal { E } \\left ( \\mathbf { H } \\right ) , \\wedge _ { s } , \\vee _ { s } , ^ { \\prime } , ^ { \\sim } , \\mathbb { O } , \\mathbb { I } \\right \\rangle , \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} N _ 4 = - 2 c _ 6 \\int _ { \\mathbb R } [ \\zeta _ k \\nu _ { n _ k } + \\varphi _ k \\nu _ { n _ k } ] D ( b _ k ) + \\varphi _ k \\nu _ { n _ k } D ( \\zeta _ k \\nu _ { n _ k } ) d x . \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} z = - ( k _ 1 a _ 1 + \\cdots + k _ M a _ M ) , \\ ; k _ 1 \\cdots k _ M \\in \\mathbb { N } , \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\dot { q } _ t ^ { \\Lambda , N } ( \\eta ) = \\int _ { \\Gamma _ 0 } \\left ( L ^ { \\dagger , \\sigma } _ \\vartheta R _ t ^ { \\Lambda , N } \\right ) ( \\eta \\cup \\xi ) \\lambda ( d \\xi ) , \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\vec { g } _ { i } = \\frac { \\sum _ { j \\in B _ { i } } \\vec { f } _ { j } } { \\norm * { \\sum _ { j \\in B _ { i } } \\vec { f } _ { j } } } . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} \\omega ^ w _ e : = 1 \\{ U _ e \\leq \\mathbb { P } _ { \\Lambda , w , \\beta , H } ^ { | \\vec { h } | } ( \\omega _ e = 1 | \\omega _ { G _ t } = \\omega ^ w _ { G _ t } ) \\} . \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } G ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) , W ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , W _ { x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , \\bar { u } ( s ) ) \\\\ = \\min \\limits _ { u \\in U } G \\left ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) , W ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , W _ { x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) , u \\right ) . \\end{array} \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} & \\Delta _ k ^ n X ^ { ( l ) } = X ^ { ( l ) } _ { T _ k } - X ^ { ( l ) } _ { T _ { k - 1 } } , \\\\ & \\Delta _ k ^ { n , l , - } X ^ { ( l ) } = X ^ { ( l ) } _ { T _ { k - 1 } } - X ^ { ( l ) } _ { \\tau _ - ^ { ( l ) } ( T _ { k - 1 } ^ n ) } , \\\\ & \\Delta _ k ^ { n , l , + } X ^ { ( l ) } = X ^ { ( l ) } _ { \\tau _ + ^ { ( l ) } ( T _ { k } ^ n ) } - X ^ { ( l ) } _ { T _ k } \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{gather*} L ( h _ 4 , 3 ) = L \\big ( \\eta ( 3 \\tau ) ^ 8 , 3 \\big ) = \\frac 8 3 L ( f _ { 3 6 } , 1 ) ^ 3 . \\end{gather*}"}
-{"id": "3587.png", "formula": "\\begin{align*} : = [ - 2 \\sqrt { d } , 2 \\sqrt { d } ] \\setminus ( - 1 / l , 1 / l ) , \\ l > 1 . \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} \\hat { \\phi } _ { \\mathbf h , p } ( x ; y ) = \\begin{cases} \\mathbf 1 _ { \\Z _ p } ( x _ 1 ) \\mathbf 1 _ { \\Z _ p } ( x _ 2 ) \\mathbf 1 _ { \\Z _ p } ( x _ 3 ) \\mathbf 1 _ { p \\Z _ p } ( y _ 1 ) \\mathbf 1 _ { \\Z _ p ^ { \\times } } ( y _ 2 ) & p \\mid N / M , \\\\ \\mathbf 1 _ { p \\Z _ p } ( x _ 1 ) \\mathbf 1 _ { \\Z _ p } ( x _ 2 ) \\mathbf 1 _ { \\Z _ p } ( x _ 3 ) \\mathbf 1 _ { p ^ 2 \\Z _ p } ( y _ 1 ) \\mathbf 1 _ { \\Z _ p ^ { \\times } } ( y _ 2 ) \\underline { \\chi } _ p ^ { - 1 } ( y _ 2 ) & p \\mid M . \\end{cases} \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\Lambda = \\| [ X , \\ln ( Y ) ] \\| _ 1 & \\leq 9 S ( p ) = 9 ( - p \\ln p - ( 1 - p ) \\ln ( 1 - p ) ) . \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} \\widetilde { A } _ { i , r } : = \\underline { A _ { i , r } } \\in \\mathcal { Y } _ t . \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\int _ 0 ^ R F ( s ) T ^ \\prime ( s ) \\ d s = F ( R ) T ( R ) - 0 - \\int _ 0 ^ R F ^ \\prime ( s ) T ( s ) \\ d s \\geq 0 . \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} v ^ a \\mapsto \\big ( \\Sigma v ^ a \\big ) ^ { A } { } _ { B } = & \\ , \\ , { \\mathbb Y } _ a { } ^ { A } { } _ { B } v ^ a + { \\mathbb Z } _ a { } ^ b { } ^ A { } _ B \\big ( ( \\nabla _ b v ^ a ) _ o + \\tfrac { 1 } { 6 } \\delta _ b { } ^ { a } \\nabla _ c v ^ c \\big ) \\\\ & - \\tfrac { 1 } { 3 } { \\mathbb W } ^ { A } { } _ { B } \\nabla _ c v ^ c - { \\mathbb X } ^ a { } ^ A { } _ B ( \\tfrac { 1 } { 3 } \\nabla _ a \\nabla _ b + P _ { a b } ) v ^ b . \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} t r _ M ( | A _ n | T ) = \\int _ { \\sigma ( A ) } | s _ n | \\ : d \\mu _ T ^ { ( A ) } \\ : . \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} a : = \\sum _ { j = 1 } ^ { \\deg ( v _ 1 ) } F _ j ( v _ 1 ) b : = \\sum _ { j = 1 } ^ { \\deg ( v _ 2 ) } F _ j ( v _ 2 ) . \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} y ^ { k + 1 } & = \\underset { y \\in C } { \\textup { a r g m i n } } \\bigg \\{ \\lambda \\langle A y ^ k , y - y ^ k \\rangle + \\dfrac { 1 } { 2 } \\| y - x ^ { k } \\| ^ 2 \\bigg \\} , \\\\ & = \\underset { y \\in C } { \\textup { a r g m i n } } \\bigg \\{ \\dfrac { 1 } { 2 } \\| y - ( x ^ { k } - \\lambda A y ^ k ) \\| ^ 2 \\bigg \\} \\\\ & = P _ C ( x ^ { k } - \\lambda A y ^ k ) . \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} s _ V ( \\sigma ( \\gamma _ 1 , \\ldots , \\gamma _ p ) ) = s _ V ( \\sigma ( 1 _ { s ( \\gamma _ 1 ) } , \\gamma _ 2 , \\ldots , \\gamma _ p ) ) , \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q \\ , | \\ , a , b ) = \\prod \\limits _ { k = 0 } ^ \\infty \\eta _ { M - 1 , N } ( q \\ , | \\ , \\hat { a } _ i , b _ 0 + k a _ i ) , \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} F S ( e ) = \\frac { p ^ { 2 e } - r _ e ^ 2 } { n } + r _ e . \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} | f _ u | = | a | E R ^ { 1 / 2 } \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} \\sigma _ { p , q } = \\frac { p + \\sqrt { p ^ { 2 } + 4 q } } { 2 } , \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\partial _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) } ( \\zeta ^ { ( j ) } _ { L _ { \\infty } / K } ) = - C _ { L _ { \\infty } / K } - U ' _ { L _ { \\infty } / K } + M _ { L _ { \\infty } / K } \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\left ( S ^ { m + \\chi ^ I } ( S ^ * ) ^ { n + \\chi ^ I } T ' \\right ) & = \\sum _ { q \\in \\N ^ N } \\langle T ' \\delta _ q , S ^ { n + \\chi ^ I } ( S ^ * ) ^ { m + \\chi ^ I } \\delta _ q \\rangle \\\\ & = \\sum _ { q \\geq m + \\chi ^ I } \\langle T ' \\delta _ q , \\delta _ { q - m + n } \\rangle \\\\ & = \\sum _ { q \\geq \\chi ^ I } T _ { n + q , m + q } ' , \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} \\lim _ { x \\to + \\infty } \\frac { \\psi ^ { ( k ) } ( x ) } { \\varphi _ 2 ^ { ( k + 1 ) } ( x ) } = 1 , \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} | \\psi ( \\chi ) | ^ 2 & = | \\psi _ { } ( \\chi ) + \\psi _ { } ( \\chi ) | ^ 2 + 2 \\left \\{ \\psi _ { } ( \\chi ) \\psi _ { } ( \\overline { \\chi } ) + \\psi _ { } ( \\chi ) \\psi _ { } ( \\overline { \\chi } ) \\right \\} + | \\psi _ { } ( \\chi ) | ^ 2 . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} C ( x ) = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n + 1 } \\binom { 2 n } { n } x ^ n = \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 x } . \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\P _ x ( \\tau _ \\vartheta > n ) & = \\P _ 0 ( \\tau _ \\vartheta > n ) + \\sum _ { y \\in E \\setminus ( E \\star x ) } \\sum _ { k = 1 } ^ n \\P _ x ( X ( t _ 1 ) = y , \\ ; t _ 1 = k ) \\P _ y ( \\tau _ \\vartheta > n - k ) \\\\ & \\geq \\P _ 0 ( \\tau _ \\vartheta > n ) + \\sum _ { y \\in E \\setminus ( E \\star x ) } \\sum _ { k = 1 } ^ n \\P _ x ( X ( t _ 1 ) = y , \\ ; t _ 1 = k ) \\P _ y ( \\tau _ \\vartheta > n ) \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} g _ { i } = \\frac { \\sum _ { j \\in \\Lambda _ { i } } f _ { j } } { \\norm { \\sum _ { j \\in \\Lambda _ { i } } f _ { j } } } = \\sqrt { \\frac { d } { k ( d + 1 - k ) } } \\sum _ { j \\in \\Lambda _ { i } } f _ { j } , \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\partial _ x M ^ { n - 1 } ( t ) = \\frac { \\langle w \\rangle _ { f ^ { n - 1 } _ t , \\varphi ' } \\langle 1 \\rangle _ { f ^ { n - 1 } _ t , \\varphi } - \\langle w \\rangle _ { f ^ { n - 1 } _ t , \\varphi } \\langle 1 \\rangle _ { f ^ { n - 1 } _ t , \\varphi ' } } { \\langle 1 \\rangle _ { f ^ { n - 1 } _ t , \\varphi } ^ 2 } \\ , , \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} \\Psi _ 2 : = \\Psi _ 1 \\circ \\Theta _ { h ' } ^ { i d } + h \\circ \\kappa _ { \\Psi _ 1 } - \\Psi _ 1 \\circ \\Phi _ 2 \\circ ( \\wedge ^ 2 \\Psi _ 1 ) , \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\Xi ( F ) = \\mathcal { P } ^ - + \\left [ \\begin{array} { c c } ( \\hat { \\lambda } + 2 \\hat { \\mu } ) g _ n ( F ) & \\hat { \\lambda } g _ t ( F ) \\\\ \\hat { \\mu } g _ t ( F ) & \\hat { \\mu } g _ n ( F ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} \\dd { x } { t } = \\sum _ { i \\to i ' \\in E } k _ { i \\to i ' } \\ , x ^ { \\tilde y ( i ) } \\big ( y ( j ) - y ( i ) \\big ) . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} & ( T _ { \\rho } \\cdot ( T _ { \\mu } \\cdot X ) ) ( A , w ) \\\\ & = R ( \\rho ) ( T _ { \\mu } \\cdot X ) ( R ( \\rho _ { w } ) ^ { - 1 } A , \\rho ( w ) ) R ( \\rho ) ^ { - 1 } \\\\ & = R ( \\rho ) R ( \\mu ) X ( R ( \\mu _ { \\rho ( w ) } ) ^ { - 1 } R ( \\rho _ { w } ) ^ { - 1 } A , \\mu ( \\rho ( w ) ) ) R ( \\mu ) ^ { - 1 } R ( \\rho ) ^ { - 1 } . \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} E _ T : = L ^ { \\infty } ( 0 , T ; L ^ 2 ( \\R ^ d ) ) \\cap L ^ 2 ( 0 , T ; \\dot H ^ 1 ( \\R ^ d ) ) . \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} F \\left ( \\frac { 1 } { p } \\right ) & = 1 + O \\left ( p ^ { - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} C _ { \\Gamma , \\chi } = \\left \\{ \\tfrac { a } { c } \\in C _ { \\Gamma } \\ ; : \\ ; q \\mid \\frac { N } { \\gcd ( c , N / c ) } \\right \\} . \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} a = I + \\sum _ { j = 1 } ^ \\infty c _ j \\mathsf { f } _ j \\otimes \\mathsf { f } _ j , \\| \\mathsf { f } _ j \\| _ \\infty = 1 , \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\hat { \\Theta } ( r ) = ( 1 , p ( r ) , K _ { 1 } ( r ) ) \\hat { X } ( r ) + \\hat { L } ( r ) , \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} f _ 1 & = X ^ { N - 1 } + a _ 1 X ^ { N - 2 } + \\cdots + a _ { 4 t - 1 } X ^ { N - 4 t } + g _ 1 , \\\\ f _ 2 & = X ^ { N - 1 } + a ' _ 1 X ^ { N - 2 } + \\cdots + a ' _ { 4 t - 1 } X ^ { N - 4 t } + g _ 2 . \\\\ \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} Z ( t ) = \\exp \\left ( \\nu \\{ F ( X ^ { 1 / 2 , \\mu } ( t ) ) - F ( \\bar { x } ) - A ( t ) \\} - \\frac { \\nu ^ 2 } { 2 } \\langle M \\rangle ( t ) \\right ) . \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} \\partial _ y \\psi ( t , x , y ) = \\hat { h } _ 1 \\big ( t , x , \\psi ( t , x , y ) \\big ) , \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} w _ i : = \\langle E _ i , \\xi \\rangle \\textrm { a n d } \\ , \\bar w _ i : = \\langle E _ i , \\star \\xi \\rangle . \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} \\dot { \\phi } ( n ) - \\dot { \\phi } ( n + 2 ) = ( n + 1 ) ^ { - N - \\frac { 1 } { 2 } } , \\quad \\forall n \\in \\N . \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} f ( x ) \\leq & m 2 ^ { 1 - s } f ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) ^ s [ f ( b ) - m f ( a ) ] = p _ { 1 } ( x ) . \\\\ g ( x ) \\leq & m 2 ^ { 1 - s } g ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) ^ s [ g ( b ) - m g ( a ) ] = p _ { 2 } ( x ) . \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} D _ r ( n ) \\equiv ( - 1 ) ^ n \\sum _ { j = r } ^ { d - 1 } ( - 1 ) ^ j { j \\choose r } ( n ) _ j \\pmod { d } \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ 2 \\theta _ k ^ 2 > C \\varepsilon ^ 2 \\sqrt { D _ \\varepsilon } , \\forall \\varepsilon \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} K = Q \\mathcal { P } ^ - Q ^ T , ~ ~ ~ ~ \\mathcal { P } ^ - = \\left [ \\begin{array} { c c } ( \\lambda ^ - + 2 \\mu ^ - ) & 0 \\\\ 0 & \\mu ^ - \\end{array} \\right ] , ~ ~ Q = [ \\bar { \\mathbf { n } } , \\bar { \\mathbf { t } } ] , \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\frac { d } { d t } g ( \\rho + t \\sigma _ 1 ) = - g ( \\rho + t \\sigma _ 1 ) L ( \\sigma _ 1 ) g ( \\rho + t \\sigma _ 1 ) . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} L ( s , \\pi _ \\infty \\boxtimes \\chi _ \\infty , \\varrho _ { 5 } ) = \\Gamma _ \\R ( s + \\epsilon ) \\Gamma _ \\C ( s + \\ell + m - 1 ) \\Gamma _ \\C ( s + \\ell - 2 ) , \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} ( v _ 1 , \\vartheta , \\partial _ \\eta w ) | _ { \\eta = 0 } = \\mathbf { 0 } , \\lim _ { \\eta \\rightarrow + \\infty } ( v _ 1 , \\vartheta , w ) ( \\tau , \\xi , \\eta ) = \\mathbf { 0 } . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} | \\nabla ^ { k , \\mathbb { C } ^ m } \\hat { g } ^ \\bullet _ \\infty ( 0 ) | _ { g _ { \\mathbb { C } ^ m } } = 1 . \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} \\sigma ^ K _ K = \\frac { 1 - \\frac { \\theta _ { K - 1 } } { \\theta _ 0 } ( 1 - \\theta _ 0 ) } { 1 + \\frac { \\theta _ { K - 1 } } { \\theta _ 0 \\mu } } \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ 4 \\log [ ( 1 - | \\zeta _ k | ^ 2 ) | f ' ( \\zeta _ k ) | ] + \\sum _ { k = 1 } ^ 4 \\sum _ { { \\substack { l = 1 \\\\ l \\neq k } } } ^ 4 \\delta _ k \\delta _ l \\log \\left | \\frac { 1 - \\overline { \\zeta } _ k \\zeta _ l } { \\zeta _ k - \\zeta _ l } \\right | \\le 2 \\log \\frac { 4 \\rho ^ 2 } { 1 - \\rho ^ 2 } . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} \\partial _ t u _ 1 + ( u _ 1 \\partial _ x + u _ 2 \\partial _ y ) u _ 1 = \\partial _ \\tau \\hat u _ 1 + \\hat u _ 1 \\partial _ \\xi \\hat u _ 1 + \\nu \\hat h _ 1 \\partial _ \\eta \\hat h _ 1 \\partial _ \\eta \\hat u _ 1 . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} g ^ { \\omega } _ s ( n ) : = \\frac { n } { e ^ { \\omega ( n ) \\cdot ( \\log n ) ^ { 1 - 1 / s } } } . \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} Q ( x ) = \\sum _ { m _ 1 , . . . , m _ N = 0 } ^ \\infty \\delta _ { \\omega _ { x _ 1 } ( m _ 1 ) } \\otimes \\cdots \\otimes \\delta _ { \\omega _ { x _ N } ( m _ N ) } \\otimes A e _ { ( m _ 1 , . . . , m _ N ) } , \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } c ( n ) q ^ n | T _ { p ^ 2 } = \\sum _ { n \\geq 1 } \\left ( c ( p ^ 2 n ) + \\left ( \\frac { ( - 1 ) ^ k n } { p } \\right ) \\chi ( p ) p ^ { k - 1 } c ( n ) + \\chi ( p ) ^ 2 p ^ { 2 k - 1 } c ( n / p ^ 2 ) \\right ) q ^ n , \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} J ( A _ r , B _ r , C _ r ) = 1 / 1 2 = 0 . 0 8 3 3 \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} \\begin{cases} - \\mu \\bigl ( 3 \\log \\varepsilon - \\kappa + \\log | u - v | \\bigr ) + O ( \\varepsilon ) , \\ ; , \\\\ - \\mu \\bigl ( 4 \\log \\varepsilon - 2 \\kappa \\bigr ) + O ( \\varepsilon ) , \\ ; \\end{cases} \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} \\mathcal { \\dot M } ( t ) \\leq 2 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } \\Delta ^ 2 _ x \\psi ( x , y ) K ( t , x , y ) \\ , d x d y & \\\\ - \\frac { 4 p } { p + 1 } \\sum _ { \\substack { \\mu , \\nu , \\iota = 1 } } ^ N \\gamma _ { \\mu \\nu } \\int _ { \\R ^ d \\times \\R ^ d } | u _ \\mu ( x ) | ^ { p + 1 } | u _ \\nu ( x ) | ^ { p + 1 } m _ { u _ \\iota } ( y ) \\Delta _ x \\psi ( x , y ) \\ , d x d y , & \\\\ \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\widehat { h } = \\begin{cases} 0 , & 1 \\leq h \\leq m ; \\\\ 1 , & m + 1 \\leq h \\leq m + n . \\end{cases} \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} ( V \\xi ) ( s , t ) = \\xi ( s t , t ) \\Delta ( t ) ^ { 1 / 2 } , \\quad \\xi \\in L ^ 2 ( G ) \\otimes L ^ 2 ( G ) , \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} & X = v v ^ T \\geq 0 \\\\ & F _ 0 = \\left ( \\frac { 1 } { 2 ^ 2 } , \\cdots , \\frac { \\mathbf 1 _ { n - 1 } \\mathbf 1 ^ T _ { n - 1 } } { ( n ) ^ 2 } , \\frac { \\mathbf 1 _ { n } \\mathbf 1 ^ T _ { n } } { ( n + 1 ) ^ 2 } \\right ) \\\\ & F _ i = \\left ( e _ { n , i } , e _ { n - 1 , i } , \\cdots , e _ { n - i + 1 , i } , 0 _ { n - i } , \\cdots , 0 _ { 1 } \\right ) \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} A _ { \\xi } ' = A _ { \\xi } \\circ C _ { \\gamma ' } . \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) z ( \\lambda ) = \\begin{bmatrix} M ( \\lambda ) & K _ 2 ( \\lambda ) ^ T \\\\ K _ 1 ( \\lambda ) & 0 \\end{bmatrix} \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T h ( \\lambda ) \\\\ - X ( \\lambda ) h ( \\lambda ) \\end{bmatrix} = 0 , \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} \\mathfrak { d } _ 2 ^ { m + 1 } \\dot { \\phi } ( n ) = ( - 1 ) ^ { m + 1 } \\int _ 0 ^ 2 \\cdots \\int _ 0 ^ 2 f ^ { ( m + 1 ) } ( n + t _ 1 + \\cdots + t _ { m + 1 } ) \\ , d t _ 1 \\cdots d t _ { m + 1 } , \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} \\varphi = \\begin{bmatrix} x ^ 2 & 0 & x ^ 4 \\\\ y ^ 2 & x ^ 2 & y ^ 4 \\\\ z ^ 2 & y ^ 2 & z ^ 4 \\\\ 0 & z ^ 2 & x ^ 3 z \\end{bmatrix} . \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} V ( f ) = \\sum _ { j = 1 } ^ { n } V _ j ( f ( \\omega _ j ) ) . \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} P V ( e ) ~ \\leq ~ 1 ~ = ~ V ( e ) . \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} & f _ 1 ( z ^ 2 ) + f _ 3 ( z ^ 2 ) z = z ^ 2 \\\\ & f _ 2 ( z ^ 2 ) z ^ * + f _ 4 ( z ^ 2 ) = \\bar z ^ 2 \\ ; . \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} r - ( \\vec z ^ * , | \\vec v | ) = G ( \\vec z ^ * ) \\| \\vec z ^ * \\| _ p \\le G ( \\vec 1 ) \\| \\vec z ^ * \\| _ p = \\frac { \\| \\vec z ^ * \\| _ p } { \\| \\vec 1 \\| _ p } ( r - \\| \\vec v \\| _ 1 ) < 0 , \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} \\nabla ^ { \\hat { g } } _ { \\mathbf { f } } \\hat { \\eta } _ \\infty = 0 . \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} S _ 1 ( z \\ , | \\ , a ) = 2 \\sin ( \\pi w / a ) . \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} S _ i & : = \\{ x \\in \\Omega : u ( x ) = u _ i \\} , i = 1 , \\dots , d , \\\\ T _ i & : = \\{ x \\in \\Omega : u _ i < u ( x ) < u _ { i + 1 } \\} , i = 1 , \\dots , d - 1 . \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} \\tilde { T } = \\left ( S _ { k + 1 } ^ { m _ { k + 1 } } ( S _ { k + 1 } ^ * ) ^ { n _ { k + 1 } } - \\frac { 1 } { q _ { k + 1 } } S _ { k + 1 } ^ * S _ { k + 1 } ^ { m _ { k + 1 } } ( S _ { k + 1 } ^ * ) ^ { n _ { k + 1 } } S _ { k + 1 } \\right ) T . \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} S ( X , X ' ) = \\Sigma _ 1 - \\Sigma _ { 2 1 } - \\Sigma _ { 2 2 } + \\Sigma _ 3 , \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} \\mathbf g _ p ( h \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } \\varpi _ p ) = \\begin{cases} \\xi _ 1 ( p ) ^ { - 2 } \\xi _ 2 ( \\gamma t ) | p ^ { - 1 } | _ p ^ { 1 / 2 } = p ^ { 1 / 2 } \\xi _ 1 ( p ) ^ { - 2 } \\underline { \\chi } _ p ( \\gamma ) \\underline { \\chi } _ p ( h ) & \\gamma \\delta \\equiv 1 \\pmod p , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ( \\tau , b ) ^ { - k } \\bigr ] & = \\prod \\limits _ { l = 0 } ^ { k - 1 } \\Bigl [ \\frac { \\Gamma \\bigl ( ( - ( l + 1 ) + b _ 0 ) / \\tau \\bigr ) \\ , \\Gamma \\bigl ( ( - ( l + 1 ) + b _ 0 + b _ 1 + b _ 2 ) / \\tau \\bigr ) } { \\Gamma \\bigl ( ( - ( l + 1 ) + b _ 0 + b _ 1 ) / \\tau \\bigr ) \\ , \\Gamma \\bigl ( ( - ( l + 1 ) + b _ 0 + b _ 2 ) / \\tau \\bigr ) } \\Bigr ] , \\ ; k < b _ 0 . \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\Phi ( t ) = \\left ( \\begin{array} { r c l } e ^ { \\frac { 1 } { 2 } t } \\cos t & & e ^ { - t } \\sin t \\\\ - e ^ { \\frac { 1 } { 2 } t } \\sin t & & e ^ { - t } \\cos t \\end{array} \\right ) , \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} \\omega _ { \\ ! D } ' ( s _ 1 , \\dots , s _ k ) _ { \\upharpoonright U _ k } = \\omega _ { \\mathfrak { U } } ( s _ 1 , \\dots s _ k ) \\cdot ( { \\pi } _ { 1 , k } ) ^ * \\xi _ { 1 } ^ { - k } \\cdot ( { \\pi } _ { 2 , k } ) ^ * \\xi _ { 2 } ^ { - ( k - 1 ) } \\cdots ( { \\pi } _ { k - 1 , k } ) ^ * \\xi _ { k - 1 } ^ { - 2 } \\cdot \\xi _ { k } ^ { - 1 } . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} \\theta _ { p + 1 } \\nabla ^ { \\rm D S } \\big ( P _ j ^ p { \\nabla } _ { \\mathfrak { U } ^ 2 } ^ j ( s _ { U ^ 2 } ) \\big ) & = P _ j ^ p \\theta _ { p + 1 } \\nabla ^ { \\rm D S } \\big ( { \\nabla } _ { \\mathfrak { U } ^ 2 } ^ j ( s _ { U ^ 2 } ) \\big ) + { \\nabla } _ { \\mathfrak { U } ^ 2 } ^ j ( s _ { U ^ 2 } ) d P _ j ^ p ( \\xi _ { p + 1 } ^ 2 ) \\\\ & = P _ j ^ p \\theta _ { p + 1 } \\beta _ { j , p } { \\nabla } _ { \\mathfrak { U } ^ 2 } ^ { j + 1 } ( s _ { U ^ 2 } ) + { \\nabla } _ { \\mathfrak { U } ^ 2 } ^ j ( s _ { U ^ 2 } ) d P _ j ^ p ( \\xi _ { p + 1 } ^ 2 ) , \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} | \\mathcal { D } _ { 2 n } ^ 1 ( 2 4 1 3 , 3 1 4 2 ) | = C ( 2 , n ) . \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} | ( f _ i & \\pm \\varphi ) ( x ) - ( f _ i \\pm \\varphi ) ( u ) | = \\\\ & = \\begin{cases} | a ^ x _ i - ( g ^ { + } _ i ( u ) - \\frac 1 2 \\pm \\frac 1 2 ) | \\leq 1 \\leq d ( x , u ) \\\\ \\\\ | ( g ^ { + } _ i ( u ) - \\frac 1 2 ) - ( g ^ { + } _ i ( u ) - \\frac 1 2 \\pm \\frac 1 2 ) | = \\frac 1 2 \\leq d ( x , u ) , \\end{cases} \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} V _ { k , r } = W _ { k , r } + O \\ ( r \\frac { N ( \\log N ) ^ { \\nu - 1 } } { x _ k } \\ ) , \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\varphi ( x _ 0 ) = 0 , d \\varphi ( x _ 0 ) = i d : T _ { x _ 0 } M \\rightarrow T _ { x _ 0 } M \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} h _ { t o p } ( R _ \\varphi ( a ) = t _ a . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} M _ { \\sigma _ d } = \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\alpha ^ q \\end{pmatrix} \\sim \\begin{pmatrix} 0 & - b \\\\ 1 & - a \\end{pmatrix} = M _ { \\sigma ' } \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} p _ G ^ 1 ( M ) = \\sum _ { A \\subseteq E } p _ G ( M \\backslash A ) . \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\sum _ { | w | \\geq 1 } \\gamma _ { a , w } ' ( t ) ~ w = - \\sum _ { | w | \\geq 1 } \\lambda _ { a , w } ( t ) ~ H ( w ) , a \\in Q _ 1 . \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} L = \\left ( \\begin{array} { c c c } \\sqrt { \\frac { \\theta q } { 2 R } } ( P - q ) & 0 & \\frac { \\theta } { 2 } \\sqrt { P + ( 1 - 2 a ) q } \\\\ 0 & \\sqrt \\frac { q } { a } ( P - q ) & \\sqrt { a q } \\theta \\\\ - \\sqrt { \\frac { \\theta q } { 2 R } } ( P - q ) & 0 & \\frac { \\theta } { 2 } \\sqrt { P + ( 1 - 2 a ) q } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\check { \\mathbb { L } } ( P _ 1 , P _ 2 , P _ 3 , P _ 4 ) = \\check { C } \\Phi _ { 1 4 } \\Psi ' _ { 2 4 } \\Psi _ { 1 3 } \\Phi _ { 2 3 } \\ ; . \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} \\bold { Q } ( a ) = \\zeta ^ { Q ( a ) } , & & \\forall a \\in A , \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} \\mathcal C : = \\begin{cases} \\Re ( s + z + u + v ) > 1 , & \\ \\Re ( s + z ' + u + w ) > 1 , \\\\ \\Re ( z + z ' + v + w ) > 1 , & \\ \\Re ( 2 s + 2 u ) > 1 \\end{cases} \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{gather*} 1 - \\l = \\left ( \\frac { \\theta _ 4 } { \\theta _ 3 } \\right ) ^ 4 = \\ ( \\frac { \\eta ( \\tau / 2 ) ^ 2 \\eta ( 2 \\tau ) } { \\eta ( \\tau ) ^ 3 } \\ ) ^ 8 . \\end{gather*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\underbrace { x _ s x _ t \\ \\cdots } _ { m _ { s t } } = \\underbrace { x _ t x _ s \\ \\cdots } _ { m _ { s t } } , \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} a : = \\sum _ { j = 1 } ^ { \\deg ( v _ 1 ) } F _ j ( v _ 1 ) b : = \\sum _ { j = 1 } ^ { \\deg ( v _ 2 ) } F _ j ( v _ 2 ) . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} I _ { 2 , 3 } = \\frac { \\log \\sqrt { 2 \\pi } } { \\zeta ( 2 ) } x + O \\left ( ( \\log x ) ^ { 2 / 3 } ( \\log \\log x ) ^ { 4 / 3 } \\right ) . \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\Delta _ { a } ( x ) = \\frac { x ^ { \\frac 1 4 + \\frac { a } { 2 } } } { \\pi \\sqrt { 2 } } \\sum _ { n \\leq N } \\frac { \\sigma _ { a } ( n ) } { n ^ { \\frac { 3 } { 4 } + \\frac { a } { 2 } } } \\cos \\left ( 4 \\pi \\sqrt { n x } - \\frac { \\pi } { 4 } \\right ) + O \\left ( x ^ { \\frac 1 2 + \\varepsilon } N ^ { - \\frac 1 2 } \\right ) , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} [ \\beta ] = [ \\beta ' ] = \\sum _ { S \\subset X } [ { \\jmath _ S } _ { ! * } \\jmath _ S ^ * \\imath _ S ^ { ! * } \\beta ' ] . \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} N \\ ; : = N _ 1 + N _ 2 \\ ; \\to \\ ; + \\infty \\ , , \\lim _ { N \\to + \\infty } \\frac { N _ j } { N } \\ ; = : \\ ; c _ j \\in ( 0 , 1 ) \\ , , j \\in \\{ 1 , 2 \\} \\ , . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} \\mu _ P = 2 ~ \\{ a _ j \\} \\left ( \\sum _ { j = 0 } ^ n \\frac { 1 } { a _ j } - 1 \\right ) , \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\langle \\ ! \\langle G , Q _ { \\alpha _ 2 \\alpha _ 1 } ( t ) k _ 0 \\rangle \\ ! \\rangle - \\langle \\ ! \\langle G , Q ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) k _ 0 \\rangle \\ ! \\rangle = \\langle \\ ! \\langle G , \\Upsilon _ 1 ( t , \\sigma ) \\rangle \\ ! \\rangle + \\langle \\ ! \\langle G , \\Upsilon _ 2 ( t , \\sigma ) \\rangle \\ ! \\rangle . \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 2 \\pi \\mathbb { V } ( \\mathbb { G } _ { n } ^ { ( N ) } ( f ) ) } } \\leqslant C _ { 2 } = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma _ { 0 } } < + \\infty . \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} | P _ 1 ^ { 2 ^ * - 2 } \\bigl ( u _ 0 + \\sum _ { j = 2 } ^ k P _ j ) | \\le C U _ 1 ^ { 2 ^ * - 2 } , \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , r p + \\sigma \\right ) = a ^ { r p } \\prod _ { s = 2 } ^ { L } \\alpha _ { s } { } ^ { r _ { s } p _ { s } } . \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} \\theta _ 1 ( \\boldsymbol { \\alpha } ; \\boldsymbol { w } ) & : = \\sum _ { n \\in \\alpha + \\Z ^ 2 } ( 2 n _ 1 + n _ 2 ) n _ 2 e ^ { \\frac { 3 \\pi i } { 2 } ( 2 n _ 1 + n _ 2 ) ^ 2 w _ 1 + \\frac { \\pi i n _ 2 ^ 2 w _ 2 } { 2 } } , \\\\ \\theta _ 2 ( \\boldsymbol { \\alpha } ; \\boldsymbol { w } ) & : = \\sum _ { \\boldsymbol { n } \\in \\boldsymbol { \\alpha } + \\Z ^ 2 } ( 3 n _ 1 + 2 n _ 2 ) n _ 1 e ^ { \\frac { \\pi i } { 2 } ( 3 n _ 1 + 2 n _ 2 ) ^ 2 w _ 1 + \\frac { 3 \\pi i n _ 1 ^ 2 w _ 2 } { 2 } } . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} c _ + + ( - 1 ) ^ { | m | + | n | } c _ - & = \\lim _ { k \\to \\infty } \\tilde { \\phi } ( m + n + 2 k \\chi ^ J ) \\\\ & = \\lim _ { k \\to \\infty } \\sum _ { \\substack { I \\subset [ N ] \\\\ | I | \\geq l } } f _ \\phi ^ I ( U ( m + k \\chi ^ J , n + k \\chi ^ J ) ) \\\\ & = f _ \\phi ^ \\varnothing ( U ( m , n ) ) + f _ \\phi ^ { \\tilde { I } } ( U ( m , n ) ) , \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} i ) B ( U , \\tilde { J } V ) = B ( \\tilde { J } U , V ) = \\tilde { J } B ( U , V ) , \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} e \\left ( \\frac { n \\overline { m \\ell _ 1 \\ell _ 3 v } } { w } \\right ) & = \\frac { 1 } { \\varphi ( w ) } \\sum _ { \\chi ( w ) } \\tau ( \\overline { \\chi } ) \\chi ( n ) \\overline { \\chi } ( m \\ell _ 1 \\ell _ 3 v ) . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} \\Psi ( v _ 1 ) - \\Phi ( v _ 1 ) = h ( t _ { V _ 1 } ( v _ 1 ) ) \\cdot 0 ^ { V _ 2 } + 0 ^ { V _ 2 } \\cdot h ( s _ { V _ 1 } ( v _ 1 ) ) ^ { - 1 } = J _ h ( v _ 1 ) . \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} \\Q _ { s + k \\gamma , x } ( X _ { t + k \\gamma } \\in \\cdot ) = \\Q _ { s , x } ( X _ { t } \\in \\cdot ) , ~ ~ \\forall k \\in \\Z _ + . \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} N _ 6 = - 2 c _ 6 \\int _ { \\mathbb R } b _ k | D | \\coth ( \\sqrt { \\mu _ 2 } | D | ) ( \\zeta _ k \\nu _ { n _ k } + \\varphi _ k \\nu _ { n _ k } ) + \\zeta _ k \\nu _ { n _ k } | D | \\coth ( \\sqrt { \\mu _ 2 } | D | ) ( \\varphi _ k \\nu _ { n _ k } ) d x , \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\widetilde h [ f ] - h [ f ] & = C _ { v _ 0 } \\sum _ { j = 1 } ^ { \\deg ( v _ 0 ) } | F _ j ( v _ 0 ) | ^ 2 - C _ { v _ 1 } \\sum _ { j = 1 } ^ { \\deg ( v _ 1 ) } | F _ j ( v _ 1 ) | ^ 2 - C _ { v _ 2 } \\sum _ { j = 1 } ^ { \\deg ( v _ 2 ) } | F _ j ( v _ 2 ) | ^ 2 \\\\ & = 0 \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} \\langle \\ ! \\langle G , q ^ { \\Lambda , N } _ t \\rangle \\ ! \\rangle = \\langle \\ ! \\langle K G , R ^ { \\Lambda , N } _ t \\rangle \\ ! \\rangle , \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} E ^ { P ^ n } \\Big [ \\ 1 _ K ( X _ 0 ) & \\int _ 0 ^ T \\| b ^ { n } ( X _ { s \\wedge \\tau _ m } ) \\| ^ p \\dd s \\Big ] \\\\ & \\leq \\begin{cases} T \\cdot \\big ( \\sup _ { k \\in \\mathbb { N } } \\sup _ { \\| x \\| \\leq m \\vee m ^ * } \\| b ^ k ( x ) \\| \\big ) ^ p , & , \\\\ T K ^ p 2 ^ { p } \\big ( 1 + C _ { p , T } \\big ) , & . \\end{cases} \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} ( - \\eta , X ) _ { H } = ( \\eta , 0 - X ) \\le \\Psi ( 0 ) - \\Psi ( X ) = - \\Psi ( X ) d t \\otimes \\P \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} \\Delta _ C ^ { \\kappa } ( h , j ) = C _ j ^ 0 - \\sum \\limits _ { \\begin{subarray} ~ i \\in [ I ] \\\\ i \\neq i _ { \\kappa } \\end{subarray} } S _ C ^ { o , h } ( i , \\kappa - 1 , j ) - f ^ { o , \\kappa , h } _ C ( 1 , j ) , \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} \\begin{cases} & a _ 0 ^ B = 1 , \\\\ & a _ 1 ^ B = \\alpha _ 1 , \\\\ & a _ 2 ^ B = 2 ^ { - 1 } ( a _ 1 ^ B \\alpha _ 1 - \\alpha _ 2 ) , \\\\ & a _ 3 ^ B = 3 ^ { - 1 } ( a _ 2 ^ B \\alpha _ 1 - a _ 1 ^ B \\alpha _ 2 + \\alpha _ 3 ) , \\\\ & \\vdots \\\\ & a _ { 4 t - 1 } ^ B = ( 4 t - 1 ) ^ { - 1 } ( a _ { 4 t - 2 } ^ B \\alpha _ 1 - a _ { 4 t - 3 } ^ B \\alpha _ 2 + \\cdots + \\alpha _ { 4 t - 1 } ) . \\end{cases} \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} F _ 4 = \\ , & a _ 1 + b + b ^ 2 + a _ 1 b ^ 2 + b ^ 3 + b ^ 4 + a _ 1 b ^ 4 + b ^ 5 + b ^ 6 + a _ 1 b ^ 6 + b ^ 7 + b ^ 8 \\\\ & + ( a _ 1 ^ 3 + a _ 1 ^ 2 b + a _ 1 ^ 3 b ^ 4 + a _ 1 ^ 2 b ^ 5 ) k + ( a _ 1 ^ 4 + a _ 1 ^ 5 + a _ 1 ^ 4 b + a _ 1 ^ 4 b ^ 2 + a _ 1 ^ 5 b ^ 2 + a _ 1 ^ 4 b ^ 3 ) k ^ 2 \\cr & + ( a _ 1 ^ 7 + a _ 1 ^ 6 b ) k ^ 3 + a _ 1 ^ 8 k ^ 4 , \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { a _ 1 } \\left | u ( x , t ) - \\frac { M _ i } { M } \\mathcal { B } _ M ( x , t ) \\right | = 0 \\mbox { u n i f o r m l y i n $ \\R ^ n $ } \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\lim _ n \\lim _ N P _ n ^ N ( f ) = f \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} | f _ { \\omega , 2 } ^ n ( z , w ) | & = \\frac { | w | } { 2 ^ { \\alpha ( n ) } } , \\\\ | f _ { \\omega , 1 } ^ n ( z , w ) | & \\le | z | \\prod _ { i = 1 } ^ { n } \\left ( 1 + \\frac { | w | } { 2 ^ { \\alpha ( i ) } } \\right ) . \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} A _ \\delta + q ^ \\delta C _ \\delta & = a _ { 2 \\delta , 1 } + a _ { 2 \\delta , - 1 } + a _ { 2 \\delta - 1 } + ( a _ { 2 \\delta , 1 } - a _ { 2 \\delta , - 1 } ) \\\\ & = 2 a _ { 2 \\delta , 1 } + a _ { 2 \\delta - 1 } \\\\ & = 0 \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\| H \\| _ { S _ 1 } = \\frac { ( 1 - r ) ^ N } { ( 1 - r ^ 2 ) ^ N } = \\frac { 1 } { ( 1 + r ) ^ N } < 1 . \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} m _ 3 \\widetilde { \\sigma } _ { n k } \\leq C N _ { n k } \\cdot 2 ^ { - 3 k } \\leq C \\cdot 2 ^ { - n } \\cdot \\| M _ 1 M _ 2 \\| \\cdot 2 ^ k \\cdot 2 ^ { - 3 k } = C \\cdot 2 ^ { - n - 2 k } \\| M _ 1 M _ 2 \\| . \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} W _ { T } \\left ( \\Omega \\right ) : = C \\left ( \\left [ 0 , T \\right ] ; H _ { 0 } ^ { 1 } \\left ( \\Omega \\right ) \\cap W ^ { 2 , \\infty } \\left ( \\Omega \\right ) \\right ) \\cap L ^ { \\infty } \\left ( 0 , T ; H ^ { 2 } \\left ( \\Omega \\right ) \\right ) \\cap C ^ { 1 } \\left ( 0 , T ; L ^ { 2 } \\left ( \\Omega \\right ) \\right ) , \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} a ( p ^ e ) = p ^ { e ( k - 1 / 2 ) } \\sum _ { i = 0 } ^ e \\alpha _ p ^ { e - 2 i } . \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} A _ r ^ { \\rm B T } & = \\begin{pmatrix} 2 . 8 9 4 4 & - 0 . 0 4 2 2 & - 1 . 4 7 2 9 \\\\ - 0 . 0 3 1 8 & 1 . 0 4 7 0 & - 0 . 2 6 1 5 \\\\ - 1 . 1 7 6 4 & - 0 . 2 3 5 5 & 4 . 1 8 9 8 \\end{pmatrix} . \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} R = \\sum _ { i , j } q ^ { i j } \\delta _ i \\otimes \\delta _ j . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} K _ 2 ( x ) = \\frac { \\sqrt { 2 } } { \\sqrt { \\pi } } \\int _ 0 ^ { + \\infty } \\frac { y e ^ { - | x | y } } { \\alpha ^ 2 + y ^ 2 } d y , \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{gather*} \\Phi _ 2 ( X , 0 ) = ( X - 5 4 0 0 0 ) ^ 3 \\end{gather*}"}
-{"id": "168.png", "formula": "\\begin{align*} \\lim \\sup _ { t \\to T _ * } \\| u ( t ) \\| _ { L ^ { \\infty } ( \\R ^ d ) } = \\infty . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} \\phi : = \\sum _ { a \\in Q _ 1 } \\sum _ { r \\geq 1 } \\frac { 1 } { r } ~ a \\cdot f _ a [ r - 1 ] \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} \\mathfrak D ( N , M ) : = \\left \\lbrace D < 0 : \\left ( \\frac { D } { p } \\right ) = w _ p p \\mid N / M , \\ , \\left ( \\frac { D } { p } \\right ) = - w _ p p \\mid M \\right \\rbrace . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\frac { R } { Z ( R ) } = \\langle a + Z , b + Z : 3 ( a + Z ) = 6 ( b + Z ) = Z \\rangle \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} X \\rtimes \\Gamma : = X \\times \\Gamma \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} P = ( 2 ) \\times _ { \\chi } ( 2 ) , \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} \\partial _ x H _ N ( X ) : = \\frac { \\partial } { \\partial X _ x } H _ N ( X ) , \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{gather*} \\Gamma _ 1 = ( \\zeta ^ 5 + \\zeta ) x ^ 2 - ( \\zeta ^ 4 + \\zeta ^ 3 - 1 ) x y + ( \\zeta ^ 5 + \\zeta ^ 3 ) y ^ 2 + \\\\ ( - \\zeta ^ 3 + \\zeta ^ 2 - \\zeta ) x z + ( \\zeta ^ 6 - \\zeta ^ 4 - \\zeta ) y z + ( \\zeta ^ 5 + \\zeta ^ 4 ) z ^ 2 . \\end{gather*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\sum _ { q } & \\Psi \\left ( \\frac { q } { Q } \\right ) \\frac { q } { \\varphi ( q ) } \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\left | L \\left ( \\frac { 1 } { 2 } , \\chi \\right ) \\right | ^ 2 \\psi _ { } ( \\chi ) \\psi _ { } ( \\overline { \\chi } ) \\\\ & = ( P _ 1 ( 1 ) P _ 3 ( 1 ) + o ( 1 ) ) \\sum _ { q } \\Psi \\left ( \\frac { q } { Q } \\right ) \\frac { q } { \\varphi ( q ) } \\varphi ^ + ( q ) . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} b ( u , v , w ) & = \\frac { 1 } { 2 } ( u \\cdot \\nabla v , w ) - \\frac { 1 } { 2 } ( u \\cdot \\nabla w , v ) \\ ; \\ ; \\ ; \\forall u , v , w \\in X . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} D _ q f \\left ( x \\right ) = \\frac { { f \\left ( { q x } \\right ) - f \\left ( x \\right ) } } { { \\left ( { q - 1 } \\right ) x } } , x \\ne 0 . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\lbrace { \\tilde z _ 1 } , { \\tilde z _ 3 } \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace { \\tilde z _ 2 } , { \\tilde z _ 4 } \\rbrace = 1 . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} \\varepsilon \\int _ { \\Omega } ^ { } u ( t ) d x + \\int _ { \\Gamma } ^ { } u _ { \\Gamma } ( t ) d \\Gamma = \\varepsilon \\int _ { \\Omega } ^ { } u _ 0 d x + \\int _ { \\Gamma } ^ { } u _ { 0 \\Gamma } d \\Gamma \\mbox { f o r a l l } t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} q ^ { 1 , h } ( \\iota , \\iota ' ) = \\begin{cases} h \\lambda ^ 0 _ { \\ell } , & i = i ' , \\ n _ { \\iota ' } = n _ { \\iota } + 1 , \\\\ n _ { \\iota } \\mu _ { i } , & i = i ' , \\ n _ { \\iota ' } = n _ { \\iota } - 1 , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\langle M \\rangle ( t ) = \\sum _ { i = 1 } ^ { d } \\int _ { 0 } ^ { t } u _ { i 1 } ( X ^ { 1 / 2 , \\mu } ( s ) ) ^ 2 \\ , d s = \\sum _ { i = 1 } ^ { d } \\int _ { 0 } ^ { t } u _ { i 2 } ( X ^ { 1 / 2 , \\mu } ( s ) ) \\ , d s . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} \\tau ^ { - \\frac { q } { \\tau } } \\frac { \\Gamma _ 2 ( 1 - q \\ , | \\tau ) \\Gamma _ 2 ( \\tau \\ , | \\tau ) } { \\Gamma _ 2 ( 1 \\ , | \\tau ) \\Gamma _ 2 ( \\tau - q \\ , | \\tau ) } = \\frac { \\Gamma ( 1 - q ) } { \\Gamma ( 1 - \\frac { q } { \\tau } ) } , \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\gamma ( n ) = \\frac { \\binom { 4 n } { 2 n } \\cdot 6 \\cdot \\binom { 4 n + 3 } { 2 n + 1 } } { \\binom { 4 n + 4 } { 2 n + 2 } } . \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\tau _ y = \\tau _ { \\gamma y } ( u _ \\gamma \\cdot u _ \\gamma ^ { - 1 } ) \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\nabla = d + p _ { - 1 } d z + \\widetilde u ( z ) d z - \\frac { \\varphi ( z ) } { h ^ \\vee } \\rho \\ , d z , \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} \\sum \\limits _ { i _ 1 = 0 } ^ { n - 1 } \\sum \\limits _ { i _ 2 = 0 } ^ { i _ 1 - 1 } \\cdots \\sum \\limits _ { i _ k = 0 } ^ { i _ { k - 1 } - 1 } 1 = \\binom { n } { k } , \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} & \\left ( \\left [ \\prod _ { i = 1 } ^ k \\left ( S _ i ^ { m _ i } ( S _ i ^ * ) ^ { n _ i } - \\frac { 1 } { q _ i } S _ i ^ * S _ i ^ { m _ i } ( S _ i ^ * ) ^ { n _ i } S _ i \\right ) \\right ] T \\right ) \\\\ & \\qquad = \\left ( \\left [ \\prod _ { i = 1 } ^ k S _ i ^ { m _ i } ( S _ i ^ * ) ^ { n _ i } \\right ] \\left [ \\prod _ { i = 1 } ^ k \\left ( I - \\frac { \\tau _ i } { q _ i } \\right ) \\right ] T \\right ) . \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] g ( u ( x ) + \\rho v ( x ) ) - g ( u ( x ) ) & = \\frac { 1 } { 2 } ( ( u _ i + u _ { i + 1 } ) ( u ( x ) + \\rho v ( x ) ) - u _ i u _ { i + 1 } ) \\\\ \\MoveEqLeft [ - 1 ] - \\frac { 1 } { 2 } ( ( u _ i + u _ { i + 1 } ) u ( x ) - u _ i u _ { i + 1 } ) \\\\ & = \\frac { \\rho } { 2 } ( u _ i + u _ { i + 1 } ) v ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} f _ o ^ { L } ( \\vec x ) = \\sum _ { i = 0 } ^ { n - 1 } ( x _ { \\sigma ( i + 1 ) } - x _ { \\sigma ( i ) } ) f ( V _ { \\sigma ( i ) } ) . \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} a - 1 - \\left \\lfloor \\frac { m } { \\ell } \\right \\rfloor \\ge a - 1 - \\frac { m } { \\ell } = \\frac { ( a - 1 ) b - 3 } { b + 2 } > - 1 . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} F ^ * _ { j , p } \\left ( \\tau \\right ) - \\frac { 1 } { \\sqrt { - i \\tau } } \\sqrt { \\frac { 2 } { p } } \\sum _ { k = 1 } ^ { p - 1 } \\sin \\left ( \\frac { \\pi k j } { p } \\right ) F ^ * _ { k , p } \\left ( - \\frac { 1 } { \\tau } \\right ) = i \\sqrt { 2 p } \\cdot r _ { f _ { j , p } } ( \\tau ) , \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} H _ { J _ { k + 2 } } = H _ { J _ { k + 1 } } \\cap { H _ { k + 2 } } . \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\mu ^ { L _ n \\cdots L _ 1 } _ { c _ 1 , c _ 2 } & = L _ n ( \\mu ^ { L _ { n - 1 } \\cdots L _ { 1 } } _ { L _ { n - 1 } \\cdots L _ 1 ( c _ 1 ) , L _ { n - 1 } \\cdots L _ 1 ( c _ 2 ) } ) \\\\ \\iota ^ { L _ n \\cdots L _ 1 } & = L _ n ( \\iota ^ { L _ { n - 1 } \\cdots L _ { 1 } } ) \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} D \\left ( p \\right ) & = \\left \\{ \\left ( x , y \\right ) : x \\sim y \\left ( x \\vee x ^ { \\prime } \\right ) \\wedge p = \\left ( y \\vee y ^ { \\prime } \\right ) \\wedge p \\right \\} \\\\ E \\left ( p \\right ) & = \\left \\{ \\left ( x , y \\right ) : x \\sim y \\left ( x \\vee x ^ { \\prime } \\right ) \\vee p = \\left ( y \\vee y ^ { \\prime } \\right ) \\vee p \\right \\} \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\lambda _ k = 0 , \\ \\ \\sum _ { k = 0 } ^ { \\infty } \\lambda _ k = + \\infty . \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} \\sup _ q \\sum _ { k = 1 } ^ \\infty q v _ { k , n } ( q ) < \\infty , \\sup _ q \\sum _ { k = 1 } ^ \\infty k v _ { k , n } ( q ) < \\infty . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\frac { w ( e ) ^ p } { \\prod _ { v \\in e } \\sum _ { f : \\ , v \\in f } w ( f ) } = \\alpha , \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} h ^ 0 ( V \\otimes L ) = k _ 0 \\hbox { f o r g e n e r a l } L \\in B ^ { k _ 0 } _ { 1 , e _ 0 } ( V ) . \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} s = w \\left ( \\begin{array} { c c } - 1 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} d '' f ( 0 ) & = \\nabla '' _ 0 s _ k ^ 1 ( 0 ) , \\\\ d d '' f ( 0 ) & = \\nabla _ 0 \\nabla '' _ 0 s _ k ^ 1 ( 0 ) \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\widetilde { Y } _ { i , r } \\widetilde { Y } _ { j , r } = \\widetilde { Y } _ { j , r } \\widetilde { Y } _ { i , r } \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} K _ { 1 } ( s ) = ( 1 - p ( s ) \\sigma _ { z } ( s ) ) ^ { - 1 } \\left [ \\sigma _ { x } ( s ) p ( s ) + \\sigma _ { y } ( s ) p ^ { 2 } ( s ) + q ( s ) \\right ] . \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} \\beta ( x , y ) ^ 2 = q ( x + y ) / q ( x ) q ( y ) ; \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} | J | ^ 2 = Q ^ 2 ( Q + \\sigma ^ 2 _ 2 ) . \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\left ( \\nabla _ { W } g \\right ) \\left ( U , V \\right ) { \\small = \\bar { g } } \\left ( h ^ { \\ell } \\left ( W , U \\right ) , V \\right ) { \\small + \\bar { g } } \\left ( h ^ { \\ell } \\left ( W , V \\right ) , U \\right ) . \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} \\widetilde { G } ^ { ( d _ 1 , d _ 2 ) , n } _ { p _ 1 , p _ 2 } ( t ) = \\begin{cases} G ^ n _ { p _ 1 , p _ 2 } ( t ) , & d _ 1 > 0 , d _ 2 > 0 , \\\\ G ^ { ( 1 ) , n } _ { p _ 1 } ( t ) , & d _ 1 > 0 , d _ 2 = 0 , \\\\ G ^ { ( 2 ) , n } _ { p _ 2 } ( t ) , & d _ 1 = 0 , d _ 2 > 0 . \\end{cases} \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} b _ { 1 } ^ { 2 } + b _ { 2 } ( b _ { 1 } u _ { 2 } + b _ { 2 } u _ { 3 } ) & = b _ { 1 } b _ { 4 } - b _ { 2 } b _ { 3 } \\\\ b _ { 1 } b _ { 3 } + b _ { 2 } ( b _ { 3 } u _ { 2 } + b _ { 4 } u _ { 3 } ) & = 0 \\\\ b _ { 3 } ^ { 2 } + b _ { 4 } ( b _ { 3 } u _ { 2 } + b _ { 4 } u _ { 3 } ) & = ( b _ { 1 } b _ { 4 } - b _ { 2 } b _ { 3 } ) u _ { 3 } . \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} C ^ \\star _ { \\Phi , d } = \\{ x + n x \\boldsymbol \\epsilon ^ 3 \\ ; | \\ ; x \\in C _ { \\Phi , d } , \\ ; n \\in \\mathbb Z , \\ ; - d \\le n \\le d \\} \\subseteq \\Q ^ \\star _ { - 1 , 4 } . \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} \\mathbb { G } ^ { ( 2 m + 2 ) } ( f ) & = \\mathbb { G } ^ { ( 2 m + 1 ) } ( f ) - \\mathbb { E } [ f | \\mathcal { B } ] ^ { t } \\cdot \\mathbb { G } ^ { ( 2 m + 1 ) } [ \\mathcal { B } ] , \\\\ \\mathbb { G } ^ { ( 2 m + 3 ) } ( f ) & = \\mathbb { G } ^ { ( 2 m + 2 ) } ( f ) - \\mathbb { E } [ f | \\mathcal { A } ] ^ { t } \\cdot \\mathbb { G } ^ { ( 2 m + 2 ) } [ \\mathcal { A } ] . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} F _ 1 ( x ) = \\begin{bmatrix} - x _ 2 - 1 \\\\ x _ 1 ^ 3 - 1 \\end{bmatrix} , \\ F _ 2 ( x ) = \\begin{bmatrix} x _ 2 - 1 \\\\ - x _ 1 ^ 3 - 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} 2 \\theta A d P _ { B S } ^ { 2 } + ( 2 \\theta A + d B \\theta ) P _ { B S } + ( B \\theta - a d ) = 0 . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} \\psi _ N ( \\omega _ 0 , y ^ 1 , \\ldots , y ^ N , u ^ 1 , \\ldots , u ^ N ) : = c ( \\omega _ 0 , u ^ 1 , \\ldots , u ^ N ) . \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} H _ { i } : = \\phi _ { i } ( \\{ P _ { 2 , i } , P _ { 3 , i } , . . . , P _ { 2 g , i } \\} ) = \\phi _ { i } ( A _ { i } [ \\ell ] ( \\overline { K } ) ) . \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\left ( 1 + x \\right ) ^ \\alpha _ q - 1 = \\left [ \\alpha \\right ] _ q \\left ( { 1 + q \\eta } \\right ) _ q ^ { \\alpha - 1 } \\left ( x - 0 \\right ) \\ge \\left [ \\alpha \\right ] _ q x \\forall q \\in \\left ( \\widehat { q } , 1 \\right ) . \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} U = Q U + R U , \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} R _ i ( \\delta ) = C _ { 0 i } \\mathsf { P } _ { 0 i } + C _ { 1 i } \\mathsf { P } _ { 1 i } \\ ; , \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} \\exp ( \\pi | r | ) \\Phi _ k ( s , 1 / 4 ) \\ll \\frac { 4 ^ k } { k + | r | } . \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { s , p , q } ( \\Omega ) } = \\left ( \\| u \\| _ { L ^ { p } ( \\Omega ) } + \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ { p } } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d x d y \\right ) ^ { 1 / p } , \\ , \\ , \\ , u \\in W ^ { s , p , q } ( \\Omega ) . \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} \\lfloor \\varphi _ 1 ( n ) \\rfloor - \\lfloor \\varphi _ 1 ( n ) - \\psi ( n ) \\rfloor = \\psi ( n ) + \\Phi \\big ( \\varphi _ 1 ( n ) - \\psi ( n ) \\big ) - \\Phi \\big ( \\varphi _ 1 ( n ) \\big ) . \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} \\tilde { \\mathsf { E } } ( \\Psi ) = \\tilde { \\mathsf { D } } \\Psi + \\mathsf { V } ( \\Psi ) . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} T _ 1 ( s , f ) & = \\sum _ { n = 2 } ^ { \\infty } \\frac { \\lambda _ f ( n ) c ( n ) } { n ^ s } e ^ { - n / X } = \\sum _ { M ^ { 1 - \\Upsilon } < n \\leq X ( \\log q ) ^ 2 } \\frac { \\lambda _ f ( n ) c ( n ) } { n ^ s } e ^ { - n / X } + \\mathcal { O } \\Big ( q ^ { - B } \\Big ) . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} m \\sum _ { k = 0 } ^ { m - 1 } \\tbinom { m - 1 } { k } ( - 1 ) ^ k a _ { n + k + 1 } = - \\sum _ { k = 1 } ^ { m } k \\tbinom { m } { k } ( - 1 ) ^ k a _ { n + k } . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} ( \\vec 1 _ S ) _ i = \\left \\{ \\begin{array} { l } 1 , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , i \\in S , \\\\ 0 , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , i \\notin S , \\end{array} \\right . \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} K ( A ) \\subseteq { K ( A _ { k + 1 } ) } \\subseteq { K ( A / H _ { J _ { k } } ) } = K ( A / H _ { J _ { k + 1 } } ) \\subseteq { K ( A ) } . \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} h _ { i j } ( x ) = \\frac { \\partial ^ { 2 } } { \\partial x _ { i } \\partial x _ { j } } h ( x ) = 4 \\alpha ^ { 2 } e ^ { - \\alpha | x - x _ { 0 } | ^ { 2 } } \\left [ ( x _ { i } - ( x _ { 0 } ) _ { i } ) ( x _ { j } - ( x _ { 0 } ) _ { j } ) - \\frac { 1 } { 2 \\alpha } \\delta _ { i j } \\right ] . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\chi ( M _ k ) = m \\chi ( M ) + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( 2 - 2 g e n u s ( \\Sigma ^ y _ \\ell ) - 2 b ^ y _ \\ell ) . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} d _ n ( f , g ) : = \\sup _ { \\Vert z \\Vert \\le \\varepsilon _ n } \\Vert f ( z ) - g ( z ) \\Vert . \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} T _ { \\gamma , \\phi } ( f ) = c \\ , \\gamma \\circ \\phi ^ { - 1 } \\cdot f \\circ \\phi ^ { - 1 } . \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} \\begin{pmatrix} S _ 1 ^ T E _ { 2 \\lfloor \\frac { 1 } { 4 } ( n - 2 k - 2 l _ 2 ) \\rfloor } & S _ 3 ^ T E _ { 2 \\lfloor \\frac { 1 } { 4 } ( n - 2 k - 2 l _ 2 + 2 ) \\rfloor } \\end{pmatrix} \\begin{pmatrix} S _ 2 \\\\ S _ 4 \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} h _ j ( r ) = \\sum _ { i = 0 } ^ { p } \\alpha _ i \\psi _ { i + j } ( r ) . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} f ( x _ 1 , \\dots , x _ { d - 1 } ) = \\int _ 0 ^ 1 \\chi _ P ( \\{ x _ 1 + t \\alpha _ 1 \\} , \\dots , \\{ x _ { d - 1 } + t \\alpha _ { d - 1 } \\} , t ) \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} V _ N \\triangleq \\max \\Big \\{ V _ { \\varepsilon } ( \\psi _ j ) + 2 \\lambda \\log | 1 + e ^ { 2 \\pi i \\psi _ j } | , \\ , j = - N / 2 \\cdots N / 2 \\Big \\} \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} \\int _ R ^ \\infty e ^ { - r } \\chi _ i ( r ) r ^ { 2 b } \\ , \\mathrm { d } r = e ^ { - R } \\sum _ { j = 0 } ^ b \\frac { 2 ^ j b ! } { ( b - j ) ! } R ^ { 2 ( b - j ) - 1 } \\chi _ { i + j + 1 } ( R ) . \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} \\sigma ( x , y , z ) & = - ( 1 + \\gamma ) ( x ^ 2 + y ^ 2 + z ^ 2 + x ^ 2 y ^ 2 + y ^ 2 z ^ 2 + z ^ 2 x ^ 2 ) \\\\ & + \\gamma [ ( x + y ) ^ 2 + ( y + z ) ^ 2 + ( z + x ) ^ 2 + ( x y + y z ) ^ 2 + ( y z + z x ) ^ 2 + ( z x + x y ) ^ 2 ] \\le 0 . \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} \\mathcal { O } \\left ( m b ^ m + \\min \\{ s , s ^ { \\ast } \\} \\ , b ^ m + \\sum _ { j = 1 } ^ { \\min \\{ s , s ^ { \\ast } \\} } ( m - w _ d ) ^ 2 b ^ { m - w _ d } \\right ) . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} \\varphi _ u ( x ) = G T _ x A _ u V ( e ) \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} X _ t = S _ t X _ 0 + \\int _ 0 ^ t S _ { t - s } b ^ n ( X _ s ) \\dd s + \\int _ 0 ^ t S _ { t - s } \\sigma ^ n ( X _ s ) \\dd W _ s , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} & \\frac { 1 } { \\mu _ { \\alpha _ i } ^ { n - \\nu - 2 } } \\left ( \\frac { 2 ( n + 1 ) } { n } c _ 2 ( X ) - c _ 1 ( X ) ^ 2 \\right ) \\cdot [ \\tilde \\omega _ i ] ^ { n - 2 } \\\\ & \\to \\binom { n - 2 } { \\nu } ( 2 \\pi ) ^ \\nu ( 3 n ) ^ { n - \\nu - 2 } \\left ( \\frac { 2 ( n + 1 ) } { n } c _ 2 ( X ) - c _ 1 ( X ) ^ 2 \\right ) \\cdot ( - c _ 1 ( X ) ) ^ k \\cdot \\alpha _ \\infty ^ { n - k - 2 } , \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} | ( f _ i & \\pm \\varphi ) ( x ) - ( f _ i \\pm \\varphi ) ( u ) | = \\left | g _ i ( x ) - \\left ( g ^ { + } _ i ( u ) - \\frac 1 2 \\pm \\frac 1 2 \\right ) \\right | = \\\\ & = \\begin{cases} | g _ i ( x ) - g ^ { + } _ i ( u ) | \\leq d ( x , u ) \\\\ \\\\ | g _ i ( x ) - g ^ { + } _ i ( u ) + 1 | = | g _ i ( x ) - g ^ { + } _ i ( v ) | \\leq d ( x , v ) = d ( x , u ) . \\end{cases} \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} \\psi _ p ( u ) = \\sum _ { j = 0 } ^ \\kappa d _ { j , p } u ^ j , \\forall p \\in \\{ 0 , . . . , \\kappa \\} , \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\ , \\longmapsto \\ , \\begin{pmatrix} \\alpha & \\gamma \\\\ 0 & \\beta \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} + \\begin{pmatrix} u \\\\ v \\end{pmatrix} \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} I ^ { ( k ) } = \\bigcap _ { J \\subset \\{ 1 , \\dots , m - 1 \\} : \\atop | J | = k - 1 } \\langle \\hat { F } _ { J \\cup \\{ m \\} } \\rangle \\cap \\langle F _ { j _ 1 } , \\dots , F _ { j _ { k - 1 } } , F _ m \\rangle . \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} [ \\beta ] = [ { \\jmath _ 1 } _ { ! * } \\jmath _ 1 ^ * \\beta ] + \\sum _ { k = 2 } ^ { n } [ { \\jmath _ { k } } _ { ! * } \\jmath _ { k } ^ * \\imath _ { k - 1 } ^ { ! * } \\cdots \\imath _ { 1 } ^ { ! * } \\beta ] . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\lbrace S _ i , S _ j \\rbrace = { f } _ { i j } ^ { \\ ; \\ ; \\ ; k } S _ k , \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} | p _ { n + 1 } ^ y ( x , y ) | & \\le | p _ { n } ^ y ( x , y _ { i - 1 } ) | + \\frac { \\alpha } { 2 } \\max _ { k } \\beta ^ i _ k \\\\ & = | p _ { n } ^ y ( x , y _ { i - 1 } ) | + \\frac { \\delta } { 2 } \\max _ { k } ( p _ n ^ x ( x _ k , y _ { i - 1 } ) - p _ n ^ x ( x _ k , y _ i ) ) \\\\ & \\le \\delta ( 2 - 2 ^ { - ( n - 1 ) } ) + \\delta 2 ^ { - n } = \\delta ( 2 - 2 ^ { - n } ) , \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} 0 \\ , \\longrightarrow \\ , ( E _ H ) \\ , \\stackrel { \\iota ' } { \\longrightarrow } \\ , ( E _ H ) / \\theta ' ( { \\mathcal F } ) \\ , \\stackrel { \\widehat { \\mathrm { d } } p } { \\longrightarrow } \\ , T X / { \\mathcal F } \\ , = \\ , { \\mathcal N } _ { \\mathcal F } \\ , \\longrightarrow \\ , 0 \\ , , \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} f _ { 3 2 } ( \\tau ) & = \\sum _ { \\mathfrak I \\in S _ 1 } \\tilde \\psi ( \\mathfrak I ) q ^ { N \\mathfrak I } = \\sum _ { m , n \\in \\Z } ( - 1 ) ^ n ( 4 m + 1 - 2 n { \\rm i } ) q ^ { ( 4 m + 1 ) ^ 2 + 4 n ^ 2 } \\\\ & = \\sum _ { m , n \\in \\Z } ( - 1 ) ^ n ( 4 m + 1 ) q ^ { ( 4 m + 1 ) ^ 2 + 4 n ^ 2 } = \\eta ( 4 \\tau ) ^ 2 \\eta ( 8 \\tau ) ^ 2 . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} \\psi _ { w _ 0 } y _ 1 ^ { c _ 1 } \\cdots y _ j ^ { c _ j } y _ { j + 1 } ^ { c _ { j + 1 } } \\cdots y _ n ^ { c _ n } \\psi _ { w _ { 0 } } y _ { \\min } = 0 . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\psi ( \\chi ) = \\psi _ { } ( \\chi ) + \\psi _ { } ( \\chi ) + \\psi _ { } ( \\chi ) , \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\frac { \\partial \\ell _ j ^ { ( k ) } } { \\partial \\zeta _ q ^ { ( p ) } } , \\ , \\frac { \\partial \\ell _ j ^ { ( k ) } } { \\partial \\varphi _ q ^ { ( p ) } } = 0 , \\mbox { f o r a l l } p > k . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} u ( \\textbf { z } , 0 ) = u _ 0 ( \\textbf { z } ) , \\textbf { z } \\in \\overline { \\Omega } , \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} \\partial _ t u - \\Delta u + u \\cdot \\nabla u + \\nabla p = 0 , \\ \\ { \\rm d i v } \\ , u = 0 , \\ \\ u ( 0 , x ) = u _ 0 ( x ) \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} I ( u ) = \\frac { 1 } { 2 } \\int _ { \\Omega } | \\nabla u | ^ 2 - \\frac { 1 } { 2 ^ { \\ast } } \\int _ { \\Omega } | u | ^ { 2 ^ { \\ast } } . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} \\sum _ { f \\in E ( G _ 1 \\times G _ 2 ) } w ( f ) & = \\frac { 1 } { r ! } \\sum _ { f \\in E ( G _ 1 \\times G _ 2 ) } w _ 1 ( \\pi _ 1 ( f ) ) w _ 2 ( \\pi _ 2 ( f ) ) \\\\ & = \\sum _ { e _ 1 \\in E ( G _ 1 ) } \\sum _ { e _ 2 \\in E ( G _ 2 ) } w _ 1 ( e _ 1 ) w _ 2 ( e _ 2 ) \\\\ & = 1 . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} { \\rm { E } } _ q ^ { - x } { \\rm { e } } _ q ^ x = 1 , { \\rm { e } } _ { 1 / q } ^ x = { \\rm { E } } _ q ^ x . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} A ( m ) = \\sum _ { j = 1 } ^ m ( | v _ j | - | v _ { m + 1 } | ) , \\ , \\ , \\ , m \\in \\{ 1 , 2 , \\ldots , n \\} , \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\Lambda , \\rho , \\beta , H } ^ { | \\vec { h } | } ( \\omega _ e = 1 | \\omega _ { G _ t } = \\omega ^ { \\rho } _ { G _ t } ) ) & \\leq \\mathbb { P } _ { \\Lambda , w , \\beta , H } ^ { | \\vec { h } | } ( \\omega _ e = 1 | \\omega _ { G _ t } = \\omega ^ { \\rho } _ { G _ t } ) \\\\ & \\leq \\mathbb { P } _ { \\Lambda , w , \\beta , H } ^ { | \\vec { h } | } ( \\omega _ e = 1 | \\omega _ { G _ t } = \\omega ^ { w } _ { G _ t } ) . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\nabla ^ k _ { \\ ! \\ ! D } s = \\sigma d ^ k \\ ! \\ ! \\left ( \\frac { s } { \\sigma } \\right ) \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} v ^ f : = \\lim _ { i \\to \\infty } \\frac { 1 } { F ( \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) ) } \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} ( \\xi \\otimes \\eta , \\delta ( X ) ) = ( { \\delta } ^ t ( \\xi \\otimes \\eta ) , X ) = ( [ \\xi , \\eta ] _ { \\tilde { \\bf g } } , X ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\forall X \\in \\bf g ; \\ ; \\ ; \\xi , \\eta \\in \\tilde { \\bf g } , \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 , \\ , x _ 1 > 0 \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\frac { \\Gamma _ 2 ( z + 1 - k \\ , | \\ , 1 , \\tau ) } { \\Gamma _ 2 ( z + 1 \\ , | \\ , 1 , \\tau ) } = & \\prod \\limits _ { j = 0 } ^ { k - 1 } \\Gamma _ 1 \\bigl ( z - j \\ , | \\ , 1 , \\tau \\bigr ) , \\\\ = & \\bigl ( \\frac { 1 } { 2 \\pi \\tau } \\bigr ) ^ { k / 2 } \\tau ^ { \\sum \\limits _ { j = 0 } ^ { k - 1 } ( z - j ) / \\tau } \\ ; \\prod \\limits _ { j = 0 } ^ { k - 1 } \\Gamma \\bigl ( \\frac { z } { \\tau } - \\frac { j } { \\tau } \\bigr ) . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} \\bar { J } ( \\bar { \\gamma } ) & = \\sqrt { 2 } \\int _ { \\bar { \\gamma } } \\bar { T } d t = \\sqrt { 2 } \\int _ { \\bar { \\gamma } } ( \\bar { U } + h ) d t = \\int _ { \\bar { \\gamma } } \\sqrt { \\bar { U } + h } d \\bar { s } = \\int d \\bar { s } _ { h , \\omega } \\\\ d \\bar { s } _ { h , \\omega } ^ { 2 } & = \\bar { T } d \\bar { s } ^ { 2 } = ( \\bar { U } + h ) d \\bar { s } ^ { 2 } = ( U + h - \\frac { \\omega ^ { 2 } } { 2 \\rho ^ { 2 } } ) d \\bar { s } ^ { 2 } \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\tilde { p } _ H \\bigl ( ( y \\star u , y ) , ( z , \\vartheta ) \\bigr ) ~ = ~ p _ H ( y \\star u , z ) \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} | X | = \\frac { 1 } { n ! \\ , \\omega _ n } \\biggl ( ( X ) + \\sum _ { ( p + 1 ) / 2 < j \\le p + 1 } ( - 1 ) ^ j \\binom { p + 1 } { j } \\int _ { \\partial X } \\frac { \\partial } { \\partial \\nu } \\Delta ^ { j - 1 } h \\ , \\mathrm { d } { s } \\biggr ) , \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} ( A v ) _ \\gamma = \\abs { \\gamma } v _ \\gamma . \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\hat { v } _ \\mu ( p ) = - \\frac { \\mu ^ 2 } { 2 \\pi } \\sum _ { j = 1 } ^ N a _ j \\frac { e ^ { - i p \\cdot y _ j } } { p ^ 2 ( p ^ 2 - \\mu ^ 2 ) } . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\psi \\in C ^ 2 ( \\R ) , \\\\ & \\psi ( - y ) = \\psi ( y ) , \\ ; \\psi ( 0 ) = 0 , \\\\ & | \\psi ' ( y ) | \\leq \\nu _ 1 , \\\\ & \\psi ( y ) \\geq \\nu _ 2 | y | - \\nu _ 3 , \\ ; \\psi '' ( y ) > 0 , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} \\eta ( x ) = \\begin{cases} 1 & \\norm { x } _ \\infty \\leq \\tfrac { 1 } { 3 2 d } , \\\\ 0 & \\norm { x } _ \\infty \\geq \\tfrac { 1 } { 1 6 d } . \\end{cases} \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} d X _ { t } ( x ) = b ( t , X _ { s , t } ( x ) ) d t + d B _ { t } , \\ X _ { s } = x \\in \\R ^ { d } , \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { n \\leqslant N } r ( n ) ^ 2 \\omega ( n ) \\mathop { \\sum \\dots \\sum } _ { \\substack { m _ 1 , \\dots , m _ s \\leqslant N / n \\\\ ( n , m _ i ) = 1 , \\ , \\ , ( m _ i , m _ j ) = 1 } } \\prod _ { i = 1 } ^ s \\frac { r ( m _ i ) \\omega ' _ i ( m _ i ) } { \\sqrt { m _ i } } \\\\ & \\qquad = \\big ( 1 + o ^ { \\star } ( 1 ) \\big ) \\prod _ p \\bigg ( 1 + r ( p ) ^ 2 \\omega ( p ) + \\frac { r ( p ) } { \\sqrt { p } } \\sum _ { i = 1 } ^ s \\omega ' _ i ( p ) \\bigg ) , \\end{aligned} \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{gather*} \\mathcal G _ { 1 , ( a ; N ) } ^ \\ast ( \\tau ) = - { \\rm i } \\pi \\ ( 1 - \\frac { 2 a } { N } \\ ) + ( - 2 \\pi ) 2 ^ { 1 / 2 } \\pi ^ { - 1 / 2 } \\\\ \\hphantom { \\mathcal G _ { 1 , ( a ; N ) } ^ \\ast ( \\tau ) = } { } \\times \\sum _ { n > 0 } \\sum _ { m \\ge 0 } \\ ( W _ 1 ( 2 \\pi n ( N m + a ) \\tau , { 0 } ) - W _ 1 ( 2 \\pi n ( N m + N - a ) \\tau , { 0 } ) \\ ) \\\\ \\hphantom { \\mathcal G _ { 1 , ( a ; N ) } ^ \\ast ( \\tau ) } { } = - { \\rm i } \\pi \\ ( 1 - \\frac { 2 a } { N } \\ ) - 2 \\pi { \\rm i } \\sum _ { n > 0 } \\frac { q ^ { n a } - q ^ { n ( N - a ) } } { 1 - q ^ { N n } } . \\end{gather*}"}
-{"id": "350.png", "formula": "\\begin{align*} \\Sigma ( s ) \\ll \\frac { ( 1 + | r | ) ^ A } { | c | ^ { 1 / 2 } } \\sum _ { n = 1 } ^ { \\infty } n ^ { 2 \\theta } \\left | 1 - \\sqrt { - n ^ 2 / ( 4 c ^ 2 ) } \\right | ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} ( 1 - p ^ { k - 1 / 2 } \\alpha _ p X ) ( 1 - p ^ { k - 1 / 2 } \\alpha _ p ^ { - 1 } X ) = 1 - a ( p ) X + p ^ { 2 k - 1 } X ^ 2 . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\limsup _ { t \\to T ( u _ 0 ) } \\| u ( t ) \\| _ { \\dot B ^ { s _ p } _ { p , q } } = \\infty . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} N ^ * : = \\ln \\left ( \\frac { F ( x _ 0 ) - F ^ * } { \\epsilon } \\right ) K ^ * . \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} \\operatorname { f i x } ( y , M ^ G ) < \\sum _ { \\sum i l _ i = m } \\frac { \\Delta ( l _ 1 , l _ 2 , l _ 3 , l _ 4 ) } { | C _ { \\overline { G L _ m ( q ) } } ( [ y ] _ \\beta ) | } < q ^ { \\frac { 1 } { 8 } m ( 2 n - 3 m ) + c m } , \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} a \\cdot _ x b = a \\cdot b \\cdot a ^ { - 1 } \\cdot \\beta _ x ( a ) . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} ( f _ 1 \\sqcup f _ 2 ) _ * ( g _ 1 \\sqcup g _ 2 ) ^ * = ( f _ 1 ) _ * ( g _ 1 ) ^ * + ( f _ 2 ) _ * ( g _ 2 ) ^ * , \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} ( u _ t ^ 2 , m _ t ) + \\int _ 0 ^ t 2 ( | \\nabla u _ r | ^ 2 , m _ r ) d r = | u _ 0 | ^ 2 _ { 0 } - 2 \\int _ 0 ^ t ( F ( u _ r ) , \\nabla ( u _ r m _ r ) ) d r . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { 3 } X _ { r } ( - 1 ) ^ { 2 } = \\frac { 1 } { 2 } H ( - 1 ) ^ { 2 } + E ( - 1 ) F ( - 1 ) + F ( - 1 ) E ( - 1 ) . \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} R _ { 1 ^ k , n - k } = \\sum _ { \\ell ( I ) = k + 1 , \\ I \\vDash n } V _ I . \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} I = 0 . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} \\widehat { K } ( y ) = \\frac { 1 } { y ^ 2 - \\ell | y | + c } \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} G ( u _ n , v _ n ) ~ \\geq ~ \\P _ { u _ n } ( X ( k _ n ) = \\hat { u } _ n ) G ( \\hat { u } _ n , \\hat { v } _ n ) \\P _ { \\hat { v } _ n } ( X ( m _ n ) = v _ n ) \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} [ ( X \\otimes b '' ) \\wedge T _ \\theta ] = [ ( X \\otimes b '' ) \\wedge d K ] + B \\otimes \\{ e \\} \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} A _ 1 ( z ) = \\frac { 1 } { 6 } \\cdot \\frac { 1 2 z \\sin z + 1 2 \\cos z - 1 2 - 3 z ^ 2 \\cos z - z ^ 3 } { 1 - \\sin z } . \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} \\begin{pmatrix} I _ { n _ 1 } \\\\ & I _ { n _ 2 - 2 n _ 1 } \\\\ I _ { n _ 1 } & & I _ { n _ 1 } \\end{pmatrix} \\in G L _ { n _ 2 } . \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} s R e s ( A ) : = \\bigcup _ { j = 1 } ^ N s R e s _ j ( A ) , \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} E _ { k , N } ^ { \\chi } ( Z , s ) : = j ( g , I ) ^ { k } E \\Big ( g , \\frac { 2 s } { 2 n + 1 } + \\frac k { 2 n + 1 } - \\frac 1 2 , f \\Big ) , \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q \\ , | \\ , b ) = \\exp \\Bigl ( \\bigl ( \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j / \\prod \\limits _ { i = 1 } ^ M a _ i \\bigr ) q \\log ( q ) + O ( q ) \\Bigr ) , \\ ; q \\rightarrow \\infty . \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\cdot v & = \\begin{bmatrix} 1 & a _ { 1 } & a _ { 3 } \\\\ 0 & 1 & 0 \\\\ 0 & a _ { 2 } & 1 \\end{bmatrix} \\cdot v \\\\ & = \\rho ^ { a _ { 1 } v _ { 2 } + \\frac { 1 } { 2 } a _ { 2 } v _ { 2 } \\left ( v _ { 2 } - 1 \\right ) + a _ { 3 } v _ { 3 } } v \\tau ^ { a _ { 2 } v _ { 2 } } . \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} N ( \\lambda ) B ( \\lambda ) = Q ( \\lambda ) \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\ell ' & = b d ' , \\\\ m ' & = a d ' , \\\\ \\frac { n ' } { d ' } & = \\frac { a _ 1 b + a b _ 2 } c , \\gcd ( n ' , d ' ) = 1 . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} & \\Big ( ( 1 - \\lambda ) I + \\lambda A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } \\Big ) - \\Big ( ( 1 - \\lambda ) I + \\lambda A ^ { \\frac { 1 } { 2 } } B ^ { - 1 } A ^ { \\frac { 1 } { 2 } } \\Big ) ^ { - 1 } \\\\ & = \\lambda ( 1 - \\lambda ) ( I - A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } ) \\Big ( ( 1 - \\lambda ) A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } + \\lambda I \\Big ) ^ { - 1 } ( I - A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} x _ i ' & = \\hat { F } _ i / x _ i , \\\\ x _ j ' & = x _ j ( j \\ne i ) . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} R = \\int _ 0 ^ T m _ { c _ { s - } } ( f ) d { G _ p } ( s ) . \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} u = \\abs { m } ^ { - \\frac { 3 } { 7 } } \\big ( \\varphi _ 1 ( X ) \\sigma _ 1 ( X ) \\big ) ^ { \\frac { 3 } { 7 } } , \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} L _ h ( x ) : = \\frac { 1 } { ( p - 1 ) ! } f ^ { ( p ) } ( x _ 0 ) [ h ] ^ { p - 1 } [ h + x ] + N _ C ( x _ 0 + h + x ) . \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ^ q ( \\delta ) \\bigr ] & = \\frac { G ( \\delta ) } { G ( q + \\delta ) } \\frac { G ^ 2 ( q + \\delta + 1 / 2 ) } { G ^ 2 ( \\delta + 1 / 2 ) } \\frac { G ( \\delta + 1 ) } { G ( q + \\delta + 1 ) } , \\\\ & = \\prod \\limits _ { k = 0 } ^ \\infty \\Bigl [ \\frac { \\delta + k } { q + \\delta + k } \\frac { ( q + \\delta + 1 / 2 + k ) ^ 2 } { ( \\delta + 1 / 2 + k ) ^ 2 } \\frac { \\delta + 1 + k } { q + \\delta + 1 + k } \\Bigr ] ^ { k + 1 } \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} L ^ { \\ast } ( f \\otimes g , s ) = \\sum _ { n \\geq 1 } \\frac { \\lambda _ f ( n ) \\lambda _ g ( n ) } { n ^ s } , \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} & \\langle \\mathbb { A } _ r \\frac { \\sigma _ k ^ i - 2 \\sigma _ k ^ { i - 1 } + \\sigma _ k ^ { i - 2 } } { \\delta _ k ^ 2 } , \\sigma _ k ^ i - \\sigma _ k ^ { i - 1 } \\rangle + \\langle \\frac { p _ k ^ i - 2 p _ k ^ { i - 1 } + p _ k ^ { i - 2 } } { \\delta _ k ^ 2 } , \\sigma _ k ^ i - \\sigma _ k ^ { i - 1 } \\rangle \\\\ & = \\langle \\frac { E u _ k ^ i - 2 E u _ k ^ { i - 1 } + E u _ k ^ { i - 2 } } { \\delta _ k ^ 2 } , \\sigma _ k ^ i - \\sigma _ k ^ { i - 1 } \\rangle \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\tilde { H } _ 1 & \\in K [ x _ 2 , x _ 3 , x _ 4 , \\ldots , x _ n ] \\mbox { , } \\\\ \\tilde { H } _ 2 & = \\tilde { H } _ 3 = \\tilde { H } _ 4 = \\cdots = \\tilde { H } _ n = 0 \\mbox { . } \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} W ( \\bar { x } , \\bar { \\lambda } ) = \\frac { 1 } { 2 \\alpha } ( \\bar { x } - \\bar { x } ^ { * } ) ^ { \\top } ( \\bar { x } - \\bar { x } ^ { * } ) + \\frac { 1 } { 2 } ( \\bar { \\lambda } - \\bar { \\lambda } ^ { * } ) ^ { \\top } ( \\bar { \\lambda } - \\bar { \\lambda } ^ { * } ) . \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} \\Pi ( y ) = & y ^ { 2 } [ \\beta _ { 0 } + \\beta _ { 1 } \\omega y + \\ \\beta _ { 2 } \\omega ^ { 2 } y ^ { 2 } + \\ \\beta _ { 3 } \\omega ^ { 3 } y ^ { 3 } + \\ \\beta _ { 4 } \\omega ^ { 4 } y ^ { 4 } ] \\\\ & + h [ \\alpha _ { 0 } + \\alpha _ { 1 } \\omega y + \\alpha _ { 2 } \\omega ^ { 2 } y ^ { 2 } ] = 0 \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} \\alpha _ n = \\frac { ( 4 \\pi ) ^ { n ( n + 1 ) / 2 } } { \\prod _ { m = 1 } ^ n ( m - 1 ) ! } . \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} p ^ * ( X _ { a , k } ) = p ^ * ( X _ { n - k } ) = \\lambda _ { b - k } \\frac { \\Delta _ { \\{ b - k , \\dots , n - k - 1 \\} } \\Delta _ { \\{ b - k + 2 , \\dots , n - k , n \\} } } { \\Delta _ { \\{ b - k + 1 , \\dots , n - k \\} } \\Delta _ { \\{ b - k + 1 , \\dots , n - k - 1 , n \\} } } . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} \\partial _ t ( \\rho X ) = - \\partial _ x ( \\rho u X ) - \\partial _ x p ( \\rho ) + \\partial _ x ( \\mu ( \\rho ) \\partial _ x u ) - \\partial _ x ( \\rho ^ 2 ( \\partial _ x u ) \\xi ' ( \\rho ) ) + \\rho f . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} Q _ 1 ^ \\vee = \\bigoplus _ { r ( V ) = 1 } Q _ V ^ \\vee \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} | w ( t ) | _ { \\alpha + 1 / 2 - \\rho , \\sigma } & \\le | w ( T _ * ) | _ { \\alpha + 1 / 2 - \\rho , \\sigma } e ^ { - t + T _ * } e ^ { K ^ { \\alpha + \\frac { 1 } { 2 } - \\rho } \\int _ { T _ * } ^ t 2 D _ 0 / \\tau d \\tau } \\\\ & \\le | w ( T _ * ) | _ { \\alpha + 1 / 2 - \\rho , \\sigma } e ^ { - t + T _ * } e ^ { 2 K ^ { \\alpha + \\frac { 1 } { 2 } - \\rho } D _ 0 \\ln t } = M _ 2 t ^ \\beta e ^ { - t } , \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\| u _ t \\| _ { L ^ 2 } ^ 2 + & 2 \\int _ 0 ^ t \\| \\nabla u _ r \\| ^ 2 _ { L ^ 2 } d r = \\| u _ 0 \\| _ { L ^ 2 } ^ 2 - 2 \\int _ 0 ^ t ( F ( u _ r ) , \\nabla u _ r ) - \\frac { 1 } { 2 } ( u _ r ^ 2 , \\mathrm { d i v } ( \\beta _ j ) ) \\dot { Z } _ r ^ j d r \\\\ & \\leq \\| u _ 0 \\| _ { L ^ 2 } ^ 2 + \\int _ 0 ^ t \\| \\nabla u _ r \\| ^ 2 _ { L ^ 2 } d r + \\int _ 0 ^ t \\| F ( u _ r ) \\| _ { L ^ 2 } ^ 2 + \\| \\mathrm { d i v } ( \\beta _ j ) \\| _ { \\infty } \\| u _ r \\| _ { L ^ 2 } ^ 2 | \\dot { Z } _ r ^ j | d r . \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} h \\nu _ { \\gamma } \\beta _ m \\varpi _ p = \\left ( \\begin{array} { c c } - y p ^ { m - 1 } & x p ^ { - m } + \\gamma y p ^ { 1 - m } \\\\ - t p ^ { m - 1 } & z p ^ { - m } + \\gamma t p ^ { 1 - m } \\end{array} \\right ) , \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { r _ { i } ^ { ( 1 ) } } ( \\frac { 2 } { 3 } \\sum _ { p > p ' > 0 } \\mathrm { m i n } \\{ n _ { p , i } , n _ { p ' , i } \\} + \\frac { 1 } { 3 } n _ { p , i } ) & = \\frac { 1 } { 3 } \\sum _ { s = 1 } ^ { k } r ^ { ( s ) ^ { 2 } } _ { i } . & \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} \\Lambda = \\left ( \\begin{matrix} \\lambda _ { 1 , 1 } & \\lambda _ { 1 , 2 } & \\lambda _ { 1 , 3 } & \\cdots & \\lambda _ { 1 , n } \\\\ & \\lambda _ { 2 , 2 } & \\lambda _ { 2 , 3 } & \\cdots & \\lambda _ { 2 , n } \\\\ & & \\lambda _ { 3 , 3 } & \\cdots & \\lambda _ { 3 , n } \\\\ & & & \\ddots & \\vdots \\\\ & & & & \\lambda _ { n , n } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} \\displaystyle q _ t = \\displaystyle \\frac { p _ t } { \\displaystyle \\prod _ { k = 2 } ^ { t - 1 } ( 1 - q _ k ) } , t = 3 , \\ldots , T _ { \\max } , \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} a ( f - b ) = 2 \\pi w . \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} g _ a ( x ) = \\epsilon ( a ) + \\frac { 1 } { a } \\frac { f ( \\frac { a - x } { \\epsilon ( a ) } ) } { f ( 1 - \\frac { a - x } { \\epsilon ( a ) } ) + f ( 1 \\frac { a - x } { \\epsilon ( a ) } ) } . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} m _ 1 = \\max \\{ m \\in \\{ 1 , 2 , \\ldots , n \\} : A ( m - 1 ) < r \\} , \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} L a ( n , \\{ \\wedge _ 4 , \\vee _ 4 \\} , P _ 2 ) = \\frac { 3 } { 2 } \\binom { n } { \\lfloor n / 2 \\rfloor } \\left ( 1 + O \\left ( \\frac { 1 } { n } \\right ) \\right ) . \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\lbrace z _ 1 , z _ 3 \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace z _ 2 , z _ 4 \\rbrace = 1 . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} C _ 2 = F \\left ( d , T , c _ 2 , c _ 3 , \\| h \\| _ { \\varrho ( h ) } , \\frac { 1 } { \\varrho ( \\gamma _ { \\infty } \\circ h ) } , \\frac { 1 } { \\| \\gamma _ { \\infty } \\circ h \\| _ { \\varrho ( \\gamma _ { \\infty } \\circ h ) } } , \\frac { 1 } { \\gamma _ 0 ^ L } , \\frac { 1 } { \\gamma _ { \\infty } ^ L } , \\frac { 1 } { c _ 1 } \\right ) ; \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} ( ( M \\ast N ) \\ast P ) ( w , z ) & = \\bigvee _ { y \\in Y } \\bigg ( \\bigvee _ { x \\in X } M ( w , x ) \\otimes N ( x , y ) \\bigg ) \\otimes P ( y , z ) \\\\ & \\cong \\bigvee _ { y \\in Y , x \\in X } M ( w , x ) \\otimes N ( x , y ) \\otimes P ( y , z ) \\\\ & \\cong \\bigvee _ { x \\in X } M ( w , x ) \\otimes \\bigg ( \\bigvee _ { y \\in Y } N ( x , y ) \\otimes P ( y , z ) \\bigg ) \\\\ & = ( M \\ast ( N \\ast P ) ) ( w , z ) . \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} & E _ 2 ( x , \\zeta ) = e ^ { i ( x _ 1 \\zeta _ 1 + x _ 2 \\zeta _ 2 ) } \\left ( \\sinh ( x _ 0 | \\zeta | _ { \\C } ) - \\frac { i ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) } { | \\zeta | _ { \\C } } \\cosh ( x _ 0 | \\zeta | _ { \\C } ) \\right ) . \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} C & : = \\cup _ k A _ k \\\\ W & : = \\{ n \\in L ^ 0 ( \\N ) \\colon \\cup _ k ( i ( n , N _ k ) \\cap A _ k ) = C \\} \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} | \\phi _ n | _ { \\alpha , \\sigma } & \\le | A \\zeta _ n | _ { \\alpha , \\sigma } + ( n - 1 ) | \\zeta _ { n - 1 } | _ { \\alpha , \\sigma } + \\sum _ { k = 1 } ^ { n - 1 } | B ( \\zeta _ k , \\zeta _ { n - k } ) | _ { \\alpha , \\sigma } \\\\ & \\le | \\zeta _ n | _ { \\alpha + 1 , \\sigma } + ( n - 1 ) | \\zeta _ { n - 1 } | _ { \\alpha + 1 , \\sigma } + K ^ \\alpha \\sum _ { k = 1 } ^ { n - 1 } | \\zeta _ k | _ { \\alpha + 1 / 2 , \\sigma } | \\zeta _ { n - k } | _ { \\alpha + 1 / 2 , \\sigma } . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} K = \\mathbb { S } ^ m : = \\Big \\{ A = ( a _ { i j } ) _ { 1 \\leq i , j \\leq m } \\in \\R ^ { m \\times m } : a _ { i j } = a _ { j i } , \\ , i , j = 1 , . . . , m , \\\\ \\sum _ { i , j = 1 } ^ m a _ { i j } \\xi _ i \\xi _ j \\geq \\alpha | \\xi | ^ 2 \\xi \\in \\R ^ m \\Big \\} , \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\nu \\beta _ 1 \\nu ' ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ) p ^ { 1 / 2 } \\underline { \\chi } _ p ( \\gamma ) \\mathcal G \\left ( \\frac { - \\gamma } { p } \\right ) \\mathbf 1 _ { \\Z _ p ^ { \\times } } ( x ) \\psi \\left ( \\frac { x ^ 2 } { \\gamma p } \\right ) \\underline { \\chi } _ p ^ { - 1 } ( x ) , \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { \\mu } { p _ 1 } + \\frac { 1 - \\mu } { p _ 2 } , \\ \\ \\mu = \\frac { p _ 1 ( p _ 2 - p ) } { p ( p _ 2 - p _ 1 ) } . \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} P _ i \\left ( \\frac { \\log ( y _ i / x ) } { \\log y _ i } \\right ) = P _ i [ x ] . \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} & \\partial _ t \\rho + \\partial _ x ( u \\rho ) = 0 , \\\\ & \\partial _ t ( \\rho u ) + \\partial _ x ( \\rho u ^ 2 ) = - \\partial _ x p ( \\rho ) + \\partial _ x ( \\mu ( \\rho ) \\partial _ x u ) + \\rho f , \\\\ & ( \\rho , u ) | _ { t = 0 } = ( \\rho _ 0 , u _ 0 ) \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} \\phi ( v _ { i + 1 } ) = \\begin{cases} \\lambda _ i v _ { i } & \\mbox { i f } i \\neq n , \\\\ ( - 1 ) ^ { a - 1 } \\lambda _ n v _ n & \\mbox { i f } i = n . \\end{cases} \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} Y = \\left [ \\begin{smallmatrix} y _ 1 ^ { 1 / 2 } \\\\ & \\ddots \\\\ & & y _ n ^ { 1 / 2 } \\end{smallmatrix} \\right ] , C = \\left [ \\begin{smallmatrix} \\cos ( \\theta _ 1 ) \\\\ & \\ddots \\\\ & & \\cos ( \\theta _ n ) \\end{smallmatrix} \\right ] , S = \\left [ \\begin{smallmatrix} \\sin ( \\theta _ 1 ) \\\\ & \\ddots \\\\ & & \\sin ( \\theta _ n ) \\end{smallmatrix} \\right ] . \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} F _ { 4 , 2 2 } ( \\tau ) = \\left ( \\begin{smallmatrix} ( u + 1 / u ) \\\\ 2 ( u - 1 / u ) \\\\ 0 \\\\ - 2 ( u - 1 / u ) \\\\ - ( u + 1 / u ) \\end{smallmatrix} \\right ) X ^ 3 Y ^ 3 + \\ldots \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} G ( 2 z \\ , | \\ , \\tau ) = ( 2 \\pi ) ^ { - z } 2 ^ { 1 + { } _ 2 S _ 0 ( 2 z ) } \\frac { G ( z \\ , | \\ , \\tau ) \\ , G ( z + 1 / 2 \\ , | \\ , \\tau ) \\ , G ( z + \\tau / 2 \\ , | \\ , \\tau ) \\ , G \\bigl ( z + ( 1 + \\tau ) / 2 \\ , | \\ , \\tau \\bigr ) } { G ( 1 / 2 \\ , | \\ , \\tau ) \\ , G ( \\tau / 2 \\ , | \\ , \\tau ) \\ , G \\bigl ( ( 1 + \\tau ) / 2 \\ , | \\ , \\tau \\bigr ) } . \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} = - 4 \\kappa \\int _ { \\R ^ d \\times \\R ^ d } \\left ( H _ { \\mu \\iota } D ^ 2 _ { x } \\phi ( | x - y | ) \\overline { H } _ { \\mu \\iota } + G _ { \\mu \\iota } D ^ 2 _ { x } \\phi ( | x - y | ) \\overline { G } _ { \\mu \\iota } \\right ) \\ , d x d y , & \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n k _ i r _ { p _ i } = 0 \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} U ( m , n ) = \\prod _ { i = 1 } ^ N U _ { i , m _ i , n _ i } \\in \\mathcal { B } ( \\ell _ 2 ( X ) ) , \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} \\frac { 1 } { ( n + 1 ) ^ { n - 1 } } c _ 1 ^ { o r b } ( X , \\Delta ) ^ n = \\frac { ( - 1 ) ^ n } { ( n + 1 ) ^ { n - 1 } } ( K _ X + \\Delta ) ^ n , \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\P _ { s , \\mu _ 1 } ( X _ t \\in B | \\tau _ { A } > t ) = \\limsup _ { t \\to \\infty } \\P _ { s , \\mu _ 2 } ( X _ t \\in B | \\tau _ { A } > t ) . \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} \\gamma ^ k \\gamma _ k = - 2 n , \\gamma _ k \\gamma ^ k = 2 n , \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\widetilde { m } _ { i j } \\ge \\left | \\sum _ { k = 1 } ^ { n } \\sigma _ k u _ { i k } \\overline { v } _ { j k } \\right | ^ 2 . \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} ( u _ 2 ^ e , h _ 2 ^ e ) | _ { y = 0 } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} \\left ( \\lambda \\begin{bmatrix} 0 \\\\ D ( \\lambda ) \\end{bmatrix} + B \\right ) \\left ( L _ \\epsilon ( \\lambda ^ \\ell ) \\otimes I _ n \\right ) + \\left ( L _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m \\right ) \\left ( \\lambda \\begin{bmatrix} 0 & - D ( \\lambda ) \\end{bmatrix} + C \\right ) \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} R _ 0 = \\{ ( ( n - 1 , r + 1 ) , ( n - 1 , r + 2 s + 1 ) ) , ( ( n - 1 , r + 3 ) , ( n - 1 , r + 2 s + 3 ) ) \\} . \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\alpha _ B \\pi ^ \\flat ( r ) \\widetilde { D } ( \\bar { r } ) & = \\alpha _ B \\pi ^ \\flat ( r ) \\cdot ( u - 1 ) = \\alpha _ C \\varphi ^ \\flat ( r ) \\cdot ( u - 1 ) = \\alpha _ C \\varphi ^ \\flat ( r ) u - \\alpha _ C \\varphi ^ \\flat ( r ) \\\\ & = \\alpha _ C \\psi ^ \\flat ( r ) - \\alpha _ C \\varphi ^ \\flat ( r ) = \\psi ^ \\sharp \\alpha _ F ( r ) - \\varphi ^ \\sharp \\alpha _ F ( r ) = D \\big ( \\alpha _ F ( r ) \\big ) \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} f ( x , v , \\lambda ) = \\sum _ { n = 0 } ^ \\infty f _ n ( x , v ) \\ , \\lambda ^ n \\ , . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} c ' \\otimes y _ { c ' } = c \\phi _ n ^ j \\otimes \\phi ^ { - j } ( y _ { c } ) = c \\otimes y _ c \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} \\mathcal R ( \\eta ) : = \\begin{cases} \\Re s > \\frac { 1 } { 2 } - \\eta , \\ \\Re z > \\frac { 1 } { 2 } - \\eta , & \\ \\Re z ' > \\frac { 1 } { 2 } - \\eta , \\\\ \\Re u > - \\eta , \\ \\Re v > - \\eta , & \\ \\Re w > - \\eta . \\end{cases} \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} b = \\frac { \\ell ' } { d ' } = b ' , a = \\frac { m ' } { d ' } = a ' . \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} Y ( A , z ) = Q ( \\rho ) Y ( R ( \\rho _ { z } ) ^ { - 1 } A , \\rho ( z ) ) Q ( \\rho ) ^ { - 1 } . \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\exp ( \\nu \\{ F ( X ^ { 1 / 2 , \\mu } ( t ) ) - F ( \\bar { x } ) \\} ) = \\frac { h ( X ^ { 1 / 2 , \\mu } ( t ) ) ^ \\nu } { h ( \\bar { x } ) ^ \\nu } . \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} \\operatorname { f i x } ( y , M ^ G ) < \\sum _ { \\sum i l _ i = m } \\frac { \\Delta ( l ) } { | C _ { G L _ m ( q ) } ( [ y ^ U ] _ \\beta ) | } < \\sum _ { \\sum i l _ i = m } c q ^ { \\frac { n m } { 4 } + ( \\sum l _ i - \\frac { m } { 4 } ) s - \\sum i l _ i ^ 2 - 2 \\sum _ { i < j } i l _ i l _ j } < q ^ { \\frac { 1 } { 4 } m ( n - m ) + c m } \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} G ( u _ \\mu , u _ \\nu ) = \\beta _ { \\mu \\nu } | u _ \\nu | ^ { p + 1 } | u _ \\mu | ^ { p - 1 } u _ \\mu + \\displaystyle { \\sum _ { \\substack { \\mu = 1 } } ^ N } \\lambda _ { \\mu \\nu } | u _ \\nu | ^ { p + 1 } | u _ \\mu | ^ { p - 1 } u _ \\mu \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\sum _ { n \\leqslant N } r ( n ) ^ 2 \\omega ( n ) & = \\prod _ p \\big ( 1 + r ( p ) ^ 2 \\omega ( p ) \\big ) - \\sum _ { n > N } r ( n ) ^ 2 \\omega ( n ) \\\\ & = \\big ( 1 + O \\big ( N ^ { - C / ( \\log \\log N ) ^ 3 } \\big ) \\big ) \\prod _ p \\big ( 1 + r ( p ) ^ 2 \\omega ( p ) \\big ) , \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} e \\ne h \\qquad | e v \\cap H _ { m + 1 } | = n | e v \\cap S _ V | . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} C ( T ) = \\sum _ { i = 1 } ^ { L - 1 } \\lceil c _ { i + 1 } - c _ i \\rceil , \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} ( \\rho , v ) ( x , 0 ) = \\left \\{ \\begin{array} { l l } ( \\rho _ - , u _ - ) , \\ \\ x < 0 , \\\\ ( \\rho _ + , u _ + ) , \\ \\ x > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\overline { M } = \\inf \\left \\{ M ' : w ( x , t ) \\leq \\frac { M ' } { M } \\mathcal { B } _ M ( x , t ) \\right \\} . \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} \\rho _ { p , q } : = I d _ 1 \\otimes \\widetilde { s h } _ { p - 1 , q } = \\Phi \\mathfrak g ^ { \\otimes p + q } \\rightarrow \\Phi \\mathfrak g ^ { \\otimes p + q } . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} Z : = \\bigcap _ { n \\geq 1 } \\overline { \\bigcup _ { i \\geq n } \\Phi _ i ^ { - 1 } ( 0 ) } , \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} \\Sigma ^ { d } ( \\psi ^ 0 \\phi ^ 0 ) \\circ \\omega ^ { d + 1 } = \\Sigma ^ { d } ( \\psi ^ 0 ) \\circ \\Sigma ^ d ( \\phi ^ 0 ) \\circ \\omega ^ { d + 1 } = \\omega ^ { d + 1 } . \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} \\Theta _ n ( q ) = \\frac { S _ n ( ( 1 - q ) A ) } { 1 - q } = \\sum _ { k = 0 } ^ { n - 1 } ( - q ) ^ k R _ { 1 ^ k , n - k } \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} l ' _ { i , j } & = \\min \\{ l ' _ { i , j - 1 } , l _ { i - 1 , j } \\} + l ' _ { i + 1 } + l _ { i , j + 1 } - \\min \\{ l ' _ { i + 1 , j } , l _ { i , j + 1 } \\} - l _ { i , j } \\\\ & = \\min \\{ l ' _ { i , j - 1 } , l _ { i - 1 , j } \\} + \\max \\{ l ' _ { i + 1 , j } , l _ { i , j + 1 } \\} - l _ { i , j } . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\Sigma _ B ( s ) = \\zeta ( 2 s ) \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { s } } \\sum _ { q = 1 } ^ { \\infty } \\frac { S ( 1 , n ^ 2 ; q ) } { q } \\varphi _ B \\left ( \\frac { 4 \\pi n } { q } \\right ) . \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} \\left ( u _ { \\lambda } \\right ) _ t & = \\nabla \\cdot \\left ( m \\ , U _ { \\lambda } ^ { m - 1 } \\nabla u _ { \\lambda } \\right ) \\qquad \\mbox { i n $ \\R ^ n \\times ( 0 , \\infty ) $ } \\\\ u _ { \\lambda } ( x , 0 ) & = u _ { 0 } \\left ( \\lambda ^ { a _ 2 } x \\right ) = u _ { 0 , \\lambda } ( x ) \\qquad \\forall x \\in \\R ^ n \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\mbox { e i t h e r } \\ ; \\ ; T _ { \\max } = \\infty \\quad \\mbox { o r e l s e } T _ { \\max } < \\infty \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\| u ( t , \\cdotp ) \\| _ { H ^ 1 _ 0 } + \\| u _ t ( t , \\cdotp ) \\| _ 2 \\rightarrow \\infty \\ ; \\ ; \\mbox { a s } \\ ; \\ ; t \\rightarrow T _ { \\max } . \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} \\boldsymbol { \\widetilde { \\gamma } } ( q A , q , d ) = P ( q , d ) \\boldsymbol { \\widetilde { \\gamma } } ( A , q , d ) \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\begin{array} { l l } h ^ { ( 1 ) } : = - \\Delta + U _ { \\rm t r a p } ^ { ( 1 ) } + c _ 1 V ^ { ( 1 ) } * | u _ 0 | ^ 2 + c _ 2 V ^ { ( 1 2 ) } * | v _ 0 | ^ 2 - \\mu ^ { ( 1 ) } , \\\\ h ^ { ( 2 ) } : = - \\Delta + U _ { \\rm t r a p } ^ { ( 2 ) } + c _ 2 V ^ { ( 2 ) } * | v _ 0 | ^ 2 + c _ 1 V ^ { ( 1 2 ) } * | u _ 0 | ^ 2 - \\mu ^ { ( 2 ) } \\end{array} \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} \\alpha _ { n , j } ^ { ( N - 1 ) } ( f ) = \\frac { 1 } { \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) } \\alpha _ { n } ^ { ( N - 1 ) } \\left ( ( f - \\mathbb { E } ( f | A _ { j } ^ { ( N ) } ) ) 1 _ { A _ { j } ^ { ( N ) } } \\right ) , \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} f _ k ( Q _ n \\cdot ( \\exp ( H ) , 1 ) , s ) = i ^ { n k } \\prod _ { j = 1 } ^ n ( e ^ { a _ j } + e ^ { - a _ j } ) ^ { - ( 2 n + 1 ) ( s + \\frac 1 2 ) } . \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} \\langle r ^ { - } _ { \\psi } ( \\nu _ c \\alpha _ n ) \\mathbf h _ p , \\mathbf h _ p \\rangle = \\chi _ { \\psi } ( p ^ n ) p ^ { ( 1 - n ) / 2 } \\varepsilon ( 1 / 2 , \\underline { \\chi } _ p ) \\underline { \\chi } _ p ( 2 ) \\int _ { \\Z _ p ^ { \\times } } \\psi ( - c p ^ { 2 n - 1 } z ^ 2 ) \\underline { \\chi } _ p ( z ) \\mathfrak G ( - 2 z p ^ { n - 1 } , \\underline { \\chi } _ p ) d z . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} d _ { 1 1 1 } \\ & = \\ - \\frac { d _ { 1 1 2 } ( l _ { 2 1 } - l _ { 1 1 } ) } { l _ { 1 2 } } - d _ { 1 2 1 } , \\\\ d _ { 2 1 1 } \\ & = \\ - \\frac { d _ { 2 2 2 } ( l _ { 2 1 } - l _ { 1 1 } ) } { l _ { 2 2 } } - d _ { 2 2 1 } , \\\\ d _ { 2 1 2 } \\ & = \\ - \\frac { l _ { 1 2 } d _ { 2 2 2 } } { l _ { 2 2 } } . \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\Phi _ { \\mathbf h _ p } ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) = \\frac { \\chi _ { \\psi } ( p ) \\underline { \\chi } _ p ( \\gamma ) p ^ { - 1 / 2 } } { p - 1 } ( p - G ( - \\gamma , p ) ) , \\Phi _ { \\pmb { \\phi } _ p } ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) = \\chi _ { \\overline { \\psi } _ p } ( p ) p ^ { - 1 / 2 } , \\Phi _ { \\breve { \\mathbf g } _ p } ( \\nu _ { \\gamma } \\beta _ 1 \\nu _ { \\delta } ) = \\underline { \\chi } _ p ( \\gamma ) . \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{gather*} \\vartheta _ 2 = X _ { 2 , 4 } + X _ { 2 , 3 } X _ { 2 , 4 } + X _ { 2 , 2 } X _ { 2 , 3 } X _ { 2 , 4 } + X _ { 2 , 1 } X _ { 2 , 2 } X _ { 2 , 3 } X _ { 2 , 4 } , \\\\ \\vartheta _ 5 = \\vartheta _ { 7 - 2 } = X _ { 3 , 2 } + X _ { 2 , 2 } X _ { 3 , 2 } + X _ { 1 , 2 } X _ { 2 , 2 } X _ { 3 , 2 } . \\end{gather*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\Delta _ T ( s , \\alpha , P ) = \\sum _ { k = 0 } ^ { T - 1 } \\left ( f ( s _ 1 + k \\alpha _ 1 , \\dots , s _ { d - 1 } + k \\alpha _ { d - 1 } ) - \\lambda ( P ) \\right ) , \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} ( \\left [ ( b y / x ) ( b \\backslash z x ) \\right ] / x ) y ^ { \\alpha } = ( b y / x ) ( b \\backslash z y ^ { \\alpha } ) \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\Pi _ N \\omega = \\sum _ { k \\in \\Lambda _ N } \\ < \\omega , \\tilde e _ k \\ > \\tilde e _ k . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} & x ( t + 1 ) = A x ( t ) + B w ( t ) , & y ( t ) = C x ( t ) + D w ( t ) , \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} x _ j = \\frac { a _ j } { 1 + a _ j ^ 2 } , y _ j = \\frac 1 { 1 + a _ j ^ 2 } , e ^ { i \\theta _ j } = \\frac { 1 - i a _ j } { ( 1 + a _ j ^ 2 ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} \\overline { M } ^ - = \\left ( \\overline { N } ^ + \\right ) ^ { - 1 } \\overline { N } ^ - , ~ ~ ~ ~ ~ ~ \\overline { M } ^ + = \\left ( \\overline { N } ^ - \\right ) ^ { - 1 } \\overline { N } ^ + . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} Q ( \\theta ) * ( z _ 1 , \\ z _ 2 , \\ z _ 3 ) = ( z _ 1 \\cdot e ^ { i ( 2 \\theta ) } , \\ z _ 2 \\cdot e ^ { i ( 2 \\theta ) } , \\ z _ 3 \\cdot e ^ { i ( 4 \\theta ) } ) \\in \\mathbb { C } ^ 3 \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) } w ' ( e ) = \\frac { 1 } { m ^ { ( p ' - p ) / ( p ' - r ) } } \\sum _ { e \\in E ( G ) } w ( e ) ^ { ( p - r ) / ( p ' - r ) } \\leq 1 . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} ( - \\Delta - \\mu ^ 2 ) v _ \\mu = \\mu ^ 2 \\sum _ { j = 1 } ^ N \\frac { a _ j } { 2 \\pi } \\log | x - y _ j | = - \\mu ^ 2 \\psi = ( H _ { \\alpha , Y } - \\mu ^ 2 ) \\psi , \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} B ( v , e ) & = \\mu B _ 1 ( v , e ) + ( 1 - \\mu ) B _ 2 ( v , e ) , \\\\ w ( e ) & = \\xi w _ 1 ( e ) + ( 1 - \\xi ) w _ 2 ( e ) . \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\frac { d + 1 } { k ( d + 1 - k ) } \\left | \\sigma - \\frac { k ^ { 2 } } { d + 1 } \\right | & = \\frac { d + 1 } { k ( d + 1 - k ) } \\left [ \\frac { k ^ { 2 } } { d + 1 } - \\sigma \\right ] \\\\ & = \\frac { k } { d + 1 - k } - \\frac { d + 1 } { k ( d + 1 - k ) } [ k - ( r - \\lambda ) ] . \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} X _ 1 & \\triangleq \\beta _ { 2 , 2 } ^ { - 1 } \\Bigl ( a , b _ 0 = x _ 1 , \\ , b _ 1 = b _ 2 = ( x _ 2 - x _ 1 ) / 2 \\Bigl ) , \\\\ X _ 2 & \\triangleq \\beta _ { 2 , 2 } ^ { - 1 } \\Bigl ( a , b _ 0 = ( x _ 1 + x _ 2 ) / 2 , \\ , b _ 1 = a _ 1 / 2 , \\ , b _ 2 = a _ 2 / 2 \\Bigr ) , \\\\ X _ 3 & \\triangleq \\beta _ { 2 , 2 } ^ { - 1 } \\Bigl ( a , b _ 0 = a _ 1 + a _ 2 , \\ , b _ 1 = b _ 2 = ( x _ 1 + x _ 2 - a _ 1 - a _ 2 ) / 2 \\Bigl ) . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} f ( x ) = \\int _ 0 ^ 1 \\chi _ P ( g _ x ( t ) ) \\ , \\mathrm { d } t + \\int _ 0 ^ 1 \\chi _ { P + ( 1 , 0 ) } ( g _ x ( t ) ) \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} t ^ { - 1 } = P _ 1 ( t ) + t ^ { - 1 } P ' _ { m , m ' } ( t ) . \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} F _ { S } ^ { \\ast } \\left ( { { x ^ { \\ast } } , { \\lambda } } \\right ) = \\left \\{ { \\begin{array} { l l } { { { \\sum \\limits _ { s \\in S } { { \\lambda _ { s } b _ { s } } } } } , } & { \\mathrm { i f } \\ ; { { { \\sum \\limits _ { s \\in S } { { \\lambda _ { s } a _ { s } ^ { \\ast } } } } } } } = x ^ { \\ast } { { - c ^ { \\ast } } } \\mathrm { a n d } { { \\lambda _ { s } \\geq 0 \\ , , \\ } \\forall } s \\in S , \\\\ { + \\infty , } & { \\mathrm { e l s e . } } \\end{array} } \\right . \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\sum _ { i , j , l } \\lambda _ { i , j , l } q ^ { r ( n _ { j } + m _ { l } ) } = 0 . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} T ^ { ( 1 ) } _ { m 0 } + c _ 1 V ^ { ( 1 ) } _ { m 0 0 0 } + c _ 2 V ^ { ( 1 2 ) } _ { m 0 0 0 } & = 0 \\\\ T ^ { ( 2 ) } _ { m 0 } + V ^ { ( 2 ) } _ { m 0 0 0 } + c _ 1 V ^ { ( 1 2 ) } _ { m 0 0 0 } & = 0 . \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} | y ^ * ( u _ i - y _ i ) | & \\le | y ^ * ( T x _ i - y _ i ) | + \\frac { \\delta } { 2 } = | y ^ * ( T ( x _ i - y _ i ) ) | + \\frac { \\delta } { 2 } \\\\ & = | \\varphi ( y ^ * ) ( x _ i - y _ i ) | + \\frac { \\delta } { 2 } < \\frac { \\delta } { 2 } + \\frac { \\delta } { 2 } = \\delta \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} \\lim _ { g \\to e ; g \\in S } \\tau _ g ^ s ( x ) = \\tau _ e ( x ) . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} G ( \\boldsymbol { v } , z ) = G ( \\boldsymbol { u } , e ^ z - 1 ) , G ( \\boldsymbol { u } , z ) = G ( \\boldsymbol { v } , \\log ( z + 1 ) ) . \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} \\frac { w ( e \\cup f ) } { B ( v , e \\cup f ) } = \\begin{dcases} \\frac { w _ 1 ( e ) } { B _ 1 ( v , e ) } , & \\ v \\in V ( G _ 1 ) , \\\\ [ 1 m m ] \\frac { w _ 2 ( f ) } { B _ 2 ( v , f ) } , & \\ v \\in V ( G _ 2 ) . \\end{dcases} \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} | \\underline m | = m _ 1 + \\ldots + m _ n . \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} x _ 1 = \\frac { 1 } { 4 } y _ 4 \\ ; , \\ ; \\ ; x _ 2 = - y _ 3 \\ ; , \\ ; \\ ; x _ 3 = \\frac { 1 } { 4 } y _ 2 \\ ; , \\ ; \\ ; x _ 4 = - \\frac { 1 } { 4 } y _ 1 , \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} s + \\varepsilon - \\max ( s , \\delta _ l ) = \\min ( \\varepsilon , s + \\varepsilon - \\delta _ l ) \\leq \\varepsilon . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} D _ s X ( t ) ( \\omega ) = \\sigma ( s , X ( s ) ( \\omega ) ) & + \\int _ s ^ t \\nabla _ x b ( r , X ( r ) ( \\omega ) ) D _ s X ( r ) ( \\omega ) d r \\\\ & + \\int _ s ^ t \\nabla _ x \\sigma ( r , X ( r ) ( \\omega ) ) D _ s X ( r ) ( \\omega ) d W ( r ) . \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\abs { G ( a / q ) } \\leq \\prod _ { \\gamma = ( 0 , \\gamma '' ) \\in \\Gamma } \\frac { 1 } { \\varphi ( q _ \\gamma ) } \\leq \\frac { 1 } { \\varphi ( q ) } , \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} g ^ { ( m ) } ( t ) = f ^ { ( m ) } ( t ) - f ^ { ( m ) } ( t + 2 ) = - \\int _ 0 ^ 2 f ^ { ( m + 1 ) } ( t + s ) \\ , d s , \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} g _ { 2 } ( M ) = \\frac { 1 } { \\left | B _ { \\frac { 1 } { 8 } d _ { 0 } ( M ) } ( M ) \\right | } \\int \\limits _ { B _ { \\frac { 1 } { 8 } d _ { 0 } ( M ) } ( M ) } g _ { 1 } ( \\widetilde { M } ) \\ , d m _ { 3 } ( \\widetilde { M } ) , \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{gather*} ( h m ) _ { [ 0 ] } \\otimes ( h m ) _ { [ 1 ] } = h ^ { 2 1 } m _ { [ 0 ] } \\otimes h ^ { 2 2 } m _ { [ 1 ] } S \\big ( h ^ 1 \\big ) . \\end{gather*}"}
-{"id": "9542.png", "formula": "\\begin{align*} q & = x _ 1 \\\\ r & = \\frac { \\sqrt { x _ { n + 1 } - \\bigl ( x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 \\bigr ) } } { \\sqrt { x _ 1 } } \\\\ t & = \\frac { x _ 1 x _ 2 } { x _ { n + 1 } - \\bigl ( x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 \\bigr ) } \\\\ s _ j & = \\frac { x _ j \\sqrt { x _ 1 } } { \\sqrt { x _ { n + 1 } - \\bigl ( x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 \\bigr ) } } , \\ , j = 3 , \\ldots , n , \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} \\begin{gathered} \\varepsilon - K \\varepsilon ^ { \\frac { H - \\delta } { 1 + \\delta - H } } > 0 , \\quad \\forall \\varepsilon < \\varepsilon ^ * . \\end{gathered} \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\widehat { u ^ 1 } _ k = ( - 1 ) ^ { k } \\widehat { u } _ k \\qquad \\forall \\ , k \\in \\left \\{ 0 , \\dotsc , 2 ^ { j + 1 } - 1 \\right \\} . \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\| M \\| _ F : = \\sqrt { \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ n | M _ { i , j } | ^ 2 } . \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} d ( x _ 1 , \\ldots , x _ n ) = \\sum _ { 1 \\leq i < j \\leq n } ( - 1 ) ^ j ( x _ 1 , \\ldots , x _ { i - 1 } , [ x _ i , x _ j ] , x _ { i + 1 } , \\ldots , \\hat { x } _ j , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} \\pm \\big ( \\widetilde { H } _ { N } - & f _ M \\widetilde { H } _ { N } f _ M - g _ M \\widetilde { H } _ { N } g _ M \\big ) \\leqslant \\frac { \\kappa _ 2 } { M ^ 2 } \\big ( \\widetilde { H } _ { N } + \\kappa _ 2 N \\big ) . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} M _ { 2 } = g ^ { - 1 } M _ { 1 } g \\ { \\rm a n d } \\ \\sigma _ { 2 } = \\sigma _ { 1 } ^ { g } \\otimes \\chi , \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\Psi _ p ( \\xi ; \\alpha _ p ) = 2 p ^ { e _ p ( 1 / 2 - k ) } a ( p ^ { e _ p } ) \\underline { \\chi } _ { p } ( \\mathfrak f _ { \\xi } ) W _ { \\tilde { \\varphi } _ p , \\xi } ( 1 ) . \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} \\psi ^ { \\lambda } ( \\zeta ) : = \\inf _ { \\eta \\in \\R ^ { d } } \\left [ \\psi ( \\eta ) + \\frac { 1 } { 2 \\lambda } | \\zeta - \\eta | ^ { 2 } \\right ] , \\quad \\zeta \\in \\R ^ { d } , \\ ; \\lambda > 0 , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} \\widetilde { P } ' ( x ) = \\frac { 1 - x - \\sqrt { x ^ 2 - 6 x + 1 } } { 2 x } , \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} C _ 1 = 2 M _ 2 , C _ 2 : = 2 M _ 3 + 2 + 4 \\Vert \\rho _ 0 \\Vert _ { L ^ 1 } . \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} & y _ { * * } ( 4 u _ { * * } ^ 2 y _ { * * } ^ 4 - 4 u _ { * * } ^ 2 + 8 y _ { * * } ^ 2 - 1 ) = 0 , \\\\ & u _ { * * } ( 2 u _ { * * } ^ 2 y _ { * * } ^ 4 + 2 u _ { * * } ^ 2 y _ { * * } ^ 2 - 8 y _ { * * } ^ 4 - 4 y _ { * * } ^ 2 + 1 ) = 0 . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} r ^ { 2 } R _ { 2 } ( r ) - ( \\alpha ^ { 2 } r ^ { 2 } - \\beta { r } + \\gamma { r ^ { \\rho + 2 } } + \\delta ) R ( r ) = 0 . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} N ( t ) = \\sum \\beta _ k e ^ { - \\eta _ k t } \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} \\liminf _ { t \\in I , t \\to \\infty } \\frac { 1 } { d ' _ t } \\prod _ { k = 0 } ^ { \\left \\lfloor \\frac { t - s } { t _ 0 } \\right \\rfloor - 1 } ( 1 - d _ { t - k } ) = 0 , \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\langle \\tau _ p ( g ) \\breve { \\mathbf g } _ p , \\breve { \\mathbf g } _ p \\rangle = \\int _ { K _ { 0 0 } } \\mathbf g _ p ( h g \\varpi _ p ) \\overline { \\mathbf g ( h \\varpi _ p ) } d h = p ^ { 1 / 2 } \\xi _ 1 ( p ) ^ 2 \\int _ { K _ { 0 0 } } \\mathbf g _ p ( h g \\varpi _ p ) \\underline { \\chi } _ p ^ { - 1 } ( h ) d h . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} f _ { N I G } ( x ; \\mu , \\alpha , \\beta , \\delta ) = \\frac { \\alpha \\delta } { \\pi } \\exp ( \\delta \\sqrt { \\alpha ^ 2 - \\beta ^ 2 } + \\beta x ) \\frac { K _ 1 ( \\alpha \\sqrt { \\delta ^ 2 + x ^ 2 } ) } { \\sqrt { \\delta ^ 2 + x ^ 2 } } , x \\in \\R , \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } e ^ { i t y ^ 2 } { u } ( y ) d y & = 2 \\pi \\int _ 0 ^ \\infty e ^ { i t r ^ 2 } M _ u ( r ) r d r \\\\ & = \\pi \\int _ { \\R } e ^ { i t r } N _ u ( r ) d r = \\sqrt { 2 \\pi } \\pi ( { \\mathcal F } ^ \\ast N _ u ) ( t ) , \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u ^ 3 + v ^ 2 + \\frac { 1 2 x y ^ 2 - x + 1 2 y ^ 3 - 3 y - 1 } { 2 ( y - 4 y ^ 3 ) } v - \\left ( \\frac { 1 2 x y ^ 2 - x + 4 y ^ 3 - y - 1 } { 4 ( y - 4 y ^ 3 ) } \\right ) ^ 2 - \\frac { 1 2 x y ^ 2 - x + 4 y ^ 3 - y - 1 } { 4 ( y - 4 y ^ 3 ) } + 1 = 0 \\\\ v ^ 2 = \\frac { 4 8 x y ^ 5 - 1 6 x y ^ 3 - 1 2 x y ^ 2 + x y + x + 4 8 y ^ 6 - 2 4 y ^ 4 - 4 y ^ 3 + 3 y ^ 2 + y + 1 } { 2 ( y - 4 y ^ 3 ) ^ 2 } \\\\ x ^ 2 + 3 y ^ 2 - 1 = 0 \\\\ \\end{array} \\right . . \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t u ( t , x ) = \\nu \\Delta u ( t , x ) - [ u ( t , x ) \\cdot \\nabla ] u ( t , x ) - \\nabla \\pi ( t , x ) \\\\ & u ( t , x ) = 0 \\\\ & u ( 0 , x ) = u _ { 0 } ( x ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} u _ 1 = & \\ \\frac { 1 } { l _ { 1 1 } + 1 } ( d _ { 1 1 1 } - l _ { 1 1 } , d _ { 2 1 1 } ) , & u _ 2 = & \\ u _ 1 \\ + \\ \\left ( \\frac { l _ { 1 1 } } { l _ { 1 1 } + 1 } , 0 \\right ) , \\\\ u _ 3 = & \\ \\frac { 1 } { 3 } ( - 1 , 0 ) , & u _ 4 = & \\ u _ 3 \\ + \\ \\left ( \\frac { 2 } { 3 } , 0 \\right ) , \\\\ u _ 5 = & \\ \\frac { 2 l _ { 0 1 } } { ( l _ { 0 1 } + 2 ) l _ { 1 1 } } ( - d _ { 1 1 1 } , - d _ { 2 1 1 } ) , & u _ 6 = & \\ u _ 5 \\ + \\ \\left ( \\frac { 2 l _ { 0 1 } } { l _ { 0 1 } + 2 } , 0 \\right ) . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} \\log \\theta _ { t _ { i - 1 } } ( M _ { i - 1 } ^ j ) > t _ { i - 1 } h _ { i - 1 } ^ j , ~ j = 1 , \\cdots , k . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } a _ k ^ { \\prime \\prime } + { n - 1 \\over r } a _ k ^ { \\prime } + \\left ( p u _ R ^ { p - 1 } ( r ) - { \\lambda _ k \\over r ^ 2 } \\right ) a _ k ( r ) = 0 \\ \\hbox { i n } \\ ( R , 1 ) , \\\\ a _ k ( R ) = a _ k ( 1 ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} & X ^ 2 + X + k + 1 + \\frac { ( C _ 1 ^ 2 + C _ 0 C _ 2 ) ( C _ 2 ^ 2 + C _ 1 C _ 3 ) } { ( C _ 1 C _ 2 + C _ 0 C _ 3 ) ^ 2 } \\\\ = \\ , & \\Bigl ( X + \\frac D { C _ 1 C _ 2 + C _ 0 C _ 3 } \\Bigr ) \\Bigl ( X + 1 + \\frac D { C _ 1 C _ 2 + C _ 0 C _ 3 } \\Bigr ) \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} \\sum _ { i = d + 2 } ^ { k + 1 } \\left ( i - \\frac { 1 - \\lambda ( d + 1 ) } { w - \\lambda } \\right ) \\rho _ { w , \\lambda } ^ i = 0 . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} & I _ + : = \\{ i \\in I : \\ m _ i \\} , \\\\ & I _ - : = \\{ i \\in I : \\ m _ i \\} . \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} G ( y , z ) ~ = ~ \\sum _ { t = 0 } ^ \\infty \\P _ { y \\star u } ( X ( t ) = z \\star u , \\ ; \\eta _ u > t ) \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\sup _ { B _ { \\frac { 1 } { 2 } \\lambda _ t } \\times Y } | \\nabla ^ { j , \\hat { g } _ t } \\hat { g } ^ \\bullet _ t | _ { \\hat { g } _ t } = \\lambda _ t ^ { - j } \\sup _ { B _ { \\frac { 1 } { 2 } } \\times Y } | \\nabla ^ { j , g _ t } g ^ \\bullet _ t | _ { g _ t } \\leq C \\lambda _ t ^ { - j } \\to 0 . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\rho _ a ( u _ 1 , \\dots , u _ n ) = d ( u _ n , u _ 1 ) . \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} \\sum _ { a \\in A } c _ a \\pi _ { \\mathcal { R } } b _ a v _ L = 0 , \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\| T \\| _ { c b } = \\sup _ { n \\geq 1 } \\left \\| _ n \\otimes T : M _ n ( \\C ) \\otimes \\mathcal { H } \\to M _ n ( \\C ) \\otimes \\mathcal { H } \\right \\| < \\infty . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} f ^ L ( \\vec x ) = & \\sum _ { i = 0 } ^ { n - 1 } ( m ( i + 1 ) x _ { \\sigma ( i + 1 ) } - m ( i ) x _ { \\sigma ( i ) } ) f ( V _ { x _ { \\sigma ( i ) } } ^ + , V _ { x _ { \\sigma ( i ) } } ^ - ) \\\\ = & \\sum _ { i = 1 } ^ { n - 1 } x _ { \\sigma ( i ) } m ( i ) ( f ( V _ { x _ { \\sigma ( i - 1 ) } } ^ + , V _ { x _ { \\sigma ( i - 1 ) } } ^ - ) - f ( V _ { x _ { \\sigma ( i ) } } ^ + , V _ { x _ { \\sigma ( i ) } } ^ - ) ) \\\\ & + x _ { \\sigma ( n ) } m ( n ) f ( V _ { x _ { \\sigma ( n - 1 ) } } ^ + , V _ { x _ { \\sigma ( n - 1 ) } } ^ - ) . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} { \\rm t r } ( C _ r ^ T C _ r P ' ) = - 2 { \\rm t r } ( A _ r '^ T Q P ) + 2 { \\rm t r } ( B _ r '^ T Q B _ r ) , \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} \\mathfrak { M } ( i q \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\int \\limits _ \\mathbb { R } e ^ { i q x } \\ , f ( x ) \\ , d x . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\int _ { \\Omega } \\left ( \\frac { \\hat { u } _ 2 ( y , k ) } { c _ 2 ^ 2 ( y ) } - \\frac { \\hat { u } _ 1 ( y , k ) } { c _ 1 ^ 2 ( y ) } \\right ) \\varphi d y = 0 , \\ \\ \\ \\ \\forall k \\in ( 0 , \\epsilon ) . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} Z : = \\left \\{ ( x , y ) \\in U \\times Y : \\left ( A ( x ) - \\mathbf Y ( y ) \\right ) ^ { \\wedge t } = 0 \\right \\} . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} X _ { t } = x - \\int _ { 0 } ^ { t } \\eta _ { s } \\ , d s + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } L ^ { b } X _ { s } \\ , d s + \\int _ { 0 } ^ { t } \\langle b \\nabla X _ { s } , d W _ { s } \\rangle \\quad \\P \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} { \\rm r . h . s . } = \\left ( \\frac { A ' _ { i - 1 , j - 1 } } { A ' _ { i , j } } \\prod _ { k = i - a - 1 } ^ { b - j - 1 } \\lambda _ k \\right ) \\left ( \\frac { A _ { i - 1 , j - 1 } } { A _ { i , j } } \\prod _ { k = i - a } ^ { b - j } \\lambda _ k \\right ) = p ^ * \\big ( L _ { i , j } ' L _ { i , j } \\big ) . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} B ( ( u , v ) , f ) & = \\begin{dcases} \\frac { B _ 1 ( u , \\pi _ 1 ( f ) ) B _ 2 ( v , \\pi _ 2 ( f ) ) } { ( r - 1 ) ! } , & \\ ( u , v ) \\in f , \\\\ 0 , & , \\end{dcases} \\\\ [ 2 m m ] w ( f ) & = \\frac { w _ 1 ( \\pi _ 1 ( f ) ) w _ 2 ( \\pi _ 2 ( f ) ) } { r ! } . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} M _ { 2 , f , \\psi } ( q ) = M _ { 2 , f , \\psi } ^ 0 ( q ) + M _ { 2 , f , \\psi } ^ { + } ( q ) , \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} \\big \\langle \\Lambda _ v P _ { v , \\rm R } F ( v ) , P _ { v , \\rm R } F ( v ) \\big \\rangle = C _ v \\sum _ { j = 1 } ^ { \\deg ( v ) } | F _ j ( v ) | ^ 2 . \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\langle I _ { \\lambda , p } ' ( u _ { n } , v _ { n } ) , ( \\varphi _ { n } , \\psi _ { n } ) \\rangle = \\langle I _ { \\lambda } ' ( u _ { n } , v _ { n } ) , ( \\varphi _ { n } , \\psi _ { n } ) \\rangle + o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} ( - \\Delta _ { p , q } ) ^ { s } u ( x ) = 2 \\lim _ { \\delta \\searrow 0 } \\int _ { \\mathbb { G } \\setminus B _ { q } ( x , \\delta ) } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d y , \\ , \\ , \\ , x \\in \\mathbb { G } , \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} Q _ 1 ( t ) = a _ { 0 , \\sup } ( t ) + \\frac { \\chi _ 1 } { 2 d _ 3 } \\big ( k { \\bar r _ 1 } + l { \\bar r _ 2 } \\big ) + \\frac { k ^ 2 } { 4 \\lambda d _ 3 } \\Big ( \\frac { \\chi _ 1 ^ 2 { \\bar r _ 1 } ^ 2 } { d _ 1 } + \\frac { \\chi _ 2 ^ 2 { \\bar r _ 2 } ^ 2 } { d _ 2 } \\Big ) + \\frac { a _ { 2 , \\sup } ( t ) { \\bar r _ 1 } + b _ { 1 , \\sup } ( t ) { \\bar r _ 2 } } { 2 } , \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\mathcal I _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { \\zeta _ p ( 2 ) ^ { 2 } ( p ^ 2 + w _ p ) ( p + w _ p ) } { p ^ 3 } \\frac { p - w _ p } { \\zeta _ p ( 2 ) ( p ^ 2 + w _ p ) } = \\zeta _ p ( 2 ) \\frac { ( p + w _ p ) ( p - w _ p ) } { p ^ 3 } = \\frac { p ^ 2 ( p ^ 2 - 1 ) } { ( p ^ 2 - 1 ) p ^ 3 } = \\frac { 1 } { p } . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} f ( u ) = u ^ { 3 } + \\beta u ^ 5 + f _ { 5 } ( u ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\beta < 0 , \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} & ( A ' ! _ \\lambda B ' ) ^ { - 1 } - ( A ' \\nabla _ \\lambda B ' ) ^ { - 1 } = \\lambda ( 1 - \\lambda ) ( B - A ) ( A ' \\sharp B ' ) ^ { - 2 } ( A ' \\nabla _ \\lambda B ' ) ^ { - 1 } ( B - A ) \\\\ & ( A ' ! _ \\lambda B ' ) ^ { - 1 } ( A ' \\nabla _ \\lambda B ' ) - I = \\lambda ( 1 - \\lambda ) ( A - B ) A '^ { - 1 } B '^ { - 1 } ( A - B ) . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} C _ 1 C _ 2 + C _ 0 C _ 3 \\ , & = a b + ( 1 + a ^ 2 + b ^ 2 ) ( Y ^ 2 + Y + k ) , \\cr C _ 2 ^ 2 + C _ 1 C _ 3 \\ , & = 1 + b + a b + ( 1 + a ^ 2 + b ^ 2 ) ( Y ^ 2 + Y + k ) . \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} \\begin{pmatrix} Z ^ 2 A _ 3 & 0 & 0 \\\\ 0 & Z ^ 4 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} y ( z \\cdot y ^ { \\alpha } x ) = \\left [ ( y \\cdot z x ) / x \\right ] \\cdot y ^ { \\alpha } x \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G _ 1 * G _ 2 ) = \\frac { ( r _ 1 + r _ 2 ) ^ { 1 - ( r _ 1 + r _ 2 ) / p } } { r _ 1 ^ { 1 - r _ 1 / p } r _ 2 ^ { 1 - r _ 2 / p } } \\lambda ^ { ( p ) } ( G _ 1 ) \\lambda ^ { ( p ) } ( G _ 2 ) . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} & \\left ( \\pi ( 1 ) , \\cdots , \\pi ( i _ 1 - 1 ) , \\pi ( i _ 2 ) , \\cdots , \\pi ( j _ 2 ) , \\pi ( j _ 1 + 1 ) , \\right . \\\\ & \\left . \\cdots , \\pi ( i _ 2 - 1 ) , \\pi ( i _ 1 ) , \\cdots , \\pi ( j _ 1 ) , \\pi ( j _ 2 + 1 ) , \\cdots , \\pi ( N ) \\right ) . \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} j _ ! ( \\theta _ { \\tilde Z } ) = \\theta . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{gather*} \\mathcal G ^ \\ast _ { k , ( a _ 1 , a _ 2 ; N ) } ( \\tau ) : = \\mathcal G _ { k , ( a _ 1 , a _ 2 ; N ) } ^ \\ast ( 0 , \\tau ) = \\lim _ { \\operatorname { R e } ( s ) > 0 \\atop s \\rightarrow 0 } \\mathcal G _ { k , ( a _ 1 , a _ 2 ; N ) } ^ \\ast ( s , \\tau ) . \\end{gather*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\int _ { \\R ^ + _ 0 } \\phi ( s ) T _ t n ( s ) \\d s : = \\int _ { \\R ^ + _ 0 } \\phi ( s + t ) n ( s ) \\d s . \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\hat { \\varphi } ( t ) = \\frac { \\pi i } { 2 \\sinh ( \\pi t ) } \\int _ 0 ^ { \\infty } ( J _ { 2 i t } ( x ) - J _ { - 2 i t } ( x ) ) \\varphi ( x ) \\frac { d x } { x } , \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} D _ q { { \\rm { E } } _ q ^ x } = { { \\rm { E } } _ q ^ { q x } } , D _ q { { \\rm { e } } _ q ^ x } = { { \\rm { e } } _ q ^ x } \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} F _ 4 = \\ , & a + a ^ 2 + a ^ 3 + a ^ 4 + b + a ^ 2 b + b ^ 2 + a b ^ 2 + b ^ 3 + b ^ 4 , \\\\ F _ 3 = \\ , & 1 + a ^ 2 + b + a ^ 2 b + b ^ 2 + b ^ 3 , \\\\ F _ 2 = \\ , & a ^ 3 + a ^ 4 + b + b ^ 2 + b ^ 3 + b ^ 4 , \\\\ F _ 1 = \\ , & a + b + a b + b ^ 3 + k + a ^ 2 k + b k + a ^ 2 b k + b ^ 2 k + b ^ 3 k , \\\\ F _ 0 = \\ , & b + b ^ 2 + a b ^ 2 + a ^ 2 b ^ 2 + a k + b k + a b ^ 2 k + b ^ 3 k + a k ^ 2 + a ^ 2 k ^ 2 \\\\ & + a ^ 3 k ^ 2 + a ^ 4 k ^ 2 + b k ^ 2 + a ^ 2 b k ^ 2 + b ^ 2 k ^ 2 + a b ^ 2 k ^ 2 + b ^ 3 k ^ 2 + b ^ 4 k ^ 2 . \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} u ( U ) = g ( U , \\tilde { J } E ) , v ( U ) = g ( U , \\tilde { J } N ) . \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\lambda _ j ( r ) = \\lambda _ { j ^ \\prime } ( r ) \\textrm { f o r a l l } r \\ge 0 . \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ { + } } \\rho _ \\varepsilon = \\infty . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} \\tau _ z = \\inf ( t \\geq 0 \\colon \\| X _ t \\| \\geq z ) . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\begin{array} { c c c } a ^ { ( * ) } a + c ^ { ( * ) } c = I , & a a ^ { ( * ) } + q ^ { 2 } c ^ { ( * ) } c = I , & c ^ { ( * ) } c = c c ^ { ( * ) } , \\\\ a c = q c a , & a c ^ { ( * ) } = q c ^ { ( * ) } a . \\\\ c ^ { ( * ) } a ^ { ( * ) } = q a ^ { ( * ) } c ^ { ( * ) } , & c a ^ { ( * ) } = q a ^ { ( * ) } c . \\end{array} \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} A _ { \\sigma , { \\tau } } = A _ { \\sigma _ 1 , { K N } } + R _ 1 , \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } ( \\mathbf { P } _ { \\mathcal { B } | \\mathcal { A } } \\mathbf { P } _ { \\mathcal { A } | \\mathcal { B } } ) ^ { k } \\cdot V _ { 1 } ( f ) = \\mathbf { 0 } _ { m _ { 1 } , 1 } , \\ ; \\lim _ { k \\rightarrow + \\infty } ( \\mathbf { P } _ { \\mathcal { A } | \\mathcal { B } } \\mathbf { P } _ { \\mathcal { B } | \\mathcal { A } } ) ^ { k } \\cdot V _ { 2 } ( f ) = \\mathbf { 0 } _ { m _ { 2 } , 1 } . \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} ( k + 2 ) ( k - m + 1 ) ( f ( k , i ) - f ( k + 2 , i ) ) = G ( k , i + 1 ) - G ( k , i ) \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\vert F ( z + h ) - F ( z ) \\vert & = \\vert L ( h ) + e \\vert \\\\ & \\leq \\vert L ( h ) \\vert + \\vert e \\vert \\\\ & \\leq \\frac 6 5 \\vert h \\vert + \\frac 6 5 w _ x ( \\vert F ( z + h ) - F ( z ) \\vert ) \\cdot \\vert F ( z + h ) - F ( z ) \\vert . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} \\mathrm { s u p p } ( \\phi ) \\subset \\mathrm { s u p p } ( \\psi _ m ) & , \\ ; \\ ; \\psi _ m = 1 ( \\phi ) ; \\\\ \\mathrm { s u p p } ( \\psi _ i ) \\subset \\mathrm { s u p p } ( \\psi _ { i - 1 } ) & , \\ ; \\ ; \\psi _ { i - 1 } = 1 ( \\psi _ i ) \\ ; \\ ; \\ ; \\forall i . \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { m } \\binom { m } { t } \\left ( k + t \\right ) ! S \\left ( j , k + t \\right ) = k ! \\sum _ { t = 0 } ^ { m - 1 } \\left ( - 1 \\right ) ^ { t } s \\left ( m , m - t \\right ) S \\left ( j + m - t , k + m \\right ) , \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} ( U _ 2 ( \\lambda ) ^ { - T } \\oplus I _ { m _ 1 } ) \\ , \\mathcal { L } ( \\lambda ) \\ , \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - X ( \\lambda ) \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ Q ( \\lambda ) \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ M _ { ( \\tau = 1 , 0 , 0 ) } ^ q \\bigr ] = \\frac { \\Gamma \\bigl ( 1 - q \\bigr ) \\Gamma \\bigl ( 3 - 2 q \\bigr ) } { \\Gamma \\bigl ( 3 - q \\bigr ) } \\prod \\limits _ { m = 1 } ^ \\infty m ^ { 2 q } \\frac { \\Gamma ^ 3 \\bigl ( 1 - q + m \\bigr ) } { \\Gamma ^ 3 \\bigl ( 1 + m \\bigr ) } \\frac { \\Gamma \\bigl ( 2 - q + m \\bigr ) } { \\Gamma \\bigl ( 2 - 2 q + m \\bigr ) } . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} D _ s X ( t ) ( \\omega ) = & \\sigma ( s , \\omega , X ( s ) ( \\omega ) ) + \\int _ s ^ t ( D _ s b ) ( r , \\omega , X ( r ) ( \\omega ) ) d r + \\int _ s ^ t ( D _ s \\sigma ) ( r , \\omega , X ( r ) ( \\omega ) ) d W ( r ) \\\\ & + \\int _ s ^ t \\nabla _ x b ( r , \\omega , X ( r ) ( \\omega ) ) D _ s X ( r ) ( \\omega ) d r + \\int _ s ^ t \\nabla _ x \\sigma ( r , \\omega , X ( r ) ( \\omega ) ) D _ s X ( r ) ( \\omega ) d W ( r ) . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} \\Q _ p ^ c \\otimes _ { \\Z _ p } \\varprojlim _ n \\mathcal { O } _ { L _ n } \\simeq \\Q _ p ^ c \\otimes _ { \\Z _ p } \\Lambda ^ { \\mathcal { O } _ { L ' } } ( \\Gamma ) \\simeq \\bigoplus _ { \\sigma ' \\in \\Sigma ( L ' ) } \\Q _ p ^ c \\otimes _ { \\Z _ p } \\Lambda ( \\Gamma ) \\simeq \\Q _ p ^ c \\otimes _ { \\Z _ p } H _ { L ' _ { \\infty } } = \\Q _ p ^ c \\otimes _ { \\Z _ p } H _ { L _ { \\infty } } . \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { \\gamma _ i } \\prod _ { x = 1 } ^ M \\frac { 1 + e _ { ( j - 1 ) M + x } g _ x ^ { ( \\ell ) } } { 2 } = \\left \\{ \\begin{array} { c l } 1 & E _ { N } ^ { ( j , M ) } \\in \\{ G ^ { ( 1 ) } , G ^ { ( 2 ) } , \\dots , G ^ { ( \\gamma _ i ) } \\} = \\mathcal { G } _ i , \\\\ 0 & E _ { N } ^ { ( j , M ) } \\not \\in \\mathcal { G } _ i , \\end{array} \\right . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} N : = \\beta + \\alpha \\delta ^ { k - 1 } k ' < N ' : = { - \\tilde { \\beta } - \\alpha ( \\delta ^ { k - 1 } k ( \\varepsilon + k \\delta ) - r ) } \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\alpha _ n = \\sum _ { i = 1 } ^ n \\ln \\left ( 1 - b ( i ; n ) \\right ) , \\beta _ n = \\sum _ { i = 1 } ^ n \\frac { b ( i ; n ) } { 1 - b ( i ; n ) } , m _ n = \\max _ { 1 \\leq i \\leq n } b ( i ; n ) . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} Y _ { s , t } ( x ) = x - \\int _ { s } ^ { t } b ( r , Y _ { r , t } ( x ) ) \\ d r - ( B _ { t } - B _ { s } ) . \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ \\infty \\left ( | \\det d f _ { T ^ { k - 1 } \\omega } ( 0 ) | + \\varepsilon \\right ) = 0 . \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} ( a _ 1 , g _ 1 ) ( a _ 2 , g _ 2 ) : = ( a _ 1 + g _ 1 . a _ 2 + \\omega ( g _ 1 , g _ 2 ) , g _ 1 g _ 2 ) \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} C _ R ( k ) = C _ R ( l ) = C _ R ( k + l ) = Z \\cup ( Z + k ) \\cup ( Z + l ) \\cup ( Z + ( k + l ) ) \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + \\Delta u = ( n + m ) u , x , t \\in \\R ^ + , \\\\ n _ { t t } + ( 1 - \\Delta ) n = | u | ^ 2 , \\\\ u ( x , 0 ) = u _ 0 \\in L ^ 2 ( \\R ^ + ) \\\\ n ( x , 0 ) = n _ 0 ( x ) \\in H ^ { s _ 1 } ( \\R ^ + ) , n _ t ( x , 0 ) = n _ 1 ( x ) \\in { H } ^ { s _ 1 - 1 } ( \\R ^ + ) , \\\\ u ( 0 , t ) = 0 , n ( 0 , t ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\Upsilon ( E _ { a , b } ) = E _ { b , a } , \\varpi ( E _ { a , b } ) = \\pm ( - q _ a ) ^ { f _ { a , b } } E _ { b , a } f _ { a , b } \\in \\mathbb N . \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} ( \\alpha \\otimes a ' , \\theta ) _ Y = ( \\alpha , \\beta ) _ X \\neq 0 . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} f ^ I ( U ( m + k \\chi ^ J , n + k \\chi ^ J ) ) = \\left \\langle \\pi _ I ( U ( m + k \\chi ^ J , n + k \\chi ^ J ) ) \\xi ^ I , \\eta ^ I \\right \\rangle \\xrightarrow [ k \\to \\infty ] { } 0 . \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} \\begin{cases} & - \\mu \\bigl ( \\log \\varepsilon - \\kappa + \\log | u - v | - \\log | u | - \\log | v | \\bigr ) + O ( \\varepsilon ) , \\ ; , \\\\ & - 2 \\mu \\bigl ( \\log \\varepsilon - \\kappa - \\log | u | \\bigr ) + O ( \\varepsilon ) , \\ ; \\end{cases} \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left \\| u ^ i \\left ( \\cdot , t \\right ) - \\frac { M _ i } { M } \\mathcal { B } _ { M } \\left ( \\cdot , t \\right ) \\right \\| _ { L ^ 1 } = 0 \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} & \\mathbb { E } \\left ( ( \\mathbb { P } _ { n } ^ { ( N ) } ( f ) - P ( f ) ) ^ { 2 } \\right ) - \\frac { 1 } { n } \\mathbb { V } \\left ( \\mathbb { G } ^ { ( N ) } ( f ) \\right ) \\\\ & = \\left \\vert \\mathbb { E } \\left ( \\mathbb { P } _ { n } ^ { ( N ) } ( f ) \\right ) - P ( f ) \\right \\vert ^ { 2 } + \\frac { 1 } { n } \\mathbb { V } \\left ( \\mathbb { R } _ { n } ^ { ( N ) } ( f ) \\right ) + \\frac { 2 } { n } \\mathrm { C o v } \\left ( \\mathbb { G } _ { n } ^ { ( N ) } ( f ) , \\mathbb { R } _ { n } ^ { ( N ) } ( f ) \\right ) . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} \\partial ^ \\alpha = \\partial ^ { \\alpha _ 0 } _ \\tau \\partial ^ { \\alpha _ 1 } _ \\xi \\partial ^ { \\alpha _ 2 } _ \\eta , ~ \\alpha = ( \\alpha _ 0 , \\alpha _ 1 , \\alpha _ 2 ) \\in \\mathbb { N } ^ 3 , \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} \\mu _ 1 & = \\frac { 3 } { 2 } + \\cos ^ 2 \\biggl ( \\frac { \\pi } { 8 } \\biggr ) + \\cos \\biggl ( \\frac { \\pi } { 8 } \\biggr ) \\cdot \\sin \\biggl ( \\frac { \\pi } { 8 } \\biggr ) \\\\ & = 2 + \\frac { 1 } { \\sqrt { 2 } } \\\\ & \\cong 2 , 7 1 . \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\mathbb { E } [ Y _ i ] = 2 ^ { - m } \\sum _ { j = \\frac { l } { 2 } } ^ l \\binom { l } { j } \\left ( 1 - \\sigma ^ j \\right ) \\left ( 1 + \\sigma ^ j \\right ) ^ { m - 1 } . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} S _ { 1 , p _ { 2 } } \\left ( p _ { 1 } , l \\right ) = \\sum _ { j = 0 } ^ { p _ { 1 } } \\binom { p _ { 1 } } { j } p _ { 2 } ^ { p _ { 1 } - j } S \\left ( j , l \\right ) = \\sum _ { t = 1 } ^ { p _ { 2 } } \\left ( - 1 \\right ) ^ { p _ { 2 } + t } s \\left ( p _ { 2 } , t \\right ) S \\left ( p _ { 1 } + t , l + p _ { 2 } \\right ) . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\langle \\ ! \\langle G , k \\rangle \\ ! \\rangle : = \\int _ { \\Gamma _ 0 } G ( \\eta ) k ( \\eta ) \\lambda ( d \\eta ) \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} \\pmb { \\phi } _ v ( x ) = \\begin{cases} \\mathbf 1 _ { \\Z _ q } ( x ) & v = q \\neq \\infty , \\\\ e ^ { - 2 \\pi x ^ 2 } & v = \\infty . \\end{cases} \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} F ( B _ k ; p ) = C _ k ( \\beta _ k ^ { ( 1 ) } ) \\cdots \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} q _ d ( k ) & : = \\left [ \\prod _ { j = 1 } ^ { d - 1 } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\left \\{ \\frac { k Y _ j z _ j } { N } \\right \\} \\right ) \\right ) \\right ] \\left [ \\prod _ { j = d + 1 } ^ { s } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\left \\{ \\frac { k z _ j ^ 0 } { N } \\right \\} \\right ) \\right ) \\right ] . \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} \\Big ( \\sum _ { g \\in G } \\rho ( g ) \\rho ( g ) \\Big ) _ { i , j } = \\sum _ { t = 1 } ^ { n _ { \\rho } } \\sum _ { g } ( \\rho ^ * ( g ^ { - 1 } ) ^ T ) _ { i , t } \\rho ( g ) _ { t , j } . \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} \\begin{aligned} L ( \\lambda ) = \\span \\left \\{ \\prod _ { \\epsilon _ a - \\epsilon _ b \\in \\Phi ^ + _ { \\bar 0 } } E _ { b , a } ^ { ( A _ { b , a } ) } \\prod _ { i = k + 1 } ^ { m n } E _ { - \\beta _ i } ^ { \\sigma _ i } \\prod _ { i = 1 } ^ { k } E _ { \\beta _ i } ^ { \\sigma _ i } . E _ { - \\beta _ k } \\mathfrak { m } _ \\lambda ^ { ( k - 1 ) } \\right \\} . \\end{aligned} \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\left | B _ n P \\left ( V _ n = k \\right ) - ( \\sqrt { 2 \\pi } ) ^ { - 1 } \\ , e ^ { - \\frac { ( k - \\lambda _ n ) ^ 2 } { 2 B _ n ^ 2 } } \\right | < C \\ , \\frac { \\sum _ { i = 1 } ^ n p _ i q _ i \\left ( p _ i ^ 2 + q _ i ^ 2 \\right ) } { B _ n ^ 3 } . \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) ^ { \\mathrm { a l g } } : = \\frac { \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) } { \\langle g , g \\rangle ^ 2 c ^ + ( f ) } \\in \\Q ( f , g ) . \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} c _ { t + 1 } = \\frac { 1 } { c _ { t - 1 } } \\left ( c _ t ^ { \\sum _ { i = 1 } ^ a k _ i + l _ i } + c _ t ^ { \\sum _ { j = a + 1 } ^ { a + b } k _ j + l _ j } \\right ) , \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} R _ 2 ^ { A B } ( \\rho _ - , v _ - ) : \\left \\{ \\begin{array} { l l } \\xi = \\lambda _ 2 = v + \\beta t + \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } , \\\\ v - v _ - = \\int _ { \\rho _ { - } } ^ { \\rho } \\frac { \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } } { \\rho } d \\rho . \\end{array} \\right . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} P ( s u p p ( T ) ) = s u p p ( P _ \\ast ( T ) ) . \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & A _ 5 & A _ 6 & A _ 7 & B _ 5 & B _ 6 & B _ 7 & 0 & C _ 3 \\end{pmatrix} \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} U ^ \\phi _ { n ^ * m } : = \\pi _ \\phi ( w ) \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} [ \\bar p _ { - 1 } , \\bar p _ 1 ] = - 2 \\bar \\rho , [ 2 \\bar \\rho , \\bar p _ { \\pm 1 } ] = \\pm 2 \\bar p _ { \\pm 1 } . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} R ^ n [ R , R ] = 0 . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ( t ) = & \\ , \\alpha ( \\tau + t ) ^ { \\alpha - 1 } + \\frac { C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } ( \\tau + t ) ^ { \\alpha m - \\beta } - \\frac { \\beta } { m - 1 } ( \\tau + t ) ^ { \\alpha - 1 } \\\\ \\leq & \\ , \\frac { 2 C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } ( \\tau + t ) ^ { \\alpha m - \\beta } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} ( \\hat { \\omega } ^ \\bullet _ \\infty ) ^ m = e ^ { F ( z _ \\infty ) } \\omega _ { \\mathbb { C } ^ m } ^ m . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\lbrace z _ 1 , z _ 3 \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace z _ 2 , z _ 4 \\rbrace = 1 . \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} H _ { I } \\oplus { W _ { i } } = H _ { I \\backslash { i } } . \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} u _ 5 \\circ u _ { 2 5 } & = u _ 4 \\circ e _ 2 \\\\ u _ 4 \\circ e _ 1 & = u _ 5 \\circ u _ { 1 5 } + e \\circ ( f _ 2 - f _ 1 ) \\circ e \\\\ u _ 5 \\circ u _ { 3 5 } & = u _ 4 \\circ e _ 3 - e \\circ g _ 1 \\circ e _ 3 \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} f ( g ( q _ 0 ) ) = \\bigwedge \\{ q \\in Q \\mid g ( q _ 0 ) \\leq g ( q ) \\} \\leq \\bigwedge \\{ q _ 0 \\} = q _ 0 , \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} M _ { 1 , 2 } & = \\frac { 2 P _ 3 ( 1 ) } { L } \\varphi ^ + ( q ) \\sum _ { \\substack { b \\leq y _ 2 \\\\ ( b , q ) = 1 } } \\frac { \\Lambda ( b ) P _ 2 [ b ] } { b } V _ 1 \\left ( \\frac { b } { q ^ { \\frac { 1 } { 2 } } } \\right ) + O ( q ^ { 1 - \\epsilon } ) . \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ^ q ( \\delta ) \\bigr ] & = \\frac { G ( \\delta ) } { G ( q + \\delta ) } \\frac { G ^ 2 ( q + \\delta + 1 / 2 ) } { G ^ 2 ( \\delta + 1 / 2 ) } \\frac { G ( \\delta + 1 ) } { G ( q + \\delta + 1 ) } , \\\\ & = \\prod \\limits _ { k = 0 } ^ \\infty \\Bigl [ \\frac { \\delta + k } { q + \\delta + k } \\frac { ( q + \\delta + 1 / 2 + k ) ^ 2 } { ( \\delta + 1 / 2 + k ) ^ 2 } \\frac { \\delta + 1 + k } { q + \\delta + 1 + k } \\Bigr ] ^ { k + 1 } \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\widetilde { P } _ { a } ( x ) = \\sum _ { n \\leq x } \\frac { ( \\mu * \\mu ) ( n ) } { n } \\Delta _ { a } \\left ( \\frac { x } { n } \\right ) \\log \\frac { x } { e } + O _ { a } \\left ( ( \\log x ) ^ 3 \\right ) . \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} \\psi _ 3 ( y ^ 3 ) = ( b ^ { - 1 } , ( b ^ { - 1 } ) ^ a , ( b ^ { - 1 } ) ^ a ) . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\pi _ { a \\circ x } ( b ) = \\lambda _ a ( \\pi _ x ( ( \\bar { a } \\circ b ) \\cdot \\bar { a } ^ { - 1 } ) ) = ( \\lambda _ a \\pi _ x \\rho _ { \\bar { a } } ) ( b ) = ( \\lambda _ a \\pi _ x \\rho _ a ^ { - 1 } ) ( b ) , \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\tau : { H } _ { q ^ 2 , F } ( r ) = 1 _ \\omega S _ { q , F } ( m | n , r ) 1 _ \\omega \\longrightarrow 1 _ { \\omega ' } S _ { q , F } ( m | n , r ) 1 _ { \\omega ' } , \\ ; t _ i \\longmapsto t _ { m + n - r + i } \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} & \\Psi ( y , \\pi ) = \\begin{pmatrix} \\phi \\left ( \\begin{pmatrix} y _ 1 \\\\ \\vdots \\\\ y _ r \\end{pmatrix} , \\frac { y _ { r + 1 } - \\pi _ { 1 } y _ r - \\ldots - \\pi _ { r } y _ 1 } { \\sigma } \\right ) \\begin{pmatrix} y _ 1 \\\\ \\vdots \\\\ y _ r \\end{pmatrix} \\\\ \\chi \\left ( \\left ( \\frac { y _ { r + 1 } - \\pi _ { 1 } y _ r - \\ldots - \\pi _ { r } y _ 1 } { \\sigma } \\right ) ^ 2 \\right ) \\end{pmatrix} . \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} _ { q / 2 } \\Bigl ( \\frac { b ( a ^ 2 + b ^ 2 ) } { a ^ 2 } \\Bigr ) = 0 . \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} X ^ k ( m ) = \\Big \\lbrace ( x _ 1 , . . . , x _ k ) \\in X ^ k \\ \\Big \\vert \\ \\mathrm { d i s t } ( x _ i , x _ j ) > 4 m ^ { - 1 / 2 } \\ , \\ \\Big \\rbrace , \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} e ^ { - ( q + b _ 0 ) t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) = & e ^ { - ( q + b _ 0 ) t } \\sum \\limits _ { p = 0 } ^ N ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ N \\exp \\bigl ( - ( b _ { k _ 1 } + \\cdots + b _ { k _ p } ) t \\bigr ) , \\\\ = & \\bigl ( \\mathcal { S } _ { N } e ^ { - x t } \\bigr ) ( q | b ) , \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) = ( 2 m - 2 i + 3 , 2 ) . \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} G _ 1 ( v , c ) = A + \\int _ 0 ^ t \\psi ( \\int _ 0 ^ \\tau v ( s ) \\ , d s + c ) \\ , d \\tau \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} \\begin{array} { l l } A ( x ^ i ) = ( \\kappa U ( \\xi ^ i , x ^ i ) + I ) ~ [ \\partial _ { x ^ i } { \\rm L o g } ~ ( U ( x ^ i , \\xi ^ i ) + \\kappa I ) ] \\end{array} \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} I ( q \\ , | \\ , a \\tau , \\tau ) \\thicksim \\sum \\limits _ { r = 1 } ^ \\infty \\frac { \\zeta ( r + 1 , \\ , 1 + a ) } { r + 1 } \\Bigl ( \\frac { B _ { r + 2 } ( q ) - B _ { r + 2 } } { r + 2 } \\Bigr ) / \\tau ^ { r + 1 } \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} & F _ C ^ { ( n ) } ( a , b ; c _ 1 , \\dots , c _ n ; z _ 1 , \\dots , z _ n ) \\\\ = & \\sum _ { m _ 1 , \\dots , m _ n \\in \\bold Z _ { \\geq 0 } } \\dfrac { ( a , m _ 1 + \\cdots + m _ n ) ( b , m _ 1 + \\cdots + m _ n ) z _ 1 ^ { m _ 1 } \\cdots z _ 1 ^ { m _ 1 } } { ( c _ 1 , m _ 1 ) \\cdots ( c _ n , m _ n ) m _ 1 ! \\cdots m _ n ! } , \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} e ^ { - \\sum _ { i \\in V } W _ i \\ell _ i } \\prod _ { e = \\{ i , j \\} \\in E } W _ { i j } I _ { 1 } \\left ( 2 W _ { i j } \\sqrt { \\ell _ { i } \\ell _ { j } } \\right ) \\prod _ { i \\in V \\setminus \\{ i _ 0 \\} } d \\ell _ i \\ , . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} F _ 0 & = - x _ 2 ^ 3 x _ 3 + x _ 1 ^ 4 + b x _ 1 ^ 2 x _ 3 ^ 2 + c x _ 1 x _ 3 ^ 3 + d x _ 3 ^ 4 , \\ ; \\textrm { w h e n } \\textrm { C h a r } ( K ) \\ne 2 , \\\\ F _ 0 & = - x _ 2 ^ 3 x _ 3 + x _ 1 ^ 4 + a x _ 1 ^ 3 x _ 3 + b x _ 1 ^ 2 x _ 3 ^ 2 + c x _ 1 x _ 3 ^ 3 + d x _ 3 ^ 4 , \\ ; \\textrm { w h e n } \\textrm { C h a r } ( K ) = 2 , \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\Delta \\hat { u } ( x , k ) + \\frac { k ^ 2 } { c ^ 2 ( x ) } \\hat { u } ( x , k ) = \\frac { i k } { 2 \\pi } \\frac { f ( x ) } { c ^ 2 ( x ) } , \\ \\ ( x , k ) \\in \\R ^ 3 \\times ( 0 , \\epsilon ) , \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} h ^ \\vee \\langle r ( x ) , \\check \\alpha _ i \\rangle = \\varphi ( x ) . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\log \\left ( \\frac { g ( t ) + T _ 0 } { C _ 0 + T _ 0 } \\right ) = \\int _ { C _ 0 } ^ { g ( t ) } \\frac { 1 } { x + T _ 0 } \\ , d x = \\int _ \\mu ^ t \\frac { S _ 0 \\ , \\dot { u } \\ , \\frac { d } { d t } \\Phi ' ( \\dot { u } ) } { g ( \\tau ) + T _ 0 } \\ , d \\tau \\leq 2 S _ 0 r _ 0 . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} ( \\hat { u _ 2 } - \\hat { u _ 1 } ) ( x , k ) = \\sum \\limits _ { m = 1 } ^ { n } p _ m ( x ) k ^ { j } + \\mathcal { O } ( k ^ { n + 1 } ) , \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} \\pi \\| f \\| _ { S _ n ^ p } & = \\pi \\left ( | f ( 0 ) | + | f ' ( 0 ) | + \\cdots + | f ^ { ( n - 2 ) } ( 0 ) | + \\| f ^ { ( n - 1 ) } \\| _ { S _ 1 ^ p } \\right ) \\\\ & \\geq | f ( 0 ) | + | f ' ( 0 ) | + \\cdots + | f ^ { ( n - 2 ) } ( 0 ) | + \\| f ^ { ( n - 1 ) } \\| _ { \\infty } \\\\ & \\geq | f ( 0 ) | + | f ' ( 0 ) | + \\cdots + | f ^ { ( n - 2 ) } ( 0 ) | + \\| f ^ { ( n - 1 ) } \\| _ { H ^ p } \\\\ & = \\| f \\| _ { S _ { n - 1 } ^ p } \\ , . \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\tilde W = \\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{pmatrix} W = \\begin{pmatrix} 1 & 0 & - 1 \\\\ 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} 1 - \\frac { \\log b _ B ( N , 2 t ) } { \\log N ! } \\leq R _ { B , o p t } ( N , t ) & \\leq 1 - \\frac { \\log b _ B ( N , t ) } { \\log N ! } , \\\\ 1 - \\frac { \\log b _ G ( N , 2 t ) } { \\log N ! } \\leq R _ { G , o p t } ( N , t ) & \\leq 1 - \\frac { \\log b _ G ( N , t ) } { \\log N ! } . \\\\ \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} \\theta _ i ' = \\omega _ i + \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\left ( \\sin ( \\theta _ j - \\theta _ i - \\alpha ) - \\sin ( \\alpha ) \\right ) , \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} \\Sigma _ { \\alpha ^ { 2 s + 1 } \\alpha ^ { 2 s } } ( t _ { l - s } - t _ { l - s + 1 } ) = \\Sigma _ { \\alpha ^ { 2 s + 1 } \\alpha ^ { 2 s } } ( 0 ) S ^ \\odot _ { \\alpha ^ { 2 s } } ( t _ { l - s } - t _ { l - s + 1 } ) . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} V _ N = 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\log Y + \\log Y ' + o ( 1 ) , \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} \\mathcal { W } _ I = \\bigcap _ { i \\in I } V _ i ^ * . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} \\vec { g } _ { 1 } = \\frac { \\vec { f } _ { 1 } + \\vec { f } _ { 2 } + \\vec { f } _ { 5 } + \\vec { f } _ { 6 } } { \\norm * { \\vec { f } _ { 1 } + \\vec { f } _ { 2 } + \\vec { f } _ { 5 } + \\vec { f } _ { 6 } } } , \\ \\dots , \\ \\vec { g } _ { 7 } = \\frac { \\vec { f } _ { 1 } + \\vec { f } _ { 4 } + \\vec { f } _ { 6 } + \\vec { f } _ { 7 } } { \\norm * { \\vec { f } _ { 1 } + \\vec { f } _ { 4 } + \\vec { f } _ { 6 } + \\vec { f } _ { 7 } } } . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} d _ 2 p _ 1 + d _ 1 q _ 2 = d _ 1 d _ 2 + r d _ 1 d _ 2 > d _ 1 d _ 2 . \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} E _ t ( \\underline { m } ) = \\sum _ { m ' } P _ { m , m ' } ( t ) L _ t ( \\underline { m ' } ) \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} w = ( w ( 1 ) , w ( 2 ) , \\dots , w ( n ) ) = ( y _ 1 , y _ 2 , \\dots , y _ { m - \\ell } , \\bar z _ \\ell , \\dots , \\bar z _ 2 , \\bar z _ 1 , v _ 1 , v _ 2 , \\dots , v _ { n - m } ) , \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n \\wedge N } c _ { n , N } ( k ) = N + 1 . \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} \\vec { g } _ { 1 } = \\frac { \\vec { f } _ { 4 } + \\vec { f } _ { 6 } + \\dots + \\vec { f } _ { 1 1 } } { \\norm * { \\vec { f } _ { 4 } + \\vec { f } _ { 6 } + \\dots + \\vec { f } _ { 1 1 } } } , \\ \\dots , \\ \\vec { g } _ { 1 1 } = \\frac { \\vec { f } _ { 3 } + \\vec { f } _ { 5 } + \\dots + \\vec { f } _ { 1 0 } } { \\norm * { \\vec { f } _ { 3 } + \\vec { f } _ { 5 } + \\dots + \\vec { f } _ { 1 0 } } } . \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} \\dot p _ N ( t ) = D \\phi _ N ( \\sigma ( t ) ) , \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} L = \\sum _ { \\ell = w } ^ \\infty t _ { \\ell } x ^ { - \\ell } , \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} \\lambda - \\mu = \\frac { \\lambda ^ 2 - \\mu ^ 2 } { \\mu + \\lambda } = \\frac { \\sigma ^ 2 \\log 2 } { \\mu + \\lambda } \\ge \\frac { \\sigma ^ 2 \\log 2 } { 2 \\lambda } , \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} \\phi _ { \\mathbf h , \\infty } ( z ) = ( x _ 2 + \\sqrt { - 1 } x _ 1 + \\sqrt { - 1 } x _ 5 - x _ 4 ) ^ { k + 1 } \\exp ( - \\pi ( x _ 1 ^ 2 + x _ 2 ^ 2 + 2 x _ 3 ^ 2 + x _ 4 ^ 2 + x _ 5 ^ 2 ) ) . \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} X _ 2 = \\sum _ { i = 0 } ^ K \\frac { ( \\nu _ i - t \\pi _ i ) ^ 2 } { t \\pi _ i } , \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} J \\colon R \\big ( [ 0 , T ] ; L ^ 2 _ \\eta ( S ; U ) \\big ) \\to R \\big ( [ 0 , T ] ; \\mathcal { L } _ 2 ( U , L ^ 2 _ \\eta ( S ; \\R ) ) \\big ) , J ( f ) ( t ) u = \\langle u , f ( t ) ( \\cdot ) \\rangle , \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} \\theta ( U ) = \\tilde { g } ( N , U ) . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} i \\in A , \\ i \\le 4 , \\ \\gamma _ i \\ne 0 \\ \\Rightarrow \\ y ^ * \\bigl ( S ( v _ i ) \\bigr ) = 1 . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} x _ { 1 2 } x _ { 3 4 } \\ast _ g x _ { \\sigma ( 1 ) \\sigma ( 2 ) } x _ { \\sigma ( 3 ) \\sigma ( 4 ) } = x _ { 1 4 5 7 } x _ { 2 3 6 8 } . \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} g = \\begin{pmatrix} \\cos \\varphi & - \\sin \\varphi \\\\ \\sin \\varphi & \\cos \\varphi \\end{pmatrix} \\in T , \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} \\gamma ( \\overline { T } ^ { \\overline { a } } ) = \\gamma ( \\overline { a } ) \\ , \\overline { T } ^ { \\overline { a } } , \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} \\partial f _ p = { \\rm c o } \\{ \\omega _ p ( \\gamma ) | \\ ; \\gamma \\textrm { i s a n } f \\textrm { - g e o d e s i c t h r o u g h } p \\} , \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} L = \\int _ { \\mathbb R } \\zeta _ k \\nu _ { n _ k } \\mathcal T ( \\varphi _ k \\mathcal H \\nu ' _ { n _ k } ) + \\zeta _ k \\nu _ { n _ k } \\mathcal T [ \\mathcal H , \\varphi _ k ] \\nu ' _ { n _ k } + \\zeta _ k \\nu _ { n _ k } \\mathcal T \\mathcal H ( \\varphi _ k ' \\nu _ { n _ k } ) d x \\equiv L _ 1 + L _ 2 + L _ 3 . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} \\frac { \\Gamma ( s + t + h ) ) } { \\Gamma ( s + t ) } = e ^ { \\log { \\Gamma ( s + t + h ) ) } - \\log { \\Gamma ( s + t ) } } \\le e ^ { c _ 3 \\cdot | h | } \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\beta _ m = \\left ( \\begin{array} { c c } 0 & p ^ { - m } \\\\ - p ^ m & 0 \\end{array} \\right ) = \\left ( \\begin{array} { c c } 0 & p ^ { - m } \\\\ p ^ m & 0 \\end{array} \\right ) \\left ( \\begin{array} { c c } - 1 & 0 \\\\ 0 & 1 \\end{array} \\right ) = p ^ { - m } \\left ( \\begin{array} { c c } 0 & 1 \\\\ p ^ { 2 m } & 0 \\end{array} \\right ) \\left ( \\begin{array} { c c } - 1 & 0 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} K _ 0 + \\dots + K _ { 2 ^ J - 1 } & = \\sum _ { j = 0 } ^ { J - 1 } \\left | \\{ 0 \\leq r < 2 ^ J - 1 \\ ; | \\ ; K _ r = 2 ^ j K _ 0 \\} \\right | 2 ^ j K _ 0 + K _ { 2 ^ J - 1 } \\\\ & = \\sum _ { i = 0 } ^ { J - 1 } 2 ^ { J - 1 - j } 2 ^ j K _ 0 + 2 ^ J K _ 0 = ( J + 2 ) 2 ^ { J - 1 } K _ 0 . \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{gather*} \\phi ( \\tau ) : = \\theta _ 2 ( 2 \\tau ) \\theta _ 2 ( 6 \\tau ) + \\theta _ 3 ( 2 \\tau ) \\theta _ 3 ( 6 \\tau ) , \\phi _ 1 ( \\tau ) : = \\frac 1 6 ( \\phi ( \\tau / 3 ) - \\phi ( \\tau ) ) . \\end{gather*}"}
-{"id": "44.png", "formula": "\\begin{align*} Z = \\bigcap _ { n \\geq 1 } \\overline { \\bigcup _ { i \\geq n } \\Phi _ i ^ { - 1 } ( 0 ) } . \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} & \\left \\langle \\rho , \\tau , \\sigma \\alpha _ { 1 } ^ { a } \\alpha _ { 3 } \\right \\rangle , \\left \\langle \\rho , \\tau , \\sigma \\alpha _ { 1 } ^ { a } \\alpha _ { 2 } ^ { b } \\alpha _ { 3 } \\right \\rangle \\cong C _ { p } ^ { 3 } \\\\ & \\ a = 0 , . . . , p - 1 , \\ b = 1 , . . . , p - 1 , \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} b _ n ( t ) = \\sum _ { k = 0 } ^ n \\dfrac { \\binom { t } { k } \\binom { n - t } { n - k } } { \\binom { k + n } { n } } \\ , s ( k + n , k ) , \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} \\epsilon ( \\pi \\otimes \\chi _ D , 1 / 2 ) = \\epsilon ( \\pi , 1 / 2 ) \\prod _ { v \\in \\Sigma ( \\pi ) } \\left ( \\frac { D } { \\pi _ v } \\right ) . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} y \\left ( t \\right ) = \\sqrt { { { \\rm { 2 } } \\mathord { \\left / { \\vphantom { { \\rm { 2 } } T } } \\right . \\kern - \\nulldelimiterspace } T } } \\sum \\limits _ { u \\in \\left \\{ { A , B } \\right \\} } { \\left | { { h _ u } } \\right | { \\mathop { \\Re } \\nolimits } \\left ( { { x _ u } \\left ( t \\right ) { e ^ { j \\left ( { 2 \\pi f _ u ^ { { \\rm { R F } } } t + \\varphi _ u ^ { { \\rm { R F } } } + { \\varphi _ { { h _ u } } } } \\right ) } } } \\right ) } + n ( t ) \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} p ^ * ( X _ { i , j } ) = \\frac { A _ { i - 1 , j - 1 } A _ { i , j + 1 } A _ { i + 1 , j } } { A _ { i - 1 , j } A _ { i , j - 1 } A _ { i + 1 , j + 1 } } . \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\lambda \\colon \\mathcal { Z } ( U , U ) \\to [ 0 , 1 ] , \\lambda ( C ) = P \\big ( ( Z u _ 1 , \\dots , Z u _ n ) \\in B \\big ) , \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} \\psi : & = Q \\phi = ( H - E - i 0 ) ^ { - 1 } P \\phi = ( H - E - i \\epsilon ) ^ { - 1 } P \\phi + \\psi _ \\epsilon , \\\\ & ~ ~ w h e r e ~ \\| \\psi _ \\epsilon \\| = o ( 1 ) , \\\\ ( H _ \\lambda - E ) \\psi & = ( H - E - i \\epsilon ) \\psi + \\lambda P \\psi + i \\epsilon \\psi \\\\ & = P \\phi + ( H - E - i \\epsilon ) \\psi _ \\epsilon + \\lambda P \\psi + i \\epsilon \\psi \\\\ & = P \\phi + ( H - E - i \\epsilon ) \\psi _ \\epsilon + \\lambda G ( E + i 0 ) \\phi + o ( 1 ) . \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} \\prod \\limits _ { m = 1 } ^ \\infty \\Bigl [ \\prod \\limits _ { j = 1 } ^ k \\frac { m - a _ j } { m - b _ j } \\Bigr ] = \\prod \\limits _ { j = 1 } ^ k \\frac { \\Gamma ( 1 - b _ j ) } { \\Gamma ( 1 - a _ j ) } \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\Big | \\sum _ { n = 1 } ^ N e ^ { 2 \\pi i m ( \\varphi _ 1 ( n ) - \\psi ( n ) ) } \\Big | & \\lesssim \\abs { m } ^ { \\frac { 1 } { 2 } } \\sup _ { X \\in [ 1 , N ] } X ^ { 1 + \\frac { 1 } { 2 } \\epsilon } \\big ( \\varphi _ 1 ( X ) \\sigma _ 1 ( X ) \\big ) ^ { - \\frac { 1 } { 2 } } \\\\ & \\lesssim \\abs { m } ^ { \\frac { 1 } { 2 } } N ^ { 1 + \\frac { 1 } { 2 } \\epsilon } \\big ( \\varphi _ 1 ( N ) \\sigma _ 1 ( N ) \\big ) ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ M _ { ( \\tau = 1 , 0 , 0 ) } ^ { - l } \\bigr ] = \\prod _ { k = 0 } ^ { l - 1 } \\frac { ( 3 + l + k ) ! } { ( k + 1 ) ! ^ 2 \\ , k ! } . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} d ( g x _ 0 , x _ 0 ) = \\max _ i \\{ \\log a _ i \\} \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} & { \\rm D } \\bar { J } ( A _ r , B _ r , C _ r ) [ ( A ' _ r , B ' _ r , C ' _ r ) ] \\\\ = & 2 { \\rm t r } ( C ' _ r ( P C _ r ^ T - X ^ T C ^ T ) ) - 2 { \\rm t r } ( C ^ T C _ r X '^ T ) + { \\rm t r } ( C _ r P ' C _ r ^ T ) , \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} W & : = U _ 1 V U _ 2 , \\\\ W ' & : = U _ 1 V ' U _ 2 , \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} { \\rm d i v } \\ , v _ { \\infty } = { \\rm c u r l } \\ , v _ { \\infty } = 0 \\ \\ \\textrm { o n } \\ \\ \\R ^ d \\times ( t _ 1 + 2 \\sigma , t _ 1 + \\kappa ) , \\ \\ \\forall \\ , 0 < \\sigma < \\kappa / 2 . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} l _ 0 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) & l _ 1 & = \\lim _ { \\substack { | n | \\to \\infty \\\\ | n | } } \\tilde { \\phi } ( n ) \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} \\Phi _ n ( t ) : = \\left \\{ \\begin{array} { l l } \\Phi \\left ( \\frac { t ^ { n } _ { k } + t ^ { n } _ { k + 1 } } { 2 } \\right ) , & t \\in ( t _ k ^ n , t _ { k + 1 } ^ n ) , \\ , k = 0 , \\ldots , N _ n - 1 , \\\\ \\Phi ( t ^ n _ k ) , & t = t ^ n _ k , \\ , k = 0 , \\ldots , N _ n , \\end{array} \\right . \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} \\mathcal { E } _ 1 ' & \\ll \\frac { W } { ( M N L _ 1 L _ 3 V ) ^ { \\frac { 1 } { 2 } } } \\sum _ { w \\asymp W } \\frac { 1 } { \\varphi ( w ) } \\sum _ { v \\asymp V } \\left | \\mathop { \\sum \\sum } _ { ( m n , v ) = 1 } \\chi ( n ) \\overline { \\chi } ( m ) \\right | \\left | \\mathop { \\sum \\sum } _ { ( \\ell _ 1 \\ell _ 3 , v ) = 1 } \\overline { \\chi } ( \\ell _ 1 \\ell _ 3 ) \\right | , \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X ( t ) = & b ( t , X ( t ) , Y ( t ) , Z ( t ) , u ( t ) ) d t + \\sigma ( t , X ( t ) , Y ( t ) , Z ( t ) , u ( t ) ) d B ( t ) , \\\\ d Y ( t ) = & - g ( t , X ( t ) , Y ( t ) , Z ( t ) , u ( t ) ) d t + Z ( t ) d B ( t ) , \\ ; t \\in \\lbrack 0 , T ] , \\\\ X ( 0 ) = & x _ { 0 } , \\ Y ( T ) = \\phi ( X ( T ) ) , \\end{array} \\right . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} \\tilde { \\pi } _ p \\simeq \\Theta ( \\pi _ p \\otimes \\chi _ D , \\overline { \\psi } _ p ^ { D } ) = \\Theta ( \\mathrm { S t } _ p \\cdot \\chi _ { u D } , \\overline { \\psi } _ p ^ { D } ) , \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} \\lambda _ N = \\frac { 3 a } { N \\ell ^ 3 } + \\frac { O ( 1 ) } { N ^ { 2 } \\ell ^ 4 } , 0 \\leqslant 1 - f \\leqslant \\frac { C \\mathbf { 1 } ( | x | \\leqslant \\ell ) } { N | x | } , | \\nabla f | \\leqslant \\frac { C \\mathbf { 1 } ( | x | \\leqslant \\ell ) } { N | x | ^ 2 } . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} I _ 1 = \\sup _ { I _ { p - b } } \\sum _ { r \\leq p } \\lambda _ { | r - p | } ^ { - 2 } \\| u _ r \\| _ 2 ^ 2 \\int _ { I _ { p - b } } \\| \\nabla u _ { < p } \\| _ \\infty d t , \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} \\sigma \\ ( A \\iota _ z , \\ , R _ \\alpha \\iota _ z : \\alpha \\in \\ker A \\iota _ z \\ ) = \\sigma \\ ( A \\iota _ z , \\ , R _ \\alpha \\iota _ z : \\alpha \\in \\mathbf R ^ { J ( z ) } \\cap \\ker A \\ ) \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} M _ k = \\begin{bmatrix} 1 & c _ { k } \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ m \\tbinom { m } { k } ( - 1 ) ^ k ( n + k + 1 ) a _ { n + k } + m & \\sum _ { k = 0 } ^ { m - 1 } \\tbinom { m - 1 } { k } ( - 1 ) ^ k a _ { n + k + 1 } \\\\ & = \\sum _ { k = 0 } ^ m \\tbinom { m } { k } ( - 1 ) ^ k ( n + k + 1 - k ) a _ { n + k } \\\\ & = ( n + 1 ) \\sum _ { k = 0 } ^ m \\tbinom { m } { k } ( - 1 ) ^ k a _ { n + k } . \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} \\tilde \\theta \\left ( t \\right ) = { \\theta _ { 1 , B } } \\left ( t \\right ) - { \\theta _ { 1 , A } } \\left ( t \\right ) = 4 \\pi f '' t + 2 { \\varphi '' _ n } \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{gather*} D = g _ q \\otimes w _ r \\otimes w _ v \\otimes z , D ' = g _ { q ' } \\otimes w _ r \\otimes w _ v \\otimes z , \\\\ \\int D = g _ q \\otimes c _ r \\otimes w _ v \\otimes z , \\int D ' = g _ { q ' } \\otimes w _ r \\otimes c _ v \\otimes z , \\end{gather*}"}
-{"id": "4082.png", "formula": "\\begin{align*} X = \\frac { \\mu ( 1 - \\theta ) } { \\theta \\| \\mathbf { h } \\| ^ { 2 } } , ~ Y = \\frac { ( 1 - \\theta ) \\sigma _ { s } ^ { 2 } } { \\xi \\theta \\| \\mathbf { h } \\| ^ { 2 } | h _ { s } | ^ { 2 } } \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} \\frac { d ^ r } { d t ^ r } \\bigl [ e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) \\bigr ] = ( - 1 ) ^ r \\sum \\limits _ { p = 0 } ^ N ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ N \\bigl ( b _ 0 + \\sum b _ { k _ j } \\bigr ) ^ r \\exp \\bigl ( - ( b _ 0 + \\sum b _ { k _ j } ) t \\bigr ) . \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} \\kappa _ i & = \\lim _ { k \\to \\infty } n _ k ^ { - 1 } \\log \\Vert M ^ { n _ k } _ \\omega v \\Vert \\\\ & \\ge \\lim _ { k \\to \\infty } n _ k ^ { - 1 } \\log \\left ( \\Vert M ^ { n _ k } _ \\omega v _ { n _ k } \\Vert - \\Vert M ^ { n _ k } _ \\omega ( v _ { n _ k } - v ) \\Vert \\right ) \\\\ & \\ge \\lim _ { k \\to \\infty } n _ k ^ { - 1 } \\log \\Vert M ^ { n _ k } _ \\omega \\Vert + \\lim _ { k \\to \\infty } n _ k ^ { - 1 } \\log \\left ( 1 - \\Vert v _ { n _ k } - v \\Vert \\right ) \\\\ & \\ge \\kappa _ \\mu , \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} & a _ 1 ^ 4 ( 1 + a _ 1 ^ 2 + b + b ^ 2 + b ^ 3 ) ^ 2 k + a _ 1 ^ 2 ( 1 + a _ 1 ^ 4 + a _ 1 ^ 4 b ^ 2 + a _ 1 ^ 2 b ^ 3 + a _ 1 ^ 2 b ^ 7 + b ^ 8 ) \\\\ = \\ , & ( a _ 1 ^ 2 k ^ 2 + a _ 1 ^ 2 b k + b + b ^ 3 ) h _ 1 + h _ 2 = 0 , \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} M = \\bigcup _ { g \\in G _ \\nu } g ( \\mathbb B _ \\rho ) \\subset \\mathbb B _ \\varepsilon . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\Omega ^ { > } & = \\{ x \\in \\R ^ { n + 1 } \\mid x _ { n + 1 } > x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 , \\ , x _ 1 > 0 \\} \\ , \\mathrm { a n d } \\\\ \\Omega ^ { > } & = \\{ x \\in \\R ^ { n + 1 } \\mid x _ { n + 1 } < x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 , \\ , x _ 1 > 0 \\} . \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} C = \\left ( \\begin{matrix} 0 & 0 & 0 & 2 \\cr 0 & 0 & - 1 & 0 \\cr 0 & 1 & 0 & 0 \\cr - 2 & 0 & 0 & 0 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} P ( Y ) = \\beta _ { 0 } + \\beta _ { 1 } Y + \\beta _ { 2 } Y ^ { 2 } + \\beta _ { 3 } Y ^ { 3 } + \\beta _ { 4 } Y ^ { 4 } = 0 . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\mathcal D = \\mathrm { d i a g } ( \\imath | \\vect z | ^ 2 , \\imath | \\vect z | ^ 2 \\cos ^ 2 \\theta , \\imath | \\vect z | ^ 2 \\sin ^ 2 \\theta ) , \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} \\xi _ j = r \\xi _ j ^ * , \\ \\xi _ j ^ * : = ( e ^ { \\iota { 2 \\pi \\over k } j } , { \\bf { 0 } } ) , { \\bf { 0 } } \\in \\mathbb { R } ^ { N - 2 } , j = 1 , 2 , . . . , k , \\ r \\in ( a , b ) \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} \\widehat { g } ( x \\cdot ( m \\otimes h ) ) & = \\sum _ x \\widehat { g } ( x _ { 1 } m \\otimes x _ { 2 } h ) = \\sum _ { x , h } ( x _ { 2 } h ) _ { 2 } g ( S ^ { - 1 } ( ( x _ { 2 } h ) _ { 1 } ) x _ { 1 } m ) \\\\ & = \\sum _ { x , h } x _ { 3 } h _ { 2 } g ( S ^ { - 1 } ( h _ { 1 } ) S ^ { - 1 } ( x _ { 2 } ) x _ { 1 } m ) = \\sum _ { x , h } x _ { 2 } h _ { 2 } g ( S ^ { - 1 } ( h _ { 1 } ) \\epsilon ( x _ { 1 } ) m ) \\\\ & = \\sum _ h x h _ 2 g ( S ^ { - 1 } ( h _ 1 ) m ) = x \\widehat { g } ( m \\otimes h ) . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} \\begin{aligned} & \\psi ( b _ 0 ) = ( a ^ { e _ 1 } , \\dots , a ^ { e _ { p - 1 } } , b ) , \\\\ & \\psi ( b _ 1 ) = ( b , a ^ { e _ 1 } , \\dots , a ^ { e _ { p - 1 } } ) , \\\\ & \\vdots \\\\ & \\psi ( b _ { p - 1 } ) = ( a ^ { e _ 2 } , a ^ { e _ 3 } , \\dots , b , a ^ { e _ 1 } ) . \\end{aligned} \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{gather*} \\frac { { \\rm d } s } { t ^ 2 } ( \\tau ) = 2 \\pi { \\rm i } \\cdot f _ { 2 7 } ( \\tau / 3 ) { { \\rm d } \\tau } . \\end{gather*}"}
-{"id": "6082.png", "formula": "\\begin{align*} & \\mathcal { F } _ { ( N ) } = \\phi _ { ( 1 ) } \\circ . . . \\circ \\phi _ { ( N ) } ( \\mathcal { F ) } , \\\\ & \\mathcal { H } _ { ( N ) } = \\bigcup \\nolimits _ { 1 \\leqslant k \\leqslant N } \\bigcup \\nolimits _ { 1 \\leqslant j \\leqslant m _ { k } } \\phi _ { ( j , k ) } \\circ \\phi _ { ( k + 1 ) } \\circ . . . \\circ \\phi _ { ( N ) } ( \\mathcal { F } ) , \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} \\theta ( S ) & = - 2 b _ 0 + 2 ( b _ 0 - b _ 1 ) + \\ldots + 2 ( b _ { k - 2 } - b _ { k - 1 } ) + ( 2 b _ { k - 1 } - 1 ) \\\\ & = - 1 \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} D _ i = \\sum _ { k = 0 } ^ n P _ i ( k ) E _ k \\quad E _ k = \\frac { 1 } { \\abs { X } } \\sum _ { i = 0 } ^ n Q _ k ( i ) D _ i \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} G _ 3 ^ L ( \\vec x ) = \\Vert \\vec x - x _ { \\sigma ( k _ 0 ) } \\vec 1 \\Vert = \\min \\limits _ { \\alpha \\in \\mathbb { R } } \\| \\vec x - \\alpha \\vec 1 \\| . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} & \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 1 & 1 & 1 \\end{pmatrix} , & & \\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 1 & 1 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} T _ { P e } = \\tau \\Gamma ( 1 + 1 / \\alpha ) , \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} \\lambda _ k ^ J = \\min _ { u \\in \\mathcal { G } } \\frac { Q ( u , u ) } { \\int _ M u ^ 2 d M } , \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} G ( u ) = - G ( - u ) \\ , , \\left \\{ \\begin{array} { l l } G ( u ) > u & \\textrm { i f } 0 < u < 1 \\ , , \\\\ G ( u ) < u & \\textrm { i f } u > 1 \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\lim _ { l \\to + \\infty } \\langle \\ ! \\langle G , k _ t ^ { \\Lambda _ n , N _ l } \\rangle \\ ! \\rangle = \\langle \\ ! \\langle G , Q ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) k _ 0 \\rangle \\ ! \\rangle , \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\abs { \\langle g _ { i } , g _ { j } \\rangle } = \\frac { d + 1 } { k ( d + 1 - k ) } \\abs * { l - \\frac { k ^ { 2 } } { d + 1 } } \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} & \\mathbb E \\left [ x ^ T _ { x _ 0 } ( t ) Q x _ { x _ 0 } ( t ) \\right ] \\\\ & \\leq \\mathbb E \\int _ 0 ^ t x ^ T _ { x _ 0 } ( s ) ( A ^ T Q + Q A + \\sum _ { k = 1 } ^ m N _ k ^ T Q N _ k + \\sum _ { i , j = 1 } ^ v H _ i ^ T Q H _ j k _ { i j } ) x _ { x _ 0 } ( s ) d s \\\\ & \\quad + \\mathbb E \\int _ 0 ^ t 2 x ^ T _ { x _ 0 } ( s ) Q B u ( s ) d s + \\int _ 0 ^ t \\mathbb E \\left [ x _ { x _ 0 } ^ T ( s ) Q x _ { x _ 0 } ( s ) \\right ] \\left \\| u ( s ) \\right \\| _ 2 ^ 2 d s + x _ 0 ^ T Q x _ 0 . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} u = \\abs { m } ^ { - \\frac { 2 } { 1 5 } } \\big ( \\varphi _ 1 ( X ) \\sigma _ 1 ( X ) \\big ) ^ { \\frac { 1 } { 5 } } , \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} u = \\abs { m } ^ { - \\frac { 2 } { 6 ( 2 ^ d - 1 ) } } \\big ( \\varphi _ 1 ( X ) \\sigma _ 1 ( X ) \\big ) ^ { \\frac { 2 } { 3 } \\cdot \\frac { 1 } { 2 ^ d } } , \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} s _ 0 : = \\frac { 1 } { 2 } \\left ( \\frac { r } { 2 } - d ( x , y _ 0 ) \\right ) . \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} g \\cdot B & : = \\left \\{ g \\cdot b \\ , \\vert \\ , \\ , b \\in B \\right \\} , & G \\cdot B & : = \\left \\{ g \\cdot b \\ , \\vert \\ , g \\in G , \\ , b \\in B \\right \\} . \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} { \\rm o r d } _ { D _ i } ( \\theta _ q ) = \\vartheta _ { p _ i } ^ t ( q ) . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} y [ C , 1 , 1 ] _ { P B } & = [ T , p , F ] [ 1 , q , E ] [ C , 1 , 1 ] _ { P B } \\\\ & = [ T , p , F ] [ 1 , q , E ] [ E , 1 , 1 ] [ D , 1 , 1 ] _ { P B } \\\\ & = [ T , p , F ] [ D , 1 , 1 ] _ { P B } \\\\ & = z \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\varphi _ { Y } ( \\alpha _ { n } ^ { ( N ) } ) = \\sqrt { n } \\left \\Vert \\mathbb { V } _ { n } ^ { ( N ) } ( Y ) - \\mathbb { V } ( Y ) \\right \\Vert . \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} \\Phi ^ * \\left ( \\frac { \\Delta _ { \\{ i - a + 1 , \\dots , i - 1 , i + 1 \\} } } { \\Delta _ { \\{ i - a + 1 , \\dots , i \\} } } \\right ) = \\vartheta _ { i - a } . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} f ( \\mu \\psi + \\mu ' \\psi ' ) - \\mu f ( \\psi ) - \\mu ' f ( \\psi ' ) = \\lim _ { \\ell \\to \\infty } \\Big ( \\phi _ \\ell ^ * ( \\mu \\psi + \\mu ' \\psi ' ) - \\mu \\phi _ \\ell ^ * \\psi - \\mu ' \\phi _ \\ell ^ * \\psi ' \\Big ) = 0 , \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} \\nabla _ { U } V = \\overset { \\mu _ { 0 } } { \\nabla } _ { U } V - C ( U , \\tilde { J } V ) \\tilde { J } E - B ( U , \\tilde { J } V ) \\tilde { J } N - C ( U , V ) E , \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\tilde { p } _ H \\bigl ( ( y , \\vartheta ) , ( z , z ' ) \\bigr ) = \\begin{cases} p _ H ( y , z ) & \\\\ p _ H ( y , \\vartheta ) & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} ( \\pi ( \\gamma ) f ) ( y ) = f ( \\gamma ^ { - 1 } y ) , \\textrm { f o r $ f \\in L ^ 2 ( \\nu ) $ } . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } \\Phi _ { t } d W _ { t } = \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { i = 0 } ^ { n - 1 } \\Phi ( B _ { t _ { i } } , t _ { i } ) ( B _ { t _ { i + 1 } } - B _ { t _ { i } } ) \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} G ( u ) = \\Bigl ( \\cos \\frac { \\pi u } { 4 A } \\Bigr ) ^ { - 1 6 A } . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} \\hat { v } _ n = \\hat { u } _ { n + 1 } , \\hat { V } _ n = \\hat { U } _ { n + 1 } \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\lambda ^ { \\textrm { o p t } } = \\frac { - ( 1 - 3 d D ) + \\sqrt { ( 1 - d D ) ^ { 2 } + 2 a d ^ { 2 } C } } { 2 d C } . \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\frac { \\partial \\bar { L } } { \\partial \\dot { \\rho } } ) = \\frac { \\partial \\bar { L } } { \\partial \\rho } \\frac { d } { d t } ( \\frac { \\partial \\bar { L } } { \\partial \\dot { \\varphi } } ) = \\frac { \\partial \\bar { L } } { \\partial \\varphi } , \\frac { d } { d t } ( \\frac { \\partial \\bar { L } } { \\partial \\dot { \\theta } } ) = \\frac { \\partial \\bar { L } } { \\partial \\theta } \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} \\widetilde { Q } * ( z _ 1 , \\ z _ 2 , \\ z _ 3 ) = ( \\bar { z } _ 1 , \\ \\bar { z } _ 2 , \\ \\bar { z } _ 3 ) \\in \\mathbb { C } ^ 3 \\\\ \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} d ^ * ( B ) = \\sup \\big \\{ \\overline { d } _ { ( F _ n ) } ( B ) \\ , : \\ , \\textrm { $ ( F _ n ) $ i s a l e f t F \\o l n e r s e q u e n c e } \\big \\} . \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} & \\hat { m } _ { ( A _ r , B _ r , C _ r ) } ( \\xi , \\eta , \\zeta ) \\\\ = & J ( A _ r , B _ r , C _ r ) + \\langle { \\rm g r a d } \\ , J ( A _ r , B _ r , C _ r ) , ( \\xi , \\eta , \\zeta ) \\rangle _ { ( A _ r , B _ r , C _ r ) } \\\\ & + \\frac { 1 } { 2 } \\langle { \\rm H e s s } \\ , J ( A _ r , B _ r , C _ r ) [ ( \\xi , \\eta , \\zeta ) ] , ( \\xi , \\eta , \\zeta ) \\rangle _ { ( A _ r , B _ r , C _ r ) } . \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} v _ k ( x ) & = \\lambda u _ k ( x ) , & x \\in ( - 1 , 0 ) , \\\\ u _ k '' ( x ) & = ( k ^ 2 \\pi ^ 2 + \\lambda ^ 2 ) u _ k ( x ) , & x \\in ( - 1 , 0 ) , \\\\ w _ k '' ( x ) & = ( k ^ 2 \\pi ^ 2 + \\lambda ) w _ k ( x ) , & x \\in ( 0 , 1 ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ N L _ { M } \\bigr ) ( n \\ , | \\ , b ) - \\bigl ( \\mathcal { S } _ N L _ { M } \\bigr ) ( 0 \\ , | \\ , b ) & = \\sum \\limits _ { i = 1 } ^ k ( - 1 ) ^ i \\binom { n } { i } \\bigl ( \\mathcal { S } _ N L _ { M - i } \\bigr ) ( 0 \\ , | \\ , b ) + \\\\ & + ( - 1 ) ^ { k + 1 } \\sum \\limits _ { i _ 1 = 0 } ^ { n - 1 } \\sum \\limits _ { i _ 2 = 0 } ^ { i _ 1 - 1 } \\cdots \\sum \\limits _ { i _ { k + 1 } = 0 } ^ { i _ { k } - 1 } \\bigl ( \\mathcal { S } _ N L _ { M - k - 1 } \\bigr ) ( i _ { k + 1 } \\ , | \\ , b ) . \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\begin{dcases} \\sum _ { e \\in E ( G ) } w ( e ) = 1 , \\\\ \\sum _ { e : \\ , v \\in e } B ( v , e ) = 1 , \\ \\ v \\in V ( G ) , \\\\ w ( e ) ^ { p - r } \\prod _ { v \\in e } B ( v , e ) = \\alpha , \\ \\ e \\in E ( G ) . \\end{dcases} \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( a ; N ) } ^ \\ast ( s , \\tau ) = \\sum _ { m \\ge 0 \\atop n \\in \\Z } \\frac { 1 } { \\ ( ( N m + a ) \\tau + n \\ ) ^ k } \\frac { \\Im ( \\tau ) ^ s } { | ( N m + a ) \\tau + n | ^ { 2 s } } \\\\ \\hphantom { \\mathcal G _ { k , ( a ; N ) } ^ \\ast ( s , \\tau ) = } { } + \\sum _ { m \\ge 0 \\atop n \\in \\Z } \\frac { ( - 1 ) ^ k } { \\ ( ( N m + ( N - a ) ) \\tau + n \\ ) ^ k } \\frac { \\Im ( \\tau ) ^ s } { | ( N m + ( N - a ) ) \\tau + n | ^ { 2 s } } . \\end{gather*}"}
-{"id": "5435.png", "formula": "\\begin{align*} p _ { n + 2 } ( x ) & = \\sum _ { m = 0 } ^ { n } \\frac { i ^ m } { 4 \\pi m ! } \\int _ { \\Omega } ( c ^ { - 2 } - 1 ) ( y ) p _ { n - m } ( y ) | x - y | ^ { m - 1 } d y \\\\ & \\ \\ \\ \\ - \\frac { i ^ { n + 2 } } { 8 \\pi ^ 2 ( n + 1 ) } \\int _ { \\Omega } c ^ { - 2 } ( y ) ( f _ 2 - f _ 1 ) ( y ) | x - y | ^ n d y . \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} \\binom { a n + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\alpha _ { s } n + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } = \\sum _ { i = 0 } ^ { r p + \\sigma } A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) \\binom { n + r p + \\sigma - i } { r p + \\sigma } . \\end{align*}"}
-{"id": "2390.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": "9445.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\alpha _ n \\nu _ c ) \\mathbf h _ p ( x ) = p ^ { ( 1 - n ) / 2 } \\chi _ { \\psi } ( p ^ n ) \\varepsilon ( 1 / 2 , \\underline { \\chi } _ p ) \\underline { \\chi } _ p ( 2 ) \\mathbf 1 _ { p ^ { - n } \\Z _ p } ( x ) \\psi \\left ( \\frac { p ^ { 2 n } x ^ 2 } { c p } \\right ) \\int _ { \\Z _ p ^ { \\times } } \\psi \\left ( \\frac { - c } { p } \\left ( y + \\frac { p ^ n x } { c } \\right ) ^ 2 \\right ) \\underline { \\chi } _ p ( y ) d y . \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} \\mu _ 1 & = \\max _ { \\gamma : \\gamma ( h ) ^ 2 = 1 } \\sum _ { v \\in V } \\biggl ( \\sum _ { h : v } \\gamma ( h ) - \\sum _ { h : v } \\gamma ( h ) \\biggr ) ^ 2 \\\\ & = \\big | \\big | + \\big | \\big | \\\\ & = \\big | \\big | + \\big | \\big | - 2 \\cdot \\big | \\big | \\\\ & = k + m - 2 k - 2 m + 2 N \\\\ & = 2 N - k - m . \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} \\phi _ { \\mathbf h } ( z ) : = \\pmb { \\phi } ( x _ 3 ) \\phi _ { \\breve { \\mathbf g } } \\left ( \\left ( \\begin{smallmatrix} x _ 2 & x _ 1 \\\\ x _ 5 & x _ 4 \\end{smallmatrix} \\right ) \\right ) . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} [ \\nabla ^ { k , g _ P } \\eta ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( p , \\delta R ) ) } \\leq \\ ; & { \\epsilon [ \\nabla ^ { k , g _ P } \\eta ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( p , R ) ) } } + C [ \\nabla ^ { k - 1 , g _ P } L ^ { g _ P } \\eta ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( p , R ) ) } \\\\ & { + } \\sum _ { j = 0 } ^ k C R ^ { - k + j - \\alpha } \\| \\nabla ^ { j , g _ P } \\eta \\| _ { L ^ \\infty ( B ^ { g _ P } ( p , \\delta R ) ) } . \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} h \\beta ^ { k + 1 } ( U ) = \\beta ^ m ( h _ { j _ m } ) \\cdots \\beta ^ k ( h _ { j _ k } ) \\beta ^ { k + 1 } ( U ) \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} \\underline { \\chi } \\left ( \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) \\right ) = \\underline { \\chi } ( d ) . \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} C _ 3 x ^ 3 + C _ 2 ( y ) x ^ 2 + C _ 1 ( y ) x + C _ 0 ( y ) = 0 , \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} x _ v = \\begin{dcases} \\bigg ( \\frac { w ( e ) } { r B ( v , e ) } \\bigg ) ^ { 1 / p } , & \\ v \\in e \\in E ( G [ S _ i ] ) , \\\\ 0 , & . \\end{dcases} \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} U = \\frac { U ^ { \\ast } ( \\varphi , \\theta ) } { \\rho } \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} x \\coloneqq \\int \\P ( z ) ^ { 1 / h ^ \\vee } d z = \\int \\prod _ { i = 1 } ^ N ( z - z _ i ) ^ { k _ i / h ^ \\vee } d z , \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\dot { x } = A ( t ) x \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} \\partial _ t \\rho _ m & = - \\partial _ x u ( y _ t ) \\rho _ m = - \\frac { w ( y _ t ) } { c _ \\mu } \\rho _ m ^ { 1 - \\alpha } - \\frac { c _ p } { c _ \\mu } \\rho _ m ^ { \\gamma - \\alpha + 1 } \\geq - \\frac { c _ p } { c _ \\mu } \\rho _ m ^ { \\gamma - \\alpha + 1 } \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\mu ( F ^ \\theta ) = \\int _ { \\Gamma _ 0 } k _ \\mu ( \\eta ) e ( \\theta ; \\eta ) \\lambda ( d \\eta ) , e ( \\theta ; \\eta ) : = \\prod _ { x \\in \\eta } \\theta ( x ) . \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} L _ { i , j } ' : = R ^ * L _ { i , j } , \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} \\Delta \\frac { \\lvert \\Phi \\rvert ^ 2 } { 2 } = \\langle \\Phi , \\Delta _ A \\Phi \\rangle - \\lvert _ A \\Phi \\rvert ^ 2 = - \\lvert _ A \\Phi \\rvert ^ 2 \\leq 0 . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} R ' _ i = \\{ ( S , S ' ) \\in \\S \\times \\S : S - S ' \\in \\S _ i \\} . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} ( \\nabla _ { U } \\varphi ) V = u ( V ) A _ { N } U + g ( A _ { E } ^ { \\ast } U , V ) \\zeta , \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\beta _ m \\nu _ { \\delta } ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ m ) p ^ { ( m + 1 ) / 2 } \\varepsilon ( 1 / 2 , \\underline { \\chi } _ p ) \\underline { \\chi } _ p ( 2 ) \\int _ { \\Z _ p ^ { \\times } } \\psi \\left ( \\frac { - \\delta y ^ 2 } { p } \\right ) \\left ( \\int _ { \\Z _ p } \\psi \\left ( \\frac { 2 ( x p ^ { 1 - m } - y ) z } { p } \\right ) d z \\right ) \\underline { \\chi } _ p ( y ) d y . \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} \\tilde \\Sigma \\setminus U = \\Sigma \\setminus U . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} g ^ { ( i ) } ( \\epsilon _ 1 , \\dots , \\epsilon _ n ) = ( \\epsilon _ 1 , \\dots , \\overset { i } { - \\epsilon _ i } , \\dots , \\epsilon _ n ) . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { \\Lambda ( n ) } { n } = \\log x + O ( 1 ) , \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\Theta _ \\nu \\left ( A , h , N ; \\tau \\right ) = ( - i ) ^ \\nu ( - i \\tau ) ^ { - \\frac 1 2 - \\nu } A ^ { - \\frac 1 2 } \\sum _ { \\substack { k \\pmod { N } \\\\ A k \\equiv 0 \\pmod { N } } } e \\left ( \\frac { A k h } { N ^ 2 } \\right ) \\Theta _ \\nu \\left ( A , k , N ; - \\frac { 1 } { \\tau } \\right ) . \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} \\overline { \\mathcal { W } } ( f _ i , f _ k ) = 0 l \\leq \\mid i - k \\mid . \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} \\Omega _ b = \\{ \\omega \\in \\Omega \\ , | \\ , M ^ n _ \\omega \\textrm { i s b o u n d e d } \\} , \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} A _ n ( x ) = \\sum _ { k = 0 } ^ n \\binom { n } { k } A _ k ( 0 ) x ^ { n - k } , \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} h ( x ) = \\frac { x ( 1 + a x ^ q + b x ^ 2 ) ^ q } { 1 + a x ^ q + b x ^ 2 } = \\frac { a ^ q x ^ 3 + x ^ 2 + b } { b x ^ 3 + x + a } . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} { \\bf G } _ { \\{ a \\} , I } = 1 - \\Delta _ { \\min , \\{ a \\} } \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} \\begin{array} { c l } \\sigma ^ j ( P _ i ) & : = ( \\alpha _ { i \\ell + j } : \\beta _ { i \\ell + j } ) , \\textrm { f o r } i \\in \\{ 0 , \\dots , \\frac { n } { \\ell } - 1 \\} , j \\in \\{ 0 , \\dots , \\ell - 1 \\} \\\\ \\sigma ^ j ( Q ) & : = ( \\gamma _ { j } : \\delta _ { j } ) , \\textrm { f o r } j \\in \\{ 0 , \\dots , \\ell - 1 \\} . \\end{array} \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\inf _ { \\pi \\in \\mathrm { I r r } ( G ) } H ( \\left | \\chi _ { \\pi } \\right | ^ 2 , \\varphi _ G ) > - \\infty ~ \\mathrm { a n d ~ } H ( \\left | \\widehat { \\chi _ { \\pi } } \\right | ^ 2 , \\widehat { \\varphi } _ { \\widehat { G } } ) = \\log ( n _ { \\pi } ^ 2 ) , \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} r U _ { 2 } ( r ) + ( 1 + \\tau ) { U _ { 1 } ( r ) } + \\Big [ - ( \\alpha ^ { 2 } + \\gamma ) r + \\beta \\Big ] U ( r ) = 0 . \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} x _ i & = x _ 0 - \\sum _ { j = 0 } ^ { i - 1 } h _ { i , j } g _ j + v _ i , { i = 0 , \\hdots , N } , \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\lambda \\left ( A \\otimes I + I \\otimes A + \\sum _ { k = 1 } ^ m N _ k \\otimes N _ k + \\sum _ { i , j = 1 } ^ v H _ i \\otimes H _ j k _ { i j } \\right ) \\subset \\mathbb C _ - , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} ( N _ { e } ^ { C _ { p ^ { n } } } \\mathbb F _ { p } ) ( C _ { p ^ { n } } / C _ { p ^ { k } } ) = \\mathbb Z / p ^ { k + 1 } , \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} \\eqref { e q : M _ 3 _ V 1 2 _ 1 } = \\langle \\Phi , U _ N \\sum _ { j = 1 } ^ { N _ 1 } \\sum _ { k = 1 } ^ { N _ 2 } Q ^ { ( 1 ) } _ { x _ j } \\otimes Q ^ { ( 2 ) } _ { y _ k } V ^ { ( 1 2 ) } ( x _ j - y _ k ) Q ^ { ( 1 ) } _ { x _ j } \\otimes P ^ { ( 2 ) } _ { y _ k } U _ N ^ * \\ , \\Phi \\rangle + \\operatorname { h . c . } . \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} \\Theta = \\Theta _ N : = \\frac { p } { m ( \\alpha ^ { ( 1 ) } ) } \\tilde { \\Theta } . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} r _ \\varepsilon ^ 2 \\sim \\varepsilon ^ 2 \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } \\sim \\varepsilon ^ 2 D _ \\varepsilon ^ { ( 1 + 4 t ) / 2 } \\sim D _ \\varepsilon ^ { - ( 1 + 4 ( s + t ) ) / 2 } D _ \\varepsilon ^ { ( 1 + 4 t ) / 2 } = D _ \\varepsilon ^ { - 2 s } . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\R ^ d ) } \\leq C \\| f \\| ^ { 1 - \\theta } _ { \\dot B ^ { - \\alpha } _ { \\infty , \\infty } ( \\R ^ d ) } \\| f \\| ^ { \\theta } _ { \\dot B ^ { \\beta } _ { q , q } ( \\R ^ d ) } \\ \\ \\textrm { w i t h } \\ \\ \\beta = \\alpha \\Big ( \\frac { p } { q } - 1 \\Big ) \\ \\ \\textrm { a n d } \\ \\ \\theta = \\frac { q } { p } . \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} L _ 3 \\leqq \\| \\nu _ { n _ k } \\| \\| \\varphi _ k \\nu _ { n _ k } \\| \\leqq C | \\varphi ' _ k | _ { \\infty } \\| \\nu _ { n _ k } \\| ^ 2 = O ( \\epsilon ) . \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} ( F | _ { k + 1 } [ g ] ) ( Z ) = J ( g , Z ) ^ { - k - 1 } F ( g Z ) , \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\frac 1 { p - 1 } < \\alpha < \\frac 1 { m - 1 } , \\beta = 1 - \\alpha ( m - 1 ) \\ , . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j \\leq 2 \\prod \\limits _ { i = 1 } ^ M a _ i . \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} \\begin{array} { l } { \\inf } \\ ; \\mathbb { E } _ { \\xi _ 2 , \\ldots , \\xi _ { T _ { \\max } } } \\Big [ { \\overline { f } } _ { T _ { \\max } } ( x _ { T _ { \\max } } , x _ { T _ { \\max } - 1 } , \\xi _ { T _ { \\max } } ) \\Big ] \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) \\ ; \\mbox { a . s . } , x _ { t } \\ ; \\mathcal { F } _ t \\mbox { - m e a s u r a b l e , } t = 1 , \\ldots , T _ { \\max } . \\end{array} \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} \\int \\limits _ T \\| x ^ * ( t ) \\| \\| \\tilde { y } _ n - x ( t ) \\| d \\mu ( t ) & = \\int \\limits _ { A _ n ^ c } \\| \\tilde { x } ^ * ( t ) \\| \\| \\tilde { y } _ n - \\tilde { x } _ n ( t ) \\| d \\mu ( t ) \\\\ & + \\int _ { A _ n } \\| y ^ * ( t ) \\| \\big ( \\| \\tilde { y } _ n - x \\| + \\| y ( t ) - x \\| \\big ) d \\mu ( t ) \\\\ & \\leq \\varepsilon ^ 2 / 6 + \\varepsilon ^ 2 / 3 + \\varepsilon ^ 2 / 6 \\leq \\varepsilon . \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} ( t _ 1 , \\dots , t _ n ) . \\theta _ q = t _ 1 ^ { \\mu _ 1 } t _ 2 ^ { \\mu _ 2 } \\cdots t _ n ^ { \\mu _ n } \\theta _ q , \\mbox { w h e r e } \\mu _ k = \\sum _ { ( i , j ) \\in F _ k } x _ { i , j } . \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\gamma ( \\sigma \\times \\sigma ^ \\prime , \\psi ) = \\sum _ { U _ n ( \\mathfrak { f } ) \\backslash G _ n ( \\mathfrak { f } ) } J _ { \\sigma , \\psi } ( g ) J _ { \\sigma ^ \\prime , \\psi ^ { - 1 } } ( g ) \\psi ( e _ 1 \\ , ^ t g ^ { - 1 } \\ , ^ t e _ n ) . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\sum _ { g \\in I } \\varepsilon _ { g f } = 0 . \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} Y = A f + \\varepsilon \\xi , \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} g ( A _ { N } U , P V ) = C ( U , P V ) g ( A _ { N } U , N ) = 0 , \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} E ( z ) & = e ^ { \\bf e } E ( z ) e ^ { - { \\bf e } } , \\\\ e ^ { - { \\bf e } } E \\otimes x ( \\zeta ) e ^ { \\bf e } & = E \\otimes x ( \\zeta ) . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} I _ 3 ( u ) = I _ 3 ( u _ 0 ) \\geq 0 . \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} ( f , g ) = \\int _ { \\Omega } \\bar f ( x ) g ( x ) d x . \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\overline { \\bigcup _ { i \\geq n } \\Phi _ i ^ { - 1 } ( 0 ) } \\subseteq \\bigcup _ { j = 1 } ^ k \\overline { B _ { c m _ n ^ { - 1 / 2 } } ( x _ j ) } , \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} M ( \\lambda ) F ( \\lambda ) = g \\otimes N ( \\lambda ) \\mbox { a n d } E ( \\lambda ) M ( \\lambda ) = h ^ T \\otimes N ( \\lambda ) , \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\mu _ { k } ( \\Lambda ) = ( E ^ { ( k ) } ) ^ { T } \\Lambda E ^ { ( k ) } & & \\mu _ { k } ( \\widetilde { B } ) = E ^ { ( k ) } \\widetilde { B } F ^ { ( k ) } . \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} R _ c = \\left \\{ a \\ ; : \\ ; \\tfrac { a } { c } \\in R _ { \\Gamma , \\chi } \\right \\} \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} s _ n ^ * { \\rm { = } } \\mathop { \\arg \\max } \\limits _ { { s _ n } = 0 , 1 } \\Pr \\left ( { \\left . { { s _ n } } \\right | { \\bf { R } } } \\right ) \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} \\xi _ k : = \\left ( \\Pi ( | u | ^ 2 ) \\middle \\vert \\frac { u _ k ^ K } { \\| u _ k ^ K \\| _ { L ^ 2 } ^ 2 } \\right ) . \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} \\delta = \\frac { \\beta - r } { 4 + \\frac { 2 } { T } } > 0 . \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} & s _ k = \\sum _ { i = 1 } ^ k \\frac { \\ 1 } { \\lambda _ i } , & k = 1 , \\cdots , n , \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} S _ { a _ { 1 } , b _ { 1 } } \\left ( p _ { 1 } + p _ { 2 } , l \\right ) = \\sum _ { m = 0 } ^ { p _ { 2 } } S _ { a _ { 1 } , b _ { 1 } } \\left ( p _ { 2 } , m \\right ) S _ { a _ { 1 } , a _ { 1 } m + b _ { 1 } } \\left ( p _ { 1 } , l - m \\right ) , \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} S _ { Y + 1 } ( n , m ) = \\dfrac { 1 } { q ^ m } \\sum _ { r = m } ^ n \\binom { r } { m } \\langle m \\rangle _ { r - m } S ( n , r ) \\left ( \\dfrac { q } { p } \\right ) ^ r , m \\leq n . \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} a = \\sum _ { \\rho \\in \\Sigma _ { n + m } } \\alpha _ { \\rho ( 1 ) } \\otimes \\cdots \\otimes \\alpha _ { \\rho ( n + m ) } , \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} u = u _ { C , F } \\colon \\int _ C F \\to \\int _ C \\kappa _ A = C \\times A \\to A . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} \\{ 0 , 1 , \\dots , M \\} = \\mathcal { P } _ 0 \\cup \\mathcal { P } _ 1 \\cup \\dots \\cup \\mathcal { P } _ K , \\mathcal { P } _ i \\cap \\mathcal { P } _ j = \\emptyset 0 \\leq i < j \\leq K , \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { 0 < \\abs { \\gamma } \\leq d } a _ \\gamma x ^ \\gamma , \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} & \\int _ 0 ^ T \\Vert ( \\tilde { v } _ 3 ) _ k ( t ) - \\xi _ k ( t ) \\Vert _ { L ^ 2 } d t = \\int _ 0 ^ T \\Vert \\int _ { t - \\delta _ k } ^ t ( \\dot { \\tilde { v } } _ 3 ) _ k ( s ) d s \\Vert _ { L ^ 2 } d t \\leq \\delta _ k ^ { 1 / 2 } C \\int _ 0 ^ T \\Vert ( \\dot { \\tilde { v } } _ 3 ) _ k \\Vert _ { L ^ 2 ( L ^ 2 ) } d t = C \\delta _ k ^ { 1 / 2 } , \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} ( T _ i \\vec x ) _ j = \\begin{cases} x _ j , & j \\ne i , \\\\ - x _ j , & j = i . \\end{cases} \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} \\left [ \\begin{smallmatrix} C & & & S \\\\ & C & S \\\\ & - S & C \\\\ - S & & & C \\end{smallmatrix} \\right ] \\left [ \\begin{smallmatrix} U \\\\ & I _ n \\\\ & & ^ t U ^ { - 1 } \\\\ & & & I _ n \\end{smallmatrix} \\right ] = p \\left [ \\begin{smallmatrix} I _ n \\\\ & I _ n \\\\ & ^ t Z & I _ n \\\\ Z & & & I _ n \\end{smallmatrix} \\right ] \\left [ \\begin{smallmatrix} C & & & S \\\\ & C & S \\\\ & - S & C \\\\ - S & & & C \\end{smallmatrix} \\right ] , \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\omega _ { j , \\ell } ( z ) = \\begin{cases} \\psi _ j ( z ) , \\quad \\ell = 0 , 0 \\leq j \\leq \\kappa , \\\\ \\omega _ { 0 , \\ell - 1 } ' ( z ) , \\ell \\geq 1 , j = 0 , \\\\ \\omega _ { j , \\ell - 1 } ' ( z ) + \\omega _ { j - 1 , \\ell - 1 } ( z ) , \\ell \\geq 1 , 1 \\leq j \\leq \\kappa + \\ell - 1 , \\\\ \\omega _ { \\kappa + \\ell - 1 , \\ell - 1 } ( z ) , \\ell \\geq 1 , j = \\kappa + \\ell . \\end{cases} \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} \\mathrm { d i m } ( H _ { J _ { k + 1 } } ) = 2 g - \\# { J _ { k + 1 } } = 2 g - ( k + 1 ) . \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} t _ { z _ 1 } ^ { 2 n } t _ { \\alpha _ 1 } ^ { - 2 n } t _ { y ^ \\prime _ 1 } ^ { 2 n } t _ { \\beta _ 1 } ^ { - 2 n } & = [ t _ { z _ 1 } ^ { 2 n } t _ { \\alpha _ 1 } ^ { - 2 n } , f _ 2 ] , \\\\ t _ { x _ 1 } ^ { 2 n } t _ { \\beta _ 1 } ^ { - 2 n } t _ { z ^ \\prime _ 1 } ^ { 2 n } t _ { \\gamma _ 1 } ^ { - 2 n } & = [ t _ { x _ 1 } ^ { 2 n } t _ { \\beta _ 1 } ^ { - 2 n } , f _ 3 ] , \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} u _ i = \\begin{pmatrix} 0 & u _ i ' & 0 & c _ i \\end{pmatrix} \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} \\partial _ t u + v \\partial _ x u + a ( t , x , v ) \\partial _ v u + ( \\lambda + c - 1 ) u - \\sigma \\partial _ { v v } u = U ( t , x , v ) e ^ { - \\lambda t } \\ , . \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} \\partial _ x u _ 1 + \\partial _ y u _ 2 = ~ & \\frac { ( 1 - a ) } { Q } h _ 1 \\big [ ( h _ 1 \\partial _ x + h _ 2 \\partial _ y ) u _ 1 + \\nu \\partial _ y ^ 2 h _ 1 \\big ] - \\frac { ( 1 - a ) } { Q } ( P _ t + P _ x u _ 1 ) \\\\ & + \\frac { a } { Q } \\big [ \\kappa \\partial _ { y } ^ 2 \\theta + \\mu ( \\partial _ y u _ 1 ) ^ 2 + \\nu ( \\partial _ y h _ 1 ) ^ 2 \\big ] , \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { s , p , q } ( \\mathbb { G } ) } = ( \\| u \\| _ { L ^ { p } ( \\mathbb { G } ) } + [ u ] _ { s , p , q } ) ^ { 1 / p } , \\ , \\ , \\ , u \\in W ^ { s , p , q } ( \\mathbb { G } ) , \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J _ { K } } } \\| X ( u _ { j } ) \\| ( t _ { j + 1 } - t _ { j } ) = 0 \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} \\frac { \\Delta _ \\rho \\Phi } { \\rho } = \\frac { \\nabla \\cdot ( \\rho \\nabla \\Phi ) } { \\rho } = \\frac { \\Gamma ( \\rho , \\Phi ) + \\rho \\Delta \\Phi } { \\rho } = \\Gamma ( \\log \\rho , \\Phi ) + \\Delta \\Phi . \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} k ( M / ( A _ i \\cup \\{ e , e ' \\} ) ) - v ( M / ( A _ i \\cup \\{ e , e ' \\} ) ) = k ( M / A _ i ) - v ( M / A _ i ) . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} \\varphi ^ { 2 } U = p \\varphi U + q ( U ) - u ( U ) \\zeta , \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} \\Gamma _ { M + 1 } \\bigl ( w \\ , | \\ , a , a _ { M + 1 } \\bigr ) = & \\prod \\limits _ { k = 1 } ^ \\infty \\frac { \\Gamma _ M ( w + k a _ { M + 1 } \\ , | \\ , a ) } { \\Gamma _ M ( x + k a _ { M + 1 } \\ , | \\ , a ) } e ^ { \\Psi _ { M + 1 } ( x , \\ , k a _ { M + 1 } \\ , | \\ , a ) - \\Psi _ { M + 1 } ( w , \\ , k a _ { M + 1 } \\ , | \\ , a ) } \\times \\\\ & \\times \\exp { \\bigl ( \\phi _ { M + 1 } ( w , x \\ , | \\ , a , a _ { M + 1 } ) \\bigr ) } \\ , \\Gamma _ { M } ( w \\ , | \\ , a ) . \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} X _ \\theta ( t ) ( \\omega ) = \\theta + \\int _ 0 ^ t \\Big [ B \\big ( s , \\omega \\big ) X _ \\theta ( s ) ( \\omega ) + b ( s , \\omega ) \\Big ] d s + \\int _ 0 ^ t \\Big [ \\Sigma \\big ( s , \\omega \\big ) X ( s ) ( \\omega ) + \\sigma ( s , \\omega ) \\Big ] d W ( s ) , \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} t _ y t _ a ^ { - 1 } t _ { t _ x ( y ) } t _ a ^ { - 1 } & = [ t _ y t _ a ^ { - 1 } , \\phi ^ \\prime ] , \\\\ t _ { t _ x ^ { 2 k + 1 } ( y ) } t _ a ^ { - 1 } & = [ t _ { t _ x ^ { 2 k + 1 } ( y ) } , \\psi ^ \\prime ] , \\\\ t _ x ^ k t _ b ^ { - k } t _ z ^ k t _ c ^ { - k } & = [ t _ x ^ k t _ b ^ { - k } , \\tau ^ \\prime ] . \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} P ( \\tau , \\xi ) - q ^ 0 ( \\tau , \\xi , \\eta ) & \\geq P ( 0 , \\xi ) - \\frac { 1 } { 2 } ( h _ { 1 , 0 } ) ^ 2 ( \\xi , \\eta ) - C \\tau ( 1 + T ^ { k - 1 } ) ( \\sqrt { M _ 0 } + \\sqrt { M _ e } ) . \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} H _ { i } = \\phi _ { i } ( \\mathrm { S p a n } \\{ P _ { 2 , i } , . . . , P _ { 2 g , i } \\} ) . \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} G _ H ( x , y ) ~ = ~ \\sum _ { n = 0 } ^ \\infty \\P _ x ( H _ n = y ) , x , y \\in E . \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } \\dot { x } _ { i } \\dot { h } _ { i } = - \\frac { \\beta } { 2 } \\sum _ { i = 1 } ^ { N } \\sum _ { j \\in \\mathcal { N } _ { i } } \\left ( \\dot { x } _ { i } - \\dot { x } _ { j } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} { \\mathbb D } ^ { ( A } { } _ { ( B } { \\mathbb D } ^ { C ) } { } _ { D ) } = & \\ , \\mathrm { t f } \\big ( { \\mathbb D } ^ { ( A } { } _ { ( B } { \\mathbb D } ^ { C ) } { } _ { D ) } \\big ) + \\tilde { A } \\delta ^ { ( A } { } _ { ( B } { \\big ( { \\mathbb D } _ o ^ 2 \\big ) } ^ { C ) } { } _ { D ) } \\\\ & + \\tilde { B } \\delta ^ { ( A } { } _ { ( B } \\delta ^ { C ) } { } _ { D ) } \\langle { \\mathbb D } , { \\mathbb D } \\rangle \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} V _ k ( \\lambda ) ^ { - 1 } = \\left [ \\begin{array} { c c c c c | c } - 1 & - \\lambda ^ \\ell & - \\lambda ^ { 2 \\ell } & \\cdots & - \\lambda ^ { ( k - 1 ) \\ell } & \\lambda ^ { k \\ell } \\\\ & - 1 & - \\lambda ^ \\ell & \\ddots & \\vdots & \\lambda ^ { ( k - 1 ) \\ell } \\\\ & & - 1 & \\ddots & - \\lambda ^ { 2 \\ell } & \\vdots \\\\ & & & \\ddots & - \\lambda ^ { \\ell } & \\lambda ^ { 2 \\ell } \\\\ & & & & - 1 & \\lambda ^ \\ell \\\\ & & & & & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} 1 - e ^ { - 2 \\beta _ P ( d ) } = p _ c ^ b ( d ) , \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} \\begin{array} { l l } ( u v ^ { - 1 } ) ' = ( u ' v - v ' u ) v ^ { - 2 } , \\end{array} \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} g _ { \\infty } = n ( X ) m ( A , 1 ) = n ( X ) \\mathrm { d i a g } ( A , { } ^ t A ^ { - 1 } ) \\in \\mathrm { S p } _ 2 ( \\R ) \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} 1 + \\sum _ n a _ n \\bar b _ n t _ n \\nu _ n = 0 . \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} 8 r = 3 + 4 r - r ^ 2 \\pm ( 1 + r ) \\sqrt { 1 + 6 r + r ^ 2 } . \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} \\sigma ^ c _ t ( f ) ( g ) = e ^ { i t c ( g ) } f ( g ) \\ \\ \\ \\ f \\in C _ c ( G ) \\ \\ g \\in G . \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} W _ { i , j } & = F _ { i , j } + F _ { i , j + 1 } - F _ { i - 1 , j } - F _ { i + 1 , j + 1 } \\\\ & = \\sharp \\{ 1 , 2 , \\dots , j \\mbox { ' s i n t h e } i ^ { \\mathrm { t h } } \\mbox { r o w } \\} - \\sharp \\{ 1 , 2 , \\dots , ( j + 1 ) \\mbox { ' s i n t h e } ( i + 1 ) ^ { \\mathrm { t h } } \\mbox { r o w } \\} . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\tilde { M } : = { \\rm S y m } _ + ( r ) \\times { \\bf R } ^ { r \\times m } . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } \\| y ^ { k } - z \\| = 0 . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u _ \\mu + ( \\Delta ^ 2 - \\kappa \\Delta ) u _ \\mu + \\displaystyle { \\sum _ { \\substack { \\nu = 1 } } ^ N } G ( u _ \\mu , u _ \\nu ) = 0 , \\\\ ( u _ \\mu ( 0 , \\cdot ) ) _ { \\mu = 1 } ^ N = ( u _ { \\mu , 0 } ) _ { \\mu = 1 } ^ N \\in H ^ 2 ( \\R ^ d ) ^ N , \\end{cases} \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} \\widehat { G } ( 0 ) \\frac { N } { w } \\sum _ { d \\mid v } \\frac { \\mu ( d ) } { d } \\sum _ { ( a , w ) = 1 } e \\left ( \\frac { a d \\overline { m \\ell _ 1 \\ell _ 3 v } } { w } \\right ) = \\widehat { G } ( 0 ) \\mu ( w ) \\frac { N } { w } \\frac { \\varphi ( v ) } { v } , \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} V _ i ^ * u = V _ i ^ * ( V _ { N + 1 } ^ * ) ^ n x = ( V _ { N + 1 } ^ * ) ^ n V _ i ^ * x = 0 , \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\sup _ { x \\in C } \\left | G ^ { N _ k } ( x , \\mu _ k ) - G ( x , \\mu ) \\right | = 0 , \\quad C \\subset \\R ^ d . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} N _ C ( y ) : = \\left \\{ \\begin{array} { l l l } \\{ z \\in Y ^ { \\ast } : \\langle z , c - y \\rangle \\leq 0 , \\ ; \\forall c \\in C \\} , & \\hbox { i f } & y \\in C \\\\ \\emptyset , & \\hbox { i f } & y \\notin C . \\end{array} \\right . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} e ^ { - \\beta _ c } = \\sum _ { n = 1 } ^ \\infty f _ n , \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} V ( x ) ~ = ~ \\sum _ { k = 0 } ^ \\infty \\P _ x ( H _ k \\in \\Z _ + ) , x \\in \\Z _ + \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} \\alpha = \\begin{pmatrix} A _ 1 & A _ 2 & B _ 1 & B _ 2 & C _ 1 \\\\ & A _ 1 & B _ 2 & \\\\ & A _ 3 & B _ 3 & & C _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} f ( x _ t ) = f ( x _ s ) + \\sum _ { | I | \\leq k _ 0 } H ^ I _ { t s } ( V _ I f ) ( x _ s ) + O \\big ( | t - s | ^ { k _ 0 + 1 } \\big ) \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\int _ M \\bar w _ i \\ , d M = 0 , \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} u _ { t } + \\nabla \\cdot \\left ( - a \\left ( x , t ; u ; \\nabla u \\right ) \\nabla u \\right ) = F \\left ( x , t ; u ; \\nabla u \\right ) \\quad Q _ { T } : = \\Omega \\times \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} T _ { \\bar { u } } ( \\Psi _ k ^ K u _ k ^ K ) - \\Psi _ k ^ K T _ { \\bar { u } } ( u _ k ^ K ) & = \\Pi \\left ( \\Psi _ k ^ K ( I - \\Pi ) ( \\bar { u } u _ k ^ K ) \\right ) \\\\ & = \\Pi \\left ( \\Psi _ k ^ K \\bar { z } \\overline { \\Pi ( \\bar { z } u \\overline { u _ k ^ K } ) } \\right ) \\\\ & = \\Pi ( \\bar { z } \\Psi _ k ^ K \\overline { K _ u ( u _ k ^ K ) } ) \\\\ & = \\sigma _ k \\ , \\Pi ( \\bar { z } | \\Psi _ k ^ K | ^ 2 \\overline { u _ k ^ K } ) \\\\ & = \\sigma _ k \\ , \\Pi ( \\bar { z } \\overline { u _ k ^ K } ) \\\\ & = 0 , \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } y ^ 2 z + A x ^ 2 z + B x ^ 2 y - a x y = 0 \\\\ A x y ^ 2 + x z ^ 2 + B y ^ 2 z - a ^ q y z = 0 \\\\ x ^ 2 y + B x z ^ 2 - a ^ { q ^ 2 } x z + A y z ^ 2 = 0 \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\partial _ x \\mathbb { E } f \\bigl ( Y ^ n _ t ( x ) \\bigr ) & = \\mathbb { E } f ' \\bigl ( Y ^ { n } _ { t } ( x ) \\bigr ) + \\frac { 1 } { 2 } \\sqrt { t } \\ , \\mathbb { E } \\bigl ( \\xi g _ 1 \\bigl ( x , \\xi \\sqrt t , Y ^ { n - 1 } _ { t } \\bigr ) \\bigr ) . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( 1 ) } = \\frac { 1 } { n } { \\displaystyle \\sum \\limits _ { j = 1 } ^ { m _ { 1 } } } \\frac { P ( A _ { j } ^ { ( 1 ) } ) } { \\mathbb { P } _ { n } ( A _ { j } ^ { ( 1 ) } ) } { \\displaystyle \\sum \\limits _ { X _ { i } \\in A _ { j } ^ { ( 1 ) } } } \\delta _ { X _ { i } } , \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} { \\rm E x p } _ x ( \\xi ) : = \\Gamma _ { ( x , \\xi ) } ( 1 ) . \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\int _ { \\Omega } c _ 1 ^ { - 2 } f _ 1 d y = \\int _ { \\Omega } c _ 2 ^ { - 2 } f _ 2 d y . \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} \\| \\xi ^ { [ N ] } \\| ^ 2 & = \\sum _ { \\lambda \\in \\Lambda } \\| f _ \\lambda \\| ^ 2 , & \\| \\eta ^ { [ N ] } \\| ^ 2 & = \\sum _ { \\lambda \\in \\Lambda } \\| g _ \\lambda \\| ^ 2 . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( N ) } ( f ) - \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( f ) & = \\sum _ { j = 1 } ^ { m _ { N } } \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( f 1 _ { A _ { j } ^ { ( N ) } } ) \\frac { P ( A _ { j } ^ { ( N ) } ) } { \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) } - \\sum _ { j = 1 } ^ { m _ { N } } \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( f 1 _ { A _ { j } ^ { ( N ) } } ) \\\\ & = \\sum _ { j = 1 } ^ { m _ { N } } \\mathbb { E } _ { n } ^ { ( N - 1 ) } ( f \\ ; | \\ ; A _ { j } ^ { ( N ) } ) \\left ( P ( A _ { j } ^ { ( N ) } ) - \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) \\right ) . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} \\ < z | z ' \\ > = \\ < q _ z , q _ { z ' } \\ > = K ( z , z ' ) . \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} M _ 2 ( \\kappa ; \\boldsymbol { u } ) : = - \\frac { 1 } { \\pi ^ 2 } \\int _ { \\mathbb { R } - i u _ 2 } \\int _ { \\mathbb { R } - i u _ 1 } \\frac { e ^ { - \\pi w _ 1 ^ 2 - \\pi w _ 2 ^ 2 - 2 \\pi i ( u _ 1 w _ 1 + u _ 2 w _ 2 ) } } { w _ 2 ( w _ 1 - \\kappa w _ 2 ) } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} \\dot { x } & = A x + B u , x ( 0 ) = x _ 0 , \\\\ y & = C x , \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\mathcal J _ 2 ( m ) = \\int _ { \\mathcal B _ 2 ^ - ( m ) } ( c , - d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h + \\int _ { \\mathcal B _ 2 ^ + ( m ) } \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\breve { J } V = \\breve { J } D V + \\breve { J } E V . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} ( X , Y ) = \\sum _ { i < j } X _ { i j } Y _ { i j } . \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} P : = k [ a , b , c ] \\quad R : = \\frac { P } { ( a ^ { p } , b ^ { p } ) } \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} X ' = \\{ x \\in \\mathbb { T } ^ { \\mathbb { Z } ^ 2 } : x _ \\mathbf { n } + x _ { \\mathbf { n } + \\mathbf { e } _ 1 } + x _ { \\mathbf { n } + \\mathbf { e } _ 2 } = 0 \\enspace \\textup { f o r e v e r y } \\enspace \\mathbf { n } \\in \\mathbb { Z } ^ 2 \\} . \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} \\frac { 1 } { z - t _ n } = - \\frac { 1 } { t _ n } - \\frac { z } { t _ n ^ 2 } - \\dots - \\frac { z ^ { N - 1 } } { t _ n ^ N } + \\frac { z ^ N } { t _ n ^ N ( z - t _ n ) } , \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} { \\bf E } [ Y ( \\tau ) ^ q ] & = \\Gamma ( 1 - q / \\tau ) , \\\\ \\Gamma _ { 2 } ( \\tau - q \\ , | \\ , \\tau ) & = \\frac { \\tau ^ { ( \\tau - q ) / \\tau - 1 / 2 } } { \\sqrt { 2 \\pi } } { \\bf E } [ Y ( \\tau ) ^ q ] \\ , \\Gamma _ 2 \\bigl ( 1 + \\tau - q \\ , | \\ , \\tau \\bigr ) . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} \\Lambda ' & ( R , m ) \\\\ & \\triangleq \\left \\{ \\alpha \\in \\Omega ^ T \\colon \\alpha = \\alpha ^ 1 + \\alpha ^ 2 + \\tfrac { \\sin ( \\pi \\theta ) } { \\pi } \\alpha ^ 3 , \\ \\alpha ^ 1 \\in \\Lambda ^ \\star ( m ) , \\alpha ^ 2 \\in \\Lambda ( R , 1 , m ) , \\alpha ^ 3 \\in \\Lambda ( R , \\theta , m ) \\right \\} . \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} \\mathfrak { h } _ + ^ { ( 1 ) } \\ ; : = \\{ u _ 0 \\} ^ { \\bot } \\subset L ^ 2 ( \\R ^ 3 ) \\ , , \\mathfrak { h } _ + ^ { ( 2 ) } \\ ; : = \\{ v _ 0 \\} ^ { \\bot } \\subset L ^ 2 ( \\R ^ 3 ) . \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} \\gamma = \\left ( \\begin{array} { c c } A & B \\\\ C & D \\end{array} \\right ) , \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\textbf { E } ^ \\rho _ { \\alpha , \\beta , \\omega , b ^ - } \\varphi ( x ) = \\int _ x ^ b ( t - x ) ^ { \\beta - 1 } E _ { \\alpha , \\beta } ^ \\rho ( \\omega ( t - x ) ^ \\alpha ) \\varphi ( t ) d t , ~ ~ x < b \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\tau '' c = \\begin{cases} \\tau c & \\tau c \\in C '' , \\\\ \\tau \\theta \\sigma \\tau c & \\tau c \\in \\{ a , \\theta a , \\sigma a , \\theta \\sigma a \\} \\tau \\theta \\sigma \\tau c \\in C '' , \\\\ \\tau ( \\theta \\sigma \\tau ) ^ 2 c & \\tau c , \\tau \\theta \\sigma \\tau c \\in \\{ a , \\sigma a , \\theta a , \\sigma \\theta a \\} , \\\\ \\end{cases} \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\langle P ( x ) , Q ( y ) \\rangle = \\sum _ { j _ 1 , . . . , j _ N = 0 } ^ { \\infty } \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } \\left ( \\vec { d } ( x , y ) + 2 ( j _ 1 , . . . , j _ N ) + 2 \\chi ^ I \\right ) . \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} w _ B \\left ( \\pi \\right ) = d _ B ( e , \\pi ) = d _ B ( \\pi , e ) . \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} d f _ p ( v ) = \\lim _ { t \\searrow 0 } \\frac { f ( \\gamma ( t ) ) - f ( \\gamma ( 0 ) ) } { t } \\leq 1 . \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} \\vert L ( h ) - h \\vert & = \\vert \\pi _ { P _ { F ( z ) } } ( L ( h ) ) - \\pi _ { P _ x } ( L ( h ) ) \\vert \\\\ & \\leq \\sphericalangle \\left ( P _ { F ( z ) } , P _ x \\right ) \\vert L ( h ) \\vert \\\\ & < \\frac 1 6 \\vert L ( h ) \\vert . \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\mathbf { P W } ^ { k , p } _ { i n t } ( \\widetilde { \\Omega } ) = \\{ \\mathbf { u } \\ , : \\ , \\mathbf { u } \\in \\mathbf { W } ^ { k , p } ( \\widetilde { \\Omega } ^ s ) , ~ s = \\pm ; ~ [ \\mathbf { u } ] _ { \\Gamma } = \\mathbf { 0 } , ~ \\textrm { a n d } ~ [ \\sigma ( \\mathbf { u } ) \\mathbf { n } ] _ { \\Gamma } = \\mathbf { 0 } \\} , \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\nabla \\bar { \\tilde { J } } ( A _ r , B _ r ) = ( - 2 ( P ^ 2 - X ^ T X ) , 4 P B _ r - 4 X ^ T B ) . \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\P _ { s , \\mu } ( X _ { t } \\in B | \\tau _ { A } > t ) & = \\limsup _ { t \\to \\infty } \\P _ { s , \\mu } ( X _ { t + u } \\in B | \\tau _ { A } > t + u ) \\\\ & = \\limsup _ { t \\to \\infty } \\P _ { s + u , \\phi _ { u + s , s } ( \\mu ) } ( X _ { t } \\in B | \\tau _ { A } > t ) \\\\ & = \\limsup _ { t \\to \\infty } \\P _ { s + u , \\nu } ( X _ { t } \\in B | \\tau _ { A } > t ) , \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} d _ { P } ^ { 2 } ( h _ { ( N ) } ^ { - } , h _ { ( N ) } ^ { + } ) = \\sum _ { j = 1 } ^ { m _ { N } } d _ { P } ^ { 2 } ( h _ { ( j , N ) } ^ { - } , h _ { ( j , N ) } ^ { + } ) \\leqslant d _ { P } ^ { 2 } ( g _ { - } , g _ { + } ) + 3 m _ { N } \\varepsilon ^ { 2 } \\leqslant 4 m _ { N } \\varepsilon ^ { 2 } . \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} Z ( s , f _ k , w _ \\lambda ) = B _ \\lambda ( s ) w _ \\lambda \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ N L _ M \\bigr ) ( q \\ , | \\ , a , b ) - \\bigl ( \\mathcal { S } _ N L _ M \\bigr ) ( 0 \\ , | \\ , a , b ) & = \\sum \\limits _ { k = 0 } ^ \\infty \\Bigl [ \\bigl ( \\mathcal { S } _ N L _ { M - 1 } \\bigr ) ( q + k a _ M \\ , | \\ , a , b ) - \\\\ & - \\bigl ( \\mathcal { S } _ N L _ { M - 1 } \\bigr ) ( k a _ M \\ , | \\ , a , b ) \\Bigr ] , \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} D & = ( t _ 2 t _ 4 ) ^ { \\binom n 2 } q ^ { 2 \\binom n 3 } \\Gamma ( t _ 1 t _ 2 q ^ { n - 1 } , t _ 3 t _ 4 q ^ { n - 1 } ) ^ n \\prod _ { m = 1 } ^ n \\prod _ { \\substack { 1 \\leq j < k \\leq 6 \\\\ ( j , k ) \\neq ( 1 , 2 ) , \\ , ( 3 , 4 ) } } \\Gamma ( t _ j t _ k q ^ { m - 1 } ) \\\\ & \\quad \\times \\prod _ { j = 1 } ^ n ( q , q ^ { j - n } t _ 1 / t _ 2 , q ^ { j - n } t _ 3 / t _ 4 ) _ { j - 1 } . \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} d X _ t = b ( X _ { t - } ) \\ , d t + \\sigma ( X _ { t - } ) \\ , d L _ t \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} \\mathbf { P C } ^ k _ { i n t } ( \\widetilde { \\Omega } ) = \\{ \\mathbf { u } \\ , : \\ , \\mathbf { u } \\in \\mathbf { C } ^ k ( \\widetilde { \\Omega } ^ s ) , ~ s = \\pm ; ~ [ \\mathbf { u } ] _ { \\Gamma } = \\mathbf { 0 } , ~ \\textrm { a n d } ~ [ \\sigma ( \\mathbf { u } ) \\mathbf { n } ] _ { \\Gamma } = \\mathbf { 0 } \\} , \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} V _ { \\lambda } ^ { r , s } = \\bigoplus _ { \\substack { \\mu \\in \\Lambda ( n , r ) \\\\ \\nu \\in \\Lambda ( n , s ) \\\\ \\lambda = \\mu - \\nu } } M ^ { \\mu } \\boxtimes M ^ { \\nu } \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\Gamma ( \\delta ) ( x ) : = \\sup _ { \\{ t , s \\in [ 0 , 1 ] : | t - s | \\le \\delta \\} } | x _ t - x _ s | . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\varepsilon _ { 2 } \\left ( r \\right ) = \\left ( \\tilde { \\sigma } _ { x } ( r ) - \\sigma _ { x } ( r ) \\right ) \\hat { X } ( r ) + \\left ( \\tilde { \\sigma } _ { y } ( r ) - \\sigma _ { y } ( r ) \\right ) \\left ( p ( r ) \\hat { X } \\left ( r \\right ) + \\nu ( r ) \\right ) . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} \\mathbf { u } ( A _ i ) = \\mathbf { u } ( X ) + \\widetilde { \\mathbf { R } } _ i ( X ) , ~ ~ ~ ~ \\emph { w i t h } ~ ~ \\widetilde { \\mathbf { R } } _ i ( X ) = \\int ^ 1 _ 0 \\frac { d } { d t } \\mathbf { u } ( Y _ i ( t , X ) ) d t . \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} \\langle ( a b ^ { p - 1 } ) ^ p \\rangle = \\langle y ^ p \\rangle ^ { g b ^ { - 1 } } , \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} g ( a ) \\circeq \\frac { \\Gamma ( a M + 1 ) } { \\prod _ { i = 1 } ^ { d + 1 } \\Gamma ( a \\gamma _ i + 1 ) } \\prod _ { i = 1 } ^ { d + 1 } x _ i ^ { a \\gamma _ i } \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} S _ m & : = C _ R ( a + m b ) \\\\ & = Z \\cup ( Z + ( a + m b ) ) \\cup ( Z + 2 ( a + m b ) ) \\cup \\dots \\cup ( Z + ( p - 1 ) ( a + m b ) ) , \\\\ & 1 \\leq m \\leq ( p - 1 ) , \\\\ S _ p & : = C _ R ( a ) = Z \\cup ( Z + a ) \\cup ( Z + 2 a ) \\cup \\dots \\cup ( Z + ( p - 1 ) a ) \\\\ S _ { p + 1 } & : = C _ R ( b ) = Z \\cup ( Z + b ) \\cup ( Z + 2 b ) \\cup \\dots \\cup ( Z + ( p - 1 ) b ) . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} \\partial _ t \\rho _ t = \\L ^ \\ast \\rho _ t , \\rho | _ { t = 0 } = \\rho _ 0 \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} \\Phi _ x ( \\alpha ; \\xi ; Z ) : = \\int _ { \\Delta ^ { ( x ) } } \\varphi _ x ( \\alpha ; \\xi ; Z ; u ) _ x u . \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\deg ( \\Phi _ i \\vert _ { \\partial B _ { 1 0 m _ i ^ { - 1 / 2 } } ( x _ j ) } ) = \\deg ( \\Phi ^ 0 _ i \\vert _ { \\partial B _ { 1 0 m _ i ^ { - 1 / 2 } } ( x _ j ) } ) = 1 . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Phi _ \\mu ( r ) = \\frac { 1 } { \\mu - 1 } r + \\frac { 1 } { \\mu - 1 } \\frac { 1 } { \\mu - 2 } ( r + 1 ) ^ { - \\mu + 2 } - \\frac { 1 } { \\mu - 1 } \\frac { 1 } { \\mu - 2 } , \\ ; \\mu \\neq 2 , \\\\ \\ , \\\\ & \\Phi _ 2 ( r ) = r - \\ln ( 1 + r ) , \\ ; r \\geq 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} | p - [ p ] _ { B ( r ) } | \\leq \\sum _ { i = 1 } ^ 4 | p _ i - [ p _ i ] _ { B ( r ) } | . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} \\nabla ^ { - \\nu } \\nabla ^ { - \\upsilon } U ( t ) = \\nabla ^ { - ( \\nu + \\upsilon ) } U ( t ) = \\nabla ^ { - \\upsilon } \\nabla ^ { - \\nu } U ( t ) , \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} S _ { M } ( w \\ , | \\ , a ) = S _ { M - 1 } ( w \\ , | \\ , \\hat { a } _ i ) \\ , S _ M \\bigl ( w + a _ i \\ , | \\ , a \\bigr ) , \\ , i = 1 \\cdots M , \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} | \\Delta _ { I , J } | ^ 2 = P _ { I , J } \\left ( \\Delta _ i ^ { ( k ) } , \\overline { \\Delta } _ i ^ { ( k ) } \\right ) _ { i , k } \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\vert \\tau _ j ( y ) - x \\vert & \\leq \\vert \\sigma _ { j } ( \\tau _ { j - 1 } ( y ) ) - \\tau _ { j - 1 } ( y ) \\vert + \\vert \\tau _ { j - 1 } ( y ) - x \\vert \\\\ & \\leq ( 1 + 3 6 C _ 1 ( m ) \\delta + 2 4 \\delta ) r _ j + r - ( 2 + 3 6 C _ 1 ( m ) \\delta + 2 4 \\delta ) \\sum \\limits _ { k = j } ^ \\infty r _ k \\\\ & \\leq r - ( 2 + 3 6 C _ 1 ( m ) \\delta + 2 4 \\delta ) \\sum \\limits _ { k = j + 1 } ^ \\infty r _ k . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { X } \\Delta _ { a } ( y ) d y = O _ { a } \\left ( X ^ { 3 / 4 + a / 2 } \\right ) , \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} F = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\frac { \\partial ^ 2 u } { \\partial t ^ 2 } - { \\rm d i v } \\ , ( a ( t , z ) | D u _ t | ^ { p - 2 } D u _ t ) + \\beta ( z ) u _ t - \\Delta u = f ( t , z , u ) + \\gamma u _ t \\ \\mbox { i n } \\ T \\times \\Omega , \\\\ u | _ { T \\times \\partial \\Omega } = 0 , \\ u ( 0 , z ) = u _ 0 ( z ) , \\ u _ t ( 0 , z ) = u _ 1 ( z ) , \\end{array} \\right \\} \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} \\sum _ { X / j < k \\leq X ' / j } e ^ { 2 \\pi i F ( j k ) } \\log ( k ) = T _ j ( X ' / j ) \\log ( X ' / j ) - \\int _ { X } ^ { X ' } T _ j ( t / j ) \\frac { { \\rm d } t } { t } , \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} B ( U , E ) = 0 , \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} p ^ * ( L _ { a , j } ) = p ^ * ( X _ { a , j } L _ { a , j + 1 } ) & = \\left ( \\lambda _ { b - j } \\frac { A _ { a , j + 1 } A _ { a - 1 , j - 1 } } { A _ { a , j } A _ { a - 1 , j } } \\right ) \\left ( \\frac { A _ { a - 1 , j } } { A _ { a , j + 1 } } \\prod _ { k = i - a } ^ { b - j - 1 } \\lambda _ k \\right ) \\\\ & = \\frac { A _ { a - 1 , j - 1 } } { A _ { a , j } } \\prod _ { k = i - a } ^ { b - j } \\lambda _ k . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} g \\in L i n ( f ) & \\Longleftrightarrow \\begin{cases} \\forall i , j \\in [ n ] , \\ : i \\geq j \\mbox { a n d } f ( i ) \\leq f ( j ) \\Longrightarrow g ( i ) \\leq g ( j ) , \\\\ \\forall i , j \\in [ n ] , \\ : g ( i ) = g ( j ) \\Longrightarrow f ( i ) = f ( j ) \\end{cases} \\\\ & \\Longleftrightarrow f \\leq g . \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} S ( q , P ) = \\prod _ { s = 1 } ^ m S \\big ( p _ s ^ { j _ s } , P _ s \\big ) , \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K \\} \\\\ = C _ { n } ^ { K } ( 2 ^ { K } - 1 ) . \\end{array} \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} v ( E ) ^ { 2 } = p v ( E ) + q - u ( \\xi ) , \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\widetilde { \\mathcal T } ( M ; \\mathbf x , \\mathbf y ) = \\sum _ { A \\subseteq E } x ^ { r ( M / A ) } y ^ { r ^ * ( M \\backslash A ^ c ) } \\prod _ { \\stackrel { \\mbox { \\rm \\tiny c o n n . c p t s } } { \\mbox { \\rm \\tiny $ M _ i $ o f $ M / A $ } } } x _ { \\bar { g } ( M _ i ) } \\prod _ { \\stackrel { \\mbox { \\rm \\tiny c o n n . c p t s } } { \\mbox { \\rm \\tiny $ M _ j $ o f $ M \\backslash A ^ c $ } } } y _ { \\bar { g } ( M _ j ) } , \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} F ( \\lambda ) = \\sum _ { 1 \\leq j \\not = k \\leq N } f _ j f _ k | y _ j - y _ k | ^ 2 \\log ( | y _ j - y _ k | ^ 2 + \\lambda ) , \\lambda \\geq 0 . \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\widetilde { A } _ { i , r } ^ { - 1 } \\widetilde { A } _ { j , s } ^ { - 1 } & = t ^ { \\alpha ( i , r ; j , s ) } \\widetilde { A } _ { j , s } ^ { - 1 } \\widetilde { A } _ { i , r } ^ { - 1 } , \\\\ \\ \\alpha ( i , r ; j , s ) & = \\begin{cases} 2 ( - \\delta _ { r - s , 2 r _ i } + \\delta _ { r - s , - 2 r _ i } ) & \\ i = j , \\\\ 2 ( - \\delta _ { r - s , - 2 } + \\delta _ { r - s , 2 } ) & \\ | i - j | = 1 , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} c ( \\xi ) = 2 ^ { - \\nu ( N ) } c ( \\mathfrak d _ { \\xi } ) \\chi ( \\mathfrak f _ { \\xi } ) \\mathfrak f _ { \\xi } ^ { k - 1 / 2 } \\prod _ { p } \\Psi _ p ( \\xi ; \\alpha _ p ) . \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} & \\alpha \\circ \\beta ( x _ 1 , \\ldots , x _ { k + l - 1 } ) \\\\ & = \\sum _ { i = 1 } ^ k ( - 1 ) ^ { ( i - 1 ) ( l - 1 ) } \\alpha ( x _ 1 , \\ldots , x _ { i - 1 } , \\beta ( x _ i , \\ldots , x _ { i + l - 1 } ) , x _ { i + 1 } , \\ldots , x _ { k + l - 1 } ) \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} G ( ( A \\times C ) ( x ) , z ) = G ( A ( 0 ) , z ) G ( C ( 0 ) , z ) e ^ { x z } . \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} h _ d ( q ) = \\sum _ { j = 0 } ^ { n / d - 1 } q ^ { d j } . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\overline { \\omega } = ( \\omega _ 0 , \\ , \\omega _ { - d } ) : w _ 0 \\oplus \\Sigma ^ { - d } w _ { - d } \\rightarrow f . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} F _ { | \\mathcal H \\times \\mathcal H } ( \\tau , \\tau ' ) = \\sum _ { n , m \\in \\Z } \\left ( \\sum _ { \\substack { r \\in \\Z , \\\\ r ^ 2 < 4 n m } } A _ F ( n , r , m ) \\right ) e ^ { 2 \\pi \\sqrt { - 1 } ( n \\tau + m \\tau ' ) } . \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} I _ v ( \\chi _ v , s ) = \\chi _ v | \\cdot | _ v ^ { ( 2 n + 1 ) s } \\ , \\rtimes \\ , \\chi _ v ^ { - n } | \\cdot | _ v ^ { - n ( 2 n + 1 ) s } . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} ( f \\cdot b ) \\ast _ g a = ( f \\otimes b ) \\ast _ g \\Delta ( a ) = \\sum ( f \\ast _ g a ^ { ( 1 ) } ) \\cdot ( b \\ast _ g a ^ { ( 2 ) } ) . \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} f _ { \\eta _ 2 } \\circ { \\chi _ 2 } - f _ { \\eta _ 1 } \\circ { \\chi _ 1 } = u _ e \\cdot q _ e . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} v _ { n } ^ { \\varepsilon } \\left ( T \\right ) = v _ { f n } ^ { \\varepsilon } = \\sum _ { j = 1 } ^ { n } \\left ( V _ { f } ^ { \\varepsilon } \\right ) _ { j n } \\phi _ { j } \\to u _ { f } ^ { \\varepsilon } \\ ; L ^ { 2 } \\left ( \\Omega \\right ) \\ ; n \\to \\infty . \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\mathcal { C } ( t , x ) = \\frac { C ( t x ) } { 1 - x C ( t x ) } , \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\widetilde { F } ' = e ^ { \\widetilde { F } } ( 1 + x \\widetilde { F } ' ) ( 1 + 2 x \\widetilde { F } ' ) , \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\mid N , \\gcd ( m , N / m ) \\leq 2 , \\\\ q \\mid \\frac { N } { \\gcd ( m , N / m ) } } } \\chi _ { \\alpha ( N , m ) } ( - 1 ) = I _ \\chi 2 ^ { \\omega ( N ) } \\Omega _ 1 ( N , q ) . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\sum _ { d \\mid v } \\mu ( d ) \\frac { N } { d w } \\sum _ { | h | \\leq W ^ { 1 + \\epsilon } d / N } \\widehat { G } \\left ( \\frac { h N } { d w } \\right ) \\sum _ { ( a , w ) = 1 } e \\left ( \\frac { a d \\overline { m \\ell _ 1 \\ell _ 3 v } } { w } + \\frac { a h } { w } \\right ) . \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} 2 \\left \\{ \\zeta _ i ^ { ( k ) } , \\varphi _ q ^ { ( p ) } \\right \\} _ { \\infty } & = \\left \\{ \\ell _ i ^ { ( k ) } , \\psi _ q ^ { ( p - 1 ) } + \\mbox { ` ` h i g h e r t e r m s '' } \\right \\} _ { \\mathfrak { k } ^ * } \\\\ & = \\sum _ { j = 1 } ^ i \\frac { \\partial } { \\partial \\psi _ j ^ { ( k ) } } \\left ( \\psi _ q ^ { ( p - 1 ) } + \\mbox { ` ` h i g h e r t e r m s '' } \\right ) . \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} { \\tilde z } _ 1 = z _ 1 , \\ ; \\ ; \\ ; { \\tilde z } _ 2 = z _ 2 , \\ ; \\ ; \\ ; { \\tilde z } _ 3 = z _ 3 , \\ ; \\ ; \\ ; { \\tilde z } _ 4 = z _ 4 . \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} \\nu _ { \\delta } ( r ) : = \\frac { 1 } { r } \\sum _ { d | r } \\frac { \\mu ( d ) } { d ^ { 1 + 2 \\delta } } . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} x = \\begin{pmatrix} H \\\\ 0 \\\\ 1 \\end{pmatrix} , y = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} \\sum _ { n = N _ { k - 1 } } ^ { N _ k - 1 } \\abs { h _ { n + 1 } ( x ) - h _ n ( x ) } \\leq \\frac { \\log ( x ) } { x \\psi ( x ) } , \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} \\Theta ( \\tau ) = \\Upsilon , \\Theta ( \\Upsilon ) = \\tau . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { ( \\phi _ { 1 + a } * \\phi ) ( n ) } { n } & = \\frac { \\zeta ( 1 - a ) } { \\zeta ^ { 2 } ( 2 ) } x + \\frac { \\zeta ( 1 + a ) } { ( 1 + a ) \\zeta ^ { 2 } ( 2 + a ) } x ^ { 1 + a } + P _ { a } ( x ) , \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} \\ell ( P ' ) = b + f + c \\ge d ( s ' t ) = \\sigma / 2 - a \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ell ( P ) = \\sigma - \\ell ( P ' ) \\le \\sigma / 2 + a . \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} N ( t ) = \\int _ 0 ^ \\infty p ( N ( t ) , s ) n ( t , s ) \\d s . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} I _ 2 + \\overline { \\Psi } ^ { - T } \\overline { K } ^ { - 1 } \\overline { L } = I _ 2 + \\sum _ { j \\in \\mathcal { I } ^ - } L ( A _ j ) \\overline { \\Psi } ^ T _ j K ^ { - 1 } = ( K + \\sum _ { j \\in \\mathcal { I } ^ - } L ( A _ j ) \\overline { \\Psi } _ j ) K ^ { - 1 } \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} \\mu _ k = \\min _ { g \\in V , ( g , g _ j ) = 0 j = k + 1 , \\ldots , N } \\frac { ( A g , g ) } { ( g , g ) } = \\max _ { g \\in V , ( g , g _ l ) = 0 l = 1 , \\ldots , k - 1 } \\frac { ( A g , g ) } { ( g , g ) } . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} r ( t ) = \\frac 1 2 \\Bigl [ e ^ { i t D } \\phi _ + + e ^ { - i t D } \\phi _ - \\Bigr ] _ { x = 0 } . \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\eta : M \\to \\R , \\eta ( x ) : = \\limsup _ { t \\to \\infty } \\{ t - d ( x , S _ p ( t ) ) \\} , \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} \\lambda _ { 1 1 } \\begin{pmatrix} A _ 2 B _ 3 ^ T + B _ 3 A _ 2 ^ T \\end{pmatrix} \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } W , \\breve { J } U ) = p \\breve { g } ( \\breve { J } W , U ) + q \\breve { g } ( W , U ) , \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} A ( x ) = L _ { \\boldsymbol { a } } I ( x ) . \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} X = p _ { B , P _ J } ^ { - 1 } ( X _ { v P _ J } ) = X _ { v w _ { 0 , P _ J } B } , \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} \\varTheta = \\varTheta _ \\lambda \\times \\varTheta _ \\beta . \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\omega J _ b \\xi + \\mathcal L _ { \\mu _ 2 } \\nu = \\frac { \\epsilon } { \\gamma } \\xi \\nu \\\\ - \\omega J _ d \\nu + ( 1 - \\gamma ) J _ c \\xi = \\frac { \\epsilon } { 2 \\gamma } \\nu ^ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} f ' ( p g , s ) = | \\chi ( p ) \\delta _ { P _ { 4 n } } ( p ) ^ { s + \\frac 1 2 } | \\ , f ' ( g , s ) \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} I _ { 1 , 2 } : = \\int _ 0 ^ t \\ ! \\frac { \\mathrm { d } s } { \\sqrt { t - s } } \\int _ { \\mathbb { T } \\times \\mathbb { R } } \\ ! \\left \\vert { \\boldsymbol { \\tilde { \\varphi } } } ^ { \\nu _ s } ( x ) \\right \\vert \\nu _ s ( \\mathrm { d } x , \\mathrm { d } v ) \\left \\vert \\int _ { \\mathbb { T } \\times \\mathbb { R } } \\ ! \\varphi ( x - y ) ( \\nu _ t - \\mu _ t ) ( \\mathrm { d } y , \\mathrm { d } w ) \\right \\vert \\ , . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} \\sum _ { | I | = j , I \\subseteq \\{ 1 , \\ldots k \\} } e ^ { t \\sum _ { i \\in I } \\lambda _ i ^ { ( k ) } ( A ) } = \\sum _ { \\substack { | I | = | J | = j \\\\ I , J \\subseteq \\{ n - k + 1 , \\ldots , n \\} } } P _ { I , J } \\left ( \\Delta _ i ^ { ( k ) } ( b _ t ) , \\overline { \\Delta } _ i ^ { ( k ) } ( b _ t ) \\right ) . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\ell \\Big ( \\int _ 0 ^ 1 \\varphi ' ( t ) y ( t ) \\ , \\d t \\Big ) = \\ell \\Big ( - \\int _ 0 ^ 1 \\varphi ( t ) z ( t ) \\ , \\d t \\Big ) \\qquad \\forall \\varphi \\in C ^ \\infty _ c ( 0 , 1 ) , \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\frac { w ( f ) } { B ( ( u , v ) , f ) } = \\frac { w _ 1 ( \\pi _ 1 ( f ) ) } { r B _ 1 ( u , \\pi _ 1 ( f ) ) } \\cdot \\frac { w _ 2 ( \\pi _ 2 ( f ) ) } { B _ 2 ( v , \\pi _ 2 ( f ) ) } . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} b \\circ ( a \\cdot _ x c ) = \\rho _ b ( a ) \\cdot _ { b \\circ x } ( b \\circ c ) \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} B _ { M , m } ( x \\ , | \\ , a ) \\triangleq \\frac { d ^ m } { d t ^ m } \\Big | _ { t = 0 } \\bigl [ f ( t ) e ^ { - x t } \\bigr ] . \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} \\begin{array} { l l } & i _ { \\Delta f } ( \\Delta g ) ( X _ 0 , X _ 1 , \\cdots , X _ { p + q } ) \\\\ & = \\sum _ { k = 0 } ^ p ( - 1 ) ^ { k q } \\sum _ { \\sigma \\in S h ( q , p - k ) } \\epsilon ( \\sigma ) \\Delta f \\circ _ k ^ \\sigma \\Delta g ( X _ 0 , X _ 1 , \\cdots , X _ { p + q } ) . \\end{array} \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\tilde { V } : = V ^ 1 - V ^ 2 = ( u _ 1 ^ 1 , \\theta ^ 1 , q ^ 1 ) - ( u _ 1 ^ 2 , \\theta ^ 2 , q ^ 2 ) , \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} s ^ h & : = s ^ { ( m ^ 2 - 1 ) \\times m } = s ( [ m , 1 ] ) ^ { m - 1 } s ( [ m - 1 , 1 ] ) = s _ 1 s _ 2 s _ 3 s _ 1 \\\\ s ^ { h v } & : = s ^ { ( m ^ 2 - m - 1 ) \\times ( m - 1 ) } = s ( [ m , 1 ] ) ^ { m - 2 } s ( [ m - 1 , 1 ] ) = s _ 1 s _ 2 s _ 1 s _ 2 s _ 3 s _ 1 , \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\omega ( e , e ) = 0 , \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{gather*} \\mathcal G _ { 1 , ( 1 ; 3 ) } ^ \\ast ( \\tau ) = - \\frac { { \\rm i } \\pi } { 3 } - 2 \\pi { \\rm i } \\sum _ { n \\ge 1 } \\frac { q ^ { n } - q ^ { 2 n } } { 1 - q ^ { 3 n } } , \\\\ \\mathcal G _ { 2 , ( 1 ; 3 ) } ^ \\ast ( \\tau ) = - \\frac { \\pi } { 3 \\Im ( \\tau ) } - ( 2 \\pi ) ^ 2 \\sum _ { n \\ge 1 } n \\frac { q ^ { n } + q ^ { 2 n } } { 1 - q ^ { 3 n } } , \\end{gather*}"}
-{"id": "2305.png", "formula": "\\begin{align*} & \\Omega _ 0 = ( - 1 ) ^ { m + 1 } { 2 m \\choose m } & & \\\\ & \\Omega _ 1 = - 2 + ( - 1 ) ^ { m + 1 } { 2 m \\choose m } & & \\\\ & \\Omega _ 2 = 4 m + ( - 1 ) ^ { m + 1 } { 2 m \\choose m } & & \\\\ \\intertext { a n d } & \\Omega _ { \\Phi ( n ) - 1 } = 2 & & \\\\ & \\Omega _ { \\Phi ( n ) - 2 } = - 2 n + 2 & & \\\\ & \\Omega _ { \\Phi ( n ) - 3 } = n ^ 2 - 3 n + 2 & & . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\alpha ' e _ { r , s } = \\sqrt { 1 - q ^ { 2 r } } e _ { r - 1 , s } , & \\gamma ' e _ { r , s } = q ^ { r } e _ { r , s + 1 } . \\end{array} \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\bar A = A _ 0 e _ 0 - A _ 1 e _ 1 - A _ 2 e _ 2 - A _ 3 e _ 3 . \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\sum _ { k = 1 } ^ { n } c _ j \\overline { c _ k } f ( x _ j - x _ k ) \\ge 0 \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} F ( z , z ' ) : = F _ 0 ( u ( z ) , u ( z ' ) ) \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} T ^ { C _ { ( L ) } } _ { i } = \\max \\{ T ^ u _ { i } , T ^ d _ { i } , T ^ { u a c } _ { i } , T ^ { d a c } _ { i } , T ^ { c ' } _ { i } \\} , \\ \\forall i , \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} M _ { n } ( x ; \\beta , c ) = \\dfrac { c ^ n \\ , ( \\beta ) _ { n } } { ( c - 1 ) ^ n } \\ , _ { 2 } F _ { 1 } \\left ( \\begin{array} { c } - n , - x \\\\ \\beta \\end{array} ; 1 - \\frac 1 c \\right ) . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\rho _ { \\{ j \\} } ( \\Delta ) & = \\frac { 1 } { 2 } \\log \\left ( \\frac { \\sigma _ { { j } } ^ 2 + \\sum \\limits _ { i \\neq j } r _ { i j } ^ 2 \\sigma _ { i } ^ 2 } { \\Delta - \\Delta _ { \\min , \\{ j \\} } } \\right ) \\\\ & = \\frac { 1 } { 2 } \\log \\left ( \\frac { \\sum \\limits _ { i = 1 } ^ { m } \\sigma _ i ^ 2 - \\Delta _ { \\min , \\{ j \\} } } { \\Delta - \\Delta _ { \\min , \\{ j \\} } } \\right ) \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} w ^ B ( t ) = w _ { 0 } ^ { B } t , \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} \\l : = \\mathrm { d e g } ( \\Lambda ) = \\sum _ { k = 1 } ^ t n _ k ( p _ k - 1 ) = \\sum _ { k = 1 } ^ t ( n - n _ k ) . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} \\sum _ { j \\geq 0 } ( 1 + 2 j ) ^ { N - 1 } \\mathfrak { d } _ 2 \\dot { \\phi } ( 2 j ) = \\infty . \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} N _ { e } ^ { G } \\mathbb F _ { p } ( G / C _ { p ^ { k } } ) = \\mathbb Z / p ^ { k + 1 } , \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} [ H _ p ] = 1 + v + \\cdots + v ^ { p - 1 } = \\Phi _ p ( v ) . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} \\alpha ^ b = ( n + 1 ) , \\alpha ^ t = ( n + 3 - m , n + 2 - m , \\dots , n + 4 - m - p , n + 2 - m - p , \\dots , 3 , 2 ) \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} v - v _ { - } = \\pm \\sqrt { \\frac { \\rho - \\rho _ { - } } { \\rho \\rho _ { - } } \\Big ( A ( \\rho ^ { n } - \\rho _ { - } ^ { n } ) - B ( \\frac { 1 } { \\rho ^ { \\alpha } } - \\frac { 1 } { \\rho _ { - } ^ { \\alpha } } ) \\Big ) } . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\lambda ( n ) = n ^ 2 - o ( n ^ 2 ) . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} f _ 1 ^ 2 - y ^ 2 f _ 3 = x g , g = x ^ { 2 a + 1 } - 2 x ^ a y z + y ^ { b + 2 } . \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} f ( x ) = & f \\bigg ( m \\bigg ( 1 - \\frac { x - m a } { b - m a } \\bigg ) a + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) b \\bigg ) \\\\ \\leq & m \\bigg ( 1 - \\frac { x - m a } { b - m a } \\bigg ) ^ s f ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) f ( b ) . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} C _ { \\gamma ' } ' = C _ { \\gamma ' } ^ 2 + c I , \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\xi ^ { d + 1 } \\circ \\rho = \\xi ^ { d + 1 } \\circ \\xi ^ d \\circ \\rho ' = 0 \\circ \\rho ' = 0 , \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} & \\delta _ H ( c _ H ) ( \\psi _ g ( x _ 1 ) , \\ldots , \\psi _ g ( x _ { n + 1 } ) ) \\\\ & = \\sum _ { 1 \\leq i < j \\leq n + 1 } ( - 1 ) ^ j c _ H ( g x _ 1 , \\ldots , g x _ { i - 1 } , [ g x _ i , g x _ j ] , \\ldots , \\widehat { g x _ j } , \\ldots , g x _ { n + 1 } ) \\\\ & = \\sum _ { 1 \\leq i < j \\leq n + 1 } ( - 1 ) ^ j A ( \\hat { g } ) c _ K ( x _ 1 , \\ldots , x _ { i - 1 } , [ x _ i , x _ j ] , x _ { i + 1 } , \\ldots , \\hat { x } _ j , \\ldots , x _ { n + 1 } ) \\\\ & = A ( \\hat { g } ) \\delta _ K ( c _ K ) ( x _ 1 , \\ldots , x _ { n + 1 } ) . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{gather*} \\| u _ j ^ H \\| _ { L ^ 2 } ^ 2 = ( s ^ 2 - \\sigma ( s ) ^ 2 ) \\prod _ { s ' \\neq s } \\frac { s ^ 2 - \\sigma ( s ' ) ^ 2 } { s ^ 2 - s '^ 2 } , \\\\ \\| u _ k ^ K \\| _ { L ^ 2 } ^ 2 = ( s ^ 2 - \\sigma ( s ) ^ 2 ) \\prod _ { s ' \\neq s } \\frac { \\sigma ( s ) ^ 2 - s '^ 2 } { \\sigma ( s ) ^ 2 - \\sigma ( s ' ) ^ 2 } , \\end{gather*}"}
-{"id": "3743.png", "formula": "\\begin{align*} K _ { e _ 1 , \\cdots , e _ n } & = K \\ , { \\rm d i a g } ( \\varpi ^ { e _ 1 } , \\cdots , \\varpi ^ { e _ n } , \\varpi ^ { - e _ 1 } , \\cdots , \\varpi ^ { - e _ n } ) K . \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} F S ( e ) = \\frac { p ^ { 2 e } - r _ e ^ 2 } { n } + \\max \\{ 1 , 2 r _ e - n + 1 \\} . \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} { \\rm o r d } _ { D _ i } ( \\theta _ q ) = x _ i ( q ) . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\varphi ( z ) \\coloneqq \\sum _ { i = 1 } ^ N \\frac { k _ i } { z - z _ i } , \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} g ( \\varphi U , V ) = g ( U , \\varphi V ) + u ( V ) \\theta ( U ) - u ( U ) \\theta ( V ) , \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} f ( u ) = u ^ { 3 } + \\beta u ^ 5 + f _ { 5 } ( u ) , \\ ; \\ ; \\ ; \\ ; \\beta < 0 , \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\overline X _ u = f _ 1 ^ { - 1 } ( \\{ \\delta \\} \\cap Z _ 1 \\cap B = F _ 1 , \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} & 0 < \\beta < 1 / 2 \\ , , \\ \\ \\gamma = 9 / ( 2 \\beta ^ { 2 } - 5 \\beta + 2 ) \\ , , \\\\ & \\frac { 1 } { \\gamma ^ 2 } \\max ( 1 , 2 + \\beta \\gamma - 2 \\sqrt { 1 + \\beta \\gamma } ) < d < \\frac { 2 + \\beta \\gamma + 2 \\sqrt { 1 + \\beta \\gamma } } { \\gamma ^ 2 } \\ , . \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( a _ { i j } ^ 2 + b _ { i j } ^ 2 + c _ { i j } ^ 2 + g _ { i j } ^ 2 \\right ) = \\left | \\sum _ { k = 1 } ^ { n } \\delta _ k u _ { i k } v _ { j k } \\right | ^ 2 + \\left | \\sum _ { k = 1 } ^ { n } \\sigma _ k u _ { i k } \\overline { v } _ { j k } \\right | ^ 2 . \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} r ( p ) = \\begin{cases} \\displaystyle \\frac { A } { \\sqrt { p } } , & A _ 0 ^ 2 \\leqslant p \\leqslant N ^ { c / A _ 0 ^ 2 } , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\begin{cases} a _ { 0 , \\sup } + \\frac { a _ { 2 , \\sup } } { 2 } \\bar r _ 1 + \\frac { b _ { 1 , \\sup } } { 2 } \\bar r _ 2 < 2 a _ { 1 , \\inf } \\underbar r _ 1 + a _ { 2 , \\inf } \\underbar r _ 2 \\cr b _ { 0 , \\sup } + \\frac { b _ { 1 , \\sup } } { 2 } \\bar r _ 2 + \\frac { a _ { 2 , \\sup } } { 2 } \\bar r _ 1 < 2 b _ { 2 , \\inf } \\underbar r _ 2 + b _ { 1 , \\inf } \\underbar r _ 1 . \\end{cases} \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} | \\phi _ n | _ { \\alpha , \\sigma } & \\le c _ n + ( n - 1 ) c _ { n - 1 } + K ^ \\alpha \\sum _ { k = 1 } ^ { n - 1 } c _ k c _ { n - k } \\\\ & \\le c _ n + ( n - 1 ) c _ { n - 1 } + K ^ \\alpha ( n - 1 ) d _ n < \\infty . \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\big | f ( t , \\omega + \\varepsilon h , X ( t ) ( \\omega ) ) - f ( t , \\omega , X ( t ) ( \\omega ) ) \\big | ^ m d t = \\int _ 0 ^ T \\Big | \\int _ 0 ^ \\varepsilon D ^ h f ( t , \\omega + r h , X ( t ) ( \\omega ) ) d r \\Big | ^ m d t . \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\partial \\beta ^ { n - 2 , \\ , n } = - \\bar \\partial \\overline { \\Omega ^ { n - 1 , \\ , n - 1 } } . \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} q _ n ^ x ( 1 , y ) & = \\int _ { - 1 } ^ 1 q _ n ( x , y ) \\ , d x = \\iint p _ n ( t , y ) \\varrho _ \\varepsilon ( x - t ) \\ , d t \\ , d x \\\\ & = \\int _ { - 1 } ^ 1 p _ n ( t , y ) \\int _ { - 1 } ^ 1 \\varrho _ \\varepsilon ( x - t ) \\ , d x \\ , d t = \\int _ { - 1 } ^ 1 p _ n ( t , y ) \\ , d t = 0 . \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} Q ^ 2 & = \\sum _ { k \\geq 1 } \\ell _ k , \\\\ | J | ^ 2 & = \\sum _ { k \\geq 1 } ( Q + \\sigma _ k ^ 2 ) \\ell _ k . \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} p ^ * \\left ( L _ { i , j } \\right ) = \\frac { A _ { i - 1 , j - 1 } } { A _ { i , j } } \\prod _ { k = i - a } ^ { b - j } \\lambda _ k , \\forall \\ , 1 \\leq i \\leq a , \\forall \\ , 1 \\leq j \\leq b , \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\begin{array} { l l } ( - u v ^ { - 1 } ) ' = [ v ' v ^ { - 1 } - u ' u ^ { - 1 } ] ( - u v ^ { - 1 } ) \\end{array} \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} C : = \\sum _ { e \\in E ( G ) } \\prod _ { v \\in e } d _ v ^ { 1 / ( p - r ) } . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} w '' ( t , x ) = - \\nabla \\mathcal { W } ( w ( t , \\cdot ) ) ( x ) + f ( t , x ) . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{gather*} D = D _ 2 Y ^ 2 + D _ 1 Y + D _ 0 , \\\\ C _ 1 C _ 2 + C _ 0 C _ 3 = E _ 2 Y ^ 2 + E _ 1 Y + E _ 0 , \\\\ ( k + 1 ) ( C _ 1 C _ 2 + C _ 0 C _ 3 ) ^ 2 + ( C _ 1 ^ 2 + C _ 0 C _ 2 ) ( C _ 2 ^ 2 + C _ 1 C _ 3 ) = F _ 4 Y ^ 4 + \\cdots + F _ 0 , \\end{gather*}"}
-{"id": "3935.png", "formula": "\\begin{align*} ( W ( \\rho ^ 0 , \\rho ^ 1 ) ) ^ 2 = \\inf _ { v _ t , \\rho _ t } ~ ~ \\big \\{ { \\int _ 0 ^ 1 \\int _ { M } v _ t ^ 2 \\rho _ t d x d t } \\colon \\frac { \\partial \\rho _ t } { \\partial t } + \\nabla \\cdot ( \\rho _ t v _ t ) = 0 , ~ \\rho _ 0 = \\rho ^ 0 , ~ \\rho _ 1 = \\rho ^ 1 \\big \\} , \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} g ( t ) \\triangleq f ( t ) e ^ { - q t } \\frac { d ^ r } { d t ^ r } \\bigl [ e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) \\bigr ] . \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} E _ { i j } 1 _ A ( T ) = \\sum _ { k \\in T ^ { i , j + 1 } } c _ { i j , \\{ k \\} } T \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} \\ell ' ( L ) = \\ell ' ( A ) + \\ell ' ( B ) = ( d ( p x ) - a ) + ( d ( x q ) - a _ 1 ) \\ge d ( p q ) + a - a _ 1 . \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} Z : = \\{ ( f , x ) \\in I \\oplus A _ { 1 } \\mid f = \\dd ( x ) \\} \\ , ; \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} C ( U , \\omega V ) = \\phi B ( U , V ) , \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} ~ ~ ^ { A B C } D ^ \\alpha _ b ( p ( t ) ~ ^ { A B R } _ { a } D ^ \\alpha x ( t ) ) + q ( t ) x ( t ) = \\lambda r ( t ) x ( t ) , ~ ~ t \\in ( a , b ) , \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} \\phi _ { t , r } = \\phi _ { t , s } \\circ \\phi _ { s , r } . \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\Psi ( \\eta ) = M ( \\eta ) + E ^ a ( \\eta ) + \\langle b \\rangle | \\eta | , M ( \\eta ) : = \\sum _ { x \\in \\eta } m ( x ) \\leq m ^ * | \\eta | , \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} A ^ { * * } = A . \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} x ^ B ( t ) = \\frac { 1 } { 2 } ( u _ + + u _ - ) t + + \\frac { 1 } { 2 } \\beta t ^ 2 , \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} | \\varphi ( 0 ) | ^ 2 + \\rho ^ 2 ( | b _ 1 | ^ 2 - 1 ) + \\sum _ { j = 2 } ^ \\infty \\rho ^ { 2 j } | b _ j | ^ 2 + 2 k \\log \\rho + 2 \\log | g ( 0 ) | \\leq \\log M . \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} b _ s ^ { } = & ( - 1 ) ^ { 2 n + 1 - m - s } q ( j ^ * \\sigma _ { 2 n + 1 - m , \\dots , p + 2 , p , \\dots , 2 n + 2 - 2 m , 1 } ^ + ) ^ \\vee \\\\ & + ( - 1 ) ^ { 2 n + 1 - m - s } q ( j ^ * \\sigma _ { 2 n + 1 - m , \\dots , p + 1 , p - 1 , \\dots , 2 n + 2 - 2 m } ^ + ) ^ \\vee \\\\ = & ( - 1 ) ^ { 2 n + 1 - m - s } q \\sigma _ { 2 n - m , \\dots , p + 1 , p - 1 , \\dots , 2 n + 1 - 2 m } ^ \\vee \\\\ & + ( - 1 ) ^ { 2 n + 1 - m - s } q ( \\tau _ { 2 n + 1 - m , \\dots , p + 1 , p - 1 , \\dots , 2 n + 2 - 2 m } ' ) ^ \\vee . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} D _ r = D _ r ( x _ 0 ) = B ( x _ 0 , r ) \\cap D \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} \\phi ( t ) = \\phi ( 0 ) + \\int _ 0 ^ t \\dot \\phi ( s ) \\ , d s , \\forall t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} X & \\triangleq \\beta _ { 2 , 2 } ^ { - 1 } \\bigl ( \\tau = 1 , b _ 0 = 1 , \\ , b _ 1 = 1 + \\alpha , \\ , b _ 2 = 1 + \\alpha \\bigr ) , \\\\ Y & \\triangleq \\beta _ { 1 , 0 } ^ { - 1 } ( \\tau = 1 , b _ 0 = 2 \\alpha + 2 ) , \\\\ Y ' & \\triangleq \\beta _ { 1 , 0 } ^ { - 1 } \\bigl ( \\tau = 1 , b _ 0 = 1 \\bigr ) \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} G ( x , y ) ~ = ~ \\sum _ { n = 0 } ^ \\infty \\P _ x ( X ( n ) = y ) . \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} \\ell ( S ) : = \\lim _ n \\frac { 1 } { n } L ( S ^ n , x ) = \\inf _ { n \\geq 1 } \\frac { 1 } { n } L ( S ^ n , x ) . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\sqrt { 1 - e ^ { - h ( x ) } } K ( x , y ) \\sqrt { 1 - e ^ { - h ( y ) } } = \\sum _ { i = 1 } ^ { + \\infty } \\lambda _ i \\varphi _ i ( x ) \\varphi _ i ( y ) \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} X _ C ( T ) = \\# \\left \\{ [ \\iota ] \\in \\binom { V } { W } ^ T \\ , \\middle \\vert \\ , T \\circ \\iota = \\iota \\circ S S \\in C \\right \\} \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} \\rho : = \\sigma _ { b - 1 } \\circ \\cdots \\circ \\sigma _ 2 \\circ \\sigma _ { 1 } , \\mbox { w h e r e } \\sigma _ i : = \\mu _ { ( 1 , i ) } \\circ \\mu _ { ( 2 , i ) } \\circ \\dots \\circ \\mu _ { ( a - 1 , i ) } . \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\frac { r ^ { p - r } } { ( \\lambda ^ { ( p ) } ( G ) ) ^ p } = \\alpha = \\frac { ( r + 1 ) ^ { p - r } } { \\big ( \\lambda ^ { ( p + 1 ) } ( G ^ { r + 1 } ) \\big ) ^ { p + 1 } } , \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} Q _ s = \\big ( \\big \\lfloor e ^ { ( s + 1 ) ^ { \\rho / 1 0 } } \\big \\rfloor \\big ) ! \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} \\rho _ f ( x ) = \\int _ H \\Delta _ G ( h ) \\Delta _ H ( h ) ^ { - 1 } f ( x h ) d \\mu _ H ( h ) \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} \\Gamma _ { M } ( z \\ , | \\ , a ) = \\Gamma _ { M - 1 } ( z \\ , | \\ , \\hat { a } _ i ) \\ , \\Gamma _ M \\bigl ( z + a _ i \\ , | \\ , a \\bigr ) , \\ , i = 1 \\cdots M , \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d h ( s ) = & \\left [ g _ { y } ( s ) h ( s ) + b _ { y } ( s ) m ( s ) + \\sigma _ { y } ( s ) n ( s ) \\right ] d s \\\\ & + \\left [ g _ { z } ( s ) h ( s ) + b _ { z } ( s ) m ( s ) + \\sigma _ { z } ( s ) n ( s ) \\right ] d B ( s ) , \\\\ h ( t ) = & 1 , \\\\ d m ( s ) = & - \\left [ g _ { x } ( s ) h ( s ) + b _ { x } ( s ) m ( s ) + \\sigma _ { x } ( s ) n ( s ) \\right ] d s + n ( s ) d B ( s ) , s \\in \\lbrack t , T ] , \\\\ m ( T ) = & \\phi _ { x } ( \\bar { x } ( T ) ) h ( T ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} 3 n - 6 = e ( T ) & = e ( T [ W \\cup S \\cup \\{ u _ 1 , u _ 2 , u _ 3 , u _ 4 \\} ] ) + e _ T ( \\{ u _ 5 , \\ldots , u _ r \\} , W \\cup S \\cup \\{ u _ 1 , u _ 2 , u _ 3 , u _ 4 \\} ) \\\\ & \\le e ( \\mathcal { T } _ { 1 1 } ) + ( 2 n - 4 ) = 2 7 + ( 2 n - 4 ) , \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} \\norm { \\lambda _ { G / H } ( T _ H ( f ) ) } = \\norm { \\lambda _ { G / H } \\circ q ( f ) } \\leq \\norm { \\lambda _ G ( f ) } \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\begin{array} { c c c } e w _ { k } = - ( n - k + 1 ) w _ { k - 1 } \\\\ f w _ { k } = ( k + 1 ) w _ { k + 1 } \\\\ h w _ { k } = ( n - k ) w _ { k } . \\end{array} \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G ) ) = r ^ { 1 - r / p } \\max _ i \\big \\{ \\alpha _ i ^ { - 1 / p } \\big \\} , \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} \\varphi _ B ( x ) = \\int _ 0 ^ { 1 } \\int _ 0 ^ { \\infty } \\xi x J _ 0 ( \\xi x ) J _ 0 ( \\xi y ) \\varphi ( y ) d y d \\xi , \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} F _ \\psi ^ \\infty ( p ) = \\lim _ { t \\rightarrow \\infty } \\frac { \\psi ( t ) } { t } \\sum _ { i = 1 } ^ n \\lambda _ i ( p ) = \\lim _ { t \\rightarrow \\infty } \\frac { \\psi ( t ) } { t } F _ { T V } ( p ) . \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} [ y _ 1 ^ U ] _ \\beta ^ T \\alpha = \\alpha y _ 1 ^ T , \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} ( \\cdot ) _ 0 : = 1 , ( \\cdot ) _ j : = \\prod _ { k = 0 } ^ { j - 1 } ( \\cdot - k ) , j \\ge 1 , [ \\cdot ] _ j : = \\frac { 1 } { j ! } ( \\cdot ) _ j , j \\ge 0 . \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} | G _ { f _ 5 } ( s ) | + | F _ { f _ 5 } ( s ) | = o ( | s | ^ 6 ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; | s | \\to 0 . \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} u _ t = \\nabla \\left ( U ^ { m - 1 } \\nabla u \\right ) \\mbox { i n t h e d i s t r i b u t i o n s e n s e i n $ \\R ^ n \\times ( t _ 0 , \\infty ) $ } \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z } \\Omega _ p ( \\alpha _ n ) \\mathrm { v o l } ( \\Gamma _ 0 \\alpha _ n \\Gamma _ 0 ) = p ^ { - 1 } ( 1 - p ^ { - 1 } ) \\left ( 1 - \\frac { 2 w _ p } { p ^ 2 + w _ p } \\right ) = p ^ { - 1 } ( 1 - p ^ { - 1 } ) \\frac { p ^ 2 - w _ p } { p ^ 2 + w _ p } . \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} P _ { T } \\left ( \\Omega \\right ) : = \\left \\{ u \\in W _ { T } \\left ( \\Omega \\right ) : \\left . u \\right | _ { \\partial \\Omega } = 0 , \\left . u \\right | _ { t = T } = 0 , \\left . u \\right | _ { t = 0 } = 0 \\right \\} . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\widetilde \\nabla \\coloneqq \\P ( z ) ^ { - \\rho / h ^ \\vee } \\nabla \\P ( z ) ^ { \\rho / h ^ \\vee } = d + p _ { - 1 } \\P ( z ) ^ { 1 / h ^ \\vee } d z + \\widetilde u ( z ) d z . \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} X _ B = \\bigg { \\{ } \\mathbf x = ( x _ n ) _ { n \\in \\Z } \\subset X : \\sup _ { n \\in \\Z } \\lVert x _ n \\rVert < \\infty \\bigg { \\} } . \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} G ( a / q ) = \\prod _ { \\gamma = ( \\gamma ' , 0 ) \\in \\Gamma } \\bigg ( \\frac { 1 } { q } \\sum _ { x = 1 } ^ q e ^ { 2 \\pi i a _ \\gamma x / q } \\bigg ) \\prod _ { \\gamma = ( 0 , \\gamma '' ) \\in \\Gamma } \\bigg ( \\frac { 1 } { \\varphi ( q ) } \\sum _ { x \\in A _ q } e ^ { 2 \\pi i a _ \\gamma x / q } \\bigg ) . \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} | P ^ { \\vec { h } } _ { \\mathbb { Z } ^ d , \\beta , H } ( \\sigma _ x = s _ 1 , \\sigma _ y = s _ 2 ) - P ^ { \\vec { h } } _ { \\mathbb { Z } ^ d , \\beta , H } ( \\sigma _ x = s _ 1 ) P ^ { \\vec { h } } _ { \\mathbb { Z } ^ d , \\beta , H } ( \\sigma _ x = s _ 2 ) | \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\Omega ( t ) : = \\int _ { - \\infty } ^ { \\infty } \\ , x \\ , u ( t , x ) d x , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Lambda ( t ) : = \\int _ { - \\infty } ^ { \\infty } \\ , x \\ , u ^ 2 ( t , x ) d x , \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ { p , q } ) ^ { s } u = \\lambda | u | ^ { p - 2 } u , \\ , \\ , x \\in \\Omega , \\\\ u ( x ) = 0 , \\ , \\ , x \\in \\mathbb { G } \\setminus \\Omega . \\end{cases} \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} w ' ( e ) ^ { p ' - r } \\prod _ { v \\in e } B ' ( v , e ) = \\frac { w ( e ) ^ { p - r } } { m ^ { p ' - p } } \\prod _ { v \\in e } B ( v , e ) = \\frac { \\alpha } { m ^ { p ' - p } } . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} [ y _ 1 ^ U ] _ \\beta = \\begin{pmatrix} J _ { l , 2 } \\\\ & I _ { m - 2 l } \\end{pmatrix} \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} \\lambda ' \\big ( ( g , q ) \\big ) & = g \\cdot \\lambda ' \\big ( ( 1 , q ) \\big ) \\\\ & = g \\cdot \\partial ' ( q ) \\\\ & = \\lambda \\big ( ( g , q ) \\big ) . \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} I ( i , j ) = \\{ \\underbrace { b - j + 1 , \\dots , b - j + i } _ { } , \\underbrace { b + i + 1 , \\dots , n } _ { } \\} . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} m + 1 = \\underbrace { n _ { 1 } + \\cdots + n _ { 1 } } _ { M _ { 1 } \\ ; \\mathrm { t i m e s } } + \\underbrace { n _ { 2 } + \\cdots + n _ { 2 } } _ { M _ { 2 } \\ ; \\mathrm { t i m e s } } + \\cdots + \\underbrace { n _ { \\ell } + \\cdots + n _ { \\ell } } _ { M _ { \\ell } \\ ; \\mathrm { t i m e s } } , \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} W _ { 1 7 0 } = F _ { 8 , 4 } \\wedge F _ { 8 , 1 0 } \\wedge F _ { 8 , 1 2 } \\wedge F _ { 8 , 1 4 } \\wedge F _ { 8 , 1 6 } \\wedge F _ { 8 , 1 8 } ^ { ( 1 ) } \\wedge F _ { 8 , 1 8 } ^ { ( 2 ) } \\wedge F _ { 8 , 2 0 } \\wedge F _ { 8 , 2 2 } \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} V _ I ^ p = V _ { i _ 1 } ^ { p _ 1 } \\cdots V _ { i _ k } ^ { p _ k } . \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\ell ' ( B ) = d ( x q ) - a _ 1 . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\varphi = \\begin{bmatrix} x & y ^ 2 & z ^ 3 \\\\ y & z ^ 2 & y z ^ 2 \\\\ z & x ^ 2 & y ^ 3 \\\\ 0 & x y + y z + x z & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} \\| w \\| _ { X _ { h } ^ { \\ast } } : = \\sup _ { v _ { h } \\in X _ { h } } \\frac { ( w , v _ { h } ) } { \\| \\nabla v _ { h } \\| } , \\ ; \\ ; \\| w \\| _ { V _ { h } ^ { \\ast } } : = \\sup _ { v _ { h } \\in V _ { h } } \\frac { ( w , v _ { h } ) } { \\| \\nabla v _ { h } \\| } . \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} x _ { n + 1 } = F _ n ( x _ n ) , n \\in \\Z , \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} x = \\begin{pmatrix} J _ { \\frac { k a } { 2 } , 2 } \\\\ & I _ { s } \\end{pmatrix} . \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} f \\ast _ g \\pi ( h ) & = \\frac { 1 } { n ! } \\sum _ { \\sigma , \\tau \\in \\Sigma _ n } g f _ { \\sigma ( 1 ) } \\wedge g ^ c h _ { \\tau ( 1 ) } \\otimes \\cdots \\otimes g f _ { \\sigma ( n ) } \\wedge g ^ c h _ { \\tau ( n ) } \\\\ \\pi ( f \\ast _ g h ) & = \\frac { 1 } { n ! } \\sum _ { \\sigma , \\tau \\in \\Sigma _ n } g f _ { \\sigma \\tau ( 1 ) } \\wedge g ^ c h _ { \\tau ( 1 ) } \\otimes \\cdots \\otimes g f _ { \\sigma \\tau ( n ) } \\wedge g ^ c h _ { \\tau ( n ) } . \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} a f _ x + b f _ y + c f _ z = 0 \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} v _ { \\infty } ( x , t ) = 0 , \\ \\ \\textrm { f o r } \\ \\ t \\in ( - 5 / 4 , 0 ] . \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} & F _ { j } ( U ^ { ( j ) } ) \\cdots F _ 3 ( U ^ { ( 3 ) } ) F _ 2 ( U ^ { ( 2 ) } ) \\underline { F _ { 1 } ( U ^ { ( 1 ) } ) F ( V _ b ) } \\\\ & = F _ { j } ( U ^ { ( j ) } ) \\cdots F _ 3 ( U ^ { ( 3 ) } ) \\underline { F _ 2 ( U ^ { ( 2 ) } ) F _ 2 ( V _ { b _ 1 } ) } F _ 1 ( U ^ { ( 1 ) ' } ) \\\\ & = F _ { j } ( U ^ { ( j ) } ) \\cdots \\underline { F _ 3 ( U ^ { ( 3 ) } ) F _ 3 ( V _ { b _ 2 } ) } F _ 2 ( U ^ { ( 2 ) ' } ) F _ 1 ( U ^ { ( 1 ) ' } ) \\\\ & = \\cdots = F _ { j + 1 } ( U ^ { ( j + 1 ) ' } ) \\cdots F _ 3 ( U ^ { ( 3 ) ' } ) F _ 2 ( U ^ { ( 2 ) ' } ) F _ 1 ( U ^ { ( 1 ) ' } ) , \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} s _ { U ^ 1 \\upharpoonright U ^ { 1 2 } } = g \\cdot s _ { U ^ 2 \\upharpoonright U ^ { 1 2 } } . \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} p _ H ( x \\star u , y \\star u ) ~ = ~ p _ H ( x , y ) + G T _ x A _ u \\ 1 _ { \\{ y \\star u \\} } ( e ) ~ \\geq ~ p _ H ( x , y ) , \\forall x , y , u \\in E , \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} S ( n , k ) = \\frac { 1 } { k ! } \\left ( T _ p ( n , k ) + \\sum _ { 0 < j \\leq k , p \\mid j } ( - 1 ) ^ { k - j } { k \\choose j } j ^ n \\right ) . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} H ^ { s , r } ( \\R ) : = H ^ s ( \\R ) \\cap L ^ 2 ( | x | ^ { 2 r } d x ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; s , \\ , r \\geq 0 . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\psi ( A \\times \\{ 0 \\} ) = A \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} d _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x ( b ) + d _ 2 ~ ^ { A B R } D _ b ^ \\alpha x ( b ) = 0 , \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{gather*} A ' _ { i - 1 , j - 2 } A _ { i - 1 , j } + A _ { i - 2 , j - 1 } A ' _ { i , j - 1 } = A ' _ { i - 1 , j - 1 } A _ { i - 1 , j - 1 } , \\\\ A ' _ { i , j - 1 } A _ { i , j + 1 } + A _ { i - 1 , j } A ' _ { i + 1 , j } = A ' _ { i , j } A _ { i , j } . \\end{gather*}"}
-{"id": "4042.png", "formula": "\\begin{align*} \\widehat { \\Lambda } _ k ( \\lambda ) : = \\left [ \\begin{array} { c c c c c } - 1 & - \\lambda & - \\lambda ^ { 2 } & \\cdots & - \\lambda ^ { k - 1 } \\\\ & - 1 & - \\lambda & \\ddots & \\vdots \\\\ & & - 1 & \\ddots & - \\lambda ^ { 2 } \\\\ & & & \\ddots & - \\lambda \\\\ & & & & - 1 \\\\ 0 & \\cdots & \\cdots & \\cdots & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} \\left [ \\prod _ { i = 1 } ^ N \\frac { d _ i - 2 } { d _ i } \\right ] \\| H \\| _ { S _ 1 } + | c _ + | + | c _ - | \\leq \\| \\phi \\| _ { c b } \\leq \\| H \\| _ { S _ 1 } + | c _ + | + | c _ - | , \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} W ^ { s , p , q } ( \\Omega ) = \\{ u \\in L ^ { p } ( \\Omega ) : u , \\left ( \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ { p } } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d x d y \\right ) ^ { 1 / p } < + \\infty \\} , \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} r _ { i c _ { \\ell _ 0 } } = \\frac { r _ { 0 c _ { \\ell _ 0 } } } { c _ { \\ell _ 0 } - | I | } \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} \\displaystyle \\mathfrak { D } ( \\mathbf { u } ) = \\frac { 1 } { 2 } ( \\nabla \\mathbf { u } + ( \\nabla \\mathbf { u } ) ^ { \\it T } ) . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} \\det \\begin{pmatrix} \\mathcal A & \\mathcal B \\\\ \\mathcal C & \\mathcal D \\\\ \\end{pmatrix} & = \\det ( \\mathcal A ) \\det ( \\mathcal D - \\mathcal { C A } ^ { - 1 } \\mathcal B ) . \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} E _ { N } ^ { \\operatorname { M F } } \\ ; & : = \\ ; \\inf \\sigma ( { H } _ { N } ^ { \\operatorname { M F } } ) , \\\\ E _ { N } ^ { \\operatorname { G P } } \\ ; & : = \\ ; \\inf \\sigma ( { H } _ { N } ^ { \\operatorname { G P } } ) \\ , , \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} I ( q \\ , | \\ , a , \\tau ) \\thicksim \\sum \\limits _ { r = 1 } ^ \\infty \\frac { \\zeta ( r + 1 , \\ , 1 + a ) } { r + 1 } \\Bigl ( \\frac { B _ { r + 2 } ( q ) - B _ { r + 2 } } { r + 2 } \\Bigr ) / \\tau ^ { r + 1 } . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{gather*} \\Gamma _ { \\C } ( s ) L ( g , s ) = \\epsilon \\big ( \\psi ^ 2 , s - 1 \\big ) \\Gamma _ { \\C } ( 3 - s ) L ( g , 3 - s ) . \\end{gather*}"}
-{"id": "184.png", "formula": "\\begin{align*} u = N S ( u _ 0 ) \\ \\ \\ \\textrm { o n } \\ \\ \\R ^ d \\times [ 0 , T ( u _ 0 ) ) . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\sigma _ u ^ \\vee = \\sigma ' _ { u ^ \\vee } \\textrm { a n d } \\tau _ v ^ \\vee = \\tau ' _ { v ^ \\vee } . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} \\tilde { g } ( U , \\tilde { J } V ) = \\tilde { g } ( \\tilde { J } U , V ) , \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} \\langle \\tilde { \\pi } _ p ( \\alpha _ n ) \\mathbf h _ p , \\mathbf h _ p \\rangle = \\int _ { \\mathcal A _ 1 ( n ) } \\mathbf h _ p ( h \\alpha _ n ) \\overline { \\mathbf h _ p ( h ) } d h + \\int _ { \\mathcal A _ 2 ( n ) } \\mathbf h _ p ( h \\alpha _ n ) \\overline { \\mathbf h _ p ( h ) } d h = : A _ 1 ( n ) + A _ 2 ( n ) . \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) \\equiv 0 \\mbox { a n d } { \\overline { f } } _ { t } ( x _ { t } , x _ { t - 1 } , \\xi _ { t } ) = - \\mathbb { E } \\Big [ \\sum \\limits _ { i = 1 } ^ { n + 1 } \\xi _ { t + 1 } ( i ) x _ { t } ( i ) \\Big ] . \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} B ( z ) = ( [ ( \\alpha _ i , \\alpha _ j ) ] _ z ) _ { i , j \\in I } , \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\partial _ t \\rho ^ { N , ( n ) } _ t = \\big ( \\L _ N ^ { ( n ) } \\big ) ^ \\ast \\rho ^ { N , ( n ) } _ t , \\rho ^ { N , ( n ) } | _ { t = 0 } = \\rho ^ { N , ( n ) } _ 0 . \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} \\widetilde { M } _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\overset { { \\rm i n \\ , l a w } } { = } L \\ , N , \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} | \\underline { \\vec z _ { i - 1 } } | \\leq | \\underline { \\vec z _ i } | , i = 1 , \\ldots , k + d - 1 , \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} ( A _ i , \\Phi _ i ) = h _ { m _ i } ( ( x _ 1 , \\ldots , x _ k ) , \\alpha , \\theta ) . \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} l _ { k } = | U _ * | ^ { 2 ^ { * } - 2 } U _ * - u _ 0 ^ { 2 * - 1 } + \\sum _ { j = 1 } ^ { k } U _ j ^ { 2 ^ { * } - 1 } . \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} \\mathcal M ( t ) = 2 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } j _ { u _ \\mu } ( x ) \\cdot \\nabla _ x \\psi ( x , y ) \\ , m _ { u _ \\iota } ( y ) \\ , d x d y . \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} ( z - w ) ^ { N } [ Y ( a , z ) , Y ( b , w ) ] = 0 , \\ \\ N \\gg 0 . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} & c _ i ( J ) : = \\left | \\left \\{ l < 2 ^ J - 1 | K _ l \\geq 2 ^ i K _ 0 \\right \\} \\right | + 1 = 1 + \\sum _ { k = i } ^ { J - 1 } 2 ^ { J - 1 - k } = 2 ^ { J - i } . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} h ~ = ~ h ( e ) G _ H \\ 1 + \\tilde { h } \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} { \\bf E } _ v ( { \\rm e } ^ { - \\beta X _ t } ) = \\exp \\left ( - \\int _ v ^ { t + v } S ( d u ) \\int _ 0 ^ { \\infty } ( 1 - { \\rm e } ^ { - \\beta x } ) n ( u , d x ) \\right ) , \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\bar { u } ' ( t ) & = \\sum _ { n = 1 } ^ { \\infty } u _ n ' ( t ) = \\sum _ { n = 2 } ^ { \\infty } \\frac { - ( n - 1 ) \\xi _ { n - 1 } } { t ^ { n } } G _ { \\alpha + 1 , \\sigma } , \\\\ A \\bar { u } ( t ) & = \\sum _ { n = 1 } ^ { \\infty } \\frac { A \\xi _ n } { t ^ { n } } G _ { \\alpha , \\sigma } . \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} \\mathbb W ^ s _ \\omega ( z _ 0 , w _ 0 ) = \\{ ( z , w ) \\in \\mathbb C ^ 2 \\ , | \\ , w = w _ 0 \\} , \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} \\hat { \\varphi } ( t ) = \\frac { \\sinh ( \\pi t + 2 i \\beta t ) } { \\sinh ( \\pi t ) } = X ^ { i t } \\exp ( - t / T ) + O ( \\exp ( - \\pi t ) ) , \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} P ( Y ) & = 6 . 4 \\cdot 1 0 ^ { - 3 } - 0 . 4 1 6 \\cdot Y + 1 . 0 2 \\cdot Y ^ { 2 } \\\\ Y _ { 1 } & \\approx 0 . 0 1 6 Y _ { 2 } \\approx 0 . 3 9 2 \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} \\langle I ' ( u ) , v \\rangle & = \\displaystyle \\int _ { \\Omega } B ( x ) | \\nabla u ( x ) | ^ { p ( x ) - 2 } \\nabla u ( x ) \\nabla v ( x ) d x + \\displaystyle \\int _ { \\Omega } A ( x ) | u ( x ) | ^ { p ( x ) - 2 } u ( x ) v ( x ) d x \\\\ & + \\displaystyle \\int _ { \\Omega } D ( x ) | u ( x ) | ^ { p ( x ) - 1 } u ( x ) v ( x ) d x + \\displaystyle \\int _ { \\Omega } C ( x ) | u ( x ) | ^ { p ( x ) - 3 } u ( x ) v ( x ) d x \\\\ & - \\int _ { \\Omega } b ( x ) | u ( x ) | ^ { q ( x ) - 2 } u ( x ) v ( x ) , \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} \\theta _ i ' = \\omega _ i + \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\sin ( \\theta _ j - \\theta _ i ) , \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} f \\colon [ 0 , T ] \\rightarrow L ^ 2 _ \\eta ( S ; U ) , f ( t ) ( \\cdot ) : = \\sum _ { j = 1 } ^ { \\infty } \\left ( \\Phi ( t ) e _ j \\right ) ( \\cdot ) e _ j , \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } | \\tilde { y } ( s , t ) | ^ 2 & = \\langle a ( t ) \\tilde { y } ( s , t ) , \\tilde { y } ( s , t ) \\rangle + \\langle b ( t ) \\tilde { y } ( s , t ) , \\tilde { y } ( s , t ) \\rangle . \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} { d '' _ { 0 } } _ { \\bar { \\lambda } \\bar { \\mu } } ( M ) = \\frac { 1 } { \\left | B _ { \\frac { 1 } { 8 } \\cdot 2 ^ { n - 1 } } ( M ) \\right | } \\int \\limits _ { B _ { \\frac { 1 } { 8 } \\cdot 2 ^ { n - 1 } } ( M ) } \\left ( \\bar { \\mu } , \\ , \\bar { \\nu } ( M ) \\right ) { d ' _ { 2 } } _ { \\bar { \\lambda } } ( \\widetilde { M } ) \\ , d S ( \\widetilde { M } ) , \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} \\rho ^ n _ { n , n e _ 0 + 1 } ( V ) \\ & = \\ n ^ 2 ( g - 1 ) + 1 - n ( n - r n e _ 0 - r + r n ( g - 1 ) ) \\\\ \\ & = \\ n ^ 2 g - n ^ 2 + 1 - n ^ 2 ( 1 - r e _ 0 + r ( g - 1 ) ) + n r \\\\ \\ & = \\ n ^ 2 \\cdot \\hbox { \\Large ( } g - ( 1 - r e _ 0 + r ( g - 1 ) ) \\hbox { \\Large ) } + n r - n ^ 2 + 1 \\\\ \\ & = \\ n ^ 2 \\cdot \\rho ^ 1 _ { 1 , e _ 0 } ( V ) + n ( r - n ) + 1 \\\\ \\ & = \\ n ( r - n ) + 1 . \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} X _ 0 = \\bigcap _ n T ^ { - n } \\left ( \\bigcap _ k X _ { 1 / k } \\right ) \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} \\tau ( x ) = \\int _ { 0 } ^ { 1 } \\exp [ s \\log x ] d s . \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} & - \\int _ 0 ^ 1 r ^ { d - 1 } ( 1 - r ^ 2 ) ^ { \\mu + 1 / 2 } \\partial _ r ^ 2 f ( r \\xi ) g ( r \\xi ) \\ , d r \\\\ & = \\int _ 0 ^ 1 r ^ { d - 1 } \\partial _ r f ( r \\xi ) \\partial _ r g ( r \\xi ) ( 1 - r ^ 2 ) ^ { \\mu + 1 / 2 } \\ , d r \\\\ & + \\int _ 0 ^ 1 r ^ { d - 1 } \\Big ( \\frac { d - 1 } { r } ( 1 - r ^ 2 ) - ( 2 \\mu + 1 ) r \\Big ) \\partial _ r f ( r \\xi ) g ( r \\xi ) ( 1 - r ^ 2 ) ^ { \\mu - 1 / 2 } \\ , d r . \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\langle B u ( t ) , u ' ( t ) \\rangle = \\frac { 1 } { 2 } \\frac { d } { d t } \\langle B u ( t ) , u ( t ) \\rangle \\ \\mbox { f o r a l m o s t a l l } \\ t \\in T . \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} h ^ 0 ( W ) = h ^ 0 ( W ( p ) ) = l - m . \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} P T _ u f ( x ) & = ~ \\sum _ { y \\in E } p ( x , y ) f ( y \\star u ) ~ \\leq ~ \\sum _ { y \\in E } p ( x \\star u , y \\star u ) f ( y \\star u ) \\\\ & \\leq ~ \\sum _ { z \\in E } p ( x \\star u , z ) f ( z ) ~ \\leq ~ f ( x \\star u ) ~ = ~ T _ u f ( x ) , \\forall x \\in E . \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} f _ { u u } = - ( a E _ u + a _ u E ) \\nu \\times \\nu _ v - a E ( \\nu _ u \\times \\nu _ v + \\nu \\times \\nu _ { u v } ) \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\lVert \\mathbf s \\rVert = \\inf \\bigl \\{ c > 0 : M _ \\phi ( \\mathbf s / c ) \\le 1 \\bigr \\} . \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { \\theta } { 2 } + \\frac { 1 - \\theta } { m } . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} B ^ { \\delta } ( x ) \\zeta : = \\sum _ { i = 1 } ^ { d } \\langle b _ { i } , \\nabla J _ { \\delta } ^ { a } x \\rangle \\zeta ^ { i } . \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\hat { u } ( x , k ) : = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { \\infty } u ( x , t ) e ^ { i k t } d t . \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} h \\beta _ 1 \\varpi _ p = \\left ( \\begin{array} { c c } - y & x p ^ { - 1 } \\\\ - t & z p ^ { - 1 } \\end{array} \\right ) = \\left ( \\begin{array} { c c } t ^ { - 1 } \\det ( h ) p ^ { - 1 } & - y \\\\ 0 & - t \\end{array} \\right ) \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & - t ^ { - 1 } p ^ { - 1 } z \\end{array} \\right ) , \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} [ M ( m ) ] = [ L ( m ) ] + \\sum _ { m ' < m } P _ { m , m ' } ( 1 ) [ L ( m ' ) ] , \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ( v ) = \\beta _ v f ' ( v ) . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} 3 & ( 1 - \\gamma _ 1 ) & + 5 2 & ( 1 - \\gamma _ 2 ) & + 7 2 0 & \\epsilon & < 3 , \\\\ 4 & ( 1 - \\gamma _ 1 ) & + 3 2 & ( 1 - \\gamma _ 2 ) & + 3 2 0 & \\epsilon & < 3 \\end{aligned} \\right . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { U } \\tilde { J } V = \\tilde { J } ( \\nabla _ { U } V ) + B ( U , V ) \\tilde { J } N . \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\tau _ { 1 } = w _ { 0 } K _ { 0 } + J \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } T _ n = \\frac { 1 } { 3 \\langle a \\rangle } \\sum _ { n \\geq 1 } \\exp \\left ( - \\alpha _ { n - 1 } ^ * - \\delta ( \\alpha _ { n - 1 } ^ * ) \\right ) . \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} E _ 3 = E _ 2 + \\Vert \\partial _ x ^ 2 \\rho _ 0 \\Vert _ { L ^ 2 } + \\Vert \\partial _ x ^ 2 u _ 0 \\Vert _ { L ^ 2 } . \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} \\mathcal W _ { \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) \\otimes \\underline { \\chi } , B } ( g _ { \\infty } ) = \\mathcal W _ { \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) , B } ( g _ { \\infty } ) = C \\mathcal W _ { \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) = C \\mathcal W _ { \\mathfrak R _ M \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) , \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ l ( - 1 ) ^ i d _ i = \\sum _ { i = 1 } ^ { m + 1 } ( - 1 ) ^ i d _ i + ( d _ { m + 2 } - d _ { m + 3 } + \\cdots ) \\ge \\sum _ { i = 1 } ^ { m + 1 } ( - 1 ) ^ i d _ i > 0 , \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\mathcal { B } _ { \\mathcal { L } } = \\{ S ( \\lambda ) v _ 1 ( \\lambda ) , \\ldots , S ( \\lambda ) v _ p ( \\lambda ) \\} , \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\vert y ( z ) - x \\vert & = \\vert x + z + f ( z ) - x \\vert \\\\ & = \\vert z + f ( z ) \\vert \\\\ & \\leq \\vert z \\vert + \\vert f ( z ) \\vert \\\\ & \\leq \\vert z \\vert + w _ x ( \\vert y ( z ) - x \\vert ) \\cdot \\vert y ( z ) - x \\vert . \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} g = \\left ( \\begin{array} { c c } y & 0 \\\\ 0 & y ^ { - 1 } \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & x \\\\ 0 & 1 \\end{array} \\right ) k \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} H _ { 1 , ( 0 , \\alpha _ 2 ) } ( \\tau ) = - 2 \\lim _ { \\alpha _ 1 \\to 0 } \\int _ { \\R ^ 2 } \\mathcal F _ { \\alpha _ 1 } ( w _ 1 ) \\mathcal F _ { \\alpha _ 2 } ( w _ 2 ) e ^ { 2 \\pi i \\tau Q ( \\boldsymbol { w } ) } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} P Q \\phi = P ( H - E - \\iota 0 ) ^ { - 1 } \\phi = G ( E + \\iota 0 ) \\phi = - \\frac { 1 } { \\lambda } \\phi . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\mathrm { c a p } \\left ( B ( x _ 0 , r ^ { k } ) \\cap \\Omega ^ c \\right ) } { r ^ { k ( n - 2 ) } } = + \\infty , \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} \\| u \\| ^ { p } _ { L ^ { p ^ { * } } ( \\mathbb { G } ) } \\leq C \\sum _ { k \\in \\mathbb { Z } , \\ , a _ { k } \\neq 0 } 2 ^ { k p } a ^ { - s p / Q } _ { k } a _ { k + 1 } \\leq C \\int _ { \\mathbb { G } } \\int _ { \\mathbb { G } } \\frac { | u ( x ) - u ( y ) | ^ { p } } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d x d y = C [ u ] ^ { p } _ { s , p , q } . \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\sum _ { r \\in \\mathcal R } \\Omega _ p ( r ) \\mathrm { v o l } ( \\Gamma _ { 0 0 } r \\Gamma _ { 0 0 } ) . \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\begin{bmatrix} \\lambda ^ 2 P _ 6 + \\lambda P _ 5 & \\lambda ^ 2 P _ 4 + \\lambda P _ 3 & \\lambda ^ 2 P _ 2 + \\lambda P _ 1 + P _ 0 \\\\ \\hline - I _ n & \\lambda ^ 2 I _ n & 0 \\\\ 0 & - I _ n & \\lambda ^ 2 I _ n \\end{bmatrix} \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} d X _ t = b ( X _ { t - } ) \\ , d t + \\sigma ( X _ { t - } ) \\ , d L _ t . \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { B _ { r _ { k + 1 } } } v \\leq \\sup _ { B _ { \\tau _ 2 r _ { k } } } v & \\leq ( 1 - \\mu ) \\left ( \\hat { C } M r _ { k } ^ { \\beta } - M r _ { k } ^ { \\alpha } \\right ) + M r _ { k } ^ { \\alpha } \\\\ & \\leq \\hat { C } M r _ { k + 1 } ^ { \\beta } \\cdot \\frac { 1 - \\mu } { \\tau _ 1 ^ { \\beta } } + \\mu M r _ { k } ^ { \\beta } \\\\ & \\leq \\hat { C } M r _ { k + 1 } ^ { \\beta } \\left ( \\frac { 1 - \\mu } { \\tau _ 1 ^ { \\beta } } + \\frac { \\mu } { \\hat { C } \\tau _ 1 ^ { \\beta } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} \\langle \\xi _ x ^ { ( n ) } , \\xi _ y ^ { ( n ) } \\rangle & = \\phi ( ( r - t p _ x ^ { ( n ) } ) ^ * ( r - t p _ y ^ { ( n ) } ) ) \\\\ & = r ^ 2 - r t \\phi ( p _ x ^ { ( n ) } ) - r t \\phi ( p _ y ^ { ( n ) } ) + t ^ 2 \\phi ( p _ x ^ { ( n ) } p _ y ^ { ( n ) } ) = 0 . \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} E ( x ) = \\sum _ { n \\leq x } \\frac { \\mu ( n ) } { n } \\Delta \\left ( \\frac { x } { n } \\right ) + O \\left ( \\log x \\right ) . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} N ^ * _ M : = \\left \\{ z \\in M \\mid \\ ; c \\in S ^ 0 \\ ; q \\gg 0 , c z ^ q \\in N ^ { [ q ] } _ M \\right \\} . \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} \\Sigma ( s ) \\ll \\frac { ( 1 + | r | ) ^ A X ^ { \\theta } } { X ^ { 1 / 4 } } \\left ( \\sum _ { n = 1 } ^ { [ 2 | c | ] } f \\left ( \\frac { n } { 2 | c | } \\right ) + \\sum _ { n = [ 2 | c | ] + 1 } ^ { \\infty } f \\left ( \\frac { n } { 2 | c | } \\right ) \\right ) . \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} f _ 0 ( M _ 0 ) = - \\frac { 1 } { 4 \\pi } \\int \\limits _ { \\sum _ { n } } \\frac { \\Delta f _ 0 ( M ) } { \\rho _ { M _ 0 } ( M ) } \\ , d m _ 3 ( M ) + O ( \\alpha _ { n } + 2 ^ { - n } ) \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\nu _ c \\alpha _ n ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ n ) p ^ { - n / 2 } \\int _ { \\Q _ p } \\psi ( - 2 x p ^ n z - c p ^ { 2 n + 1 } z ^ 2 ) \\mathfrak G ( 2 z , \\underline { \\chi } _ p ^ { - 1 } ) d z . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} & \\tau _ 0 = 2 \\\\ & \\tau _ { i + 1 } = \\tau _ { i } + 2 ( - 1 ) ^ { i + 1 } { n \\choose i + 1 } . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} \\frac { ( 1 + z _ s ) a _ s q ^ N } { 1 - a _ s q ^ N } \\prod _ { k = 1 } ^ { s - 1 } \\frac { 1 + a _ k z _ k q ^ N } { 1 - a _ k q ^ N } . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\overline { \\bigcup _ { i \\geq n } \\Phi _ i ^ { - 1 } ( 0 ) } \\subseteq \\left ( \\bigcup _ { j = 1 } ^ l \\overline { B _ { c m _ n ^ { - 1 / 2 } } ( x _ j ) } \\right ) \\cup \\left ( \\bigcup _ { i \\geq n } \\bigcup _ { j = l + 1 } ^ k \\overline { B _ { c m _ n ^ { - 1 / 2 } } ( m _ i x _ j ) } \\right ) , \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} P \\left ( V _ n = k \\right ) = \\sum _ { \\substack { B \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | B | = k } } \\prod _ { i \\in B } b ( i ; n ) \\ , \\prod _ { j \\in B ^ c } ( 1 - b ( j ; n ) ) \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} s = \\max _ { y \\in \\eta } | \\eta \\cap K _ { 2 r } ( y ) | . \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} p _ { ( N _ { 0 } ) } = \\min _ { 0 \\leqslant N \\leqslant N _ { 0 } } p _ { N } = \\min _ { 0 \\leqslant N \\leqslant N _ { 0 } } \\min _ { 1 \\leqslant j \\leqslant m _ { N } } P ( A _ { j } ^ { ( N ) } ) > 0 . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} N _ { k + 1 } : = \\begin{bmatrix} Q _ 1 \\ , | \\ , Q _ 2 V _ 2 \\end{bmatrix} , \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( 1 ; 3 ) } ^ \\ast ( \\tau ) : = \\lim _ { \\Re ( s ) > 0 \\atop s \\rightarrow 0 } \\sum _ { ( m , n ) \\in \\Z } \\frac 1 { ( ( 3 m + 1 ) \\tau + n ) ^ k } \\frac { \\Im ( \\tau ) ^ s } { | ( 3 m + 1 ) \\tau + n | ^ { 2 s } } \\end{gather*}"}
-{"id": "6107.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( \\Upsilon _ { N } ( f , g ) - 2 \\Gamma _ { N } ( f , g ) \\right ) = - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ { N } \\Phi _ { k } ^ { ( N ) } ( f ) ^ { t } \\cdot \\mathbb { V } \\left ( \\mathbb { G } [ \\mathcal { A } ^ { ( k ) } ] \\right ) \\cdot \\Phi _ { k } ^ { ( N ) } ( g ) , \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} C _ { s } = \\log \\bigg ( 1 + \\frac { p _ { s } | h _ { s } | ^ { 2 } } { \\sigma _ { s } ^ { 2 } } \\bigg ) , \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} \\sigma \\left ( \\frac { L ^ S ( r , \\pi \\boxtimes \\chi , \\varrho _ { 5 } ) } { ( 2 \\pi i ) ^ { 3 r } G ( \\chi ) ^ 3 C ( \\pi ) } \\right ) = \\frac { L ^ S ( r , { } ^ \\sigma \\ ! \\pi \\boxtimes { } ^ \\sigma \\ ! \\chi , \\varrho _ { 5 } ) } { ( 2 \\pi i ) ^ { 3 r } G ( { } ^ \\sigma \\ ! \\chi ) ^ 3 C ( { } ^ \\sigma \\ ! \\pi ) } , \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} \\| K ^ { ( 1 ) } \\| _ { \\operatorname { H S } } ^ 2 = \\int \\d x \\d y | K ^ { ( 1 ) } ( x , y ) | ^ 2 \\leqslant C ^ { ( 1 ) } + C ^ { ( 1 ) } \\| \\nabla u _ 0 \\| ^ 2 _ 2 + C ^ { ( 1 ) } \\| \\nabla v _ 0 \\| _ 2 ^ 2 < + \\infty , \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{gather*} \\mathcal R | _ { M '' } = U - \\sum _ { i \\ne 1 } ^ n V _ i { \\ss } _ { 1 i } \\ , , U = \\prod _ { l \\ne 1 } ^ n \\sigma _ \\mu ( x _ { 1 l } ) \\ , , V _ i = \\sigma _ { \\xi _ { 1 2 } } ( x _ { 1 i } ) \\prod _ { l \\ne 1 , i } \\sigma _ \\mu ( x _ { i l } ) \\ , , \\\\ \\mathcal R ^ + | _ { M '' } = U ^ + - \\sum _ { i \\ne 1 } ^ n V _ i ^ + { \\ss } _ { 1 i } ^ + \\ , , U ^ + = \\prod _ { l \\ne 1 } ^ n \\sigma _ \\mu ( x _ { 1 l } ^ + ) \\ , , V _ i ^ + = \\sigma _ { \\xi _ { 1 n } ^ + } ( x _ { 1 i } ^ + ) \\prod _ { l \\ne 1 , i } \\sigma _ \\mu ( x _ { l i } ) \\ , . \\end{gather*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\sphericalangle \\big ( P _ y , P ( y , r _ { x , i } ) \\big ) & \\leq \\sum \\limits _ { l = 0 } ^ { \\infty } \\sphericalangle \\big ( P ( y , r _ { x , i + l } ) , P ( y , r _ { x , i + l + 1 } ) \\big ) \\\\ & \\leq \\tilde C C ( m ) \\sum _ { l = i } ^ \\infty ( \\delta _ { x , l + 1 } + 2 C \\delta _ { x , l } ) , \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} \\phi _ t ( f _ t ( x _ 1 , \\ldots , x _ n ) ) = g _ t ( \\phi _ t ( x _ 1 ) , \\ldots , \\phi _ t ( x _ n ) ) \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} & \\alpha _ { n } ^ { ( N ) } ( f ) = \\sum _ { j = 1 } ^ { m _ { N } } \\frac { P ( A _ { j } ^ { ( N ) } ) } { \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) } \\alpha _ { n } ^ { ( N - 1 ) } ( \\phi _ { ( j , N ) } f ) = \\alpha _ { n } ^ { ( N - 1 ) } ( \\phi _ { ( N ) } f ) + \\Gamma _ { n } ^ { ( N ) } ( f ) , \\\\ & \\Gamma _ { n } ^ { ( N ) } ( f ) = \\sum _ { j = 1 } ^ { m _ { N } } q _ { n } ( j , N ) \\alpha _ { n } ^ { ( N - 1 ) } \\left ( \\phi _ { ( j , N ) } f \\right ) , q _ { n } ( j , N ) = \\frac { P ( A _ { j } ^ { ( N ) } ) } { \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) } - 1 . \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} ( a , b ) \\circledcirc ( c , x ) = a \\circ ( b * ( c , x ) ) , \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} \\Omega _ { n o r } & = \\bigcup _ n \\{ \\omega \\in \\Omega \\ , | \\ , \\mathcal F _ \\omega \\subset \\overline B _ n ( 0 , r _ n ) \\} \\\\ & = \\bigcup _ n \\bigcap _ m \\{ \\omega \\in \\Omega \\ , | \\ , f ^ m _ \\omega \\in \\overline B _ n ( 0 , r _ n ) \\} . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} & ( H + \\lambda P - E ) \\phi = 0 \\\\ \\Rightarrow & \\phi + \\lambda ( H - E - \\iota \\epsilon ) ^ { - 1 } P \\phi + \\iota \\epsilon ( H - E - \\iota \\epsilon ) ^ { - 1 } \\phi = 0 \\qquad \\forall \\epsilon > 0 . \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} S ^ 2 ( h ) = \\omega h \\omega ^ { - 1 } . \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} 2 n + 4 \\le e ( G ) & = e _ G ( U , S ) + e ( G [ S \\cup W ] ) \\\\ & \\le r s + 3 ( n - r ) - 8 \\\\ & = 3 n - ( 3 - s ) r - 8 \\\\ & \\le 3 n - ( 3 - s ) ( n + 2 s - 1 5 ) - 8 , \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} x ^ 1 : = f ^ i ( x ) , x ^ 2 : = f ^ { W ^ { i - 1 } } ( x ^ 1 ) , x ^ 3 : = f ^ i ( x ^ 2 ) . \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} \\tilde { X } ^ i = C ^ { i l } X _ l \\ ; , \\ ; \\ ; [ \\tilde { X } ^ i , \\tilde { X } ^ j ] = \\tilde { f } ^ { i j } _ { \\ ; \\ ; \\ ; k } \\tilde { X } ^ k , \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\Phi _ { N _ { 0 } - k } ^ { ( N _ { 0 } ) } = \\Phi _ { N _ { 1 } - k } ^ { ( N _ { 1 } ) } , \\quad 0 \\leqslant k < N _ { 0 } , \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} B _ \\lambda ( s ) = i ^ { n k } \\ , \\bigg ( \\prod _ { j = 1 } ^ n \\beta ( k - j , s ) \\bigg ) C _ k ( s ) \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} \\tilde f = 2 f '' = { f _ { 1 , B } } - { f _ { 1 , A } } = f _ B ^ { { \\rm { R F } } } - \\Delta f - f _ A ^ { { \\rm { R F } } } + \\Delta f = f _ B ^ { { \\rm { R F } } } - f _ A ^ { { \\rm { R F } } } \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} V _ 1 ( x ) & = 1 + O ( x ^ { \\frac { 1 } { 3 } } ) , \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} \\displaystyle w ( x , \\tau ) = ( \\psi w ) ( x , \\tau ) & \\leq ( \\psi w ) ( x _ 1 , t _ 1 ) \\\\ \\displaystyle & \\leq w ( x _ 1 , t _ 1 ) \\\\ \\displaystyle & \\leq \\frac { C } { \\tau - ( t _ 0 - T ) } + ( N - 1 ) K + C ( \\delta ) D ^ { \\alpha - 1 } | \\nabla q | + \\alpha D ^ { \\alpha - 1 } ( q ^ + ) \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\tilde { \\phi } ( d ( x _ 1 , y _ 1 ) , . . . , d ( x _ N , y _ N ) ) , \\quad \\forall x , y \\in X . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} f = f _ { n _ { m + 1 } - 1 } ( q _ 1 , \\dots , q _ { m } ) q _ { m + 1 } ^ { n _ { m + 1 } - 1 } + \\cdots + f _ { 1 } ( q _ 1 , \\dots , q _ { m } ) q _ { m + 1 } + f _ { 0 } ( q _ 1 , \\dots , q _ { m } ) , \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ \\frac { t ^ 2 \\ , \\bigl ( e ^ { - t / 2 ( 1 + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) ) } - 1 \\bigr ) } { ( e ^ { t / 2 } - 1 ) ( e ^ { t \\tau / 2 } - 1 ) } \\Bigr ] = \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ \\frac { ( 2 t ) ^ 2 \\ , \\bigl ( e ^ { - t ( 1 + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) ) } - 1 \\bigr ) } { { ( e ^ t - 1 ) ( e ^ { t \\tau } - 1 ) } } \\Bigr ] , \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} \\bigg | \\xi _ \\gamma - \\frac { a _ \\gamma } { q _ \\gamma } \\bigg | = \\bigg | \\xi _ \\gamma - \\frac { a ' _ \\gamma } { q ' } \\bigg | \\leq N _ n ^ { - \\abs { \\gamma } } ( \\log N _ n ) ^ { \\beta / d } \\leq \\frac { 1 } { 3 2 d } N _ n ^ { - \\abs { \\gamma } } \\cdot 2 ^ { \\chi \\sqrt { \\log N _ n } } , \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} W _ { \\mathbf h , \\xi } ( 1 ) = e ^ { - 2 \\pi \\xi } 2 ^ { - \\nu ( N ) } c ( \\mathfrak d _ { \\xi } ) \\chi ( \\mathfrak f _ { \\xi } ) \\mathfrak f _ { \\xi } ^ { k - 1 / 2 } \\prod _ p \\Psi _ p ( \\xi ; \\alpha _ p ) = 2 ^ { - \\nu ( N ) } c ( \\mathfrak d _ { \\xi } ) \\mathfrak f _ { \\xi } ^ { k - 1 / 2 } \\prod _ v W _ { v , \\xi } ( 1 ) , \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\| H \\| _ { S _ 1 } = ( H ) = ( 1 - r ) ^ N \\sum _ { j \\geq 0 } \\tbinom { N - 1 + j } { N - 1 } r ^ { 2 j } . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} \\sum _ { i _ { 1 } + i _ { 2 } + \\cdots + i _ { r } = N } \\frac { a _ { 1 } ^ { i _ { 1 } } a _ { 2 } ^ { i _ { 2 } } \\cdots a _ { r } ^ { i _ { r } } q ^ { \\binom { i _ 1 } { 2 } + \\binom { i _ 2 } { 2 } + \\cdots + \\binom { i _ r } { 2 } } } { ( q ; q ) _ { i _ { 1 } } ( q ; q ) _ { i _ { 2 } } \\cdots ( q ; q ) _ { i _ { r } } } . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} c _ { n , N } ( k ) = \\sum _ { m = k } ^ { n \\wedge N } \\binom { N + 1 } { m + 1 } \\binom { m } { k } ( - 1 ) ^ { m - k } , k = 0 , 1 , \\ldots , n \\wedge N . \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} \\tilde { \\phi } ( m + n ) = f _ \\phi ( U ( m , n ) ) , \\quad \\forall m , n \\in \\N ^ N , \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} \\begin{aligned} \\widetilde { Q } * E _ 1 = E _ 1 , \\widetilde { Q } * E _ 2 = - E _ 2 , \\widetilde { Q } * \\mathbb { E } _ 1 = \\mathbb { E } _ 1 , \\widetilde { Q } * \\mathbb { E } _ 2 = - \\mathbb { E } _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n ^ { - 1 } \\log \\Vert M ^ n _ \\omega \\Vert = \\kappa _ \\mu . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} \\Phi _ { \\breve { \\mathbf g } _ p } ( g ) : = \\frac { \\langle \\tau _ p ( g ) \\breve { \\mathbf g } _ p , \\breve { \\mathbf g } _ p \\rangle } { | | \\breve { \\mathbf g } _ p | | ^ 2 } , \\Phi _ { \\mathbf h _ p } ( g ) : = \\frac { \\langle \\tilde { \\pi } _ p ( g ) \\mathbf h _ p , \\mathbf h _ p \\rangle } { | | \\mathbf h _ p | | ^ 2 } , \\Phi _ { \\pmb { \\phi } _ p } ( g ) : = \\frac { \\langle \\omega _ p ( g ) \\pmb { \\phi } _ p , \\pmb { \\phi } _ p \\rangle } { | | \\pmb { \\phi } _ p | | ^ 2 } \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} \\widetilde { c } _ { j i } ( - 2 k ) & = \\begin{cases} 1 & \\ ( \\ast ) , \\\\ 0 & \\end{cases} \\ \\ i \\in I \\setminus \\{ n \\} , & \\widetilde { c } _ { j n } ( - 2 k - 1 ) & = \\begin{cases} 1 & \\ ( \\ast \\ast ) \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} ( 2 Q + \\sigma _ 1 ^ 2 ) ( \\rho _ 1 ^ 2 - \\rho _ 2 ^ 2 ) ^ 2 & = \\left | \\rho _ 1 e ^ { i \\varphi _ 1 } ( \\rho _ 1 ^ 2 - \\sigma _ 2 ^ 2 ) - \\rho _ 2 e ^ { i \\varphi _ 2 } ( \\rho _ 2 ^ 2 - \\sigma _ 2 ^ 2 ) \\right | ^ 2 \\\\ & = \\left | \\rho _ 1 e ^ { i \\varphi _ 1 } ( \\rho _ 1 ^ 2 - \\rho _ 2 ^ 2 ) + ( \\rho _ 1 e ^ { i \\varphi _ 1 } - \\rho _ 2 e ^ { i \\varphi _ 2 } ) ( \\rho _ 2 ^ 2 - \\sigma _ 2 ^ 2 ) \\right | ^ 2 . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} ( y , Y , v , V ) = \\mathbb { L } _ { V _ x V _ u } ^ { - 1 } ( t ) ( x , X , u , U ) \\mathbb { L } _ { V _ x V _ u } ( t ) \\ ; , \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} ( \\alpha _ h , \\beta _ h u ^ h ) _ X = ( \\alpha _ h u ^ h , \\beta _ h ) _ X = ( \\alpha , \\beta _ h ) _ X \\neq 0 . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} \\left ( - n + \\frac { n } { c _ { \\ell _ 0 } } \\right ) r _ { k c _ { \\ell _ 0 } } + \\sum _ { i \\neq k } \\frac { n } { c _ { \\ell _ 0 } } r _ { i c _ { \\ell _ 0 } } = 0 \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} \\log \\mathfrak { M } ( q \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = q \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\ , B ( t ) + \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ \\bigl ( e ^ { t q } - 1 - t q \\bigr ) \\ , \\mathfrak { f } ( t ) \\Bigr ] + \\frac { q ^ 2 } { 2 } \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\ , \\mathfrak { g } ( t ) . \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} C ( x ) = ( 0 , 0 , C ^ 0 ) \\cup ( 1 , 1 , C ^ 2 ) \\cup ( 1 , 0 , C ^ 1 ) \\cup ( 0 , 1 , C ^ 3 ) \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} R _ { \\Lambda } : = c \\Lambda \\quad \\varepsilon _ { \\Lambda } : = \\varepsilon _ { 1 / 2 , R _ { \\Lambda } } = \\min \\{ 4 ^ { - 1 } C _ { R _ { \\Lambda } } ^ { - 1 } R _ { \\Lambda } , \\varepsilon _ 0 \\} . \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} L _ { 2 , n } ^ { 2 , k } \\cong _ { h t } \\begin{cases} S ^ 2 & \\textnormal { i f } 0 < k < n \\\\ \\bigvee _ { i = 1 } ^ { e - 1 } S ^ 1 & \\textnormal { i f } k = n \\\\ e \\textnormal { p o i n t s } & \\textnormal { i f } k = n + 1 \\\\ \\emptyset & \\textnormal { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\sum _ { ( \\sigma , \\omega ) } \\hat { \\mathbb { P } } _ { \\Lambda , \\rho , \\beta , H } ^ { \\vec { h } } ( \\sigma , \\omega ) = 1 . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} u _ \\lambda ( 0 ) = u _ 0 \\mbox { a . e . \\ i n } \\Omega , u _ { \\Gamma , \\lambda } ( 0 ) = u _ { 0 \\Gamma } \\mbox { a . e . \\ o n } \\Gamma . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} h \\beta _ 1 \\nu _ { \\delta } \\varpi _ p = \\left ( \\begin{array} { c c } t ^ { - 1 } \\det ( h ) p ^ { - 1 } & \\ast \\\\ 0 & - t \\end{array} \\right ) \\left ( \\begin{array} { c c } \\delta & 1 \\\\ 1 - \\delta t ^ { - 1 } p ^ { - 1 } z & - t ^ { - 1 } p ^ { - 1 } z \\end{array} \\right ) . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\beta ( i , r ; j , s ) = 2 \\delta _ { i j } ( - \\delta _ { r - s , - r _ i } + \\delta _ { r - s , r _ i } ) . \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} ( \\Phi \\circ \\Psi ) _ 1 = \\Phi _ 1 \\circ \\Psi _ 1 \\hbox { a n d } ( \\Phi \\circ \\Psi ) _ 2 = \\Phi _ 1 \\circ \\Psi _ 2 + \\Phi _ 2 \\circ ( \\wedge ^ 2 \\Psi _ 1 ) \\ , . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} \\phi _ { p } \\left ( x \\right ) = \\prod _ { j = 1 } ^ { d } e ^ { 2 \\pi i p _ { j } x _ { j } } , \\quad \\mu _ { p } = \\sum _ { j = 1 } ^ { d } \\left ( 2 \\pi p _ { j } \\right ) ^ { 2 } , p _ { j } \\in \\mathbb { N } , 1 \\le j \\le d , \\ ; i = \\sqrt { - 1 } . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ N L _ { M } \\bigr ) ( k \\ , | \\ , b ) = \\bigl ( \\mathcal { S } _ N L _ { M } \\bigr ) ( 0 \\ , | \\ , b ) - \\sum \\limits _ { l = 0 } ^ { k - 1 } \\bigl ( \\mathcal { S } _ N L _ { M - 1 } \\bigr ) ( l \\ , | \\ , b ) . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} P _ 0 ( X ) = 1 , P _ n ( X ) = ( X + 1 ) P _ { n - 1 } ( X + 1 ) - P _ { n - 1 } ( X ) , n > 0 . \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} X = \\begin{bmatrix} 1 + x _ i x _ j \\end{bmatrix} . \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} r _ { \\alpha _ n ^ { k - 1 } , n } = r _ { \\alpha _ m ^ { k - 1 } , \\alpha _ m ^ { k } } = R _ { \\alpha _ m ^ { k } , m } - R _ { \\alpha _ m ^ { k - 1 } , m } \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} \\| U ( t ) \\| ^ 2 _ { L ^ 2 } + 2 \\int _ { 0 } ^ t \\| \\nabla U ( \\tau ) \\| ^ 2 _ { L ^ 2 } d \\tau = \\| u _ { 0 , 2 } \\| ^ 2 _ { L ^ 2 } + 2 \\int _ 0 ^ t \\int _ { \\R ^ d } V \\otimes U : \\nabla U d x d \\tau \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} Q ( D ) \\ , = \\ , \\frac { I - e ^ { - \\frac 1 2 D ^ * D } } { D ^ * D } D ^ * \\ , \\equiv \\ , \\int _ 0 ^ 1 e ^ { - \\frac s 2 D D ^ * } D ^ * \\ , d s . \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} F ( \\vec y ) - F ( \\vec x ) = & F ( \\vec y ' ) - F ( \\vec x ) = \\frac { 1 } { \\| \\vec x \\| _ \\infty } ( I ( \\vec y ' ) - I ( \\vec x ) ) \\\\ = & \\frac { 1 } { \\| \\vec x \\| _ \\infty } ( g ( 1 ) - g ( 0 ) ) \\le 0 , \\forall \\ , \\vec y \\in U . \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} g ( A _ { E } ^ { \\ast } U , P V ) = B ( U , P V ) , \\ \\ \\ g \\ ( A _ { E } ^ { \\ast } U , N ) = 0 , \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} B [ N _ G ] \\cap \\bigl ( B ^ { \\Sigma } + B ^ { \\Sigma ' } + B ^ { G } \\bigr ) & = B [ N _ G ] \\cap \\bigl ( I _ G B + B ^ { \\Sigma } + B ^ { G } \\bigr ) \\\\ & = I _ G B + \\bigl ( B [ N _ G ] \\cap \\bigl ( B ^ { \\Sigma } + B ^ { G } \\bigr ) \\bigr ) \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} \\bigotimes _ { i = 1 , } ^ { 2 q - 1 } ( U _ i ^ * U _ { i + 1 } L ) & = \\bigotimes _ { i = 1 , } ^ { 2 q - 1 } \\left ( ( U _ i ^ * ) ^ { r _ { i + 1 } } ( U _ { i + 1 } ) L ^ { r _ { i + 1 } } \\right ) \\\\ & = P ( U ^ * _ 1 ) ^ { r _ 2 + 1 } ( U _ 3 ^ * ) ^ { r _ 4 + 1 } \\ldots ( U _ { 2 q - 1 } ^ * ) ^ { r _ { 2 q } + 1 } U _ { 2 q + 1 } ^ * \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} e ( G , N ) = \\sum _ { i = 1 } ^ { s } \\left \\lvert \\mathrm { O r b } ( N _ { i } ) \\right \\rvert = \\sum _ { i = 1 } ^ { s } \\dfrac { \\left \\lvert \\mathrm { A u t } ( G ) \\right \\rvert } { \\left \\lvert \\mathrm { S t a b } ( N _ { i } ) \\right \\rvert } = \\sum _ { G _ { N } } \\dfrac { \\left \\lvert \\mathrm { A u t } ( G ) \\right \\rvert } { \\left \\lvert \\mathrm { A u t } _ { \\mathcal { B } r } ( G _ { N } ) \\right \\rvert } . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} J _ s ( t ) = & J ( t ) J ( s ) ^ { - 1 } \\\\ = & J ( s ) J ( s ) ^ { - 1 } + \\int _ s ^ t \\nabla _ x b ( r , \\omega , X ( r ) ( \\omega ) ) J ( r ) J ( s ) ^ { - 1 } d r \\\\ & + \\int _ s ^ t \\nabla _ x \\sigma ( r , \\omega , X ( r ) ( \\omega ) ) J ( r ) J ( s ) ^ { - 1 } d W ( r ) \\\\ = & I _ d + \\int _ s ^ t \\nabla _ x b ( r , \\omega , X ( r ) ( \\omega ) ) J _ s ( r ) d r + \\int _ s ^ t \\nabla _ x \\sigma ( r , \\omega , X ( r ) ( \\omega ) ) J _ s ( r ) d W ( r ) . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} \\Big | f ( t ) - \\sum _ { n = 1 } ^ N \\phi _ n t ^ { - \\mu _ n } \\Big | _ { \\alpha , \\sigma } & \\le \\Big | f ( t ) - \\sum _ { n = 1 } ^ { N _ * } \\phi _ n t ^ { - \\mu _ n } \\Big | _ { \\alpha , \\sigma } + \\Big | \\sum _ { n = N + 1 } ^ { N _ * } \\phi _ n t ^ { - \\mu _ n } \\Big | \\\\ & = \\mathcal O ( t ^ { - \\mu _ { N _ * } - \\delta } ) + \\mathcal O ( t ^ { - \\mu _ { N + 1 } } ) t \\to \\infty , \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} f _ 1 = x ^ { a + 1 } - y z , f _ 2 = y ^ { b + 1 } - x ^ a z , f _ 3 = z ^ 2 - x y ^ b \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} T : = K \\langle y _ { 1 } , \\dots , y _ { c } \\mid \\dd ( y _ { i } ) = z _ { i } \\rangle \\ , . \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\dot { x } = [ A ( t ) + B ( t ) ] x \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} ( 2 \\theta _ 2 P _ 3 ( 1 ) \\widetilde { P _ 2 } ( 1 ) + o ( 1 ) ) \\varphi ^ + ( q ) . \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} S ( n , k ) = \\dfrac { \\Delta ^ k I _ n ( 0 ) } { k ! } , k = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { 2 j + \\frac { k } { \\beta } + \\abs { \\sigma } < q } \\rho ^ { 2 j + \\frac { k } { \\beta } } \\left ( a _ { j , k } ^ \\sigma \\cos k \\theta + b _ { j , k } ^ \\sigma \\sin k \\theta \\right ) \\xi ^ \\sigma + O ( d ( x ) ^ q ) , \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\pi _ { \\rho } ^ { o , \\kappa , h } ( n ) = 0 ; \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\widehat { d _ 1 } ( G ( A , B ) ) = \\widehat { d _ 1 } ( G ( M ^ T A M , M ^ T B M ) ) . \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} B = S - S ( A + S ) ^ { - 1 } S \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\sigma _ x ^ { \\eta } = \\sigma _ x ^ { \\eta ^ { \\prime } } x \\in ( \\partial _ { e x } V _ { \\tau } ) \\cap \\bar { \\Lambda } . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} \\mathcal { H } _ 1 ( 0 , 0 , \\delta , t ) = \\frac { \\zeta ( 4 + 4 \\delta ) } { \\zeta ( 2 + 2 \\delta ) | \\zeta ( 2 + 2 \\delta + 2 i t ) | ^ 2 } . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} D \\psi _ \\lambda ( \\xi _ 1 ) = D \\psi _ \\lambda ( \\xi _ 2 ) . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} a ^ x = ( 1 + p b ) ^ x = \\sum _ { j = 0 } ^ x { x \\choose j } p ^ j b ^ j = \\sum _ { j = 0 } ^ { + \\infty } b ^ j ( x ) _ j \\frac { p ^ j } { j ! } , \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} l _ { A , 1 } & \\ ; : = e _ A : ( A , A ^ \\bot ) \\to ( a ) , \\\\ l _ { A , 2 } & \\ ; : = r _ { A ^ \\bot } : ( A ^ \\bot , 1 _ x ) \\to ( A ^ \\bot ) , \\\\ l _ { A , 3 } & \\ ; : = l _ A : ( 1 _ x , A ) \\to ( A ) , \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} V _ { n } = X _ { 1 , n } + X _ { 2 , n } + \\cdots + X _ { n , n } \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} I = 0 . \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} 2 \\gamma \\sum _ { k = k _ 0 } ^ N \\lambda _ k \\| y ^ { k } - z \\| ^ 2 \\leq \\sigma _ { k _ 0 } - \\sigma _ { N + 1 } \\leq \\sigma _ { k _ 0 } , \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\frac { F _ 4 } { E _ 2 ^ 2 } = 1 + \\frac 1 { 1 + a + b } . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} X _ { \\underline { m } } = \\begin{cases} x ^ 2 = x \\oplus x , \\vert \\underline { m } \\vert = 1 , \\\\ x , \\vert \\underline { m } \\vert = 2 \\end{cases} \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\lambda _ k = + \\infty ; \\lim _ { k \\to \\infty } \\lambda _ k = 0 . \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} \\widetilde { \\omega } _ { i , j } \\coloneqq \\begin{cases} - \\omega _ { i , j } i = 1 \\cr \\omega _ { i , j } \\end{cases} \\ ; . \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ { \\beta _ j } f _ j ( \\rho ) \\leq \\epsilon \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ { \\beta _ j } f _ j ( R ) + \\sum _ { \\ell = 1 } ^ { m } A _ \\ell ( R - \\rho ) ^ { \\gamma _ { \\ell } } \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} p ^ * \\left ( L ' _ { i , j } \\right ) = \\frac { A ' _ { i - 1 , j - 1 } } { A ' _ { i , j } } \\prod _ { k = i - a } ^ { b - j } \\lambda ' _ k = \\frac { A ' _ { i - 1 , j - 1 } } { A ' _ { i , j } } \\prod _ { k = i - a - 1 } ^ { b - j - 1 } \\lambda _ k . \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} b = d > 0 , \\ ; a = 0 , \\ ; c < 0 . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} f ^ I \\left ( U \\left ( m + k \\chi ^ J , n + k \\chi ^ J \\right ) \\right ) = f ^ I \\left ( U \\left ( m , n \\right ) \\right ) , \\quad \\forall k \\in \\N . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} E _ { + \\infty } ( { \\bf { h } } _ { n _ k } ) = E _ { + \\infty } ( { \\bf { w } } _ { k } ) + E _ { + \\infty } ( { \\bf { h } } _ { k , 1 } ) + E _ { + \\infty } ( { \\bf { h } } _ { k , 2 } ) + N _ 3 + N _ 4 , \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} H _ { 3 , p } & = \\Big \\langle r _ { ( \\alpha , \\alpha - 1 , - 1 ) } , r _ { ( - \\alpha , \\alpha - 1 , 1 ) } , r _ { ( 1 , - \\alpha , \\alpha - 1 ) } \\Big \\rangle , \\\\ H _ { 4 , p } & = \\Big \\langle r _ { ( \\alpha , \\alpha - 1 , - 1 , 0 ) } , r _ { ( - \\alpha , \\alpha - 1 , 1 , 0 ) } , r _ { ( 1 , - \\alpha , \\alpha - 1 , 0 ) } , r _ { ( 1 , 0 , - \\alpha , \\alpha - 1 ) } \\Big \\rangle , \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} x ^ { - 1 } ( y y ^ { \\alpha } ) & = ( \\left [ ( x ^ { - 1 } \\cdot y x ) \\right ] / x ) y ^ { \\alpha } & \\\\ & = ( \\left [ ( x ^ { - 1 } y \\cdot x ) \\right ] / x ) y ^ { \\alpha } & \\\\ & = ( x ^ { - 1 } y ) y ^ { \\alpha } \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} L = \\sum _ { j = 1 } ^ { N } r ^ { j - 1 } r _ { d } ^ { N - 1 } = \\left ( \\frac { 1 - r ^ N } { 1 - r } \\right ) r _ d ^ { N - 1 } . \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} a _ 1 ^ 8 b ^ 3 ( 1 + b ) ^ 4 ( 1 + a _ 1 ^ 2 + a _ 1 b + a _ 1 b ^ 2 + b ^ 4 ) ^ 4 = d _ 1 h _ 1 + d _ 2 h _ 2 = 0 , \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} w ^ B ( t ) = w _ 0 ^ B t , \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} \\left ( { x - a } \\right ) _ q ^ n = \\prod \\limits _ { j = 0 } ^ { n - 1 } { \\left ( { x - q ^ j a } \\right ) } , \\ , \\ , \\ , \\ , \\ , \\left ( { x - a } \\right ) _ q ^ { ( 0 ) } = 1 , \\ , \\ , \\ , \\ , \\ , \\left ( { x - a } \\right ) _ q ^ { - n } = \\frac { 1 } { { \\left ( { x - q ^ { - n } a } \\right ) _ q ^ n } } . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} F _ u ( x ) = E _ { x \\star u , x } \\left ( V ( H ^ u _ { { \\cal T } _ \\vartheta } ) , \\ ; { \\cal T } _ \\vartheta < { \\cal T } _ \\vartheta ^ u \\leq + \\infty \\right ) \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} \\widetilde K ( t , x , y ) = - \\Delta _ x m _ { u _ \\mu } ( t , x ) \\Delta _ y m _ { u _ \\iota } ( t , y ) - 2 \\nabla _ x m _ { \\nabla u _ \\mu } ( t , x ) \\nabla _ y m _ { u _ \\iota } ( t , y ) . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\nabla _ \\beta ( f \\star g ) = ( \\nabla _ \\beta f ) \\star g = f \\star ( \\nabla _ \\beta g ) \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\| \\eta _ T ( t ) e ^ { i t \\Delta } u _ 0 \\| _ { X ^ { 0 , b } } & \\lesssim T ^ { \\frac 1 2 - b } \\| u _ 0 \\| _ { L ^ 2 } \\\\ \\| \\eta _ T ( t ) e ^ { \\pm i t D } \\phi \\| _ { Y ^ { s _ 1 , b } } & \\lesssim T ^ { \\frac 1 2 - b } \\Bigl ( \\| n _ 0 \\| _ { H ^ { s _ 1 } } + \\| n _ 1 \\| _ { H ^ { s _ 1 - 1 } } \\Bigr ) . \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} L ( \\alpha ) \\circ \\tau = \\tau \\circ L ( \\beta ) . \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\mathrm { d i m } ( H _ { J _ { k + 1 } } ) = \\mathrm { d i m } ( H _ { J _ { k } } ) + \\mathrm { d i m } ( H _ { k + 1 } ) - 2 g = 2 g - k + 2 g - 1 - 2 g = 2 g - k - 1 , \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\lim _ { \\rho \\searrow 0 } \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } ( | V ( f , \\pi _ n ) _ T - V ( f _ \\rho , \\pi _ n ) _ T | > \\varepsilon ) = 0 ~ ~ ~ \\forall \\varepsilon > 0 . \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} \\langle \\xi _ 1 , \\xi _ 2 \\rangle _ S : = { \\rm t r } ( S ^ { - 1 } \\xi _ 1 S ^ { - 1 } \\xi _ 2 ) , \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\eta _ s ( b _ 1 , b _ 2 ) = \\eta _ s ( b _ 2 , b _ 1 ) \\ ; \\ ; \\mbox { f o r a l l $ b _ 2 \\in A $ . } \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\mathbb { V } _ { 1 } & = M o d \\left ( E q \\left ( \\mathbb { A O L } \\right ) \\cup \\left \\{ \\right \\} \\right ) \\\\ \\mathbb { V } _ { 2 } & = M o d \\left ( E q \\left ( \\mathbb { A O L } \\right ) \\cup \\left \\{ \\right \\} \\right ) \\\\ \\mathbb { V } _ { 3 } & = M o d \\left ( E q \\left ( \\mathbb { A O L } \\right ) \\cup \\left \\{ \\right \\} \\right ) \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} \\begin{cases} \\underbar r _ 1 - \\epsilon \\le u ^ * ( x , t ; t _ 0 , u ^ * ( \\cdot , t _ 0 ) , v ^ * ( \\cdot , t _ 0 ) ) \\le \\bar r _ 1 + \\epsilon \\cr \\underbar r _ 2 - \\epsilon \\le v ^ * ( x , t ; t _ 0 , u ^ * ( \\cdot , t _ 0 ) , v ^ * ( \\cdot , t _ 0 ) ) \\le \\bar r _ 2 + \\epsilon \\end{cases} \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } x _ j ^ { \\downarrow } \\leq \\sum _ { j = 1 } ^ { k } y _ j ^ { \\downarrow } , 1 \\leq k \\leq m . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} W ^ { ( 1 ) } = & \\frac { 1 } { 2 } ( b _ 1 + \\beta ) y ^ 5 + \\left ( - 5 b _ 1 + \\frac { \\gamma } { 2 } \\right ) y ^ 9 + ( 1 2 0 6 4 - 3 6 2 b _ 1 - 2 7 \\beta - 5 \\gamma ) y ^ { 1 3 } \\\\ & + \\left ( 7 3 4 9 7 6 - 1 8 4 7 b _ 1 + 1 6 0 \\beta + \\frac { 4 5 \\gamma } { 2 } \\right ) y ^ { 1 7 } + \\cdots . \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} \\mathcal M = 1 - \\frac { \\sqrt { \\mu } } { \\gamma \\sqrt { \\mu _ 2 } } - \\frac { \\sqrt { \\mu } } { \\gamma } | D | \\Big [ \\coth ( \\sqrt { \\mu _ 2 } | D | ) - \\frac { 1 } { \\sqrt { \\mu _ 2 } | D | } \\Big ] , \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} F _ 1 & = X _ 1 X _ 2 - Y _ 1 Z _ 2 , \\\\ F _ 2 & = Y _ 1 Y _ 2 - X _ 2 Z _ 1 , \\\\ F _ 3 & = X _ 1 Y _ 2 - Z _ 1 Z _ 2 . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ { p , q } ) ^ { s } u ( x ) = \\omega | u ( x ) | ^ { p - 2 } u ( x ) , \\ , \\ , x \\in \\Omega , \\\\ u ( x ) = 0 , \\ , \\ , \\ , \\ , x \\in \\mathbb { G } \\setminus \\Omega , \\end{cases} \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} \\tau _ { s } ( g ) = f ( g \\iota ( g ) ^ { - 1 } ) , \\iota ( g ) = s \\circ p ( g ) . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} = \\ , ( { \\rm a d } ( E _ G ) ) - ( { \\rm a d } ( E _ H ) ) + { \\rm d e g r e e } ( { \\mathcal N } ^ * _ { \\mathcal F } ) \\ , . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} \\mathbf { S L } _ \\alpha \\ L = L \\ \\mathbf { S L } _ \\alpha , \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\begin{array} { l l } \\int _ { \\mathbb R } \\zeta _ k \\nu _ { n _ k } | D | c o t h ( \\sqrt { \\mu _ 2 } | D | ) ( \\varphi _ k \\nu _ { n _ k } ) d x & = \\int _ { \\mathbb R } \\zeta _ k \\nu _ { n _ k } \\Big [ | D | \\coth ( \\sqrt { \\mu _ 2 } | D | ) - \\frac { 1 } { \\sqrt { \\mu _ 2 } } \\Big ] ( \\varphi _ k \\nu _ { n _ k } ) d x \\\\ \\\\ & = \\int _ { \\mathbb R } \\zeta _ k \\nu _ { n _ k } \\mathcal T | D | ( \\varphi _ k \\nu _ { n _ k } ) d x \\equiv L . \\end{array} \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} C _ { \\epsilon } = X \\backslash \\cup _ { j = 1 } ^ k B _ { \\epsilon } ( x _ j ) , \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\nabla ^ n \\left ( f ( x ) g ( x ) \\right ) = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\nabla ^ k f ( x ) \\nabla ^ { n - k } g ( x - k ) , \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\mathfrak { M } ( q \\ , | \\ , \\mu , \\lambda _ 1 , \\lambda _ 2 ) = \\int \\limits _ 0 ^ \\infty y ^ q \\ , f _ { ( \\mu , \\lambda _ 1 , \\lambda _ 2 ) } ( \\log y ) \\ , \\frac { d y } { y } = { \\bf E } \\bigl [ M _ { ( \\mu , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\det _ { 1 \\leq j , k \\leq n } \\left ( \\int f _ j ( x ) g _ k ( x ) \\ , d \\mu ( x ) \\right ) = \\frac 1 { n ! } \\int \\prod _ { 1 \\leq j < k \\leq n } ( x _ j - x _ k ) ^ { 2 } \\ , d \\mu ( x _ 1 ) \\dotsm d \\mu ( x _ n ) . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} h _ { ( j , N ) } ^ { - } & = 1 _ { A _ { j } ^ { ( N ) } } g _ { - } - 1 _ { A _ { j } ^ { ( N ) } } \\mathbb { E } \\left ( g _ { + } \\ ; | \\ ; A _ { j } ^ { ( N ) } \\right ) \\\\ & \\leqslant \\phi _ { ( j , N ) } f \\leqslant 1 _ { A _ { j } ^ { ( N ) } } g _ { + } - 1 _ { A _ { j } ^ { ( N ) } } \\mathbb { E } \\left ( g _ { - } \\ ; | \\ ; A _ { j } ^ { ( N ) } \\right ) = h _ { ( j , N ) } ^ { + } , \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\sum _ { v ' \\in \\tilde { E } } \\tilde { p } _ H ( v , v ' ) = 1 . \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\left \\langle \\tau _ { \\infty } \\left ( \\left ( \\begin{array} { c c } e ^ t & \\\\ & e ^ { - t } \\end{array} \\right ) \\right ) v _ 0 , v _ 0 \\right \\rangle = \\mathrm { c o s h } ( t ) ^ { - ( k + 1 ) } , \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\sum _ { H } n _ H \\cdot \\chi _ H = 0 \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} J ^ { \\varepsilon } ( u ) : = \\left \\{ x \\in p ^ { - 1 } ( \\mathbb { R } ) \\ , : \\ , p ( x ) \\leq F _ { u } ( x , 0 _ { u } ) + \\varepsilon \\right \\} , \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} \\mathcal { C } _ { s } ^ { m + 1 } = - \\sum _ { n = s } ^ { m } \\binom { m + 1 } { m - n + 1 } N _ { \\mathrm { d } \\ , m - n + 1 } \\mathcal { C } _ { s } ^ { n } \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} C _ { s , \\pi } = \\frac { 1 } { c _ 1 c _ 2 } \\frac { \\sup _ { z \\in E _ s } \\P _ { s , z } ( \\tau _ { A } > v _ s ) } { d ' _ { v _ s } \\P _ { s , \\pi } ( \\tau _ { A } > v _ s ) } , \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } e ^ { y q } \\frac { d } { d y } \\exp \\Bigl ( - e ^ { - \\beta y } X \\Bigr ) d y = X ^ { \\frac { q } { \\beta } } \\ , \\Gamma ( 1 - \\frac { q } { \\beta } ) , \\ ; \\Re ( q ) < 0 , \\ , X > 0 . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\ell ' = b _ 1 c _ 1 + b c _ 2 \\geq b _ 1 c _ 1 + b ' d ' > b ' d ' = \\ell ' . \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} \\{ \\hat { x } , \\hat { X } \\} = \\hat { x } \\hat { X } \\ ; . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} \\frac { G ( z ) } { A ( z ) } R ( z ) = \\bigg ( \\sum _ n \\frac { c _ n \\mu _ n ^ { 1 / 2 } } { z - t _ n } \\bigg ) \\cdot \\bigg ( \\sum _ n \\frac { G ( t _ n ) \\bar c _ n } { A ' ( t _ n ) \\mu _ n ^ { 1 / 2 } ( z - t _ n ) } \\bigg ) - \\frac { G ( z ) } { A ( z ) } \\sum _ n \\frac { | c _ n | ^ 2 } { z - t _ n } . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { k } { Y _ s } - a Y _ s \\ge \\frac { k } { \\varepsilon } - a \\varepsilon . \\end{gathered} \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\| \\partial _ v f ^ n _ t - \\partial _ v f ^ { n - 1 } _ t \\| _ { L ^ 1 } & \\le \\int _ 0 ^ t \\ ! \\mathrm { d } s \\ , \\| ( G ( M ^ { n - 1 } ( s ) - G ( M ^ { n - 2 } ( s ) ) \\partial _ { v v } f ^ { n - 1 } _ s \\| _ { L ^ 1 } \\\\ & + \\int _ 0 ^ t \\ ! \\mathrm { d } s \\ , \\| \\partial _ x f ^ n _ s - \\partial _ x f ^ { n - 1 } _ s \\| _ { L ^ 1 } \\ , . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\psi ^ { \\prime \\prime } + { ( n - 1 ) ( 1 - R ) \\over t ( 1 - R ) + 2 R - 1 } \\psi ^ { \\prime } + ( 1 - R ) ^ 2 \\left ( p w _ R ^ { p - 1 } ( t ) - { \\lambda _ k \\over \\ ( t ( 1 - R ) + 2 R - 1 \\ ) ^ 2 } \\right ) \\psi = \\lambda _ k \\psi , \\ \\hbox { i n } \\ ( 1 , 2 ) , \\\\ \\psi ( 1 ) = \\psi ( 2 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} A _ 2 ( n ) = p ^ { 3 n / 2 } ( - 1 ) ^ n \\chi _ { \\psi } ( p ^ n ) \\int _ { \\mathcal A _ 2 ( n ) } ( c , p ^ n ) _ p \\chi _ { \\psi } ( c ) e ( h ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h , \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} L ( f , s ) & : = \\prod _ p L _ p ( f , s ) = \\prod _ { p } \\prod _ { i = 1 } ^ 2 \\Bigl ( \\ , 1 - \\frac { \\alpha _ { f , i } ( p ) } { p ^ s } \\ , \\Bigr ) ^ { - 1 } \\\\ & = \\prod _ { p } \\Bigl ( 1 - \\frac { \\lambda _ { f } ( p ) } { p ^ s } + \\frac { \\chi _ r ( p ) } { p ^ { 2 s } } \\Bigr ) ^ { - 1 } = \\sum _ { n \\geq 1 } \\frac { \\lambda _ { f } ( n ) } { n ^ s } , \\Re s > 1 . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} I _ f ( \\tau ) : = \\int _ { - \\overline \\tau } ^ { i \\infty } \\frac { f ( w ) } { ( - i ( \\tau + w ) ) ^ { 2 - k } } d w . \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} \\int \\limits _ G F ( g ) \\ , d ' g & = 2 ^ n \\int \\limits _ A \\int \\limits _ { N _ 1 } \\int \\limits _ { N _ 2 } \\int \\limits _ K F ( a n _ 1 n _ 2 k ) \\ , d k \\ , d n _ 2 \\ , d n _ 1 \\ , d a \\\\ & = 2 ^ n \\int \\limits _ A \\int \\limits _ { N _ 1 } \\int \\limits _ { N _ 2 } F ( n _ 2 a n _ 1 ) \\det ( a ) ^ { - ( n + 1 ) } \\ , d n _ 2 \\ , d n _ 1 \\ , d a . \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} C _ { \\Phi , d } = \\left \\{ | k | \\prod _ { i = 1 } ^ s | h _ i | ^ { e _ i } \\ , \\middle | \\ , k \\in K ( \\Phi ) , \\ ; e _ 1 \\dotsc e _ s \\in { \\mathbb Z } , \\ ; \\sum _ { r = 1 } ^ s \\lvert e _ r \\rvert < d \\right \\} \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} X = Z _ { \\lambda _ 1 , \\lambda _ 2 , \\varepsilon } ( \\beta ) \\frac { e ^ { \\kappa } \\ , \\Gamma ( 1 - \\beta ^ 2 ) } { 2 \\pi } \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} U _ { L _ n / K } ' - [ ( \\mathcal { P } _ 0 ) _ { \\Gamma _ n } , \\alpha _ n , H _ { L _ n } ] = U _ { L _ n / K } - [ \\mathcal { O } _ { L _ n } , \\rho _ n , H _ { L _ n } ] . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} \\sigma ^ - ( L \\mathbf { c } _ 0 ) ( F ) = \\left [ \\begin{array} { c c } \\bar { n } _ 1 ^ 2 ( \\lambda ^ - + \\mu ^ - ) + \\mu ^ - & \\bar { n } _ 1 \\bar { n } _ 2 ( \\lambda ^ - + \\mu ^ - ) \\\\ \\bar { n } _ 1 \\bar { n } _ 2 ( \\lambda ^ - + \\mu ^ - ) & \\bar { n } _ 2 ^ 2 ( \\lambda ^ - + \\mu ^ - ) + \\mu ^ - \\end{array} \\right ] \\left [ \\begin{array} { c } c ^ 1 _ 0 \\\\ c ^ 2 _ 0 \\end{array} \\right ] : = K \\mathbf { c } _ 0 . \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{gather*} N ( k ) : = \\sum _ { ( m , n ) \\in \\Z ^ 2 \\setminus \\{ ( 0 , 0 ) \\} } \\frac { 1 } { \\big ( m + n \\sqrt { - 1 } \\big ) ^ { 4 k } } = \\sum _ { ( m , n ) \\in \\Z ^ 2 \\setminus \\{ ( 0 , 0 ) \\} } \\frac { \\big ( m - n \\sqrt { - 1 } \\big ) ^ { 4 k } } { \\big ( m ^ 2 + n ^ 2 \\big ) ^ { 4 k } } \\end{gather*}"}
-{"id": "2671.png", "formula": "\\begin{align*} K ( x ) = 2 \\pi { \\rm i } \\sum _ { k \\in \\Z ^ 2 _ 0 } \\frac { k ^ \\perp } { | k | ^ 2 } { \\rm e } ^ { 2 \\pi { \\rm i } k \\cdot x } = - 2 \\pi \\sum _ { k \\in \\Z ^ 2 _ 0 } \\frac { k ^ \\perp } { | k | ^ 2 } \\sin ( 2 \\pi k \\cdot x ) . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} \\forall ( t : T ) \\ldotp \\exists ( s : S ) \\ldotp f ( s ) = t \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} \\tau _ g ^ s = \\sum _ { [ w ] \\in W _ G / W _ K } \\tau _ { w g w ^ { - 1 } } . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} a _ { p , s } = P ^ { \\hat { I } } ( L ) , \\ ; \\ ; \\hat { I } = ( I \\setminus \\{ s \\} ) \\cup \\{ p \\} . \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} \\mathcal { E } _ n : = \\bigcup _ { \\substack { 1 \\leq a \\leq n , \\\\ ( a , n ) = 1 } } \\left ( \\frac { a - \\psi ( n ) } { n } , \\frac { a + \\psi ( n ) } { n } \\right ) . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} & \\mathcal O _ t v = 0 , & t \\in \\mathbb N \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} q _ { \\gamma _ 0 } = Q = q \\leq \\prod _ { \\atop { \\gamma \\in \\Gamma } { \\abs { \\gamma } \\geq 2 } } q _ \\gamma \\leq q _ { \\gamma _ 0 } ^ d , \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} \\abs { Y } \\le \\begin{cases} q ^ { m ( m - d + 2 ) / 2 } & , \\\\ q ^ { ( m + 1 ) ( m - d + 1 ) / 2 } & , \\\\ q ^ { ( m - 1 ) ( m - d + 2 ) / 2 } & , \\\\ q ^ { m ( m - d + 1 ) / 2 } & . \\end{cases} \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} \\mathcal { K } _ { t , \\mathcal { Q } ^ { \\flat } } ( = K _ t ( \\mathcal { C } _ { \\mathcal { Q } ^ { \\flat } } ) ) : = \\bigoplus _ { m \\in \\mathbb { B } ^ { \\xi , \\flat } } \\mathbb { Z } [ t ^ { \\pm 1 / 2 } ] F _ t ( m ) \\subset \\mathcal { K } _ { t } . \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} C ^ { e v } : = \\bigoplus _ { i \\in \\Z } C ^ { 2 i } , C ^ { o d d } : = \\bigoplus _ { i \\in \\Z } C ^ { 2 i + 1 } . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { m - 1 } 2 ( m - k ) = m ( m + 1 ) . \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u \\in S ( \\lambda , \\Lambda , f ) & & ~ ~ \\mbox { i n } ~ ~ \\Omega ; \\\\ & u = g & & ~ ~ \\mbox { o n } ~ ~ \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\sigma _ j ( y ) & : = y - \\sum _ { z \\in Z _ j } \\varphi _ z ( y ) \\cdot \\pi _ { L ( z , 1 2 r _ j ) } ^ \\perp ( y - z ) \\\\ \\intertext { a n d } \\tau _ j ( y ) & : = ( \\sigma _ j \\circ t _ { j - 1 } ) ( y ) . \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} g ^ { p ^ { n - 1 } } = ( a b ^ i ) ^ { k p ^ { n - 1 } } . \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 0 } ^ { 2 n + 1 - m } a _ i t ^ i \\right ) \\equiv \\left ( \\sum _ { i = 0 } ^ { n + 1 - m } ( - 1 ) ^ i \\beta _ i t ^ { 2 i } + \\sum _ { i = n + 1 } ^ { 2 n + 1 - m } ( - 1 ) ^ i \\beta _ i t ^ { 2 i } \\right ) \\left ( \\sum _ { i = 0 } ^ m \\delta _ i t ^ i + \\sum _ { i \\geq 2 n + 3 - m } \\delta _ i t ^ i \\right ) . \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} \\alpha - { } ^ a \\alpha = ( 1 , a ) - ( g , a ) \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} S ( T , X ) = \\sum _ { t _ j \\le T } X ^ { i t _ j } , \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\lambda = \\ell k ^ 2 , \\ \\ell \\in [ \\eta , \\eta ^ { - 1 } ] \\ \\hbox { f o r s o m e $ \\eta > 0 $ s m a l l e n o u g h . } \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} \\left ( u \\alpha _ { 1 } \\right ) \\left ( w \\alpha _ { 3 } \\right ) & = \\rho ^ { w _ { 2 } } u w \\alpha _ { 1 } \\alpha _ { 3 } \\ \\\\ \\left ( w \\alpha _ { 3 } \\right ) \\left ( u \\alpha _ { 1 } \\right ) & = \\rho ^ { u _ { 3 } + w _ { 3 } u _ { 2 } - u _ { 3 } w _ { 2 } } u w \\alpha _ { 1 } \\alpha _ { 3 } , \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} B ( v , e ) & = \\eta B _ 1 ( v , e ) + ( 1 - \\eta ) B _ 2 ( v , e ) , \\\\ w ( e ) & = \\xi w _ 1 ( e ) + ( 1 - \\xi ) w _ 2 ( e ) . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\deg ( z ( \\lambda ) ) = \\max \\{ \\deg ( N _ 1 ( \\lambda ) ^ T h ( \\lambda ) ) \\ , , \\ , \\deg ( X ( \\lambda ) h ( \\lambda ) ) \\} \\ , . \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\left ( \\frac { | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 } { \\langle g , g \\rangle ^ 4 } \\right ) ^ { \\sigma } = \\frac { | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } ^ { \\sigma } , g ^ { \\sigma } \\times g ^ { \\sigma } \\rangle | ^ 2 } { \\langle g ^ { \\sigma } , g ^ { \\sigma } \\rangle ^ 4 } , \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} d S _ \\nu : = \\{ d f ^ n _ \\omega ( 0 ) \\ , | \\ , \\omega \\in \\Omega , \\ , n \\in \\mathbb N \\} , \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} \\| S _ { t _ 0 } \\mu _ 1 - S _ { t _ 0 } \\mu _ 2 \\| _ { \\mathrm { T V } } \\leq 2 ( 1 - \\alpha ) = ( 1 - \\alpha ) \\| \\mu _ 1 - \\mu _ 2 \\| _ { \\mathrm { T V } } . \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\sum _ { \\mu = 1 } ^ N \\int _ { \\R } \\int _ { \\R ^ d \\times \\R ^ d } \\frac { \\nabla _ x | u _ \\mu ( t , x ) | ^ { 2 } \\nabla _ y | u _ \\mu ( t , y ) | ^ 2 } { | x - y | ^ 3 } \\ , d x \\ , d y \\ , d t & \\\\ + \\sum _ { \\mu = 1 } ^ N \\gamma _ { \\mu \\mu } \\int _ { \\R } \\int _ { \\R ^ d \\times \\R ^ d } \\frac { | u _ \\mu ( t , x ) | ^ { 2 p + 2 } | u _ \\mu ( t , y ) | ^ 2 } { | x - y | } \\ , d x \\ , d y \\ , d t \\leq C \\sum _ { \\mu = 1 } ^ N \\| u _ { \\mu , 0 } \\| ^ 4 _ { H ^ 2 _ x } . & \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\varepsilon _ { f g } = \\# \\{ g \\rightarrow f \\} - \\# \\{ f \\rightarrow g \\} . \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} h _ { \\ell - n + t } ( y _ 1 , \\cdots , y _ n ) = \\sum _ { l _ 1 + \\cdots + l _ n = \\ell - n + t } y _ 1 ^ { l _ 1 } \\cdots y _ n ^ { l _ n } = 0 . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} & 4 \\left \\| u \\right \\| _ { 2 } ^ 2 \\geq \\\\ & x _ + ^ T \\left ( A ^ T \\Sigma ^ { - 1 } + \\Sigma ^ { - 1 } A + \\sum _ { k = 1 } ^ m N _ k ^ T \\Sigma ^ { - 1 } N _ { k } + \\sum _ { i , j = 1 } ^ v H ^ T _ { i } \\Sigma ^ { - 1 } H _ { j } k _ { i j } \\right ) x _ + + 4 x _ + ^ T \\Sigma ^ { - 1 } B u . \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} S _ { L _ i } ( x ) \\cap F ^ 1 = S _ { \\widehat { L } _ i } ( x ) \\cap F ^ 1 \\enspace \\enspace \\frac { 1 } { 3 } | S _ { \\widehat { L } _ i } ( x ) \\cap F ^ 2 | \\leq | S _ { L _ i } ( x ) \\cap F ^ 2 | \\leq \\frac { 1 } { 2 } | S _ { \\widehat { L } _ i } ( x ) \\cap F ^ 2 | . \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} d = N - ( 2 - 2 + 1 ) ( n - 2 + 1 ) = N - n + 1 \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} ( J ( f ) ( t ) ) ( u ) = \\sum _ { j = 1 } ^ { \\infty } \\Phi ( t ) e _ j ( \\cdot ) \\langle u , e _ j \\rangle = \\sum _ { j = 1 } ^ { \\infty } \\Phi ( t ) \\big ( \\langle u , e _ j \\rangle e _ j \\big ) ( \\cdot ) = \\Phi ( t ) \\left ( u \\right ) ( \\cdot ) , \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ { k } \\frac 1 { | y - \\xi _ j | ^ { N - 2 } } \\le \\frac C { | y - \\xi _ 1 | ^ { N - 2 - \\tau } } \\sum _ { j = 2 } ^ { k } \\frac 1 { | \\xi _ j - \\xi _ 1 | ^ \\tau } \\le \\frac { C k } { | y - \\xi _ 1 | ^ { N - 2 - \\tau } } \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} d _ { 2 n + 2 - m } ^ { } = \\sum _ { p = 1 } ^ { 2 n + 1 - m } ( - 1 ) ^ { p + 1 } \\tau _ p ' * d _ { 2 n + 2 - p - m } . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} \\displaystyle \\int _ { \\Omega } B ( x ) & | \\nabla u ( x ) | ^ { p ( x ) - 2 } \\nabla u ( x ) \\nabla v ( x ) d x + \\displaystyle \\int _ { \\Omega } A ( x ) | u ( x ) | ^ { p ( x ) - 2 } u ( x ) v ( x ) d x \\\\ & + \\displaystyle \\int _ { \\Omega } D ( x ) | u ( x ) | ^ { p ( x ) - 1 } u ( x ) v ( x ) d x + \\displaystyle \\int _ { \\Omega } C ( x ) | u ( x ) | ^ { p ( x ) - 3 } u ( x ) v ( x ) d x \\\\ & - \\int _ { \\Omega } b ( x ) | u ( x ) | ^ { q ( x ) - 2 } u ( x ) v ( x ) d x = 0 , \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J _ { K } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) = 0 \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ l ( - 1 ) ^ i d _ i = \\sum _ { i = 1 } ^ { m + 1 } ( - 1 ) ^ i d _ i - ( d _ { m + 2 } - d _ { m + 3 } + \\cdots ) \\le \\sum _ { i = 1 } ^ { m + 1 } ( - 1 ) ^ i d _ i < 0 , \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\sigma / 2 } f ( x ) = \\partial _ \\nu u ( x , 0 ) = - \\lim _ { y \\rightarrow 0 } y ^ { 1 - \\sigma } u _ y ( x , y ) , \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} 0 . 3 2 4 7 = ( s ) \\int _ { 1 } ^ { 2 } \\frac { 1 } { x ^ 4 } d \\mu \\leq m i n \\{ 0 . 4 8 0 2 , 2 - 1 \\} = 0 . 4 8 0 2 . \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\frac { \\Gamma ( m + 1 ) } { m ^ { 1 / 2 } \\Gamma ( m + 1 / 2 ) } = 1 + \\frac { 1 } { 8 m } + O ( m ^ { - 2 } ) , \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} A _ { 1 1 } ( N ) = \\begin{bmatrix} N _ { n } \\\\ * & \\ddots \\\\ * & * & N _ { n } \\end{bmatrix} , \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\psi _ { n , c } ( 1 ) \\geq \\left ( \\chi _ n ( c ) \\right ) ^ { 1 / 4 } \\sqrt { \\frac { \\pi } { 2 \\mathbf K ( \\sqrt { q _ n } ) } } \\left ( 1 - \\frac { 3 } { ( 1 - q _ n ) \\sqrt { \\chi _ n ( c ) } } \\right ) , q _ n = \\frac { c ^ 2 } { \\chi _ n ( c ) } , \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} \\begin{array} { l l l } I ( U _ * ) & = I ( u _ 0 ) + \\displaystyle \\frac { 1 } { 2 } \\int _ \\Omega | D V | ^ 2 - \\frac { k } { 2 ^ * } \\int _ S V ^ { 2 ^ * } + k \\int _ S V ^ { 2 ^ * - 1 } u _ 0 \\\\ & \\quad - \\displaystyle \\int _ { \\Omega } u _ 0 ^ { 2 ^ * - 1 } V + k \\int _ { \\Omega _ 1 \\setminus S } u _ 0 ^ { 2 ^ * - 1 } V + O ( \\frac { k } { \\lambda ^ { \\frac { N - 2 } { 2 } ( 1 + \\delta ) } } ) . \\end{array} \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} i i ) B ( \\tilde { J } U , \\tilde { J } V ) = p B ( U , \\tilde { J } V ) + q B ( U , V ) . \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} \\sum _ { k \\in J } b _ { k i } \\lambda _ { k j } = d _ { i } \\delta _ { i j } \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} \\Omega _ { a } : = \\left \\{ \\omega \\in \\Omega \\ , \\Big | \\ , \\forall n \\in \\mathbb N \\textrm { a n d } \\forall \\alpha \\in \\Omega , \\ , \\exists k _ j \\ , : \\ , f ^ n _ { T ^ { k _ j } \\omega } \\to f ^ n _ \\alpha \\right \\} \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} \\xi = r _ { d } ^ { N - 1 } \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} & - \\\\ & \\phantom { } = 2 \\pi w ( P ) - j \\left ( \\varepsilon + v \\delta \\right ) \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\ell ' = b _ 2 c _ 2 + b _ 1 c > b _ 1 c \\geq b ' c ' + b ' _ 1 c ' > b _ 2 ' c _ 2 ' + b ' _ 1 c ' = \\ell ' . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} ( 1 + v ^ T A ^ { - 1 } u ) & = \\left ( 1 - \\frac { r _ 2 } { r } - \\frac { 1 } { r } - \\frac { r _ 4 } { r } - \\ldots - \\frac { 1 } { r } \\right ) \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} B _ \\lambda ( s ) = i ^ { n k } \\bigg ( \\prod _ { j = 1 } ^ n \\beta ( \\ell _ j , s ) \\bigg ) C _ k ( s ) , \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} h _ { k } = \\max _ { \\vec x \\in \\mathbb { R } ^ { n l } \\setminus K } \\frac { F ^ L ( \\vec x ) } { G ^ L ( \\vec x ) } , \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} y \\cdot b y ^ { \\alpha } & = y \\cdot b ( b ^ { - 1 } \\cdot b y ^ { \\alpha } ) & \\\\ y ^ { \\alpha } & = b ^ { - 1 } \\cdot b y ^ { \\alpha } \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} [ \\underline { V } ] : = \\underline { \\chi _ q ( V ) } \\in \\mathcal { Y } _ t . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\frac { 1 } { d _ j ( A ) } = \\underset { { \\mathcal { M } \\subset \\mathbb { C } ^ { 2 n } } \\atop { \\dim \\mathcal { M } = j } } { \\max } \\ , \\ , \\ , \\underset { { x \\in \\mathcal { M } } \\atop { \\langle x , A x \\rangle = 1 } } { \\min } \\ , \\ , \\ , \\langle x , i J x \\rangle , \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } \\abs { \\Delta L _ n ( t ) } ^ 2 = \\sup _ { t \\in [ 0 , T ] } \\abs { \\Delta Y _ n ( t ) } ^ 2 \\le 4 \\sup _ { t \\in [ 0 , T ] } \\abs { Y _ n ( t ) } ^ 2 , \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} a ( \\mathfrak f _ { \\xi , N } ) = 2 ^ { - \\nu ( N ) } \\mathfrak f _ { \\xi , N } ^ { k - 1 / 2 } \\prod _ { p \\mid N } \\Psi _ p ( \\xi ; \\alpha _ p ) . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} \\log _ p | G _ n | = t p ^ { n - 2 } + 1 , \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal { I } } \\Phi _ { i , T } ( X ) = I _ 2 , ~ ~ ~ \\sum _ { i \\in \\mathcal { I } } \\partial _ { x _ j } \\Phi _ { i , T } ( X ) = \\mathbf { 0 } _ { 2 \\times 2 } , ~ ~ ~ \\sum _ { i \\in \\mathcal { I } } \\partial _ { x _ j x _ k } \\Phi _ { i , T } ( X ) = \\mathbf { 0 } _ { 2 \\times 2 } , ~ ~ j , k = 1 , 2 . \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} \\gamma : = \\frac { \\ell } { \\gcd \\{ \\ell , n , m \\} } . \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} & P ( A _ { i } ^ { ( k ) } ) ( 1 - P ( A _ { i } ^ { ( k ) } ) ) P ( A _ { j } ^ { ( l ) } \\mid A _ { i } ^ { ( k ) } ) - \\sum _ { j \\neq m \\leqslant m _ { k } } P ( A _ { i } ^ { ( k ) } ) P ( A _ { m } ^ { ( k ) } ) P ( A _ { j } ^ { ( l ) } \\mid A _ { m } ^ { ( k ) } ) \\\\ & = P ( A _ { j } ^ { ( l ) } \\cap A _ { i } ^ { ( k ) } ) - P ( A _ { i } ^ { ( k ) } ) \\sum _ { 1 \\leqslant m \\leqslant m _ { k } } P ( A _ { j } ^ { ( l ) } \\cap A _ { m } ^ { ( k ) } ) \\\\ & = \\mathrm { C o v } \\left ( \\mathbb { G } ( A _ { i } ^ { ( k ) } ) , \\mathbb { G } ( A _ { j } ^ { ( l ) } ) \\right ) . \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} V _ n = \\{ \\zeta _ 1 , \\zeta _ 2 , \\cdots , \\zeta _ { \\Phi ( n ) } \\} , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ { j } f _ j ( \\rho ) \\leq \\epsilon \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ j f _ j ( R ) + \\epsilon ( R - \\rho ) ^ { k + \\alpha } [ \\nabla ^ k \\sigma ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( p , R ) ) } + C _ \\epsilon \\| \\sigma \\| _ { L ^ \\infty ( B ^ { g _ P } ( p , R ) ) } . \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} u _ { \\lambda } \\left ( x , t \\right ) = \\lambda ^ { a _ 1 } u \\left ( \\lambda ^ { a _ 2 } x , \\lambda t \\right ) \\mbox { a n d } U _ { \\lambda } \\left ( x , t \\right ) = \\lambda ^ { a _ 1 } U \\left ( \\lambda ^ { a _ 2 } x , \\lambda t \\right ) \\left ( \\lambda > 0 \\right ) \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} I ( \\vec x ) & = \\sum _ { i < j } w _ { i j } | x _ i - x _ j | \\\\ & = \\sum _ { i < j } w _ { i j } ( | x _ i | + | x _ j | + | | x _ i | - | x _ j | | - | x _ i + x _ j | ) \\\\ & = \\sum _ { i = 1 } ^ n d _ i | x _ i | + \\sum _ { i < j } w _ { i j } | | x _ i | - | x _ j | | - \\sum _ { i < j } w _ { i j } | x _ i + x _ j | \\\\ & = \\| \\vec x \\| + \\hat { I } ( \\vec x ) - I ^ + ( \\vec x ) , \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} \\Gamma _ \\R ( s ) = \\pi ^ { - s / 2 } \\Gamma \\Big ( \\frac s 2 \\Big ) , \\Gamma _ \\C ( s ) = 2 ( 2 \\pi ) ^ { - s } \\Gamma ( s ) . \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} x ^ { q ^ 2 + q - 1 } + A x ^ { q ^ 3 - q ^ 2 + q } + B x = a . \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} Q ( \\rho ) = \\exp \\left ( - \\sum _ { j < 0 } v _ { j } L _ { j } \\right ) v _ { 0 } ^ { - L _ { 0 } } \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} I ( q \\ , | \\ , a , \\tau ) = \\log \\frac { G ( 1 + a + \\tau \\ , | \\ , \\tau ) } { G ( 1 - q + a + \\tau \\ , | \\ , \\tau ) } - q \\log \\Bigl [ \\Gamma \\bigl ( 1 + \\frac { a } { \\tau } \\bigr ) \\Bigr ] + \\frac { ( q ^ 2 - q ) } { 2 \\tau } \\psi \\bigl ( 1 + \\frac { a } { \\tau } \\bigr ) , \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} D _ r ( n ) = r D _ { r - 1 } ( n - 1 ) + ( n - 1 ) D _ r ( n - 2 ) + ( n + r - 1 ) D _ r ( n - 1 ) . \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} E ^ { 2 } _ { p , q } = H _ { p } \\big ( \\pi _ { q } ( E _ { \\bullet } \\big ) \\Rightarrow \\pi _ { p + q } \\big ( | E _ { \\bullet } | \\big ) , \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} \\begin{array} { c c c c } A _ { 0 } \\xi _ { k } = - c _ { k + 1 } \\xi _ { k + 2 } & A _ { 1 } \\xi _ { k } = d _ { k } \\xi _ { k } & A _ { 2 } \\xi _ { k } = - q c _ { k } \\xi _ { k - 2 } & \\\\ \\end{array} \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\vert e \\vert & = \\vert \\pi _ { P _ x } ^ \\perp ( e ) \\vert \\\\ & \\leq \\sphericalangle \\left ( P _ x , P _ { F ( z ) } \\right ) \\vert e \\vert + \\vert \\pi _ { P _ { F ( z ) } } ^ \\perp ( e ) \\vert \\\\ & \\leq C _ 2 ( m ) \\tilde \\delta _ { x , k } \\vert e \\vert + \\vert \\pi _ { P _ { F ( z ) } } ^ \\perp ( e ) \\vert . \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} M _ { n } ( 0 ; \\beta , c ) = \\dfrac { ( \\beta ) _ { n } c ^ n } { ( c - 1 ) ^ n } . \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} ( \\omega _ N \\otimes \\omega _ N ) ( x , y ) = \\sum _ { k , l \\in \\Lambda _ N } \\ < \\omega , \\tilde e _ k \\ > \\ < \\omega , \\tilde e _ l \\ > \\tilde e _ k ( x ) \\tilde e _ l ( y ) . \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\frac { n } { m \\ell _ 1 \\ell _ 3 v w } \\ll \\frac { N } { M L _ 1 L _ 3 V W } \\ll \\frac { N } { Q } \\ll \\frac { 1 } { Q ^ { \\frac { 1 } { 2 } } } = o ( 1 ) , \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} \\widetilde { c } _ { j i } ( r ) = 0 \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} \\lim _ { t \\to \\pm \\infty } \\left \\| u _ { \\mu } ( t , \\cdot ) - e ^ { i t ( \\Delta ^ 2 - \\kappa \\Delta ) } u _ { \\mu , 0 } ^ \\pm ( \\cdot ) \\right \\| _ { H ^ 2 } = 0 . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d \\sum _ { a _ i = 0 } ^ { p ^ e - 1 } ( \\xi _ { g , e , i } ) ^ { a _ i } = \\eta _ { g , c } p ^ { c e } \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} \\nabla _ { a + 1 } ^ { - \\nu } \\nabla { U ( t ) } = \\nabla \\nabla _ { a } ^ { - \\nu } U ( t ) - \\frac { ( t - a + 1 ) ^ { \\overline { \\nu - 1 } } } { \\Gamma ( \\nu ) } U ( a ) \\quad ( \\nu > 0 ) . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} b y \\cdot ( b \\backslash ( b z \\cdot y ^ { \\alpha } ) ) = ( b y \\cdot z ) y ^ { \\alpha } \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\Phi _ n ^ { \\min } ( p ^ e , \\chi ; m ) = \\Phi _ { m , n } ( \\chi ) = \\begin{cases} \\overline { \\chi ( m ) } + \\chi ( n m ) & s = e , \\\\ - 1 & p \\mid m , \\ : e = 1 s = 0 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\widetilde { \\Psi } ( 1 - s ; \\rho ( w _ { n , m } ) \\widetilde { W } , \\widetilde { W ^ \\prime } ) = \\gamma ( s , \\pi \\times \\pi ^ \\prime , \\psi ) \\Psi ( s ; W , W ^ \\prime ) , \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} U _ { 0 , 1 } & \\le \\ ( \\# \\Psi ( N , x ) \\sum _ { n \\le N } \\tau _ \\nu ^ { 2 } ( n ) \\ ) ^ { 1 / 2 } \\\\ & \\ll N \\exp { \\ ( - \\ , \\frac { \\log { N } } { 4 \\log { x } } \\ ) } ( \\log { N } ) ^ { ( \\nu ^ 2 - 1 ) / 2 } . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } w _ R ^ { \\prime \\prime } + { ( n - 1 ) ( 1 - R ) \\over t ( 1 - R ) + 2 R - 1 } w _ R ^ { \\prime } + ( 1 - R ) ^ 2 w _ R ^ { p } ( t ) = 0 \\ \\hbox { i n } \\ ( 1 , 2 ) , \\\\ w _ R ( 1 ) = w _ R ( 2 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\partial _ t u _ t = \\Delta u _ t + \\mathrm { d i v } ( F ( u _ t ) ) + \\beta _ j \\nabla u _ t \\dot Z ^ j \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\mu ( P , a ) = a . \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} ( f _ 1 \\sqcup f _ 2 ) _ * ( t d ( T _ { g _ 1 \\sqcup g _ 2 } ) \\cap ( g _ 1 \\sqcup g _ 2 ) ^ * ) = ( f _ 1 ) _ * ( t d ( T _ { g _ 1 } ) \\cap ( g _ 1 ) ^ * ) + ( f _ 2 ) _ * ( t d ( T _ { g _ 2 } ) \\cap ( g _ 2 ) ^ * ) . \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} \\frac { 1 } { \\varphi ( q ) } \\sum _ { x \\in A _ q } e ^ { 2 \\pi i a x / q } = \\frac { \\mu ( q / \\gcd ( a , q ) ) } { \\varphi ( q / \\gcd ( a , q ) ) } , \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} I ^ A { } _ B : = \\left ( \\begin{array} { c } v ^ a \\\\ \\mu ^ a { } _ { b } \\ \\ \\mid \\ \\ \\varphi \\\\ \\rho _ a \\end{array} \\right ) = v ^ c { \\mathbb Y } _ c { } ^ { A } { } _ { B } + \\mu ^ d { } _ { c } { \\mathbb Z } _ d { } ^ { c } { } ^ A { } _ { B } + \\varphi { \\mathbb W } ^ { A } { } _ { B } + \\rho _ c { \\mathbb X } ^ { c } { } ^ A { } _ { B } , \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} ( t _ 1 , \\dots , t _ n ) . F = t _ 1 ^ { \\mu _ 1 } \\cdots t _ n ^ { \\mu _ n } F . \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{gather*} h ^ 1 m _ { \\langle 0 \\rangle } \\otimes h ^ 2 m _ { \\langle 1 \\rangle } = \\big ( h ^ 2 m \\big ) _ { \\langle 0 \\rangle } \\otimes \\big ( h ^ 2 m \\big ) _ { \\langle 1 \\rangle } S ^ 2 \\big ( h ^ 1 \\big ) . \\end{gather*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\Delta ( l ) = \\sum _ { \\substack { 0 \\le l _ 1 \\le \\frac { l } { 2 } \\\\ 0 \\le l _ 2 \\le \\frac { m - 2 l } { 2 } } } \\Delta ( l , l _ 1 , l _ 2 ) . \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\omega _ { \\log } ( a _ { I _ 1 } \\tau ^ { ( r + k ) I _ 1 } , \\dots , a _ { I _ k } \\tau ^ { ( r + k ) I _ k } ) = \\omega _ { \\log , I _ 1 , \\ldots , I _ k } ( a _ { I _ 1 } , \\ldots , a _ { I _ k } ) \\cdot ( p \\circ { \\pi } _ { 0 , k } ) ^ * \\tau ^ { r ( I _ 1 + \\cdots + I _ k ) } . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} Q _ i & = \\prod _ { a \\in \\mathbb { C } ^ { \\times } } ( 1 - z a ) ^ { q _ { i , a } } & R _ i & = \\prod _ { a \\in \\mathbb { C } ^ { \\times } } ( 1 - z a ) ^ { r _ { i , a } } . \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { M , N } ( a , b ) ^ { k } \\bigr ] & = \\exp \\Bigl ( - \\sum \\limits _ { l = 0 } ^ { k - 1 } \\bigl ( \\mathcal { S } _ N \\log \\Gamma _ { M - 1 } \\bigr ) ( l \\ , | \\ , \\hat { a } _ i , b ) \\Bigr ) , \\\\ { \\bf E } \\bigl [ \\beta _ { M , N } ( a , b ) ^ { - k } \\bigr ] & = \\exp \\Bigl ( \\sum \\limits _ { l = 0 } ^ { k - 1 } \\bigl ( \\mathcal { S } _ N \\log \\Gamma _ { M - 1 } \\bigr ) ( - ( l + 1 ) \\ , | \\ , \\hat { a } _ i , b ) \\Bigr ) , \\ ; k < b _ 0 . \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{gather*} \\mathcal R | _ { M '' } = U + \\sum _ { i \\ne 1 } ^ n V _ i { \\ss } _ { 1 i } \\ , , \\mathcal R ^ + | _ { M '' } = U ^ + + \\sum _ { i \\ne 1 } ^ n V _ i ^ + { \\ss } _ { 1 i } ^ + \\ , , \\quad \\\\ U = \\prod _ { l \\ne 1 } ^ n a _ { 1 l } \\ , , V _ i = b _ { 1 i } \\prod _ { l \\ne 1 , i } a _ { i l } \\ , , U ^ + = \\prod _ { l \\ne 1 } ^ n a _ { 1 l } ^ + \\ , , V _ i ^ + = b _ { 1 i } ^ + \\prod _ { l \\ne 1 , i } a _ { l i } \\ , . \\end{gather*}"}
-{"id": "2154.png", "formula": "\\begin{align*} N ( w ) = N ( v ) \\sqcup v ( N ( u ) ) . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} z ^ 2 _ { m i n } > 3 ( X _ u - x _ i ) ^ 2 + 3 ( Y _ u - y _ i ) ^ 2 , \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} G ( x , y ) ~ = ~ \\sum _ { t = 0 } ^ \\infty \\P _ x ( X ( t ) = y ) ~ = ~ \\sum _ { t = 0 } ^ \\infty \\P _ x ( X ( t ) = y , \\ ; t < \\tau _ \\vartheta ) , x , y \\in E , \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} \\mathcal B ( \\Theta ) = \\sigma \\ ( A , \\ , R _ \\alpha , \\ , Z : \\alpha \\in \\ker A \\ ) \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) = \\frac { \\left ( - 1 \\right ) ^ { i } } { \\left ( r ! \\right ) ^ { p } \\prod _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\right ) ^ { p _ { s } } } \\sum _ { k = 0 } ^ { r p + \\sigma } \\binom { r p + \\sigma - k } { r p + \\sigma - i } \\left ( - 1 \\right ) ^ { k } k ! S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} \\| P _ \\pi u _ B \\| _ { L ^ 2 ( \\nu ) } \\geq \\big \\langle u _ B , 1 \\big \\rangle _ { L ^ 2 ( \\nu ) } = \\sqrt { \\beta } . \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\Sigma = \\R \\cup i \\R \\cup \\Sigma _ \\infty , \\Sigma _ \\infty = \\{ | z | = R \\} \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} ( q _ 0 + q _ 2 + . . . + q _ { m - 2 } + q _ m ) ( x ) & = - \\frac { i } { 2 \\pi } L ^ { \\frac { n + 3 } { 2 } } ( f _ 2 - f _ 1 ) ( x ) + \\frac { i ^ { m + 1 } } { 8 \\pi ^ 2 m ! } L ^ { \\frac { n - m + 1 } { 2 } } ( f _ 2 - f _ 1 ) ( y ) | x - y | ^ { m - 1 } d y . \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} f _ 2 \\circ u _ 1 & = u _ 4 \\circ f _ 1 \\circ e _ 1 \\\\ g _ 2 \\circ u _ 1 & = u _ 4 \\circ g _ 1 \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} n = q _ 1 q _ 2 \\cdots q _ r = p _ 1 ^ { n _ 1 } p _ 2 ^ { n _ 2 } \\cdots p _ s ^ { n _ s } , \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} X ( t ) = \\exp \\Big ( W ( t ) - \\frac { t } { 2 } \\Big ) \\Big [ 1 & - \\int _ 0 ^ t \\int _ 0 ^ r \\exp \\big ( \\frac { r } { 2 } - W ( r ) \\big ) g ( u ) d u d r \\\\ & + \\int _ 0 ^ t \\int _ 0 ^ r \\exp \\big ( \\frac { r } { 2 } - W ( r ) \\big ) g ( u ) d W ( u ) d r \\Big ] . \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} \\left ( \\int _ G g ( \\theta _ { e _ 1 , e _ 2 } ) \\ , d g \\right ) ( e ) ( x ) & : = \\int _ M \\left ( \\int _ G \\theta _ { g ( e _ 1 ) ( x ) , g ( e _ 2 ) ( y ) } \\ , d g \\right ) e ( y ) \\ , d \\mu \\\\ & = \\int _ M \\int _ G g ( e _ 1 ) ( x ) \\langle g B ^ s e _ 2 ( y ) , B ^ s e ( y ) \\rangle _ { E _ y } \\ , d g \\ , d \\mu . \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ s _ { R } + j s _ { I } ~ | ~ s _ { R } , s _ { I } \\in \\{ \\pm 1 , \\pm 3 , \\ldots , \\pm ( 2 L - 1 ) \\} \\} , \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\Psi _ y \\circ \\Psi _ x ^ { - 1 } ( z ) & = \\Psi _ y ( \\rho _ x \\rho _ z , \\theta _ x + \\theta _ z , \\xi _ x + \\rho _ x \\xi _ z ) \\\\ & = ( \\frac { \\rho _ x } { \\rho _ y } \\rho _ z , \\theta _ z + \\theta _ x - \\theta _ y , \\frac { \\rho _ x \\xi _ z + \\xi _ x - \\xi _ y } { \\rho _ y } ) . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} \\nabla \\bar { J _ 3 } ( U ) = & 2 A U { \\rm s y m } ( \\nabla _ { A _ r } \\bar { J } ( U ^ T A U , U ^ T B , C U ) ) \\\\ & + B ( \\nabla _ { B _ r } \\bar { J } ( U ^ T A U , U ^ T B , C U ) ) ^ T \\\\ & + C ^ T \\nabla _ { C _ r } \\bar { J } ( U ^ T A U , U ^ T B , C U ) . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\begin{pmatrix} X \\\\ P & Y \\\\ Q & R & X \\end{pmatrix} \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} A _ { n } & = \\left \\{ r _ { n } ^ { ( N ) } \\leqslant ( d _ { 0 } + \\varepsilon ) v _ { n } \\right \\} , B _ { n } = \\left \\{ \\left \\Vert \\mathbb { G } _ { n } ^ { ( N ) } \\right \\Vert _ { \\mathcal { F } } \\leqslant a _ { n } \\right \\} , \\\\ C _ { n , k } & = \\left \\{ \\theta ^ { k - 1 } a _ { n } < \\left \\Vert \\mathbb { G } _ { n } ^ { ( N ) } \\right \\Vert _ { \\mathcal { F } } \\leqslant \\theta ^ { k } a _ { n } \\right \\} . \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , 0 \\right ) = \\left ( r ! \\binom { b } { r } \\right ) ^ { p } \\prod \\limits _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\binom { \\beta _ { s } } { r _ { s } } \\right ) ^ { p _ { s } } , \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} 0 = F ( 0 ) \\ge F ( P ) - P : D F ( P ) \\ge \\nu _ 1 | P | - \\nu _ 2 - P : D F ( P ) , \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} D ( m , n ) = \\frac { \\max ( n \\psi ( m ) , m \\psi ( n ) ) } { ( m , n ) } . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} X _ t = Y _ t ^ 2 = \\Biggl ( \\sqrt { X _ 0 } + \\frac { 1 } { 2 } \\int _ 0 ^ t \\biggl ( \\frac { k } { Y _ s } - a Y _ s \\biggr ) d s + \\frac { \\sigma } { 2 } B _ t ^ H \\Biggr ) ^ 2 . \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\mathcal W _ { \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) , B } = \\begin{cases} 2 ^ { - \\nu ( N ) } c ( \\mathfrak d _ { \\xi } ) \\mathfrak f _ { \\xi } ^ { k - 1 / 2 } \\zeta _ { \\Q } ( 2 ) ^ { - 1 } \\prod _ v \\mathcal W _ { B , v } & \\xi > 0 , \\\\ 0 & \\xi \\leq 0 , \\end{cases} \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} h \\cdot ( x \\otimes y ) = \\sum _ h h _ 1 x \\otimes h _ 2 y . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} e _ n = \\left \\{ \\begin{array} { c l } \\left ( \\frac { f ( n ) } { p } \\right ) & ( f ( n ) , p ) = 1 \\\\ + 1 & p \\mid f ( n ) , \\end{array} \\right . \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} \\left ( Y + ( x _ 0 ^ 3 + y _ 0 ^ 3 - z _ 0 ^ 3 ) \\right ) \\left ( Y - ( x _ 0 ^ 3 + y _ 0 ^ 3 - z _ 0 ^ 3 ) \\right ) = - 4 x _ 0 ^ 3 y _ 0 ^ 3 , \\\\ \\left ( Y + ( x _ 0 ^ 3 - y _ 0 ^ 3 + z _ 0 ^ 3 ) \\right ) \\left ( Y - ( x _ 0 ^ 3 - y _ 0 ^ 3 + z _ 0 ^ 3 ) \\right ) = - 4 x _ 0 ^ 3 z _ 0 ^ 3 , \\\\ \\left ( Y + ( x _ 1 ^ 3 + y _ 1 ^ 3 - z _ 1 ^ 3 ) \\right ) \\left ( Y - ( x _ 1 ^ 3 + y _ 1 ^ 3 - z _ 1 ^ 3 ) \\right ) = - 4 x _ 1 ^ 3 y _ 1 ^ 3 , \\\\ \\left ( Y + ( - x _ 2 ^ 3 + y _ 2 ^ 3 + z _ 2 ^ 3 ) \\right ) \\left ( Y - ( - x _ 2 ^ 3 + y _ 2 ^ 3 + z _ 2 ^ 3 ) \\right ) = - 4 y _ 2 ^ 3 z _ 2 ^ 3 . \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} j ( r ) = j _ n ( r ) = \\int _ 0 ^ \\infty ( 4 \\pi t ) ^ { - n / 2 } e ^ { - \\frac { r ^ 2 } { 4 t } } \\mu ( d t ) , \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} | I _ { 1 , f , \\psi , L } ^ { + } ( q ) | \\geqslant \\tfrac 1 2 | I _ { 1 , f , \\psi , L } ( q ) | = \\tfrac 1 2 c _ f \\eta | \\hat { \\psi } ( 1 ) | \\log q + O _ { f , \\psi , \\eta } ( 1 ) . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} d s ^ { 2 } = 2 T d t ^ { 2 } = \\sum \\limits _ { i } m _ { i } ( d x _ { i } ^ { 2 } + d y _ { i } ^ { 2 } + d z _ { i } ^ { 2 } ) \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\psi _ { \\varepsilon , \\zeta } ^ i ( x ) = \\psi _ \\zeta ^ i ( \\varepsilon ^ { - 1 } x ) . \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} f ~ ~ v \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ' ( v ) = 0 . \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} c - d + 2 ( e - f ) = 0 , \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} \\max _ { 0 \\le i _ 1 < i _ 2 < \\ldots < i _ { | P | - 1 } \\le n } \\sum _ { 1 \\le l < j \\le | P | - 1 } \\binom { n } { i _ j } \\binom { i _ j } { i _ l } . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\widetilde { \\zeta } ( 2 ) = \\frac { \\pi ^ 2 } { 8 } . \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} \\mathcal G _ 0 ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * g ^ * : \\mathcal G _ 0 ( Y ) \\to \\mathcal G _ 0 ( X ) , \\\\ \\mathcal H ^ { T o d d } _ * ( X \\xleftarrow f M \\xrightarrow g Y ) & : = f _ * \\bigl ( t d ( T _ g ) \\cap g ^ * \\bigr ) : \\mathcal H ^ { T o d d } _ * ( Y ) \\to \\mathcal H ^ { T o d d } _ * ( X ) . \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} { m + n \\choose m } ( \\hat \\omega _ \\infty ^ \\bullet ) ^ m = c _ \\infty e ^ { G ( z _ \\infty ) } \\omega _ { \\mathbb { C } ^ m } ^ m . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\widetilde { C } ( z ) _ { j i } = \\sum _ { r \\in \\mathbb { Z } } \\widetilde { c } _ { j i } ( r ) z ^ r \\in \\mathbb { Z } ( ( z ^ { - 1 } ) ) . \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\mathcal I _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { L ( 1 , \\pi _ p , \\mathrm { a d } ) L ( 1 , \\tau _ p , \\mathrm { a d } ) } { \\zeta _ { \\Q _ p } ( 2 ) L ( 1 / 2 , \\pi _ p \\times \\mathrm { a d } \\tau _ p ) } \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) . \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} \\frac { d } { d t } K _ u & = B _ u K _ u - K _ u B _ u , \\\\ \\frac { d } { d t } H _ u & = B _ u H _ u - H _ u B _ u + i \\bar { J } ( u \\vert \\cdot ) u , \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} \\lbrace { \\tilde z _ 1 } , { \\tilde z _ 3 } \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace { \\tilde z _ 2 } , { \\tilde z _ 4 } \\rbrace = 1 . \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { E ^ { \\operatorname { M F } } _ { N } } { N } = e _ { \\operatorname { H } } . \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} u ( \\textbf { z } , t ) = \\bigtriangleup u ( \\textbf { z } , t ) = 0 , ( \\textbf { z } , t ) \\in \\partial \\Omega \\times \\bar { J } , \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} S _ I ^ { m _ I } = \\prod _ { i \\in I } S _ i ^ { m _ i } \\in \\mathcal { B } ( \\ell _ 2 ( \\N ^ I ) ) . \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} \\varphi _ 1 ^ { ( d + 1 ) } ( t k ) ( t k ) ^ { d + 1 } - \\varphi _ 1 ^ { ( d + 1 ) } ( t k ' ) ( t k ' ) ^ { d + 1 } = \\big ( \\varphi _ 1 ^ { ( d + 2 ) } ( x ) x ^ { d + 1 } + ( d + 1 ) \\varphi _ 1 ^ { ( d + 1 ) } ( x ) x ^ d \\big ) ( k - k ' ) t , \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} \\sup _ { k : | k - n p | \\leq \\varphi ( n ) } \\left | \\frac { P \\left ( V _ n = k \\right ) } { \\frac { 1 } { \\sqrt { 2 \\ , \\pi n p ( 1 - p ) } } e ^ { - ( k - n p ) ^ 2 / ( 2 n p ( 1 - p ) ) } } - 1 \\right | \\longrightarrow 0 \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} ( S ^ { ( 2 ) } _ { 0 , 2 } ) ^ 3 = \\Sigma ^ 2 ( S ^ { ( 2 ) } _ { 0 , 1 } ) ^ 4 = \\Sigma ^ 2 . \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\phi _ { \\mathbf h , \\infty } ( x ) = ( x _ 2 + \\sqrt { - 1 } x _ 1 + \\sqrt { - 1 } x _ 5 - x _ 4 ) ^ { k + 1 } e ^ { - \\pi ( x _ 1 ^ 2 + x _ 2 ^ 2 + 2 x _ 3 ^ 2 + x _ 4 ^ 2 + x _ 5 ^ 2 ) } , \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\int _ { Z _ G ( g ) } c ^ g ( h m ) \\ , d h = 1 , \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} [ \\tau ( - w ) ] ^ \\alpha [ - w - \\tau ( - w ) ] ^ \\beta = \\sum _ { | \\alpha | + | \\beta | \\le | \\delta | \\le N \\cdot ( | \\alpha | + | \\beta | ) } E _ \\delta ( \\tau , w ) w ^ \\delta \\ , , \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} j = \\begin{cases} i & \\ i \\neq n - 1 , \\\\ n - 1 , n & \\ i = n - 1 . \\end{cases} \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} \\left ( \\psi _ { n , c } ( 1 ) \\right ) ^ 2 \\geq \\frac { \\pi } { 3 2 } \\frac { \\sqrt { \\chi _ n ( c ) } } { \\mathbf K ( \\sqrt { q _ n } ) } = \\frac { \\pi } { 3 2 } \\frac { c } { \\sqrt { q _ n } \\mathbf K ( \\sqrt { q _ n } ) } . \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} \\epsilon _ { u _ i } ' & = M _ { k ' } \\epsilon _ { u _ i } M _ i ^ { - 1 } = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Z ^ 2 } x \\mu ( x ) = 0 . \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} f ( u \\ast v ) = ( f u ) \\ast ( f v ) , \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} C = \\left ( \\begin{matrix} q & 0 & 0 & 0 \\cr 0 & a & 0 & 0 \\cr 0 & 0 & 0 & b \\cr 0 & 0 & - 1 & 0 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} V _ d g ( z ) : = d \\ , g ( d z ) , U _ d g ( z ) : = \\frac { 1 } { d } \\sum _ { j = 0 } ^ { d - 1 } g \\left ( \\frac { z + j } { d } \\right ) . \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} h \\nu _ { \\gamma } \\varpi _ p = \\left ( \\begin{array} { c c } p ^ { - 1 } t ^ { - 1 } \\det ( h ) & y \\\\ 0 & t \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & 0 \\\\ t ^ { - 1 } p ^ { - 1 } z + \\gamma & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q \\ , | \\ , b ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\Bigl [ \\bigl ( \\mathcal { S } _ N \\ , \\exp ( - x t ) \\bigr ) ( q \\ , | \\ , b ) - \\bigl ( \\mathcal { S } _ N \\ , \\exp ( - x t ) \\bigr ) ( 0 \\ , | \\ , b ) \\Bigr ] f ( t ) d t / t ^ { M + 1 } \\Bigr ) . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , q _ { 1 } + q _ { 2 } } \\left ( p _ { 2 } , l \\right ) = \\sum _ { m = 0 } ^ { p _ { 2 } + q _ { 2 } } S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , q _ { 2 } } \\left ( p _ { 2 } , m \\right ) S _ { a _ { 2 } , a _ { 2 } m + b _ { 2 } } \\left ( q _ { 1 } , l - m \\right ) . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} & ~ w ( e \\cup f ) ^ { p - ( r _ 1 + r _ 2 ) } \\cdot \\prod _ { v \\in e \\ , \\cup f } B ( v , e \\cup f ) \\\\ = & ~ \\big ( w _ 1 ( e ) w _ 2 ( f ) \\big ) ^ { p - ( r _ 1 + r _ 2 ) } \\cdot \\prod _ { v \\in e } B ( v , e \\cup f ) \\prod _ { u \\in f } B ( u , e \\cup f ) \\\\ = & ~ w _ 1 ( e ) ^ { p - r _ 1 } \\prod _ { v \\in e } B _ 1 ( v , e ) \\cdot w _ 2 ( f ) ^ { p - r _ 2 } \\prod _ { u \\in f } B _ 2 ( u , f ) \\\\ = & ~ \\alpha _ 1 \\alpha _ 2 . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| y ^ { k + 1 } - x ^ { k } \\| = 0 . \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} P _ \\ast ( T _ { \\tilde \\Psi } ) = S _ a + b K \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\limsup _ { t \\nearrow T _ { \\max } \\left ( t _ 0 , \\overline { u } _ 0 , \\underline { u } _ 0 , \\overline { v } _ 0 , \\underline { v } _ 0 \\right ) } \\left ( | \\overline { u } ( t _ 0 + t ) | + | \\underline { u } ( t _ 0 + t ) | + | \\overline { v } ( t _ 0 + t ) | + | \\underline { v } ( t _ 0 + t ) | \\right ) = \\infty . \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\eta \\Pi = \\begin{pmatrix} P & P & P & \\cdots & P & 1 \\\\ P & L ' _ { 1 , 1 } & L ' _ { 1 , 2 } & \\cdots & L ' _ { 1 , n - a } & 1 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ P & L ' _ { a , 1 } & L ' _ { a , 2 } & \\cdots & L ' _ { a , n - a } & 1 \\\\ 1 & 1 & 1 & \\cdots & 1 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} M ^ { k + 1 } ( i , j ) = \\sum _ { i ' = 1 } ^ N M ^ k ( i , i ' ) \\ , M ( i ' , j ) . \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} F _ 1 & = \\langle \\nabla \\cdot ( a - 1 ) \\cdot w , | w | ^ { q - 2 } ( - b \\cdot w ) \\rangle + \\langle ( - \\mu u - b \\cdot w + f ) , | w | ^ { q - 2 } ( - b \\cdot w ) \\rangle \\\\ & = \\langle \\nabla a \\cdot w , | w | ^ { q - 2 } ( - b \\cdot w ) \\rangle \\\\ & + c \\langle \\mathsf { f } \\cdot ( \\mathsf { f } \\cdot \\nabla w ) , | w | ^ { q - 2 } ( - b \\cdot w ) \\rangle \\\\ & + \\langle ( - \\mu u - b \\cdot w + f ) , | w | ^ { q - 2 } ( - b \\cdot w ) \\rangle . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} U f ( z ) = \\int \\limits _ { \\R } A ( z , x ) f ( x ) d x , \\quad \\forall f \\in L ^ 2 ( \\R ) , \\forall z \\in \\C , \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} T _ { m \\chi _ { [ a , b ] } } f = m ( a ) S _ { [ a , b ] } f + \\int _ a ^ b ( S _ { [ t , b ] } f ) m ' ( t ) d t . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} J ( t ) = I _ d + \\int _ 0 ^ t \\nabla _ x b \\Big ( s , \\omega , X _ x ( s ) ( \\omega ) \\Big ) J ( s ) d s + \\int _ 0 ^ t \\nabla _ x \\sigma \\Big ( s , \\omega , X _ x ( s ) ( \\omega ) \\Big ) J ( s ) d W ( s ) . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\mu ( n ) = \\begin{cases} ( - 1 ) ^ k & m _ 1 = \\ldots = m _ k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} \\tau ( n g ) = f ( n ) + \\tau ( g ) \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} B _ { 1 } & = \\{ 2 , 5 , 6 \\} , & B _ { 5 } & = \\{ 2 , 7 , 8 \\} , \\\\ B _ { 2 } & = \\{ 3 , 5 , 7 \\} , & B _ { 6 } & = \\{ 3 , 6 , 8 \\} , \\\\ B _ { 3 } & = \\{ 4 , 5 , 8 \\} , & B _ { 7 } & = \\{ 4 , 6 , 7 \\} . \\\\ B _ { 4 } & = \\{ 2 , 3 , 4 \\} , & & \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} \\mathcal { K } _ { t , \\mathcal { Q } ' _ { 2 n - 1 } } ( = K _ t ( \\mathcal { C } _ { \\mathcal { Q } ' _ { 2 n - 1 } } ) ) : = \\bigoplus _ { m \\in \\mathbb { B } ^ { \\xi ' } } \\mathbb { Z } [ t ^ { \\pm 1 / 2 } ] E _ t ^ { \\mathrm { A } } ( m ) = \\bigoplus _ { m \\in \\mathbb { B } ^ { \\xi ' } } \\mathbb { Z } [ t ^ { \\pm 1 / 2 } ] L _ t ^ { \\mathrm { A } } ( m ) \\subset \\mathcal { K } ^ { \\mathrm { A } } _ t . \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{gather*} r ( [ n , \\phi ] ) = \\alpha _ n ( \\phi ) , s ( [ n , \\phi ] ) = \\phi , \\\\ M ( [ n , \\phi ] , [ m , \\psi ] ) = [ n m , \\psi ] , I ( [ n , \\phi ] ) = [ n ^ * , \\alpha _ n ( \\phi ) ] . \\end{gather*}"}
-{"id": "9248.png", "formula": "\\begin{align*} ( x , y ) = { } ^ t x Q y x , y \\in V . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} a ( \\xi ) = \\begin{pmatrix} a _ { 1 } ( t , s ) \\\\ a _ { 2 } ( t , s ) \\end{pmatrix} : = \\begin{pmatrix} h _ { 1 } ( t - s ) & 0 \\\\ 0 & h _ { 2 } ( t - s ) \\end{pmatrix} , b : \\equiv \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\lim _ { \\mu _ 2 \\to + \\infty } \\widehat { \\delta } ( \\mathcal Z _ { \\mu _ 2 } , \\mathcal T _ 0 ) = 0 . \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} { \\rm c h } [ { M } ] \\left ( - \\frac { 1 } { \\tau } \\right ) = \\int _ { \\Omega } S _ { M , \\nu } { \\rm c h } [ { M _ { \\nu } } ] ( \\tau ) d \\nu + { \\sum _ { j \\in \\mathcal { D } } \\alpha _ { M , j } { \\rm c h } [ { M _ j } ] ( \\tau ) } , \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} \\lim \\limits _ { A \\rightarrow 0 } \\int _ { x _ 1 ^ { A } } ^ { x _ 2 ^ { A } } \\rho _ * ^ { A } d x = w _ 0 ^ { B } t , \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} \\sum _ { V \\in e } \\pm u _ e = 0 , \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\tilde { \\phi } ( m + n ) = c _ + + ( - 1 ) ^ { | m | + | n | } c _ - + \\left ( S ^ m ( S ^ * ) ^ n T ' \\right ) , \\quad \\forall m , n \\in \\N ^ N , \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} f ( z ) = \\frac { \\int Q ( z ) P ( z ) } { Q ( z ) } = \\frac { K ( z ) } { Q ( z ) } , \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\tilde { \\mathbb C } ^ 2 & = \\mathbb C ^ 2 , \\\\ \\tilde { \\mathbb C } ^ 4 & = \\Big \\{ \\vect z \\in \\mathbb C ^ 4 \\ ; : \\ ; | z _ 1 | ^ 2 + | z _ 2 | ^ 2 = | z _ 3 | ^ 2 + | z _ 4 | ^ 2 \\Big \\} , \\\\ \\tilde { \\mathbb C } ^ 8 & = \\Big \\{ \\vect z \\in \\mathbb C ^ 8 \\ ; : \\ ; \\sum _ { j = 1 } ^ 4 | z _ j | ^ 2 = \\sum _ { j = 1 } ^ 4 | z _ { j + 4 } | ^ 2 , z _ 7 \\overline z _ 1 + z _ 5 \\overline z _ 3 = z _ 8 \\overline z _ 2 + z _ 6 \\overline z _ 4 \\Big \\} . \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} v = E _ { { i _ { { s } } } , h } ^ { ( a _ s ) } E _ { { i _ { { s - 1 } } } , h } ^ { ( a _ { s - 1 } ) } \\cdots E _ { i _ 1 , h } ^ { ( a _ 1 ) } . \\mathfrak { m } _ \\lambda \\neq 0 , \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} ( a + k ) ( a ) _ k = a ( a + 1 ) _ k , \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} u ^ m _ r ( r , t ) = - \\frac { C ^ m m } { a ( m - 1 ) } \\zeta ^ m ( t ) \\eta ( t ) F ^ { \\frac 1 { m - 1 } } \\ , ; \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} \\rho _ 1 ( \\vect z ) = \\frac { 1 } { 2 } ( z _ 2 ^ 2 - z _ 1 ^ 2 ) , \\rho _ 2 ( \\vect z ) = \\frac { \\imath } { 2 } ( z _ 1 ^ 2 + z _ 2 ^ 2 ) , \\rho _ 3 ( \\vect z ) = z _ 1 z _ 2 . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} \\log ( | \\Omega _ 0 | ) & \\ge ( d / 2 ) \\log \\left ( \\frac { p } { 5 d } \\right ) , \\sum _ { j = 1 } ^ p \\mathbf 1 _ { w _ j \\ne w _ j ' } = \\| w - w ' \\| ^ 2 > d , \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} \\mu _ b ( \\alpha ) : = \\sum _ { h = 1 } ^ \\infty r _ { \\alpha } ( h ) = \\sum _ { a = 0 } ^ \\infty \\frac { 1 } { b ^ { a \\alpha } } \\sum _ { k = b ^ a } ^ { b ^ { a + 1 } - 1 } 1 = \\sum _ { a = 0 } ^ \\infty \\frac { ( b - 1 ) b ^ a } { b ^ { a \\alpha } } = \\frac { b ^ { \\alpha } ( b - 1 ) } { b ^ \\alpha - b } . \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} r _ i ( x ) = v _ { m _ i i } ( x ) ^ { 2 \\ell _ i / m _ i - 1 } \\exp \\int \\frac { \\Im ( v _ { ( m _ i - 1 ) i } ( x ) ) } { ( m _ i / 2 ) v _ { m _ i i } ( x ) } d x . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} r < \\sup _ { P _ X } \\Big ( \\min \\{ I _ P ( X ; Y ) , \\min _ z H _ Q ( Y | Z = z ) - H _ P ( Y | X ) \\} \\Big ) \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} b x ^ 3 + x + a = 0 . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} u ( x ) \\mapsto ( { \\mathcal F } { \\mathcal R } { \\mathcal F } ^ \\ast u ) ( x ) = \\frac { i } { 2 \\pi } \\int _ { \\R } \\frac { u ( y ) } { x - y - i 0 } d y \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} L _ { i , j } : = \\prod _ { \\substack { k \\geq i \\\\ l \\geq j } } X _ { k , l } . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} m ~ = ~ \\sum _ { x \\in \\Z ^ d } x \\mu ( x ) \\not = 0 m \\in { \\cal C } . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\tau _ x R _ x ^ 2 = \\frac { L R _ x ^ 2 } { 2 \\theta _ { N , N } ^ 2 } . \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} V _ 0 = \\langle e _ { \\frac { k b } { 8 } + 1 } , \\dots , e _ { \\frac { 7 k b } { 8 } } \\rangle , \\ V _ 1 = \\langle e _ { \\frac { 3 k b } { 8 } + 1 } , \\dots , e _ { \\frac { 5 k b } { 8 } } \\rangle , \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} \\widehat { K _ 2 } ( y ) = \\frac { 1 } { | y | + \\alpha } \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} \\mathcal { T S } _ 0 ^ { o p } & = \\{ B _ 1 , B _ 2 , \\ldots , B _ { n - 1 } , B _ n \\} \\\\ & = \\{ A _ { n - 1 } , A _ { n - 2 } , \\ldots , A _ 2 , A _ 1 , A _ n \\} \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} { \\tilde z } _ 1 = z _ 1 , \\ ; \\ ; \\ ; { \\tilde z } _ 2 = z _ 2 , \\ ; \\ ; \\ ; { \\tilde z } _ 3 = z _ 3 , \\ ; \\ ; \\ ; { \\tilde z } _ 4 = z _ 4 . \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\psi _ { s } \\psi _ w = \\begin{cases} \\psi _ { s w } , & \\ell ( s w ) = \\ell ( w ) + 1 , \\\\ 0 , & \\ell ( s w ) = \\ell ( w ) - 1 , \\end{cases} \\psi _ { w } \\psi _ s = \\begin{cases} \\psi _ { w s } , & \\ell ( w s ) = \\ell ( w ) + 1 , \\\\ 0 , & \\ell ( w s ) = \\ell ( w ) - 1 . \\end{cases} \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} ( \\hat u _ { 1 , 0 } , \\hat \\theta _ 0 , \\hat h _ { 1 , 0 } ) ( x , \\eta ) ~ = ~ ( u _ { 1 , 0 } , \\theta _ 0 , h _ { 1 , 0 } ) \\big ( x , \\eta ( x , y ) \\big ) . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} ( L ^ H \\gamma , \\gamma ) _ H = ( \\delta \\delta ^ * \\gamma , \\gamma ) _ H = ( \\delta ^ * \\gamma , \\delta ^ * \\gamma ) _ V \\geq 0 . \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} ( \\mathcal { S } _ N \\log S _ M ) ( q \\ , | \\ , a , b ) = ( - 1 ) ^ { N + M } ( \\mathcal { S } _ N \\log \\Gamma _ M ) ( - q \\ , | \\ , a , \\bar { b } ) - ( \\mathcal { S } _ N \\log \\Gamma _ M ) ( q \\ , | \\ , a , b ) , \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} \\dd Y _ t = ( A Y _ t + b ( Y _ t ) ) \\dd t + \\dd W _ t , \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} E _ { a / c } ( V z , s , \\chi ) = \\chi _ { \\alpha ( N , c ) } ( - 1 ) E _ { k ( a / c ) } ( z , s , \\chi ) , \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} I _ { 1 , 3 } = \\log \\sqrt { 2 \\pi } \\ x + O ( 1 ) . \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} \\lbrace { \\tilde z _ 1 } , { \\tilde z _ 3 } \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace { \\tilde z _ 2 } , { \\tilde z _ 4 } \\rbrace = 1 . \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} L ^ { } ( x ; \\phi ) & = \\sum _ { n \\leq x } \\frac { ( \\phi * \\phi ) ( n ) } { n } \\log \\frac { n } { e } + \\frac { 1 } { 2 } \\sum _ { n \\leq x } \\frac { ( \\phi * \\Lambda ) ( n ) } { n } \\\\ & + \\log \\sqrt { 2 \\pi } \\sum _ { n \\leq x } \\frac { \\phi ( n ) } { n } + { \\Theta } \\sum _ { n \\leq x } \\frac { ( \\phi * \\phi _ { - 1 } ) ( n ) } { n } \\\\ & : = I _ { 2 , 1 } + I _ { 2 , 2 } + I _ { 2 , 3 } + I _ { 2 , 4 } , \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} c : = - i \\cosh \\beta = a - i b , \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\varepsilon } ^ { \\beta } u = \\sum _ { p \\in \\mathbb { N } } \\left ( \\frac { 1 } { T } \\log \\left ( 1 + \\gamma ^ { - 1 } \\left ( T , \\beta \\right ) e ^ { \\overline { M } T \\mu _ { p } } \\right ) - \\overline { M } \\mu _ { p } \\right ) \\left \\langle u , \\phi _ { p } \\right \\rangle \\phi _ { p } . \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} ( \\rho , u ) ( x , 0 ) = ( \\rho _ \\pm , u _ \\pm ) , \\ \\ \\ \\ \\pm x > 0 , \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} & \\left | F ( x , t , | y ' | ) y ' - F ( x , t , | y | ) y \\right | = | h ( 1 ) - h ( 0 ) | = \\left | \\int _ 0 ^ 1 h ' ( \\tau ) d \\tau \\right | \\\\ & \\qquad = \\left | \\int _ 0 ^ 1 F ( x , t , | \\gamma ( \\tau ) | ) ( y ' - y ) + F _ z ( x , t , | \\gamma ( \\tau ) | ) \\frac { \\gamma ( \\tau ) ( y ' - y ) } { | \\gamma ( \\tau ) | } \\gamma ( \\tau ) d \\tau \\right | \\\\ & \\qquad \\le | y ' - y | \\int _ 0 ^ 1 F ( x , t , | \\gamma ( t ) | ) + F _ z ( x , t , | \\gamma ( \\tau ) | ) | \\gamma ( \\tau ) | d \\tau . \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} \\begin{cases} s _ { i _ 0 } \\ge p _ { i _ 0 } - q _ { i _ 0 } = \\bar { p } _ { i _ 0 } \\ge \\bar { p } _ { j _ 1 } = s ^ \\sigma _ { j _ 1 } \\ge 0 , i _ 0 \\in S ^ + ( \\vec x ^ k ) , \\\\ s _ { i _ 0 } \\le p _ { i _ 0 } + q _ { i _ 0 } = \\bar { p } _ { i _ 0 } \\le \\bar { p } _ { j _ 2 } = s ^ \\sigma _ { j _ 2 } \\le 0 , i _ 0 \\in S ^ - ( \\vec x ^ k ) , \\end{cases} \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} x _ { \\sigma ( \\iota ) } = w _ { j _ k , i _ { \\iota } } \\bigl ( 1 + \\frac { \\ell ( i _ { \\iota } ) } { \\mu _ { i _ { \\iota } } } \\bigr ) \\Biggl ( \\frac { y _ { \\sigma ( \\iota ) } } { w _ { j _ k , i _ { \\iota } } \\bigl ( 1 + \\frac { \\lambda _ { \\ell ( i _ { \\iota } ) } } { \\mu _ { i _ { \\iota } } } \\bigr ) } - \\frac { y _ { \\sigma ( \\iota _ { i ^ * _ k } ) } } { w _ { j _ k , i ^ * _ k } \\bigl ( 1 + \\frac { \\lambda _ { \\ell ( i ^ * _ k ) } } { \\mu _ { i ^ * _ k } } \\bigr ) } \\Biggr ) ; \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} h ( x ) = x ^ c L ( x ) , \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} F ( m ) = \\binom { N - 1 } { m } m ! \\sum \\limits _ { k = 0 } ^ m ( - 1 ) ^ { m - k } \\frac { ( k + 1 ) } { ( m - k ) ! } . \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} \\int _ { r = R } ^ \\infty e ^ { - r } \\chi _ i ( r ) \\ , \\dd r = \\left [ - \\frac { e ^ { - r } \\chi _ { i + 1 } ( r ) } { r } \\right ] ^ \\infty _ { r = R } = \\tfrac { 1 } { R } e ^ { - R } \\chi _ { i + 1 } ( R ) , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} C _ { f , g } : = \\frac 1 { 6 \\sqrt { 1 0 } } C _ { f , g } ^ 0 , C _ { f , g } ^ 0 : = \\frac { n _ { f , g , 3 , 1 } + 2 n _ { f , g , 2 , 2 } + n _ { f , g , 1 , 3 } } { ( n _ { f , g , 4 , 2 } + 2 n _ { f , g , 3 , 3 } + n _ { f , g , 2 , 4 } ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ \\exp \\bigl ( q \\log \\beta _ { M , N } ( b ) \\bigr ) \\Bigr ] = \\eta _ { M , N } ( q \\ , | \\ , b ) , \\ ; \\Re ( q ) > - b _ 0 . \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} ( K G ) ( \\gamma ) = \\sum _ { \\eta \\Subset \\gamma } G ( \\eta ) , \\gamma \\in \\Gamma , \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} \\mathcal { Q } _ t ( x _ { t - 1 } , 1 ) = ( 1 - q _ t ) \\sum _ { j = 1 } ^ { M _ { t } } p _ { t j } \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 1 ) + q _ t \\sum _ { j = 1 } ^ { M _ t } p _ { t j } \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 0 ) , \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} A u ( x ) = \\lim _ { t \\downarrow 0 } \\frac { P _ t ^ D u ( x ) - u ( x ) } { t } . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} B _ { \\rm b s } ^ { \\star } ( \\Gamma _ 0 ) = \\{ G \\in B _ { \\rm b s } ( \\Gamma _ 0 ) : \\left ( K G \\right ) ( \\gamma ) \\geq 0 \\ \\ { \\rm f o r } \\ \\ { \\rm a l l } \\ \\ \\gamma \\in \\Gamma \\} . \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} m _ k \\leq \\frac { u } { c _ 2 V ( d _ D ) } \\leq M _ k ~ ~ ~ D _ { r _ k } = D _ { r _ k } ( x _ 0 ) \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} \\pi _ i & = \\frac { | \\{ e _ j : \\ ( i - 1 ) M < j \\leq i M , e _ j = + 1 \\} | } { M } \\\\ & = \\frac { 1 } { M } \\sum _ { j = ( i - 1 ) M + 1 } ^ { i M } \\frac { 1 } { 2 } ( e _ j + 1 ) = \\frac { 1 } { 2 M } \\sum _ { j = ( i - 1 ) M + 1 } ^ { i M } e _ j + \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} g ( u , v ) & = ( v , u - 2 v ) , \\\\ g ( v , u ) & = ( u , v - 2 u ) , \\\\ g ( u , - v ) & = ( - v , u + 2 v ) \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} M ' _ s ( t ) = & \\sigma ( s , X ( s ) ) + \\int _ 0 ^ t U ( s , r ) d r + \\int _ 0 ^ t V ( s , r ) d W ( r ) \\\\ & + \\int _ 0 ^ t \\nabla _ x b ( r , X ( r ) ) M ' _ s ( r ) d r + \\int _ 0 ^ t \\nabla _ x \\sigma ( r , X ( r ) ) M ' _ s ( r ) d W ( r ) , \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} T ^ \\perp = \\{ \\tau \\in \\{ - , 0 , + \\} ^ n \\mid \\tau \\cdot \\rho = 0 \\rho \\in T \\} \\ , . \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} d ( x _ i , y _ i ) = m _ i + n _ i , \\forall \\ , i \\in \\{ 1 , . . . , N \\} . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\tilde W = \\begin{pmatrix} 1 & 0 & - 1 \\\\ 0 & 1 & 0 \\end{pmatrix} W = \\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} \\displaystyle \\lim \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\mathbb { E } [ | \\sum _ { \\substack { j \\in J _ { K } } } \\Phi ( \\frac { B _ { t _ { j } } + B _ { t _ { j + 1 } } } { 2 } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) | ^ 2 ] = 0 \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} \\sup _ { x = ( z , y ) \\in B _ { \\frac { 1 } { 4 } } \\times Y } \\sup _ { x ' \\in B ^ { g _ { z , t } } ( x , \\frac { 1 } { 8 } ) } \\frac { | { g } _ t ^ \\bullet ( x ) - \\mathbf { P } ^ { g _ { z , t } } _ { x ' x } ( { g } _ t ^ \\bullet ( x ' ) ) | _ { g _ { z , t } ( x ) } } { d ^ { g _ { z , t } } ( x , x ' ) ^ \\alpha } \\leq C ' . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} \\tau _ \\vartheta ~ = ~ \\inf \\{ t \\geq 1 : X ( t ) = \\vartheta \\} . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{gather*} \\phi ( \\tau ) = 4 \\frac { \\eta ( 4 \\tau ) ^ 2 \\eta ( 1 2 \\tau ) ^ 2 } { \\eta ( 2 \\tau ) \\eta ( 6 \\tau ) } + \\frac { \\eta ( 2 \\tau ) ^ 5 \\eta ( 6 \\tau ) ^ 5 } { \\eta ( \\tau ) ^ 2 \\eta ( 4 \\tau ) ^ 2 \\eta ( 3 \\tau ) ^ 2 \\eta ( 1 2 \\tau ) ^ 2 } , \\end{gather*}"}
-{"id": "3022.png", "formula": "\\begin{align*} \\ell ( G ) = \\ell ( \\ker \\phi ) + \\ell ( \\phi ( G ) ) . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} F _ \\mu ( P ) = \\Phi _ \\mu \\big ( | P | \\big ) , \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} & K _ 0 = K _ 0 K _ 1 = 2 ^ 1 K _ 0 K _ 2 = K _ 0 K _ 3 = 2 ^ 2 K _ 0 \\\\ & K _ 4 = K _ 0 K _ 5 = 2 ^ 1 K _ 0 K _ 6 = K _ 0 K _ 7 = 2 ^ 3 K _ 0 \\\\ & K _ 8 = K _ 0 K _ 9 = 2 ^ 1 K _ 0 K _ { 1 0 } = K _ 0 K _ { 1 1 } = 2 ^ 2 K _ 0 \\\\ & K _ { 1 2 } = K _ 0 K _ { 1 3 } = 2 ^ 1 K _ 0 K _ { 1 4 } = K _ 0 K _ { 1 5 } = 2 ^ 4 K _ 0 \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\left | \\left \\langle \\mathbf { P } _ { \\varepsilon } ^ { \\beta } v _ { n } ^ { \\varepsilon } \\left ( s \\right ) , \\phi _ { j } \\right \\rangle - \\left \\langle \\mathbf { P } _ { \\varepsilon } ^ { \\beta } w _ { n } ^ { \\varepsilon } \\left ( s \\right ) , \\phi _ { j } \\right \\rangle \\right | \\le C C _ { 1 } \\log \\left ( \\gamma \\left ( T , \\beta \\right ) \\right ) \\sum _ { k = 1 } ^ { n } \\left | V _ { k } ^ { \\varepsilon } - W _ { k } ^ { \\varepsilon } \\right | . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\phi p = \\tilde { p } + p _ 3 + p _ 4 , \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} & d _ { s } = \\inf _ { t \\geq 0 , x _ 1 , x _ 2 \\in E _ { s - t _ 0 } } \\frac { \\P _ { s , v _ { s , x _ 1 , x _ 2 } } ( \\tau _ { A } > t + s ) } { \\sup _ { x \\in E _ { s } } \\P _ { s , x } ( \\tau _ { A } > t + s ) } ; \\\\ & d ' _ { s } = \\inf _ { t \\geq 0 } \\frac { \\P _ { s , v _ { s } } ( \\tau _ { A } > s + t ) } { \\sup _ { x \\in E _ { s } } \\P _ { s , x } ( \\tau _ { A } > s + t ) } . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} c _ q ( n ) & = \\sum _ { \\substack { a ( q ) \\\\ ( a , q ) = 1 } } e \\left ( \\frac { a n } { q } \\right ) . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| v ( \\cdot , t + t _ 0 ; t _ 0 , u _ 0 , v _ 0 ) - v ^ { * * } ( \\cdot , t + t _ 0 ) \\| _ { L ^ 2 ( \\Omega ) } = \\lim _ { t \\to \\infty } \\| \\phi ( t + t _ 0 ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } = 0 . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\int _ M w _ i \\ , d M = 0 , \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} \\epsilon _ { u _ i } = \\begin{bmatrix} 1 & \\lambda _ { u _ i } \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} \\limsup _ n a _ { n , k } \\leq \\sum _ { i = 1 } ^ r w _ i + \\left ( 1 - \\sum _ { i = 1 } ^ r v _ i \\right ) . \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { B _ { r _ { k + 1 } } } v \\leq \\sup _ { B _ { \\tau _ 2 r _ { k } } } v & \\leq ( 1 - \\mu ) \\left ( 4 M r _ { k } ^ { \\beta } - M r _ { k } ^ { \\alpha } \\right ) + M r _ { k } ^ { \\alpha } \\\\ & \\leq 4 M r _ { k + 1 } ^ { \\beta } \\cdot \\frac { 1 - \\mu } { \\tau _ 1 ^ { \\beta } } + \\mu M r _ { k } ^ { \\beta } \\\\ & \\leq 4 M r _ { k + 1 } ^ { \\beta } \\left ( \\frac { 1 - \\mu } { \\tau _ 1 ^ { \\beta } } + \\frac { \\mu } { 4 \\tau _ 1 ^ { \\beta } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\partial ( q _ 1 \\ast q _ 2 ) & = ( 1 , q _ 1 \\ast q _ 2 ) \\\\ & = ( 1 , q _ 1 ) \\ast ( \\phi ( e _ { q _ 1 } ) , q _ 2 ) \\\\ & = ( 1 , q _ 1 ) \\ast \\big ( \\phi ( e _ { q _ 1 } ) \\cdot ( 1 , q _ 2 ) \\big ) \\\\ & = \\partial ( q _ 1 ) \\ast \\big ( \\phi ( e _ { q _ 1 } ) \\cdot \\partial ( q _ 2 ) \\big ) , \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} n = \\lfloor m / 2 \\rfloor \\quad c = q ^ { m ( m - 1 ) / ( 2 n ) } . \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} g _ { \\nu } ( x ) \\equiv \\left \\{ \\begin{aligned} & 0 ~ ~ & & \\mbox { o n } ~ ~ \\partial B _ 1 \\cap B ( e _ 1 , \\nu / 2 ) ; \\\\ & 1 ~ ~ & & \\mbox { o n } ~ ~ \\partial B _ 1 \\backslash B ( e _ 1 , \\nu ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} S _ { \\Diamond } ( L ) = S _ { B } ( L ) = S _ { K } ( L ) . \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} \\pi _ { \\mathcal { R } } b v _ L = 0 . \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} f _ \\delta : \\Omega - D \\rightarrow \\R , \\ , f _ \\delta ( x ) : = \\left \\{ \\begin{aligned} f ( x ) , & \\ , \\ , | f ( x ) | \\leq \\delta ^ { - 1 } , \\\\ \\delta ^ { - 1 } , & \\ , \\ , | f ( x ) | > \\delta ^ { - 1 } \\end{aligned} \\right . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} F _ { 4 , 2 0 } ( \\tau ) = \\left ( \\begin{smallmatrix} 0 \\\\ ( u - 1 / u ) \\\\ 0 \\\\ - ( u - 1 / u ) \\\\ 0 \\end{smallmatrix} \\right ) X ^ 3 Y ^ 3 + \\ldots \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} X _ { \\ell _ i ^ { ( k ) } } = \\begin{pmatrix} \\left \\{ \\zeta _ j , \\zeta _ i \\right \\} _ t & \\left \\{ \\varphi _ j , \\zeta _ i \\right \\} _ t \\\\ [ 0 . 3 c m ] \\left \\{ \\zeta _ j , \\varphi _ i \\right \\} _ t & \\left \\{ \\varphi _ j , \\varphi _ i \\right \\} _ t \\\\ \\end{pmatrix} _ { i , j } \\begin{pmatrix} \\dfrac { \\partial \\ell _ i ^ { ( k ) } } { \\partial \\zeta _ j } \\\\ [ 0 . 5 c m ] \\dfrac { \\partial \\ell _ i ^ { ( k ) } } { \\partial \\varphi _ j } \\\\ \\end{pmatrix} _ { j } \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} \\mu ^ { ( 1 ) } \\ ; & : = \\ ; \\langle u _ 0 , ( - \\Delta + U _ { \\rm t r a p } ^ { ( 1 ) } ) u _ 0 \\rangle + c _ 1 \\langle u _ 0 , V ^ { ( 1 ) } * | u _ 0 | ^ 2 u _ 0 \\rangle + c _ 2 \\langle u _ 0 , V ^ { ( 1 2 ) } * | v _ 0 | ^ 2 u _ 0 \\rangle , \\\\ \\mu ^ { ( 2 ) } \\ ; & : = \\ ; \\langle v _ 0 , ( - \\Delta + U _ { \\rm t r a p } ^ { ( 2 ) } ) v _ 0 \\rangle + c _ 2 \\langle v _ 0 , V ^ { ( 2 ) } * | v _ 0 | ^ 2 v _ 0 \\rangle + c _ 1 \\langle u _ 0 , V ^ { ( 1 2 ) } * | v _ 0 | ^ 2 u _ 0 \\rangle \\ , . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\lambda \\alpha = \\alpha ( y _ 2 + 1 ) ^ T , \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t _ i } \\nabla _ { \\partial _ \\sigma } = \\big [ ( g ^ { - 1 } p _ { - i } g ) _ + , \\nabla _ { \\partial _ \\sigma } \\big ] \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} \\mathcal { H } = \\mathcal { H } _ u \\oplus \\mathcal { H } _ s \\ , \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} M ( t ; t \\mid \\hat { u } _ n , \\hat { U } _ n ) \\begin{pmatrix} 1 \\\\ \\hat { x } _ { n - 1 } ^ { - 1 } \\end{pmatrix} = 0 \\ ; . \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} Q ^ { ( m ) } _ k ( 2 s + 1 ) = Q ^ { ( m ) } _ k ( 2 s - 1 ) - c q ^ { 2 ( n - s + 1 ) } \\ , Q ^ { ( m - 2 ) } _ { k - 2 } ( 2 s - 1 ) . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} h \\left ( D _ 1 \\otimes \\cdots \\otimes D _ n \\right ) = h \\left ( M _ 1 \\otimes \\cdots \\otimes M _ n \\right ) = \\sum _ { i = 1 } ^ n h _ i \\in H _ 1 ( { \\bf Z } , { \\bf a } ) . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} y = \\begin{pmatrix} J _ { \\frac { k b } { 4 } , 4 } \\\\ & I _ { s } \\end{pmatrix} . \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} \\| R \\| _ { S _ 1 } = \\left ( \\sum _ { j \\geq 1 } ( 1 + j ) ^ { 2 \\alpha } | a _ { j - 1 } | ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } = \\| H ' e _ 0 \\| _ 2 < \\infty . \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} \\tilde { \\psi } ( n ) = \\tilde { \\phi } ( n ) - \\left ( c _ + + ( - 1 ) ^ { | n | } c _ - \\right ) , \\quad \\forall n \\in \\N ^ N , \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\hat { \\partial } f ( x ) : = & \\Bigg \\{ x ^ * \\in X ^ \\ast \\mid \\liminf \\limits _ { h \\to 0 } \\frac { f ( x + h ) - f ( x ) - \\langle x ^ * , h \\rangle } { \\| h \\| } \\geq 0 \\Bigg \\} . \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\varepsilon ( \\boldsymbol { \\alpha } ) : = & \\begin{cases} - 2 \\qquad & \\boldsymbol { \\alpha } \\in \\left \\{ \\left ( 1 - \\frac { 1 } { p } , \\frac { 2 } { p } \\right ) , \\left ( \\frac { 1 } { p } , 1 - \\frac { 2 } { p } \\right ) \\right \\} , \\\\ 1 & . \\end{cases} \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} \\chi ( [ T , \\Delta _ n ^ 2 , T ] ) = c \\omega _ { 1 , n } ( \\Delta _ n ^ 2 ) + d \\sum _ { i = 1 } ^ { n - 1 } \\omega _ { i , i + 1 } ( \\Delta _ n ^ 2 ) = c + ( n - 1 ) d \\ne 0 \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\mathfrak v _ i \\vec { \\mathfrak u _ i } ^ T = \\vec { \\mathfrak u _ i } ^ T \\mathfrak V _ i \\ \\ \\vec { \\mathfrak u _ i } ^ T \\mathfrak V _ j = \\vec 0 ^ T , \\ \\ \\forall i \\neq j . \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k - 1 } \\mu _ { j , t } \\leq C _ k . \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G ) \\leq \\frac { r ^ { 1 - r / p } } { \\alpha ^ { 1 / p } } = ( r C ) ^ { 1 - r / p } = \\Bigg ( r \\sum _ { e \\in E ( G ) } \\prod _ { v \\in e } d _ v ^ { 1 / ( p - r ) } \\Bigg ) ^ { ( p - r ) / p } . \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} x _ { \\sigma ( \\iota ) } = \\frac { y _ { \\sigma ( \\iota ) } } { w _ { j _ k , i _ { \\iota } } \\bigl ( 1 + \\frac { \\lambda _ { \\ell ( i _ { \\iota } ) } } { \\mu _ { i _ { \\iota } } } \\bigr ) } . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} v _ N ' & = - A v _ N + \\frac 1 { t ^ { \\mu _ { N + 1 } } } \\Big ( - \\sum _ { \\stackrel { 1 \\le m , j \\le N , } { \\mu _ m + \\mu _ j = \\mu _ { N + 1 } } } B ( \\xi _ m , \\xi _ j ) + \\phi _ { N + 1 } + \\chi _ { N + 1 } \\Big ) \\\\ & - \\sum _ { n = 1 } ^ N \\frac 1 { t ^ { \\mu _ n } } \\Big ( A \\xi _ n + \\sum _ { \\stackrel { 1 \\le m , j \\le N , } { \\mu _ m + \\mu _ j = \\mu _ n } } B ( \\xi _ m , \\xi _ j ) - \\phi _ n - \\chi _ n \\Big ) + h _ { N + 1 , 4 } , \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{gather*} L ( g , s ) = L \\big ( \\psi ^ 2 , s - 1 \\big ) = \\prod _ { v \\ne \\mathfrak P , \\infty } \\frac { 1 } { 1 - ( a - b { \\rm i } ) ^ 2 N v ^ { - s } } = \\sum _ { \\mathfrak I \\in S _ 1 } \\tilde \\psi ( \\mathfrak I ) ^ 2 ( N \\mathfrak I ) ^ { - s } \\end{gather*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\widetilde { Y } _ { i , r } \\widetilde { Y } _ { j , s } = t ^ { \\gamma ( i , r ; j , s ) } \\widetilde { Y } _ { j , s } \\widetilde { Y } _ { i , r } , \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} F _ 2 = F _ 1 \\circ h _ 2 . \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} h ( \\cdot ) = \\sqrt n \\hat \\lambda \\| \\cdot \\| _ { S _ 1 } , \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} H = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 1 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\dot { z } = J g ( t , y ( t ) ) z \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} D _ { k } : = A _ { k } \\setminus A _ { k + 1 } = \\{ 2 ^ { k } < f \\leq 2 ^ { k + 1 } \\} \\ , \\ , \\ , \\ , \\ , \\ , d _ { k } = | D _ { k } | . \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} \\mathcal { H } ^ u & = \\bigoplus _ { I \\subset [ N ] } \\mathcal { H } _ I ^ u , & \\mathcal { H } ^ s & = \\bigoplus _ { I \\subset [ N ] } \\mathcal { H } _ I ^ s . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\int _ { \\Gamma } ^ { } u _ { \\Gamma } ( t ) d \\Gamma = \\int _ { \\Gamma } ^ { } u _ { 0 \\Gamma } d \\Gamma \\mbox { f o r a l l } t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} \\left ( { 1 + x } \\right ) _ q ^ \\alpha = \\frac { { \\left ( { 1 + x } \\right ) _ q ^ \\infty } } { { \\left ( { 1 + q ^ \\alpha x } \\right ) _ q ^ \\infty } } \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} 4 ^ y = \\frac { 2 \\sqrt { \\pi } \\ , \\Gamma ( 2 y ) } { \\Gamma ( y ) \\Gamma ( y + 1 / 2 ) } , y \\in ( 0 , \\infty ) , \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} \\| w _ k ' \\| ^ 2 + w _ k ' ( 0 ) \\overline { w _ k ( 0 ) } = \\langle h _ k - q _ k ( s ) ^ 2 w _ k , w _ k \\rangle . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} U \\left ( \\dfrac { X + i P } { \\sqrt { 2 } } \\right ) U ^ { - 1 } = \\mathbb { P } \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} h ( x ) : = e ^ { - \\alpha | x - x _ { 0 } | ^ { 2 } } - e ^ { - \\alpha R ^ { 2 } } , \\quad \\forall ~ \\alpha > 0 , x \\in \\Omega . \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} f _ j ^ { ( k ) } ( \\ell , \\zeta , \\varphi ) = \\sum _ { \\substack { | I | = j \\\\ I \\subseteq \\{ 1 , \\ldots k \\} } } e ^ { t L _ I ( \\ell ) } - \\sum _ { \\substack { | I | = | J | = j \\\\ I , J \\subseteq \\{ n - k + 1 , \\ldots , n \\} } } P _ { I , J } \\left ( \\zeta , \\varphi \\right ) , \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} d X _ { t } ^ { n } = u ^ { n } ( t , X _ { t } ^ { n } ) d t + d B _ { t } , \\ X _ { 0 } = x \\in \\R ^ { d } . \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb { R } ^ N } \\frac { \\lambda _ 1 } { \\| A ^ \\top b \\| _ { \\infty } } \\sum _ { j = 1 } ^ m \\log ( 1 + \\exp ( b _ j a _ j ^ \\top x ) ) + \\norm { x } _ 1 \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} a _ i = \\frac { m _ 0 } { \\alpha } | v _ i | = \\frac { m _ 0 | v _ i | } { \\sum _ { j = 1 } ^ { m _ 0 } \\abs { v _ j } - r } , \\ , \\ , \\ , \\ , i = 1 , 2 , \\ldots , n . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} P d _ { l _ 1 , k } = e _ { l _ 2 , n - 2 k } P , \\\\ d _ { l _ 1 , k } Q + Q d _ { l _ 1 , k } = R P , \\\\ P ^ T E e _ { l _ 2 , n - 2 k } = d _ { l _ 1 , k } ^ T F R , \\\\ Q ^ T F d _ { l _ 1 , k } + d _ { l _ 1 , k } ^ T F Q = P ^ T E P . \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} ( 1 - \\gamma ) \\xi = \\omega J ^ { - 1 } _ c J _ b \\nu + \\frac { \\epsilon } { 2 \\gamma } J ^ { - 1 } _ c ( \\nu ^ 2 ) \\equiv G _ 0 ( \\nu ) . \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } \\xi , \\xi ) = \\breve { g } ( \\xi , \\breve { J } \\xi ) = 0 . \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} \\alpha ^ { p } = 0 \\quad \\beta ^ { p } = 0 \\ , . \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} \\Delta ( h ) : = \\sum _ h h _ 1 \\otimes h _ 2 . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\underset { f \\ \\textrm { i s o d d } } { \\sum _ { f \\in H _ 2 ( q ) } } \\omega _ f = \\frac { 1 } { 2 } + \\mathcal { O } \\left ( \\frac { 1 } { q } \\right ) . \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} A _ \\delta + q ^ \\delta C _ \\delta = 0 . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} G _ p ^ n ( t ) \\overset { \\mathbb { P } } { \\longrightarrow } \\kappa _ { p / 2 } \\int _ 0 ^ t ( \\theta _ s ) ^ { p / 2 - 1 } d s = : G _ p ( t ) \\forall t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} p ^ s _ i ( s ' , q ) : = \\mathbb { P } \\bigg [ q + \\sum _ { j \\in [ \\epsilon n _ i ] \\setminus [ s ' ] } W ' _ j \\geq \\epsilon ^ { 1 / 2 } \\bigg ] . \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\partial _ t Z ^ { n , m } _ t + v \\partial _ x Z ^ { n , m } _ t + B ^ { n , m } _ t \\partial _ v Z ^ { n , m } _ t - \\partial _ v ( v Z ^ { n , m } _ t ) - \\sigma \\partial _ { v v } Z ^ { n , m } _ t = \\tilde R ^ { n , m } _ t \\ , , \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\Delta ( \\cdot ) _ j = j \\ , ( \\cdot ) _ { j - 1 } , \\Delta [ \\cdot ] _ j = [ \\cdot ] _ { j - 1 } . \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} \\mu ^ { ( J ) } \\coloneqq u _ 1 \\quad m ^ { ( J ) } \\coloneqq 2 \\cdot \\left ( 2 ^ J - u _ 1 \\right ) = 2 m \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} h ' ( a ) = - M \\psi ( a M + 1 ) + \\sum _ { i = 1 } ^ { d + 1 } \\gamma _ i \\psi ( a \\gamma _ i + 1 ) - \\sum _ { i = 1 } ^ { d + 1 } \\gamma _ i \\log x _ i , \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} \\langle ( \\mathrm { i d } \\otimes V _ M U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , ( \\mathrm { i d } \\otimes V _ M ) g \\times g \\rangle = M ^ { 2 - k } \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle , \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} \\| W _ { f , \\varphi } \\| ^ 2 = \\| W _ { f , \\varphi } ^ * \\| ^ 2 \\geq \\frac { \\| W _ { f , \\varphi } ^ * K _ p \\| ^ 2 } { \\| K _ p \\| ^ 2 } . \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\psi ( x ) = - \\sum _ { j = 1 } ^ N \\frac { a _ j } { 2 \\pi } \\log | x - y _ j | \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} \\operatorname { f i x } ( S , \\Omega ) = | \\{ \\omega \\in \\Omega : \\omega t = \\omega \\ \\forall t \\in S \\} | , \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} T _ x P ^ n = P T _ x P ^ { n - 1 } + A _ x P ^ { n - 1 } = P ^ n T _ x + \\sum _ { k = 1 } ^ { n } P ^ k A _ x P ^ { n - k } . \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} \\mathbb { Y } : = \\bigl \\{ Y ^ { ( k ) } = \\bigl \\{ Y _ t ^ { ( k ) } , t \\ge 0 \\bigr \\} , k > 0 \\bigr \\} , \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} \\begin{array} { l } a = a ^ * : \\mathbb R ^ d \\rightarrow \\mathbb { R } ^ d \\otimes \\mathbb { R } ^ d , \\\\ \\sigma I \\leq a ( x ) \\leq \\xi I . \\end{array} \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\left ( 1 - y \\right ) ^ \\alpha _ q - 1 = - \\left [ \\alpha \\right ] _ q \\left ( { 1 - q \\eta } \\right ) _ q ^ { \\alpha - 1 } \\left ( y - 0 \\right ) \\ge - \\left [ \\alpha \\right ] _ q y \\forall q \\in \\left ( \\widehat { q } , 1 \\right ) . \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} D _ { n + 1 , t } ^ 2 D _ n ^ 2 - D _ { n , t } ^ 2 D _ { n + 1 } ^ 2 = 2 D _ { n + 1 } ^ 1 D _ n ^ 3 , \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} \\rho _ { 1 3 4 } ( S _ 2 ) ^ { 2 ( k + 1 ) } & = \\rho _ { 1 3 4 } ( S _ 2 ) ^ { 2 k } \\rho _ { 1 3 4 } ( S _ 2 ) ^ { 2 } \\\\ & = ( S _ 2 ( S _ 2 S _ 1 ) ^ k S _ 2 ^ * + S _ 1 ( S _ 1 S _ 2 ) ^ k S _ 1 ^ * ) ( S _ 2 ( S _ 2 S _ 1 ) S _ 2 ^ * + S _ 1 ( S _ 1 S _ 2 ) S _ 1 ^ * ) \\\\ & = S _ 2 ( S _ 2 S _ 1 ) ^ { k + 1 } S _ 2 ^ * + S _ 1 ( S _ 1 S _ 2 ) ^ { k + 1 } S _ 1 ^ * \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} H _ 1 & = \\left \\{ l ( p ) = 0 \\right \\} \\\\ H _ 2 & = \\left \\{ f ( p ) = 0 \\right \\} \\\\ H _ 3 & = \\left \\{ { \\det } _ 3 J ( l , f ) ( p ) = 0 \\right \\} \\\\ H _ 4 & = \\left \\{ p _ 2 = 0 \\right \\} \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} \\varphi _ x ( u _ g ) = 0 \\ \\ \\ \\ \\mu \\ \\ x \\in G ^ { ( 0 ) } \\ \\ \\ \\ g \\in G ^ x _ x \\setminus c ^ { - 1 } ( 0 ) . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} \\pi _ x ( a ) : = \\beta _ x ( a ) ^ { - 1 } \\cdot a , \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\partial _ { x _ j } ( I _ { h , T } \\mathbf { u } ( X ) - \\mathbf { u } ( X ) ) = - \\partial _ { x _ j } \\mathbf { u } ( X ) + \\sum _ { i \\in \\mathcal { I } } \\partial _ { x _ j } \\Phi _ { i , T } ( X ) \\widetilde { \\mathbf { R } } _ i ( X ) , \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\int _ \\Gamma \\left ( K G \\right ) ( \\gamma ) \\mu ( d \\gamma ) = \\langle \\ ! \\langle G , k _ \\mu \\rangle \\ ! \\rangle \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} \\nu ( \\{ 0 \\} ) = P r o b ( H _ x = 0 ) < p _ c ^ s ( d ) , \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} \\sum _ { j _ 1 + \\cdots + j _ m = n } \\dfrac { n ! } { j _ 1 ! \\cdots j _ m ! } A _ { j _ 1 } ( x _ 1 ) \\cdots A _ { j _ m } ( x _ m ) , n , m = 1 , \\ldots , \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} F _ 2 ^ L ( \\vec x ) & = \\frac 1 4 \\sum _ { i < j } w _ { i j } ( 2 ( | x _ i | + | x _ j | ) - \\big | x _ i + x _ j + | x _ i | - | x _ j | \\big | - \\big | x _ i + x _ j - | x _ i | + | x _ j | \\big | ) \\\\ & = \\frac 1 4 \\sum _ { i < j } w _ { i j } ( 2 | x _ i | + 2 | x _ j | - 2 \\max \\{ | x _ i + x _ j | , \\big | | x _ i | - | x _ j | \\big | \\} ) \\\\ & = \\frac { 1 } { 2 } \\| \\vec x \\| - \\frac { 1 } { 2 } \\sum _ { i < j } \\abs { x _ i + x _ j } = \\frac { 1 } { 2 } \\| \\vec x \\| - \\frac { 1 } { 2 } I ^ + ( \\vec x ) , \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} H _ { 1 , \\boldsymbol { \\alpha } } ( \\tau ) = \\int _ { \\R ^ 2 } \\cot \\left ( \\pi i w _ 1 + \\pi \\alpha _ 1 \\right ) \\cot \\left ( \\pi i w _ 2 + \\pi \\alpha _ 2 \\right ) e ^ { 2 \\pi i \\tau Q ( \\boldsymbol { w } ) } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\dot { \\phi } ( d ( x , y ) ) , \\quad \\forall x , y \\in X . \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} { \\bf E } [ M _ { ( \\tau = 1 , 0 , 0 ) } ^ q ] = \\frac { G ( 4 - 2 q ) } { G ( 1 - q ) \\ , G ^ 2 ( 2 - q ) \\ , G ( 4 - q ) } \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} \\mathcal { H } ^ 1 _ T ( M ) : = \\{ \\star w \\in \\Omega ^ 1 ( M , \\partial M ) ; d w = 0 \\ \\mbox { a n d } \\ \\delta w = 0 \\} . \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} D _ s X ( t ) = J _ s ( t ) A ( s , t ) , \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\tau _ i = \\Omega _ { \\Phi ( n ) - 1 - i } \\ ; \\ ; \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} \\epsilon ( g _ 1 , g _ 2 ) = ( x ( g _ 1 ) x ( h \\alpha _ n ) , x ( g _ 2 ) x ( h \\alpha _ n ) ) _ p = \\begin{cases} ( d c , d p ^ n ) _ p & c \\neq 0 , \\\\ 1 & c = 0 . \\end{cases} \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\epsilon ( \\alpha ) = 1 , & \\epsilon ( \\gamma ) = 0 , \\end{array} \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} \\tau ' b = \\begin{cases} \\tau b & \\tau b \\in C ' , \\\\ \\tau ^ 2 b & \\tau b \\in \\{ a , \\theta a , \\sigma a , \\theta \\sigma a \\} \\tau ^ 2 b \\in C ' , \\\\ \\tau ^ 3 b & \\tau b , \\tau ^ 2 b \\in \\{ a , \\theta a , \\sigma a , \\theta \\sigma a \\} . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\frac { \\mathcal { A } \\big ( \\big \\{ \\lambda _ f ( n ) \\big \\} ; q \\big ) } { \\mathcal { A } ( q ) } = \\frac { 1 } { \\zeta ( 2 ) } \\sum _ { f \\in H _ 2 ( q ) } \\omega _ f \\cdot L ( 1 , \\operatorname { s y m } ^ 2 f ) \\lambda _ f ( n ) + \\mathcal O \\left ( \\frac { ( \\log q ) ^ 5 } { q } \\right ) . \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} = d ( H _ { \\mathcal { F } } ( g ) ( \\alpha ) , H _ { \\mathcal { F } } ( h ) ( \\alpha ) ) < \\varepsilon \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} G ( z + \\tau \\ , | \\ , \\tau ) = ( 2 \\pi ) ^ { \\frac { ( \\tau - 1 ) } { 2 } } \\tau ^ { - \\frac { 1 } { 2 } } e ^ { \\gamma \\frac { ( z - z ^ 2 ) } { 2 \\tau } } \\prod \\limits _ { m = 1 } ^ \\infty ( m \\tau ) ^ { z - 1 } e ^ { \\frac { ( z ^ 2 - z ) } { 2 m \\tau } } \\frac { \\Gamma ( 1 + m \\tau ) } { \\Gamma ( z + m \\tau ) } , \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} g ^ { ( k ) } ( 0 ) & = ( \\mathcal { S } _ { N } B ^ { ( f ) } _ k ) ( q \\ , | \\ , b ) . \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} - 2 L _ { - 2 } + \\frac { \\kappa } { 2 } L _ { - 1 } ^ { 2 } + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { \\ell } \\tau _ { i } H _ { i } ( - 1 ) ^ { 2 } , \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\dot { u } ( t ) = \\psi \\left ( \\int _ 0 ^ t \\lambda f ( \\tau , u , \\dot { u } ) \\ , d \\tau + c \\right ) \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} & P _ B ( H ^ \\omega - z ) ^ { - 1 } P _ B \\\\ & = \\left [ P _ { B } H ^ \\omega P _ { B } - z P _ { B } - P _ { B } H _ 0 ( I - P _ { B } ) ( \\tilde { H } ^ \\omega - z ) ^ { - 1 } ( I - P _ { B } ) H _ 0 P _ { B } \\right ] ^ { - 1 } . \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} H : = z t \\dd _ t \\prod _ { i = 1 } ^ { n - 1 } ( z t \\dd _ t - \\gamma _ { m + i } z ) - t \\prod _ { j = 1 } ^ m ( z t \\dd _ t - \\gamma _ j z ) . \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} E _ { a / c } ( z , s , \\chi ) = \\frac { \\gcd ( c ^ 2 , N ) ^ { s } } { 2 N ^ s } \\sum _ { \\langle \\gamma , \\delta \\rangle } \\chi ( a \\alpha + b \\gamma ) \\frac { y ^ { s } } { | \\gamma z + \\delta | ^ { 2 s } } , \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\| u _ 1 ^ 0 ( \\tau , \\xi , \\eta ) - U ( \\tau , \\xi ) \\| _ { \\mathcal { H } ^ k ( \\Omega ) } ^ 2 + \\| q ^ 0 ( \\tau , \\xi , \\eta ) - Q ( \\tau , \\xi ) \\| _ { \\mathcal { H } ^ k ( \\Omega ) } ^ 2 \\leq C _ 0 ( 1 + \\tau ^ k ) ( M _ 0 + M _ e ) . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} J _ 3 ( \\lambda ) = \\left [ \\begin{array} { c c c } \\lambda & 1 & 0 \\\\ 0 & \\lambda & 1 \\\\ 0 & 0 & \\lambda \\end{array} \\right ] . \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\tau = { \\pi _ 0 \\over \\pi _ 1 } \\ , , \\ ; k _ 0 = \\sqrt { \\pi _ 0 \\pi _ 1 } ( 2 \\alpha - 1 ) \\ , , \\ ; k _ 1 = \\sqrt { \\pi _ 0 \\pi _ 1 } ( 2 \\alpha - 1 ) \\ , . \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} \\forall 1 \\leq k \\leq 2 ^ n - r , f ^ { i _ 1 i _ 2 \\dots i _ k } ( x ^ p ) = x ^ q . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} T \\leq \\frac { 1 } { 4 \\| p \\| _ { \\infty } } , C : = 2 \\| n _ 0 \\| _ { \\mathrm { T V } } , \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} B ( V , \\mathcal { L } , \\sigma ) = \\bigoplus _ { i \\ge 0 } H ^ 0 \\left ( V , ~ \\mathcal { L } \\otimes \\mathcal { L } ^ { \\sigma } \\otimes \\dots \\otimes \\mathcal { L } ^ { \\sigma ^ { i - 1 } } \\right ) \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} N = \\underbrace { 1 + 1 + \\cdots + 1 } _ { N \\ ; \\mathrm { n o n \\ ; e m p t y \\ ; b o x e s } } . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} P _ 1 ( x ) & = 4 . 8 6 x + 0 . 2 9 x ^ 2 - 0 . 9 6 x ^ 3 + 0 . 9 7 4 x ^ 4 - 0 . 1 7 x ^ 5 , \\\\ P _ 2 ( x ) & = - 3 . 1 1 x - 0 . 3 x ^ 2 + 0 . 8 7 x ^ 3 - 0 . 1 8 x ^ 4 - 0 . 5 3 x ^ 5 , \\\\ P _ 3 ( x ) & = 4 . 8 6 x + 0 . 0 6 x ^ 2 . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} \\Q _ { s , \\mu } ( \\cdot ) : = \\int _ { E _ s } \\Q _ { s , x } ( \\cdot ) \\mu ( d x ) . \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} F _ \\bullet : = ( \\left \\{ 0 \\right \\} = F _ 0 \\subset K = F _ 1 \\subset F _ 2 \\subset \\dots \\subset F _ { 2 n } \\subset F _ { 2 n + 1 } = V ) , \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} \\chi ( M _ k ) = m \\chi ( M ) + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( \\chi ( \\Sigma ^ y _ { \\ell } ) - b ^ y _ { \\ell } ) \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} \\sum _ i \\int _ { \\partial M } ( \\bar w _ i \\eta ( \\bar w _ i ) + I I ^ { \\partial W } ( N , N ) \\bar w _ i ^ 2 ) \\ , d s = 2 \\int _ { \\partial M } H ^ { \\partial W } \\| \\xi \\| ^ 2 \\ , d s . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} I _ l : = [ - 2 \\sqrt { d } , 2 \\sqrt { d } ] \\setminus ( - 1 / l , 1 / l ) , \\ l > 1 . \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} h \\Lambda = \\epsilon ( h ) \\Lambda , & & \\forall h \\in H _ n . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } W _ { t } ( t , x ) + \\inf \\limits _ { u \\in U } \\{ G ( t , x , W ( t , x ) , W _ { x } ( t , x ) , W _ { x x } \\left ( t , x \\right ) , u ) \\} = 0 , \\\\ W ( T , x ) = \\phi ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} \\Phi _ N ^ \\dagger : \\mathbb { F } _ { n + b } [ \\lambda ] ^ { t \\times r } & \\longrightarrow \\mathbb { F } _ { b } [ \\lambda ] ^ { ( s + t ) \\times r } \\\\ Q ( \\lambda ) & \\longrightarrow \\Phi _ N ^ \\dagger [ Q ] ( \\lambda ) = B ( \\lambda ) , \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} | k | _ S = \\sup \\{ | k | _ \\sigma \\ , : \\ , \\sigma \\in S \\big \\} , \\textrm { f o r $ k \\in K $ } . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} r _ i ( x ) = \\Im \\left [ a _ { t _ i i } ( x ) ^ { \\left ( \\frac { 2 \\ell _ i n _ i } { m _ i } - n _ i + \\frac { 2 n _ i ( m _ i + t _ i ) - 2 t _ i } { m _ i } \\right ) } h _ { k _ i i } ( x ) ^ { - 2 } \\exp \\int \\frac { 2 n _ i } { m _ i } \\frac { a _ { ( t _ i - 1 ) i } ( x ) } { a _ { t _ i i } ( x ) } d x \\right ] \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} S ^ \\varepsilon ( x ^ * ) \\ = \\ \\mathcal { A } ^ \\varepsilon ( x ^ * ) . \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} \\mathcal { K } _ { t } ( = K _ t ( \\mathcal { C } _ { \\bullet } ) ) : = \\bigcap _ { i \\in I } \\mathcal { K } _ { i , t } . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} z ^ 2 _ { m i n } > 3 ( Y _ u - y _ i ) ^ 2 - ( X _ u - x _ i ) ^ 2 , \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\frac { w ( e _ 1 ) } { B ( v , e _ 1 ) } = \\frac { w ( e _ 2 ) } { B ( v , e _ 2 ) } = \\cdots = \\frac { w ( e _ d ) } { B ( v , e _ d ) } = r x _ v ^ p . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} F _ \\mu ( P ) = \\widetilde { F } _ \\mu \\big ( | P | \\big ) \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\iota ( D D ' ) = \\iota ( D ) + \\iota ( D ' ) + m ( h ' , \\partial h ) . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} \\psi ( ( a b ^ i ) ^ p ) = \\big ( a ^ { i \\alpha } ( b ^ i ) ^ { a ^ { i e _ 1 } } , a ^ { i \\alpha } ( b ^ i ) ^ { a ^ { i ( e _ 1 + e _ 2 ) } } , \\dots , b ^ i a ^ { i \\alpha } , a ^ { i \\alpha } b ^ i \\big ) . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} x ^ B ( t ) = \\sigma _ { 0 } ^ { B } t + \\frac { 1 } { 2 } \\beta t ^ 2 , \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ A _ { j } ^ { ( N + 1 ) } \\mid \\mathcal { A } ^ { ( l ) } \\right ] = \\left ( \\frac { P ( A _ { j } ^ { ( N + 1 ) } \\cap A _ { 1 } ^ { ( l ) } ) } { P ( A _ { 1 } ^ { ( l ) } ) } , . . . , \\frac { P ( A _ { j } ^ { ( N + 1 ) } \\cap A _ { m _ { l } } ^ { ( l ) } ) } { P ( A _ { m _ { l } } ^ { ( l ) } ) } \\right ) ^ { t } , \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} { \\mathcal { E } } ( u , v ) : = J ( u ) + \\frac { \\gamma } { 2 ( 1 + \\hat { \\delta } ) } \\| v - { \\mathcal { L } } u \\| ^ { 2 } \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} f _ n ( V ^ n ) : = \\left \\{ \\begin{array} { c c } \\max \\{ j : \\Psi ( V ^ n , C _ j ( n ) ) = 1 \\} & \\mbox { i f } \\exists \\ , j \\mbox { s . t . } \\Psi ( V ^ n , C _ j ( n ) ) = 1 \\\\ 1 & \\mbox { o t h e r w i s e } \\end{array} \\right . . \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\leq e _ 1 \\leq \\ldots \\leq e _ n } } { \\rm v o l } ( K _ { e _ 1 , \\cdots , e _ n } ) Y ^ { e _ 1 + \\ldots + e _ n } = \\frac { 1 - Y } { 1 - q ^ n Y } \\prod _ { i = 1 } ^ n \\frac { 1 + q ^ i Y } { 1 - q ^ { n + i } Y } , \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & \\mathcal { H } ( s , x , y , z , u , p , q , P ) \\\\ & = p b ( s , x , y , z + \\Delta ( s ) , u ) + q \\sigma ( s , x , y , z + \\Delta ( s ) , u ) + g ( s , x , y , z + \\Delta ( s ) , u ) \\\\ & \\ \\ + \\frac { 1 } { 2 } P ( \\sigma ( s , x , y , z + \\Delta ( s ) , u ) - \\sigma ( s , \\bar { x } ( s ) , \\bar { y } ( s ) , \\bar { z } ( s ) , \\bar { u } ( s ) ) ) ^ { 2 } \\ , \\end{array} \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} X _ v : = \\sum _ { o \\prec u \\preceq v } \\eta _ u . \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} & \\| \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i + \\rho ( U ^ k ) ^ * \\rho ( S _ \\alpha S _ \\alpha ^ * ) \\rho ( U ) ^ k e _ i \\| ^ 2 = \\\\ & = \\| \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i + \\rho ( U ^ k ) ^ * \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i \\| ^ 2 = \\\\ & = \\| \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i \\| ^ 2 + \\| \\rho ( U ^ k ) ^ * \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i \\| ^ 2 = 2 \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} \\partial _ t v = - \\frac { 1 } { 2 } \\sigma _ { i } ^ k \\sigma _ { j } ^ k \\partial _ { i } \\partial _ { j } v + \\mathrm { d i v } ( \\beta _ j v ) \\dot { Z } _ t ^ j \\enskip , \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} \\lambda ^ { ( p ' ) } ( G ) \\geq \\frac { r ^ { 1 - r / p ' } } { ( \\alpha ' ) ^ { 1 / p ' } } = \\bigg ( \\frac { ( \\lambda ^ { ( p ) } ( G ) ) ^ { p ( p ' - r ) } } { \\Delta ^ { r ( p ' - p ) } } \\bigg ) ^ { 1 / ( p ' ( p - r ) ) } , \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} ( w ' ( e \\cup \\{ v _ e \\} ) ) ^ { ( p + 1 ) - ( r + 1 ) } \\prod _ { v \\in e \\ , \\cup \\{ v _ e \\} } B ' ( v , e \\cup \\{ v _ e \\} ) = \\alpha . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} 3 & ( 1 - \\gamma _ 1 ) & + 6 2 & ( 1 - \\gamma _ 2 ) & + 3 6 0 & \\epsilon & < 3 , \\\\ 4 & ( 1 - \\gamma _ 1 ) & + 3 2 & ( 1 - \\gamma _ 2 ) & + 1 6 0 & \\epsilon & < 3 \\end{aligned} \\right . \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\mathfrak { f } \\mathfrak { f } ^ { \\dagger } + \\mathfrak { f } ^ { \\dagger } \\mathfrak { f } = 1 , \\mathfrak { f } ^ 2 = ( \\mathfrak { f } ^ { \\dagger } ) ^ 2 = 0 , \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} ( \\partial + \\Gamma ) ^ { k + 1 } ( ( \\partial ^ { | \\beta | } \\eta ) ( \\tilde { \\mathbf { x } } _ i ) ( \\tilde { \\mathbf { x } } - \\tilde { \\mathbf { x } } _ i ) ^ \\beta ) = ( \\partial ^ { | \\beta | } \\eta ) ( \\tilde { \\mathbf { x } } _ i ) \\sum \\partial ^ { a _ 1 } \\Gamma \\cdots \\partial ^ { a _ \\ell } \\Gamma \\cdot \\partial ^ b ( \\tilde { \\mathbf { x } } - \\tilde { \\mathbf { x } } _ i ) ^ \\beta , \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\zeta _ 1 ^ { ( 1 ) } = \\frac { 1 } { t } \\ln ( a _ 2 ) , \\ , \\zeta _ 1 ^ { ( 2 ) } = \\frac { 1 } { t } \\ln ( | b | ) , \\ , \\zeta _ 2 ^ { ( 2 ) } = \\frac { 1 } { t } \\ln ( a _ 1 a _ 2 ) , \\ , \\varphi _ 1 ^ { ( 2 ) } = { \\rm A r g } ( b ) . \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} F _ 1 ^ L ( \\vec x ) & = \\sum _ { i < j } w _ { i j } \\int _ 0 ^ { \\norm { \\vec x } } \\chi _ { x _ i \\le t < x _ j } + \\chi _ { x _ j \\le t < x _ i } + \\chi _ { x _ i < - t \\le x _ j } + \\chi _ { x _ j < - t \\le x _ i } d t \\\\ & = \\sum _ { i < j } w _ { i j } \\int _ { - \\norm { \\vec x } } ^ { \\norm { \\vec x } } \\chi _ { x _ i < t < x _ j } + \\chi _ { x _ j < t < x _ i } d t \\\\ & = \\sum _ { i < j } w _ { i j } \\vert x _ i - x _ j \\vert = I ( \\vec x ) , \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} F S ( e ) = \\left \\{ \\begin{array} { l l } \\frac { p ^ { 3 e } - 1 } { 3 } + 1 & \\mbox { f o r } e \\\\ \\\\ \\frac { p ^ { 3 e } - 8 } { 3 } + 1 & \\mbox { f o r } e \\end{array} \\right . \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} \\frac { 1 } { p } \\sum _ { \\ell = 1 } ^ { p } \\frac { 1 } { s + ( r + \\ell ) / p } \\leq \\frac { 1 } { p + 1 } \\sum _ { \\ell = 1 } ^ { p + 1 } \\frac { 1 } { s + ( r + \\ell ) / ( p + 1 ) } . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\| H \\| _ { S _ 1 } & \\leq \\sum _ { k = 0 } ^ N \\tbinom { N } { k } ( n - k + 1 ) ^ N \\\\ & \\leq ( n + 1 ) ^ N \\sum _ { k = 0 } ^ N \\tbinom { N } { k } \\\\ & = 2 ^ N ( 1 + n ) ^ N . \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\lvert \\Phi \\rvert & \\geq m - \\Lambda ( N ) ^ { - 1 } \\rho ^ { - 1 } + o ( \\rho ^ { - 1 } ) , \\quad \\\\ \\phi & = m - 2 \\Lambda ( N ) ^ { - 1 } \\rho ^ { - 1 } + o ( \\rho ^ { - 1 } ) , \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) \\begin{bmatrix} \\Lambda _ \\epsilon ( \\lambda ^ \\ell ) \\otimes I _ n \\\\ - ( \\widehat { \\Lambda } _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m ) ( \\lambda ) M ( \\lambda ) ( \\Lambda _ \\epsilon ( \\lambda ^ \\ell ) \\otimes I _ n ) \\end{bmatrix} = \\begin{bmatrix} e _ { \\eta + 1 } \\\\ 0 \\end{bmatrix} \\otimes P ( \\lambda ) , \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\alpha ' \\circ \\phi = \\phi \\circ \\alpha . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} \\iota ( a ) \\circledcirc ( c , \\bar { c } \\circ x ) = ( a \\cdot _ x c , \\overline { a \\cdot _ x c } \\circ x ) . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\left | \\sum _ { t = 0 } ^ e \\left ( ( \\nabla ^ { k - 1 , \\tilde { g } _ i } _ { \\mathbf { b } } L _ { \\mathbf { b } } ^ { \\tilde { g } _ i } ( \\tilde \\eta _ i ' ) ^ { t } ) ( x ) - \\mathbf { P } ^ { \\tilde { g } _ i } _ { x ' x } [ ( \\nabla ^ { k - 1 , \\tilde { g } _ i } _ { \\mathbf { b } } L _ { \\mathbf { b } } ^ { \\tilde { g } _ i } ( \\tilde \\eta _ i ' ) ^ { t } ) ( x ' ) ] \\right ) \\right | _ { \\tilde { g } _ i ( x ) } < \\frac { 1 } { i } | z - z ' | _ { \\R ^ d } ^ \\alpha + C \\lambda _ i ^ \\alpha , \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} f _ 2 ( M _ 1 \\otimes M _ 2 ) = A ^ * _ { \\mathcal { C R } } \\Big ( f _ 1 ( M _ 1 M _ 2 ) + f _ 1 ( M _ 1 ) f _ 1 ( M _ 2 ) \\Big ) . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\ln L = & \\lim _ { n \\rightarrow \\infty } g ( n ) \\ln f ( n ) \\\\ = & \\lim _ { n \\rightarrow \\infty } \\frac { 2 n } { a n ^ { \\alpha } + \\sqrt { a n ^ { \\alpha } } } \\left ( \\sqrt { a n ^ { \\alpha } } \\ln \\left ( 1 + \\frac { 1 } { \\sqrt { a n ^ { \\alpha } } } \\right ) - 1 \\right ) \\\\ \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 0 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; { \\overline { f } } _ t ( x _ t , x _ { t - 1 } , \\xi _ { t j } ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ { t j } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} G \\varphi ( x ) ~ = ~ \\sum _ { y \\in E } G ( x , y ) \\varphi ( y ) , x \\in E , \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} \\phi _ i ^ { \\pm } ( z ) : = \\sum _ { r = 0 } ^ { \\infty } \\phi _ { i , \\pm r } ^ { \\pm } z ^ { \\pm r } = k _ i ^ { \\pm 1 } \\exp \\left ( \\pm ( q _ i - q _ i ^ { - 1 } ) \\sum _ { r > 0 } h _ { i , \\pm r } z ^ { \\pm r } \\right ) . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\delta ( u , v ) _ { s t } & = \\int _ s ^ t ( \\dot { \\mu } _ r , v _ r ) + ( u _ r , \\dot { \\nu _ r } ) d r + ( u _ s , \\bar { \\Gamma } _ { s t } ^ { 1 } v _ s + \\bar { \\Gamma } _ { s t } ^ { 2 } v _ s ) + ( u , v ) _ { s t } ^ { \\natural } \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} & ( b - m a ) ^ 2 \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } f ( a ) } { f ( b ) - m f ( a ) } \\bigg ) ^ \\frac { 1 } { s } . \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } g ( a ) } { g ( b ) - m g ( a ) } \\bigg ) ^ \\frac { 1 } { s } \\\\ & + ( b - m a ) ( m a - a ) \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } f ( a ) } { f ( b ) - m f ( a ) } \\bigg ) ^ { \\frac { 1 } { s } } \\\\ & + ( b - m a ) ( m a - a ) \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } g ( a ) } { g ( b ) - m g ( a ) } \\bigg ) ^ \\frac { 1 } { s } + ( m a - a ) ^ 2 = \\beta . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} ( u _ 1 , u _ 2 ) = ( \\pm 1 , 0 ) , \\ ( 0 , \\pm 1 ) \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} { } H _ { v , [ a , b ] } = ( P _ { [ a , b ] } ) ^ { * } H _ { v } P _ { [ a , b ] } \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} M ^ T A M = \\left [ \\begin{array} { c c } D & O \\\\ O & D \\end{array} \\right ] , \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} h _ { m + 1 } \\circ G _ m = A _ m \\circ h _ m \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\gamma ( \\sigma \\times \\sigma ^ \\prime , \\psi ) = \\sum _ { \\substack { h \\in U _ m ( \\mathfrak { f } ) \\backslash G _ m ( \\mathfrak { f } ) \\\\ x \\in M _ { m , n - m - 1 } ( \\mathfrak { f } ) } } J _ { \\sigma , \\psi } \\begin{pmatrix} & 1 & \\\\ & & I \\\\ h & & x \\end{pmatrix} J _ { \\sigma ^ \\prime , \\psi ^ { - 1 } } ( h ) . \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\begin{aligned} S _ { q , F } ( m | n , r ) = \\oplus _ { \\lambda \\in \\Lambda ( m | n , r ) } S _ { q , F } ( m | n , r ) 1 _ \\lambda . \\end{aligned} \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} \\alpha _ { 0 } & = - 2 d ^ { 2 } , \\beta _ { 0 } = a ^ { 2 } - J _ { 1 } d , \\alpha _ { 1 } = - 4 d e , \\beta _ { 1 } = \\frac { d e } { 4 } + 2 a b - J _ { 1 } e , \\\\ \\alpha _ { 2 } & = - 2 e ^ { 2 } , \\beta _ { 2 } = 1 + 2 a c + b ^ { 2 } + \\frac { e ^ { 2 } } { 4 } , \\beta _ { 3 } = 2 b c , \\beta _ { 4 } = c ^ { 2 } \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} & F ( k ) = \\sum _ { k = r \\ell _ 1 n _ 1 \\ell _ 2 n _ 2 } \\mu ^ 2 ( r \\ell _ 1 n _ 1 ) \\mu ^ 2 ( r \\ell _ 2 n _ 2 ) h ( r ) f _ 1 ( \\ell _ 1 ) f _ 2 ( \\ell _ 2 ) g _ 1 ( n _ 1 ) g _ 2 ( n _ 2 ) \\prod _ { p | ( \\ell _ 1 , \\ell _ 2 ) } t ( p ) . \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} j _ { k , n } = \\begin{pmatrix} I _ k \\\\ & I _ { n - 2 k } \\\\ I _ k & & I _ k \\end{pmatrix} \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\mathop { \\sum _ c \\sum _ d \\sum _ n \\sum _ r \\sum _ s } _ { ( r d , s c ) = 1 } & b _ { n , r , s } g ( c , d ) e \\left ( n \\frac { \\overline { r d } } { s c } \\right ) \\\\ & \\ll _ { \\epsilon , g _ 0 } ( C D N R S ) ^ { \\epsilon } K ( C , D , N , R , S ) \\| b _ { N , R , S } \\| _ 2 , \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { 1 } { 3 \\cdot 2 ^ d } ( 1 - \\gamma _ 1 ) + \\bigg ( 1 + \\frac { 1 } { 6 ( 2 ^ d - 1 ) } \\bigg ) & ( 1 - \\gamma _ 2 ) + 1 2 \\epsilon & < \\frac { 1 } { 3 \\cdot 2 ^ d } , \\\\ & ( 1 - \\gamma _ 2 ) + 8 \\epsilon & < \\frac { 1 } { 4 \\cdot 2 ^ d } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} \\phi _ { \\mathbf g , \\infty } \\left ( \\left ( \\begin{smallmatrix} x _ 1 & x _ 2 \\\\ x _ 3 & x _ 4 \\end{smallmatrix} \\right ) \\right ) = ( x _ 1 + \\sqrt { - 1 } x _ 2 + \\sqrt { - 1 } x _ 3 - x _ 4 ) ^ { k + 1 } \\mathrm { e x p } ( - \\pi \\mathrm { t r } ( x ^ t x ) ) . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} S ( q , P ) = S \\big ( q _ 1 , q _ 2 ^ { - 1 } P ( q _ 2 { \\ : \\cdot \\ : } ) \\big ) S \\big ( q _ 2 , q _ 1 ^ { - 1 } P ( q _ 1 { \\ : \\cdot \\ : } ) \\big ) . \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} T ^ { { C } _ { ( L ) } } _ { i } = \\max \\{ T ^ { u } _ { i } , T ^ { d } _ { i } , T ^ { u a c } _ { i } , T ^ { d a c } _ { i } , T ^ { c ' } _ { i } \\} , \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} Z _ { \\lambda , \\varepsilon } ( \\beta ) = \\sum \\limits _ { j = - N / 2 } ^ { N / 2 } | 1 + e ^ { 2 \\pi i \\psi _ j } | ^ { 2 \\lambda } e ^ { - \\beta V _ \\varepsilon ( \\psi _ j ) } . \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} \\langle F _ 1 , F _ 2 \\rangle = \\overline { \\langle \\bar { F _ 1 } , \\bar { F _ 2 } \\rangle } . \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} \\begin{aligned} \\prod _ { t = s ' } ^ { { s } } { ( s - s ' ) + \\lambda _ { i _ { t } } - ( t - s ' ) \\brack 1 } _ q = \\prod _ { t = 1 } ^ { { s - s ' + 1 } } { \\lambda _ { i _ { s ' + t - 1 } } + ( s - s ' + 1 ) - t \\brack 1 } _ q \\neq 0 , \\end{aligned} \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} R ( r ) = B e ^ { - \\eta { r } } r ^ { ( \\frac { 1 + \\tau } { 2 } ) } \\Big [ e ^ { 2 \\eta { r } } r ^ { - ( 1 + \\tau + \\mathrm { b } ) } \\Big ] _ { - ( 1 + \\mathrm { b } E ^ { - 1 } ) } \\quad \\Bigg ( \\mathrm { b } = \\Big ( \\frac { \\beta - \\eta ( 1 + \\tau ) } { 2 \\eta } \\Big ) \\Bigg ) , \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} \\left [ \\frac { c a ^ q t ^ { k q } } { f ^ { l q } t ^ { ( d - 1 ) l q } } \\right ] = 0 \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\frac { f ( \\gamma _ i ( 0 ) ) - f ( p ) } { d ( p , \\gamma _ i ( 0 ) ) } = g _ { w _ \\infty } ( w _ \\infty , v ^ f ) . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} J _ { 2 , 3 } = \\frac { \\log \\sqrt { 2 \\pi } } { ( 1 + a ) \\zeta ( 2 + a ) } x ^ { 1 + a } + O _ { a } ( 1 ) , \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} [ X \\xleftarrow p V \\xrightarrow s Y ; E ] \\circ _ { \\oplus } [ & Y \\xleftarrow q W \\xrightarrow t Z ; F ] \\\\ & : = [ ( X \\xleftarrow p V \\xrightarrow s Y ) \\circ ( Y \\xleftarrow q W \\xrightarrow s Z ) ; \\widetilde { q } ^ * E \\oplus \\widetilde { s } ^ * F ] , \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\rho ( t , s ) = ( x _ 1 ( t ' ) / s ^ { \\ell _ 1 } , \\dots , x _ n ( t ' ) / s ^ { \\ell _ n } ) \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} f _ ! ( A ' ) \\coloneqq \\{ b \\in B \\mid \\mbox { t h e r e e x i s t s } a \\in A ' \\mbox { s u c h t h a t } f ( a ) = b \\} \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{gather*} \\sum _ { m \\equiv a \\mod N } \\frac 1 { m ^ s } = \\frac 1 { N ^ s } \\sum _ { m \\in \\Z } \\frac 1 { \\ ( m + \\frac a N \\ ) ^ s } , \\end{gather*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\Pr \\left ( { \\left . { { s _ n } } \\right | { { \\bf { R } } _ q } } \\right ) = \\int { \\Pr \\left ( { \\left . { { s _ n } , { { \\tilde \\theta } _ n } , \\vartheta } \\right | { { \\bf { R } } _ q } } \\right ) d { { \\tilde \\theta } _ n } } d \\vartheta \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\sum _ n \\sum _ i 2 ^ { L - i } \\binom { L } { i } \\binom { N } { n - i } = \\sum _ n \\sum _ k \\binom { L } { k } \\binom { N + k } { n } = 3 ^ L 2 ^ N . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} \\partial p ( x ) = \\displaystyle \\bigcup _ { u \\in I ( x ) } \\mathrm { p r o j } ^ u _ { X ^ { \\ast } } \\left ( \\partial F _ { u } \\right ) ( x , 0 _ { u } ) , \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} H _ K ( \\beta _ P ( d ) + \\epsilon ) = 0 \\epsilon > 0 . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} V _ i ( C _ r ) = r \\cdot V _ i ( C ) . \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} k ( M \\ ! / \\ ! F ) \\ ! - \\ ! v ( M \\ ! / \\ ! F ) + \\ ! k ( M \\backslash F ^ c ) = 1 = k ( M \\ ! / \\ ! \\ ; \\emptyset ) \\ ! - \\ ! v ( M \\ ! / \\ ! \\ ; \\emptyset ) + \\ ! k ( M \\backslash \\emptyset ^ c ) , \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} T _ \\theta = B \\otimes b + B ' \\otimes b ' + \\varsigma + d K \\in \\ \\mathcal D ' ( X \\times E ) \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} ^ { C } L _ 1 \\overline { x } ( t ) = \\overline { \\lambda } r ( t ) \\overline { x } ( t ) , \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} ~ ^ { A B C } D _ b ^ \\alpha f ( t ) = ~ ^ { A B R } D _ b ^ \\alpha f ( t ) - \\frac { B ( \\alpha ) } { 1 - \\alpha } f ( b ) E _ \\alpha \\left ( - \\frac { \\alpha } { 1 - \\alpha } ( b - t ) ^ \\alpha \\right ) . \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { ( r - 1 ) / 2 } x _ { 2 i - 1 } x _ { 2 i } + x _ r ^ 2 . \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} T ( \\alpha , \\alpha ' ) = \\frac { \\alpha - \\alpha ' } { 2 \\langle b \\rangle + \\upsilon + \\langle a \\rangle e ^ { \\alpha } } . \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\Phi \\circ \\mathbb P F ^ k ( \\omega , [ v ] ) = \\log \\frac { \\Vert M ^ n _ \\omega v \\Vert } { \\Vert v \\Vert } \\le \\log \\Vert M ^ n _ \\omega \\Vert . \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} U _ N H _ N ^ { \\operatorname { M F } } U ^ * _ N = \\sum _ { j = 0 } ^ 4 M _ j . \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} i n d e x ( \\Sigma ^ y _ { \\ell } \\cap B _ R ( 0 ) ) = i n d e x ( \\Sigma ^ y _ { \\ell } ) \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} \\widetilde { F } _ n = \\sum \\limits _ { r \\geq 0 } \\frac { 1 } { r ! } \\sum \\limits _ { \\sum \\limits _ { i = 0 } ^ r k _ i = n } \\widetilde { P } _ { k _ { 0 } } \\widetilde { F } _ { k _ { 1 } } \\cdots \\widetilde { F } _ { k _ { r } } { n \\choose k _ 0 , k _ 1 , \\hdots , k _ r } ( k _ 0 ) ^ r . \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} M _ k : = \\frac { \\phi \\left ( v _ { k - a + 1 } \\right ) \\wedge \\dots \\wedge \\phi \\left ( v _ k \\right ) } { v _ { k - a + 1 } \\wedge \\dots \\wedge v _ k } . \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { ( \\phi _ { 1 + a } * \\phi _ { - 1 } ) ( n ) } { n } & = \\frac { \\zeta ( 3 + a ) } { ( 1 + a ) \\zeta ^ { 2 } ( 2 + a ) } x ^ { 1 + a } + O _ { a } \\left ( \\log x \\right ) . \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} \\eta _ k = \\max \\{ 1 , \\| g ( y ^ k ) \\| \\} , \\ \\ \\lambda _ k = \\dfrac { \\beta _ k } { \\eta _ k } \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\P ( z ) \\coloneqq \\prod _ { i = 1 } ^ N ( z - z _ i ) ^ { k _ i } , \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} L ( l ) : = \\sum _ { m = 1 } ^ { l } \\log m = l \\log l - l + \\frac 1 2 \\log l + \\log \\sqrt { 2 \\pi } + \\frac { \\vartheta } { 1 2 l } \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} Y _ t \\triangleq Y ^ 1 _ t \\ 1 \\{ t < \\tau _ 1 \\circ Y ^ 1 \\} + \\sum _ { k = 2 } ^ \\infty Y ^ k _ t \\ 1 \\{ \\tau _ { k - 1 } \\circ Y ^ { k - 1 } \\leq t < \\tau _ k \\circ Y ^ k \\} , t \\geq 0 , \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} J _ 1 = \\int _ { I _ q } \\sum _ { r \\leq q } \\lambda _ r ^ { \\frac { 5 } { 2 } } \\| u _ r \\| _ 2 \\sum _ { | r - q | \\leq 2 } \\| u _ r \\| _ 2 , \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} \\frac { d } { d t } X _ i \\cdot X _ i ^ { - 1 } = \\Omega _ i + \\frac 1 2 \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\left ( f ( B X _ j X _ i ^ { - 1 } A ^ { - 1 } ) - f ( A X _ i X _ j ^ { - 1 } B ^ { - 1 } ) - f ( B A ^ { - 1 } ) - f ( A B ^ { - 1 } ) \\right ) . \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} \\mathcal A : = \\ ! \\left \\{ \\left ( 0 , 2 \\right ) , \\left ( p , p + 2 \\right ) , \\left ( p - 1 , p - 1 \\right ) , \\left ( - 1 , - 1 \\right ) , \\left ( p + 1 , p - 1 \\right ) , \\left ( 1 , - 1 \\right ) \\right \\} \\ ! , \\varepsilon _ 1 ( \\boldsymbol { A } ) : = \\varepsilon \\left ( \\frac { A _ 1 - A _ 2 } { 2 p } , \\frac { A _ 2 } { p } \\right ) . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} E _ u ( \\rho _ j ) & = \\left \\lbrace \\frac { f } { D } H _ u \\left ( u _ j ^ H \\right ) \\ ; \\middle | \\ ; f \\in \\mathbb { C } _ { m - 1 } [ z ] \\right \\rbrace , \\\\ F _ u ( \\rho _ j ) & = \\left \\lbrace \\frac { g } { D } H _ u \\left ( u _ j ^ H \\right ) \\ ; \\middle | \\ ; g \\in \\mathbb { C } _ { m - 2 } [ z ] \\right \\rbrace , \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} & \\mathbb { P } [ \\sum _ { j = 1 } ^ k d ^ { e _ j } _ { i } = 1 ] = \\mathbb { P } [ d ^ { e _ k } _ { i } = 1 ] \\mathbb { P } [ \\sum _ { j = 1 } ^ { k - 1 } d ^ { e _ j } _ { i } = 0 ] \\\\ & + \\mathbb { P } [ d ^ { e _ k } _ { i } = 0 ] \\mathbb { P } [ \\sum _ { j = 1 } ^ { k - 1 } d ^ { e _ j } _ { i } = 1 ] = \\frac { 1 } { 2 } ( 1 - \\sigma ) \\frac { 1 } { 2 } ( 1 + \\sigma ^ { k - 1 } ) \\\\ & + \\frac { 1 } { 2 } ( 1 + \\sigma ) \\frac { 1 } { 2 } ( 1 - \\sigma ^ { k - 1 } ) = \\frac { 1 } { 2 } ( 1 - \\sigma ^ k ) \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} x = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\end{pmatrix} y = \\begin{pmatrix} \\epsilon \\\\ 0 \\\\ 1 \\end{pmatrix} z = \\begin{pmatrix} 0 \\\\ \\epsilon \\\\ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} g ( M ) = \\left \\{ \\begin{array} { r l } f \\left ( M _ { k n } \\right ) , \\ , M \\in \\omega _ { k n } \\\\ 0 , \\ , M \\in \\mathbb { R } ^ { 3 } \\setminus \\bigcup _ { n = 0 } ^ { \\infty } \\widetilde { \\Omega } _ { n } \\end{array} \\right . \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} \\displaystyle \\Delta _ f u - \\frac { m + n - 2 } { 4 ( m + n - 1 ) } \\mathcal { R } ^ m _ f u - c _ 1 u ^ { \\frac { m + n } { m + n - 2 } } e ^ { \\frac { f } { m } } + c _ 2 u ^ { \\frac { m + n + 2 } { m + n - 2 } } = 0 , \\ \\ \\ m > 0 \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} v _ i = \\frac { \\lambda _ 1 } { 4 \\| A ^ \\top b \\| _ { \\infty } } \\sum _ { i = 1 } ^ m ( b _ j A _ { i j } ) ^ 2 , \\enspace i = 1 , \\dots , n . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\sigma ( x , y , z ) & \\le - 2 \\gamma [ ( x ^ 2 + y ^ 2 + z ^ 2 + x ^ 2 y ^ 2 + y ^ 2 z ^ 2 + z ^ 2 x ^ 2 ) \\\\ & \\phantom { \\le - 2 \\gamma [ } - ( x ^ 2 y z + x y ^ 2 z + x y z ^ 2 + x y + y z + z x ) ] \\\\ & = - \\gamma [ ( x - y ) ^ 2 + ( y - z ) ^ 2 + ( z - x ) ^ 2 + ( x y - y z ) ^ 2 + ( y z - z x ) ^ 2 + ( z x - x y ) ^ 2 ] \\le 0 . \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} u _ n = \\sum _ { k = 0 } ^ n s ( n , k ) v _ k \\quad \\Leftrightarrow v _ n = \\sum _ { k = 0 } ^ n S ( n , k ) u _ k . \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} \\sigma & = k - r + \\lambda \\\\ \\tau & = \\frac { k } { v } ( 2 k - v ) \\\\ \\Sigma & = \\frac { 2 k \\lambda } { r } - ( k - r + \\lambda ) . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} \\begin{array} { c c } \\underset { \\underset { i = 1 , \\cdots , N } { x _ { i } } } { } \\sum _ { i = 1 } ^ { N } f _ { i } ( x _ { i } ) \\\\ \\ , x _ { i } = x _ { j } \\ , \\ , g _ { i } ( x _ { i } ) \\leq 0 , \\end{array} \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} M ( t , x ) : = \\frac { \\langle w \\rangle _ { f _ t , \\varphi } } { \\langle 1 \\rangle _ { f _ t , \\varphi } } = \\frac { \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } y \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } w \\ , f _ t ( y , w ) \\ , \\varphi ( x - y ) \\ , w } { \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } y \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } w \\ , f _ t ( y , w ) \\ , \\varphi ( x - y ) } \\ , . \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} v _ p ( k ! S ( a p ^ n ( p - 1 ) , k ) ) = \\left \\lfloor \\frac { k - 1 } { p - 1 } \\right \\rfloor + \\tau _ p ( k ) , \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} w _ \\infty : = \\lim _ { i \\to \\infty } \\dot \\gamma _ i ( 0 ) \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} \\int _ M \\bar w _ i = \\int _ M w _ i \\phi _ k = \\int _ M \\bar w _ i \\phi _ k = 0 , \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\gamma \\circ ( \\mu ' + \\nu ' ) \\circ \\omega ^ 0 = \\gamma \\circ ( \\mu + \\nu ) = 1 _ { W ^ 0 } . \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} ( 1 - \\tau ) ^ { \\beta _ k } \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ { \\beta _ j } f _ j ( \\rho ) \\leq P _ 0 \\leq \\liminf _ { i \\to \\infty } ( \\delta ^ { i + 1 } P _ { i + 1 } + Q _ 0 + \\delta Q _ 1 + \\cdots + \\delta ^ { i } Q _ { i } ) . \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} Z ^ \\circ = \\{ Q \\in \\Q : \\langle Q , S \\rangle = 1 \\ ; \\} . \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} a \\cdot _ x ( b \\circ c ) = b \\circ ( \\rho _ { \\bar { b } } ( a ) \\cdot _ { \\bar { b } \\circ x } c ) , \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} 0 = E _ { i } ^ { ( M ) } \\centerdot v = \\sigma ( E _ { i } ^ { ( M ) } ) \\ , v = F _ { 2 n - i } ^ { ( M ) } \\ , v , M > 0 , \\ 1 \\leq i \\leq 2 n - 1 . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\gamma ( \\sigma \\times \\sigma ^ \\prime , \\psi ) & \\sum _ { h \\in U _ m ( \\mathfrak { f } ) \\backslash G _ m ( \\mathfrak { f } ) } W \\begin{pmatrix} h & \\\\ & I \\end{pmatrix} W ^ \\prime ( h ) \\\\ & = \\sum _ { \\substack { h \\in U _ m ( \\mathfrak { f } ) \\backslash G _ m ( \\mathfrak { f } ) \\\\ x \\in M _ { m , n - m - 1 } ( \\mathfrak { f } ) } } W \\begin{pmatrix} & 1 & \\\\ & & I \\\\ h & & x \\end{pmatrix} W ^ \\prime ( h ) . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} T : = T _ { \\tilde \\Psi } - [ e ] \\otimes b K \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} { \\bf G } _ { \\{ j \\} } { \\bf \\Sigma } _ { \\{ j \\} } = \\big ( 1 + \\sum \\limits _ { i \\neq j } \\frac { r _ { i j } ^ 2 \\sigma _ { i } ^ 2 } { \\sigma _ j ^ 2 } \\big ) \\sigma _ { { j } } ^ 2 = \\sigma _ { { j } } ^ 2 + \\sum \\limits _ { i \\neq j } r _ { i j } ^ 2 \\sigma _ { i } ^ 2 \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} \\Phi ( x ) : = L _ { t \\bar { h } } ^ { - 1 } ( r ( t \\bar { h } + x ) ) . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} \\beta \\left ( u \\alpha _ { 1 } \\right ) ^ { b _ { 1 } b } \\left ( v \\alpha _ { 3 } \\right ) ^ { b _ { 2 } b } \\beta ^ { - 1 } & = \\rho ^ { \\kappa _ { 1 } } \\left ( b \\beta \\beta _ { 0 } \\beta ^ { T } \\right ) \\cdot \\sigma \\alpha _ { 1 } , \\\\ \\beta \\left ( u \\alpha _ { 1 } \\right ) ^ { b _ { 3 } b } \\left ( v \\alpha _ { 3 } \\right ) ^ { b _ { 4 } b } \\beta ^ { - 1 } & = \\rho ^ { \\kappa _ { 2 } } \\left ( b \\beta \\beta _ { 0 } \\beta ^ { T } \\right ) \\cdot \\tau \\alpha _ { 3 } \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} \\varphi \\circ f ( z ) = d f ( 0 ) \\cdot \\varphi . \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} \\mathfrak { M } ( - n \\ , | \\tau , \\ , \\lambda _ 1 , \\ , \\lambda _ 2 ) = \\prod \\limits _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma ( 1 + \\lambda _ 1 + \\frac { ( j + 1 ) } { \\tau } ) \\ , \\Gamma ( 1 + \\lambda _ 2 + \\frac { ( j + 1 ) } { \\tau } ) \\Gamma ( 1 - \\frac { 1 } { \\tau } ) } { \\Gamma ( 1 + \\lambda _ 1 + \\lambda _ 2 + \\frac { ( j + 1 ) } { \\tau } ) \\ , \\Gamma ( 1 + \\frac { j } { \\tau } ) } . \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} \\mathfrak R _ p F _ { | [ \\gamma ] } = \\sum F _ { | [ u _ j \\gamma ] } = \\sum F _ { | [ \\gamma ' u _ i ] } = \\chi ( \\gamma ' ) \\sum F _ { | [ u _ i ] } = \\chi ( \\gamma ) \\mathfrak R _ p F . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} c _ { i , j } ^ { ' } z _ { 1 , 1 } z _ { i , j } = c _ { i , j } z _ { i , 1 } z _ { 1 , j } , \\ ; \\ ; 2 \\leq i \\leq q , \\ ; 2 \\leq j \\leq k + 1 - q , \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} \\underset { \\underset { i = 1 , \\cdots , N } { x _ { i } } } { \\min } \\sum _ { i = 1 } ^ { N } f _ { i } ( x _ { i } ) \\\\ \\ , g _ { i } ( x _ { i } ) \\leq 0 , \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} N = \\underbrace { 1 + 1 + \\cdots + 1 } _ { N - M \\ ; \\mathrm { n o n \\ ; e m p t y \\ ; b o x e s } } + \\underbrace { 1 + \\cdots + 1 } _ { \\ ; M \\ ; \\mathrm { i d e n t i c a l \\ ; o b j e c t s } } , \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} \\omega ( n ) = | a _ f ( n ) | ^ 2 , \\omega ' ( n ) = \\overline { a _ f ( n ) } \\lambda _ f ( n ) ; \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} B = B ^ G + \\sum _ { \\begin{subarray} { l } \\widetilde { \\Sigma } < D \\\\ \\vert \\widetilde { \\Sigma } \\rvert = 2 \\end{subarray} } B ^ { \\widetilde { \\Sigma } } . \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} S ( z ) G ( z ) = A ( z ) \\sum _ n \\frac { \\bar c _ n G ( t _ n ) } { A ' ( t _ n ) \\mu _ n ^ { 1 / 2 } ( z - t _ n ) } \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} X _ i ' X _ i ^ { - 1 } = \\Omega _ i + F _ i ( X ) . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} \\begin{aligned} u _ k ( x ) & = a _ k ( s ) \\sinh ( p _ k ( s ) ( x + 1 ) ) + U _ { k , s } ( x ) , \\quad & x \\in ( - 1 , 0 ) , \\\\ w _ k ( x ) & = b _ k ( s ) \\sinh ( q _ k ( s ) ( 1 - x ) ) + W _ { k , s } ( x ) , & x \\in ( 0 , 1 ) , \\end{aligned} \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} [ \\beta ] = \\sum _ \\textrm { s t r a t a $ S $ } [ { \\imath _ S } _ * { \\jmath _ S } _ { ! * } \\jmath _ S ^ * \\imath _ S ^ { ! * } \\beta ] \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} I I ^ { \\partial W } ( X , Y ) = \\langle - D _ X \\nu , Y \\rangle , \\textrm { f o r } X , Y \\in T \\partial W , \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} \\Vert f ^ n _ \\omega ( z ) - f ^ n _ \\omega ( w ) \\Vert = \\left \\Vert f ^ { n - n _ { k _ i } } _ { T ^ { n _ { k _ i } } \\omega } ( f ^ { n _ { k _ i } } _ \\omega ( z ) ) - f ^ { n - n _ { k _ i } } _ { T ^ { n _ { k _ i } } \\omega } ( f ^ { n _ { k _ i } } _ \\omega ( w ) ) \\right \\Vert < \\varepsilon _ i . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} w _ { t } + \\nabla \\cdot \\left ( - a \\left ( x , t ; w ; \\nabla w \\right ) \\nabla w \\right ) & = F \\left ( x , t ; u ; \\nabla u \\right ) - F \\left ( x , t ; v ; \\nabla v \\right ) \\\\ & + \\nabla \\cdot \\left ( a \\left ( x , t ; u ; \\nabla u \\right ) \\nabla u \\right ) - \\nabla \\cdot \\left ( a \\left ( x , t ; v ; \\nabla v \\right ) \\nabla v \\right ) \\\\ & - \\nabla \\cdot \\left ( a \\left ( x , t ; w ; \\nabla w \\right ) \\nabla w \\right ) , \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} E ^ { ( r ) } _ { i } 1 _ { \\lambda } = \\frac { 1 } { r ! } E ^ { r } _ i 1 _ { \\lambda } F ^ { ( r ) } _ { i } 1 _ { \\lambda } = \\frac { 1 } { r ! } F ^ { r } _ i 1 _ { \\lambda } \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 1 - c _ { i j } } ( - 1 ) ^ { k } \\left [ \\begin{array} { c } 1 - c _ { i j } \\\\ k \\end{array} \\right ] _ { v _ i } f _ { i } ^ { k } f _ { j } f _ { i } ^ { 1 - c _ { i j } - k } = 0 \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} \\mathbf { V } ( e , u _ 1 - 1 , \\ldots , u _ n - 1 ) ^ { S } / / \\mathbf { k } ^ * = \\mathbf { V } ( e ) ^ { S } / / \\mathrm { P G L ( \\mathbf { 1 } ) } \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n b ^ 2 ( i ; n ) \\geq m _ n ^ 2 \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{gather*} K _ { v } ^ 2 ( y _ k ^ K ) = \\sigma _ k ^ 2 y _ k ^ K , \\\\ K _ { v } ( y _ k ^ K ) = \\sigma _ k e ^ { i \\psi ^ \\infty } \\Psi _ k ^ K ( 0 ) y _ k ^ K . \\end{gather*}"}
-{"id": "8337.png", "formula": "\\begin{align*} N ( u ) = \\frac { \\| \\bar \\Sigma ^ { 1 / 2 } u \\| } { \\theta _ 1 } , \\| u \\| \\le \\frac { \\| \\bar \\Sigma ^ { 1 / 2 } u \\| } { \\theta _ 1 } \\quad \\frac { \\| u _ { T ^ c } \\| _ 1 } { \\sqrt { s _ 1 } } \\le \\frac { \\| \\bar \\Sigma ^ { 1 / 2 } u \\| } { \\theta _ 1 } \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} T _ { n , m } = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) , \\quad \\forall m , n \\in \\N ^ N , \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\tilde { w } _ r = \\frac { c _ 2 } { C _ 3 } V ( \\Psi ) + \\eta _ r , \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\mu \\Bigl ( \\bigcup _ { r = 1 } ^ { 2 ^ q } \\Gamma _ r \\Bigr ) = 1 . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\exists \\ , r ^ * \\in [ 0 , r _ { \\max } ] \\ , \\ , \\ , \\lim \\limits _ { k \\to + \\infty } r ^ k = r ^ * , \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} \\int \\partial + \\partial \\int = 1 . \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} \\ddot \\rho - L ( \\dot \\rho ) L ( \\rho ) ^ { \\dagger } \\dot \\rho + \\frac { 1 } { 2 } L ( \\rho ) \\big ( \\nabla _ G L ( \\rho ) ^ { \\dagger } \\dot \\rho \\circ \\nabla _ G L ( \\rho ) ^ { \\dagger } \\dot \\rho \\big ) = 0 . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} & F _ n ( t ) = \\phi _ n t ^ { - \\mu _ n } , \\bar F _ n ( t ) = \\sum _ { j = 1 } ^ n F _ j ( t ) , \\tilde F _ n ( t ) = f ( t ) - \\bar F _ n ( t ) , \\\\ & u _ n ( t ) = \\xi _ n t ^ { - \\mu _ n } , \\bar u _ n ( t ) = \\sum _ { j = 1 } ^ n u _ j ( t ) , \\quad v _ n = u ( t ) - \\bar u _ n ( t ) . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\bar { F } ( u ) ( x ) : = F ( u ( x ) ) , x \\in \\R ^ d . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\ell ( S ) = \\lim \\frac { 1 } { n } L ( S ^ n ) = \\inf _ { n \\geq 1 } \\frac { 1 } { n } L ( S ^ n ) \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} [ z ] : = \\{ z ' \\in Z | z ' \\equiv z \\} ~ ~ ~ ( z \\in Z ) \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\frac { \\partial U _ { L } } { \\partial \\lambda } = \\frac { a d C } { 1 + d ( \\lambda C - D ) - d D } - 2 \\lambda C + 2 D = 0 . \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) } w ' ( e ) = \\sum _ { e \\in E ( G ) } w ( e ) = 1 \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} \\lim _ { \\mu \\rightarrow \\infty } \\| u _ \\mu - u \\| _ { L ^ 2 ( \\Omega , \\R ^ N ) } = 0 . \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} H = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 2 \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} U ( t , r ) ~ U ( r , s ) = U ( t , s ) , \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} S _ { 2 } = c \\big \\langle ( \\nabla _ r \\mathsf { f } ) \\cdot w , ( \\mathsf { f } \\cdot \\nabla w _ r ) | w | ^ { q - 2 } \\big \\rangle & + c ( q - 2 ) \\big \\langle ( \\nabla _ r \\mathsf { f } ) \\cdot w , w _ r | w | ^ { q - 3 } \\mathsf { f } \\cdot \\nabla | w | \\big \\rangle . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\mathbb { U } _ { b ^ r } & = \\langle \\langle g \\rangle \\rangle _ { b ^ r } \\cup ( - 1 ) \\langle \\langle g \\rangle \\rangle _ { b ^ r } , \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} \\| \\mathbf { R } ^ s _ { i 3 } \\| _ { 0 , T _ { \\ast } ^ s } \\leqslant C h ^ 2 | \\mathbf { u } | _ { 2 , T } , ~ i \\in \\mathcal { I } ^ { s ' } , s = \\pm . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\int \\Phi \\ , d m = \\kappa _ j , \\quad m \\left ( \\mathbb P \\Omega _ j \\right ) = 1 . \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} r - \\sum _ { j \\ne i } z _ j ^ * ( | v _ j | - | v _ i | ) & \\ge r - \\sum _ { j = 1 } ^ { i - 1 } z _ j ^ * ( | v _ j | - | v _ i | ) \\\\ & \\ge r - \\sum _ { j = 1 } ^ { i - 1 } ( | v _ j | - | v _ i | ) = r - A ( i - 1 ) > 0 . \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} F _ k ( z , z ' ) : = \\frac { K _ k ( z , z ' ) - 1 } { \\beta _ k } = \\frac { e ^ { \\beta _ k F ( z , z ' ) } - 1 } { \\beta _ k } = F ( z , z ' ) + \\beta _ k F ( z , z ' ) ^ 2 + O ( \\beta _ k ) ^ 2 \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} & \\left | \\Phi ( X ( t ) , A , p ( t ) ) - \\Phi ( X ( 0 ) , A , q ) - t \\left . \\frac { \\partial \\Phi ( X ( u ) , A , p ( u ) ) } { \\partial u } \\right | _ { u = 0 } \\right | \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ = \\left | \\int _ 0 ^ t ( t - v ) \\frac { \\partial ^ 2 \\Phi ( X ( v ) , A , p ( v ) ) } { \\partial v ^ 2 } d v \\right | \\le K t ^ 2 \\ , . \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\langle \\tilde { P } ( u ) , \\tilde { Q } ( v ) \\rangle & = \\langle P ( \\psi ( u _ 1 ) , . . . , \\psi ( u _ N ) ) , Q ( \\psi ( v _ 1 ) , . . . , \\psi ( v _ N ) ) \\rangle \\\\ & = \\dot { \\phi } ( d ( \\psi ( u _ 1 ) , \\psi ( v _ 1 ) ) + \\cdots + d ( \\psi ( u _ N ) , \\psi ( v _ N ) ) ) \\\\ & = \\dot { \\phi } \\left ( \\tfrac { 1 } { 2 } d ( u _ 1 , v _ 1 ) + \\cdots + \\tfrac { 1 } { 2 } d ( u _ N , v _ N ) \\right ) \\\\ & = \\dot { \\phi } \\left ( \\tfrac { 1 } { 2 } d ( u , v ) \\right ) . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} \\displaystyle \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) u ( x , t ) = 0 . \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} V ^ { ( l ) } ( g , \\pi _ n ) _ T = \\sum _ { i : t _ { i , n } ^ { ( l ) } \\leq T } g ( \\Delta _ { i , n } ^ { ( l ) } X ^ { ( l ) } ) , l = 1 , 2 , \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} 0 & = \\ddot { \\rho } + \\frac { \\dot { \\rho } ^ { 2 } } { \\rho } - \\frac { 1 } { \\rho } ( \\frac { 1 } { \\rho } U ^ { \\ast } + 2 h ) \\\\ 0 & = \\ddot { \\gamma } ^ { \\ast } + P \\dot { \\gamma } ^ { \\ast } + Q \\mathbf { \\nu } ^ { \\ast } + R \\nabla U ^ { \\ast } \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{gather*} m = \\varepsilon ( m _ { [ 1 ] } ) m _ { [ 0 ] } \\varepsilon ( \\alpha ) . \\end{gather*}"}
-{"id": "7219.png", "formula": "\\begin{align*} \\inf _ { z \\in \\R ^ n } \\big ( Q ^ { - \\frac 1 2 } ( c + A ^ \\top \\lambda ) \\big ) ^ \\top z + \\| z \\| = \\min _ { z \\in \\R ^ n } \\big ( Q ^ { - \\frac 1 2 } ( c + A ^ \\top \\lambda ) \\big ) ^ \\top z + \\| z \\| = 0 \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{gather*} L ( h _ 3 , 2 ) = L \\big ( \\eta ( 2 \\tau ) ^ 3 \\eta ( 6 \\tau ) ^ 3 , 2 \\big ) = \\frac { 2 ^ { 1 / 3 } } { 4 8 } B ( 1 / 3 , 1 / 3 ) ^ 2 = \\frac { 3 } { 2 } L ( f _ { 3 6 } , 1 ) ^ 2 . \\end{gather*}"}
-{"id": "6044.png", "formula": "\\begin{align*} d _ { K } ( Q _ { n } \\mid \\mid Q ) = \\sum _ { i = 1 } ^ { n } Q _ { n } ( \\left \\{ X _ { i } \\right \\} ) \\log \\left ( \\frac { Q _ { n } ( \\left \\{ X _ { i } \\right \\} ) } { Q ( \\left \\{ X _ { i } \\right \\} ) } \\right ) . \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} \\{ e \\} \\lhd G _ 0 \\lhd G _ 1 \\lhd \\cdots \\lhd G _ n = G \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta u + \\mu u = f _ { 1 } ( u ) + \\lambda v , & x \\in \\mathbb { R } ^ { N } , \\\\ - \\Delta v + \\nu v = f _ { 2 } ( v ) + \\lambda u , & x \\in \\mathbb { R } ^ { N } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} K ( z , z ' ) = K ( [ z ] , [ z ' ] ) = K ( [ w ] , [ z ' ] ) = K ( w , z ' ) , \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} x _ 1 = y _ 4 \\ ; , \\ ; \\ ; x _ 2 = y _ 3 \\ ; , \\ ; \\ ; x _ 3 = - y _ 2 + y _ 4 \\ ; , \\ ; \\ ; x _ 4 = - y _ 1 , \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} \\sum _ { \\ell \\leqslant L } \\mu ^ 2 ( \\ell ) \\frac { | \\lambda _ f ( \\ell ) | ^ 2 } { \\ell } = c _ f \\log L + O _ f ( 1 ) . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} \\lambda _ n ( c ) \\leq \\beta ^ { - 1 } ( n ) = \\exp \\left ( - \\frac { n - \\frac { 2 c } { \\pi } } { K \\big ( \\log ( c ) \\big ) ^ 2 } \\right ) , n \\geq \\frac { 2 c } { \\pi } + K ( \\log 2 ) \\big ( \\log ( c ) \\big ) ^ 2 . \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { 3 } \\Biggl ( & E \\otimes e ^ { - 2 h _ { t } ( \\zeta ) } e _ { t } ^ { r } ( \\zeta ) \\\\ & + H \\otimes ( e ^ { - 2 h _ { t } ( \\zeta ) } f _ { t } ( \\zeta ) e _ { t } ^ { r } ( \\zeta ) + h _ { t } ^ { r } ( \\zeta ) ) \\\\ & + F \\otimes ( f _ { t } ^ { r } ( \\zeta ) - e ^ { - 2 h _ { t } ( \\zeta ) } f _ { t } ( \\zeta ) ^ { 2 } e _ { t } ^ { r } ( \\zeta ) - 2 f _ { t } ( \\zeta ) h _ { t } ^ { r } ( \\zeta ) ) \\Biggr ) d B _ { t } ^ { ( r ) } \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + { \\rm c o n s t } ) } \\ , { \\bf E } \\bigl [ M ^ q _ { ( \\tau = 1 , \\alpha , \\alpha ) } \\bigr ] , \\ ; N \\rightarrow \\infty . \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} x _ * = x _ 0 + \\sum _ { j = 0 } ^ N \\nu _ { j } v _ j + r _ * . \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\mathcal { A } = C ^ * ( U _ 1 ) \\otimes _ { } \\cdots \\otimes _ { } C ^ * ( U _ N ) \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} L ^ G _ { \\lambda - \\rho _ n } : = G \\times _ T \\C _ { \\lambda - \\rho _ n } \\to G / T . \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} \\varphi ( u , v ) = \\tilde { \\varphi } ( d ( u _ 1 , v _ 1 ) , . . . , d ( u _ N , v _ N ) ) , \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} \\check { C } ^ \\bullet ( x _ 1 , \\ldots , x _ t ; M ) : 0 \\to M \\to \\bigoplus _ { i = 1 } ^ t M _ { x _ i } \\to \\cdots \\to M _ { x _ 1 \\ldots x _ t } \\to 0 \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} & x ^ k = \\dfrac { ( \\varphi - 1 ) y ^ k + x ^ { k - 1 } } { \\varphi } \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{gather*} a ^ 3 = b ^ 3 + c ^ 3 . \\end{gather*}"}
-{"id": "5218.png", "formula": "\\begin{align*} V _ N = & 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\mathcal { N } ( 0 , \\ , 4 \\log 2 ) + \\log X _ 1 + \\log X _ 2 + \\log X _ 3 + \\\\ & + \\log Y + \\log Y ' + o ( 1 ) , \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} D & = \\Gamma ( t _ 1 t _ 2 q ^ { n - 1 } , t _ 3 t _ 4 q ^ { n - 1 } , t _ 5 t _ 6 ) ^ n \\prod _ { m = 1 } ^ n \\prod _ { \\substack { 1 \\leq j < k \\leq 6 \\\\ ( j , k ) \\neq ( 1 , 2 ) , \\ , ( 3 , 4 ) , \\ , ( 5 , 6 ) } } \\Gamma ( t _ j t _ k q ^ { m - 1 } ) \\\\ & \\quad \\times \\det _ { 1 \\leq j , k \\leq n } \\left ( ( t _ 1 t _ 3 q ^ { k - 1 } , t _ 1 t _ 4 q ^ { n - k } ) _ { j - 1 } ( t _ 2 t _ 3 q ^ { k - 1 } , t _ 2 t _ 4 q ^ { n - k } ) _ { n - j } \\right ) . \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { \\sigma } } e ^ u = & \\int _ { \\Omega _ { \\sigma } } - \\Delta u = \\sum _ { x \\in \\Omega _ { \\sigma } } \\sum _ { y \\in V : y \\sim x } w _ { x y } ( u ( x ) - u ( y ) ) \\\\ = & \\sum _ { x \\in \\Omega _ { \\sigma } } \\sum _ { y \\in \\Omega _ { \\sigma } : y \\sim x } w _ { x y } ( u ( x ) - u ( y ) ) + \\sum _ { x \\in \\Omega _ { \\sigma } } \\sum _ { y \\not \\in \\Omega _ { \\sigma } : y \\sim x } w _ { x y } ( u ( x ) - u ( y ) ) . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N f ^ a _ { i } \\leq f _ { A } . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} ( p + \\frac { 1 } { 2 } h ^ 2 _ 1 ) ( t , x , y ) \\equiv \\big ( R \\rho ^ e \\theta ^ e + \\frac { 1 } { 2 } ( h ^ e _ 1 ) ^ 2 \\big ) ( t , x , 0 ) = : P ( t , x ) . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\mathbf { f } _ c = \\mathbf { f } + \\mathbf { f } _ u . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} \\chi ( g _ { 1 } , g _ { 2 } ) = \\frac { 1 } { 4 \\pi ^ { 2 } } \\int _ { 0 } ^ { 2 \\pi } \\left ( h _ { 1 } \\circ h _ { 2 } ( x ) - h _ { 1 } ( x ) - h _ { 2 } ( x ) \\right ) \\ , d x \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\nu _ i = \\sum _ { \\substack { 1 \\leq j \\leq t \\\\ \\psi ( E _ N ^ { ( j , M ) } ) \\in \\mathcal { P } _ i } } 1 , \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } = & \\beta ^ { - 1 } _ { 2 2 } ( a = ( 1 , 1 ) , b _ 0 = 1 , \\ , b _ 1 = 1 + \\lambda _ 1 , \\ , b _ 2 = 1 + \\lambda _ 2 ) \\times \\\\ & \\times \\beta _ { 1 , 0 } ^ { - 1 } ( a = 1 , b _ 0 = \\lambda _ 1 + \\lambda _ 2 + 2 ) , \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} \\int _ { \\mathcal L _ 0 \\cap \\mathcal R _ 0 } ( c d , d ) _ p \\chi _ { \\psi } ( d ) \\chi _ { \\delta } ( d ) | d | _ p ^ { - 3 / 2 } \\mathbf h _ p ( h ) d h = \\mathrm { v o l } ( \\mathcal L _ 0 \\cap \\mathcal R _ 0 ) = ( 1 - p ^ { - 1 } ) ^ 2 . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} C ( a ) \\circeq \\frac { \\Gamma ( a M + 1 ) } { \\prod _ { i = 1 } ^ { d + 1 } \\Gamma ( a \\gamma _ i + 1 ) } , a \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\Psi ( \\eta \\cup \\xi ) = \\Psi ( \\eta ) + \\Psi ( \\xi ) + 2 \\sum _ { x \\in \\eta } \\sum _ { y \\in \\xi } a ( x - y ) . \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} & = ( p ^ { 3 } - 1 ) ( p ^ { 3 } - p ) ( p ^ { 3 } - p ^ { 2 } ) \\times \\\\ & \\left ( \\dfrac { 1 } { ( p - 1 ) p ^ { 3 } } + \\dfrac { \\frac { p - 1 } { 2 } - 1 } { ( p - 1 ) ^ { 2 } p ^ { 2 } } + \\dfrac { \\frac { p - 1 } { 2 } } { ( p ^ { 2 } - 1 ) p ^ { 2 } } + \\dfrac { 1 } { ( p ^ { 2 } - 1 ) ( p ^ { 2 } - p ) p ^ { 2 } } + \\dfrac { p - 1 } { ( p - 1 ) p ^ { 2 } } \\right ) \\\\ & = ( p ^ { 3 } - 1 ) ( p ^ { 2 } - 2 ) p ^ { 2 } . \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} P e = \\frac { L ^ 2 } { D T _ \\mathrm { m a x } } . \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom { n } { k } B _ k ( x ) B _ { n - k } ( y ) = - n ( x + y - 1 ) B _ { n - 1 } ( x + y ) - ( n - 1 ) B _ n ( x + y ) , \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} H _ t ^ N : = \\langle S _ t ^ N , \\psi _ k \\rangle \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} g ( \\tau ) & = \\sum _ { \\mathfrak I \\in S _ 1 } \\tilde \\psi ( \\mathfrak I ) ^ 2 q ^ { N \\mathfrak I } = \\sum _ { m , n \\in \\Z } ( 4 m + 1 - 2 n { \\rm i } ) ^ 2 q ^ { ( 4 m + 1 ) ^ 2 + 4 n ^ 2 } \\\\ & = \\sum _ { m , n \\in \\Z } \\big ( ( 4 m + 1 ) ^ 2 - 4 n ^ 2 \\big ) q ^ { ( 4 m + 1 ) ^ 2 + 4 n ^ 2 } = \\eta ( 4 \\tau ) ^ 6 . \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} ( L ^ V f , f ) _ V = ( \\delta ^ * \\delta f , f ) _ V = ( \\delta f , \\delta f ) _ H \\geq 0 . \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { - \\infty } ^ { \\infty } x u ( t , x ) d x = \\int _ { - \\infty } ^ { \\infty } f ( u ( t , x ) ) d x . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} \\sum _ { c \\mid N , \\gcd ( c , N / c ) = 1 } \\chi _ { N / c } ( - 1 ) = \\prod _ { p \\mid N } ( 1 + \\chi _ p ( - 1 ) ) = I _ \\chi 2 ^ { \\omega ( N ) } . \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} G A _ u V ( x ) = \\sum _ { n = 0 } ^ \\infty P ^ n A _ u V ( x ) ~ = ~ \\varphi _ u ( x ) + \\sum _ { m = 1 } ^ n P _ H ^ m \\varphi _ u ( x ) ~ = ~ G _ H \\varphi _ u ( x ) \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathbb E \\left [ x ^ T ( t ) x ( t ) \\right ] = 2 \\mathbb E \\left [ x ^ T ( t ) a ( t ) \\right ] + \\sum _ { i , j = 1 } ^ v \\mathbb E \\left [ b _ i ^ T ( t ) b _ j ( t ) \\right ] k _ { i j } . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} u _ { * * } ^ 2 = \\frac { 8 y _ { * * } ^ 4 + 4 y _ { * * } ^ 2 - 1 } { 2 y _ { * * } ^ 2 ( y ^ 2 _ { * * } + 1 ) } . \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\theta _ 0 = \\textbf { 1 } , \\ E _ i K _ { \\nu ^ + _ i } ( \\theta ) = c ^ + _ i \\cdot \\theta \\ \\ \\mathrm { a n d } \\ \\ \\bar { \\theta } _ 0 = \\bar { \\textbf { 1 } } , \\ \\bar { E } _ i \\bar { K } _ { \\nu ^ - _ i } ( \\bar { \\theta } ) = c ^ - _ i \\cdot \\bar { \\theta } . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} R ( \\rho _ { z } ) = \\left ( \\begin{array} { c c } v _ { 0 } ( w ) ^ { - 2 } & 0 \\\\ - \\frac { c } { 2 } v _ { 0 } ( w ) ^ { - 2 } v _ { 2 } ( w ) & 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} A u ( x ) & = \\lim _ { t \\downarrow 0 } \\frac { P _ t u ( x ) - u ( x ) } { t } = \\lim _ { t \\downarrow 0 } \\frac { 1 } { t } \\left ( \\int _ { \\R ^ n } u ( x + y ) p ( t , | y | ) d y - u ( x ) \\right ) \\\\ & = \\lim _ { t \\downarrow 0 } \\int _ { \\R ^ n } \\left ( \\frac { u ( x + y ) + u ( x - y ) } { 2 } - u ( x ) \\right ) \\frac { p ( t , | y | ) } { t } d y . \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\begin{aligned} u _ t ( r , t ) & = C \\zeta ' ( t ) F ^ { \\frac 1 { m - 1 } } - \\frac C { m - 1 } \\zeta ( t ) F ^ { \\frac 1 { m - 1 } - 1 } \\frac { \\eta ' ( t ) } { \\eta ( t ) } \\frac { r } { a } \\eta ( t ) \\\\ & = C \\zeta ' ( t ) F ^ { \\frac 1 { m - 1 } } - \\frac { C } { m - 1 } \\zeta ( t ) \\frac { \\eta ' ( t ) } { \\eta ( t ) } F ^ { \\frac 1 { m - 1 } - 1 } + \\frac C { m - 1 } \\zeta ( t ) \\frac { \\eta ' ( t ) } { \\eta ( t ) } F ^ { \\frac 1 { m - 1 } } \\ , ; \\end{aligned} \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} \\max _ { 1 \\leqslant N \\leqslant N _ { 0 } } \\max _ { 1 \\leqslant j \\leqslant m _ { N } } \\left \\vert \\alpha _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) \\right \\vert \\leqslant b _ { n } = 2 \\sigma _ { \\mathcal { F } } \\kappa _ { N _ { 0 } } \\sqrt { L \\circ L ( n ) } . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} \\Delta _ { \\min , A } = 1 - \\sum \\limits _ { i = 1 } ^ { k - 1 } \\gamma ( a _ { i + 1 } - a _ { i } ) , \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\int _ X | R i c ( \\tilde \\omega _ i ) + \\tilde \\omega _ i | ^ 2 _ { \\tilde \\omega _ i } \\tilde \\omega _ i ^ n & = \\int _ X | 3 n \\mu _ { \\alpha _ i } \\omega _ { i } | ^ 2 _ { \\tilde \\omega _ i } \\tilde \\omega _ i ^ n \\\\ & \\le 9 n ^ 3 \\int _ X \\tilde \\omega _ i ^ n \\\\ & \\to 9 n ^ 3 ( 2 \\pi ) ^ n \\int _ X c _ 1 ( K _ X ) ^ n \\\\ & = 0 \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} ( f _ \\lambda \\odot g _ \\lambda ) h = \\langle h , g _ \\lambda \\rangle f _ \\lambda , \\qquad \\forall \\ , h \\in \\ell _ 2 ( \\N ^ N ) . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} t = 2 \\left ( \\left \\lfloor \\frac { m + 1 } { 2 } \\right \\rfloor - \\frac { d - 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} M _ { ( n ) , s } ( t ) = \\sum _ { k = 1 } ^ n M _ k ( t ) \\otimes e _ k ( s ) \\ 1 _ { [ 0 , t ) } ( s ) . \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } V , W ) = \\breve { g } ( V , \\breve { J } W ) = 0 , \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} g _ n \\colon [ 0 , T ] \\to U , g _ n ( t ) : = \\sum \\limits _ { k = 0 } ^ { N _ n - 1 } g ( t ^ n _ k ) \\ 1 _ { [ t ^ n _ k , t ^ n _ { k + 1 } ) } ( t ) + \\ 1 _ { \\{ T \\} } ( t ) g ( T ) . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} w ( x , t ) \\equiv w ( r ( x ) , t ) : = \\begin{cases} u ( x , t ) & \\textrm { i n } \\ ; \\ ; ( M \\setminus B _ 1 ) \\times ( 0 , T ) \\ , , \\\\ v ( x , t ) & \\textrm { i n } \\ ; \\ ; B _ 1 \\times ( 0 , T ) \\ , , \\end{cases} \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} \\begin{aligned} \\| A f \\| _ { \\infty } & \\leq 2 \\| f \\| _ { ( 2 ) } \\sup _ { | x | \\leq R } \\left ( | b ( x ) | + | Q ( x ) | + \\int _ { y \\neq 0 } \\min \\{ | y | ^ 2 , 1 \\} \\ , \\nu ( x , d y ) \\right ) \\\\ & + \\| f \\| _ { \\infty } \\sup _ { | x | > R } \\nu ( x , \\overline { B ( - x , R ) } ) . \\end{aligned} \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} X _ t ( x _ { t - 1 } , \\xi _ t ) : = \\{ x _ t \\in \\mathbb { R } ^ n : A _ { t } x _ { t } + B _ { t } x _ { t - 1 } = b _ t , \\ , x _ t \\geq 0 \\} , \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} & \\big ( ( 1 - \\lambda ) I + \\lambda A ^ { \\frac { 1 } { 2 } } B ^ { - 1 } A ^ { \\frac { 1 } { 2 } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ( ( 1 - \\lambda ) I + \\lambda A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } ) \\big ( ( 1 - \\lambda ) I + \\lambda A ^ { \\frac { 1 } { 2 } } B ^ { - 1 } A ^ { \\frac { 1 } { 2 } } \\big ) ^ { \\frac { 1 } { 2 } } - I \\\\ & = \\lambda ( 1 - \\lambda ) ( I - A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } ) A ^ { \\frac { 1 } { 2 } } B ^ { - 1 } A ^ { \\frac { 1 } { 2 } } ( I - A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} V ' _ { k _ { \\ell } x } ( M ) = \\frac { 1 } { 4 \\pi r _ \\ell } \\int \\limits _ { \\partial B _ { r _ \\ell } ( { O } ) } \\ , V ' _ { k _ { \\ell } x } ( P ) \\frac { r ^ 2 _ { \\ell } - \\| { O } M \\| ^ 2 } { \\| M P \\| ^ 3 } \\ , d m _ { 2 } ( P ) , \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} v _ { n } ^ { \\varepsilon } \\left ( x , t \\right ) = \\sum _ { j = 1 } ^ { n } V _ { j n } ^ { \\varepsilon } \\left ( t \\right ) \\phi _ { j } \\left ( x \\right ) \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { 4 ^ { p - 4 } ( 1 + 4 ^ { 1 / ( p - 2 ) } ) ^ { 2 ( p - 2 ) } } . \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} w '' = \\Delta w - k w ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} \\mathcal I _ v ^ { \\sharp } ( \\mathbf h , \\mathbf g , \\pmb { \\phi } ) = \\frac { L ( 1 , \\pi _ v , \\mathrm { a d } ) L ( 1 , \\tau _ v , \\mathrm { a d } ) } { L ( 1 / 2 , \\pi _ v \\times \\mathrm { a d } \\tau _ v ) } \\alpha _ v ^ { \\sharp } ( \\mathbf h , \\mathbf g , \\pmb { \\phi } ) , \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} \\Psi _ { V , W } ( v \\otimes w ) = q ^ { \\deg ( v ) \\deg ( w ) } w \\otimes v , \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\dot { \\rho } ^ { 2 } + \\frac { \\rho ^ { 2 } } { 8 } v ^ { 2 } + \\frac { \\omega ^ { 2 } } { 2 \\rho ^ { 2 } } - \\frac { U ^ { \\ast } } { \\rho } = h \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} \\langle H ^ { - 1 } M H u , H ^ { - 1 } M H v \\rangle _ E & = \\langle M H ^ { - 1 } u , M H ^ { - 1 } v \\rangle _ \\mathcal G \\\\ & = \\langle H ^ { - 1 } u , H ^ { - 1 } v \\rangle _ \\mathcal G \\\\ & = \\langle u , v \\rangle _ E \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Z ^ d } \\tilde \\mu _ { k } ( x ) ~ = ~ \\sum _ { x \\in \\Z ^ d } \\mu _ k ( x ) \\exp ( \\langle \\alpha _ { k } , y - x \\rangle ) ~ = ~ R _ k ( \\alpha _ { k } ) ~ = ~ 1 . \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} \\int \\partial E = \\int \\left ( D + D ' \\right ) = g _ q \\otimes c _ r \\otimes w _ v \\otimes z + g _ { q ' } \\otimes w _ r \\otimes c _ v \\otimes z , \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} ( 1 + b ) ^ 3 ( 1 + a + b ) ( a ^ 2 + b + b ^ 2 ) = 0 . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} ( v _ { ( m _ i - 1 ) i } ( x ) + ( - 1 ) ^ { m _ i } v _ { ( m _ i - 1 ) i } ( x ) ) ^ * r _ i ( x ) + ( 2 \\ell _ i - m _ i ) v _ { m _ i i } ' ( x ) r _ i ( x ) = m _ i r _ i ' ( x ) v _ { m _ i i } ( x ) . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} K _ { n } ( x , y ) : = \\sum _ { k = 0 } ^ { n - 1 } \\frac { p _ { k } ( x ) p _ { k } ( y ) } { \\langle { \\bf u } , p ^ 2 _ k \\rangle } . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} \\dim ( \\langle W _ { i - 4 l _ 4 - 1 } , w _ i , w _ i \\theta \\rangle ) = \\dim ( W _ { i - 4 l _ 4 - 1 } ) + 2 , \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} \\varphi _ H ( x ) = \\int _ 1 ^ { \\infty } \\int _ 0 ^ { \\infty } \\xi x J _ 0 ( \\xi x ) J _ 0 ( \\xi y ) \\varphi ( y ) d y d \\xi , \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} [ u ] ^ { p } _ { s , p , q } = \\int _ { \\mathbb { G } } \\int _ { \\mathbb { G } } \\frac { | u ( x ) - u ( y ) | ^ { p } } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d x d y < + \\infty . \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} M _ i = \\begin{bmatrix} 1 & c _ i \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} C = \\begin{bmatrix} \\cos \\bigl ( \\frac { ( i - j ) \\pi } { n } \\bigr ) \\end{bmatrix} . \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} U _ { r } = \\frac { 1 } { r } V _ { r } + O \\ ( N x ^ { - 1 } ( \\log N ) ^ { \\nu - 1 } \\ ) \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\mathcal { O } _ { E _ { \\tau } } [ G _ n ] _ { \\varphi } = \\mathcal { O } _ { L _ { \\tau , n } } [ G _ n ] _ { \\varphi } . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} M _ X = \\begin{array} { c | c c c c } \\nearrow & A & B & C & D \\\\ \\hline A & 0 & \\infty & 3 & \\infty \\\\ B & 2 & 0 & \\infty & 5 \\\\ C & \\infty & 3 & 0 & \\infty \\\\ D & \\infty & \\infty & 6 & 0 \\end{array} \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} E _ { \\mu _ 2 } ( { \\bf { h } } _ { n _ k } ) = E _ { \\mu _ 2 } ( { \\bf { w } } _ { k } ) + E _ { \\mu _ 2 } ( { \\bf { h } } _ { k , 1 } ) + E _ { \\mu _ 2 } ( { \\bf { h } } _ { k , 2 } ) + N _ 5 + N _ 6 + N _ 7 , \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} L r _ l - q _ 2 r _ l = ( q _ 2 - q _ 1 ) ( 1 + r _ l ( q _ 1 ) ) . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{gather*} u ( \\tau ) = ( \\eta ( 2 \\tau ) / \\eta ( \\tau ) ) ^ { 2 4 } \\end{gather*}"}
-{"id": "904.png", "formula": "\\begin{align*} \\big ( \\sigma _ 1 \\sigma _ 2 + \\sigma _ 4 \\sigma _ 5 \\big ) \\sigma _ 3 = \\sigma _ 1 \\sigma _ 4 + \\sigma _ 2 \\sigma _ 5 \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} \\begin{pmatrix} v _ { i - 1 } \\\\ \\vdots \\\\ v _ { i - a } \\end{pmatrix} = \\Phi _ i \\begin{pmatrix} v _ i \\\\ \\vdots \\\\ v _ { i - a + 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\mathcal W _ { \\mathfrak R _ M \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) = \\mathcal W _ { \\mathbf F _ { \\chi } , B } ( g _ { \\infty } ) . \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} { F _ { S } } \\left ( { x , { \\mu } } \\right ) = \\left \\{ { \\begin{array} { l l } \\left \\langle c ^ { \\ast } , x \\right \\rangle { , } & { \\mathrm { i f } \\ ; \\ \\left \\langle a _ { s } ^ { \\ast } , x \\right \\rangle \\leq b _ { s } - { \\mu _ { s } } , \\ ; \\forall s \\in S , } \\\\ { + \\infty , } & { \\mathrm { e l s e , } } \\end{array} } \\right . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{gather*} P R ^ 1 ( R m ) _ { [ 0 ] } \\otimes Q \\big ( R ^ 2 ( R m ) _ { [ 1 ] } \\kappa \\big ) ^ 1 S ^ 2 ( P ) \\otimes R \\big ( R ^ 2 ( R m ) _ { [ 1 ] } \\kappa \\big ) ^ 2 S ^ 2 ( Q ) \\\\ \\qquad { } = R ^ 1 \\big ( Q R ^ 1 m _ { [ 0 ] } \\big ) _ { [ 0 ] } \\otimes R ^ 2 \\big ( Q R ^ 1 m _ { [ 0 ] } \\big ) _ { [ 1 ] } \\kappa S ^ 2 ( P ) \\otimes R R ^ 2 m _ { [ 1 ] } \\kappa , \\end{gather*}"}
-{"id": "6086.png", "formula": "\\begin{align*} d _ { P } ^ { 2 } ( h _ { ( j , N ) } ^ { - } , h _ { ( j , N ) } ^ { + } ) & = \\int _ { A _ { j } ^ { ( N ) } } \\left ( g _ { + } - g _ { - } + \\mathbb { E } ( g _ { + } - g _ { - } \\ ; | \\ ; A _ { j } ^ { ( N ) } ) \\right ) ^ { 2 } d P \\\\ & = P ( 1 _ { A _ { j } ^ { ( N ) } } ( g _ { + } - g _ { - } ) ^ { 2 } ) + P ( A _ { j } ^ { ( N ) } ) \\mathbb { E } ( g _ { + } - g _ { - } \\ ; | \\ ; A _ { j } ^ { ( N ) } ) ^ { 2 } \\\\ & + 2 \\mathbb { E } ( g _ { + } - g _ { - } \\ ; | \\ ; A _ { j } ^ { ( N ) } ) P ( 1 _ { A _ { j } ^ { ( N ) } } ( g _ { + } - g _ { - } ) ) . \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} Q ( x , e ) ~ = ~ Q ( x , e ) f _ n ( e ) ~ \\leq ~ f _ n ( x ) ~ \\leq ~ \\frac { 1 } { Q ( e , x ) } f _ n ( e ) ~ = ~ \\frac { 1 } { Q ( e , x ) } , \\forall n \\geq 0 , \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} \\ell ' & = b _ 1 c _ 1 + b _ 1 c _ 2 + b _ 2 c _ 2 = b _ 1 c + b _ 2 c _ 2 = b _ 1 c _ 1 + b c _ 2 , \\\\ m ' & = a _ 1 c _ 1 + a _ 2 c _ 1 + a _ 2 c _ 2 = a c _ 1 + a _ 2 c _ 2 = a _ 1 c _ 1 + a _ 2 c , \\\\ n ' & = a _ 1 b _ 1 + a _ 1 b _ 2 + a _ 2 b _ 2 = a _ 1 b + a _ 2 b _ 2 = a _ 1 b _ 1 + a b _ 2 , \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} \\frac { H ( w ) } { F ( w ) } \\int \\frac { G _ 1 ( z ) F ( z ) \\overline { F _ 2 ( z ) } } { z - w } d \\nu ( z ) = \\int \\frac { G _ 1 ( z ) H ( z ) \\overline { F _ 2 ( z ) } } { z - w } d \\nu ( z ) \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} \\partial \\Gamma ^ { n - 2 , \\ , n } _ \\omega = - \\bar \\partial \\omega ^ { n - 1 } . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} F ( q \\ , | \\ , \\beta , \\lambda _ 1 , \\lambda _ 2 ) = F ( q \\ , \\big | \\ , \\frac { 1 } { \\beta } , \\lambda _ 1 , \\lambda _ 2 ) , \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} n _ j > n _ 1 \\tfrac { \\binom { M d _ j } { M d _ 1 } } { \\binom { d _ j } { M d _ 1 } } , \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} \\mathcal { D } ( W ) \\otimes _ { \\mathcal Z ( W ) } \\mathcal { F } ( W ) \\cong \\bigoplus _ { i = 1 } ^ r M _ { n _ i } ( \\mathcal F _ i ( W ) ) , \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} \\max _ { f \\in F _ { A \\Gamma } ( n ) } \\lambda ( f ) = 2 ^ n - 1 . \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} h s = \\left ( \\begin{array} { c c } - b & a \\\\ - d & c \\end{array} \\right ) , h \\beta _ m = h s \\alpha _ m = \\left ( \\begin{array} { c c } - b p ^ m & a p ^ { - m } \\\\ - d p ^ m & c p ^ { - m } \\end{array} \\right ) , \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} \\sup _ { x \\in \\real ^ d } \\int _ { \\real ^ d } \\big | K _ \\rho ( x , y ) \\big | ^ 2 \\ d y \\underset { \\rho \\rightarrow 0 } { = } \\mathcal { O } \\big ( \\lambda ( \\rho ) \\big ) . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} { m _ { { v _ { n - 1 } } \\nearrow } } & = { m _ { \\searrow { v _ { n - 1 } } } } { m _ { \\uparrow { v _ { n - 1 } } } } \\\\ { m _ { \\searrow { v _ n } } } & = \\int d { { \\tilde \\theta } _ { n - 1 } } \\sum \\limits _ { { s _ { n - 1 } } } { { m _ { { v _ { n - 1 } } \\nearrow } } } \\psi \\left ( { { v _ { n { \\rm { - 1 } } } } , { v _ n } } \\right ) \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\displaystyle \\mathbb { E } \\sum _ { \\substack { j \\in J _ { K } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) = \\displaystyle \\sum _ { \\substack { j \\in J _ { K } } } \\mathbb { E } \\Phi ( B _ { t _ { j } } , t _ { j } ) \\mathbb { E } ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) = 0 \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} \\chi ( M _ k ) = m \\chi ( M ) + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( \\chi ( \\Sigma ^ y _ { \\ell } ) - b ^ y _ { \\ell } ) = m \\chi ( M ) - 2 \\sum _ { y \\in \\mathcal { Y } } J _ y \\geq m \\chi ( M ) - 2 j . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} L _ k = h _ k ^ { - 1 } ( \\{ \\eta \\} ) \\cap Z _ k \\cap B , \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\mu ( A ) = \\sum \\limits _ { j = 1 } ^ { l } \\lim _ { r \\downarrow 0 } \\mu ( A \\cap B _ r ( x _ j ) ) \\leq 4 \\pi k \\mathcal { H } ^ 0 ( { S } ) . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} \\begin{aligned} & x ( t + 1 ) = A x ( t ) + B w ( t ) , & x ( 0 ) = 0 \\\\ & y ( t ) = C x ( t ) + D w ( t ) + a ( t ) , \\end{aligned} \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\eta _ \\eth ( c ) ( t ) = \\sum _ { j = 0 } ^ N c _ j \\mathcal { H } _ \\eth ^ j ( t ) = \\left ( \\eth \\right ) ^ { \\frac { 1 } { 4 } } \\sum _ { j = 0 } ^ N \\mathcal { H } _ 1 ^ j ( \\sqrt { \\eth } t ) = \\left ( \\eth \\right ) ^ { \\frac { 1 } { 4 } } \\eta _ 1 ( c ) ( \\sqrt { \\eth } t ) \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\partial f _ n ( M ) = \\partial A ^ * _ { \\mathcal { C R } } \\left ( f _ 1 X _ n ( M ) - U _ n ( M ) \\right ) = \\left ( f _ 1 X _ n - U _ n \\right ) ( M ) , \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} _ A ^ { \\ast } F _ A = [ _ A \\Phi , \\Phi ] , \\quad \\Delta _ A \\Phi = 0 , \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\| \\phi \\| _ { c b } \\leq \\| A \\| _ { S _ 2 } \\| B \\| _ { S _ 2 } = \\| T \\| _ { S _ 1 } . \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} P ( V _ { n } = k ) = \\sum \\limits ^ { n } _ { \\substack { l = 0 } } ( - 1 ) ^ { l } \\binom { k + l } { k } S _ { k + l , n } . \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\mathcal { P } } x ^ { | \\lambda | } \\prod _ { h \\in \\mathcal { H } ( \\lambda ) } \\dfrac { 1 } { h ^ { 2 } } = e ^ { x } , \\\\ \\sum _ { \\mu \\in \\mathcal { D } } x ^ { | \\mu | } \\prod _ { h \\in \\mathcal { H } ( \\mu ) } \\dfrac { 1 } { 2 ^ { \\ell ( \\mu ) } h ^ { 2 } } = e ^ { \\frac { x } { 2 } } , \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } ( 1 - { \\rm e } ^ { - \\beta \\ , x } ) n ( u , d x ) = \\lim _ { w \\rightarrow u - } \\frac { 1 - { \\bf E } _ w ( { \\rm e } ^ { - \\beta X _ u } ) } { S ( u ) - S ( w ) } . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} \\dd { x } { t } = \\sum _ { i \\to i ' \\in E } k _ { i \\to i ' } \\ , x ^ { y ( i ) } \\big ( y ( i ' ) - y ( i ) \\big ) . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} ( 1 - \\gamma ) \\mathcal L _ { \\mu _ 2 } \\nu = \\frac { \\epsilon } { 2 \\gamma } \\nu G _ 0 ( \\nu ) + \\omega J _ b G _ 0 ( \\nu ) \\equiv G _ 1 ( v ) . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} Z _ k : = Z \\cap p ^ { - 1 } ( W _ k ) \\cap V _ { k - 1 } \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\big ( \\left \\| u ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - u ^ { * * } ( t ) \\right \\| _ { C ^ 0 ( \\bar \\Omega ) } + \\left \\| v ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - v ^ { * * } ( t ) \\right \\| _ { C ^ 0 ( \\bar \\Omega ) } \\big ) = 0 , \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} X _ { ( \\lambda ) } \\cdot X _ { ( \\mu ) } = X _ { \\left ( \\begin{smallmatrix} \\lambda & 0 \\\\ 0 & \\mu \\end{smallmatrix} \\right ) } \\ ; \\ ; { X _ { ( \\lambda ) } } ^ 2 = X _ { ( \\lambda ) } + ( q + 1 ) q X _ { \\left ( \\begin{smallmatrix} \\lambda & 0 \\\\ 0 & \\lambda \\end{smallmatrix} \\right ) } \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} \\Lambda _ 1 & = 2 \\left | \\sum _ m \\sum _ { \\substack { i < j \\\\ i , j \\in I _ m \\cup I _ { m + 1 } } } ( f ( y _ i ) - f ( y _ j ) ) ( x _ { i j } h _ { j i } - x _ { j i } h _ { i j } ) \\right | \\\\ \\Lambda _ 2 & = 2 \\left | \\sum _ m \\sum _ { i \\in I _ m } \\sum _ { l \\geq m + 2 } \\sum _ { j \\in I _ l } ( f ( y _ i ) - f ( y _ j ) ) ( x _ { i j } h _ { j i } - x _ { j i } h _ { i j } ) \\right | \\\\ \\Lambda _ 3 & = 2 \\left | \\sum _ { m = 1 } ^ { k - 1 } \\sum _ { \\substack { i < j \\\\ i , j \\in I _ m } } ( f ( y _ i ) - f ( y _ j ) ) ( x _ { i j } h _ { j i } - x _ { j i } h _ { i j } ) \\right | . \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} \\mathcal O & = \\hat \\theta ^ { - 1 } \\mathcal O ^ * = \\hat \\theta ^ { - 1 } \\sigma \\ ( \\pi ' , R _ \\alpha \\circ \\pi , \\ , Z \\circ \\pi ; \\alpha \\in \\ker A \\ ) \\\\ & = \\sigma \\ ( \\pi ' ( \\hat \\theta ) , R _ \\alpha \\circ \\pi ( \\hat \\theta ) , \\ , Z \\circ \\pi ( \\hat \\theta ) ; \\alpha \\in \\ker A \\ ) = \\sigma \\ ( A X , \\ , R _ \\alpha ( \\theta ) , \\ , Z ( \\theta ) ; \\alpha \\in \\ker A \\ ) . \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} \\tilde \\mu _ { t , x } ( J ) : = \\sigma _ N ( J \\cap J ( t , x ) ) / \\sigma _ N ( J ( t , x ) ) , ( t , x ) \\in Y ; \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} [ \\rho _ g ] [ \\rho _ h ] & = [ \\rho _ { g + h } ] \\ , , & [ \\rho _ \\pm ] [ \\rho _ g ] & = [ \\rho _ g ] [ \\rho _ \\pm ] = [ \\rho _ \\pm ] \\ , , & [ \\rho _ \\pm ] [ \\rho _ \\pm ] & = \\bigoplus _ { g \\in G } [ \\rho _ g ] \\ , . \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\eta _ r ( x , y ) = \\eta _ r ( x _ 0 t ^ 0 , y _ { 2 n _ 0 } t ^ { 2 n _ 0 } ) + \\eta _ r ( x _ 0 t ^ 0 , y ' ) + \\eta _ r ( x ' , y ) \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} | \\lambda _ i ^ j ( P ) - \\nu _ { i - 1 } ^ j ( P ) | < \\delta _ { i - 1 } ~ \\textrm { f o r a l l } ~ \\lambda _ i ^ j \\in \\mathfrak { M } _ { \\sigma } ( M _ i ^ j ) ~ \\textrm { a n d } ~ P \\in \\bigcup _ { t \\leq t _ { i - 1 } } B l _ t ( M _ { i - 1 } ) ( j = 1 , \\cdots , k ) . \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\langle A ^ * B e _ { ( k _ 1 , . . . , k _ N ) } , e _ { ( m _ 1 , . . . , m _ N ) } \\rangle = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } \\left ( \\vec { d } ( x , y ) + 2 ( j _ 1 , . . . , j _ N ) + 2 \\chi ^ I \\right ) , \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} f ( t ) \\coloneqq \\frac { \\binom { k t + i } { k a } ^ j } { \\binom { j t + i } { j a } ^ { k } } . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\ \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ { j } f _ j ( \\rho ) \\leq \\epsilon \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ j f _ j ( R ) + C _ \\epsilon ( R - \\rho ) ^ { k + \\alpha } [ \\nabla ^ k \\sigma ] _ { C ^ { \\alpha } ( B ^ { g _ P } ( p , R ) ) } + C _ \\epsilon \\| \\sigma \\| _ { L ^ \\infty ( B ^ { g _ P } ( p , R ) ) } . \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} \\varphi ( s ) - \\frac { 1 } { 2 } P ( r ) ( \\hat { X } ( r ) ) ^ { 2 } = o ( \\left \\vert x ^ { \\prime } - \\bar { X } ^ { t , x ; \\bar { u } } ( s ) \\right \\vert ^ { 2 } ) , P a . s . . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\breve { g } ( \\breve { J } \\xi , \\xi ) = \\breve { g } ( \\xi , \\breve { J } \\xi ) = 0 . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} f _ 1 ( x , y ) & = y _ 1 ^ 2 - y _ 2 ^ 2 - x _ 1 \\geq 0 \\\\ f _ 2 ( x , y ) & = y _ 1 \\cdot y _ 2 - x _ 2 \\geq 0 \\\\ y \\geq 0 , & \\langle f ( x , y ) , y \\rangle = 0 \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} S _ { a , b } \\left ( p , k \\right ) = \\sum _ { j = 0 } ^ { p } \\binom { p } { j } a ^ { j } b ^ { p - j } S \\left ( j , k \\right ) . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} u + v = b + \\frac { b + \\frac { \\alpha } { d _ 1 } + \\frac { \\beta } { d _ 2 } } { r } . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} X _ { i } = \\frac { 1 } { 2 } [ E _ i , [ E _ i , X _ { i + 1 } ] ] Y _ { i } = \\frac { 1 } { 2 } [ F _ i , [ F _ i , Y _ { i + 1 } ] ] \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} \\varphi ^ { \\ast } \\nabla _ A = \\pi ^ { \\ast } \\nabla _ { \\infty } + a , \\quad \\quad \\lvert \\nabla _ { \\infty } ^ j a \\rvert = O ( \\rho ^ { - 1 - j - \\eta } ) , \\quad \\forall j \\in \\mathbb { N } _ 0 \\eta > 0 , \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} ( \\alpha _ i , \\gamma _ j ) = ( \\alpha _ j , \\gamma _ i ) \\ \\mathrm { f o r \\ a n y } \\ 1 \\leq i , j \\leq n . \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} d = \\gcd ( \\ell , m ) . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} \\mu _ 1 & = \\max _ { f } \\frac { ( \\delta f , \\delta f ) _ H } { ( f , f ) _ V } \\\\ & = \\max _ { f } \\frac { \\sum _ { h \\in H } \\delta f ( h ) ^ 2 } { \\sum _ { v \\in V } \\deg v \\cdot f ( v ) ^ 2 } \\\\ & = \\max _ { f : \\sum _ { v \\in V } \\deg v \\cdot f ( v ) ^ 2 = 1 } \\sum _ { h \\in H } \\delta f ( h ) ^ 2 \\\\ & = \\max _ { f : \\sum _ { v \\in V } \\deg v \\cdot f ( v ) ^ 2 = 1 } \\sum _ { h \\in H } \\biggl ( \\sum _ { v _ i h } f ( v _ i ) - \\sum _ { v ^ j h } f ( v ^ j ) \\biggr ) ^ 2 \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} L ^ G _ { \\nu } & : = G \\times _ T \\C _ { \\nu } \\to G / T ; \\\\ L ^ { G _ c } _ { \\nu } & : = G _ c \\times _ T \\C _ { \\nu } \\to G _ c / T . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! T _ { s } \\ ! = \\ ! ( 1 \\ ! - \\ ! \\theta ) R \\ ! = \\ ! ( 1 \\ ! - \\ ! \\theta ) \\bigg [ \\ ! \\log \\bigg ( 1 \\ ! + \\ ! \\frac { p _ { s } | h _ { s } | ^ { 2 } } { \\sigma _ { s } ^ { 2 } } \\bigg ) \\ ! - \\ ! \\log ( 1 \\ ! + \\ ! \\rho _ { e } ) \\ ! \\bigg ] . \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} E _ N = ( e _ 1 , \\dots , e _ N ) e _ n = \\left ( \\frac { n } { p } \\right ) \\ n = 1 , \\dots , N , \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} g | _ { \\ell } [ \\gamma ] = \\chi ( \\gamma ) g \\gamma \\in \\Gamma _ 0 ( N ) , \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) = 2 ^ { k + 1 - \\nu ( M ) } C ( N , M , \\chi ) \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } \\frac { | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 } { \\langle g , g \\rangle ^ 2 } , \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow + \\infty } \\displaystyle \\int _ { \\Omega } C ( x ) ( | u _ { n } ( x ) | ^ { p ( x ) - 3 } u _ { n } ( x ) - | u _ { 0 } ( x ) | ^ { p ( x ) - 3 } u ( x ) ) ( u _ { n } ( x ) - u _ { 0 } ( x ) ) d x = 0 , \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\gamma _ { d , \\alpha } = \\min \\Big \\{ \\frac { 2 } { d } , \\frac { 2 } { \\alpha ' } \\Big \\} , \\ \\ \\ \\theta _ { d , \\alpha } = \\min \\Big \\{ \\frac { 1 } { d } , \\frac { 1 } { \\alpha } - \\frac { 1 } { 2 } \\Big \\} . \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} M _ 2 \\left ( \\kappa ; \\sqrt { v } \\boldsymbol { N } \\right ) = - \\frac 1 { \\pi ^ 2 } e ^ { - \\pi v \\left ( N _ 1 ^ 2 + N _ 2 ^ 2 \\right ) } \\int _ { \\R ^ 2 } \\frac { e ^ { - \\pi v w _ 1 ^ 2 - \\pi v w _ 2 ^ 2 } } { \\left ( w _ 2 - i N _ 2 \\right ) \\left ( w _ 1 - \\kappa w _ 2 - i \\left ( N _ 1 - \\kappa N _ 2 \\right ) \\right ) } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} \\theta = \\min \\{ \\| A ^ \\top u + c ^ \\top w + C ^ \\top v \\| : \\| u \\| ^ 2 + \\| v \\| ^ 2 + \\| w \\| ^ 2 = 1 , v \\geq 0 \\} . \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\phi ( \\sigma ) = ( m m p ( 0 , n , 0 , 0 ) + 1 , m m p ( 0 , n - 1 , 0 , 0 ) + 1 , \\hdots , m m p ( 0 , 1 , 0 , 0 ) + 1 ) . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} D _ r ( n ) = \\sum _ { j = r } ^ n { j \\choose r } \\frac { n ! } { ( n - j ) ! } ( - 1 ) ^ { n - j } , \\mbox { } n \\geq r . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} Z ( K ) & = \\int D \\Xi \\ , e ^ { - \\frac { 1 } { 2 } \\Xi \\mathsf { Q } \\Xi + K Q \\Psi } \\ , , & K & = \\begin{pmatrix} k \\\\ \\bar { k } \\\\ j \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} \\int _ { \\Gamma } ^ { } u ( t ) d \\Gamma = \\int _ { \\Gamma } ^ { } u _ { 0 \\Gamma } d \\Gamma \\mbox { f o r a l l ~ } t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\partial _ { e x } \\Lambda : = \\{ z \\in \\mathbb { Z } ^ d : z \\notin \\Lambda , z \\Lambda \\} , \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\nabla \\mathbf { u } ^ + ( \\widetilde { Y } _ i ) ( A _ i - \\widetilde { Y } _ i ) & = ( ( A _ i - \\widetilde { Y } _ i ) ^ T \\otimes I _ 2 ) \\textrm { V e c } ( \\nabla \\mathbf { u } ^ + ( \\widetilde { Y } _ i ) ) \\\\ & = ( 1 - \\tilde { t } _ i ) ( ( A _ i - X ) ^ T \\otimes I _ 2 ) M ^ - ( \\widetilde { Y } _ i ) \\textrm { V e c } ( \\nabla \\mathbf { u } ^ - ( \\widetilde { Y } _ i ) ) . \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} h ( x ) ~ \\geq ~ Q ( x , x \\star u ^ { \\star n } ) h ( x \\star u ^ { \\star n } ) ~ = ~ Q ( x , x \\star u ^ { \\star n } ) T _ u ^ n h ( x ) , \\forall x \\in E \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} e ( N ( 1 ) ) & = ( - 1 ) ^ { n - 1 } \\frac { p ^ { ( n - 1 ) } ( 1 ) } { ( n - 1 ) ! } \\\\ & = ( - 1 ) ^ { n - 1 } \\left [ \\sum _ { t = 1 } ^ { n - 1 } ( - 1 ) ^ t \\cdot t \\cdot \\sum _ { 1 \\leq j _ 1 \\leq \\cdots \\leq j _ { t + 1 } \\leq n } { d _ { j _ 1 } + \\cdots + d _ { j _ { t + 1 } } - 1 \\choose n - 1 } \\right ] + ( - 1 ) ^ { n - 1 } \\\\ & = \\sum _ { t = 1 } ^ { n - 1 } \\left [ ( - 1 ) ^ { n - 1 + t } \\cdot t \\cdot \\sum _ { 1 \\leq j _ 1 \\leq \\cdots \\leq j _ { t + 1 } \\leq n } { d _ { j _ 1 } + \\cdots + d _ { j _ { t + 1 } } - 1 \\choose n - 1 } \\right ] + 1 , \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} M ( t , x ) = \\frac { \\mathbb { E } [ v _ t ' \\ , \\varphi ( x - x _ t ' ) ] } { \\mathbb { E } \\ , \\varphi ( x - x _ t ' ) } \\ , , \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} _ a D _ { t } ^ { \\nu } f ( t ) = [ f ( t ) ] _ { \\nu } = \\frac { 1 } { \\Gamma ( n - \\nu ) } \\frac { d ^ { n } } { d t ^ { n } } \\int _ a ^ t \\frac { f ( s ) } { ( t - s ) ^ { \\nu + 1 - n } } d s , \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} g _ 2 = x ^ { a - a _ 1 } g _ 1 \\pm y ^ { b _ 1 c + b _ 2 } z ^ { ( c _ 2 - 1 ) c } . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\omega _ x ( k ) = \\omega _ y ( l ) \\iff \\exists \\ , j \\in \\N , \\ k = k _ 0 + j , \\ m = m _ 0 + j . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} { \\bf u } = T _ G { \\bf Q } T _ G \\Big [ { \\bf f } - \\Re ( { \\bf u } { \\bf D } ) { \\bf u } \\Big ] - T _ G { \\bf Q } T _ G { \\bf D } p , \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\pi _ F \\colon X \\to X _ F , \\ \\ \\pi _ F ( x ) = \\begin{cases} t _ i & { } \\ x = t _ i \\\\ ( t _ i , t _ { i + 1 } ) _ o & { } \\ x \\in ( t _ i , t _ { i + 1 } ) _ o , \\end{cases} \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} \\delta ( a ) = \\delta ( a ) _ 1 + t \\cdot \\delta ( a ) _ 2 , a \\in Q _ 1 \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} d ( a , b ) = \\det _ { 1 \\leq j , k \\leq n } \\big ( ( a z _ k ^ \\pm ) _ { j - 1 } ( b z _ k ^ \\pm ) _ { n - j } \\big ) . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} c _ r = \\sum _ { k : \\frac { m n } { k } | r } \\sum _ { i j = k } \\frac { 1 } { k } S _ i T _ j . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} \\Omega _ { \\rho } ( j ) = M _ { \\bar { \\rho } \\cup \\{ j \\} } ( j ) \\enspace \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\sqcup _ k N _ k | A _ k : = W | C \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} \\P _ { u _ n } ( X ( k _ n ) = \\hat { u } _ n ) ~ \\geq ~ \\delta ^ { \\kappa | u _ n - \\hat { u } _ n | } \\P _ { \\hat { v } _ n } ( X ( m _ n ) = v _ n ) ~ \\geq ~ \\delta ^ { \\kappa | v _ n - \\hat { v } _ n | } . \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} S ^ \\pm ( \\vec x ) & = \\{ i \\in V : x _ i = \\pm \\| \\vec x \\| _ \\infty \\} , \\\\ S ^ < ( \\vec x ) & = \\{ i \\in V : | x _ i | < \\| \\vec x \\| _ \\infty \\} . \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\hat X _ { k + 1 } ^ n = \\hat X _ k ^ n - \\frac { \\beta } { n } \\sum _ { i = 1 } ^ n \\nabla _ x f ( \\hat X _ k ^ n , W _ { k , i } ) = : \\hat T ^ n _ k ( \\hat X _ k ^ n ) , \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k \\right ) = \\sum _ { j _ { 1 } = 0 } ^ { p _ { 1 } } \\sum _ { j _ { 2 } = 0 } ^ { p _ { 2 } } \\binom { p _ { 1 } } { j _ { 1 } } \\binom { p _ { 2 } } { j _ { 2 } } a _ { 1 } ^ { j _ { 1 } } a _ { 2 } ^ { j _ { 2 } } b _ { 1 } ^ { p _ { 1 } - j _ { 1 } } b _ { 2 } ^ { p _ { 2 } - j _ { 2 } } S \\left ( j _ { 1 } + j _ { 2 } , k \\right ) . \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} G ( u ) - \\frac { 1 } { 3 } F ( u ) = \\frac { \\beta } { 6 } u ^ 6 \\Big ( 1 - \\frac { 5 } { 3 } \\Big ) + G _ { f _ 5 } ( u ) - \\frac { 1 } { 3 } F _ { f _ 5 } ( u ) , \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\gamma ( i , r ; j , s ) = \\widetilde { c } _ { j i } ( - r _ j - r + s ) - \\widetilde { c } _ { j i } ( r _ j - r + s ) , \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\bar { I } ^ { A } { } _ { R } I ^ { R } { } _ { B } = & \\ , \\ , { \\mathbb Y } _ { r } { } ^ A { } _ B \\big ( v ^ s ( \\nabla _ s \\bar { v } ^ r ) _ o + \\tfrac { 1 } { 6 } v ^ r ( \\nabla _ t \\bar { v } ^ t ) - \\tfrac { 1 } { 3 } v ^ r ( \\nabla _ t \\bar { v } ^ t ) \\big ) \\\\ & + \\mathrm { \" l o w e r \\ , \\ , a d j o i n t \\ , \\ , t r a c t o r \\ , \\ , s l o t s \" } \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} y ^ { ( 0 ) } = 1 \\quad y ^ { ( i ) } y ^ { ( j ) } = \\frac { ( i + j ) ! } { i ! j ! } \\ , y ^ { ( i + j ) } i , j \\ge 0 \\ , . \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\| I _ { h , T } \\mathbf { u } - \\mathbf { u } \\| _ { 0 , T _ { \\ast } ^ s } + h | I _ { h , T } \\mathbf { u } - \\mathbf { u } | _ { 1 , T _ { \\ast } ^ s } + h ^ 2 | I _ { h , T } \\mathbf { u } - \\mathbf { u } | _ { 2 , T _ { \\ast } ^ s } \\leqslant C h ^ 2 ( | \\mathbf { u } | _ { 1 , T } + | \\mathbf { u } | _ { 2 , T } ) , ~ s = \\pm , ~ \\forall T \\in \\mathcal { T } _ h ^ i . \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\begin{bmatrix} B _ k ( \\lambda ) & B _ { k - 1 } ( \\lambda ) & \\cdots & B _ { 1 } ( \\lambda ) \\end{bmatrix} & ( \\Lambda _ { k - 1 } ( \\lambda ^ \\ell ) \\otimes I _ n ) = \\\\ & ( \\Lambda _ { k - 1 } ( \\lambda ^ \\ell ) ^ T \\otimes I _ m ) \\begin{bmatrix} B _ s ( \\lambda ) \\\\ B _ { k - 1 } ( \\lambda ) \\\\ \\vdots \\\\ B _ 1 ( \\lambda ) \\end{bmatrix} = P ( \\lambda ) , \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} \\phi _ { \\mathbf h , 2 } ( z ) = \\pmb { \\phi } _ 2 ( x _ 3 ) \\phi _ { \\breve { \\mathbf g } , 2 } ( X ) = \\mathbf 1 _ { 2 \\Z _ 2 } ( x _ 1 ) \\mathbf 1 _ { 2 \\Z _ 2 } ( x _ 2 ) \\mathbf 1 _ { \\Z _ 2 } ( x _ 3 ) \\mathbf 1 _ { 2 \\Z _ 2 } ( x _ 4 ) \\mathbf 1 _ { 2 \\Z _ 2 } ( x _ 5 ) . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\mathfrak { N } = \\sum _ { m = 0 } ^ { N - 1 } \\binom { N - 1 } { m } = 2 ^ { N - 1 } , \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} \\alpha = \\begin{pmatrix} A _ 1 & A _ 2 & B _ 1 \\\\ & A _ 1 \\\\ & A _ 3 & B _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} a _ i = \\frac { \\abs { ( X \\times X ) \\cap R _ i } } { \\abs { X } } , \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} M '' _ { U _ { ( i ) } } ( t ) & = \\mathsf { E } U _ { ( i ) } ^ 2 e ^ { U _ { ( i ) } t } = \\mu _ i ^ 2 \\bigl ( e ^ t - 1 \\bigr ) ^ 2 M _ { U _ { ( i ) } } ( t ) + \\mu _ i e ^ t M _ { U _ { ( i ) } } ( t ) , \\\\ \\frac { \\mathsf { E } U _ { ( i ) } ^ 2 e ^ { ( k + 1 ) \\beta ^ \\top U } } { M _ { k , \\beta } } & = \\mu _ i ^ 2 \\bigl ( e ^ { ( k + 1 ) \\beta _ i } - 1 \\bigr ) ^ 2 + \\mu _ i e ^ { ( k + 1 ) \\beta _ i } \\le \\textrm { c o n s t } \\cdot e ^ { 2 ( k + 1 ) \\cdot | \\beta _ i | } , \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} B R ( 0 ) ^ { 1 / 2 } \\phi = B R ( 0 ) ^ { 1 / 2 } \\lim _ { n \\rightarrow \\infty } \\psi _ n = B \\lim _ { n \\rightarrow \\infty } R ( 0 ) ^ { 1 / 2 } \\psi _ n , \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{gather*} { \\bf E } \\bigl [ M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] = \\frac { G ( 2 - q + \\lambda _ 1 ) } { G ( 2 + \\lambda _ 1 ) } \\frac { G ( 2 - q + \\lambda _ 2 ) } { G ( 2 + \\lambda _ 2 ) } \\frac { G ( 1 ) } { G ( - q + 1 ) } \\frac { G ( 2 + \\lambda _ 1 + \\lambda _ 2 ) } { G ( 2 - q + \\lambda _ 1 + \\lambda _ 2 ) } . \\end{gather*}"}
-{"id": "6577.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } F _ 0 ( t ) = \\int _ { \\Omega } u _ { t t } \\phi _ 0 \\ , d x = \\int _ { \\Omega } | u | ^ p \\phi _ 0 \\ , d x . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} P \\left ( B _ { n , N _ { 0 } } ^ { c } \\right ) \\leqslant \\ \\sum _ { N = 1 } ^ { N _ { 0 } } m _ { N } \\left ( 1 - p _ { N } \\right ) ^ { n } \\leqslant S _ { N _ { 0 } } \\left ( 1 - p _ { ( N _ { 0 } ) } \\right ) ^ { n } . \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { c c c c c } a _ 1 & a _ 2 & \\hdots & a _ { d - 1 } & a _ d \\\\ 0 & a _ 1 & a _ 2 & \\hdots & a _ { d - 1 } \\\\ & & \\ddots & \\ddots & \\vdots \\\\ 0 & \\hdots & 0 & a _ 1 & a _ 2 \\\\ 0 & \\hdots & 0 & 0 & a _ 1 \\\\ \\end{array} \\right ] . \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} ( a \\otimes b ) ( a ' \\otimes b ' ) = ( - 1 ) ^ { | a ' | | b | } a a ' \\otimes b b ' \\ , . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} N ^ p H ^ { 2 p + k } ( X ) = M ^ p H ^ { 2 p + k } ( X ) \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} v = v ( t ) = d s / d t \\geq 0 \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} 2 b _ k \\leq \\sum \\limits _ { p + q = k } h ^ { p , \\ , q } _ { B C } + \\sum \\limits _ { p + q = k } h ^ { p , \\ , q } _ A \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } \\Phi ' ( \\dot { u } ) = \\lambda f ( t , u , \\dot { u } ) , \\ t \\in [ 0 , 1 ] \\\\ u \\in ( B C ) \\end{cases} \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} \\displaystyle \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) u ( x , t ) + q ( x , t ) u ^ \\alpha ( x , t ) = 0 , \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} \\dot { u } ( 0 ) = \\psi ( c _ 2 ) , \\dot { u } ( 1 ) = \\psi ( \\int _ 0 ^ 1 v ( s ) \\ , d s + c _ 2 ) \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} ( \\tau _ { 1 } , \\ldots , \\tau _ { k } ) \\to ( \\tau _ { 1 , 1 } , \\tau _ { 1 , 2 } , \\ldots , \\tau _ { q , k + 1 - q } ) , \\ ; \\ ; \\ ; \\ ; \\tau _ { i , j } = \\frac { \\tau _ { i } \\tau _ { j } } { \\tau _ { 1 } } , \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} \\dfrac { ( - a q ; q ) _ { \\infty } } { ( a q ; q ) _ { \\infty } } = 1 + \\sum _ { N = 1 } ^ { \\infty } \\left ( \\frac { ( - q ; q ) _ { N - 1 } ( - a q ; q ) _ { N - 1 } } { ( q ; q ) _ { N - 1 } ( a q ; q ) _ { N - 1 } } a ^ { N } q ^ { N ^ { 2 } } + \\dfrac { ( - q ; q ) _ { N } ( - a q ; q ) _ { N } } { ( q ; q ) _ { N } ( a q ; q ) _ { N } } a ^ { N } q ^ { N ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\lim _ n P ^ n T _ u g ( x ) ~ \\leq ~ \\frac { 1 } { \\delta } \\lim _ n P ^ n g ( x ) ~ = ~ 0 , \\forall x \\in E . \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} \\lambda _ \\infty ( S ) = \\ell ( S ) . \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\Gamma \\setminus \\Gamma _ 1 = ( n + \\Gamma _ 1 ) \\setminus \\Gamma _ 1 . \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} \\ln M ' & = N ' ( R ( D - \\gamma _ { N ' } ) + \\gamma _ { N ' } ) + \\ln \\ln N ' \\\\ & = N ' R ( D ) + O ( \\gamma _ { N ' } ) , \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\kappa \\psi _ j ( u ) \\overline { z } ^ j = \\sum _ { j = 0 } ^ \\kappa \\overline { \\psi _ j ( z ) } u ^ j , \\quad \\forall z , u \\in \\C . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\Big ( \\| u ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - u ^ { * * } ( \\cdot , t ) \\| _ { C ^ 0 ( \\bar \\Omega ) } + \\| v ( \\cdot , t ; t _ 0 , u _ 0 , v _ 0 ) - v ^ { * * } ( \\cdot , t ) \\| _ { C ^ 0 ( \\bar \\Omega ) } \\Big ) = 0 , \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} E _ 2 \\ , & = 1 + a ^ 2 + b ^ 2 , \\\\ E _ 1 \\ , & = 1 + a ^ 2 + b ^ 2 , \\\\ E _ 0 \\ , & = a b + k + a ^ 2 k + b ^ 2 k . \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\tilde { \\bar { \\nabla } } _ { X } Y = \\bar { \\nabla } _ X Y + \\eta ( Y ) X - g ( X , Y ) \\xi . \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} \\displaystyle \\int \\limits _ T | f ( t , x _ n ( t ) ) - f ( t , x ( t ) ) | d \\mu ( t ) & = \\int \\limits _ { A ^ + _ n } f ( t , x _ n ( t ) ) - f ( t , x ( t ) ) d \\mu ( t ) & \\\\ & + \\int \\limits _ { A ^ - _ n } f ( t , x ( t ) ) - f ( t , x _ n ( t ) ) d \\mu ( t ) \\\\ & \\leq \\delta / 2 + \\delta / 4 < \\delta ; \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u ( t , x ) = A u ( t , x ) , \\\\ u ( 0 , x ) = f ( x ) , \\end{cases} \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} & \\int _ G \\int _ G \\mu ^ { - 1 / 2 } ( g ' ) \\mu ^ { - 1 / 2 } ( g ) \\langle \\sqrt { \\mathfrak { c } } Q g ( e ) , \\sqrt { \\mathfrak { c } } Q ( g ' ( e ) ) \\rangle _ { H ^ 0 } \\cdot ( g ( h ) , g ' ( h ) ) _ H \\ , d g \\ , d g ' \\\\ & = \\int _ G \\int _ G \\mu ^ { - 1 / 2 } ( g ) \\mu ^ { - 1 / 2 } ( g ' ) \\langle Q \\mathfrak { c } Q g ( e ) , g ' ( e ) \\rangle _ { H ^ 0 } \\cdot ( g ( h ) , g ' ( h ) ) _ H \\ , d g ' \\ , d g . \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} ( v _ 1 , \\vartheta , \\partial _ \\eta w ) | _ { \\eta = 0 } = \\mathbf { 0 } , \\lim _ { \\eta \\rightarrow + \\infty } ( v _ 1 , \\vartheta , w ) ( \\tau , \\xi , \\eta ) = \\mathbf { 0 } . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} \\begin{cases} ( | u ' ( x ) | ^ { p - 2 } u ' ( x ) ) ' + \\omega ( x ) u ( x ) = 0 , \\ , \\ , x \\in ( a , b ) , \\ , \\ , \\ , 1 < p < \\infty , \\\\ u ( a ) = u ( b ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} N _ { \\mathrm { c } \\ , m + 1 } = N _ { m + 1 } - N _ { \\mathrm { d } \\ , m + 1 } - \\sum _ { n = 1 } ^ { m } \\binom { m + 1 } { m - n + 1 } N _ { \\mathrm { d } \\ , m - n + 1 } N _ { \\mathrm { c } n } . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} L _ { m , n } ^ { t , k } : = H _ k \\cap M _ { m , n } ^ t \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} d _ { 2 n + 2 - m } ^ { } = & \\left ( ( - 1 ) ^ { 2 n + 2 - m } + ( - 1 ) ^ { 2 n + 1 - 2 m } \\right ) \\frac { q } { 2 } \\\\ = & \\begin{cases} 0 & \\\\ - q & \\end{cases} \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} \\eqref { e q - t o o } \\times ( v / m j ) & < ( m j v ^ m - m ^ 2 j v ^ 2 w + m j v w ^ 2 + m v - 2 w ) w ^ 2 b ^ 2 \\\\ & + ( m j v ^ m w ^ 2 - m ^ 2 j v ^ 2 w ^ m + m j v w ^ 4 + 2 v ^ 2 w - m v w ^ 2 ) b ^ 2 \\\\ & - 2 ( m j v ^ m w - m ^ 2 j v ^ 2 w ^ 2 + m j v w ^ m + v ^ 2 - w ^ 2 ) w b ^ 2 = 0 . \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} g ( x ) = & g \\bigg ( m \\bigg ( 1 - \\frac { x - m a } { b - m a } \\bigg ) a + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) b \\bigg ) \\\\ \\leq & m \\bigg ( 1 - \\frac { x - m a } { b - m a } \\bigg ) ^ s g ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) g ( b ) . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} P = \\sum _ { J : | J | = k - 1 } H _ J \\hat { F } _ { J \\cup \\{ m \\} } . \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{gather*} L ( f _ { 3 6 } , 1 ) = - 2 \\pi { \\rm i } \\frac { \\rm i } { \\sqrt 3 } \\frac { \\phi ( \\tau _ 0 ) } { 6 } = \\frac 1 { 2 ^ { 4 / 3 } \\cdot 3 } B ( 1 / 3 , 1 / 3 ) , \\end{gather*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\Big ( \\Delta _ f - \\frac { \\partial } { \\partial t } \\Big ) h ( x , t ) + | \\nabla h ( x , t ) | ^ 2 + q ( D e ^ { h } ) ^ { \\alpha - 1 } = 0 . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} I ( x ) & \\equiv x _ 1 I _ 1 + \\ldots + x _ n I _ n \\\\ & = \\{ x _ 1 y _ 1 + \\ldots + x _ n y _ n \\in \\R \\mid y _ 1 \\in I _ 1 , \\ldots , y _ n \\in I _ n \\} \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} E _ { k , N } ^ { \\chi } ( Z , s ) = E ( z , s + k / 2 ; k , \\chi , N ) . \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\big ( P ( B ) \\big ) ^ N \\left ( k ^ d \\lambda ^ k \\right ) = \\sum _ { s = 0 } ^ d P ( \\lambda ) ^ { N + s - d } A _ { d , N , s } \\left ( k ^ s \\lambda ^ k \\right ) \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} A ^ { ( 3 , 1 ) } _ { ( 2 , 2 ) } & = \\{ \\begin{bmatrix} 2 & 1 \\\\ 0 & 1 \\\\ \\end{bmatrix} , \\begin{bmatrix} 1 & 2 \\\\ 1 & 0 \\\\ \\end{bmatrix} \\} A ^ { ( 3 , 1 ) } _ { ( 3 , 1 ) } = \\{ \\begin{bmatrix} 3 & 0 \\\\ 0 & 1 \\\\ \\end{bmatrix} , \\begin{bmatrix} 2 & 1 \\\\ 1 & 0 \\\\ \\end{bmatrix} \\} . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} M = \\begin{pmatrix} c & s & a \\\\ - s & c & b \\\\ \\epsilon & \\delta & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} | F f ( l ) | \\le \\sum _ { i = 1 } ^ M | c _ i | \\frac { \\sqrt { M } } { \\sqrt { a _ 1 ^ i \\cdots a _ d ^ i } } | F \\chi _ { R _ i } ( l ) | \\le \\sqrt { M / A ^ d } \\left ( \\sum _ { i = 1 } ^ M | F \\chi _ { R _ i } ( l ) | ^ 2 \\right ) ^ \\frac 1 2 . \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} ( U _ { \\mathbf { \\tau } } ^ { \\ast } , U _ { \\mathbf { \\nu } } ^ { \\ast } ) ^ { T } = \\left ( \\begin{array} [ c ] { c c } J _ { \\varphi } & J _ { \\theta } \\sin \\varphi \\\\ - J _ { \\theta } \\sin \\varphi & J _ { \\varphi } \\end{array} \\right ) ( U _ { \\varphi } ^ { \\ast } , \\frac { 1 } { \\sin \\varphi } U _ { \\theta } ^ { \\ast } ) ^ { T } \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\breve { J } W = S W + L W , \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} L _ { i } ( x _ { i } , \\lambda _ { i } ) = f _ { i } ( x _ { i } ) + \\lambda _ { i } g _ { i } ( x _ { i } ) . \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{gather*} L \\big ( \\psi _ v ^ 2 , s - 1 \\big ) = \\frac { 1 } { 1 - ( a - b { \\rm i } ) ^ 2 N v ^ { - s } } . \\end{gather*}"}
-{"id": "3055.png", "formula": "\\begin{align*} ( a , g ) ^ { - 1 } = ( - g ^ { - 1 } . a - g ^ { - 1 } . \\omega ( g , g ^ { - 1 } ) , g ^ { - 1 } ) \\ ; \\ ; \\mbox { f o r $ ( a , g ) \\in A \\times _ \\omega G $ . } \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} \\tau ( w ) = \\sum _ { 1 \\le | \\alpha _ 0 | \\le N - 1 } c _ { \\alpha _ 0 } ( \\tau ) w ^ { \\alpha _ 0 } + \\sum _ { | \\beta _ 0 | = N } c _ { \\beta _ 0 } ( \\tau , w ) w ^ { \\beta _ 0 } . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} { \\left ( { { M ^ { \\varepsilon } } { F _ { u } } } \\right ) \\left ( { { x ^ { \\ast } } , y _ { u } ^ { \\ast } } \\right ) = \\left \\{ { \\begin{array} { l l } { \\left ( { { M ^ { \\varepsilon } } { f _ { u } } } \\right ) \\left ( { x ^ { \\ast } } \\right ) , } & { \\mathrm { i f } \\ \\ y _ { u } ^ { \\ast } = 0 _ { u } ^ { \\ast } , } \\\\ \\emptyset , & . \\end{array} \\ ; } \\right . } \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} R ( \\alpha ) ~ = ~ \\sum _ { x \\in \\Z ^ d } \\mu ( x ) e ^ { \\langle \\alpha , x \\rangle } , \\alpha \\in \\R ^ d , \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} \\left ( u _ { \\epsilon , M } \\right ) _ { t } & = \\nabla \\left ( m \\ , \\left ( U _ { M } ^ { m - 1 } + \\epsilon \\right ) \\nabla u _ { \\epsilon , M } \\right ) \\mbox { i n $ \\R ^ n \\times \\left ( 0 , \\infty \\right ) $ } \\\\ u _ { \\epsilon , M } ( x , 0 ) & = u _ { 0 , M } ( x ) \\forall x \\in \\R ^ n . \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} ( \\omega ^ \\bullet _ t ) ^ { m + n } = e ^ F \\omega _ t ^ { m + n } = \\binom { m + n } { n } e ^ { - n t + F } \\omega _ { \\mathbb { C } ^ m } ^ m \\wedge \\omega _ Y ^ n \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } \\left ( 1 + \\gamma _ d \\ , \\omega \\left ( \\left \\{ \\frac { k Y _ d z _ d } { N } \\right \\} \\right ) \\right ) q _ d ( k ) & = \\sum _ { k = 0 } ^ { N - 1 } q _ d ( k ) + \\gamma _ d \\sum _ { k = 0 } ^ { N - 1 } \\omega \\left ( \\left \\{ \\frac { k Y _ d z _ d } { N } \\right \\} \\right ) q _ d ( k ) \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} f _ y ^ { ( k ) } ( 0 , 0 ) = 0 , k = 1 , 2 , \\ldots , p - 1 . \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { 3 } \\alpha _ 2 + \\frac { 2 } { 3 } \\alpha _ 1 , \\ \\alpha ' = \\frac { 2 } { 3 } \\alpha _ 2 + \\frac { 1 } { 4 } \\alpha _ 1 . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} \\bigg \\vert 2 \\ , \\zeta _ i ^ { ( k ) } \\circ \\Delta _ t \\circ { \\rm g w } _ t ( A ) - \\ell _ i ^ { ( k ) } ( A ) \\bigg \\vert & = \\bigg \\vert \\frac { 2 } { t } \\ln \\vert \\Delta _ i ^ { ( k ) } ( b _ t ) \\vert - \\ell _ i ^ { ( k ) } ( A ) \\bigg \\vert \\\\ & = \\bigg \\vert \\frac { 1 } { t } \\ln \\vert \\Delta _ i ^ { ( k ) } ( b _ t ) \\vert ^ 2 - \\frac { 1 } { t } \\ln \\left ( \\prod _ { j = 1 } ^ i \\lambda _ { j } ^ { ( k ) } ( b _ t b _ t ^ * ) \\right ) \\bigg \\vert \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} d _ 1 \\textbf { E } ^ 1 _ { \\overline { \\alpha , 1 } , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x ( b - 1 ) + d _ 2 ~ ^ { A B R } \\nabla _ b ^ \\alpha x ( b - 1 ) = 0 , \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} \\pm \\sum \\limits _ { V _ i ^ \\pm \\ni j } \\lambda _ i = \\pm \\sum \\limits _ { i = 0 } ^ { p _ j } \\lambda _ i = \\pm t _ { p _ j + 1 } = x _ j , \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\begin{cases} V _ { t } ^ \\pm ( \\tilde { \\vec x } ) = V _ { t - \\alpha } ^ \\pm ( \\vec x ) , & \\mbox { i f } t \\geq \\alpha , \\\\ V _ t ^ \\pm ( \\tilde { \\vec x } ) = V _ 0 ^ \\pm ( \\vec x ) , & \\mbox { i f } t \\in [ 0 , \\alpha ) . \\end{cases} \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} R ( \\rho _ { w } ) = \\exp \\left ( - \\sum _ { j > 0 } v _ { j } ( w ) L _ { j } \\right ) v _ { 0 } ( w ) ^ { - L _ { 0 } } . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & T _ 2 & - T _ 1 \\\\ - T _ 2 & 0 & T _ 0 \\\\ T _ 1 & - T _ 0 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} \\mathcal F _ { \\alpha _ 2 } ( w _ 2 ) & = \\left ( \\mathcal F _ { \\alpha _ 2 } ( w _ 2 ) - \\mathcal F _ { \\alpha _ 2 } \\left ( w _ 2 \\pm \\frac { 3 w _ 1 } { 2 } \\right ) \\right ) + F _ { \\alpha _ 2 } \\left ( w _ 2 \\pm \\frac { 3 w _ 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} ( | u _ n ( t , x ) | ^ q ) _ { n = 1 } ^ \\infty \\ . \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} \\mathbb { G } ^ { ( \\infty ) } ( f ) = \\mathbb { G } ( f ) - S _ { 1 , } ( f ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { A } ] - S _ { 2 , } ( f ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { B } ] , f \\in \\mathcal { F } \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} V \\delta _ i = \\tbinom { N + i - 1 } { N - 1 } ^ { - \\frac { 1 } { 2 } } \\sum _ { | m | = i } \\delta _ { m } . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} L ( \\bar { x } , \\bar { \\lambda } ) = \\sum _ { i = 1 } ^ { N } f _ { i } ( x _ { i } ) + \\lambda _ { i } g _ { i } ( x _ { i } ) , \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} \\begin{cases} \\mathrm { d } x _ t = v _ t \\mathrm { d } t \\ , , \\\\ \\mathrm { d } v _ t = \\big [ G ( M ( t , x _ t ) ) - v _ t \\big ] \\mathrm { d } t + \\sqrt { 2 \\sigma } \\mathrm { d } W _ t \\ , , \\end{cases} \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\begin{gathered} d Y _ t = \\frac { 1 } { 2 } \\biggl ( \\frac { k } { Y _ t } - a Y _ t \\biggr ) d t + \\frac { \\sigma } { 2 } d B _ t ^ H , Y _ 0 > 0 , \\end{gathered} \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} T _ * < \\infty \\ \\Rightarrow \\ \\limsup _ { t \\to T _ * } \\| u ( t ) \\| _ { X } = \\infty . \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} ( p \\cdot \\widetilde { C } _ { \\lambda } ( \\nu ) ) \\cap \\Gamma = \\emptyset p \\in \\Gamma . \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} \\sum _ { \\substack { B \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | B | = k } } \\prod _ { i \\in B } b ( i ; n ) \\geq \\frac { \\prod \\limits _ { j = 0 } ^ { k - 1 } \\left ( \\lambda _ n - j m _ n \\right ) } { k ! } , \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\alpha _ m * \\beta _ n = ( m + 1 ) \\alpha _ m \\beta _ n \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} T _ d ^ + : = \\left \\{ ( A , \\Sigma ^ + , B ) \\mid ( A , B ) \\in I _ d ^ + , \\Sigma ^ + \\in X ^ + , A \\subset \\Sigma ^ + \\subset B \\right \\} , \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} \\underbar r _ 2 = \\frac { b _ { 0 , \\inf } - b _ { 1 , \\sup } \\bar r _ 1 - l \\frac { \\chi _ 2 } { d _ 3 } \\bar r _ 1 } { b _ { 2 , \\sup } - l \\frac { \\chi _ 2 } { d _ 3 } } . \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} & \\left \\langle \\rho , \\tau ^ { x _ { 3 } } \\alpha _ { 1 } , \\sigma \\alpha _ { 2 } \\alpha _ { 3 } ^ { ( 1 + x _ { 3 } ) x _ { 3 } ^ { - 1 } } \\right \\rangle \\cong C _ { p } ^ { 3 } , \\ \\left \\langle \\rho , \\tau ^ { x _ { 3 } } \\alpha _ { 1 } , \\sigma \\alpha _ { 2 } \\alpha _ { 3 } ^ { a } \\right \\rangle \\cong M _ { 1 } \\\\ & \\ a = 0 , . . . , p - 1 , \\ x _ { 3 } = 1 , . . . , p - 1 \\ \\ a - ( 1 + x _ { 3 } ) x _ { 3 } ^ { - 1 } \\not \\equiv 0 \\ \\mathrm { m o d } \\ p . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} C _ { L _ { \\infty } / K } : = \\chi _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) } ( K _ { L _ { \\infty } } ^ { \\bullet } , \\phi _ { \\infty } ^ { - 1 } ) \\in K _ 0 ( \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) ) . \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} g _ { \\infty } = \\left ( \\begin{array} { c c } z \\mathbf 1 _ 2 & 0 \\\\ 0 & z \\mathbf 1 _ 2 \\end{array} \\right ) \\left ( \\begin{array} { c c } \\mathbf 1 _ 2 & X \\\\ 0 & \\mathbf 1 _ 2 \\end{array} \\right ) \\left ( \\begin{array} { c c } A & 0 \\\\ 0 & { } ^ t A ^ { - 1 } \\end{array} \\right ) \\left ( \\begin{array} { c c } \\alpha & \\beta \\\\ - \\beta & \\alpha \\end{array} \\right ) , \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} M _ { 1 , 2 } & = \\frac { P _ 3 ( 1 ) } { L } \\sum _ { w \\mid q } \\varphi ( w ) \\mu ( q / w ) \\mathop { \\sum \\sum } _ { \\substack { b \\leq y _ 2 \\\\ b \\equiv \\pm n ( w ) \\\\ ( b , q ) = 1 } } \\frac { \\Lambda ( b ) P _ 2 [ b ] } { ( b n ) ^ { \\frac { 1 } { 2 } } } V _ 1 \\left ( \\frac { n } { q ^ { \\frac { 1 } { 2 } } } \\right ) . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} \\delta c ( g _ { 1 } , \\ldots , g _ { p + 1 } ) = g _ { 1 } . c ( g _ { 2 } , \\ldots , g _ { p + 1 } ) & + \\sum _ { i = 1 } ^ { p } ( - 1 ) ^ { i } c ( g _ { 1 } , \\ldots , g _ { i } g _ { i + 1 } , \\ldots , g _ { p + 1 } ) \\\\ & + ( - 1 ) ^ { p + 1 } c ( g _ { 1 } , \\ldots , g _ { p } ) . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} c _ 1 \\left ( \\sum _ { i = 1 } ^ n \\lambda _ i ( p ) \\right ) - n c _ 2 \\leq F _ \\psi ( p ) \\leq c _ 3 \\left ( \\sum _ { i = 1 } ^ n \\lambda _ i ( p ) \\right ) + n c _ 4 . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} T M _ { 1 2 } + M _ { 1 2 } T ^ * & = \\cos \\theta M _ { 1 2 } , \\\\ T M _ { 1 q } + M _ { 1 q } T ^ * & = \\frac 1 2 ( 1 + \\cos \\theta ) M _ { 1 q } + \\frac 1 2 \\sin \\theta M _ { 2 q } , q > 2 , \\\\ T M _ { 2 q } + M _ { 2 q } T ^ * & = \\frac 1 2 ( 1 + \\cos \\theta ) M _ { 2 q } - \\frac 1 2 \\sin \\theta M _ { 1 q } , q > 2 , \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} { \\rm g w } _ t \\colon \\mathcal { H } \\to A N , \\ , { \\rm g w } _ t ( A ) = h ^ { - 1 } ( \\gamma ( t A ) ) , \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} ( \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) \\otimes \\underline { \\chi } ) ( g ) = \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) ( g ) \\underline { \\chi } ( \\nu ( g ) ) , \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} n S _ { n } = S _ { n - 1 } \\Psi _ 1 + S _ { n - 2 } \\Psi _ 2 + \\cdots + \\Psi _ n . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} g _ s ( y _ 1 , y _ 2 ) = \\int _ { \\Gamma _ 0 } e ^ { \\alpha | \\eta | } | G _ { s } ( \\eta \\cup \\{ y _ 1 , y _ 2 \\} ) | \\lambda ( d \\eta ) . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} C ^ { p } ( G ; A ) = \\{ c : G ^ { p } \\to A \\} , \\delta : C ^ { p } ( G ; A ) \\to C ^ { p + 1 } ( G ; A ) , \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} \\nu _ n : = \\prod _ { \\tau \\in \\Sigma ( K ) } \\gamma _ { \\tau , n } ( 1 ) \\in \\mathcal { O } _ { \\Q _ { p , m } } [ \\Gamma _ { n + n _ 0 } ] ^ { \\times } \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} ( \\lceil j / 2 \\rceil - 1 ) ( j - \\lceil j / 2 \\rceil ) = \\frac { 1 } { 4 } ( \\tau ( C , p ) - 1 ) . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ f | \\mathcal { A } ^ { ( k ) } \\right ] = \\left ( \\mathbb { E ( } f | A _ { 1 } ^ { ( k ) } ) , \\dots , \\mathbb { E } ( f | A _ { m _ { k } } ^ { ( k ) } ) \\right ) ^ { t } , \\quad \\mathbb { G } \\left [ \\mathcal { A } ^ { ( k ) } \\right ] = \\left ( \\mathbb { G } ( A _ { 1 } ^ { ( k ) } ) , \\dots , \\mathbb { G } ( A _ { m _ { k } } ^ { ( k ) } ) \\right ) ^ { t } , \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} q _ G ( M ) = \\sum _ { A \\subseteq E } ( - 1 ) ^ { | A ^ c | } ( 2 ^ d ) ^ { | A | - | V | + k ( M \\backslash A ^ c ) } m ^ { | A | - | V | + k _ { \\mathrm { o } } ( M \\backslash A ^ c ) } , \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} { \\rm m u l t } _ \\lambda ( \\mathbf { m } ) = \\sum _ { \\sigma \\in S _ { n - 1 } } \\varepsilon ( \\sigma ) Q \\Big ( \\sum _ { j = 1 } ^ { n - 1 } ( \\ell _ 2 + 1 + m ' - \\ell _ { j + 1 } ) e ' _ j - \\sigma \\Big ( \\sum _ { j = 1 } ^ { n - 1 } j e ' _ j \\Big ) \\Big ) . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\Lambda _ 1 & = x + ( \\zeta ^ 5 + \\zeta ) y + ( \\zeta ^ 5 + \\zeta ^ 4 + \\zeta ^ 2 + 1 ) z , \\\\ \\Lambda _ 2 & = x - ( \\zeta ^ 5 + \\zeta ^ 4 + \\zeta ^ 3 + \\zeta + 1 ) y + ( \\zeta ^ 5 + \\zeta ^ 3 + \\zeta ^ 2 + 1 ) z , \\\\ \\Lambda _ 3 & = x + ( \\zeta ^ 5 + 1 ) y - ( \\zeta ^ 3 + \\zeta ^ 2 + \\zeta ) z , \\\\ \\Lambda _ 4 & = x + ( \\zeta ^ 2 + 1 ) y + ( \\zeta ^ 3 + \\zeta ^ 2 + \\zeta + 1 ) z . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} \\beta _ m = p ^ m \\left ( \\begin{array} { c c } 0 & p ^ { - 2 m } \\\\ - 1 & 0 \\end{array} \\right ) = p ^ m \\left ( \\begin{array} { c c } 0 & p ^ { - 2 m } \\\\ 1 & 0 \\end{array} \\right ) \\left ( \\begin{array} { c c } - 1 & 0 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} ( f \\star g ) ( x ) : = \\int _ G f ( x - y ) g ( y ) d \\lambda ( y ) , ( f \\star \\nu ) ( x ) : = \\int _ G f ( x - y ) d \\nu ( y ) \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} \\partial _ i ^ 2 & = 0 , 1 \\leq i < n , \\\\ \\partial _ { i + 1 } \\partial _ i \\partial _ { i + 1 } & = \\partial _ i \\partial _ { i + 1 } \\partial _ i , 1 \\leq i < n - 1 , \\\\ \\partial _ { i } \\partial _ { j } & = \\partial _ j \\partial _ { i } , 1 \\leq i < j - 1 < n - 1 . \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} A _ { \\kappa _ 1 , { K N } } A _ { \\sigma _ 1 , { K N } } = I + R _ 2 , \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} ( r _ { f _ 1 } - r _ { f _ 2 } ) r _ { u _ { 3 5 } } & = r _ g r _ { e _ 2 } r _ { u _ { 3 5 } } / r _ { u _ { 2 5 } } - r _ { g _ 1 } r _ { e _ 1 } r _ { u _ { 3 5 } } / r _ { u _ { 1 5 } } \\\\ & = r _ { g _ 2 } r _ { u _ 5 } r _ { u _ { 3 5 } } / r _ { u _ 4 } - r _ { g _ 1 } r _ { u _ 5 } r _ { u _ { 3 5 } } / r _ { u _ 4 } \\\\ & = ( r _ { g _ 2 } - r _ { g _ 1 } ) r _ { e _ 3 } \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} D ( z , z ' ) - D ( z , a ) - D ( a , z ' ) + D ( a , a ) = 0 . \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} \\alpha ( t s ) = \\alpha ( s t ^ { k } ) = s ^ { p - 2 } t ^ { n k } , \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r w _ r \\leq \\liminf _ { n } a _ { n , k } , \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\triangleq \\lim \\limits _ { \\tau \\downarrow 1 } \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) \\ , M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} \\begin{cases} \\underbar r _ 1 ^ k - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 1 ^ k + \\epsilon \\cr \\underbar r _ 2 ^ k - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 2 ^ k + \\epsilon \\end{cases} \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} G ( \\tau \\ , | \\ , \\tau ) = G ( 1 + \\tau \\ , | \\ , \\tau ) . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } } \\frac { \\sqrt { n } } { v _ { n } } \\left \\vert \\mathbb { E } \\left ( \\mathbb { P } _ { n } ^ { ( N ) } ( f ) \\right ) - P ( f ) \\right \\vert = \\sup _ { f \\in \\mathcal { F } } \\left \\vert \\frac { 1 } { v _ { n } } \\mathbb { E } \\left ( \\mathbb { R } _ { n } ^ { ( N ) } ( f ) \\right ) \\right \\vert \\leqslant \\mathbb { E } \\left ( \\frac { r _ { n } ^ { ( N ) } } { v _ { n } } \\right ) . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} \\stackrel { * } { \\bar { R } } ( X , Y , Z , W ) & = \\stackrel { * } { R } ( X , Y , Z , W ) - g \\big ( \\stackrel { * } { h } ( X , W ) , \\stackrel { * } { h } ( Y , Z ) \\big ) \\\\ & + g \\big ( \\stackrel { * } { h } ( X , Z ) , \\stackrel { * } { h } ( Y , W ) \\big ) , \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} \\bigl | \\bigl ( ( 1 - \\varepsilon ) \\partial _ t u _ { \\lambda , \\varepsilon } , 0 \\bigr ) \\bigr | _ { L ^ 2 ( 0 , T ; \\boldsymbol { V } ^ * ) } & = \\bigl | ( 1 - \\varepsilon ) \\partial _ t u _ { \\lambda , \\varepsilon } \\bigr | _ { L ^ 2 ( 0 , T ; H ) } \\\\ & \\le ( 1 - \\varepsilon ) C ^ * M _ 2 \\\\ & \\le C ^ * M _ 2 . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\langle \\tilde { \\pi } _ p ( \\beta _ m ) \\mathbf h _ p , \\mathbf h _ p \\rangle = \\int _ { \\mathcal B _ 1 ( m ) } \\mathbf h _ p ( h \\beta _ m ) \\overline { \\mathbf h _ p ( h ) } d h + \\int _ { \\mathcal B _ 2 ( m ) } \\mathbf h _ p ( h \\beta _ m ) \\overline { \\mathbf h _ p ( h ) } d h = : B _ 1 ( m ) + B _ 2 ( m ) . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} \\mathbf { z } _ i ^ T \\mathbf { G } \\mathbf { z } _ i = \\textrm { T r } ( \\mathbf { G } \\mathbf { Z } _ i ) , \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} F _ Z = ( f _ 1 , f _ 2 , \\dots , f _ Z ) \\in \\{ - 1 , + 1 \\} ^ Z , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} S _ { d , k } ( n , W ) = \\sum _ { i = d + 1 } ^ { k + 1 } S _ { d , k } ( n - i , W - 1 ) \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} \\theta = \\beta \\otimes \\omega + \\beta _ i \\otimes a + \\beta ' \\otimes a ' + \\gamma \\otimes 1 \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\deg ( N _ 1 ( \\lambda ) ^ T \\ , g ( \\lambda ) ) = \\deg ( N _ 1 ( \\lambda ) ) + \\deg ( g ( \\lambda ) ) \\ , , \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} q _ { x _ 0 } ( x ) = \\begin{cases} 0 , & x \\leq x _ 0 \\\\ q ( x ) & x > x _ 0 \\end{cases} \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\omega } \\ , \\int _ { - \\infty } ^ { \\infty } f ( u ( t , x ) ) d x \\ , d t = 0 . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{gather*} L ( f _ { 3 6 } , s ) = \\sum _ { m , n \\in \\Z } \\frac { ( 3 m + 1 ) - n \\sqrt { - 3 } } { \\big ( ( 3 m + 1 ) ^ 2 + 3 n ^ 2 \\big ) ^ s } = \\sum _ { m , n \\in \\Z } \\ ! \\frac { 1 } { \\big ( ( 3 m + 1 ) \\ ! + n \\sqrt { - 3 } \\big ) \\big | ( 3 m + 1 ) \\ ! + n \\sqrt { - 3 } \\big | ^ { 2 s - 2 } } . \\end{gather*}"}
-{"id": "6000.png", "formula": "\\begin{align*} t . \\theta _ q = t ^ { P ^ t ( q ) } \\theta _ q \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} ( \\frac { u ^ { n + 1 } _ { h } - u ^ { n } _ { h } } { \\Delta t } , v _ { h } ) = - b ( u ^ { n } _ { h } , u ^ { n + 1 } _ { h } , v _ { h } ) - \\nu ( u ^ { n + 1 } _ { h } , v _ { h } ) + ( f ^ { n + 1 } , v _ { h } ) \\ ; \\ ; \\forall v _ { h } \\in V _ { h } . \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} z ^ { \\alpha } b \\cdot b ^ { \\alpha } & = z ^ { \\alpha } x ^ { \\alpha } \\cdot x & \\\\ z ^ { \\alpha } b & = ( z ^ { \\alpha } x ^ { \\alpha } \\cdot x ) / b ^ { \\alpha } & \\\\ b & = ( z ^ { \\alpha } x ^ { \\alpha } \\cdot x ) / ( z ^ { \\alpha } \\backslash b ^ { \\alpha } ) \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} p ^ * ( X _ g ) : = \\prod _ { f \\in \\tilde { I } } A _ f ^ { \\tilde { \\varepsilon } _ { f g } } , \\forall \\ , g \\in I . \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} c _ \\chi c _ { \\tilde \\chi } & = \\begin{cases} \\frac 1 2 ( 1 + \\varepsilon ) & \\chi = \\tilde \\chi ^ { - 1 } \\ , , \\\\ c _ { \\chi \\tilde \\chi } & \\ , , \\end{cases} \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\norm { x _ i - x _ * } ^ 2 & = \\norm { x _ 0 - \\sum _ { k = 0 } ^ { i - 1 } h _ { i , k } g _ k + v _ i - x _ 0 - \\sum _ { j = 0 } ^ N \\nu _ { j } v _ j - r _ * } ^ 2 \\\\ & = \\norm { \\hat x _ i - \\hat x _ * } ^ 2 + \\norm { v _ i - \\sum _ { j = 0 } ^ N \\nu _ { j } v _ j } ^ 2 \\\\ & \\geq \\norm { \\hat x _ i - \\hat x _ * } ^ 2 , { i = 0 , \\hdots , N } , \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} \\varphi _ 0 = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { \\infty } J _ 0 ( y ) \\varphi ( y ) d y , \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} \\chi _ p ( a \\alpha + b \\gamma ) = \\begin{cases} \\chi _ p ( a ) \\overline { \\chi _ p ( \\delta ) } & e _ p \\leq 2 f _ p , \\\\ \\chi _ p ( - \\gamma / c ) & e _ p > 2 f _ p . \\end{cases} \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} V ' _ { 3 2 } \\Lambda _ 3 = \\binom { 2 } { 2 } V ' _ { 3 2 3 } + \\binom { 3 } { 2 } V ' _ { 3 1 4 } + \\binom { 4 } { 2 } V ' _ { 3 5 } + \\binom { 5 } { 2 } V ' _ { 2 6 } + \\binom { 6 } { 2 } V ' _ { 1 7 } + \\binom { 7 } { 2 } V ' _ { 8 } , \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} \\begin{cases} { \\displaystyle \\lim _ { \\varepsilon \\to 0 ^ { + } } t ^ { \\varepsilon } = 0 } , \\\\ t ^ { \\varepsilon } = \\gamma ^ { - C _ { 1 } t ^ { \\varepsilon } } \\left ( T , \\beta \\right ) , \\end{cases} \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} A _ \\infty : = \\bigcap _ { t \\in I } A _ t \\ne \\emptyset . \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} \\left | \\left \\{ x \\in B _ { R } : u ( x , t ) > \\left ( 1 - \\frac { 1 } { 2 ^ { s _ 2 + 1 } } \\right ) \\omega \\right \\} \\right | \\leq \\left | \\mathcal { S } \\right | + N \\nu \\left | B _ R \\right | . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c c } & & - 1 \\\\ & Q _ 1 \\\\ - 1 \\end{array} \\right ) , Q _ 1 = \\left ( \\begin{array} { c c c } 0 & 0 & 1 \\\\ 0 & 2 & 0 \\\\ 1 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} g ( x ) = \\dfrac { x } { \\sqrt [ a ] { 1 + c x ^ a } } \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} x _ i ( \\tau , \\lambda _ i ) / \\tau = x _ i ( 1 / \\tau , \\tau \\lambda _ i ) , \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\begin{dcases} \\sum _ { e \\in E ( G _ i ) } w _ i ( e ) = 1 , \\\\ \\sum _ { e \\in E ( G _ i ) : \\ , v \\in e } B _ i ( v , e ) = 1 , \\ \\ v \\in V ( G _ i ) , \\\\ w _ i ( e ) ^ { p - r } \\prod _ { v \\in e } B _ i ( v , e ) = \\alpha _ i , \\ \\ e \\in E ( G _ i ) . \\end{dcases} \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} { { { \\sum \\limits _ { s \\in S _ { r } } { { \\lambda _ { s } } } } } } \\left ( { { { { { { a _ { s } ^ { \\ast } , } b _ { s } } } } } } \\right ) { = - } \\left ( c ^ { \\ast } , v _ { r } \\right ) , \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} k _ \\mu ( \\eta ) = \\int _ { \\Gamma _ \\Lambda } R ^ \\Lambda _ \\mu ( \\eta \\cup \\xi ) \\lambda ( d \\xi ) , \\eta \\in \\Gamma _ \\Lambda . \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} \\lbrace f , g \\rbrace = { \\bf P } ^ { i j } \\frac { \\partial f } { \\partial x _ i } \\frac { \\partial f } { \\partial x _ j } , \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} \\big | F ^ { ( d + 1 ) } ( t ) \\big | = j ^ { d + 1 } \\abs { m } \\cdot \\big | \\varphi _ 1 ^ { ( d + 1 ) } ( j t ) - \\tau \\psi ^ { ( d + 1 ) } ( j t ) \\big | \\simeq \\frac { j ^ { d + 1 } \\abs { m } \\varphi _ 1 ( j X ) \\sigma _ 1 ( j X ) } { ( j X ) ^ { d + 1 } } . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} A ^ { * * } \\phi ( \\psi ) = ( A ^ * \\psi ) ^ * \\phi = A \\phi ( \\psi ) , \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} \\lambda _ { G / H } \\circ q ( f ) = \\int _ G f ( g ) \\lambda _ { G / H } ( g H ) \\ , d g \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} \\psi ( b ) = ( a ^ { e _ 1 } , \\dots , a ^ { e _ { d - 1 } } , b ) . \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} e _ { i j } = \\begin{cases} \\delta _ { i j } & \\ j \\neq k , \\\\ - 1 & \\ i = j = k , \\\\ \\max ( 0 , - b _ { i k } ) & \\ i \\neq j = k , \\end{cases} & & f _ { i j } = \\begin{cases} \\delta _ { i j } & \\ i \\neq k , \\\\ - 1 & \\ i = j = k , \\\\ \\max ( 0 , b _ { k j } ) & \\ i = k \\neq j . \\end{cases} \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} 2 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } \\nabla _ x m _ { u _ \\mu } ( t , x ) \\nabla _ y m _ { u _ \\iota } ( t , y ) \\Delta ^ 2 _ x \\psi ( x , y ) \\ , d x d y & \\\\ = 2 \\int _ { \\R ^ d \\times \\R ^ d } \\sum _ { \\mu = 1 } ^ N \\nabla _ x m _ { u _ \\mu } ( t , x ) \\sum _ { \\iota = 1 } ^ N \\nabla _ y m _ { u _ \\iota } ( t , y ) \\Delta ^ 2 _ x \\psi ( x , y ) \\ , d x d y & \\\\ = - 2 \\int _ { \\R ^ d \\times \\R ^ d } \\nabla _ x \\zeta ( t , x ) \\nabla _ y \\zeta ( t , y ) \\Delta ^ 2 _ x \\psi ( x , y ) , \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} u _ { i , r } ( m ) = c _ { ( i , r ) } \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} \\chi ( M _ n ( a ) ) = & 2 \\sum _ { s = 1 } ^ m ( - 1 ) ^ { m + s } s { n \\choose m - s } \\\\ = & ( - 1 ) ^ { m + 1 } { 2 m \\choose m } . \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} \\partial _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) } ( x _ { L _ { \\infty } / K } ) = - C _ { L _ { \\infty } / K } - U ' _ { L _ { \\infty } / K } + M _ { L _ { \\infty } / K } \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c c c } B _ k ( \\lambda ) & B _ { k - 1 } ( \\lambda ) & B _ { k - 2 } ( \\lambda ) & \\cdots & B _ 1 ( \\lambda ) \\\\ \\hline - I _ n & \\lambda ^ \\ell I _ n \\\\ & - I _ n & \\lambda ^ \\ell I _ n \\\\ & & \\ddots & \\ddots \\\\ & & & - I _ n & \\lambda ^ \\ell I _ n \\end{array} \\right ] = \\left [ \\begin{array} { c } M _ 1 ^ \\ell ( \\lambda ) \\\\ \\hline \\phantom { \\Big { ( } } L _ { k - 1 } ( \\lambda ^ \\ell ) \\otimes I _ n \\phantom { \\Big { ( } } \\end{array} \\right ] \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} B _ { n , k } = \\frac { 1 } { n + 1 } \\binom { 2 n + 2 } { n - k } \\binom { n + k } { n } . \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} ( b - a ) ^ 2 \\bigg ( \\frac { \\beta - f ( a ) } { f ( b ) - f ( a ) } \\bigg ) . \\bigg ( \\frac { \\beta - g ( a ) } { g ( b ) - g ( a ) } \\bigg ) = \\beta . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} g _ k ( x ) = \\tilde { g } _ { k } ( T _ k x ) = g _ { \\nu } ( T _ k x / r _ k ) . \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} J = \\begin{pmatrix} & 1 \\\\ 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} J _ { s , i } = \\begin{pmatrix} I _ { s } \\\\ I _ { s } & I _ { s } \\\\ & I _ { s } & I _ { s } \\\\ & & \\ddots \\\\ & & & I _ { s } & I _ { s } \\end{pmatrix} . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} ( f \\ast g ) ( q ) = f ( q ) \\ast g ( q ) , \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} | u ( t ) | _ { \\alpha + 1 / 2 , \\sigma } & = \\mathcal O ( t ^ { - \\mu _ 1 } ) , \\\\ | B ( u ( t ) , u ( t ) ) | _ { \\alpha , \\sigma } & = \\mathcal O ( t ^ { - 2 \\mu _ 1 } ) . \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} \\Gamma _ { ( S , \\xi ) } ( t ) & = \\phi _ { S ^ { 1 / 2 } } ( \\Gamma _ { ( I _ r , { \\rm D } \\phi _ { S ^ { - 1 / 2 } } ( \\xi ) ) } ( t ) ) \\\\ & = S ^ { \\frac { 1 } { 2 } } \\exp ( t S ^ { - \\frac { 1 } { 2 } } \\xi S ^ { - \\frac { 1 } { 2 } } ) S ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} \\lambda ^ { ( p + 1 ) } ( G * K _ 1 ) = \\left ( \\frac { ( r + 1 ) ^ { p - r } } { r ^ { p + 1 - r } } \\right ) ^ { 1 / ( p + 1 ) } \\lambda ^ { ( p + 1 ) } ( G ) , \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} \\tau ^ { ' } = \\frac { 1 } { m ( m + 1 ) } \\sum \\limits _ { i , j = 1 } ^ { m + 1 } { R } ^ { ' } ( E _ i , E _ j , E _ j , E _ i ) . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} ( X _ u , Y _ u ) ^ { n + 1 } = [ ( X _ u , Y _ u ) ^ n + \\gamma . \\triangledown F ( ( X _ u , Y _ u ) ^ n ) ) ] ^ + . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\nabla = d + p _ { - 1 } d z - \\sum _ { i = 1 } ^ N \\frac { \\lambda _ i } { z - z _ i } d z + \\sum _ { j = 1 } ^ m \\frac { \\alpha _ { c ( j ) } } { z - w _ j } d z . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\begin{cases} d u ( t ) + A u ( t ) \\ d t = B ( u ( t ) ) \\ d t , & t > 0 \\\\ u ( 0 ) = u _ { 0 } \\end{cases} \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} \\sigma \\ ( A \\theta , \\mathcal O \\ ) & = \\sigma \\ ( A \\theta , \\ , \\sigma \\ ( A X , \\ , R _ \\alpha ( \\theta ) , \\ , Z ( \\theta ) : \\alpha \\in \\ker A \\ ) \\ ) \\\\ & = \\sigma \\ ( A \\theta , \\ , A X , \\ , R _ \\alpha ( \\theta ) , \\ , Z ( \\theta ) : \\alpha \\in \\ker A \\ ) = \\sigma \\ ( A X , \\ , \\theta \\ ) \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} \\psi _ 0 ( z ) f ( z ) + \\psi _ n ( z ) f ^ { ( n ) } ( z ) = V f ( z ) = \\lambda f ( z ) , \\quad \\forall z \\in \\C . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} ( u _ t ^ 2 , m _ t ) + \\int _ 0 ^ t 2 ( | \\nabla u _ r | ^ 2 , m _ r ) d r = ( u _ 0 ^ 2 , m _ 0 ) - 2 \\int _ 0 ^ t ( F ( u _ r ) , \\nabla ( u _ r m _ r ) ) d r . \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} F ( k ) = \\sum _ { k = r m n } \\mu ^ 2 ( r m ) \\mu ^ 2 ( r n ) h ( r ) f ( m ) g ( n ) . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} ( \\alpha + | D | ) \\nu = \\gamma \\alpha ( P ( \\nu ) + c H ( \\nu ) ) \\equiv G _ 3 ( \\nu ) \\in L ^ 2 ( \\mathbb R ) \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\operatorname { G C D } \\left \\{ k _ i , l _ i \\mid i = 1 , \\ldots , a + b \\right \\} = 2 ^ r , \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} [ n ] _ \\nu = \\frac { 1 - \\nu ^ { n } } { 1 - \\nu } = 1 + \\nu + \\ldots + \\nu ^ { n - 1 } , & & { n \\brack k } _ \\nu = \\frac { [ n ] _ \\nu ! } { [ k ] _ \\nu ! [ n - k ] _ \\nu ! } . \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\Lambda ( X ) = \\left \\{ \\begin{array} { c c } \\Lambda ^ - ( X ) & \\ ; X \\in \\overline { T } ^ - , \\\\ \\Lambda ^ + ( X ) & \\ ; X \\in \\overline { T } ^ + . \\end{array} \\right . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} \\sum _ { e : \\ , v \\in e } B ( v , e ) = \\frac { \\sum _ { e : \\ , v \\in e } \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G [ S ] ) x _ v ^ p } = 1 . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } W ( t , x ) = & \\underset { u ( \\cdot ) \\in \\mathcal { U } ^ { w } [ t , T ] } { \\inf } J ( t , x ; u ( \\cdot ) ) . \\end{array} \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} \\tau _ g ^ s ( f ) : = \\sum _ { g ' } \\tau _ { g ' } ( f ) = \\sum _ { g ' } \\int _ { G / Z _ G ( g ' ) } f ( h g ' h ^ { - 1 } ) d h ( Z _ G ( g ' ) ) , \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { k } d _ j ( A ^ t ) \\leq \\prod _ { j = 1 } ^ { k } d _ j ^ t ( A ) . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t \\varphi ( t ) & = \\log \\frac { ( \\hat \\omega - t R i c ( \\hat \\omega ) + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ( t ) ) ^ n } { \\hat \\omega ^ n } \\\\ \\varphi ( 0 ) & = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\mathcal { O } _ { E _ { \\tau } } [ G _ n ] _ { \\varphi } : = \\left \\{ y \\in \\mathcal { O } _ { E _ { \\tau } } [ G _ n ] \\mid ( \\phi \\otimes 1 ) y = y ( 1 \\otimes \\phi ) \\right \\} \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} a = \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} b = \\begin{pmatrix} \\epsilon \\\\ 0 \\\\ 1 \\end{pmatrix} c = \\begin{pmatrix} 0 \\\\ \\epsilon \\\\ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\gamma ( t ) : = C ^ { p - 1 } \\zeta ^ p ( t ) \\ , . \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\mathcal { Z } _ { N , w } & = \\{ z \\in \\{ 1 , 2 , \\ldots , b ^ { m - w } - 1 \\} : \\gcd ( z , b ) = 1 \\} = \\mathbb { U } _ { b ^ { m - w } } , \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} P ( \\{ \\omega \\mbox { s u c h t h a t } ( \\omega _ 0 , \\omega _ 1 ) \\in A \\} ) = \\gamma ( A ) . \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} m ^ { \\varphi ^ { * } } \\in \\arg \\max \\limits _ { m } \\Biggl \\{ ( \\lambda _ { \\ell } \\tilde { r } _ { i } - \\nu _ { \\ell } ) \\frac { \\sum \\limits _ { n = 0 } ^ { m - 1 } \\frac { ( \\lambda _ { \\ell } ) ^ n } { n ! ( \\mu _ { i } ) ^ { n } } } { \\sum \\limits _ { n = 0 } ^ { m } \\frac { ( \\lambda _ { \\ell } ) ^ n } { n ! ( \\mu _ { i } ) ^ n } } \\Biggr \\} \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\tau \\circ \\sigma _ 1 ( n ) & = \\tau \\circ \\sigma _ 1 ^ k \\circ \\sigma _ 2 ( m ) = \\sigma _ 2 ^ k \\circ \\sigma _ 1 ( m ) \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} a _ i \\vee ( \\bigwedge _ { j \\neq i } a _ j ) = 1 \\quad \\forall i \\in I . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} { \\tilde x _ u } \\left ( t \\right ) = \\sqrt { { { \\rm { 2 } } \\mathord { \\left / { \\vphantom { { \\rm { 2 } } T } } \\right . \\kern - \\nulldelimiterspace } T } } { \\mathop { \\Re } \\nolimits } \\left ( { { x _ u } \\left ( t \\right ) { e ^ { j \\left ( { 2 \\pi f _ u ^ { { \\rm { R F } } } t + \\varphi _ u ^ { { \\rm { R F } } } } \\right ) } } } \\right ) \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} J ( j , k ) : = \\{ b - k , \\underbrace { b - k + 2 , \\dots , b - k + j } _ , \\underbrace { b + j + 1 , \\dots , n } _ \\} . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} \\theta ( x ) = \\begin{cases*} 1 & i f \\ ( x > 0 \\ ) , \\\\ 0 & i f \\ ( x < 0 \\ ) \\end{cases*} \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} 0 \\ , & = _ { q / 2 } \\Bigl ( \\frac { E _ 2 ^ 2 D _ 1 ^ 2 + E _ 1 E _ 2 ^ 2 D _ 1 + F _ 3 ^ 2 + E _ 1 E _ 2 F _ 3 + E _ 1 ^ 2 F _ 4 } { E _ 1 ^ 2 E _ 2 ^ 2 } \\Bigr ) \\cr & = _ { q / 2 } \\Bigl ( \\frac { D _ 1 ^ 2 } { E _ 1 ^ 2 } + \\frac { D _ 1 } { E _ 1 } + \\frac { F _ 3 ^ 2 } { E _ 1 ^ 2 E _ 2 ^ 2 } + \\frac { F _ 3 } { E _ 1 E _ 2 } + \\frac { F _ 4 } { E _ 2 ^ 2 } \\Bigr ) \\cr & = _ { q / 2 } \\Bigl ( \\frac { F _ 4 } { E _ 2 ^ 2 } \\Bigr ) . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} \\Sigma _ a & = \\sum _ { \\substack { 2 \\le k \\le \\frac { n } { 2 } \\\\ k \\ } } | a _ { k , n } ^ { H ^ F } | i _ { 2 \\times 2 } ( C _ { H ^ F } ( a _ { k , n } ) ) < \\sum _ { 2 \\le k \\le \\frac { n } { 2 } } q ^ { \\frac { 1 } { 8 } n ^ 2 + \\frac { 1 } { 8 } ( n - 2 k ) ^ 2 + k ( n - k ) + d ' n } . \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ { + } } \\varrho \\left ( \\beta \\right ) = \\infty . \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\bar { M } = M / S O ( 2 ) M ^ { \\ast } = M ^ { 1 } / S O ( 2 ) \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} | U _ n | = \\frac { 1 } { 2 } \\left ( - 4 ^ m + \\frac { ( 2 m + 1 ) ! } { ( m ! ) ^ 2 } \\right ) , \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} ( x , y , z ) = ( t ^ 4 , t ^ 5 + s t ^ 6 , t ^ 7 ) . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} R ( r ) = D e ^ { - \\eta { r } } r ^ { ( \\frac { 1 - \\tau } { 2 } ) } \\Big [ e ^ { 2 \\eta { r } } r ^ { - ( 1 - \\tau + \\mathrm { d } ) } \\Big ] _ { - ( 1 + \\mathrm { d } E ^ { - 1 } ) } \\quad \\Bigg ( \\mathrm { d } = \\Big ( \\frac { \\beta - \\eta ( 1 - \\tau ) } { 2 \\eta } \\Big ) \\Bigg ) . \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} H _ 1 = \\begin{cases} X ( t _ 1 ) & \\\\ \\vartheta , & . \\end{cases} \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} \\sum _ { e \\in E ( G ) } w ( e ) = \\frac { r } { \\lambda ^ { ( p ) } ( G ) } \\sum _ { e \\in E ( G ) } \\prod _ { u \\in e } x _ u = \\frac { \\lambda ^ { ( p ) } ( G ) } { \\lambda ^ { ( p ) } ( G ) } = 1 . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ \\infty \\theta _ { B _ { R _ x } ( x ) } ( r _ { x , k } ) & \\leq \\frac { C ^ { \\frac 1 2 } } { C ^ { \\frac 1 2 } - 1 } \\cdot \\sum \\limits _ { k = 0 } ^ \\infty \\int \\limits _ { R _ x / C ^ { \\frac { k + 1 } 2 } } ^ { R _ x / C ^ { \\frac { k } 2 } } \\frac { \\theta _ { B _ { R _ x } ( x ) } ( r ) } r \\ d r \\\\ & \\leq \\frac { C ^ { \\frac 1 2 } } { C ^ { \\frac 1 2 } - 1 } \\int \\limits _ { 0 } ^ { R _ x } \\frac { \\theta _ { B _ { R _ x } ( x ) } ( r ) } r \\ d r \\\\ & < \\infty . \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} \\kappa _ s = 2 0 d \\big ( \\lfloor \\rho ^ { - 1 } ( s + 1 ) ^ { \\rho / 1 0 } \\rfloor + 1 \\big ) . \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{gather*} \\Phi = X \\otimes Y \\otimes Z , \\Phi ^ { - 1 } = P \\otimes Q \\otimes R , \\end{gather*}"}
-{"id": "3624.png", "formula": "\\begin{align*} \\mathcal { Q } _ t ( x _ { t - 1 } , 1 ) = ( 1 - q _ t ) \\sum _ { j = 1 } ^ { M _ { t } } p _ { t j } \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 1 ) + q _ t \\sum _ { j = 1 } ^ { M _ t } p _ { t j } \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 0 ) , \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} \\delta _ { V ^ \\vee } ( \\sigma ) ( \\gamma ) = - 0 _ \\gamma \\cdot \\sigma ( s ( \\gamma ) ) ^ { - 1 } - \\sigma ( t ( \\gamma ) ) \\cdot 0 _ \\gamma \\ ; , \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} A \\# _ t B = A ^ { 1 / 2 } \\left ( A ^ { - 1 / 2 } B A ^ { - 1 / 2 } \\right ) ^ t A ^ { 1 / 2 } , 0 \\leq t \\leq 1 . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} \\vartheta _ n = X _ { 0 , 0 } = \\vartheta _ { p _ n } . \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } = - A x + B u , \\\\ y = C x , \\end{cases} \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} x ^ { p ^ e } \\in \\bigcap _ { n \\ge 1 } ( x _ 1 , \\ldots , x _ t , x _ { t + 1 } ^ n , \\ldots , x _ d ^ n ) ^ { [ p ^ e ] } = ( x _ 1 , \\ldots , x _ t ) ^ { [ p ^ e ] } \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} & \\left | j + \\hat \\theta ^ { \\mathrm f } _ { i , k - 1 } + \\eta _ { i , k } - \\theta _ { i , k } \\right | ^ 2 \\\\ & = \\left ( 1 + \\Im \\{ \\eta _ { i , k } \\} \\right ) ^ 2 + \\left ( \\hat \\theta ^ { \\mathrm f } _ { i , k - 1 } + \\Re \\{ \\eta _ { i , k } \\} - \\theta _ { i , k } \\right ) ^ 2 . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} a _ n = \\tilde a _ n \\qquad \\mu - a . e . \\ o n \\ E _ m , \\forall m \\in \\N \\cup \\{ \\infty \\} , \\ m > n . \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} | N | ^ { r } N _ { w , H } ( h N ) = \\sum _ { y \\in h N } N _ { w , G } ( y ) \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} G ( x ) = \\int _ 0 ^ x f ( s ) d s . \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 0 } ^ { 2 n + 1 - m } ( - 1 ) ^ i a _ i t ^ i \\right ) \\left ( \\sum _ { i \\geq 0 } \\delta _ i t ^ i \\right ) = 1 , \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\substack { n \\longrightarrow + \\infty } } C _ { n } ^ { K ( n ) } ( 2 ^ { K ( n ) } - 1 ) = \\displaystyle \\lim _ { \\substack { n \\longrightarrow + \\infty } } C _ { n } ^ { n ^ r } ( 2 ^ { n ^ r } - 1 ) = + \\infty \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} E _ { k , N } ^ { \\chi } ( Z , s ; h ) = j ( g , I ) ^ { k } E \\Big ( g , \\frac { 2 s + k } { 2 n + 1 } - \\frac 1 2 , f ^ { ( h ) } \\Big ) = j ( g , I ) ^ { k } E \\Big ( g h ^ { - 1 } , \\frac { 2 s + k } { 2 n + 1 } - \\frac 1 2 , f \\Big ) . \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} & \\left | ( f ^ { \\star } _ 0 ( x _ \\ell ) - V ^ { \\star } _ { k _ \\ell } ( x _ \\ell ) + ( f ^ { \\star } _ 0 ( - x _ \\ell ) - V ^ { \\star } _ { k _ \\ell } ( - x _ \\ell ) ) - 2 \\ , ( f ^ { \\star } _ 0 ( 0 ) - V ^ { \\star } _ { k _ \\ell } ( 0 ) ) ) \\right | \\leq \\\\ & \\leq 4 C ' _ 1 \\ , \\omega ( \\delta _ { k _ \\ell } ) . \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} L u ( x ) = A u ( x ) \\mbox { f o r a n y } x \\in U . \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\langle P ( x ) , P ( y ) \\rangle = ( - 1 ) ^ { d ( x , x _ 0 ) + d ( y , x _ 0 ) } . \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} P _ { x , y } ( q ) = P _ { \\phi ( x ) , \\phi ( y ) } ( q ) \\ x , y \\in [ i d , w ] . \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\check { \\nu } ( x ) = \\int _ { \\widehat G } \\gamma ( x ) d \\nu ( \\gamma ) ( x \\in G ) . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} \\log ( M _ x ) = f ( x ) : = \\varphi ( x ) - \\varphi ( T x ) , \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} P _ \\varepsilon ( t ) ( \\omega ) = & \\int _ 0 ^ t \\sigma ( s , \\omega + \\varepsilon h , X ( s ) ( \\omega + \\varepsilon h ) ) \\dot { h } ( s ) d s \\\\ & + \\int _ 0 ^ t \\Big [ b ( s , \\omega + \\varepsilon h , X ( s ) ( \\omega + \\varepsilon h ) ) - b ( s , \\omega , X ( s ) ( \\omega ) ) \\Big ] d s \\\\ & + \\int _ 0 ^ t \\Big [ \\sigma ( s , \\omega + \\varepsilon h , X ( s ) ( \\omega + \\varepsilon h ) ) - \\sigma ( s , \\omega , X ( s ) ( \\omega ) ) \\Big ] d W ( s ) . \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} w ( \\alpha ) = ( S _ { \\alpha _ 2 } , S _ { \\alpha _ 3 } , { \\rm i m } ( \\alpha _ 1 ) , { \\rm i m } ( \\alpha _ 4 ) ) . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} ~ ^ { A B R } _ { a } \\nabla ^ { \\alpha } ( p ( t ) ~ ~ ^ { A B R } \\nabla _ b ^ \\alpha x ( t ) ) + q ( t ) x ( t ) = \\lambda r ( t ) x ( t ) , t \\in \\mathbb { N } _ { a + 1 , b - 1 } , \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\bigoplus _ { i = 1 } ^ g P _ { d _ i , m _ i } \\to N \\to 0 , \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} r ^ { k + 1 } = \\frac { I ( \\vec x ^ { k + 1 } ) } { \\norm { \\vec x ^ { k + 1 } } } \\ge \\frac { ( \\vec x ^ { k + 1 } , \\vec s ^ k ) } { \\norm { \\vec x ^ { k + 1 } } } \\ge \\sum _ { i \\in \\iota } | s _ i ^ k | \\ge r ^ k , \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} u _ { n } : = \\max \\{ \\min \\{ u ( x ) , n \\} , - n \\} , \\ , \\ , \\ , f o r \\ , \\ , a n y \\ , \\ , x \\in \\mathbb { G } . \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} \\frac { 1 } { r } > \\frac { d - 2 } { 2 d } \\ ; \\frac { 1 } { r } = \\frac { \\alpha } { 2 } + \\frac { \\left ( 1 - \\alpha \\right ) \\left ( d - 2 \\right ) } { 2 d } . \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} Z = \\lbrace x _ 1 , \\ldots , x _ l \\rbrace . \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\alpha = \\begin{pmatrix} A _ 1 & A _ 2 & A _ 3 & A _ 4 & B _ 1 \\\\ & & A _ 1 & A _ 2 \\\\ & & A _ 5 & A _ 6 & B _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} \\mathcal { A } ( t , x ) = \\frac { 1 - 2 x - \\sqrt { 1 - 4 x + 4 x ^ 2 - 4 x ^ 2 t } } { 2 t x } , \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\mathcal { Q } _ t ^ k ( x _ { t - 1 } ) = \\displaystyle \\max _ { 0 \\leq j \\leq N _ { t } ^ k } \\ ; \\theta _ t ^ j + \\langle \\beta _ t ^ j , x _ { t - 1 } \\rangle . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} \\Lambda ( s , f ) = \\left ( \\frac { \\sqrt { q } } { 2 \\pi } \\right ) ^ s \\Gamma \\left ( s + \\frac { 1 } { 2 } \\right ) L ( s , f ) , \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} T _ { l } ^ { k } = \\sum _ { j = 0 } ^ { n _ { l } - 1 } \\binom { k } { j } N _ { l } ^ { j } D _ { l } ^ { k - j } , l = 1 , 2 , k \\geqslant n _ { l } . \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\mathbb { G } ^ { ( 2 m ) } ( f ) & = \\mathbb { G } ( f ) - S _ { 1 , } ^ { ( m - 1 ) } ( f ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { A } ] - S _ { 2 , } ^ { ( m - 2 ) } ( f ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { B } ] , \\\\ \\mathbb { G } ^ { ( 2 m + 1 ) } ( f ) & = \\mathbb { G } ( f ) - S _ { 1 , } ^ { ( m - 1 ) } ( f ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { A } ] - S _ { 2 , } ^ { ( m - 1 ) } ( f ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { B } ] . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} \\big \\langle \\Lambda _ v P _ { v , \\rm R } F ( v ) , P _ { v , \\rm R } F ( v ) \\big \\rangle = \\frac { 1 } { \\beta _ v } \\bigg | \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ( v ) \\bigg | ^ 2 . \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} \\rho _ { \\alpha } ( \\theta ) = \\dfrac { \\theta ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } e ^ { - \\theta } , \\theta > 0 . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( N ) } ( f ) = \\sum _ { j = 1 } ^ { m _ { N } } \\frac { P ( A _ { j } ^ { ( N ) } ) } { \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) } \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( f 1 _ { A _ { j } ^ { ( N ) } } ) , \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} D ^ { ( l ) } _ { i , j } = \\begin{cases} 1 & i + j = l \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} \\mathbb { E } _ 2 = \\frac { \\sqrt { 8 } } { 8 } ( e _ 1 \\otimes e _ 1 \\otimes e _ 1 \\otimes e _ 2 + e _ 1 \\otimes e _ 1 \\otimes e _ 2 \\otimes e _ 1 + e _ 1 \\otimes e _ 2 \\otimes e _ 1 \\otimes e _ 1 + e _ 2 \\otimes e _ 1 \\otimes e _ 1 \\otimes e _ 1 \\\\ - e _ 2 \\otimes e _ 2 \\otimes e _ 2 \\otimes e _ 1 - e _ 2 \\otimes e _ 2 \\otimes e _ 1 \\otimes e _ 2 - e _ 2 \\otimes e _ 1 \\otimes e _ 2 \\otimes e _ 2 - e _ 1 \\otimes e _ 2 \\otimes e _ 2 \\otimes e _ 2 ) \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} \\nabla _ { U } V = \\overset { \\mu _ { 0 } } { \\nabla } _ { U } V + \\overset { \\mu _ { 0 } } { h } ( U , V ) , \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\dot { u } _ t = L ^ { \\Delta , \\sigma } _ { \\alpha _ 2 , u } u _ t , u _ t | _ { t = 0 } = u _ 0 \\in \\mathcal { U } _ { \\sigma , \\alpha _ 1 } \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} J _ 0 ^ n ( \\omega ) : = J _ 0 ( \\omega ) { \\upharpoonright _ { [ 1 , n ] } } . \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} g ^ p = ( a b ^ i ) ^ { k p } , \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} & x ^ k = \\dfrac { ( \\varphi - 1 ) y ^ k + x ^ { k - 1 } } { \\varphi } , \\\\ & y ^ { k + 1 } = { \\rm a r g m i n } \\Big \\{ \\lambda f ( y ^ k , y ) + \\dfrac { 1 } { 2 } \\| y - x ^ k \\| ^ 2 : y \\in C \\Big \\} . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} \\mathcal { C } _ { n } ^ { m } = \\sum _ { i = 1 } ^ { m - n } ( - 1 ) ^ { i } \\sum _ { a _ { 1 } , \\cdots , a _ { i } = 1 } ^ { \\infty } \\delta _ { a _ { 1 } + \\cdots + a _ { i } , m - n } \\prod _ { j = 1 } ^ { i } N _ { \\mathrm { d } a _ { j } } \\binom { m } { m - a _ { 1 } } \\binom { m - a _ { 1 } } { m - a _ { 1 } - a _ { 2 } } \\cdots \\binom { m - a _ { 1 } - \\cdots - a _ { i - 1 } } { m - a _ { 1 } - \\cdots - a _ { i - 1 } - a _ { i } } , \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\int _ { 0 } ^ { T } X ( t ) d t & = \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { j = 0 } ^ { n - 1 } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) \\\\ & = \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J \\setminus J _ { K } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) \\\\ \\end{array} \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} A _ { \\kappa , { \\tau } } A _ { \\sigma , { \\tau } } = I + R \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} & \\left ( ( A _ i - \\overline { X } _ i ) ^ T \\otimes ( \\Phi ^ { s } _ i ( X ) ) \\right ) ( \\overline { M } ^ s - I _ 4 ) - \\left ( ( A _ i - \\overline { X } ' _ i ) ^ T \\otimes ( \\Phi ^ { s } _ i ( X ) ) \\right ) ( \\overline { M } ^ s - I _ 4 ) \\\\ = & \\| \\overline { X } _ i - \\overline { X } ' _ i \\| \\left [ \\Phi ^ { s } _ i , \\Phi ^ { s } _ i \\right ] ~ \\left [ \\alpha _ 1 , \\alpha _ 2 \\right ] ( \\overline { M } ^ s - I _ 4 ) = \\mathbf { 0 } , ~ ~ ~ s = \\pm , \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\chi ( M ) : = v ( M ) - e ( M ) + f ( M ) . \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} ( \\dot { x } ) ^ 2 = 4 x ^ 2 ( Q - x ) ^ 2 | \\xi _ 1 | ^ 2 | \\xi _ 2 | ^ 2 - \\left ( | J | ^ 2 - | \\xi _ 1 | ^ 2 x ^ 2 - | \\xi _ 2 | ^ 2 ( Q - x ) ^ 2 \\right ) ^ 2 . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\varphi _ n ( x , y ) = \\begin{cases} 1 & d ( x , y ) = n \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\sphericalangle \\big ( L ( x , r ) , P _ y \\big ) & \\leq \\lim _ { r \\to 0 } C _ 1 ( m ) \\frac 2 { 1 - 2 \\tilde \\delta _ { x , k ( r ) } } \\left ( \\delta ( r ) + 2 \\tilde \\delta _ { x , k ( r ) } \\right ) = 0 . \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} \\phi _ t ( t _ { n - \\theta } ) = & \\frac { ( 3 - 2 \\theta ) \\phi ( t _ { n } ) - ( 4 - 4 \\theta ) \\phi ( t _ { n - 1 } ) + ( 1 - 2 \\theta ) \\phi ( t _ { n - 2 } ) } { 2 \\Delta t } + O ( \\Delta t ^ 2 ) \\\\ = & \\frac { ( 3 - 2 \\theta ) \\phi ^ { n } - ( 4 - 4 \\theta ) \\phi ^ { n - 1 } + ( 1 - 2 \\theta ) \\phi ^ { n - 2 } } { 2 \\Delta t } \\\\ \\triangleq & \\mathcal { D } _ t \\phi ^ { n - \\theta } , ~ n \\geq 2 . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} g ( t ) \\triangleq f ( t ) e ^ { - q t } e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) . \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} \\exp ( \\pi | r | ) \\Psi _ k ( s , x ) \\ll ( 1 - x ) ^ { 1 / 4 } \\frac { ( 1 + | r | ) ^ { 2 k - 2 a - 1 } } { 2 ^ { 2 k } k ^ { a + 1 / 2 } } \\left ( \\frac { x } { 1 - x } \\right ) ^ { k - a } , \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} \\begin{pmatrix} P _ 1 \\\\ P _ 3 & P _ 2 \\\\ P _ 4 & P _ 5 & P _ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\mathbf { A } = [ \\mathbf { a } _ 1 , \\mathbf { a } _ 2 , \\dots , \\mathbf { a } _ m ] , \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} \\mathfrak V _ i q ( \\mathfrak V _ i ) & = W ( x ) \\mathfrak U ^ * \\mathfrak c _ i q ( \\mathfrak v _ i ) E _ { i i } \\mathfrak U \\\\ & = W ( x ) \\mathfrak U ^ * \\mathfrak c _ i \\mathfrak t _ i \\mathfrak h _ i E _ { i i } \\mathfrak U \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} L _ V ( c ) = - 0 _ \\gamma ^ V \\cdot c ^ { - 1 } \\ ; , \\ ; \\ ; \\ ; R _ V ( c ' ) = c ' \\cdot 0 _ \\gamma ^ V \\ ; , \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { + \\infty } h ( t , s ) \\d s & = \\int _ { 0 } ^ { + \\infty } [ p ( N _ * , s ) - p ( N ( t ) , s ) ] n ( t , s ) \\d s \\\\ & + \\int _ { 0 } ^ { + \\infty } \\delta _ { 0 } ( x ) \\int _ { 0 } ^ { + \\infty } [ p ( N ( t ) , s ) - p ( N _ * , s ) ] n ( t , s ) \\d s \\d x \\\\ & = \\int _ { 0 } ^ { + \\infty } [ p ( N _ * , s ) - p ( N ( t ) , s ) ] n ( t , s ) \\d s + \\int _ { 0 } ^ { + \\infty } [ p ( N ( t ) , s ) - p ( N _ * , s ) ] n ( t , s ) \\d s \\\\ & = 0 , \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} g \\tau ^ { r } = \\left ( \\sigma \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\right ) \\tau ^ { r } = \\rho ^ { r a _ { 3 } } \\sigma \\tau ^ { r } \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } = \\rho ^ { r \\left ( a _ { 3 } - 1 \\right ) } \\tau ^ { r } g , \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\mathbb { E } ( \\nu _ i ) = \\mathbb { E } \\left ( \\sum _ { \\substack { 1 \\leq j \\leq t \\\\ \\psi ( E _ N ^ { ( j , M ) } ) \\in \\mathcal { P } _ i } } 1 \\right ) = t \\cdot \\mathbb { E } \\left ( \\sum _ { \\substack { G \\in \\Phi _ M \\\\ \\psi ( G ) \\in \\mathcal { P } _ i } } 1 \\right ) = t \\pi _ i . \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} v ^ m _ r ( r , t ) = - \\frac { C ^ m m } { a ( m - 1 ) } \\zeta ^ m ( t ) \\eta ( t ) r G ^ { \\frac 1 { m - 1 } } \\ , ; \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} \\lambda = \\frac { 3 } { 8 } M _ { i i k k } ^ { ( 2 ) } - \\frac { 1 } { 4 } M _ { i k i k } ^ { ( 2 ) } , \\mu = \\frac { 1 } { 4 } M _ { i k i k } ^ { ( 2 ) } - \\frac { 1 } { 8 } M _ { i i k k } ^ { ( 2 ) } v = \\frac { 1 } { 4 } \\epsilon _ { i j } M _ { i k j k } ^ { ( 2 ) } \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) = 2 ^ { k + 1 } \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } \\frac { | \\langle F _ { | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 } { \\langle g , g \\rangle ^ 2 } , \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\beta = \\frac { 1 } { 1 + \\lambda / ( 1 + M ) \\sqrt { n } } \\in \\left ( 0 , 1 \\right ) , K = \\beta p _ { N } , K ^ { \\prime } = p _ { N } ( 1 - \\beta ) , \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} A ^ * p & = z - y _ \\gamma \\Omega , \\\\ p & = 0 \\quad \\quad \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} d _ M = { \\left ( \\begin{array} { c c c c } w _ { 2 , 1 } & w _ { 2 , 2 } & \\cdots & w _ { 2 , n - 1 } \\\\ w _ { 3 , 1 } & w _ { 3 , 2 } & \\cdots & w _ { 3 , n - 1 } \\\\ \\vdots & \\vdots & \\cdots & \\vdots \\\\ w _ { n , 1 } & w _ { n , 2 } & \\cdots & w _ { n , n - 1 } \\end{array} \\right ) , } \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\phi \\ , = \\sum _ \\alpha f ^ \\alpha _ 0 \\otimes f ^ \\alpha _ 1 \\otimes \\ldots \\otimes f ^ \\alpha _ q , f ^ \\alpha _ i \\in C ^ \\infty ( M ) , \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} ( J ^ \\natural - J _ { z _ 0 } ) | _ { ( z , y ) } = A ( z _ 0 , z , y ) \\circledast d z \\otimes \\partial _ y + B ( z _ 0 , z , y ) \\circledast d y \\otimes \\partial _ y , \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\wedge ^ 1 F _ 1 \\otimes S ( - s _ 1 ) \\xrightarrow { \\begin{bmatrix} T _ 0 \\\\ T _ 1 \\\\ T _ 2 \\end{bmatrix} } \\wedge ^ 1 F _ 0 \\otimes S ( - s _ 1 ) \\xrightarrow { \\begin{bmatrix} 0 & T _ 2 & - T _ 1 \\\\ - T _ 2 & 0 & T _ 0 \\\\ T _ 1 & - T _ 0 & 0 \\end{bmatrix} } F _ 0 ^ * \\xrightarrow { \\begin{bmatrix} T _ 0 & T _ 1 & T _ 2 \\end{bmatrix} } F _ 1 ^ * . \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} | C ( p ) n | = \\sqrt { \\sum _ { i = 1 } ^ k \\left ( n _ i - \\frac { 1 } { 2 } v _ { i + k } n _ { 2 k + 1 } \\right ) ^ 2 + \\left ( n _ { k + i } + \\frac { 1 } { 2 } v _ i n _ { 2 k + 1 } \\right ) ^ 2 } \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} & \\textrm { i ) } \\sup _ { N \\in \\mathbb { N } } \\mathbb { E } \\left \\vert H _ t ^ N - H _ u ^ N \\right \\vert ^ p \\leq C \\left \\vert t - u \\right \\vert ^ { \\alpha } \\mbox { f o r s o m e $ C > 0 $ a n d $ p , \\alpha > 1 $ ; } \\\\ & \\textrm { i i ) } \\lim _ { K \\to \\infty } \\sup _ { N \\in \\mathbb { N } } \\mathbb { P } \\left ( \\left \\vert H _ 0 ^ N - { a } \\right \\vert > K \\right ) = 0 \\mbox { f o r s o m e $ a \\in \\mathbb { R } $ . } \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} \\omega \\left ( \\left ( \\begin{smallmatrix} \\det ( h _ 1 h _ 2 ) & 0 \\\\ 0 & 1 \\end{smallmatrix} \\right ) , ( h _ 1 , h _ 2 ) \\right ) \\phi _ { \\mathbf g , p } = \\underline { \\chi } _ p ( h _ 1 h _ 2 ) \\phi _ { \\mathbf g , p } . \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} w ( 0 , t ) = w ( \\pi , t ) = 0 \\ , , w ( x , 0 ) = \\left \\{ \\begin{array} { l l l } 1 & { \\rm i f } & x \\in ( ( 1 - \\beta ) \\pi ) , ( 1 + \\beta ) \\pi ) \\\\ 0 & & { \\rm o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} e _ n \\ , h ( \\ , e _ n \\ , a \\ , e _ n \\ , ) \\ , e _ n \\ = \\ e _ n \\ , a \\ , e _ n \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\Phi ^ t _ i = \\Phi ^ 0 _ i + t \\phi _ i : C _ { 1 0 m _ i ^ { - 1 / 2 } } \\rightarrow \\mathfrak { s u } ( 2 ) \\backslash \\lbrace 0 \\rbrace , \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} \\frac { F ( a X , Y ) } { \\left ( \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { ( a \\delta _ j X - \\gamma _ j Y ) } \\right ) ^ { t } } & = \\frac { F ( X , Y ) } { \\left ( \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { ( \\delta _ j X - \\gamma _ j Y ) } \\right ) ^ { t } } \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} C _ { 1 } & = \\frac { 1 } { 2 } a _ { 1 } a _ { 3 } a _ { 1 } ' ( a _ { 1 } ' - 1 ) + \\frac { 1 } { 2 } a _ { 2 } a _ { 4 } a _ { 3 } ' ( a _ { 3 } ' - 1 ) + a _ { 3 } a _ { 1 } ' a _ { 2 } a _ { 3 } ' \\\\ C _ { 2 } & = \\frac { 1 } { 2 } a _ { 1 } a _ { 3 } a _ { 2 } ' ( a _ { 2 } ' - 1 ) + \\frac { 1 } { 2 } a _ { 2 } a _ { 4 } a _ { 4 } ' ( a _ { 4 } ' - 1 ) + a _ { 3 } a _ { 2 } ' a _ { 2 } a _ { 4 } ' . \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} B ( \\lambda ) = B _ 0 ( \\lambda ) + K ( \\lambda ) ^ T X ( \\lambda ) , \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} w _ * : = \\left \\{ \\begin{array} { l l } \\min \\{ M _ * \\log { n } , M _ * ^ { 1 / 2 } n ^ { 1 / 2 } / \\Delta \\} & k = 2 , \\\\ M _ * \\log { n } & , \\end{array} \\right . \\nu : = \\left \\{ \\begin{array} { l l } \\epsilon ^ { 1 / 3 } & k = 1 , \\\\ n ^ { - 1 / 1 0 } & . \\end{array} \\right . \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{gather*} \\Gamma _ { \\C } ( s + 1 ) L \\big ( \\psi ^ 2 , s \\big ) = \\epsilon \\big ( \\psi ^ 2 , s \\big ) \\Gamma _ { \\C } ( 2 - s ) L \\big ( \\psi ^ { - 2 } , 1 - s \\big ) , \\end{gather*}"}
-{"id": "5366.png", "formula": "\\begin{align*} B = \\frac { q } { \\gcd ( N / M , q ) } . \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} ( M ^ { \\ell , \\ell - 1 } _ e ) ^ T ( M ^ \\ell _ e ) ^ { - 1 } M ^ { \\ell , \\ell - 1 } _ e = R _ e ^ T M ^ \\ell _ e ( M ^ \\ell _ e ) ^ { - 1 } M ^ { \\ell } _ e R _ e = R ^ T _ e M ^ \\ell _ e R _ e = M ^ { \\ell - 1 } _ e . \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} \\Phi _ { \\mathbf h _ p } ( \\nu \\beta _ 1 \\nu ' ) = \\frac { \\chi _ { \\psi } ( p ) \\underline { \\chi } _ p ( \\gamma ) p ^ { 3 / 2 } } { p - 1 } \\cdot \\mathcal G \\left ( \\frac { - \\gamma } { p } \\right ) \\left ( \\mathcal G \\left ( \\frac { \\gamma ^ { - 1 } } { p } \\right ) - \\frac { 1 } { p } \\right ) . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} [ \\gamma _ { a , b } ( t ) ] _ { a , b \\in Q _ 1 ^ { ( i j ) } } = e ^ { N _ { i j } ( t ) } , i , j \\in Q _ 0 , \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} \\Pr \\left ( { \\left . { { s _ n } } \\right | { \\bf { R } } } \\right ) = \\int { \\Pr \\left ( { \\left . { { s _ n } , { { \\tilde \\theta } _ n } } \\right | { \\bf { R } } } \\right ) d { { \\tilde \\theta } _ n } } \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} \\Phi ( n ) : = \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { m } \\varphi ( 2 s + 1 ) , \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\bar { R } ( X , Y , Z , W ) & = R ( X , Y , Z , W ) - g \\big ( h ( X , W ) , h ( Y , Z ) \\big ) \\\\ & + g \\big ( h ( X , Z ) , h ( Y , W ) \\big ) , \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to \\infty } \\lvert \\Phi \\rvert = m . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} f ^ n _ \\omega = f _ { T ^ { n - 1 } \\omega } \\circ \\cdots \\circ f _ \\omega , \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} v _ { \\lambda } : = \\underbrace { v _ 1 \\otimes \\cdots \\otimes v _ 1 } _ \\textrm { $ \\lambda _ 1 $ t i m e s } \\otimes \\underbrace { v _ 2 \\otimes \\cdots \\otimes v _ 2 } _ \\textrm { $ \\lambda _ 2 $ t i m e s } \\otimes \\cdots \\otimes \\underbrace { v _ n \\otimes \\cdots \\otimes v _ n } _ \\textrm { $ \\lambda _ n $ t i m e s } \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} & \\rho _ { 1 3 4 } ( S _ 1 ) ^ { 2 k } = \\tilde S _ 1 ^ { 2 k } = S _ 2 ( S _ 1 S _ 2 ) ^ k S _ 2 ^ * + S _ 1 ( S _ 2 S _ 1 ) ^ k S _ 1 ^ * \\\\ & \\rho _ { 1 3 4 } ( S _ 2 ) ^ { 2 k } = \\tilde S _ 2 ^ { 2 k } = S _ 2 ( S _ 2 S _ 1 ) ^ k S _ 2 ^ * + S _ 1 ( S _ 1 S _ 2 ) ^ k S _ 1 ^ * \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\frac { 1 } { 2 } ( u _ r ^ 2 , \\nabla ( u _ r m _ r ) ) d r = - \\int _ 0 ^ t \\frac { 1 } { 3 } ( u ^ 3 _ r , \\nabla m _ r ) d r \\lesssim \\| \\nabla m \\| _ { \\infty } \\int _ 0 ^ t \\| u _ r \\| _ { L ^ 3 } ^ 3 d r . \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{gather*} m = \\varepsilon ( m _ { \\langle 1 \\rangle } ) m _ { \\langle 0 \\rangle } . \\end{gather*}"}
-{"id": "4103.png", "formula": "\\begin{align*} \\int \\partial + \\partial \\int = 1 . \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} \\begin{cases} v ^ { 1 / 2 } \\mapsto t ^ { - 1 / 2 } , & \\\\ \\hat { X } _ { ( i , r ) } \\mapsto \\widetilde { A } _ { i , r - r _ i } ^ { - 1 } & \\ ( i , r ) \\in ( \\overline { I } _ { \\xi } ^ { \\mathrm { t w } , \\flat } ) _ e , \\\\ X ' _ { \\Pi _ { \\imath } } \\mapsto \\underline { \\widetilde { Y } _ { \\Pi _ { \\imath } } } & \\ \\imath \\in I _ { \\mathrm { A } } . \\end{cases} \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\rho _ s ( \\tau ) = | \\{ p \\in \\mathcal { P } : \\ ; r _ { p , s } \\leq \\tau \\} | / | \\mathcal { P } | . \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} h = \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) , h \\alpha _ n = \\left ( \\begin{array} { c c } a p ^ n & b p ^ { - n } \\\\ c p ^ n & d p ^ { - n } \\end{array} \\right ) , \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} q ^ { 2 n - 2 } t _ 1 t _ 2 t _ 3 t _ 4 t _ 5 t _ 6 = p q . \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} t _ { s _ { 1 , 1 } } ^ { 2 m + 1 } = \\prod _ { i = 1 } ^ { m } { } _ { t _ { c _ 1 } ^ { - 2 i } } ( t _ { t _ { c _ 1 } ( z _ 1 ) } ^ { - 1 } t _ { a _ 1 } t _ { z _ 1 } ^ { - 1 } t _ { a _ 1 } ) \\cdot t _ { t _ { c _ 1 } ^ { - 2 m - 1 } ( z _ 1 ) } ^ { - 1 } t _ { a _ 1 } ^ { 2 m + 1 } t _ { c _ 1 } ^ { - 2 m - 1 } t _ { a _ 1 } \\cdot t _ { d _ { g - 1 } } ^ { 2 m + 1 } t _ { d _ 1 } ^ { 2 m + 1 } . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} z ^ 2 + z + k = 0 . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} G ( A _ 1 , \\ldots , A _ m ) = \\underset { k \\rightarrow \\infty } { \\lim } \\ , S _ k . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} i \\widetilde { \\mathcal { G } } _ { \\alpha \\beta } ( x , y ) = \\sum _ { m = 0 } ^ { \\infty } \\left ( - \\frac { i } { \\hbar } \\right ) ^ { m } \\frac { 1 } { m ! } \\int _ { - \\infty } ^ { \\infty } d t _ { 1 } \\cdots \\int _ { - \\infty } ^ { \\infty } d t _ { m } \\langle \\phi _ { 0 } | T [ \\hat { H } _ { 1 } ( t _ { 1 } ) \\cdots \\hat { H } _ { 1 } ( t _ { m } ) \\hat { \\psi } _ { \\mathrm { H } \\alpha } ( x ) \\hat { \\psi } _ { \\mathrm { H } \\beta } ^ { \\dagger } ( y ) ] | \\phi _ { 0 } \\rangle , \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} S ^ n ( x , r ) = \\left ( T ^ n x , r + \\sum _ { i = 0 } ^ { n - 1 } f \\circ T ^ i x \\right ) = ( T ^ n x , s _ n ( x , r ) ) . \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\sigma _ \\lambda = j ^ * \\sigma _ { \\lambda + 1 ^ m } ^ + \\tau _ \\mu ' = j ^ * \\sigma _ \\mu ^ + , \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\Big ( | M _ i | + | R _ i | \\Big ) & \\ge \\sum _ { i = 1 } ^ k \\Big ( \\gamma _ { t 2 } ( H ) - | P _ i | \\Big ) = \\gamma _ { t 2 } ( G ) \\gamma _ { t 2 } ( H ) - | D | \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} \\nabla \\nabla ^ { - \\nu } U ( t ) = \\nabla ^ { - ( \\nu - 1 ) } U ( t ) , \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\dot { x } + 3 \\ddot { y } - 2 z = 0 \\\\ \\ddot { y } + 5 \\dot { z } = 0 \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} M _ 0 = \\{ m _ { s / 2 , r + 1 } ^ { ( n - 1 ) } , m _ { s / 2 , r + 3 } ^ { ( n - 1 ) } \\} , \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} \\delta ' ( \\imath , r ) : = & \\delta ( \\imath , r ) - \\delta _ { ( \\imath , r ) , ( \\imath _ 0 , r _ 0 + \\frac { 1 } { 2 } ) } - \\delta _ { ( \\imath , r ) , ( \\imath _ 0 , r _ 0 - \\frac { 1 } { 2 } ) } - \\delta _ { ( \\imath , r ) , ( \\imath _ 0 - 1 , r _ 0 ) } \\\\ & + \\delta _ { ( \\imath , r ) , ( \\imath _ 0 - 1 , r _ 0 + \\frac { 1 } { 2 } ) } + \\delta _ { ( \\imath , r ) , ( \\imath _ 0 - 1 , r _ 0 - \\frac { 1 } { 2 } ) } + \\delta _ { ( \\imath , r ) , ( \\imath _ 0 , r _ 0 ) } \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} | u ( t ) | \\leq r _ 0 = \\max \\{ M , | A / \\alpha | , | B / a | \\} . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} \\vec { g } _ { i } = \\frac { \\sum _ { j \\in B _ { i } } \\vec { f } _ { j } } { \\norm * { \\sum _ { j \\in B _ { i } } \\vec { f } _ { j } } } . \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\varphi ( x ) = \\varphi _ H ( x ) + \\varphi _ B ( x ) , \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} f ( m ) = f ( \\underbrace { 1 + 1 + \\dots + 1 } _ { m } ) = \\underbrace { f ( 1 ) + f ( 1 ) + \\dots + f ( 1 ) } _ { m } = \\underbrace { 1 _ R + 1 _ R + \\dots + 1 _ R } _ { m } . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} | K ^ { c } \\cap B _ { q } ( x , \\delta ) | = | B _ { q } ( x , \\delta ) | - | K \\cap B _ { q } ( x , \\delta ) | = | K | - | K \\cap B _ { q } ( x , \\delta ) | = | K \\cap B _ { q } ^ { c } ( x , \\delta ) | , \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} H _ 1 ( \\beta ) \\begin{cases} \\infty , & \\beta \\in [ 0 , \\beta _ P ( 2 ) ] , \\\\ \\in ( 0 , \\infty ) , & \\beta \\in ( \\beta _ P ( 2 ) , \\beta _ c ( 2 ) ) . \\end{cases} \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\gamma ( q A , q , d ; \\mathbf { s } ) = \\sum _ { \\lambda = 0 } ^ { q - 1 } \\gamma ( A , q , d ; \\mathbf { s } M _ \\lambda ) \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} \\partial _ { x _ j x _ k } ( I _ { h , T } \\mathbf { u } ( X ) - \\mathbf { u } ( X ) ) = - \\partial _ { x _ j x _ k } \\mathbf { u } ( X ) + \\sum _ { i \\in \\mathcal { I } } \\partial _ { x _ j x _ k } \\Phi _ { i , T } ( X ) \\widetilde { \\mathbf { R } } _ i ( X ) , \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} \\int _ M \\sigma _ 2 ( x ) ( - \\Delta ^ { \\dagger } _ \\rho ) \\sigma _ 3 ( x ) d x = \\int _ M \\sigma _ 2 ( x ) \\big ( g ( \\rho + t \\sigma _ 1 ) \\sigma _ 3 \\big ) ( x ) d x . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} [ [ r , r ] ] = [ r _ { 1 2 } , r _ { 1 3 } ] + [ r _ { 1 2 } , r _ { 2 3 } ] + [ r _ { 1 3 } , r _ { 2 3 } ] = 0 ; \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} \\log | \\det ( M ^ n _ \\omega ) | = \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i \\omega ) . \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { r } ( - a _ { j } q ; q ) _ { N - 1 } . \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} F ( x , t ) = { \\bf 1 } _ { \\omega ( t ) } g ( x , t ) \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} - \\hat b ^ \\top \\lambda ^ * = c ^ \\top x ^ * + \\sqrt { ( x ^ * ) ^ \\top Q ( x ^ * ) } = \\alpha c ^ \\top \\bar x + | \\alpha | \\sqrt { \\bar x ^ \\top Q \\bar x } = \\alpha \\big ( c ^ \\top \\bar x - \\sqrt { \\bar x ^ \\top Q \\bar x } \\big ) \\ ; . \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} A _ { d , N + 1 , s } = \\sum _ { r = s } ^ d P ( \\lambda ) ^ { r - s - 1 } A _ { d , N , r } Q _ { r , s } ( \\lambda ) . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} C _ 1 ( t ) = U _ 1 ^ { - 1 } ( - 5 + t , - 2 - t , - 1 , 3 , 4 , 1 ; x ) , C _ 2 ( t ) = U _ 2 ^ { - 1 } ( - 4 + t , - 2 , - 2 - t , 3 , 4 , 1 ; x ) . \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) = 2 ^ { k + 1 } N ^ { - 1 } \\prod _ { p \\mid N } ( p + 1 ) ^ 2 \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } \\frac { | \\langle F _ { | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 } { \\langle g , g \\rangle ^ 2 } . \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} T _ H ( f ) ( x H ) = \\int _ H f ( x h ) \\ , d \\mu _ H ( h ) . \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} \\Sigma ^ + = \\{ e _ j - e _ i : 1 \\leq i < j \\leq n \\} \\ , \\cup \\ , \\{ - e _ i - e _ j : 1 \\leq i \\leq j \\leq n \\} \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} E _ { 2 i } \\cdot P _ { 2 j } + P _ { 2 i } \\cdot E _ { 2 j } = 0 , \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} \\mu _ { n + 1 } : = \\mu _ n + R _ { X _ n } . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} | B _ { i + 1 } ( x , y ) | \\leq | B _ 1 ( x , y ) | \\left ( \\sup _ { z \\in B _ 1 ( x , y ) } | B _ i ( x , z ) | \\right ) \\leq N N ^ i = N ^ { i + 1 } . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} p _ { t } ( \\mu ; x ) & = \\alpha f ( x ) + ( 1 - \\alpha ) g ( x ) \\\\ & \\geq \\alpha p _ { t } ( f ' ( 1 ) ; x ) + ( 1 - \\alpha ) p _ { t } ( g ' ( 1 ) ; x ) \\\\ & \\geq p _ { t } ( \\alpha f ' ( 1 ) + ( 1 - \\alpha ) g ' ( t ) ; x ) . \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} \\| A _ { \\cdot } \\| = \\Big ( \\sum _ { i = 1 } ^ d \\sum _ { j = 1 } ^ m \\int _ 0 ^ T | a ^ { ( i , j ) } ( u ) | ^ 2 d u \\Big ) ^ { 1 / 2 } = \\Big ( \\int _ 0 ^ T | A ( u ) | ^ 2 d u \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} S = \\ker W , \\tilde S = \\ker \\tilde W . \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} J ( u , v ) ~ = ~ \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\log G ( u _ n , v _ n ) \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , 0 , 0 ) } = \\frac { \\pi } { 3 2 } \\ , L \\ , \\ , X _ 2 \\ , X _ 3 \\ , Y , \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\widetilde { \\nabla } _ \\alpha = d + \\Bigg ( p _ { - 1 } - \\frac { \\varphi _ \\alpha ( t _ \\alpha ) } { h ^ \\vee } \\rho + \\sum _ { i \\in E } v _ { \\alpha , i } ( t _ \\alpha ) p _ i \\Bigg ) d t _ \\alpha \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} O _ 1 = \\left [ \\begin{array} { c c } X & - Y \\\\ Y & X \\end{array} \\right ] , O _ 2 = \\left [ \\begin{array} { c c } Z & - W \\\\ W & Z \\end{array} \\right ] . \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} V _ { h } : = \\{ v _ { h } \\in X _ { h } : ( q _ { h } , \\nabla \\cdot v _ { h } ) = 0 , \\forall q _ { h } \\in Q _ { h } \\} . \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} f ( u , v ) = \\int \\dfrac { 1 } { \\hat { H } } \\Big ( \\big ( \\lambda \\nu _ u + F \\nu \\times \\nu _ u - E \\nu \\times \\nu _ v \\big ) \\ , d u + \\big ( \\lambda \\nu _ v + G \\nu \\times \\nu _ u - F \\nu \\times \\nu _ v \\big ) \\ , d v \\Big ) . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} ( A , B ; C , D ) = ( \\lambda _ A \\cdot A , \\lambda _ B \\cdot B ; \\lambda _ C \\cdot C , \\lambda _ D \\cdot D ) . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ 0 } E _ 1 ( x , \\zeta ) = - i ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) E _ 1 ( x , \\zeta ) & = - \\left ( e _ 1 \\frac { \\partial } { \\partial x _ 1 } + e _ 2 \\frac { \\partial } { \\partial x _ 2 } \\right ) E _ 1 ( x , \\zeta ) \\\\ & = - \\left ( \\frac { \\partial } { \\partial x _ 1 } E _ 1 e _ 1 + \\frac { \\partial } { \\partial x _ 2 } E _ 1 e _ 2 \\right ) . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\frac { \\partial x } { \\partial z _ 1 } \\bigg | _ { t , 0 , z _ 2 } = \\frac { \\partial x } { \\partial z _ 1 } \\bigg | _ { t , l , z _ 2 } = \\frac { \\partial x } { \\partial z _ 1 } \\bigg | _ { t , z _ 1 , 0 } = \\frac { \\partial x } { \\partial z _ 1 } \\bigg | _ { t , z _ 1 , l } = 0 . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\mathcal { H } _ \\varnothing & = \\bigcap _ { q \\in \\N ^ N } V _ { [ N ] } ^ q \\mathcal { H } , & \\mathcal { H } _ { [ N ] } & = \\bigoplus _ { p \\in \\N ^ N } V _ { [ N ] } ^ p \\left ( \\mathcal { W } _ { [ N ] } \\right ) . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\| \\phi \\| _ { c b } = \\| M _ \\phi \\| . \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\sigma ( A ) ( x , \\xi ) \\sim \\sum _ { j \\ge 0 } ^ \\infty \\sum _ { k = 0 } ^ q \\sigma _ { p - j , k } ( A ) ( x , \\xi ) \\log ^ k | \\xi | \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{gather*} B ( a , b ) : = \\int _ 0 ^ 1 x ^ { a - 1 } ( 1 - x ) ^ { b - 1 } { \\rm d } x . \\end{gather*}"}
-{"id": "1215.png", "formula": "\\begin{align*} \\quad \\gamma \\left ( T , \\beta \\right ) \\ge 1 , \\quad \\lim _ { \\beta \\to 0 ^ { + } } \\gamma \\left ( t , \\beta \\right ) = \\infty \\quad t \\in ( 0 , T ] . \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\phi ( n ) = 1 - \\frac { 1 } { e } . \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { n - \\delta + 1 } { n - r \\brack \\delta - 1 } A ' _ r = { n \\brack \\delta - 1 } ( c ^ { n - \\delta + 1 } - \\abs { Y } ) . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} \\phi = 1 \\ \\ & \\textrm { o n } \\ \\ B ( 3 \\rho / 4 ) , \\ \\ \\ \\textrm { s u p p } \\ , \\phi \\subset B ( \\rho ) , \\\\ | \\nabla \\phi | & \\leq C \\rho ^ { - 1 } , \\ \\ | \\nabla ^ 2 \\phi | \\leq C \\rho ^ { - 2 } . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} D ( D + C _ 1 C _ 2 + C _ 0 C _ 3 ) = ( k + 1 ) ( C _ 1 C _ 2 + C _ 0 C _ 3 ) ^ 2 + ( C _ 1 ^ 2 + C _ 0 C _ 2 ) ( C _ 2 ^ 2 + C _ 1 C _ 3 ) . \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n - 1 } \\big ( r _ i d _ { i + 1 } - r _ { i + 1 } d _ i + r _ i r _ { i + 1 } t \\big ) + ( 1 - g ) \\left ( \\sum _ { i = 1 } ^ { n - 1 } r _ i r _ { i + 1 } - \\sum _ { i = 1 } ^ n r _ i ^ 2 \\right ) + \\min _ { 1 \\leq i \\leq n } \\{ h ^ 0 ( \\emph { E n d } U _ i ) \\} . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} ( \\mathcal { S } _ N h ) ( q \\ , | \\ , b ) \\triangleq \\sum \\limits _ { p = 0 } ^ N ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ N h \\bigl ( q + b _ 0 + b _ { k _ 1 } + \\cdots + b _ { k _ p } \\bigr ) . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{gather*} \\Gamma _ 2 ( x _ 1 + x _ 2 - 2 q \\ , | \\ , a ) = 2 ^ { - B _ { 2 , 2 } ( x _ 1 + x _ 2 - 2 q \\ , | \\ , a ) / 2 } \\Gamma _ 2 ( ( x _ 1 + x _ 2 ) / 2 - q \\ , | \\ , a ) \\times \\\\ \\times \\Gamma _ 2 ( ( x _ 1 + x _ 2 + a _ 1 ) / 2 - q \\ , | \\ , a ) \\times \\Gamma _ 2 ( ( x _ 1 + x _ 2 + a _ 2 ) / 2 - q \\ , | \\ , a ) \\times \\\\ \\times \\Gamma _ 2 ( ( x _ 1 + x _ 2 + a _ 1 + a _ 2 ) / 2 - q \\ , | \\ , a ) . \\end{gather*}"}
-{"id": "3495.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( a _ 1 , a _ 2 ; N ) } ^ \\ast ( s , \\tau ) = \\sum _ { ( m , n ) \\in \\Z ^ 2 \\atop ( m , n ) \\equiv ( a _ 1 , a _ 2 ) \\mod N } \\frac 1 { ( m \\tau + n ) ^ k } \\frac { \\Im ( \\tau ) ^ s } { | m \\tau + n | ^ { 2 s } } . \\end{gather*}"}
-{"id": "1656.png", "formula": "\\begin{align*} j ^ * \\sigma _ { p } ^ + = \\begin{cases} \\sigma _ { p } ' & \\\\ \\sigma _ { p } ' + \\tau _ { 1 ^ { p - 2 n - 2 + 2 m } } & \\end{cases} \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} ( d - e ) + ( f - g ) + 2 ( j - k ) = 0 , \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} f ^ L ( \\lambda \\vec x ) & = \\int _ 0 ^ { \\lambda \\| x \\| _ \\infty } f ( V _ t ^ + ( \\lambda \\vec x ) , V _ t ^ - ( \\lambda \\vec x ) ) d t \\\\ & = \\int _ 0 ^ { \\| x \\| _ \\infty } \\lambda f ( V _ { s } ^ + ( \\vec x ) , V _ { s } ^ - ( \\vec x ) ) d s = \\lambda f ^ L ( \\vec x ) . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\Omega _ p ( \\alpha _ n ) \\mathrm { v o l } ( \\Gamma _ 0 \\alpha _ n \\Gamma _ 0 ) = ( - w _ p ) ^ n p ^ { - 2 | n | - 1 } ( 1 - p ^ { - 1 } ) , \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\displaystyle \\lim \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\mathbb { E } [ | \\sum _ { \\substack { j \\in J _ { K } } } \\Phi ( \\frac { B _ { t _ { j } } + B _ { t _ { j + 1 } } } { 2 } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) | ^ 2 ] = 0 \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\left ( \\frac { c f } { a ^ 2 } \\right ) ^ { \\binom n 2 } q ^ { 2 \\binom n 3 } \\prod _ { j = 1 } ^ n \\frac { ( q ^ { j - n } b / c , q ^ { n - j } b c / a , q ^ { j - n } e / f , q ^ { n - j } e f / a ) _ { j - 1 } } { ( b , b / a , c , c / a , e , e / a , f , f / a ) _ { j - 1 } } . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} \\phi ' _ i & = \\phi _ i + \\phi _ j \\psi _ { i j } \\\\ & = ( A _ p z ^ p + \\ldots + A _ 0 ) + ( B _ q z ^ q + \\ldots + B _ 0 ) ( C _ r z ^ r + \\ldots + C _ 0 ) \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} Q \\left ( R , \\theta _ 0 ^ { - \\alpha _ 0 } R ^ 2 \\right ) = B _ { R } \\times \\left ( - \\theta _ 0 ^ { - \\alpha _ 0 } R ^ 2 , 0 \\right ) \\left ( \\theta _ 0 = \\frac { \\omega } { 4 } , \\ , \\ , \\alpha _ 0 = \\beta ( m - 1 ) \\right ) \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} { \\rm g r a d } \\ , f ( S ) = S { \\rm s y m } ( \\nabla \\bar { f } ( S ) ) S . \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} M _ { 1 , 1 } & = \\frac { 2 P _ 3 ( 1 ) } { L q ^ { 1 / 2 } } \\mathop { \\sum \\sum } _ { \\substack { v w = q \\\\ ( v , w ) = 1 } } \\mu ^ 2 ( v ) \\varphi ( w ) \\mathop { \\sum \\sum } _ { \\substack { b \\leq y _ 2 \\\\ ( b n , q ) = 1 } } \\frac { \\Lambda ( b ) P _ 2 [ b ] } { ( b n ) ^ { \\frac { 1 } { 2 } } } V _ 1 \\left ( \\frac { n } { q ^ { \\frac { 1 } { 2 } } } \\right ) \\cos \\left ( \\frac { 2 \\pi b n \\overline { v } } { w } \\right ) , \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\vert \\tau ( y ) - \\tau _ k ( y ) \\vert & < \\varepsilon + \\sum _ { i = k } ^ { j - 1 } \\vert \\tau _ { i + 1 } ( y ) - \\tau _ i ( y ) \\vert \\\\ & \\leq \\varepsilon + \\sum _ { i = k } ^ { \\infty } \\vert \\tau _ { i + 1 } ( y ) - \\tau _ i ( y ) \\vert . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} \\int _ { d f ( 0 ) \\neq 0 } \\log \\Vert d f ( 0 ) \\Vert d \\nu & = \\sum _ { n = 1 } ^ { \\infty } \\int _ { \\delta _ n < \\Vert d f ( 0 ) \\Vert \\le \\delta _ { n - 1 } } \\log \\Vert d f ( 0 ) \\Vert d \\nu \\\\ & = \\lim _ { N \\to \\infty } U _ { \\delta _ N } \\\\ & \\ge 0 . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\mathrm { d i m } _ \\nu ( H _ n ) = \\prod _ { k = 1 } ^ t ( 1 + \\nu ^ { n _ k } + \\cdots + \\nu ^ { ( p _ k - 1 ) n _ k } ) = \\prod _ { k = 1 } ^ t \\dfrac { 1 - \\nu ^ n } { 1 - \\nu ^ { n _ k } } . \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) = ( 2 m - 2 \\Phi ( n ) + 2 i + 1 , 2 m - 2 \\Phi ( n ) + 2 i ) . \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} M ^ T B M = \\begin{bmatrix} E & 0 \\\\ 0 & D \\end{bmatrix} , \\quad D = B '' - U E ^ { - 1 } U ^ T . \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} Q _ 1 ( \\lambda ) = \\begin{bmatrix} \\lambda ^ 2 P _ 5 + \\lambda P _ 4 + P _ 3 - \\lambda ( I _ n + P _ 8 P _ 0 ) & \\lambda ^ 2 P _ 8 & \\lambda ^ 2 P _ 7 + \\lambda P _ 6 & I _ n \\\\ P _ 0 & - \\lambda I _ n & \\lambda ^ 2 I _ n & 0 \\\\ \\lambda P _ 2 + P _ 1 & I _ n & 0 & - \\lambda ^ 2 I _ n \\\\ \\lambda ^ 2 I _ n & 0 & - I _ n & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} F _ \\varepsilon ( u ) = \\int _ 0 ^ t e ^ { - t / \\varepsilon } \\bigg ( \\int _ { \\mathbb { R } ^ n } \\frac { \\varepsilon ^ 2 | u '' ( t , x ) | } { 2 } \\ , d x + \\mathcal { W } ( u ( t , \\cdot ) ) - \\int _ { \\mathbb { R } ^ n } f _ \\varepsilon ( t , x ) u ( t , x ) \\ , d x \\bigg ) \\ , d t , \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} u _ { t } + \\gamma \\bigtriangleup \\sigma - \\bigtriangleup u + f ( u ) = g ( \\textbf { z } , t ) , \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} 2 n \\left ( \\binom { 2 n } { n } \\frac { 1 } { 4 ^ n } \\right ) ^ 2 & = \\frac { 1 } { 2 } \\prod _ { j = 2 } ^ n \\left ( 1 + \\frac { 1 } { 4 j ( j - 1 ) } \\right ) , \\\\ ( 2 n + 1 ) \\left ( \\binom { 2 n } { n } \\frac { 1 } { 4 ^ n } \\right ) ^ 2 & = \\prod _ { j = 1 } ^ n \\left ( 1 - \\frac { 1 } { 4 j ^ 2 } \\right ) \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} p ( \\rho ) = - \\gamma \\sqrt \\rho \\quad \\mu ( \\rho ) = 3 \\nu \\rho . \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} K _ { n } ( x , y ) = \\frac { 1 } { \\langle { \\bf u } , p ^ 2 _ { n - 1 } \\rangle } \\frac { p _ { n } ( x ) p _ { n - 1 } ( y ) - p _ { n } ( y ) p _ { n - 1 } ( x ) } { x - y } . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\Omega _ 0 ( x ) = \\partial _ 1 \\left ( \\frac { \\mathcal { R } ( x _ 0 , x _ 1 ) } { \\mathcal { S } ( x _ 0 , x _ 1 ) } \\right ) _ { x _ 0 = x _ 1 = x } \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\theta _ { 1 } & = \\left \\{ \\left ( x , y \\right ) \\in L ^ { 2 } : b \\leq x , y \\leq b ^ { \\prime } x = y \\right \\} \\\\ \\theta _ { 2 } & = \\left \\{ \\left ( x , y \\right ) \\in L ^ { 2 } : a \\leq x , y \\leq b b ^ { \\prime } \\leq x , y \\leq a ^ { \\prime } x = y \\right \\} \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} - \\Delta p _ { \\rho } = \\partial _ i \\partial _ j ( u _ i u _ j \\phi _ { \\rho } ) \\ \\ \\textrm { o n } \\ \\ \\R ^ d \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} \\rho ^ { * } = \\tau _ 1 , \\tau _ 2 ^ { \\tau _ 1 } , \\ldots , \\tau _ n ^ { \\tau _ { n - 1 } \\ldots \\tau _ 1 } \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} x _ { n + 1 } = \\sum _ { i = 1 } ^ p x _ i ^ 2 - \\sum _ { i = p + 1 } ^ n x _ i ^ 2 \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} \\psi : H & \\longrightarrow N \\\\ g & \\longmapsto g _ { 1 } \\stackrel { \\mathrm { d e f } } { = } g ( 1 _ { N } ) . \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} F ^ * ( x ) = \\sum \\limits _ { n \\geq 1 } F _ n ^ * \\frac { x ^ n } { n ! } . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} I ( \\lambda , x ) : = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\Delta ) } \\tilde { \\varphi } ( \\lambda , w ) \\Gamma ( 1 - s - w ) \\sin \\left ( \\pi \\frac { s + w } { 2 } \\right ) x ^ w d w \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} E _ j ^ { \\Gamma } ( z , s , \\chi ) = E _ j ( z , s , \\chi ) + \\chi ( V ^ { - 1 } ) E _ j ( V z , s , \\chi ) . \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\mathbf { 1 } _ { ( n , w ) = 1 } = \\sum _ { \\substack { f \\mid n \\\\ f \\mid w } } \\mu ( f ) . \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} Y _ { j , k } : = 1 + X _ { j , k } ( 1 + X _ { j - 1 , k } ( \\dots ( 1 + X _ { 1 , k } ) \\dots ) ) . \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\eta _ \\infty ( x ) = \\lim _ { t \\to \\infty } e ^ { \\lambda _ \\infty t } \\P _ x ( \\tau _ { A _ \\infty } > t ) . \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} \\widetilde h [ f ] - h [ f ] = \\frac { 1 } { \\beta _ { v _ 0 } } | a + b | ^ 2 - \\frac { 1 } { \\beta _ { v _ 2 } } | b | ^ 2 = \\frac { 1 } { \\beta _ { v _ 2 } } | b | ^ 2 - \\frac { 1 } { \\beta _ { v _ 2 } } | b | ^ 2 = 0 . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} U ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) = \\Sigma _ { \\alpha _ 2 \\alpha _ 1 } ^ \\sigma ( t ) + \\sum _ { l = 1 } ^ \\infty \\int _ { 0 } ^ { t } \\int _ 0 ^ { t _ 1 } . . . \\int _ 0 ^ { t _ { l - 1 } } \\Pi ^ { l , \\sigma } _ { \\alpha _ 2 \\alpha _ 1 } ( t , t _ 1 , t _ 2 , . . . , t _ l ) d t _ l . . . d t _ 1 , \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} R _ 0 = \\{ ( ( n - 1 , r + 1 ) , ( n - 1 , r + 2 s + 3 ) ) , ( ( n - 1 , r + 3 ) , ( n - 1 , r + 2 s + 1 ) ) \\} . \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} r = \\prod _ { \\substack { p , \\\\ u _ p = v _ p } } p ^ { u _ p } , s = \\prod _ { \\substack { p , \\\\ u _ p \\neq v _ p } } p ^ { \\min ( u _ p , v _ p ) } , t = \\prod _ { \\substack { p , \\\\ u _ p \\neq v _ p } } p ^ { \\max ( u _ p , v _ p ) } . \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\mathcal { Q } _ { t } ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } ) = \\mathbb { E } _ { \\xi _ t , D _ t } \\Big [ \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } , \\xi _ t , D _ t ) | D _ { t - 1 } , \\xi _ { [ t - 1 ] } \\Big ] \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} T ^ * K _ z ^ { [ m ] } = \\sum _ { j = 0 } ^ { \\kappa + m } \\overline { \\omega _ { j , m } ( z ) } K _ z ^ { [ j ] } . \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} \\nu ' ( C ) = \\frac { 1 } { 2 } \\dim H ^ 1 ( F , \\C ) _ { - 1 } + \\frac { 1 } { 4 } \\left ( ( d - 2 ) ^ 2 - \\tau ( C ) + \\sum _ { j } \\nu _ j \\right ) . \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} \\mathrm { C L } \\circ \\overline { ( \\cdot ) } = \\sigma ' \\circ \\mathrm { C L } . \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { c } _ { \\jmath \\imath } ( - 2 + r ^ { ( \\jmath ) } + 4 k ) = \\delta _ { 0 , k } & \\ 1 \\leq \\jmath \\leq n - 1 , \\\\ \\widetilde { c } _ { 2 n - \\jmath , \\imath } ( - 2 + r ^ { ( \\jmath ) } + 4 k ) = \\delta _ { 0 , k } & \\ n + 1 \\leq \\jmath \\leq 2 n - 1 , \\\\ \\widetilde { c } _ { n , \\imath } ( - 1 + r ^ { ( n ) } + 2 k ) = \\delta _ { 0 , k } & \\ \\jmath = n , \\\\ \\end{cases} \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} V _ N = & 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\mathcal { N } ( 0 , \\ , 4 \\log 2 ) + \\log X _ 1 + \\log X _ 2 + \\log X _ 3 + \\\\ & + \\log Y + \\log Y ' + o ( 1 ) , \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} O r d _ { 2 ^ { \\beta } } ( a b ^ { - 1 } ) = 2 ~ ~ ~ \\Rightarrow ~ ~ ~ a b ^ { - 1 } \\equiv - 1 ( { \\rm m o d } ~ 2 ^ { \\beta } ) ~ ~ ~ ~ { \\rm i . e . } ~ ~ ~ 2 ^ { \\beta } \\mid a + b . \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} K ( [ z ] , [ z ' ] ) = K ( z , z ' ) = K ( w , z ' ) = K ( w , w ' ) = K ( [ w ] , [ w ' ] ) . \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} \\omega _ { k n } = \\left ( B _ { 2 \\Lambda _ { n } } \\left ( M _ { k n } \\right ) \\cap { \\Omega _ { n } } \\right ) \\setminus \\bigcup _ { \\nu = 0 } ^ { k - 1 } B _ { 2 \\Lambda _ { n } } \\left ( M _ { \\nu n } \\right ) , \\ ; 1 \\leq k \\leq 2 ^ { n } \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} e \\in \\N \\mapsto \\psi _ i ^ { ( \\alpha ) } ( e ) = \\sum _ { J \\in \\Gamma _ i } g _ J \\theta _ J ^ { ( \\alpha ) } ( e ) . \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} \\begin{aligned} & \\big \\| \\big ( [ \\partial ^ \\alpha , f ] ~ g \\big ) ( \\tau , \\cdot ) \\big \\| _ { L ^ 2 ( \\Omega ) } \\lesssim \\| f ( \\tau ) \\| _ { \\mathcal { H } ^ k ( \\Omega ) } \\cdot \\| g ( \\tau ) \\| _ { \\mathcal { H } ^ { k - 1 } ( \\Omega ) } . \\end{aligned} \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} \\epsilon _ a = \\begin{bmatrix} r _ a & \\lambda _ a \\\\ 0 & r _ a \\end{bmatrix} \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} \\begin{aligned} w _ t = \\nabla \\cdot \\left ( m \\ , \\mathcal { B } _ { M } ^ { m - 1 } \\nabla w \\right ) \\forall ( x , t ) \\in \\R ^ n \\times \\left ( 0 , \\infty \\right ) \\end{aligned} \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} L ^ \\Delta = A ^ \\Delta + B ^ \\Delta = A ^ \\Delta _ \\upsilon + B ^ \\Delta _ \\upsilon , \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} g \\sim g ^ { ( 2 n - 2 ) } \\epsilon ^ { 2 n - 2 } = - \\frac { h _ n } { h _ { n - 1 } } \\epsilon ^ { 2 n - 2 } , \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} \\nabla ^ g & = d - ( d g ) g ^ { - 1 } + g p _ { - 1 } g ^ { - 1 } d z + g u g ^ { - 1 } d z \\\\ & = d + \\Big ( p _ { - 1 } - { h ^ \\vee } ^ { - 1 } \\varphi \\ , \\rho + r ( z ) + \\frac { \\langle r ( z ) , \\check \\alpha _ i \\rangle - { h ^ \\vee } ^ { - 1 } \\varphi ( z ) } { z - x } \\check e _ i \\Big ) d z . \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} e _ { i j } v & = \\sum _ { k \\in T ^ { j } _ v } v _ { i _ 1 } \\otimes \\cdots \\otimes v _ { i _ { k - 1 } } \\otimes v _ i \\otimes v _ { i _ { k + 1 } } \\otimes \\cdots \\otimes v _ { i _ d } \\\\ \\intertext { H e n c e , } e _ { i j } T _ v & = T _ { e _ { i j } v } = \\sum _ { k \\in T ^ { j } _ v } c _ { i , \\{ k \\} } T _ v \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} \\tilde { w } _ r ( x ) = \\frac { c _ 2 } { C _ 3 } V ( \\Psi ( x ) ) \\ge c _ 3 V ( d _ D ( x ) ) = c _ 3 V ( 4 r - | x | ) . \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} p _ i ( \\mathfrak d _ i ) q ( p _ i ( \\mathfrak d _ i ) ) E _ { i i } & = \\mathfrak h \\mathfrak v _ i q ( \\mathfrak v _ i ) \\mathfrak h _ i ^ { - 1 } E _ { i i } \\\\ & = \\mathfrak h \\mathfrak b _ i \\mathfrak c _ i q ( \\mathfrak v _ i ) \\mathfrak h _ i ^ { - 1 } E _ { i i } \\\\ & = \\mathfrak h \\mathfrak b _ i \\mathfrak c _ i \\mathfrak t _ i E _ { i i } \\\\ & = \\mathfrak h \\mathfrak U W ( x ) \\mathfrak U ^ * \\mathfrak c _ i \\mathfrak t _ i E _ { i i } . \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { k a - 1 } \\frac { 1 } { k t + i - \\ell } \\geq \\sum _ { \\ell = 0 } ^ { j a - 1 } \\frac { 1 } { j t + i - \\ell } . \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} 2 k + 3 = \\frac { u } { v } . \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} \\left \\vert \\pi _ { x + L } \\left ( \\tau ( z ) \\right ) - z \\right \\vert & = \\left \\vert \\pi _ { x + L } \\left ( \\tau ( z ) - z \\right ) \\right \\vert \\\\ & \\leq \\vert \\tau ( z ) - z \\vert \\\\ & \\leq \\frac 5 { 1 4 4 } r . \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} u _ 1 = & \\ \\frac { 1 } { 3 } ( d _ { 1 1 2 } , d _ { 2 1 2 } ) , & u _ 2 = & \\ \\frac { 1 } { 2 } ( d _ { 1 1 2 } + 1 , d _ { 2 1 2 } ) , \\\\ u _ 3 = & \\ \\left ( - \\frac { 1 } { 2 } , 0 \\right ) , & u _ 4 = & \\ \\left ( \\frac { 1 } { 3 } , 0 \\right ) \\\\ u _ 5 = & \\ \\frac { 1 } { 3 } ( - d _ { 1 1 2 } - 1 , - d _ { 2 1 2 } ) , & u _ 6 = & \\ \\frac { 1 } { 2 } ( - d _ { 1 1 2 } , - d _ { 2 1 2 } ) . \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} t _ 0 = 0 , t _ { k + 1 } = \\begin{cases} \\inf \\{ n > t _ k : X ( n ) \\not \\in E + X ( t _ k ) \\} & \\\\ + \\infty & \\end{cases} \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} R _ { 1 } & = \\rho _ { 0 } ^ { 3 } \\frac { v _ { 0 } v _ { 1 } } { 2 } = \\rho _ { 0 } ^ { 3 } \\left [ \\varphi _ { 1 } \\varphi _ { 2 } + \\frac { 1 } { 4 } f _ { 0 } \\varphi _ { 1 } \\theta _ { 1 } ^ { 2 } + g _ { 0 } \\theta _ { 1 } \\theta _ { 2 } \\right ] \\\\ & = \\varphi _ { 1 } ( \\rho _ { 0 } ^ { 3 } \\varphi _ { 2 } ) + \\theta _ { 1 } ( g _ { 0 } \\rho _ { 0 } ^ { 3 } \\theta _ { 2 } ) + \\frac { 1 } { 4 } f _ { 0 } \\rho _ { 0 } ^ { 3 } \\varphi _ { 1 } \\theta _ { 1 } ^ { 2 } \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} \\prod _ p \\Psi _ p ( \\xi ; X ) \\neq 0 \\iff \\xi \\in \\Z , \\ , \\left ( \\frac { - \\xi } { p } \\right ) = w _ p p \\mid N / M , \\ , \\left ( \\frac { - \\xi } { p } \\right ) = - w _ p p \\mid M . \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} \\mathcal { G } _ 1 ( t ) & : = \\{ ( a ^ \\ast , b ^ \\ast ) \\in K ( t ) \\mathbb { B } \\times C ( t ) : x _ n ^ \\ast ( t ) = a ^ \\ast + b ^ \\ast \\} , \\\\ \\mathcal { G } _ 2 ( t ) & : = \\{ ( a ^ \\ast , b ^ \\ast ) \\in K ( t ) \\mathbb { B } \\times C ( t ) : y _ n ^ \\ast ( t ) = a ^ \\ast + b ^ \\ast \\} , \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} h _ 1 ( t , x , y ) ~ = ~ \\partial _ y \\psi ( t , x , y ) . \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} \\rho D _ t u = \\partial _ x w . \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} \\left ( \\mathbf { P } _ { \\mathcal { A } ^ { ( k ) } | \\mathcal { A } ^ { ( l ) } } \\right ) _ { i , j } = P ( A _ { j } ^ { ( k ) } | A _ { i } ^ { ( l ) } ) = \\frac { P ( A _ { j } ^ { ( k ) } \\cap A _ { i } ^ { ( l ) } ) } { P ( A _ { i } ^ { ( l ) } ) } , 1 \\leqslant i \\leqslant m _ { l } , 1 \\leqslant j \\leqslant m _ { k } . \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} f ' ( z ) + ( 1 - 2 B ( z ) ) f ( z ) = P ( z ) . \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} L ( f , u , v , w ) & = \\frac { T ( 1 + 2 u ) T ( 1 + v + w ) } { T ( 1 + u + v ) T ( 1 + u + w ) } D ( u , v , w ) \\\\ & = \\eta _ 3 ( u , v , w ) \\frac { ( u + v ) ( u + w ) } { u ( v + w ) } , \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\eta ( i ) = \\tfrac 1 2 ( 1 + \\kappa ( i ) ) , \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} V _ n ( 0 , \\ell ) + \\ldots + V _ n ( \\ell - 1 , \\ell ) = 0 . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} { } ^ { \\jmath } B : = \\{ \\jmath ' \\mid \\ \\jmath \\ \\ \\jmath ' \\ \\ \\Upsilon _ { \\Xi } \\} , \\\\ B ^ { \\jmath } : = \\{ \\jmath ' \\mid \\ \\jmath ' \\ \\ \\jmath \\ \\ \\Upsilon _ { \\Xi } \\} . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{gather*} \\vartheta _ n = X _ { 0 , 0 } , \\vartheta _ a = X _ { a , b } , \\\\ \\vartheta _ i = \\sum _ { j = 1 } ^ b X _ { i , b } X _ { i , b - 1 } \\cdots X _ { i , j } , \\\\ \\vartheta _ i = \\sum _ { j = 1 } ^ a X _ { a , n - i } X _ { a - 1 , n - i } \\cdots X _ { j , n - i } , \\end{gather*}"}
-{"id": "4459.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } U _ { \\lambda } \\left ( x , t \\right ) \\ , d x = \\int _ { \\R ^ n } \\lambda ^ { a _ 1 } U \\left ( \\lambda ^ { a _ 2 } x , \\lambda t \\right ) \\ , d x = \\int _ { \\R ^ n } U \\left ( y , \\lambda t \\right ) \\ , d y = M < \\infty \\forall \\lambda > 0 , \\ , \\ , t \\geq 0 \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} \\sum _ { i + j + k = n } f _ { i + 1 + k } \\left ( 1 ^ { \\otimes i } \\otimes m _ j \\otimes 1 ^ { \\otimes k } \\right ) = \\sum _ { i _ 1 + \\cdots + i _ s = n } m ' _ s \\left ( f _ { i _ 1 } \\otimes f _ { i _ 2 } \\otimes \\cdots \\otimes f _ { i _ s } \\right ) . \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\langle P ( x ) , Q ( y ) \\rangle , \\quad \\forall x , y \\in X ^ N . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} \\tilde { \\tau } = \\frac { 1 } { m ( m + 1 ) } \\sum \\limits _ { i , j = 1 } ^ { m + 1 } \\tilde { R } ( E _ i , E _ j , E _ j , E _ i ) . \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\partial _ t g + v \\partial _ x g + b ( t , x , v ) \\partial _ v g - \\partial _ v ( v g ) + c \\ , g - \\sigma \\partial _ { v v } g = U ( t , x , v ) \\ , , \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\rho _ t + ( \\rho u ) _ x = 0 , \\\\ ( \\rho u ) _ t + ( \\rho u ^ 2 - \\frac { B } { \\rho ^ { \\alpha } } ) _ x = \\beta \\rho , \\end{array} \\right . \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} \\Delta _ { J \\cup \\{ i , j \\} } \\Delta _ { J \\cup \\{ k , l \\} } + \\Delta _ { J \\cup \\{ i , l \\} } \\Delta _ { J \\cup \\{ j , k \\} } = \\Delta _ { J \\cup \\{ i , k \\} } \\Delta _ { J \\cup \\{ j , l \\} } . \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} \\mathbf F _ { \\chi } \\left ( \\left ( \\begin{array} { c c } z \\mathbf 1 _ 2 & 0 \\\\ 0 & z \\mathbf 1 _ 2 \\end{array} \\right ) g \\left ( \\begin{array} { c c } \\alpha & \\beta \\\\ - \\beta & \\alpha \\end{array} \\right ) \\right ) = \\det ( \\mathbf k ) ^ { k + 1 } \\mathbf F _ { \\chi } ( g ) \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{gather*} | \\xi _ 1 | ^ 2 = 2 Q + \\sigma _ 1 ^ 2 \\\\ | \\xi _ 2 | ^ 2 ( Q - x ) = ( 2 Q + \\sigma _ 2 ^ 2 ) ( Q - x ) - Q ^ 2 \\end{gather*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\bigcup _ { j \\in [ n ] } \\mathcal { L } _ j = \\Phi _ { n , d } ( \\mathcal { T } ) \\mbox { f o r } \\mathcal { L } _ j = \\{ ( u , v ) \\mid u _ j = 1 \\} \\enspace . \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} S ( f ) : = \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } v \\left [ f ( v ) \\log f ( v ) + \\frac { v ^ 2 } { 2 \\sigma } f ( v ) \\right ] + \\frac { 1 } { \\sigma } V ( \\langle w \\rangle _ f ) \\ , , \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} L _ { - 2 } e ^ { \\pm \\Lambda } = L _ { - 1 } ^ { 2 } e ^ { \\pm \\Lambda } = \\left ( \\frac { 1 } { 4 } \\alpha ( - 1 ) ^ { 2 } \\pm \\frac { 1 } { 2 } \\alpha ( - 2 ) \\right ) e ^ { \\pm \\Lambda } \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} t E _ i = \\sum _ { j = 1 } ^ n \\lambda _ { i , j } [ D ] ^ j \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} V ( \\dot { \\bar { x } } , \\dot { \\bar { \\lambda } } ; \\sigma ) = \\frac { 1 } { 2 \\alpha } \\sum _ { i = 1 } ^ { N } \\dot { x } _ { i } ^ { 2 } + \\frac { 1 } { 2 } \\sum _ { i = 1 , i \\notin \\sigma } ^ { N } \\dot { \\lambda } _ { i } ^ { 2 } . \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} E _ j ^ { \\Gamma } ( z , 1 - s , \\chi ) = \\sum _ { \\ell \\in R _ { \\Gamma , \\chi } } \\varphi _ { j , \\ell } ^ { \\Gamma } ( 1 - s ) E _ \\ell ^ { \\Gamma } ( z , s , \\chi ) . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} g ( \\phi X , \\phi Y ) = g ( X , Y ) - \\eta ( X ) \\eta ( Y ) , \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} m _ t ( x ) = E \\Big [ \\exp \\Big \\{ \\int _ t ^ T \\mathrm { d i v } \\beta _ j ( \\phi _ { t , s } ( x ) ) d \\tilde { \\Z } _ s ^ j \\Big \\} \\Big ] . \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\Big | \\sum _ { m = 1 } ^ M e ^ { 2 \\pi i \\tilde { P } ( m ) } \\Big | & \\leq C M ( \\log M ) \\bigg ( \\frac { 1 } { q ' } + \\frac { 1 } { M } + \\frac { q ' } { M ^ d } \\bigg ) ^ { \\frac { 1 } { 2 d ^ 2 - 2 d + 1 } } . \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} e \\ne v \\qquad | h e \\cap V _ { m + 1 } | = n | h e \\cap S _ H | \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\boldsymbol { \\lambda } } ( f \\ast _ { \\boldsymbol { \\lambda } } g ) = \\mathcal { L } _ { \\boldsymbol { \\lambda } } ( f ) \\mathcal { L } _ { \\boldsymbol { \\lambda } } ( g ) , \\mathcal { B } _ { \\boldsymbol { \\lambda } } ( f g ) = \\mathcal { B } _ { \\boldsymbol { \\lambda } } ( f ) \\ast _ { \\boldsymbol { \\lambda } } \\mathcal { B } _ { \\boldsymbol { \\lambda } } ( g ) , \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\sup _ { \\{ k : \\ , k ^ 2 \\leq \\ > \\phi ( n ) \\} } \\left | \\frac { P \\left ( V _ n = k \\right ) } { e ^ { - \\lambda } \\lambda ^ { k } / k ! } - 1 \\right | \\longrightarrow 0 . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} p ^ { \\mathbb { T } } ( w ) = \\lim _ { t \\to \\infty } \\frac { 1 } { t } \\ln \\left ( p ( e ^ { t w + \\sqrt { - 1 } \\phi } ) \\right ) , \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} \\tilde { P } ( x ) = \\sum _ { \\atop { \\gamma \\in \\Gamma } { \\abs { \\gamma } \\geq 2 } } a _ \\gamma x ^ \\gamma . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} W _ { k , r } ^ { ( i ) } : = L ( m _ { k , r } ^ { ( i ) } ) , \\ m _ { k , r } ^ { ( i ) } : = \\prod _ { s = 1 } ^ { k } Y _ { i , r + 2 r _ i ( s - 1 ) } . \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} T ( \\{ \\Gamma _ 1 \\} _ { D R } ) = [ [ \\alpha ^ { n - 2 , \\ , n } + \\beta ^ { n - 2 , \\ , n } ] _ { \\bar \\partial } ] _ { d _ 1 } \\in E _ 2 ^ { n - 2 , \\ , n } ( X ) . \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} U _ n ^ 1 : = N S ( v _ { 0 , n } ^ 1 ) , \\ \\ \\ U _ n ^ 2 = v _ n - U _ n ^ 1 . \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} e _ m : = \\frac { 1 } { 4 } \\frac { \\cosh ^ 4 ( 2 m r ) + ( 3 2 \\ , { m } ^ { 4 } { r } ^ { 4 } - 2 ) \\cosh ^ 2 ( 2 m r ) - 3 2 \\sinh ( 2 m r ) \\cosh ( 2 m r ) m ^ 3 r ^ 3 + 1 6 m ^ 4 r ^ 4 + 1 } { \\sinh ^ 4 ( 2 \\ , m r ) r ^ 4 } . \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} T _ { \\lambda } : = \\begin{tabular} { | c | } \\hline $ 1 , \\ldots , \\lambda _ 1 $ \\\\ \\hline $ \\lambda _ 1 + 1 , \\ldots , \\lambda _ 1 + \\lambda _ 2 $ \\\\ \\hline $ \\vdots $ \\\\ \\hline $ \\lambda _ 1 + \\cdots + \\lambda _ { n - 1 } + 1 , \\ldots , d $ \\\\ \\hline \\end{tabular} \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq b \\leq \\theta , \\\\ ( b , t ) = 1 } } 1 \\ll \\theta \\prod _ { \\substack { p \\mid t , \\\\ p \\leq \\theta } } \\left ( 1 - \\frac { 1 } { p } \\right ) = \\theta \\frac { \\varphi ( t ) } { t } \\prod _ { \\substack { p \\mid t , \\\\ p > \\theta } } \\left ( 1 - \\frac { 1 } { p } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{gather*} C _ 3 = 1 + a _ 1 + a _ 1 b + b ^ 2 + a _ 1 ^ 2 k , \\\\ C _ 2 ( Y ) = 1 + a _ 1 + a _ 1 b + b ^ 2 + ( 1 + b ^ 2 + a _ 1 ^ 2 k ) Y , \\\\ C _ 1 ( Y ) = b + a _ 1 b + b ^ 2 + k + a _ 1 k + a _ 1 b k + b ^ 2 k + a _ 1 ^ 2 k ^ 2 + ( 1 + a _ 1 + b ^ 2 + a _ 1 ^ 2 k ) Y , \\\\ C _ 0 ( Y ) = b + a _ 1 b + b ^ 2 + a _ 1 k + a _ 1 ^ 2 k ^ 2 + ( a _ 1 + b + b ^ 2 + k + a _ 1 ^ 2 k + b ^ 2 k + a _ 1 ^ 2 k ^ 2 ) Y . \\end{gather*}"}
-{"id": "5577.png", "formula": "\\begin{align*} ( \\ast ) & = \\begin{cases} [ i + 1 \\leq k + j \\leq 2 n - i - 1 \\ \\ 1 - i \\leq k - j \\leq i - 1 \\\\ \\ ( k + j ) - ( i + 1 ) , ( k - j ) - ( 1 - i ) \\in 2 \\mathbb { Z } _ { \\geq 0 } ] , \\\\ o r \\\\ [ 2 n - i \\leq k + j ] . \\end{cases} \\\\ ( \\ast \\ast ) & = [ k + j - n \\in 2 \\mathbb { Z } _ { \\geq 0 } ] . \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\mathbb { E } _ { n } ^ { ( N ) } ( f | A ) = \\frac { \\mathbb { P } _ { n } ^ { ( N ) } ( f 1 _ { A } ) } { \\mathbb { P } _ { n } ^ { ( N ) } ( A ) } , \\quad \\mathbb { E } ( f | A ) = \\frac { P ( f 1 _ { A } ) } { P ( A ) } , \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( 1 ; 3 ) } ^ \\ast ( \\tau ) = \\frac { ( - 2 \\pi { \\rm i } ) ^ k } { ( k - 1 ) ! } \\sum _ { n \\ge 1 } n ^ { k - 1 } \\left ( \\frac { q ^ { n } + ( - 1 ) ^ k q ^ { 2 n } } { 1 - q ^ { 3 n } } \\right ) . \\end{gather*}"}
-{"id": "134.png", "formula": "\\begin{align*} d ( q , \\ ; ^ { b _ 0 } { M } _ { f } ) = b _ 0 - f ( q ) . \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} \\varTheta _ \\lambda : = \\bigl \\{ & f : [ 0 , \\tau ] \\to \\mathbb { R } | \\ ; f ( t ) \\geq 0 , \\forall t \\in [ 0 , \\tau ] \\ \\\\ & \\big | f ( t ) - f ( s ) \\big | \\leq L | t - s | , \\ \\forall t , s \\in [ 0 , \\tau ] \\bigr \\} , \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} \\alpha _ 2 ( \\mathbf h _ 2 , \\breve { \\mathbf g } _ 2 , \\pmb { \\phi } _ 2 ) = \\alpha _ 2 ( \\tilde { \\pi } _ 2 ( t ( 2 ) ) \\mathbf h _ 2 , \\tau _ 2 ( t ( 2 ) ) \\breve { \\mathbf g } _ 2 , \\omega _ 2 ( t ( 2 ) ) \\pmb { \\phi } _ 2 ) = \\alpha _ 2 ( \\tilde { \\pi } _ 2 ( t ( 2 ) ) \\mathbf h _ 2 , \\mathbf g _ 2 , \\omega _ 2 ( t ( 2 ) ) \\pmb { \\phi } _ 2 ) , \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} g _ 0 = l _ 0 g _ 1 = A _ 0 y ^ 2 + A _ 1 y z + A _ 2 z ^ 2 , \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} 0 = ( 1 - 2 x _ n ) ( x _ { n + 1 } + x _ 1 x _ { n } + \\frac 1 2 \\sum _ { i = 2 } ^ { n - 1 } x _ i ^ 2 ) + x _ n x _ 2 ^ 2 . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} f _ 1 ^ c = q f _ 3 + x ^ k g , k = a _ 1 c _ 2 , \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} C _ a ( e _ i ) : = \\begin{cases} e _ { i - 1 } & , \\\\ ( - 1 ) ^ { a - 1 } e _ n & . \\end{cases} \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} & L ^ 1 ( \\sqrt { 1 + v ^ 2 } \\ , \\mathrm { d } x \\ , \\mathrm { d } v ) : = \\left \\{ f \\colon { \\mathbb { T } } _ x \\times \\mathbb { R } _ v \\to \\mathbb { R } \\ : \\colon \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } x \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } v \\ , \\left \\vert f \\right \\vert \\sqrt { 1 + v ^ { 2 } } < \\infty \\right \\} \\ , , \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{gather*} \\operatorname { H o m } ^ l ( M , N ) = \\operatorname { H o m } _ k ( M , N ) , h \\cdot \\varphi = h ^ 1 \\varphi \\big ( S \\big ( h ^ 2 \\big ) { - } \\big ) . \\end{gather*}"}
-{"id": "2121.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | h _ { n + 1 } - h _ n \\big | \\Big \\| _ { \\ell ^ 1 } \\lesssim N _ 1 ^ { - k } \\big ( \\vartheta _ B ( N _ 2 ) - \\vartheta _ B ( N _ 1 ) \\big ) , \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} \\frac { \\mathcal { A } \\big ( \\{ \\alpha _ f \\} ; q \\big ) } { \\mathcal { A } ( q ) } = \\frac { 1 } { \\zeta ( 2 ) } \\mathcal { A } \\big ( \\{ \\omega _ f \\cdot \\omega _ f ( x ) \\alpha _ f \\} ; q \\big ) + \\mathcal { O } ( q ^ { - \\varsigma } ) . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{gather*} \\theta _ 2 ( \\tau ) : = \\sum _ { n \\in \\mathbb Z } q ^ { ( 2 n + 1 ) ^ 2 / 8 } , \\theta _ 3 ( \\tau ) : = \\sum _ { n \\in \\mathbb Z } q ^ { n ^ 2 / 2 } , \\theta _ 4 ( \\tau ) : = \\sum _ { n \\in \\mathbb Z } ( - 1 ) ^ n q ^ { n ^ 2 / 2 } , \\end{gather*}"}
-{"id": "6515.png", "formula": "\\begin{align*} & \\int _ { \\Omega _ 1 \\setminus S } | U _ * | ^ { 2 ^ * } \\\\ = & \\int _ { \\Omega _ 1 } u _ 0 ^ { 2 ^ * } - 2 ^ * \\int _ { \\Omega _ 1 \\setminus S } u _ 0 ^ { 2 ^ * - 1 } V + O ( \\frac { 1 } { \\mu ^ { \\frac { N - 2 } { 2 } ( 1 + \\delta ) } } ) . \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} H _ { t , n } ( \\chi ) = \\alpha \\chi \\ ! \\left ( \\frac { t + 2 ^ { r / 2 } \\omega } { 2 } \\right ) 2 ^ { e - 3 } \\begin{cases} 3 & e \\ge 3 , r > e , \\\\ 1 + 2 ( - 1 ) ^ e & e \\ge 3 , r = e , \\\\ 1 - 2 ( - 1 ) ^ d & e \\ge 3 , r = e - 1 , 2 \\nmid e , \\\\ 2 & e = 2 , r \\geq e , \\\\ - 1 & e = 2 , r < e , \\\\ 4 & e = 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { d , c } ^ { ( N ) } = \\begin{cases} ( - 1 ) ^ { \\bar E _ { a , b } \\bar E _ { d , c } } E _ { d , c } E _ { a , b } - ( - 1 ) ^ { \\bar E _ { a , b } \\bar E _ { d , c } } K _ { a , b } E _ { d , b } , & M = N = 1 ; \\\\ \\displaystyle \\sum _ { t = 0 } ^ { \\min ( M , N ) } ( - 1 ) ^ t q _ { b } ^ { - t ( M - 1 - t ) } E _ { d , b } ^ { ( t ) } E _ { d , c } ^ { ( N - t ) } K _ { a , b } ^ { t } E _ { a , b } ^ { ( M - t ) } , & . \\end{cases} \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} g \\cdot a = s ( g ) a s ( g ) ^ { - 1 } \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} M ^ - ( \\widetilde { X } ) = \\left ( N ^ + ( \\widetilde { X } ) \\right ) ^ { - 1 } N ^ - ( \\widetilde { X } ) , ~ ~ ~ ~ ~ ~ M ^ + ( \\widetilde { X } ) = \\left ( N ^ - ( \\widetilde { X } ) \\right ) ^ { - 1 } N ^ + ( \\widetilde { X } ) , \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} u _ { \\lambda } & \\to u _ { \\infty } \\mbox { l o c a l l y i n $ L ^ { 1 } $ } \\\\ \\nabla u _ { \\lambda } & \\to \\nabla u _ { \\infty } \\mbox { l o c a l l y i n $ L ^ 2 $ w i t h w e i g h t $ \\mathcal { B } _ M ^ { m - 1 } $ } \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\mathbb { P } [ \\omega _ i = 1 ] & = \\frac { 1 } { 2 } ( 1 - \\sigma ) \\\\ \\mathbb { P } [ \\omega _ i = 0 ] & = \\frac { 1 } { 2 } ( 1 + \\sigma ) . \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\overline X _ u \\cong _ { h t } L _ { m , n } ^ { t , m \\cdot n - N } \\vee \\bigvee _ { i = 1 } ^ r S ^ d , \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} { } ^ \\sigma ( \\pi ^ { - 2 m _ 0 n } \\Delta ^ { m _ 0 } _ { k - 2 m _ 0 } F ) = \\Delta ^ { m _ 0 } _ { k - 2 m _ 0 } ( \\pi ^ { - 2 m _ 0 n } \\ { } ^ \\sigma \\ ! F ) . \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} B ( X ) = \\ , & X ^ 3 ( 1 + b + a _ 1 z ) + X ^ 2 ( 1 + b + a _ 1 k + z + a _ 1 z + b z ) \\cr & + X ( b + k + a _ 1 k + b k + z + a _ 1 z + b z + a _ 1 k z ) \\cr & + b + a _ 1 k + a _ 1 k ^ 2 + a _ 1 z + b z + k z + b k z . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} \\begin{cases} x ^ 0 \\in C , \\\\ y ^ { n } = \\underset { y \\in C } { \\textup { a r g m i n } } \\bigg \\{ \\lambda _ n f ( x ^ n , y ) + \\dfrac { 1 } { 2 } \\| y - x ^ n \\| ^ 2 \\bigg \\} , \\\\ x ^ { n + 1 } = \\underset { y \\in C } { \\textup { a r g m i n } } \\bigg \\{ \\lambda _ n f ( y ^ n , y ) + \\dfrac { 1 } { 2 } \\| y - x ^ { n } \\| ^ 2 \\bigg \\} , \\end{cases} \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\widetilde T _ d : = \\begin{bmatrix} \\Delta & - 1 & * & \\dots & * \\\\ & \\ddots & \\ddots & \\ddots & \\vdots \\\\ & & \\ddots & \\ddots & * \\\\ & & & \\Delta & - 1 \\\\ & & & & \\Delta \\end{bmatrix} = \\Delta I + \\left [ t _ { j k } \\right ] _ { j , k = 0 , \\dots , d } . \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} v w _ 1 = w _ 2 v \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} \\P _ { x \\star u } ( \\nu _ u > n ) = \\P _ x ( { \\cal T } _ \\varphi > n ) . \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} ( x ) _ n = \\sum _ { k = 0 } ^ n s ( n , k ) x ^ k , x ^ n = \\sum _ { k = 0 } ^ n S ( n , k ) ( x ) _ k , \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} e ' & = ( U _ { j - 1 } ^ * U ''' _ j L ) + ( U _ { j - 1 } ^ * U _ { j + 1 } L ) \\\\ & = - 2 d _ { j - 1 } + d ''' _ j + d _ { j + 1 } + 3 t \\\\ & < - 2 d _ { j - 1 } + \\frac { d } { r } + d _ { j + 1 } + 3 t . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\mu _ T ^ { ( A ) } ( E ) = | | P _ E ^ { ( A ) } \\psi | | ^ 2 = \\mu ^ { ( A ) } _ \\psi ( E ) \\ : . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} \\abs { \\mathcal K ( t ) - \\mathcal K ( s _ n ) } = o ( 1 ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; t \\uparrow \\ , \\infty , \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} F ( v ) = f ( p ) + v , \\ \\ \\ \\hat { F } ( v ) = \\hat { f } ( p ) + \\overline { \\tau } v , \\ \\ \\ v \\in \\Lambda _ p , \\ \\ p \\in M ^ n . \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\widehat { K _ 3 } ( y ) = \\frac { 1 } { 1 + \\frac { \\beta - 1 } { \\gamma } \\sqrt { \\mu } | y | c o t h ( \\sqrt { \\mu _ 2 } | y | ) } = \\theta \\frac { s i n h ( \\sqrt { \\mu _ 2 } y ) } { \\sqrt { \\mu _ 2 } y c o s h ( \\sqrt { \\mu _ 2 } y ) + \\theta s i n h ( \\sqrt { \\mu _ 2 } y ) } . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} \\tau ^ { - l } \\frac { \\Gamma \\bigl ( 2 + 2 l + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) \\bigr ) } { \\Gamma \\bigl ( 2 + l + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) \\bigr ) } = \\prod \\limits _ { j = 0 } ^ { l - 1 } \\bigl ( 1 + \\lambda _ 1 + \\lambda _ 2 + ( l + j + 2 ) / \\tau \\bigr ) . \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} y & = x + \\pi _ { P _ x } ( y - x ) + \\pi _ { P _ x } ^ \\perp ( y - x ) \\\\ & = x + z + \\pi _ { P _ x } ^ \\perp ( y - x ) . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} \\int _ { \\mathbb { C P } ^ { n - 1 } } H _ \\omega ( x ) \\omega _ { F S } ^ { n - 1 } = \\frac { 2 } { n ( n + 1 ) } S ^ \\omega ( x ) \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} D ^ 2 F ( P ) ( Q , Q ) = \\frac { 1 } { | P | } \\widetilde { F } ' \\big ( | P | \\big ) \\left [ | Q | ^ 2 - \\frac { ( P : Q ) } { | P | ^ 2 } \\right ] + \\widetilde { F } '' \\big ( | P | \\big ) \\frac { ( P : Q ) ^ 2 } { | P | ^ 2 } , \\ ; \\ ; P , Q \\in \\R ^ { N \\times n } , \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\mathfrak { b } ^ * ( B _ i ) = \\sum _ { \\mu \\vdash k } \\mbox { l c m } ( \\mu ) E _ { i : \\mu } . \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G ) & = \\Big ( \\frac { r ^ { p - r } } { \\alpha } \\Big ) ^ { 1 / p } = ( r C ) ^ { ( p - r ) / p } \\\\ & = \\Bigg ( \\sum _ { i = 1 } ^ k \\big ( \\lambda ^ { ( p ) } ( G _ i ) \\big ) ^ { p / ( p - r ) } \\Bigg ) ^ { ( p - r ) / p } , \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} \\langle x _ 1 , \\dots , x _ n ; y , z \\rangle : = p _ { n , 1 } ( x _ 1 , \\dots , x _ n ; z ; y ) - p _ { n , 1 } ( x _ 1 , \\dots , x _ n ; y ; z ) . \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} [ q ] ^ + = a r g ~ m i n _ { w \\in Q } | | w - q | | _ 2 \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} \\dfrac { \\partial } { \\partial v } \\Big ( \\dfrac { 1 } { \\hat { H } } \\big ( \\lambda \\nu _ u + F \\nu \\times \\nu _ u - E \\nu \\times \\nu _ v \\big ) \\Big ) = \\dfrac { \\partial } { \\partial u } \\Big ( \\dfrac { 1 } { \\hat { H } } \\big ( \\lambda \\nu _ v + G \\nu \\times \\nu _ u - F \\nu \\times \\nu _ v \\big ) \\Big ) , \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} F _ { \\omega _ { n _ 1 } ^ { j _ 1 } , \\dots , \\omega _ { n _ { m } } ^ { j _ { m } } } ( q _ { m + 1 } ) = f ( \\omega _ { n _ 1 } ^ { j _ 1 } , \\dots , \\omega _ { n _ { m } } ^ { j _ { m } } , q _ { m + 1 } ) \\in \\mathbb { Z } [ q _ { m + 1 } ] , \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} & 1 + d \\bigg ( \\frac { - ( 1 - 3 d D ) + \\sqrt { ( 1 - d D ) ^ { 2 } + 2 a d ^ { 2 } C } } { 2 d } - 2 D \\bigg ) \\\\ & = 1 + d \\bigg ( \\frac { - ( 1 - d D ) + \\sqrt { ( 1 - d D ) ^ { 2 } + 2 a d ^ { 2 } C } } { 2 d } \\bigg ) \\\\ & > 1 + d \\bigg ( \\frac { - ( 1 - d D ) + | 1 - d D | } { 2 d } \\bigg ) \\geq 1 . \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} Z _ j = j = Z _ { 2 m + 1 - j } \\textrm { f o r } j = 0 , 1 , \\ldots , m . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\sum _ { n _ 1 = 0 } ^ \\infty \\sum _ { m _ 1 = 0 } ^ { n _ 1 } \\cdots \\sum _ { n _ N = 0 } ^ \\infty \\sum _ { m _ N = 0 } ^ { n _ N } \\langle \\eta _ { m _ 1 } ^ + ( x _ 1 ) , \\eta _ { n _ 1 - m _ 1 } ^ - ( & y _ 1 ) \\rangle \\cdots \\langle \\eta _ { m _ N } ^ + ( x _ N ) , \\eta _ { n _ N - m _ N } ^ - ( y _ N ) \\rangle \\\\ & \\langle B e _ { ( m _ 1 , . . . , m _ N ) } , A e _ { ( n _ 1 - m _ 1 , . . . , n _ N - m _ N ) } \\rangle . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} Q = Q ( t , x , y ) : = P ( t , x ) + \\frac { 1 } { 2 } ( 1 - 2 a ) h _ 1 ^ 2 ( t , x , y ) , \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } \\Phi ' ( \\dot { u } ) = f ( t , u , \\dot { u } ) , \\ t \\in [ 0 , 1 ] \\\\ u \\in ( B C ) \\end{cases} \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\tilde { \\phi } ( d ( x _ 1 , y _ 1 ) , . . . , d ( x _ N , y _ N ) ) , \\qquad \\forall \\ x , y \\in X , \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} \\left \\{ E _ A \\prod _ { a = 1 } ^ { m + n } \\left ( K _ a ^ { \\sigma _ a } { K _ a \\brack \\mu _ a } \\right ) F _ A \\ , \\bigg | \\ , A \\in P ( m | n ) , \\sigma _ a \\in \\{ 0 , 1 \\} , \\mu \\in \\mathbb N ^ { m + n } \\right \\} . \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\epsilon ( \\alpha , 0 ) & = \\epsilon ( 0 , \\alpha ) = 1 , \\\\ \\epsilon ( \\alpha , \\beta ) & = ( - 1 ) ^ { ( \\alpha | \\beta ) + | \\alpha | ^ { 2 } | \\beta | ^ { 2 } } \\epsilon ( \\beta , \\alpha ) \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} X ^ { ( N ) } _ t : = \\frac { 1 } { N ^ { 1 / 2 } } X _ { \\lfloor N t \\rfloor } + \\frac { 1 } { N ^ { 1 / 2 } } ( N t - \\lfloor N t \\rfloor ) ( X _ { \\lfloor N t \\rfloor + 1 } - X _ { \\lfloor N t \\rfloor } ) . \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} | d _ n | = \\bigg | \\frac { G ( t _ n ) c _ n } { A ' ( t _ n ) \\mu _ n ^ { 1 / 2 } } \\bigg | = \\bigg | \\frac { a _ n b _ n t _ n ^ 2 \\nu _ n c _ n } { \\mu _ n ^ { 1 / 2 } } \\bigg | = | a _ n t _ n \\nu _ n ^ { 1 / 2 } c _ n | . \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} \\| P ( x ) \\| ^ 2 & = \\sum _ { k \\geq 0 } \\| P _ k ( x ) \\| ^ 2 \\| \\tilde { B } e _ k \\| ^ 2 \\\\ & = \\sum _ { k \\geq 0 } | \\mathcal { A } ( x , k ) | \\| B D e _ k \\| ^ 2 \\\\ & \\leq M \\sum _ { k \\geq 0 } \\tbinom { N - 1 + k } { N - 1 } \\tbinom { N - 1 + k } { N - 1 } ^ { - 1 } \\| B e _ k \\| ^ 2 \\\\ & = M \\| B \\| _ { S _ 2 } ^ 2 . \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} Q ^ { 1 / d } \\leq q _ { \\gamma _ 0 } = Q < Q ^ { \\abs { \\gamma _ 0 } - 1 / d } . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} \\gamma _ { g , h } ( x ) : = \\frac { \\omega ( g , h , x ) \\omega ( ^ { g h } x , g , h ) } { \\omega ( g , \\ ^ h x , h ) } , & & \\mu _ g ( x , y ) : = \\frac { \\omega ( ^ g x , g , y ) } { \\omega ( ^ g x , ^ g y , g ) \\omega ( g , x , y ) } . \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } a _ { 1 1 } & a _ { 1 2 } & a _ { 1 3 } \\\\ a _ { 1 2 } & a _ { 2 2 } & a _ { 2 3 } \\\\ a _ { 1 3 } & a _ { 2 3 } & a _ { 3 3 } \\\\ \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { p _ { 2 } + q _ { 2 } } S _ { a , b } ^ { a , b + 1 , q _ { 2 } } \\left ( p _ { 2 } , m \\right ) S _ { a , a m + b } ^ { a , a m + b + 1 , q _ { 1 } } \\left ( p _ { 1 } , l - m \\right ) = S _ { a , b } ^ { a , b + 1 , q _ { 1 } + q _ { 2 } } \\left ( p _ { 1 } + p _ { 2 } , l \\right ) . \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { p - w _ p } { p ^ 2 + w _ p } \\zeta _ p ( 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} C \\langle y \\mid \\dd ( y ) = z _ { f } \\otimes 1 \\rangle \\cong C \\langle y \\mid \\dd ( y ) = z _ { g } \\otimes 1 \\rangle \\ , . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\int _ { \\Omega } F _ n \\varphi c ^ { - 2 } d x = 0 , \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} E ( u _ { 1 } , \\dots , u _ { N } ) \\\\ = \\int _ { \\R ^ d } \\sum _ { \\mu = 1 } ^ N \\abs { \\Delta { u _ { \\mu } } } ^ 2 + \\kappa \\int _ { \\R ^ d } \\sum _ { \\mu = 1 } ^ N \\abs { \\nabla { u _ { \\mu } } } ^ 2 + \\sum _ { \\mu , \\nu = 1 } ^ N ( \\beta _ { \\mu \\nu } + N \\lambda _ { \\mu \\nu } ) \\int _ { \\R ^ d } \\frac { | u _ \\mu u _ \\nu | ^ { p + 1 } } { p + 1 } \\ , d x . & \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} \\int _ t ^ { t + 1 } | A ^ { 1 / 2 } u ( \\tau ) | _ { \\alpha , \\sigma } ^ 2 d \\tau & \\le | u ( t ) | _ { \\alpha , \\sigma } ^ 2 + 3 c _ 1 ^ 2 \\int _ t ^ { t + 1 } ( 1 + \\tau ) ^ { - 2 \\lambda } d \\tau \\\\ & \\le 2 c _ * ^ 2 ( 1 + t ) ^ { - 2 \\lambda } + 3 c _ 1 ^ 2 ( 1 + t ) ^ { - 2 \\lambda } \\\\ & = \\Big ( 2 c _ * ^ 2 + \\frac { c _ * ^ 2 } { M _ 2 } \\Big ) ( 1 + t ) ^ { - 2 \\lambda } . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} T _ { v _ 1 \\otimes v _ 1 \\otimes v _ 1 \\otimes v _ 2 \\otimes v _ 4 } & = \\begin{tabular} { | c | } \\hline 1 , 2 , 3 \\\\ \\hline 4 \\\\ \\hline \\\\ \\hline 5 \\\\ \\hline \\end{tabular} \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} & \\mathbf { E } ^ s _ i ( X ) = \\left ( ( A _ i - \\widetilde { Y } _ i ) ^ T \\otimes I _ 2 \\right ) \\left ( M ^ s ( \\widetilde { Y } _ i ) - \\overline { M } ^ s \\right ) \\emph { V e c } ( \\nabla \\mathbf { u } ^ s ( X ) ) , \\\\ & \\mathbf { F } ^ s _ i ( X ) = - \\left ( ( \\widetilde { Y } _ i - \\overline { X } _ i ) ^ T \\otimes I _ 2 \\right ) \\left ( \\overline { M } ^ s - I _ 4 \\right ) \\emph { V e c } ( \\nabla \\mathbf { u } ^ s ( X ) ) . \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { \\psi ( n ) \\varphi ( n ) } { n ( \\log n ) ^ \\varepsilon } = \\infty . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } f ( x ) d x = \\int _ { 0 } ^ { \\infty } \\int _ { \\omega _ { Q } } f ( r y ) r ^ { Q - 1 } d \\sigma ( y ) d r . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} h ^ n _ t ( x , v ) : = Y _ t ^ { n , 1 } = f ^ n _ t ( x , v ) \\sqrt { 1 + v ^ 2 } \\ , , \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} i n d e x ( \\Sigma \\cap B _ R ( 0 ) ) = i n d e x ( \\Sigma ) \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , g e n u s ( \\Sigma \\cap B _ R ( 0 ) ) = g e n u s ( \\Sigma ) . \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} \\Gamma [ n ] ( t ) : = T _ t n _ 0 + \\int _ 0 ^ t T _ { t - \\tau } ( A [ n ] ( \\tau , , ) ) ( s ) d \\tau \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\sigma _ i \\otimes t _ i \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} F ( B _ { k - 1 } + ( T + 1 ) \\delta _ k ; p ) = C _ k ( T ) \\cdot F ( B _ { k - 1 } + T \\delta _ k ; p ) , T = 0 , 1 , \\dots , \\beta ^ { ( 1 ) } _ k - 2 . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} S _ Y ( n , m ) = \\binom { n } { m } \\langle m \\rangle _ { n - m } , m \\leq n . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} | \\alpha _ { f , i } ( p ) | & \\leq p ^ \\theta \\\\ \\Re \\mu _ { f , i } & \\leq \\theta , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} \\psi | _ { y = 0 } = 0 , \\psi | _ { y \\rightarrow + \\infty } \\rightarrow + \\infty . \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} { \\bf C o v } \\bigl ( S ( f _ i ) , \\ , S ( f _ j ) \\bigr ) = \\langle f _ i , \\ , f _ j \\rangle . \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} \\sigma ( t ) : = \\zeta ' ( t ) + \\frac { C ^ { m - 1 } m } { a ( m - 1 ) } ( N - 1 ) k \\coth ( k ) \\zeta ^ m ( t ) \\eta ( t ) + \\frac { \\zeta ( t ) } { m - 1 } \\frac { \\eta ' ( t ) } { \\eta ( t ) } \\ , , \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} t _ { s _ h } ^ { 2 m } = \\prod _ { i = 1 } ^ m { } _ { t _ x ^ { 2 i - 1 } } ( t _ y t _ a ^ { - 1 } t _ { t _ x ( y ) } t _ a ^ { - 1 } ) \\cdot t _ x ^ { 2 m } t _ b ^ { - 2 m } t _ z ^ { 2 m } t _ c ^ { - 2 m } , \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} u _ { * } ^ 2 v _ { * } ^ 2 = \\frac { - 4 ( 1 - 4 y _ { * } ^ 4 ) } { y _ { * } ^ 4 + y _ { * } ^ 2 } . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } \\Phi ( \\frac { B _ { t _ { j } } + B _ { t _ { j + 1 } } } { 2 } , t _ { j } ) [ B _ { t _ { j + 1 } } - B _ { t _ { j } } ] | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K ( n ) \\} \\\\ = C _ { n } ^ { K ( n ) } ( 2 ^ { K ( n ) } - 1 ) . \\end{array} \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{gather*} K _ { v } ^ 2 ( v _ k ^ K ) = \\sigma _ k ^ 2 v _ k ^ K , \\\\ K _ { v } ( v _ k ^ K ) = \\sigma _ k e ^ { i \\psi ^ \\infty } \\Psi _ k ^ K ( 0 ) v _ k ^ K . \\end{gather*}"}
-{"id": "6771.png", "formula": "\\begin{align*} f ^ i ( x ) : = ( x _ 1 , \\dots , f _ i ( x ) , \\dots , x _ n ) . \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { \\atop { ( x ' , x '' ) \\in \\Omega ^ \\sigma } { x \\equiv r \\bmod q } } \\exp \\big ( 2 \\pi i P ( x ' , p ^ \\sigma x '' ) / p ^ j \\big ) \\\\ & \\qquad \\qquad = \\exp \\big ( 2 \\pi i \\tilde { P } ( r ' , p ^ \\sigma r '' ) / p ^ j \\big ) \\sum _ { \\atop { x \\in \\Omega ^ \\sigma } { x \\equiv r \\bmod q } } \\prod _ { \\atop { \\gamma \\in \\Gamma } { \\abs { \\gamma } = 1 } } \\exp \\big ( 2 \\pi i b _ \\gamma x ^ \\gamma / q _ \\gamma \\big ) , \\end{aligned} \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} \\mathcal D = 1 + \\frac { \\beta } { \\gamma } \\sqrt { \\mu } | D | , \\mathcal B = \\frac { 1 } { \\gamma } \\Big ( 1 + \\frac { \\beta - 1 } { \\gamma } \\sqrt { \\mu } | D | \\Big ) , \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} ( \\tau _ { i , j } \\pi ) _ { k , l } : = \\begin{cases} \\pi _ { k , l } & , \\\\ \\max \\left \\{ \\pi _ { i , j + 1 } , \\pi _ { i + 1 , j } \\right \\} + \\min \\left \\{ \\pi _ { i - 1 , j } , \\pi _ { i , j - 1 } \\right \\} - \\pi _ { i , j } & . \\end{cases} \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} \\left | \\mathsf { E } _ \\hslash ^ { p , q , d } ( \\xi _ { r ( \\hslash ) } ) ^ 2 - \\mathsf { E } _ \\eth ^ { p , q , d } ( \\xi _ { r ( \\eth ) } ) ^ 2 \\right | & \\leq \\frac { \\varepsilon } { 2 } + | \\mathsf { F } _ \\hslash ^ { p , q , d } ( \\xi _ { r ( \\hslash ) } ) ^ 2 - \\mathsf { F } _ \\eth ^ { p , q , d } ( \\xi _ { r ( \\eth ) } ) ^ 2 | \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ^ j } { \\partial \\tilde { \\mathbf { x } } ^ j } ( \\tilde { g } _ { t } ( \\tilde { \\mathbf { x } } ) _ { a b } - \\delta _ { a b } ) \\right | \\leq \\frac { \\epsilon _ t ^ { - 2 } } { 1 0 0 } | \\tilde { \\mathbf { x } } | ^ { 2 - j } \\ , \\ ; { \\rm f o r } \\ ; \\ , | \\tilde { \\mathbf { x } } | \\leq 2 \\epsilon _ t \\ ; \\ , { \\rm a n d } \\ ; \\ , j = 0 , 1 . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} \\dfrac { d } { d x } \\left ( \\dfrac { x } { \\sqrt [ a ] { 1 + c x ^ a } } \\right ) = ( 1 + c x ^ a ) ^ { - \\frac { 1 + a } { a } } > 0 , ( x > 0 ) \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} x _ { k + 1 } = ( 1 - \\theta _ k ) x _ k + \\frac { \\theta _ k } { \\theta _ 0 } z _ { k + 1 } - \\frac { \\theta _ k } { \\theta _ 0 } ( 1 - \\theta _ 0 ) z _ k \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} u _ { t } ^ { \\varepsilon } + \\nabla \\cdot \\left ( - a \\left ( x , t ; u ^ { \\varepsilon } ; \\nabla u ^ { \\varepsilon } \\right ) \\nabla u ^ { \\varepsilon } \\right ) - \\mathbf { Q } _ { \\varepsilon } ^ { \\beta } u ^ { \\varepsilon } = F _ { \\ell ^ { \\varepsilon } } \\left ( x , t ; u ^ { \\varepsilon } ; \\nabla u ^ { \\varepsilon } \\right ) \\quad Q _ { T } , \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} Q _ 2 ( t ) = b _ { 0 , \\sup } ( t ) + \\frac { \\chi _ 2 } { 2 d _ 3 } \\big ( k { \\bar r _ 1 } + l { \\bar r _ 2 } \\big ) + \\frac { l ^ 2 } { 4 \\lambda d _ 3 } \\Big ( \\frac { \\chi _ 1 ^ 2 { \\bar r _ 1 } ^ 2 } { d _ 1 } + \\frac { \\chi _ 2 ^ 2 { \\bar r _ 2 } ^ 2 } { d _ 2 } \\Big ) + \\frac { a _ { 2 , \\sup } ( t ) { \\bar r _ 1 } + b _ { 1 , \\sup } ( t ) { \\bar r _ 2 } } { 2 } . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\vert \\tilde y - y \\vert & = \\left \\vert \\pi _ { L ( z _ 0 , 1 2 r _ j ) } ^ \\perp ( y - z _ 0 ) \\right \\vert \\\\ & \\leq \\left \\vert \\pi _ { L ( z _ 0 , 1 2 r _ j ) } ^ \\perp ( y - w ' ) \\right \\vert + \\left \\vert \\pi _ { L ( z _ 0 , 1 2 r _ j ) } ^ \\perp ( w ' - z _ 0 ) \\right \\vert \\\\ & \\leq \\vert y - w ' \\vert + 1 2 r _ j \\delta . \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} { \\bf E } [ \\beta _ { 2 , 1 } ( a , b , \\bar { b } ) ^ q ] = \\frac { S _ 2 ( b _ 0 \\ , | \\ , a ) } { S _ 2 ( q + b _ 0 \\ , | \\ , a ) } \\frac { S _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , a ) } { S _ 2 ( b _ 0 + b _ 1 \\ , | \\ , a ) } . \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} { 2 m \\choose m + s + 1 } + { 2 m \\choose m - s } = & { 2 m \\choose m - s - 1 } + { 2 m \\choose m - s } \\\\ = & { 2 m + 1 \\choose m - s } . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} 0 \\le \\mu _ { 0 } < \\mu _ { 1 } \\le \\mu _ { 2 } \\le . . . , \\ ; \\lim _ { p \\to \\infty } \\mu _ { p } = \\infty . \\end{align*}"}
-{"id": "2392.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": "8848.png", "formula": "\\begin{align*} \\vert w - x \\vert & \\leq \\vert w - \\tilde y \\vert + \\vert \\tilde y - x \\vert \\\\ & < 1 2 r _ j \\delta + r - 2 r _ j ( 1 + 6 \\delta ) + 2 r _ j \\\\ & = r . \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} ( L L ^ * - 1 ) \\psi & = R ( 0 ) ^ { 1 / 2 } B ^ 2 R R ( 0 ) ^ { 1 / 2 } ( B ^ 2 + f ) \\psi - \\psi \\\\ & = - R ( 0 ) ^ { 1 / 2 } f R ( 0 ) ^ { 1 / 2 } \\psi . \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} F = 1 + x F ^ 2 \\cdot \\frac { 1 } { 1 - t ( F - 1 ) } \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} S _ A ^ { o , h } ( i _ { \\kappa } , \\kappa ) = \\sum \\limits _ { k \\in [ h ] } \\pi ^ { o , \\kappa , h } _ { i _ { \\kappa } , k } ( n _ { \\kappa } ) \\alpha ^ { \\varphi ( o , \\kappa ) , h } _ { i _ { \\kappa } , k } ( n _ { \\kappa } ) = 1 , \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ ( e ^ { - t q } - 1 ) e ^ { - b _ 0 t } \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } + q e ^ { - t } \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j } { \\prod \\limits _ { i = 1 } ^ M a _ i } \\Bigr ] \\Bigr ) . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\mathbb { G } ( f ) \\right ) = 0 , \\mathrm { C o v } ( \\mathbb { G } ( f ) , \\mathbb { G } ( g ) ) = P ( f g ) - P ( f ) P ( g ) . \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} c ( n ^ 2 | D | ) = c ( | D | ) \\sum _ { \\substack { d \\mid n , \\\\ d > 0 } } \\mu ( d ) \\left ( \\frac { D } { d } \\right ) \\chi ( d ) d ^ { k - 1 } a _ { \\chi } ( n / d ) . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\Phi = ( h , f ) : B \\cap Z \\cap \\Phi ^ { - 1 } ( D \\times D ' ) \\to D \\times D ' \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} \\varphi ( \\gamma _ d ( x ) y ) = N _ { C / A } ( x ) \\varphi ( y ) . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} k ( M \\backslash A ^ c ) - k ( M ) - f ( M \\backslash A ^ c ) + k ( M / A ) = 0 k ( M / A ) - k ( M ) - v ( M / A ) + k ( M \\backslash A ^ c ) = 0 . \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} a x ^ 3 + x ^ 2 + b = x ^ 3 ( b x ^ 3 + x + a ) ^ q = 0 . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} \\langle \\mathcal L _ k ^ i , u \\rangle = \\langle f _ k ^ i , \\bar u \\rangle + \\langle g _ k ^ i , u _ 3 \\rangle , \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\sum i n _ i ^ 2 + 2 \\sum _ { i < j } i n _ i n _ j \\ge \\frac { 1 } { 4 } ( \\sum i n _ i ) ^ 2 = \\frac { 1 } { 4 } n ^ 2 , \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} W ^ C _ { 8 2 } = & 1 + ( 3 2 8 0 + 2 c ) y ^ { 1 4 } + ( 3 6 2 4 4 + 1 2 8 b - 2 c ) y ^ { 1 6 } \\\\ & + ( 5 1 4 3 4 5 - 8 1 9 2 a - 8 9 6 b - 2 6 c ) y ^ { 1 8 } + \\cdots , \\\\ W ^ S _ { 8 2 } = & a y ^ 5 + ( - 1 8 a + b ) y ^ 9 + ( 1 5 3 a - 1 6 b - c ) y ^ { 1 3 } \\\\ & + ( 3 0 4 3 8 4 - 8 1 6 a + 1 2 0 b + 1 4 c ) y ^ { 1 7 } + \\cdots , \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} J _ { 1 , 2 } & = \\frac 1 2 \\sum _ { l \\leq x } \\frac { \\Lambda ( l ) } { l } \\sum _ { d \\leq x / l } d ^ a \\\\ & = - \\frac { \\zeta ' ( 2 + a ) } { 2 ( 1 + a ) \\zeta ( 2 + a ) } x ^ { 1 + a } + O _ { a } ( \\log x ) . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} d _ B ( \\pi _ 1 , \\pi _ 2 ) & = \\frac { 1 } { 2 } \\lvert A ( \\pi _ 1 ) \\Delta A ( \\pi _ 2 ) \\rvert \\\\ & = \\frac { 1 } { 2 } \\lvert \\nu ( \\pi _ 1 ) \\Delta \\nu ( \\pi _ 2 ) \\rvert \\\\ & > \\frac { 1 } { 2 } ( 2 \\cdot 2 t ) = 2 t , \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} \\theta ( \\mathbf Y _ M \\mathbf G , \\phi _ { \\breve { \\mathbf g } } ) = C _ 1 { \\breve { \\mathbf g } } , \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} P ( V _ n = k ) \\leq P ( V _ n = 0 ) \\ > \\frac { \\beta _ n ^ k } { k ! } . \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom { n } { k } B _ k ( x _ 1 ) B _ { n - k } ( x _ 2 ) = B _ n ( 2 ; x ) = 2 \\sum _ { k = 0 } ^ n \\dfrac { ( - 1 ) ^ k H _ { k + 1 } } { k + 2 } \\Delta ^ k I _ n ( x ) . \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} \\delta v _ { s t } = \\delta \\nu _ { s t } + B _ { s t } ^ 1 v _ s + B _ { s t } ^ 2 v _ s + v _ { s t } ^ { \\natural } , \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} \\eta _ { 2 , 2 } ( q \\ , | \\ , 1 , b ) = \\prod \\limits _ { k = 0 } ^ \\infty \\Bigl [ \\frac { b _ 0 + k } { q + b _ 0 + k } \\frac { q + b _ 0 + b _ 1 + k } { b _ 0 + b _ 1 + k } \\frac { q + b _ 0 + b _ 2 + k } { b _ 0 + b _ 2 + k } \\frac { b _ 0 + b _ 1 + b _ 2 + k } { q + b _ 0 + b _ 1 + b _ 2 + k } \\Bigr ] ^ { k + 1 } . \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} \\| V ^ * T V \\| _ { S _ 1 } = ( V ^ * | T | V ) = ( | T | ) = \\| T \\| _ { S _ 1 } . \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\lambda _ a ( b ) = a ^ { - 1 } \\cdot ( a \\circ b ) , \\rho _ a ( b ) = ( a \\circ b ) \\cdot a ^ { - 1 } . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\mathrm { c h } ( c ' ( \\theta _ 1 ) ) = \\Sigma y _ 3 - \\frac { 1 } { 6 } \\Sigma y _ 7 , \\quad \\mathrm { c h } ( c ' ( \\theta _ 2 ) ) = 2 \\Sigma y _ 7 . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} | a + b | ^ 2 & = \\left | \\left \\langle \\binom { \\sqrt { y } a } { \\frac { b } { \\sqrt { y } } } , \\binom { \\frac { 1 } { \\sqrt { y } } } { \\sqrt { y } } \\right \\rangle \\right | ^ 2 \\leq \\left ( y | a | ^ 2 + \\frac { | b | ^ 2 } { y } \\right ) \\left ( \\frac { 1 } { y } + y \\right ) \\\\ & = \\left ( 1 + y ^ 2 \\right ) | a | ^ 2 + \\left ( \\frac { 1 } { y ^ 2 } + 1 \\right ) | b | ^ 2 . \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} ( h \\cdot f ) ( m ) = \\sum _ h h _ 2 f ( S ^ { - 1 } ( h _ 1 ) m ) = \\sum _ h f ( h _ 2 S ^ { - 1 } ( h _ 1 ) m ) = \\epsilon ( h ) f ( m ) , \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} I _ \\lambda \\leqq E _ { + \\infty } ( d _ k { \\bf { h } } _ { k , 2 } ) = d ^ 2 _ k E _ { + \\infty } ( { \\bf { h } } _ { k , 2 } ) = E _ { + \\infty } ( { \\bf { h } } _ { k , 2 } ) + O ( \\epsilon ) . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} \\nabla _ x \\left ( \\int _ 0 ^ t P ^ { \\kappa } _ s f ( x ) \\ , d s \\right ) = \\int _ 0 ^ t \\nabla _ x P ^ { \\kappa } _ s f ( x ) \\ , d s \\ , . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} ( \\tau - \\tau ' ) ( n g ) = f ( n ) + \\tau ( g ) - f ( n ) - \\tau ' ( g ) = ( \\tau - \\tau ' ) ( g ) \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} & \\hat x _ 0 : = x _ 0 , \\\\ & \\hat x _ i : = x _ 0 - \\sum _ { j = 0 } ^ { i - 1 } h _ { i , j } g _ j , { i = 0 , \\hdots , N } , \\\\ & \\hat x _ * : = x _ 0 + r _ * . \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} B ^ { n , m } _ t = G ( M ^ { n - 1 } ( t , x ) ) + 2 \\sigma \\frac { A _ m ( v ) } { \\sqrt { 1 + v ^ { 2 m } } } \\ , , \\ ; \\ ; R ^ { n , m } _ t = \\left [ A _ m ( v ) ( B ^ { n , m } _ t - v ) - \\sigma A _ m ' ( v ) \\right ] f ^ { n } _ t \\ , , \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} ( \\chi _ N ( D ) - 1 ) f ( x ) & = { ( 2 \\pi ) ^ { - 1 } } \\int _ { \\R ^ n } \\Big ( N ^ n { ( \\mathcal { F } ^ { - 1 } { \\chi } ) } ( N ( x - y ) ) f ( y ) - f ( x ) \\Big ) d y \\\\ & = { ( 2 \\pi ) ^ { - 1 } } \\int _ { \\R ^ n } { ( \\mathcal { F } ^ { - 1 } { \\chi } ) } ( y ) ( f ( x + N ^ { - 1 } y ) - f ( x ) ) d y \\\\ & = { ( 2 \\pi ) ^ { - 1 } } \\int _ { \\R ^ n } N ^ { - 1 } y { ( \\mathcal { F } ^ { - 1 } { \\chi } ) } ( y ) \\cdot \\left ( \\int _ 0 ^ 1 \\nabla f ( x + \\theta N ^ { - 1 } y ) d \\theta \\right ) d y \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} & s _ { h , j } : = \\rho _ k ^ { j - 1 } ( s _ h ) , & & & \\\\ & X _ { i , j } : = { } _ { \\rho _ k ^ { j - 1 } } ( X _ i ) , & & Y _ { i , j } : = { } _ { \\rho _ k ^ { j - 1 } } ( Y _ i ) & \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\mathbb { P } ( ) & = \\frac { - 2 4 \\ln ( 2 ) + \\left ( 4 - 3 \\ln ( 2 ) \\right ) \\pi + 1 2 G } { 4 \\left ( - 2 + 3 \\ln ( 2 ) \\right ) \\pi } \\\\ & = 0 . 3 9 0 3 3 3 8 8 7 0 . . . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\sum _ { f \\in H _ 2 ( q ) } \\omega _ f = 1 \\ ; + \\ ; \\mathcal O \\left ( q ^ { - 3 / 2 } \\right ) . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} E _ { a } ( x ) = \\sum _ { n \\leq x } \\frac { \\mu ( n ) } { n } \\Delta _ { a } \\left ( \\frac { x } { n } \\right ) + O _ { a } ( 1 ) , \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} \\mathbb { W } _ { { H } _ { \\sigma , k } ; L _ \\sigma } : = { \\rm S p a n } \\{ \\omega _ { \\ ! D _ \\sigma } ( s _ 1 ' , \\ldots , s _ k ' ) \\mid s ' _ 1 , \\ldots , s ' _ k \\in H ^ 0 ( H _ \\sigma , L _ \\sigma ) \\} . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} & f ^ { \\star } _ 0 ( x _ \\ell ) + f ^ { \\star } _ 0 ( - x _ \\ell ) - 2 \\ , f ^ { \\star } _ 0 ( 0 ) = 2 ( f ^ { \\star } _ 0 ( x _ \\ell ) - f ^ { \\star } _ 0 ( 0 ) ) = \\\\ & = 2 f ^ { \\star } _ 0 ( x _ \\ell ) \\ge 2 \\widetilde { C ' } \\omega ( x _ \\ell ) = 2 \\widetilde { C ' } \\omega ( A _ \\ell \\delta _ { k _ \\ell } ) . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} \\alpha _ { n } ^ { ( N ) } ( f ) = \\sqrt { n } ( \\mathbb { P } _ { n } ^ { ( N ) } ( f ) - P ( f ) ) . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow 0 } g '' _ 0 ( x ; a , b ) & = \\lim _ { x \\downarrow 0 } g _ 1 ' ( x ; a , b ) + 4 a ^ 3 f ^ { ( 4 ) } ( A ) \\int _ { - 1 } ^ 1 \\int _ 0 ^ s \\ , d u \\ , s \\ , d s \\\\ & \\qquad + 8 a ^ 5 f ^ { ( 5 ) } ( A ) \\int _ { - 1 } ^ 1 \\int _ 0 ^ s \\int _ 0 ^ u \\ , d v \\ , u \\ , d u \\ , s \\ , d s \\\\ & = 4 a f ^ { ( 3 ) } ( A ) + \\frac { 1 6 a ^ 3 f ^ { ( 4 ) } ( A ) } { 3 } + \\frac { 1 6 a ^ 5 f ^ { ( 5 ) } ( A ) } { 1 5 } . \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} & \\int _ { \\Omega } n \\min \\limits _ { 1 \\le k \\le n } | x - \\alpha ( k , \\omega ) | \\ , d P ( \\omega ) \\\\ & = \\int _ 0 ^ \\infty P ( \\{ \\omega : n \\min \\limits _ { 1 \\le k \\le n } | x - \\alpha ( k , \\omega ) | \\ge t \\} ) \\ , d t = \\int \\limits _ 0 ^ { n / 2 } ( 1 - 2 t / n ) ^ n \\ , d t = \\frac n { 2 ( n + 1 ) } . \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) = \\frac { \\left ( r ! \\right ) ^ { p } \\prod _ { s = 2 } ^ { l } \\left ( r _ { s } ! \\right ) ^ { p _ { s } } } { k ! } \\sum _ { j = 0 } ^ { k } \\left ( - 1 \\right ) ^ { j } \\binom { k } { j } \\binom { \\left ( k - j \\right ) a + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\left ( k - j \\right ) \\alpha _ { s } + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } . \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\sum _ { U _ n ( \\mathfrak { f } ) \\backslash G _ n ( \\mathfrak { f } ) } J _ { \\sigma , \\psi } ( g ) J _ { \\sigma ^ \\prime , \\psi ^ { - 1 } } ( g ) = 0 , \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda u ^ { \\ , \\beta } \\leq U \\mbox { l o c a l l y i n $ \\R ^ n \\times \\left ( 0 , \\infty \\right ) $ } . \\end{aligned} \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} h ( r ) = \\begin{cases} 1 & r \\in [ 0 , R ) \\\\ \\sum _ { i = 0 } ^ { p } e ^ { R - r } ( R / r ) ^ { 2 i } \\tilde \\alpha _ i \\chi _ { i } ( r ) & r \\in [ R , \\infty ) \\end{cases} \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} \\widetilde { \\Phi } ^ T ( \\widetilde { D } ( ( i , r + 2 r _ i ) , 0 ) ^ { - 1 } \\widetilde { D } ( ( i , r ) , 0 ) ) = t ^ { \\alpha ( i , r ) / 2 } \\widetilde { Y } _ { i , r } \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} X _ 0 & = G , \\\\ X _ 1 & = \\{ g G _ i \\ , : \\ , g \\in G , \\ i \\in I \\} . \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{gather*} f ( \\tau ) = q \\frac 1 { y ^ 2 } \\frac { { \\rm d } x } { { \\rm d } q } = \\frac { 2 ^ { 1 / 3 } } 3 \\eta ( \\tau ) ^ 4 . \\end{gather*}"}
-{"id": "2707.png", "formula": "\\begin{align*} t _ { i , n } = t _ { i , n } ^ { ( 1 ) } = t _ { i , n } ^ { ( 2 ) } , i \\in \\mathbb { N } _ 0 , ~ n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} \\delta \\tau ( g , h ) = g \\cdot \\tau ( h ) - \\tau ( g h ) + \\tau ( g ) \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\alpha ( t s ) = \\alpha ( s t ^ { p + 1 } ) = s ^ { p - 1 } t ^ { ( p + 1 ) y } \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} \\sum _ { y \\in g N } N _ { w , G } ( y ) = \\sum _ { y \\in h N } N _ { w , G } ( y ) \\ \\ \\ \\ ( * ) \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} e ^ i _ { \\ ; j } e _ k ^ { \\ ; j } = \\delta ^ i _ k , \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} F ( z , z ' ) ( \\ell ) : = \\int _ L d \\mu ( \\ell ' ) F _ 0 ( u ( z , \\ell ) , u ( z ' , \\ell ' ) ) \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} W ^ C _ { 5 8 } = & 1 + ( 3 1 9 - 2 4 \\beta - 2 \\gamma ) y ^ { 1 0 } + ( 3 1 3 2 + 1 5 2 \\beta + 2 \\gamma ) y ^ { 1 2 } + \\cdots , \\\\ W ^ S _ { 5 8 } = & \\beta y ^ 5 + \\gamma y ^ 9 + ( 2 4 1 2 8 - 5 4 \\beta - 1 0 \\gamma ) y ^ { 1 3 } \\\\ & + ( 1 4 6 9 9 5 2 + 3 2 0 \\beta + 4 5 \\gamma ) y ^ { 1 7 } + \\cdots , \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\psi ( z + 1 ) = \\psi ( z ) + \\frac { 1 } { z } , z > 0 , \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} \\underbar r _ 1 = \\frac { a _ { 0 , \\inf } - a _ { 2 , \\sup } \\bar r _ 2 - k \\frac { \\chi _ 1 } { d _ 3 } \\bar r _ 1 } { a _ { 1 , \\sup } - k \\frac { \\chi _ 1 } { d _ 3 } } , \\bar r _ 1 = \\frac { a _ { 0 , \\sup } - a _ { 2 , \\inf } \\underbar r _ 2 - k \\frac { \\chi _ 1 } { d _ 3 } \\underbar r _ 1 } { a _ { 1 , \\inf } - k \\frac { \\chi _ 1 } { d _ 3 } } , \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} W ^ 1 = W _ 1 \\oplus \\cdots \\oplus W _ t . \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} & \\psi \\in C ^ 2 \\big ( [ 0 , \\infty ) \\big ) , \\psi ( 0 ) = 0 , \\\\ & \\psi ' ( y ) \\leq \\nu _ 1 , \\\\ & \\psi ( y ) \\geq \\nu _ 2 y - \\nu _ 3 , \\\\ & \\psi '' ( y ) > 0 \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\mathcal { H } = \\bigoplus _ { I \\subset [ N ] } \\mathcal { H } _ I , \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} & e ^ { ( n - \\nu - 2 ) t } \\int _ X | R i c ( \\omega ( t ) ) + \\omega ( t ) | ^ 2 _ { \\omega ( t ) } \\omega ( t ) ^ n \\\\ & = e ^ { ( n - \\nu - 2 ) t } \\partial _ t \\left ( \\int _ X S ( t ) \\omega ( t ) ^ n \\right ) + e ^ { ( n - \\nu - 2 ) t } \\int _ X ( S ( t ) + 1 ) ( S ( t ) + n ) \\omega ( t ) ^ n \\\\ & = \\partial _ t \\left ( e ^ { ( n - \\nu - 2 ) t } \\int _ X S ( t ) \\omega ( t ) ^ n \\right ) + e ^ { ( n - \\nu - 2 ) t } \\int _ X ( S ( t ) ^ 2 + ( \\nu + 3 ) S ( t ) + n ) \\omega ( t ) ^ n . \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} K K ^ { T } = ( r - \\lambda ) I + \\lambda J . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} A \\sharp _ { \\lambda } B & = A ^ { \\frac { 1 } { 2 } } ( A ^ { \\frac { - 1 } { 2 } } B A ^ { \\frac { - 1 } { 2 } } ) ^ { \\lambda } A ^ { \\frac { 1 } { 2 } } , \\\\ P _ { t } ( \\lambda ; A , B ) & = A ^ { \\frac { 1 } { 2 } } \\left [ 1 - \\lambda + \\lambda ( A ^ { \\frac { - 1 } { 2 } } B A ^ { \\frac { - 1 } { 2 } } ) ^ { t } \\right ] ^ { \\frac { 1 } { t } } A ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} g ' = x ^ { \\tilde n - a _ 1 r _ 1 - a r _ 2 } y ^ { b _ 1 r _ 2 - r _ 1 b _ 2 } z ^ { r _ 1 c + r _ 2 c _ 2 } - z ^ { \\tilde \\ell } \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} x = \\sum _ { i = 0 } ^ { p ^ n - 1 } \\sum _ { h \\in H } z _ { i , h } h \\gamma _ n ^ i ( a _ n ' ) \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} W _ { 1 3 5 } = F _ { 6 , 8 } \\wedge F _ { 6 , 1 0 } \\wedge F _ { 6 , 1 2 } \\wedge F _ { 6 , 1 6 } \\wedge F _ { 6 , 1 8 } \\wedge F _ { 6 , 2 4 } \\wedge F _ { 6 , 2 6 } \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\begin{array} { c c c c } f ( \\alpha ) = q ^ { - 2 } \\alpha & f ( \\gamma ) = \\gamma & f ( \\alpha ^ { * } ) = q ^ { 2 } \\alpha ^ { * } & f ( \\gamma ^ { * } ) = \\gamma ^ { * } . \\\\ \\end{array} \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} \\xi ( g , h ) & = \\frac { \\tilde \\omega ( \\tau , g , h ) \\tilde \\omega ( g ^ { - 1 } , h ^ { - 1 } , \\tau ) } { \\tilde \\omega ( g ^ { - 1 } , \\tau , h ) } \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\Lambda , \\rho , \\beta , H } ^ { \\vec { h } } ( \\omega _ e = 1 | \\omega _ { G _ t } = \\omega ^ { \\rho } _ { G _ t } ) \\leq \\mathbb { P } _ { \\Lambda , \\rho , \\beta , H } ^ { | \\vec { h } | } ( \\omega _ e = 1 | \\omega _ { G _ t } = \\omega ^ { \\rho } _ { G _ t } ) . \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} \\begin{array} { l l } K ( x ^ i ) ~ u _ { \\xi ^ i } = ( I + \\kappa U ( \\xi ^ i , x ^ i ) ) ~ \\partial _ { x ^ i } { \\rm L o g } ~ ( U ( x ^ i , \\xi ^ i ) + \\kappa I ) ~ u _ { \\xi ^ i } , \\end{array} \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\frac { d } { d t } U ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) = L ^ { \\Delta , \\sigma } _ { \\alpha _ 2 , u } U ^ \\sigma _ { \\alpha _ 2 \\alpha _ 1 } ( t ) , \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\gamma ( i , r ; j , s ) = \\widetilde { c } _ { j i } ( - r _ j - r + s ) + \\widetilde { c } _ { j i } ( r _ j + r - s ) - \\widetilde { c } _ { j i } ( r _ j - r + s ) - \\widetilde { c } _ { j i } ( - r _ j + r - s ) . \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} & x _ i : = x ^ { r _ i } , \\displaystyle [ n ] _ { x } : = \\frac { x ^ n - x ^ { - n } } { x - x ^ { - 1 } } \\ n \\in \\mathbb { Z } , \\\\ & \\left [ \\begin{array} { c } n \\\\ k \\end{array} \\right ] _ { x } : = \\begin{cases} \\displaystyle \\frac { [ n ] _ { x } [ n - 1 ] _ { x } \\cdots [ n - k + 1 ] _ { x } } { [ k ] _ { x } [ k - 1 ] _ { x } \\cdots [ 1 ] _ { x } } & n \\in \\mathbb { Z } , k \\in \\mathbb { Z } _ { > 0 } , \\\\ 1 & n \\in \\mathbb { Z } , k = 0 , \\end{cases} \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + { \\rm c o n s t } ) } \\ , { \\bf E } \\bigl [ M ^ q _ { ( \\tau = 1 , \\lambda , \\lambda ) } \\bigr ] , \\ ; N \\rightarrow \\infty . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} \\psi _ 3 ( [ b , g ] ) & = \\big ( [ a , a ^ { i _ 0 } b ^ { i _ 1 } a ^ { 2 i _ 2 } ] , [ a ^ 2 , a ^ { 2 i _ 0 + i _ 1 } b ^ { i _ 2 } ] , [ b , b ^ { i _ 0 } a ^ { 2 i _ 1 + i _ 2 } ] \\big ) \\\\ & = \\big ( [ a , a ^ { i _ 0 } b ^ { i _ 1 } a ^ { 2 i _ 2 } ] , \\ b _ 2 ^ { - i _ 2 } b _ 0 ^ { i _ 2 } , \\ b _ 0 ^ { - 1 } b _ { 2 i _ 1 + i _ 2 } \\big ) . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} Q ( \\theta ) = ( \\cos \\theta ) \\oplus \\left ( \\begin{array} { c c } \\frac 1 2 ( 1 + \\cos \\theta ) & - \\frac 1 2 \\sin \\theta \\\\ \\frac 1 2 \\sin \\theta & \\frac 1 2 ( 1 + \\cos \\theta ) \\end{array} \\right ) ^ { \\oplus ( d - 2 ) } \\oplus I _ { ( d - 2 ) ( d - 3 ) / 2 } . \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} \\tau _ g ( m _ { v , v } ) = \\frac { 1 } { d _ { \\pi } } \\overline { \\tau _ g ( [ \\pi ] ) } . \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} A _ { k , n , m } = \\left \\{ \\begin{array} { c c c } \\alpha ^ { k } \\gamma ^ { * n } \\gamma ^ { m } , & \\\\ \\alpha ^ { * ( - k ) } \\gamma ^ { * n } \\gamma ^ { m } , & \\end{array} \\right \\} \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\nu \\beta _ 1 \\nu ' ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ) p ^ { 1 / 2 } \\int _ { \\Z _ p } \\psi \\left ( \\frac { z ( 2 x - \\gamma z ) } { p } \\right ) \\left ( \\int _ { \\Q _ p } \\psi ( - y ( 2 z + \\gamma ^ { - 1 } y p ) ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) d y \\right ) d z . \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} \\phi u = T _ u \\phi + T _ { \\phi } u + R ( u , \\phi ) , \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} T ^ N ( u ^ n ) & \\in W , \\ n = 1 , \\dots , m - 1 \\\\ T ^ N ( u ^ m ) & \\in V . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} \\mathcal O ^ * = \\sigma \\ ( \\pi ' , \\ , R _ \\alpha \\circ \\pi , \\ , Z \\circ \\pi ; \\alpha \\in \\ker A \\ ) . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} \\int _ X \\varphi _ k d \\lambda = a _ k ^ 2 \\varepsilon _ k \\frac { 1 } { 2 ^ k } \\le \\frac { 1 } { 2 ^ k } . \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} \\vert \\Delta _ { I , J } \\vert ^ 2 = P _ { I , J } \\left ( \\Delta _ i ^ { ( k ) } , \\overline { \\Delta } _ i ^ { ( k ) } \\right ) _ { i , k } . \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} \\stackrel { * } { \\tau } = \\frac { 1 } { m ( m + 1 ) } \\sum \\limits _ { i , j = 1 } ^ { m + 1 } \\stackrel { * } { R } ( E _ i , E _ j , E _ j , E _ i ) . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} \\mathrm { T r } ( h ) = \\langle h , \\Lambda \\rangle , & & \\forall h \\in H _ n , \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} { \\mathcal { M } } ^ 2 = \\begin{pmatrix} a ^ 2 + | { \\bf a } | ^ 2 & a { } ^ t { \\bf a } + { } ^ t { \\bf a } K \\\\ a { \\bf a } + K { \\bf a } & { \\bf a } \\otimes { \\bf a } + K ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} m _ \\Sigma ( h ) = \\mathbb { E } [ h ( Z ) ] , ~ Z \\sim N ( 0 , \\Sigma ) , \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} ( v _ t ^ 2 , m _ t ) + \\int _ 0 ^ t 2 ( | \\nabla v _ r | ^ 2 , m _ r ) d r = ( v _ 0 ^ 2 , m _ 0 ) - 2 \\int _ 0 ^ t ( F ( u ^ 1 _ r ) - F ( u ^ 2 _ r ) , \\nabla ( v _ r m _ r ) ) d r . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( N ) } = \\arg \\min \\left \\{ d _ { K } ( \\mathbb { P } _ { n } ^ { ( N - 1 ) } \\mid \\mid Q ) : Q ( \\mathcal { A } ^ { ( N ) } ) = P ( \\mathcal { A } ^ { ( N ) } ) , ( Q ) = \\left \\{ X _ { 1 } , . . . , X _ { n } \\right \\} \\right \\} . \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} J _ { \\varphi } ^ { \\prime } = \\frac { 1 } { v _ { 0 } ^ { 2 } } ( 2 \\varphi _ { 2 } - v _ { 1 } J _ { \\varphi } ) , J _ { \\theta } ^ { \\prime } = \\frac { 1 } { v _ { 0 } ^ { 2 } } ( 2 \\theta _ { 2 } - v _ { 1 } J _ { \\theta } ) \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\psi ( b a b a ) = M _ { \\epsilon } ^ { - 1 } \\Gamma _ 0 M _ { \\epsilon } \\cdot \\Gamma _ i \\cdot M _ { \\epsilon } ^ { - 1 } \\Gamma _ 0 M _ { \\epsilon } \\cdot \\Gamma _ i = M _ { \\epsilon } ^ { - 1 } \\Gamma _ 0 \\Gamma _ i \\Gamma _ 0 \\Gamma _ i M _ { \\epsilon } . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} \\left | \\left \\langle F \\left ( \\cdot , s ; e ^ { \\rho _ { \\varepsilon } \\left ( T - s \\right ) } v _ { n } ^ { \\varepsilon } \\right ) - F \\left ( \\cdot , s ; e ^ { \\rho _ { \\varepsilon } \\left ( T - s \\right ) } w _ { n } ^ { \\varepsilon } \\right ) , \\phi _ { j } \\right \\rangle \\right | \\le C L _ { F } e ^ { \\rho _ { \\varepsilon } \\left ( T - s \\right ) } \\sum _ { k = 1 } ^ { n } \\left | V _ { k } ^ { \\varepsilon } - W _ { k } ^ { \\varepsilon } \\right | , \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{gather*} L ( f _ \\xi , s ) = L \\ ( \\xi , s - \\frac { k - 1 } { 2 } \\ ) . \\end{gather*}"}
-{"id": "625.png", "formula": "\\begin{align*} L ^ T B L = \\begin{bmatrix} a & 0 \\\\ 0 & C \\end{bmatrix} , \\quad C = B ' - a ^ { - 1 } u u ^ T . \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{gather*} F ( V _ N ) \\cdots F ( V _ 0 ) = F _ { N + 1 } ( U _ { N + 1 } ) \\cdots F _ 1 ( U _ 1 ) , \\\\ F ( V _ N ' ) \\cdots F ( V _ 0 ' ) = F _ { N + 1 } ( U _ { N + 1 } ' ) \\cdots F _ 1 ( U _ 1 ' ) . \\end{gather*}"}
-{"id": "6914.png", "formula": "\\begin{align*} \\zeta ( t ) : = ( T - t ) ^ { - \\alpha } [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } } \\ , , \\eta ( t ) : = [ - \\log ( T - t ) ] ^ { - \\beta } \\ , \\ , \\ ; \\ ; \\ ; t \\in [ 0 , T ) \\ , , \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} ( a b ^ 2 v ) ^ 3 = b _ 2 ^ 2 b _ 1 ^ 2 b _ 0 ^ 2 v ^ { a ^ 2 } v ^ { a } v . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} b _ n ( m ) = \\dfrac { s ( m + n , m ) } { \\binom { m + n } { n } } . \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} \\| \\tau _ i ( T ) \\| _ { S _ 1 } = ( \\tau _ i ( | T | ) ) = ( | T | ) = \\| T \\| _ { S _ 1 } . \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\partial _ t \\omega ( t ) ^ n = - ( S ( t ) + n ) \\omega ( t ) ^ n , \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} \\lvert \\Phi \\rvert = m - c \\rho ^ { - 1 } + o ( \\rho ^ { - 1 } ) \\quad , \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\theta ( y ) : = \\left ( \\frac { r } { 2 } - d ( x , y ) \\right ) ^ 3 e ( A , \\Phi ) ( y ) . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} & = \\alpha L ^ 2 \\sum _ { L ^ 2 \\leqslant p \\leqslant \\exp ( \\log ^ 2 L ) } \\frac { \\omega ( p ) } { p \\log p } + O \\bigg ( \\alpha ^ 2 L ^ 2 \\sum _ { L ^ 2 \\leqslant p \\leqslant \\exp ( \\log ^ 2 L ) } \\frac { \\omega ( p ) } p \\bigg ) \\\\ & \\leqslant \\alpha a _ { \\omega } \\cdot \\frac { L ^ 2 } { 2 \\log L } + O _ { \\omega } \\left ( \\alpha \\frac { L ^ 2 } { \\log ^ 2 L } + \\alpha ^ 2 L ^ 2 \\log \\log L \\right ) , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } i \\displaystyle \\frac { \\partial \\psi } { \\partial t } = ( - \\Delta ) ^ { s _ { 1 } } \\psi + V _ { 1 } ( x ) \\psi - f _ { 1 } ( \\psi ) - \\lambda ( x ) \\phi , & x \\in \\mathbb { R } ^ { N } , \\ t \\geq 0 , \\\\ i \\displaystyle \\frac { \\partial \\phi } { \\partial t } = ( - \\Delta ) ^ { s _ { 2 } } \\phi + V _ { 2 } ( x ) \\phi - f _ { 2 } ( \\phi ) - \\lambda ( x ) \\psi , & x \\in \\mathbb { R } ^ { N } , \\ t \\geq 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\log | \\phi ( z ) | & = \\Re e \\big ( h ( z ) \\big ) \\\\ & = \\Re e ( a _ 0 ) + \\left ( \\Re e ( a _ 1 ) x - \\Im m ( a _ 1 ) y \\right ) \\\\ & \\quad + \\left ( \\Re e ( a _ 2 ) \\left ( x ^ 2 - y ^ 2 \\right ) - 2 \\Im m ( a _ 2 ) x y \\right ) + o \\left ( x ^ 2 + y ^ 2 \\right ) \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\sum _ { d = 1 } ^ { \\infty } \\gamma _ j ^ { \\frac { 1 } { \\alpha - 2 \\delta } } b ^ { w _ d } < \\infty . \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} N ( u ) ( t ) = f ( t , u , \\dot { u } ) . \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} E _ { \\mu _ 2 } ( \\xi , \\nu ) = \\int _ { \\mathbb R } \\frac { ( 1 - \\gamma ) } { 2 } \\xi J _ c \\xi + \\frac 1 2 \\nu \\mathcal L _ { \\mu _ 2 } v - \\omega \\xi J _ b \\nu d x \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) ( 2 \\pi ) ^ { - \\frac { q } { \\tau } } \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) ( 2 \\pi ) ^ { - q } \\times \\\\ & \\times \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} u _ \\delta ( t ) = \\sigma _ { 0 } + \\beta t , \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} M ^ { n - 1 } ( t , x ) : = \\frac { \\langle w \\rangle _ { f ^ { n - 1 } _ t , \\varphi } } { \\langle 1 \\rangle _ { f ^ { n - 1 } _ t , \\varphi } } = \\frac { 1 } { \\rho ^ { n - 1 } ( t , x ) } \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } y \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } w \\ , f ^ { n - 1 } _ t ( y , w ) \\ , \\varphi ( x - y ) \\ , w \\ , , \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} J _ { c _ j } ( k , \\ell ) \\coloneqq \\begin{cases} 1 & c _ j | k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } J _ { 0 } ( x ) \\exp ( i x \\cosh \\beta ) x d x = - \\frac { \\cosh \\beta } { \\sinh \\beta } . \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\Delta _ { l } ^ { k } = \\mathbf { 0 } _ { m _ { l } - 1 , m _ { l } - 1 } , l = 1 , 2 . \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} ( \\Omega ^ 1 _ { A / k } ) _ { \\rm { t o r s } } \\ , : = \\ , \\left \\{ \\ , m \\in \\Omega ^ 1 _ { A / k } \\ , \\vert \\ , { \\rm { t h e r e \\ , i s \\ , a \\ , r e g u l a r \\ , e l e m e n t } } \\ , a \\in A \\ , { \\rm { s u c h \\ , t h a t } } \\ , a m = 0 \\ , \\right \\} . \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} X _ { t \\wedge \\tau _ m } = S _ { t \\wedge \\tau _ m } X _ 0 + \\left ( G _ 1 b ^ { n } ( X _ { \\cdot \\wedge \\tau _ m } ) \\right ) ( t \\wedge \\tau _ m ) + \\tfrac { \\sin ( \\pi \\theta ) } { \\pi } \\big ( G _ \\theta Y ^ { n , m , \\theta } \\big ) ( t \\wedge \\tau _ m ) , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} R ^ D f ( x ) = \\int _ { D \\backslash \\{ x \\} } G ^ D ( x , y ) f ( y ) d y = \\int _ 0 ^ \\infty P _ t ^ D f ( x ) d t . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\sigma _ { p } ^ + \\cup \\sigma _ { 1 ^ m } ^ + = \\begin{cases} \\sigma _ { p + 1 , 1 ^ { m - 1 } } ^ + & \\\\ \\sigma _ { p + 1 , 1 ^ { m - 1 } } ^ + + \\sigma _ { 2 n + 2 - m , 1 ^ { p - 2 n - 2 + 2 m } } ^ + & \\end{cases} \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} P \\in B l _ { t _ { i - 1 } } ( M _ { i - 1 } ) , Q \\in B l _ { \\overline { r _ i } } ( M _ i ^ j ) \\Rightarrow P \\prec Q ( j = 1 , \\cdots , k ) . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} E _ { a , b } ^ { ( M ) } E _ { c , d } ^ { ( N ) } = \\begin{cases} E _ { a , d } + q _ c ^ { - 1 } E _ { c , d } E _ { a , b } , & M = N = 1 ; \\\\ \\sum _ { t = 0 } ^ { \\min ( M , N ) } q _ { b } ^ { - ( N - t ) ( M - t ) } E _ { c , d } ^ { ( N - t ) } E _ { a , d } ^ { ( t ) } E _ { a , b } ^ { ( M - t ) } , & . \\end{cases} \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} [ \\imath _ S ^ { ! * } \\beta ' ] = [ \\beta ' ] - \\jmath _ { ! * } \\jmath ^ * [ \\beta ' ] = [ \\beta ] - \\jmath _ { ! * } \\jmath ^ * [ \\beta ] = [ \\imath _ S ^ { ! * } \\beta ] \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} F _ { 8 , 1 7 } = \\frac { 2 0 2 5 } { 2 } \\nu ( \\xi _ 7 ) \\chi _ 5 ^ 3 , F _ { 8 , 2 3 } = - \\frac { 3 8 2 7 2 5 } { 3 2 } \\nu ( \\xi _ 8 ) \\chi _ 5 ^ 3 \\ , . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} t _ { i , n } = t _ { i - 1 , n } + \\theta ^ n _ { t _ { i - 1 , n } } \\varepsilon _ { i , n } \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} | J _ 1 | \\le & C \\sum _ { j = 1 } ^ k U _ j \\\\ \\le & \\frac { C } { \\lambda ^ { \\frac { N - 2 } 2 } | y - \\xi _ 1 | ^ { N - 2 } } + \\frac { C } { \\lambda ^ { \\frac { N - 2 } 2 } | y - \\xi _ 1 | ^ { N - 2 - \\tau } } \\sum _ { j = 2 } ^ k \\frac 1 { | \\xi _ j - \\xi _ 1 | ^ \\tau } \\\\ \\le & \\frac { C } { \\lambda ^ { \\frac { N - 2 } 2 - \\tau } | y - \\xi _ 1 | ^ { N - 2 - \\tau } } . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} L _ 1 & : = t _ { \\alpha _ 1 } ^ { - 1 } t _ { \\delta _ 1 } ^ { - 1 } t _ { \\gamma _ 1 } ^ { - 1 } t _ { \\beta _ 1 } ^ { - 1 } t _ { x _ 1 } t _ { y _ 1 } t _ { z _ 1 } , \\\\ L & : = t _ { d _ 1 } t _ y t _ z t _ \\delta ^ { - 1 } t _ \\gamma ^ { - 1 } t _ { d _ 2 } ^ { - 1 } t _ { s _ { 1 , 1 } } ^ { - 1 } . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t \\omega ( t ) & = - R i c ( \\omega ( t ) ) - \\omega ( t ) \\\\ \\omega ( 0 ) & = \\omega _ 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\theta _ n ( \\vec { p } ) : = \\mathbb { P } _ { \\Lambda _ n , w , \\beta , H } ( 0 \\overset { \\mathbb { Z } ^ d } { \\longleftrightarrow } \\partial _ { i n } \\Lambda _ n ) . \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} p _ H ( x , y ) = G A _ { x } \\ 1 _ { \\{ y \\} } ( e ) ~ = ~ \\begin{cases} \\sum _ { z \\in E } G ( e , z ) p ( z \\star x , y ) & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} & x ^ 2 - m x y + y ^ 2 - ( a ^ 2 - m a b + b ^ 2 ) \\\\ & = m j ( m j u ^ 4 - m ^ 2 j u ^ m v + m j u ^ 2 v ^ 2 - m u ^ 2 + 2 u v ) a ^ 2 \\\\ & + m j ( m j u ^ 2 v ^ 2 - m ^ 2 j u v ^ m + m j v ^ 4 - 2 u v + m v ^ 2 ) b ^ 2 \\\\ & - 2 m j ( m j u ^ m v - m ^ 2 j u ^ 2 v ^ 2 + m j u v ^ m - u ^ 2 + v ^ 2 ) a b \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} D + D '' = \\partial F _ 1 , D '' + D ' = \\partial \\left ( F _ 2 + \\cdots + F _ k \\right ) \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\chi _ { 1 } = \\frac { q ^ { 2 } } { 1 - q ^ { 2 } } ( f _ { 1 } - \\epsilon ) \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} M ( \\vec x ^ * ) : = \\{ \\vec x \\ , \\ , \\ , | \\ , \\ , \\ , \\| \\vec x \\| _ \\infty = \\| \\vec x ^ * \\| _ \\infty , \\ ; \\ ; \\ ; S ^ \\pm ( \\vec x ^ * ) \\subset S ^ \\pm ( \\vec x ) \\} \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\begin{aligned} & x ( 1 ) = v , \\\\ & w ( t ) = 0 , t \\geq 1 \\\\ & a _ i ( t ) = \\begin{cases} - C _ i x ( t ) & i \\in \\mathcal K _ 1 \\\\ 0 & \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} [ y ^ U ] _ \\beta = \\begin{pmatrix} J _ { l _ 4 , 4 } \\\\ & J _ { l _ 3 , 3 } \\\\ & & J _ { l _ 2 , 2 } \\\\ & & & I _ { l _ 1 } \\end{pmatrix} \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\left \\langle \\mathbf { K } \\left [ 1 - e ^ { - h _ p } \\right ] ( e _ n ) , e _ n \\right \\rangle = \\int _ { E \\times E } \\sqrt { 1 - e ^ { - h _ p ( x ) } } K ( x , y ) \\sqrt { 1 - e ^ { - h _ p ( y ) } } e _ n ( x ) e _ n ( y ) d x d y \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d p ( s ) = & - \\left \\{ g _ { x } ( s ) + g _ { y } ( s ) p ( s ) + g _ { z } ( s ) K _ { 1 } ( s ) + b _ { x } ( s ) p ( s ) + b _ { y } ( s ) p ^ { 2 } ( s ) \\right . \\\\ & \\left . + b _ { z } ( s ) K _ { 1 } ( s ) p ( s ) + \\sigma _ { x } ( s ) q ( s ) + \\sigma _ { y } ( s ) p ( s ) q ( s ) + \\sigma _ { z } ( s ) K _ { 1 } ( s ) q ( s ) \\right \\} d s \\\\ & + q ( s ) d B ( s ) , \\\\ p ( T ) = & \\phi _ { x } ( \\bar { X } ( T ) ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = N _ 1 } ^ { N _ 2 - 1 } \\big | H _ { n + 1 } f - H _ n f \\big | \\Big \\| _ { \\ell ^ p } \\leq C _ p N _ 1 ^ { - k } \\big ( \\vartheta _ B ( N _ 2 ) - \\vartheta _ B ( N _ 1 ) \\big ) \\| f \\| _ { \\ell ^ p } . \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\mathbf { c } ^ T = \\left [ \\mathbf { c } _ 1 ^ T , \\cdots , \\mathbf { c } ^ T _ { | \\mathcal { I } ^ - | } \\right ] , ~ ~ ( \\mathbf { v } ^ { - } ) ^ T = \\left [ \\mathbf { v } _ 1 ^ T , \\cdots , \\mathbf { v } ^ T _ { | \\mathcal { I } ^ - | } \\right ] , ~ ~ ( \\mathbf { v } ^ { + } ) ^ T = \\left [ \\mathbf { v } _ { \\abs { \\mathcal { I } ^ - } + 1 } ^ T , \\mathbf { v } _ { \\abs { \\mathcal { I } ^ - } + 2 } ^ T \\cdots , \\mathbf { v } ^ T _ { | \\mathcal { I } | } \\right ] . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\partial _ x F ( u ) : = u \\partial _ x f ( u ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; F ( 0 ) = 0 . \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} V ( x ) ~ = ~ \\lim _ { n \\to \\infty } { \\P _ x ( \\tau _ \\vartheta > n ) } / { \\P _ 0 ( \\tau _ \\vartheta > n ) } ~ = ~ { \\P _ x ( \\tau _ \\vartheta + \\infty ) } / { \\P _ 0 ( \\tau _ \\vartheta = \\infty ) } , \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} \\epsilon = \\begin{cases} ( d , p ^ n ) _ p & c = 0 , \\\\ ( c , d p ^ n ) _ p & c d \\neq 0 \\mathrm { o r d } _ p ( c ) , \\\\ ( c , p ^ n ) _ p & . \\end{cases} \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} E ( x , \\zeta ) = e ^ { i ( x _ 1 \\zeta _ 1 + x _ 2 \\zeta _ 2 ) - x _ 0 | \\zeta | _ { \\C } } \\left ( 1 + \\frac { i ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) } { | \\zeta | _ { \\C } } \\right ) . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} h = h ( 0 ) V + \\tilde { h } \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} u _ \\gamma ( x ) = \\begin{cases} u _ i & p _ \\gamma ( x ) \\in Q _ i ^ \\gamma 1 \\leq i \\leq d , \\\\ \\frac 1 \\gamma \\left ( p _ \\gamma ( x ) - \\tfrac \\alpha 2 ( u _ i + u _ { i + 1 } ) \\right ) & p _ \\gamma ( x ) \\in Q _ { i , i + 1 } 1 \\leq i < d . \\end{cases} \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} f ( x , y ) + f ( y , x ) & = \\Big \\langle \\dfrac { y } { 1 + \\| y \\| ^ 2 } - \\dfrac { x } { 1 + \\| x \\| ^ 2 } , x - y \\Big \\rangle \\\\ & = \\bigg \\langle \\dfrac { \\sqrt { 2 t } } { 2 } - \\dfrac { 2 \\sqrt { 3 t } } { 5 } , \\sqrt { 3 t } - \\sqrt { 2 t } \\bigg \\rangle \\\\ & = \\dfrac { 1 } { 2 } \\bigg ( \\dfrac { \\sqrt { 2 } } { 2 } - \\dfrac { 2 \\sqrt { 3 } } { 5 } \\bigg ) ( \\sqrt { 3 } - \\sqrt { 2 } ) > 0 . \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} \\tilde S _ 2 \\tilde U & = S _ 2 S _ 1 S _ 1 ^ * U ^ { - 2 } + S _ 1 ^ 2 S _ 2 ^ * U ^ { - 2 } \\\\ & = S _ 2 U ^ { - 2 } S _ 1 S _ 1 ^ * + S _ 1 U S _ 2 S _ 2 ^ * U ^ { - 2 } \\\\ & = S _ 2 U ^ { - 2 } U S _ 2 S _ 1 ^ * + S _ 1 U U ^ { - 2 } S _ 2 S _ 2 ^ * \\\\ & = S _ 2 U ^ * S _ 2 S _ 1 ^ * + S _ 1 U ^ * S _ 2 S _ 2 ^ * \\\\ & = U ^ { - 2 } S _ 2 S _ 2 S _ 1 ^ * + U ^ { - 2 } S _ 1 S _ 2 S _ 2 ^ * \\\\ & = U ^ { - 2 } \\tilde S _ 1 \\ ; . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} \\overline { V } ( p , f , \\pi _ n ) _ T \\overset { \\mathbb { P } } { \\longrightarrow } \\kappa _ { p / 2 } \\int _ 0 ^ T m _ { c _ s } ( f ) ( \\theta _ s ) ^ { p / 2 - 1 } d s = \\int _ 0 ^ T m _ { c _ s } ( f ) d G _ p ( s ) \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} O _ F ( u ) = ( t _ i , t _ { i + 1 } ) _ { \\infty } . \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} a _ { u } ( x , y ) = \\begin{cases} p ( x \\star u , y ) - p ( x , y \\star u ^ { - 1 } ) , & \\\\ p ( x \\star u , y ) & \\end{cases} \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} P _ 0 \\left ( \\sup _ { 0 \\leq s \\leq T } X _ s \\geq b \\middle | X _ T = a \\right ) = \\exp \\left ( - \\frac { 2 b ( b - a ) } { T } \\right ) \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} \\pi _ I ( U ( m + k \\chi ^ J , n + k \\chi ^ J ) ) = \\pi _ I ( U ( m , n ) ) , \\quad \\forall k \\in \\N , \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} A G ^ T - B C ^ T = I , \\ , \\ , A B ^ T - B A ^ T = 0 , \\ , \\ , C G ^ T - G C ^ T = 0 . \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} \\langle u _ k ' , v _ k ' \\rangle - u _ k ' ( 0 ) \\overline { v _ k ( 0 ) } = \\langle i s \\ , f _ k + g _ k - p _ k ( s ) ^ 2 u _ k , v _ k \\rangle . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} \\varphi _ 0 ( e ) = \\frac { 1 } { 2 4 } \\sum _ { g \\neq I } \\sum _ { 0 \\leq a , b < p ^ e } ( \\xi _ { g , e , 1 } ) ^ { a } ( \\xi _ { g , e , 2 } ) ^ { b } , \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} \\xi _ { p , i } ( R ) : = R ^ { 2 p + 2 } \\chi _ i ( R ) + \\tilde \\xi _ { p , i } ( R ) \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} { \\bf G } _ { A } = { \\bf I } + { \\bf \\Sigma } _ { A } ^ { - 1 } { { \\bf \\Sigma } _ { { A } { { A } ^ c } } } { \\bf \\Sigma } _ { { A } { { A } ^ c } } ^ { T } { \\bf \\Sigma } _ { A } ^ { - 1 } , \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} & \\partial _ x B = 0 \\ , , \\\\ & \\partial _ t B + 2 B \\partial _ x A + A \\partial _ x B - 2 \\sigma + 2 B = 0 \\ , , \\\\ & \\partial _ t A + A B \\partial _ x A - G ( M ) + A = 0 \\ , , \\\\ & - \\frac { \\int _ { \\mathbb { T } } \\mathrm { d } x \\ , B ^ { - 1 / 2 } \\partial _ t B } { 2 \\int _ { \\mathbb { T } } \\mathrm { d } x \\ , \\sqrt B } + \\frac { \\sigma } { B } - 1 = 0 \\ , . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} \\P _ { \\alpha _ \\infty } ( \\tau _ \\infty > t ) = e ^ { - \\lambda _ \\infty t } , \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} ( 1 - \\frac { \\epsilon } { 3 } ) ^ { 2 } P ( V _ { n } = k ) \\leq P ( \\tilde { V } _ { n } = k ) \\leq ( 1 + \\frac { \\epsilon } { 3 } ) ^ { 3 } P ( V _ { n } = k ) . \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\lambda _ n ( c ) \\sim \\left ( \\frac { e c } { 4 ( n + \\frac 1 2 ) } \\right ) ^ { 2 n + 1 } = \\lambda _ n ^ W ( c ) . \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} i \\mathcal { G } _ { \\alpha \\beta } ( x , y ) = \\sum _ { m = 0 } ^ { \\infty } \\left ( - \\frac { i } { \\hbar } \\right ) ^ { m } \\frac { 1 } { m ! } \\int _ { - \\infty } ^ { \\infty } d t _ { 1 } \\cdots \\int _ { - \\infty } ^ { \\infty } d t _ { m } \\langle \\phi _ { 0 } | T [ \\hat { H } _ { 1 } ( t _ { 1 } ) \\cdots \\hat { H } _ { 1 } ( t _ { m } ) \\hat { \\psi } _ { \\mathrm { H } \\alpha } ( x ) \\hat { \\psi } _ { \\mathrm { H } \\beta } ^ { \\dagger } ( y ) ] | \\phi _ { 0 } \\rangle _ { \\mathrm { c o n n e c t e d } } . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} \\psi ^ { - 1 } \\sigma ( \\exp L _ \\alpha \\hat \\iota _ z : \\alpha \\in \\mathbf R ^ { J ( z ) } \\cap \\ker A ) = \\sigma ( R _ \\alpha \\iota _ z : \\alpha \\in \\mathbf R ^ { J ( z ) } \\cap \\ker A ) \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} \\hat { s } _ { i , t } = ( y _ { i , t } / d ) , \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\mathbb E \\left [ x ^ T ( t ) P ^ { - 1 } x ( t ) \\right ] = & 2 \\int _ 0 ^ t \\mathbb E \\left [ x ^ T ( s ) P ^ { - 1 } \\left ( A x ( s ) + B u ( s ) + \\sum _ { k = 1 } ^ m N _ k x ( s ) u _ k ( s ) \\right ) \\right ] d s \\\\ & + \\int _ 0 ^ t \\mathbb E \\left [ x ^ T ( s ) \\sum _ { i , j = 1 } ^ v H _ i ^ T P ^ { - 1 } H _ j k _ { i j } x ( s ) \\right ] d s . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} J _ { 1 } & = 2 u _ { 0 } - \\mathfrak { S } _ { 0 } J _ { 2 } = \\frac { 2 u _ { 1 } - \\mathfrak { S } _ { 1 } } { \\mathfrak { S } _ { 0 } } \\\\ J _ { 3 } & = \\frac { 2 u _ { 1 } } { w _ { 0 } } J _ { 4 } = \\frac { - K _ { 1 } } { 2 K _ { 0 } w _ { 0 } } \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} [ { \\mathbb D } , { \\mathbb D } ] _ A { } ^ { B } = ( \\mathrm { t r } ( { \\mathbb D } \\wedge { \\mathbb D } ) ) _ A { } ^ { B } = { \\mathbb D } ^ { B } { } _ { R } { \\mathbb D } ^ { R } { } _ { A } - \\ , { \\mathbb D } ^ { R } { } _ { A } { \\mathbb D } ^ { B } { } _ { R } , \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} y _ i & = \\left ( 1 - \\frac { 1 } { \\theta _ { i , N } } \\right ) x _ { i - 1 } + \\frac { 1 } { \\theta _ { i , N } } x _ 0 \\\\ d _ i & = \\left ( 1 - \\frac { 1 } { \\theta _ { i , N } } \\right ) f ' ( x _ { i - 1 } ) + \\frac { 2 } { \\theta _ { i , N } } \\sum _ { j = 0 } ^ { i - 1 } \\theta _ { j , N } f ' ( x _ j ) \\\\ x _ { i } & = y _ i - \\frac { 1 } { L } d _ i \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\dd { x } { t } = Y A _ k \\ , x ^ { \\tilde { Y } } , \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} A _ 1 ( n ) = - p ^ { 1 - 3 n / 2 } ( - 1 ) ^ n \\chi _ { \\psi } ( p ^ n ) \\int _ { \\mathcal A _ 1 ( n ) } ( d , p ^ n ) _ p \\chi _ { \\psi } ( d ) e ( h ) \\chi _ { \\delta } ( d ) | d | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h , \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} \\hat { \\omega } ( g , h ) \\hat { \\phi } = ( \\omega ( g , h ) \\phi ) { \\hat { } } . \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} \\det \\ , M ^ T A M \\ge \\prod _ { j = 1 } ^ { k } \\ , d _ j ^ { \\prime \\ , 2 } \\ge \\prod _ { j = 1 } ^ { k } \\ , d _ j ^ 2 ( A ) . \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} { \\bf E } [ M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } ^ { - n } ] = \\prod \\limits _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma ( 2 + \\lambda _ 1 + j ) \\ , \\Gamma ( 2 + \\lambda _ 2 + j ) } { \\Gamma ( 2 + \\lambda _ 1 + \\lambda _ 2 + j ) \\ , \\Gamma ( 1 + j ) } . \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} \\log { \\bf { E } } \\Bigl [ \\Bigl ( \\int _ \\mathcal { D } \\varphi ( x ) \\ , M _ { \\beta } ( d x ) \\Bigr ) ^ n \\Bigr ] = n \\log \\bar { \\varphi } + \\sum \\limits _ { p = 1 } ^ \\infty c _ p ( n ) \\mu ^ p , \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} S : = \\{ \\gamma \\in \\Gamma : \\gamma Y _ 0 \\cap Y \\neq \\emptyset \\} S _ 0 : = \\{ \\gamma \\in \\Gamma \\mid \\gamma Y _ 0 \\cap Y _ 0 \\neq \\emptyset \\} . \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} Y _ \\varepsilon ( t ) ( \\omega ) = A _ \\varepsilon ( \\omega ) + \\int _ 0 ^ t \\nabla _ x b ( s , \\omega , X ( s ) ( \\omega ) ) Y _ \\varepsilon ( s ) ( \\omega ) d s + \\int _ 0 ^ t \\nabla _ x \\sigma ( s , \\omega , X ( s ) ( \\omega ) ) Y _ \\varepsilon ( s ) ( \\omega ) d W ( s ) , \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Z ^ d } x \\mu ( x ) ~ = ~ 0 , \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} \\mathcal { Q } _ { t } ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } ) = \\mathbb { E } _ { \\xi _ t , D _ t } \\Big [ \\mathfrak { Q } _ t ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , D _ { t - 1 } , \\xi _ t , D _ t ) | D _ { t - 1 } , \\xi _ { [ t - 1 ] } \\Big ] \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\beta _ { 2 , 2 } ( \\delta ) \\triangleq \\beta _ { 2 , 2 } \\bigl ( a = ( 1 , 1 ) , \\ , b _ 0 = \\delta , \\ , b _ 1 = b _ 2 = 1 / 2 \\bigr ) . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{cases} & \\partial _ x F _ { f _ 5 } ( u ) = u \\ , \\partial _ x f _ 5 ( u ) , \\ ; \\ ; \\ ; \\ ; \\ ; F _ { f _ 5 } ( 0 ) = 0 , \\\\ \\\\ & G _ { f _ 5 } ( u ) = \\int _ 0 ^ u f _ 5 ( s ) d s . \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} C _ { a b c d e f g j k } = C _ { a b c e d g f k j } . \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\| x \\| : = \\sum _ k \\sqrt { x _ 1 ^ 2 + \\ldots + x _ { n _ k } ^ 2 } \\bigg | A _ k \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} P _ 0 ( x ) = \\sum _ { 0 < \\abs { \\gamma } \\leq \\ell } \\xi _ \\gamma x ^ \\gamma , \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\Omega _ K : = \\left \\{ \\omega \\in \\Omega \\ , \\big | \\ , \\sup _ n \\log \\Vert M ^ n _ \\omega \\Vert \\le K \\right \\} . \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} V _ r ' ( z ) = V _ r ( z ) E ( z ) + E _ r \\frac { z ^ { r - 1 } } { ( r - 1 ) ! } . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} N _ { m } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { l = 0 } ^ { \\infty } \\delta _ { m , l + n } \\binom { m } { n } N _ { \\mathrm { c } \\ , l } N _ { \\mathrm { d } n } , \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} X ' \\cap L ( F ' _ { \\ell } ) = \\emptyset . \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} \\frac { 1 } { F ( z ) } = R ( z ) + \\sum _ n \\frac { 1 } { F ' ( t _ n ) } \\cdot \\bigg ( \\frac { 1 } { z - t _ n } + \\frac { 1 } { t _ n } + \\cdots + \\frac { z ^ { k - 1 } } { t _ n ^ k } \\bigg ) . \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} \\lambda g = \\Delta ^ { - 1 } ( c ^ { - 2 } g ) \\ \\ \\hbox { i n } \\ \\ \\Omega \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} \\mathcal { B } = \\{ b v _ L : b \\in B \\} \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} \\dot { \\mu } _ t = L ^ * \\mu _ t , \\mu _ t | _ { t = 0 } = \\mu _ 0 , \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} & { \\rm g r a d } \\ , \\tilde { J } ( A _ r , B _ r ) = ( - 2 A _ r { \\rm s y m } ( P ^ 2 - X ^ T X ) A _ r , \\\\ & \\quad \\quad \\quad \\quad \\quad 4 P B _ r - 4 X ^ T B ) , \\\\ & { \\rm H e s s } \\ , \\tilde { J } ( A _ r , B _ r ) [ ( A ' _ r , B ' _ r ) ] \\\\ = & ( - 2 A _ r { \\rm s y m } ( P ' P + P P ' - X '^ T X - X ^ T X ' ) A _ r \\\\ & \\ , - 2 { \\rm s y m } ( A ' _ r { \\rm s y m } ( P ^ 2 - X ^ T X ) A _ r ) , \\\\ & \\ , \\ , 4 ( P ' B _ r + P B ' _ r ) - 4 X '^ T B ) , \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ N x ^ n \\bigr ) ( q \\ , | \\ , b ) = \\bigl ( \\mathcal { S } _ N x ^ n \\bigr ) ( 0 \\ , | \\ , b ) , \\ ; n \\leq N . \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } \\right \\vert _ { t = 0 } L ^ \\circ _ a ( 1 + t b ) & = \\left . \\frac { d } { d t } \\right \\vert _ { t = 0 } \\sum _ { m \\geq 0 } \\alpha _ m ( 1 + t b ) ^ { m + 1 } \\\\ & = \\sum _ { m \\geq 0 } ( m + 1 ) \\alpha _ m b = a * b = \\left . \\frac { d } { d t } \\right \\vert _ { t = 0 } L ^ * _ a ( 1 + t b ) . \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} C _ { 2 } : = \\frac { 1 } { 2 } + 2 L _ F ^ 2 . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} | q _ 1 ' - p _ 2 | & = | p _ 1 - p _ 2 | - | p _ 1 - q _ 1 ' | \\\\ & \\leqslant | u _ 1 x _ 0 - u _ 2 x _ 0 | - | v x _ 0 - x _ 0 | + 3 6 \\delta \\leqslant 3 6 \\delta , \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} M _ { } & = 2 P _ 1 ( 1 ) P _ 3 ( 1 ) \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\epsilon ( \\chi ) L \\left ( \\frac { 1 } { 2 } , \\overline { \\chi } \\right ) - 2 P _ 1 ( 1 ) P _ 3 ( 1 ) \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\sum _ n \\frac { \\chi ( n ) } { n ^ { \\frac { 1 } { 2 } } } F ( n ) , \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} \\mathbb W ^ s _ \\omega ( z ) = \\{ w \\in U _ \\omega \\ , | \\ , g ( w ) = g ( z ) \\} \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} n _ i = r _ i \\left ( q + 2 q _ 0 + 1 \\right ) + m _ i q _ 0 + s _ i \\quad r _ i = \\left \\lfloor \\frac { n _ i } { q + 2 q _ 0 + 1 } \\right \\rfloor \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} l - \\frac { k ^ { 2 } } { d + 1 } = \\frac { d + 1 } { 4 } - \\frac { d + 1 } { 4 } = 0 , \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} \\phi _ { k n } ( M ) = \\gamma _ { k n } \\ , \\Lambda ^ { - 2 } _ { n } \\chi _ { k n } ( M ) \\omega ( \\Lambda _ { n } ) , \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} B ( t , x ) : = 2 \\sqrt { \\frac 2 \\mu } \\partial _ x \\arctan \\left ( \\frac { \\mathcal G ( t , x ) } { \\mathcal F ( t , x ) } \\right ) , \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} F ( x , \\zeta ) = E ( x , \\zeta ) M ( x , \\zeta ) , \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} \\varphi _ \\psi ( f ) = \\psi ( f | _ { ( \\ker N ) \\boxtimes Y _ 0 } ) \\ \\ \\ \\ f \\in C _ c ( \\Gamma \\boxtimes Y ) \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} \\mathcal { C } _ { s } ^ { m } = \\sum _ { k = 1 } ^ { m - s } ( - 1 ) ^ { k } \\sum _ { a _ { 1 } , \\cdots , a _ { k } = 1 } ^ { \\infty } \\delta _ { a _ { 1 } + \\cdots + a _ { k } , m - s } \\frac { m ! } { ( m - a _ { 1 } - \\cdots - a _ { k } ) ! } \\prod _ { j = 1 } ^ { k } \\frac { [ 2 ( a _ { j } ) ] ! } { a _ { j } ! } , \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} L _ t ( m ) ^ T = \\sum _ { m ' : \\mathcal { Y } } a [ m ; m ' ] \\underline { m ' } \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} F ( P ) = \\widetilde { F } \\big ( | P | \\big ) \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 0 } ^ { 2 n - 1 } a _ i t ^ i \\right ) \\left ( \\sum _ { i \\geq 0 } ( - 1 ) ^ i \\delta _ i t ^ i \\right ) = 1 . \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} & \\tau _ k ( x ) - \\tau _ k ( 0 ) = \\frac { 3 } { ( 1 + x ^ 2 ) ^ 2 } ( 2 { { d } _ { k } } ( x - x ^ 3 ) - { \\omega _ { k } } x ^ 2 ) , \\\\ & \\tau ' _ k ( x ) = \\frac { 6 } { ( 1 + x ^ 2 ) ^ 3 } ( { { d } _ { k } } ( 1 - 6 x ^ 2 + x ^ 4 ) - { \\omega _ { k } } ( x - x ^ 3 ) ) , \\\\ & \\tau '' _ k ( x ) = \\frac { 6 } { ( 1 + x ^ 2 ) ^ 4 } [ 2 { { d } _ { k } } ( - 9 x + 1 4 x ^ 3 - x ^ 5 ) - { \\omega _ { k } } ( 1 - 8 x ^ 2 + 3 x ^ 4 ) ] . \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} I _ X \\ast M ( x , y ) & = \\bigvee _ { x ' \\in X } I _ X ( x , x ' ) \\otimes M ( x ' , y ) \\\\ & = \\left ( I _ X ( x , x ) \\otimes M ( x , y ) \\right ) \\vee \\left ( \\bigvee _ { x ' \\in X , x ' \\ne x } I _ X ( x , x ' ) \\otimes M ( x ' , y ) \\right ) \\\\ & = \\left ( I \\otimes M ( x , y ) \\right ) \\vee \\left ( \\bigvee _ { x ' \\in X , x ' \\ne x } 0 \\otimes M ( x ' , y ) \\right ) \\\\ & = M ( x , y ) \\vee 0 = M ( x , y ) . \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\nabla _ { U } \\zeta = - \\varphi A _ { N } U + \\tau ( U ) \\zeta , \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} f _ \\phi ( W ) = \\gamma ( M _ \\phi ( W ) ) , \\qquad \\forall \\ , W \\in \\mathcal { A } . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\Phi _ N = \\Phi ^ { \\operatorname { g s } } , \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} L ( \\pi \\otimes \\pi ' , s ) & = \\prod _ p L _ p ( \\pi \\otimes \\pi ' , s ) = \\prod _ p \\prod _ { i = 1 } ^ { d } \\prod _ { j = 1 } ^ { e } \\Bigl ( \\ , 1 - \\frac { \\alpha _ { \\pi \\otimes \\pi ' , ( i , j ) } ( p ) } { p ^ s } \\ , \\Bigr ) ^ { - 1 } \\\\ & = \\sum _ { n \\geq 1 } \\frac { \\lambda _ { \\pi \\otimes \\pi ' } ( n ) } { n ^ s } , \\ \\Re s > 1 \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} C & \\leq \\frac { 1 } { 1 - r _ 0 ^ { - 2 } } e ^ { ( \\frac { 1 } { r _ 0 - r _ 0 ^ { - 1 } } ) ^ 6 } = \\frac { 1 } { 1 - r _ 0 ^ { - 2 } } e ^ { \\frac { 1 } { ( N ^ 2 - 4 ) ^ 3 } } \\leq \\frac { 1 } { 1 - \\frac { 4 } { N ^ 2 } } e ^ { \\frac { 1 } { 1 2 5 } } \\leq \\frac { 9 } { 5 } e ^ { \\frac { 1 } { 1 2 5 } } \\leq 2 . \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ M \\frac { D _ { { \\textrm { i n } } } ( i j ) x _ { i j } } { \\eta ^ u _ i c ^ u _ i } \\leq D ^ u _ { i } , \\ \\forall i , \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q + a _ i \\ , | \\ , a , \\ , b ) = \\eta _ { M , N } ( q \\ , | \\ , a , \\ , b ) \\ , \\exp \\bigl ( - ( \\mathcal { S } _ N \\log \\Gamma _ { M - 1 } ) ( q \\ , | \\ , \\hat { a } _ i , b ) \\bigr ) . \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} \\lim _ \\ell \\ < z | \\psi _ \\ell = \\lim _ \\ell \\sum _ k \\alpha _ { \\ell k } \\ < z | u ( x + h _ { \\ell k } ) \\ > = \\lim _ \\ell \\sum _ k \\alpha _ { \\ell k } K ( z , u ( x + h _ { \\ell k } ) ) = \\lim _ \\ell A ( x ) K ( z , u ( x ) ) . \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} { \\mathcal O } \\left ( \\sum _ { j = 0 } ^ { L - 1 } 2 ^ { j } \\log 2 ^ { j } \\right ) = { \\mathcal O } \\left ( 2 ^ { L } ( L - 2 ) \\right ) . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} d \\star ( \\sigma d u ) = 0 . \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ Y ^ { q / \\tau } ( 1 / \\tau ) \\bigr ] = \\frac { \\Gamma ( 1 - q ) } { \\Gamma ( 1 - q / \\tau ) } \\ , { \\bf E } \\bigl [ Y ^ q ( \\tau ) \\bigr ] . \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} f _ \\phi ^ \\varnothing ( U ( m , n ) ) = c _ + + ( - 1 ) ^ { | m | + | n | } c _ - . \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\delta ( c \\cup d ) & = \\{ \\delta _ H ( c _ H \\cup d _ H ) \\} \\\\ & = \\{ \\delta _ H ( c _ H ) \\cup d _ H + ( - 1 ) ^ { | c _ H | } c _ H \\cup \\delta _ H ( d _ H ) \\} ~ ~ \\mbox { ( b y t h e n o n - e q u i v a r i a n t c a s e ) } \\\\ & = \\{ \\delta _ H ( c _ H ) \\cup d _ H \\} + ( - 1 ) ^ { | c _ H | } \\{ c _ H \\cup \\delta _ H ( d _ H ) \\} \\\\ & = \\delta ( c ) \\cup d + ( - 1 ) ^ { | c | } c \\cup \\delta ( d ) ~ ~ \\mbox { ( s i n c e $ ( - 1 ) ^ { | c _ H | } = ( - 1 ) ^ { | c | } $ ) } . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\Lambda ^ + ( X ) = \\Lambda _ + ( X ) , \\ ; \\ ; \\ ; \\Lambda ^ - ( X ) = \\Lambda _ - ( X ) \\overline { M } ^ + . \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} & \\langle D \\phi _ N ( \\sigma ^ N ( t ) ) , \\sigma ^ N ( t ) - \\varrho ( t ) \\rangle \\\\ & = - \\langle \\dot u _ 3 ^ N ( t ) - \\dot w _ 3 ( t ) , \\ddot u _ 3 ^ N ( t ) \\rangle + \\langle E \\dot w ( t ) , \\sigma ^ N ( t ) - \\varrho ( t ) \\rangle - \\langle \\mathbb A _ r \\dot \\sigma ^ N ( t ) , \\sigma ^ N ( t ) - \\varrho ( t ) \\rangle \\\\ & \\leq C ( \\| \\ddot u _ 3 ^ N ( t ) \\| _ { L ^ 2 } + \\| \\dot \\sigma ^ N ( t ) \\| _ { L ^ 2 } + 1 ) , \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} | f ( \\phi ( 0 ) ) | & = | f ( - r ) | \\le | f _ 0 | + | f _ + ( - r ) | + | f _ - ( - r ) | \\le | f _ 0 | + \\sum _ { n > 0 } r ^ n | f _ n | + \\sum _ { n < 0 } r ^ n | f _ n | \\\\ & \\le | f _ 0 | + r \\Big ( \\sum _ { n \\ne 0 } \\frac { r ^ { 2 ( n - 1 ) } } { \\omega ( n ) ^ 2 } \\Big ) ^ { 1 / 2 } \\Big ( \\sum _ { n \\ne 0 } | f _ n | ^ 2 \\omega ( n ) ^ 2 \\Big ) ^ { 1 / 2 } \\\\ & \\le | f _ 0 | + r \\ , d \\ , \\| f - f _ 0 \\| _ { 2 , \\omega } , \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\langle ( \\mathcal L _ { \\mu _ 2 } - | \\omega | J _ b ) \\nu , \\nu \\rangle & = \\Big ( \\frac { 1 } { \\gamma } - | \\omega | - \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } \\frac { 1 } { \\sqrt { \\mu _ 2 } } - \\eta \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } \\Big ) \\| \\nu \\| ^ 2 + ( \\alpha - \\mu b | \\omega | - \\frac { 1 } { 4 \\eta } \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } ) \\| \\nu ' \\| ^ 2 \\\\ & = r \\| \\nu \\| ^ 2 + \\theta \\| \\nu ' \\| ^ 2 , \\end{array} \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} \\widehat { I } ^ { [ \\beta ] } = \\left . \\left \\langle \\frac { x ^ { \\beta } } { m } ~ \\right | ~ m \\in \\mathcal { G } ( I ) \\right \\rangle = \\left \\langle \\frac { x ^ { \\beta } } { m _ 1 } , \\frac { x ^ { \\beta } } { m _ 2 } , \\ldots , \\frac { x ^ { \\beta } } { m _ p } \\right \\rangle ~ ~ \\mbox { w h e r e $ x ^ \\beta = x _ 1 ^ { \\beta _ 1 } \\cdots x _ n ^ { \\beta _ n } $ . } \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} ( \\Lambda _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m ) M ( \\lambda ) ( \\Lambda _ { \\epsilon } ( \\lambda ^ \\ell ) \\otimes I _ n ) = P ( \\lambda ) . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\Vert w - z \\Vert & = \\Vert ( w - z ) \\chi _ N \\Vert = \\Vert ( y - x - z ) \\chi _ N \\Vert \\\\ & = \\Vert ( x - y + z ) \\chi _ N \\Vert \\le s + t . \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} b = \\sum _ { \\tau \\in \\Sigma _ m } \\beta _ { \\tau ( 1 ) } \\otimes \\cdots \\otimes \\beta _ { \\tau ( m ) } . \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} R _ { L / K } = T _ { L / K } + C _ { L / K } + U _ { L / K } - M _ { L / K } , \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} \\phi _ n ( w , u ) = & \\displaystyle \\sum _ { i = 1 } ^ { \\infty } \\eta _ i \\left ( \\int \\limits _ T \\ell ( w ( t ) - x ^ n _ i ( t ) ) d \\mu ( t ) \\right ) + \\sum _ { i = 1 } ^ { \\infty } \\eta _ i \\ell ( u - y ^ n _ i ) \\\\ = & \\displaystyle \\int \\limits _ T \\left ( \\sum _ { i = 1 } ^ { \\infty } \\eta _ i \\ell ( w ( t ) - x ^ n _ i ( t ) ) \\right ) d \\mu ( t ) + \\sum _ { i = 1 } ^ { \\infty } \\eta _ i \\ell ( u - y ^ n _ i ) . \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} \\gamma ( x ) = \\hat { W } ^ * ( 1 \\otimes x ) \\hat { W } . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} \\sum _ { M ^ g \\in M ^ G } \\frac { | x ^ G \\cap M ^ g | } { | x ^ G | } \\frac { | y ^ G \\cap M ^ g | } { | y ^ G | } & = | G : M | \\frac { | x ^ G \\cap M | } { | x ^ G | } \\frac { | y ^ G \\cap M | } { | y ^ G | } \\\\ & = \\operatorname { f p r } ( x , M ^ G ) \\operatorname { f i x } ( y , M ^ G ) . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} & f ( u ) = u ^ 2 , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; f ( u ) = u ^ 4 , \\\\ & \\\\ & f ( u ) = u ^ 2 \\pm \\mu u ^ 3 + \\epsilon ^ 2 u ^ 4 , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; 0 \\leq \\mu < 2 \\epsilon , \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} \\psi ( ( a b ^ i ) ^ { k p } ) = \\big ( ( a ^ { \\alpha } b ) ^ { k i } u _ 1 , ( a ^ { \\alpha } b ) ^ { k i } u _ 2 , \\dots , ( a ^ { \\alpha } b ) ^ { k i } u _ p \\big ) , \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} i n d e x ( M _ k ) \\geq \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } i n d e x ( M _ k \\cap B ^ { ( N , g ) } _ { R r ^ { y , \\ell } _ k } ( p ^ { y , \\ell } _ k ) ) , \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} f _ 2 \\ , = \\ , a S T + b S ^ 2 + c T ^ 2 \\ , = \\ , ( u S + v T ) \\cdot ( u ' S + v ' T ) , \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} d _ 1 = \\overline { d _ 1 } , d _ 2 = \\overline { d _ 2 } , \\widetilde { d } _ 1 = \\overline { \\widetilde { d } _ 1 } , \\widetilde { d } _ 2 = \\overline { \\widetilde { d } _ 2 } \\ ; . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} \\big \\langle \\widetilde \\Lambda _ { v _ 2 } \\widetilde P _ { v _ 2 , \\rm R } \\widetilde F ( v _ 2 ) , \\widetilde P _ { v _ 2 , \\rm R } \\widetilde F ( v _ 2 ) \\big \\rangle = \\alpha _ { v _ 2 } | \\widetilde f ( v _ 1 ) | ^ 2 \\leq 0 \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} g \\left ( \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) z \\right ) = \\chi ( d ) ( c z + d ) ^ { \\ell } g ( z ) \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) \\in \\Gamma _ 0 ( N ) . \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} f _ i ( x ) = \\bigwedge _ { j \\in N ^ - ( i ) } x _ j , \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} U _ i ( \\lambda ) = \\begin{bmatrix} K _ i ( \\lambda ) \\\\ \\widehat { K } _ i ( \\lambda ) \\end{bmatrix} \\mbox { a n d } U _ i ( \\lambda ) ^ { - 1 } = \\begin{bmatrix} \\widehat { N } _ i ( \\lambda ) ^ T & N _ i ( \\lambda ) ^ T \\end{bmatrix} . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} M : = \\max _ { \\overline { B } _ { r / 2 } ( x ) } \\theta \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} c _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } x ( a ) + c _ 2 ~ ^ { A B R } D _ b ^ \\alpha x ( a ) = 0 , \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} { \\rm m u l t } _ \\lambda ( \\mathbf { m } ) & = \\sum _ { \\substack { \\sigma \\in S _ n \\\\ \\sigma ( 1 ) = 1 } } \\varepsilon ( \\sigma ) Q \\Big ( \\sum _ { j = 2 } ^ n ( \\ell _ 1 + 1 - \\ell _ j ) e _ j - \\sigma \\Big ( \\sum _ { j = 2 } ^ n j e _ j \\Big ) \\Big ) \\\\ & = \\sum _ { \\substack { \\sigma \\in S _ n \\\\ \\sigma ( 1 ) = 1 } } \\varepsilon ( \\sigma ) Q \\Big ( \\sum _ { j = 1 } ^ { n - 1 } ( \\ell _ 1 - \\ell _ { j + 1 } ) e _ { j + 1 } - \\sigma \\Big ( \\sum _ { j = 1 } ^ { n - 1 } j e _ { j + 1 } \\Big ) \\Big ) . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\iota _ z ^ { - 1 } \\mathcal O ^ { * * } \\subset \\left \\{ B \\in \\mathcal B ( \\Theta _ z ) \\vert G \\cdot B = B \\right \\} . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\mu ^ { ( J ) } \\coloneqq t _ 1 \\quad m ^ { ( J ) } \\coloneqq 2 \\cdot \\left ( 2 ^ { J - 1 } - t _ 1 \\right ) = 2 m , \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\sum _ { n \\leqslant N } r ( n ) ^ 2 \\omega ( n ) = \\big ( 1 + o ^ { \\star } ( 1 ) \\big ) \\prod _ p \\big ( 1 + r ( p ) ^ 2 \\omega ( p ) \\big ) , \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} \\langle \\pi _ k ( h ) w _ k , w _ k \\rangle = \\begin{cases} \\displaystyle \\frac { \\mu _ n ( h ) ^ { n k / 2 } \\ , 2 ^ { n k } } { \\det { ( A + D + i ( C - B ) ) ^ k } } & \\mu ( h ) > 0 , \\\\ [ 2 e x ] 0 & \\mu ( h ) < 0 . \\end{cases} \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} H _ N ( X ) : = \\sum _ { x = 0 } ^ N \\varphi _ { a } ( x ) + \\frac { 1 } { 2 } \\sum _ { x = 1 } ^ { N } ( X _ x - X _ { x - 1 } ) ^ 2 + \\frac { 1 } { 2 } X _ 0 ^ 2 + \\frac { 1 } { 2 } X _ N ( N ) ^ 2 . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} & \\left | \\frac { 1 } { 4 \\pi } \\int \\limits _ { \\beta _ { k n } } \\ , \\frac { \\Delta f _ 0 ( M ) } { \\rho _ { M _ 0 } ( M ) } \\ , d m _ 3 ( M ) \\right | \\\\ & \\leq C \\int \\limits _ { B _ { 2 \\Lambda _ { n - 2 } ( M _ { k , n - 2 } ) } } \\ , \\frac { \\omega ( d ( M ) ) } { d ^ { 2 } ( M ) \\cdot 2 ^ { - n + 1 } } \\ , d m _ 3 ( M ) \\\\ & \\leq C \\omega ( 2 ^ { - n + 2 } ) \\leq C \\omega ( 2 ^ { - n } ) \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} N _ { \\mathrm { c } \\ , m + 1 } = N _ { m + 1 } - N _ { \\mathrm { d } \\ , m + 1 } - \\sum _ { n = 1 } ^ { m } \\sum _ { r = 1 } ^ { n } \\binom { m + 1 } { m - n + 1 } N _ { \\mathrm { d } \\ , m - n + 1 } \\mathcal { C } _ { r } ^ { n } \\left ( N _ { r } - N _ { \\mathrm { d } r } \\right ) . \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\| f \\| _ { * * } = \\sup _ { y \\in \\R ^ { N } } \\Big ( \\sum _ { j = 1 } ^ { k } \\frac { 1 } { ( 1 + \\lambda | y - \\xi _ { j } | ) ^ { \\frac { N + 2 } 2 + \\tau } } \\Big ) ^ { - 1 } \\lambda ^ { - \\frac { N + 2 } { 2 } } | f ( y ) | , \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{gather*} F \\in C ^ 2 ( \\R ^ { N \\times n } ) \\ F ( 0 ) = 0 ; \\\\ \\nu _ 1 | P | - \\nu _ 2 \\le F ( P ) \\le \\nu _ 3 | P | + \\nu _ 4 ; \\\\ 0 \\leq D ^ 2 F ( P ) ( Q , Q ) \\leq \\nu _ 5 \\frac { 1 } { 1 + | P | } | Q | ^ 2 . \\end{gather*}"}
-{"id": "4970.png", "formula": "\\begin{align*} \\mathsf { n } : = \\gamma _ \\Delta - \\gamma _ { \\Delta _ 1 } = \\tau _ \\Delta - \\tau _ { \\Delta _ 1 } \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} \\Phi ( y ( t ) + C _ 0 ) - \\Phi ( C _ 0 ) = \\int ^ { y ( t ) + C _ 0 } _ { C _ 0 } \\frac { 1 } { g ( k ) } \\ d k \\leq \\int _ 0 ^ t \\Psi ( s ) \\ d s . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} \\Lambda _ k ( \\lambda ) ^ T : = \\begin{bmatrix} \\lambda ^ { k } & \\cdots & \\lambda & 1 \\end{bmatrix} \\in \\mathbb { F } [ \\lambda ] ^ { 1 \\times ( k + 1 ) } , \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\| \\theta \\| ^ 2 \\geq r _ { \\varepsilon _ 0 } ^ 2 \\mbox { a n d } \\mathbb { P } _ \\theta ( \\Delta _ { \\alpha , \\varepsilon _ 0 } = 0 ) > \\beta . \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} P ^ { ( A ) } _ { \\sigma ( A ) } = I , P ^ { ( A ) } _ { E \\cap F } = P ^ { ( A ) } _ E P ^ { ( A ) } _ F \\mbox { a n d } P ^ { ( A ) } _ { \\cup _ { n = 1 } ^ \\infty E _ n } x = \\sum _ { n = 1 } ^ \\infty P ^ { ( A ) } _ { E _ n } x \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} \\lim \\limits _ { | p | \\rightarrow \\infty } | f ( p ) | ^ 2 e ^ { | \\varphi ( p ) | ^ 2 - | p | ^ 2 } = 0 , \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} T _ { x , y } = \\langle T \\delta _ y , \\delta _ x \\rangle , \\quad \\forall x , y \\in X . \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} & \\sum _ { \\substack { n m > N \\\\ ( n , m ) = 1 } } \\frac { r ( n ) ^ 2 r ( m ) \\omega ( n ) \\omega ' ( m ) } { \\sqrt { m } } \\\\ & \\qquad \\leqslant N ^ { - \\alpha } \\sum _ { \\substack { n , m \\geq 1 \\\\ ( n , m ) = 1 } } \\big ( r ( n ) ^ 2 \\omega ( n ) n ^ { \\alpha } \\big ) \\big ( r ( m ) \\omega ' ( m ) m ^ { \\alpha - 1 / 2 } \\big ) \\\\ & \\qquad = N ^ { - \\alpha } \\prod _ p \\big ( 1 + r ( p ) ^ 2 \\omega ( p ) p ^ { \\alpha } + r ( p ) \\omega ' ( p ) p ^ { \\alpha - 1 / 2 } \\big ) . \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} S _ 1 & : = C _ R ( a ) = Z \\cup ( Z + a ) , \\\\ S _ 2 & : = C _ R ( b ) = C _ R ( c ) = C _ R ( b + c ) = Z \\cup ( Z + b ) \\cup ( Z + c ) \\cup ( Z + ( b + c ) ) , \\\\ S _ 3 & : = C _ R ( a + b ) = Z \\cup ( Z + ( a + b ) ) , \\\\ S _ 4 & : = C _ R ( a + c ) = Z \\cup ( Z + ( a + c ) ) \\\\ S _ 5 & : = C _ R ( a + b + c ) = Z \\cup ( Z + ( a + b + c ) ) . \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} \\mathcal E _ 2 ( \\tau ) = \\mathcal K _ { ( 1 , 3 ) } ( \\tau ) . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} C _ { n , k } = \\frac { n - k + 1 } { n + 1 } \\binom { n + k } { n } . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} \\int _ { Q ( r ) } u \\cdot \\nabla \\varphi | u | ^ 2 d x d t = \\int _ { Q ( r ) } u \\cdot \\nabla \\varphi ( | u | ^ 2 - [ | u | ^ 2 ] _ { B ( r ) } ) d x d t \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} [ x ^ U ] _ \\beta = \\begin{pmatrix} J _ { l , 2 } \\\\ & I _ { m - 2 l } \\end{pmatrix} . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} H ( \\xi ^ * \\xi , \\tau ) = \\lim _ { p \\rightarrow 1 } \\frac { p } { 1 - p } \\log ( \\left \\| \\xi ^ * \\xi \\right \\| _ p ) = \\lim _ { p \\rightarrow 2 } \\frac { 2 p } { 2 - p } \\log ( \\left \\| \\xi \\right \\| _ p ) . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\lim _ { \\rho \\rightarrow 0 } \\limsup _ { n \\rightarrow \\infty } \\mathbb { P } ( | V ( f , \\pi _ n ) _ T - V ( f _ \\rho , \\pi _ n ) _ T | > \\delta ) = 0 \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} R ^ { F ^ H } \\check { M } ^ \\beta _ A = z ^ { N - d } \\check { N } ^ \\beta _ A . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { i = 1 } ^ N m _ i ( t ) e ^ { - | x - x _ i ( t ) | } , \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} & ( \\overline { K } + \\overline { L } ~ \\overline { \\Psi } ^ { - T } ) \\mathbf { c } = \\mathbf { b } , \\\\ ~ ~ ~ ~ ~ ~ & \\mathbf { b } = \\overline { K } \\mathbf { v } ^ - - \\overline { L } ~ \\overline { \\Psi } ^ { + T } \\mathbf { v } ^ + . \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} \\Theta ( \\pi \\otimes \\chi _ D , \\psi ^ D ) = \\begin{cases} \\tilde { \\pi } ^ { \\epsilon } & L ( \\pi \\otimes \\chi _ D , 1 / 2 ) \\neq 0 , \\\\ 0 & L ( \\pi \\otimes \\chi _ D , 1 / 2 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle { \\inf } \\ ; \\mathbb { E } _ { \\xi _ 2 , \\ldots , \\xi _ { T _ { \\max } } } \\Big [ \\displaystyle { \\sum _ { t = 1 } ^ { T _ { \\max } } } \\ ; f _ { t } ( x _ { t } , x _ { t - 1 } , \\xi _ t ) \\Big ] \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) \\ ; \\mbox { a . s . } , x _ { t } \\ ; \\mathcal { F } _ t \\mbox { - m e a s u r a b l e , } t = 1 , \\ldots , T _ { \\max } , \\end{array} \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} M _ \\beta [ \\varphi ] ( \\mathcal { D } ) = \\int \\limits _ \\mathcal { D } \\varphi ( x ) \\ , M _ \\beta ( d x ) \\ , d x , \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\sum _ n h ( n ) \\frac { | \\rho _ j ( n ) | ^ 2 } { \\cosh ( \\pi t _ j ) } = \\frac { 2 } { \\zeta ( 2 ) } N + \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 / 2 ) } \\tilde { h } ( s ) \\frac { L ( u _ { j } \\otimes u _ { j } , s ) } { \\cosh ( \\pi t _ j ) } d s . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} d ( x , y ) = \\left ( d ' ( x , y ) \\wedge 1 \\right ) \\vee \\left | 1 / d ' ( x , \\mathbb { C } ) - 1 / d ' ( y , \\mathbb { C } ) \\right | \\ ; , \\ ; x , y \\in \\mathbb { X } \\ , . \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} h _ 1 = \\ , & 1 + a _ 1 ^ 4 + a _ 1 ^ 2 b + a _ 1 ^ 4 b ^ 2 + a _ 1 ^ 2 b ^ 5 + b ^ 8 + ( a _ 1 ^ 2 + a _ 1 ^ 6 + a _ 1 ^ 2 b ^ 2 + a _ 1 ^ 2 b ^ 4 + a _ 1 ^ 2 b ^ 6 ) k \\\\ & + ( a _ 1 ^ 4 + a _ 1 ^ 6 b + a _ 1 ^ 4 b ^ 4 ) k ^ 2 + ( a _ 1 ^ 6 + a _ 1 ^ 6 b ^ 2 ) k ^ 3 , \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} \\frac { n _ { l ' } } { n _ { l } } = \\frac { ( j ' + 1 ) 2 ^ { k + 1 } } { r _ { k } 2 ^ { k } } = \\frac { 2 ( j ' + 1 ) } { r _ { k } } \\le 1 + \\frac { 2 } { r _ { k } } < 1 + \\varepsilon \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { X } \\Delta _ { a } ( y ) d y & = O _ { a } \\left ( X ^ { 3 / 4 + a / 2 } \\right ) \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\Xi ' _ { \\imath } & : = \\begin{cases} \\xi ( n - 1 ) + \\xi ( n ) & \\ \\flat = > \\ \\imath = n , \\\\ 2 \\Xi _ { \\imath } & , \\end{cases} & \\Pi _ { \\imath } & : = \\begin{cases} ( \\imath , \\Xi ' _ { \\imath } ) & \\ 1 \\leq \\imath \\leq n , \\\\ ( 2 n - \\imath , \\Xi ' _ { \\imath } ) & \\ n + 1 \\leq \\imath \\leq 2 n - 1 . \\end{cases} \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} J ( A _ r , B _ r , C _ r ) : = | | G - G _ r | | _ { H ^ 2 } ^ 2 , \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} G _ p = \\left \\{ x \\in G \\colon \\mbox { $ x $ i s d i v i s i b l e a n d } \\ ; \\lim _ { n \\to \\infty } x ^ { p ^ n } = e \\right \\} \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\sigma ^ 2 = \\rho ^ { q } = 1 , \\sigma \\rho = \\rho ^ { - 1 } \\sigma . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} U _ { i , m , n } & = \\left ( 1 - \\frac { 1 } { q _ i } \\right ) ^ { - 1 } \\left ( U _ i ^ { m } ( U _ i ^ * ) ^ { n } - \\frac { 1 } { q _ i } U _ i ^ * U _ i ^ { m } ( U _ i ^ * ) ^ { n } U _ i \\right ) \\\\ & = \\begin{cases} \\left ( 1 - \\frac { 1 } { q _ i } \\right ) ^ { - 1 } \\left ( U _ i ^ { m } ( U _ i ^ * ) ^ { n } - \\frac { 1 } { q _ i } U _ i ^ { m - 1 } ( U _ i ^ * ) ^ { n - 1 } \\right ) & m , n \\geq 1 \\\\ U _ i ^ { m } ( U _ i ^ * ) ^ { n } & \\min \\{ m , n \\} = 0 , \\end{cases} \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} \\begin{array} { l l } \\psi ( t , x ) = - 2 \\mu ^ { - 1 / 2 } ~ \\partial _ x \\log u ( t , x ) = - 2 \\mu ^ { - 1 / 2 } ( \\partial _ x u ( t , x ) ) ~ u ( t , x ) ^ { - 1 } , \\end{array} \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\pi _ { \\mathfrak { k } ^ * } = \\sum _ { 1 \\leq i \\leq k < n } \\frac { \\partial } { \\partial \\lambda _ i ^ { ( k ) } } \\wedge \\frac { \\partial } { \\partial \\psi _ i ^ { ( k ) } } . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{gather*} \\frac { \\Gamma _ 2 ( z \\ , | \\ , a _ 1 , a _ 2 ) } { \\Gamma _ 2 ( z + a _ 1 \\ , | \\ , a _ 1 , a _ 2 ) } = \\frac { a _ 2 ^ { z / a _ 2 - 1 / 2 } } { \\sqrt { 2 \\pi } } \\Gamma \\bigl ( \\frac { z } { a _ 2 } \\bigr ) , \\\\ \\frac { \\Gamma _ 2 ( z \\ , | \\ , a _ 1 , a _ 2 ) } { \\Gamma _ 2 ( z + a _ 2 \\ , | \\ , a _ 1 , a _ 2 ) } = \\frac { a _ 1 ^ { z / a _ 1 - 1 / 2 } } { \\sqrt { 2 \\pi } } \\Gamma \\bigl ( \\frac { z } { a _ 1 } \\bigr ) , \\end{gather*}"}
-{"id": "6891.png", "formula": "\\begin{align*} u _ t \\ , = \\ , \\Delta ( u ^ m ) + u ^ p \\textrm { i n } \\ ; \\ ; M \\times ( 0 , T ) \\ , , \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\psi ( a b a b ) = \\Gamma _ i \\cdot M _ { \\epsilon } ^ { - 1 } \\Gamma _ 0 M _ { \\epsilon } \\cdot \\Gamma _ i \\cdot M _ { \\epsilon } ^ { - 1 } \\Gamma _ 0 M _ { \\epsilon } = M _ { \\epsilon } ^ { - 1 } \\Gamma _ i \\Gamma _ 0 \\Gamma _ i \\Gamma _ 0 M _ { \\epsilon } \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\Phi ^ 0 _ i : X \\backslash \\cup _ { j = 1 } ^ k B _ { 1 0 m _ i ^ { - 1 / 2 } } ( x _ j ) \\rightarrow \\mathfrak { s u } ( 2 ) , \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} p = p ( \\rho , \\theta ) = R \\rho \\theta , \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} d ( x , \\alpha ( t _ 0 ) ) \\geq f ( \\alpha ( t _ 0 ) ) - f ( x ) \\geq f ( \\alpha ( t _ 0 ) ) - f ( \\alpha ( a ) ) = d ( \\alpha ( a ) , \\alpha ( t _ 0 ) ) . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\check { \\omega } ( g , h ) \\check { \\varphi } = ( \\omega ( g , h ) \\varphi ) \\check { } . \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} L ( x , \\alpha , \\beta ) = f ( x ) + \\alpha ^ \\top h ( x ) + \\beta ^ \\top g ( x ) \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} M _ \\theta : & = \\mathrm { R e p ( Q ) } / / _ \\theta \\mathrm { P G L ( \\mathbf { 1 } ) } \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } _ r = - A _ r x _ r + B _ r u , \\\\ y _ r = C _ r x _ r \\end{cases} \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { c l } d \\hat { \\varphi } ( r ) = & - \\left [ A ( r ) \\hat { \\varphi } ( r ) + C ( r ) \\hat { \\nu } ( r ) + I ( r ) \\right ] d r + \\hat { \\nu } ( r ) d B \\left ( r \\right ) , \\\\ \\hat { \\varphi } ( T ) = & \\varepsilon _ { 8 } ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J \\setminus J _ { K } } } \\Phi ( \\frac { B _ { t _ { j } } + B _ { t _ { j + 1 } } } { 2 } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) = \\int _ 0 ^ T \\Phi _ { t } \\circ d B _ { t } . \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\langle p , l \\rangle _ * = \\epsilon \\in \\mathbb { I } . \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} \\mathcal W _ { \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) , B } ( g _ { \\infty } ) = 2 ^ { - \\nu ( N ) } \\zeta _ { \\Q } ( 2 ) ^ { - 1 } c ( \\mathfrak d _ { \\xi } ) \\mathfrak f _ { \\xi } ^ { k - 1 / 2 } \\mathcal W _ { B , \\infty } ( g _ { \\infty } ) \\prod _ p \\mathcal W _ { B , p } ( 1 ) , \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} \\lim \\limits _ { A \\rightarrow 0 } \\int _ { x _ 1 ^ { A } ( t ) } ^ { x _ 2 ^ { A } ( t ) } \\rho _ * ^ { A } d x = = \\lim \\limits _ { A \\rightarrow 0 } \\int _ { 0 } ^ { t } \\rho _ * ^ { A } ( \\sigma _ 2 ^ { A } - \\sigma _ 1 ^ { A } ) d t = w _ { 0 } ^ { B } t . \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\frac { \\partial G ( \\vec z ^ * ) } { \\partial z _ i } = \\frac { 1 } { \\| \\vec z ^ * \\| _ p } \\left ( { \\frac { \\alpha + \\beta } { m _ 0 + \\gamma } ( z _ i ^ * ) ^ { p - 1 } - | v _ i | } \\right ) , \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\nabla _ { U } Z = - \\overset { \\mu _ { 0 } } { A } _ { Z } U + \\nabla _ { U } ^ { \\bot } Z , \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} & \\sum \\limits _ { k = 0 } ^ { N _ n - 1 } \\left ( \\tau ( t ^ n _ { k + 1 } ) - \\tau ( t ^ n _ k ) \\right ) L ( t ^ n _ { k + 1 } ) ( u ) \\\\ & = \\sum \\limits _ { k = 0 } ^ { N _ n - 1 } \\tau ' ( \\xi _ k ^ n ) ( t ^ n _ { k + 1 } - t ^ n _ k ) L ( t ^ n _ { k + 1 } ) ( u ) \\\\ & = \\sum \\limits _ { k = 0 } ^ { N _ n - 1 } \\tau ' ( \\xi _ k ^ n ) ( t ^ n _ { k + 1 } - t ^ n _ k ) L ( \\xi _ { k } ^ n ) ( u ) - \\sum \\limits _ { k = 0 } ^ { N _ n - 1 } \\tau ' ( \\xi _ k ^ n ) ( t ^ n _ { k + 1 } - t ^ n _ k ) \\big ( L ( \\xi _ k ^ n ) ( u ) - L ( t ^ n _ { k + 1 } ) ( u ) \\big ) . \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} \\nabla _ { U } E = - A _ { E } ^ { \\ast } U - \\tau ( V ) E . \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} d _ n ^ 2 = \\| M _ { n } ( x ; \\beta , c ) \\| ^ { 2 } = \\left \\langle { \\bf u } ^ { \\tt M } , M ^ 2 _ { n } ( x ; \\beta , c ) \\right \\rangle = \\frac { ( \\beta ) _ { n } c ^ { n } \\ , n ! } { ( 1 - c ) ^ { \\beta + 2 n } } . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} \\bigg | \\prod _ { j = 1 } ^ n \\sin ( \\theta _ j ) ^ { j - n } \\cos ( \\theta _ j ) ^ { 1 - j } \\bigg | = \\bigg ( \\prod _ { j = 1 } ^ n a _ j \\bigg ) ^ { - \\frac { n + 1 } 2 } \\bigg ( \\prod _ { j = 1 } ^ n a _ j ^ j \\bigg ) \\bigg ( \\prod _ { j = 1 } ^ n \\frac 1 { a _ j + a _ j ^ { - 1 } } \\bigg ) ^ { \\frac { 1 - n } 2 } . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} L ( x ) = f _ n x ^ { q ^ n } + \\sum _ { i = \\delta } ^ { n - 1 } \\left ( f _ i x ^ { q ^ i } + f _ i ^ { q ^ { 2 n - i } } x ^ { q ^ { 2 n - i } } \\right ) . \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} u _ n = \\sum _ { i = 1 } ^ { n - 1 } m _ i e ^ { - ( x - x _ i ) } + \\sum _ { i = n } ^ N m _ i e ^ { - ( x _ i - x ) } , n = 1 , \\ldots , N + 1 \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { L ^ { \\infty } \\left ( 0 , T ; X \\right ) } = \\underset { t \\in \\left ( 0 , T \\right ) } { } \\left \\Vert u \\left ( t \\right ) \\right \\Vert _ { X } < \\infty \\quad p = \\infty . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} \\frac { \\theta } { \\pi } | e ^ { \\sigma _ 0 | x | } J _ c ^ { - 1 } g ( x ) | & \\leqq \\int _ { \\mathbb R } e ^ { ( \\sigma _ 0 - \\theta ) | x - y | } e ^ { \\sigma _ 0 | y | } | g ( y ) | d y \\\\ & \\leqq s u p _ { y \\in \\mathbb R } | e ^ { \\sigma _ 0 | y | } g ( y ) | \\int _ { \\mathbb R } e ^ { ( \\sigma _ 0 - \\theta ) | p | } d p < \\infty . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} & \\overline { V } ^ { ( l ) } ( p , g , \\pi _ n ) _ T = ( r ( n ) ) ^ { p / 2 - 1 } \\sum _ { i : t _ { i , n } ^ { ( l ) } \\leq T } g ( \\Delta _ { i , n } ^ { ( l ) } X ^ { ( l ) } ) , \\\\ & G ^ { ( l ) , n } _ p ( t ) = ( r ( n ) ) ^ { p / 2 - 1 } \\sum _ { i : t _ { i , n } ^ { ( l ) } \\leq t } | \\mathcal { I } _ { i , n } ^ { ( l ) } X ^ { ( l ) } | ^ { p / 2 } . \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\frac { f ( \\gamma _ i ( 0 ) ) - f ( p ) } { d ( p , \\gamma _ i ( 0 ) ) } = g _ { w _ \\infty } ( w _ \\infty , v ^ f ) . \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} \\partial _ { \\Lambda ( \\mathcal { G } ) , \\mathcal { Q } ^ c ( \\mathcal { G } ) } ( x _ { L _ { \\infty } / K } ) = - C _ { L _ { \\infty } / K } - U ' _ { L _ { \\infty } / K } + M _ { L _ { \\infty } / K } \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} U ^ { \\ast } & = u _ { 0 } + u _ { 1 } s + u _ { 2 } s ^ { 2 } + . . \\\\ U _ { \\tau } ^ { \\ast } & = \\tau _ { 0 } + \\tau _ { 1 } s + \\tau _ { 2 } s ^ { 2 } + . . . ( n o t e : \\tau _ { k - 1 } = k u _ { k } ) \\\\ U _ { \\nu } ^ { \\ast } & = w _ { 0 } + w _ { 1 } s + w _ { 2 } s ^ { 2 } + . . . \\\\ K ^ { \\ast } & = K _ { 0 } + K _ { 1 } s + K _ { 2 } s ^ { 2 } + . . . \\\\ \\mathfrak { S } & = \\mathfrak { S } _ { 0 } + \\mathfrak { S } _ { 1 } s + \\mathfrak { S } _ { 2 } s ^ { 2 } + . . . \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} S ( n , m ) : = S ( n , m ; 0 ) , m \\leq n , \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} C _ { k } ( E _ N ) = \\max _ { M , D } \\left | \\sum _ { n = 1 } ^ M e _ { n + d _ 1 } e _ { n + d _ 2 } \\dots e _ { n + d _ { k } } \\right | , \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} B = \\bigl \\{ \\mathbf s \\in \\mathcal S ( \\Z ) : M _ \\phi ( c \\mathbf s ) < + \\infty \\ \\ c > 0 \\bigr \\} \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { \\sigma _ { - 1 } ( n ) } { n } = O ( \\log x ) . \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\deg ( v ) } F _ j ( v ) = 0 . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} n h _ n = h _ { n - 1 } p _ 1 + h _ { n - 2 } p _ 2 + \\cdots + p _ n \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } b ( U , n ) ^ { 1 / n } = 1 \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\int _ { \\Omega } c ^ { - 2 } g \\varphi d x = 0 , \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} \\nabla _ { U } \\xi = - \\varphi A _ { N } U + \\tau ( U ) \\xi + A _ { N } U v ( E ) , \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} \\left | \\exp \\left ( 2 i \\pi \\eth ( x w - z y ) \\right ) - 1 \\right | & = 2 | \\sin ( \\eth \\pi ( x w - z y ) ) | \\leq 2 \\pi | \\eth | | x w - y z | \\\\ & \\leq 2 \\pi | \\eth | \\| ( x , y ) \\| _ 2 \\| ( z , w ) \\| _ 2 \\leq 2 \\pi k | \\eth | \\| ( x , y ) \\| \\| ( z , w ) \\| _ 2 \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\mathfrak R _ p \\mathbf F : = \\sum _ { j = 0 } ^ { p - 1 } \\Pi ( u _ j ) \\mathbf F , \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} B ( v , e \\cup f ) & = \\begin{cases} B _ 1 ( v , e ) \\cdot w _ 2 ( f ) , & \\ v \\in e , \\\\ B _ 2 ( v , f ) \\cdot w _ 1 ( e ) , & \\ v \\in f , \\\\ 0 , & , \\end{cases} \\\\ w ( e \\cup f ) & = w _ 1 ( e ) \\cdot w _ 2 ( f ) . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , \\Big | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) ( 2 \\pi ) ^ { - \\frac { q } { \\tau } } \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) ( 2 \\pi ) ^ { - q } \\times \\\\ & \\times \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} x = \\begin{pmatrix} J _ { \\frac { k a } { 4 } , 2 } \\\\ & J _ { \\frac { k a } { 4 } , 2 } ^ T \\\\ & & I _ { s } \\end{pmatrix} . \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} \\| f ( \\tau ) \\| _ { H ^ k ( \\Omega ) } : = \\left ( \\sum _ { \\alpha _ 1 + \\alpha _ 2 \\leq k } \\| \\partial ^ { \\alpha _ 1 } _ \\xi \\partial ^ { \\alpha _ 2 } _ \\eta f ( \\tau , \\cdot ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } \\right ) ^ \\frac { 1 } { 2 } < \\infty . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} \\mu _ f ( x ) & = \\sigma \\sqrt { 2 \\log ( x ) - 5 \\log \\log ( x ) - \\log ( 8 \\pi ) + 2 f ( x ) } \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\theta ( s ) = \\int _ { 0 } ^ { s } \\theta _ { + } ^ { \\prime } ( r ) \\ , d r , s \\ge 0 . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} ( x , y , z ) = ( t ^ 5 , t ^ 7 + s _ 2 t ^ { 1 1 } + s _ 3 t ^ { 1 6 } , t ^ { 1 3 } + s _ 4 t ^ { 1 6 } ) \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} \\Delta ^ m p _ n ( x ) = 0 , m = n + 1 , n + 2 , \\ldots , \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} \\sum _ { k > D _ \\varepsilon } b _ k ^ 2 \\theta _ k ^ 2 \\leq b _ { D _ \\varepsilon } ^ 2 \\left ( \\sum _ { k > D _ \\varepsilon } \\theta _ k ^ 2 \\right ) < b _ { D _ \\varepsilon } ^ 2 \\left ( r _ \\varepsilon ^ 2 - C _ { \\min } ( \\alpha , \\beta ) \\varepsilon ^ 2 \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } \\right ) . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\mathcal { W } ( v ) = \\frac { 1 } { 2 } \\| v \\| _ { \\dot { H } ^ m } ^ 2 + \\sum _ { 0 \\leq k < m } \\frac { \\lambda _ k } { p _ k } \\int _ { \\mathbb { R } ^ n } | \\nabla ^ k v ( x ) | ^ { p _ k } \\ , d x \\qquad ( \\lambda _ k \\geq 0 , \\ , p _ k > 1 ) . \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} & \\nabla \\bar { J } ( A _ r , B _ r , C _ r ) \\\\ = & 2 ( - Q P - Y ^ T X , Q B _ r + Y ^ T B , C _ r P - C X ) . \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} F _ i ( X ) = \\frac 1 2 \\sum _ { j = 1 } ^ n \\gamma _ { i j } ( f ( X _ j X _ i ^ { - 1 } ) - f ( X _ i X _ j ^ { - 1 } ) ) , \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} { S _ { n _ 1 ^ p } } \\left ( f \\right ) = \\frac { 1 } { { 2 T } } { S _ { w _ 1 ^ i } } \\left ( { f - f ' } \\right ) + \\frac { 1 } { { 2 T } } { S _ { w _ 1 ^ i } } \\left ( { f + f ' } \\right ) + \\frac { 1 } { { 2 T } } { S _ { w _ 1 ^ q } } \\left ( { f - f ' } \\right ) + \\frac { 1 } { { 2 T } } { S _ { w _ 1 ^ q } } \\left ( { f + f ' } \\right ) \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} \\langle I _ A , v \\rangle = Z _ A v \\mathrm { f o r \\ ; a l l \\ ; } v \\in V . \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} G = ( \\Z / 3 \\Z ) \\ast ( \\Z / 3 \\Z ) \\ast ( \\Z / 3 \\Z ) . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} R _ Q ( \\lambda ) : = \\begin{bmatrix} \\widehat { K } _ 1 ( \\lambda ) & 0 \\end{bmatrix} R _ \\mathcal { L } ( \\lambda ) \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\overset { { \\rm i n \\ , l a w } } { = } 2 \\pi \\ , 2 ^ { - \\bigl [ 6 + 2 ( \\lambda _ 1 + \\lambda _ 2 ) \\bigr ] } \\ , L \\ , X _ 1 \\ , X _ 2 \\ , X _ 3 \\ , Y . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} & \\lim _ { k \\to \\infty } \\| y ^ k - y ^ { k + 1 } \\| = \\lim _ { k \\to \\infty } \\| x ^ k - y ^ { k } \\| = 0 . \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} & v _ { s , x _ 1 , x _ 2 } = \\min _ { j = 1 , 2 } \\phi _ { s , s - t _ 0 } ( \\delta _ { x _ j } ) ; \\\\ & v _ { s } = \\min _ { x \\in E _ { s - t _ 0 } } \\phi _ { s , s - t _ 0 } ( \\delta _ { x } ) , \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} A _ m & = m + 1 - ( a _ { m - 1 } - a _ { m - 2 } + \\cdots + ( - 1 ) ^ { m - 1 } a _ 0 ) < m + 1 , \\\\ A _ m & = m + 1 - m + ( a _ { m - 2 } - a _ { m - 3 } + \\cdots + ( - 1 ) ^ m a _ 0 ) > 1 . \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\frac { | \\{ S ^ t : t \\in T , S ^ t \\le R \\} | } { | S ^ T | } = \\frac { \\operatorname { f i x } ( S , \\Omega ) } { | \\Omega | } . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\frac { | \\hat \\eta _ t ( \\hat { x } _ t ) - \\mathbf { P } ^ { \\hat { g } _ t } _ { \\hat { x } ' _ t \\hat { x } _ t } ( \\hat \\eta _ t ( \\hat { x } ' _ t ) ) | _ { \\hat { g } _ t ( \\hat { x } _ t ) } } { d ^ { \\hat { g } _ t } ( \\hat { x } _ t , \\hat { x } ' _ t ) ^ \\alpha } = 1 . \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{gather*} \\mathcal G \\big ( \\psi _ { \\mathfrak P } ^ 2 \\big ) : = \\psi _ { \\mathfrak P } ^ 2 ( 1 ) { \\rm e } ^ { 2 \\pi { \\rm i } \\operatorname { T r } _ { K _ { \\mathfrak P } / \\Q _ 2 } ( 1 / \\pi _ { \\mathfrak P } ^ 4 ) } + \\psi _ { \\mathfrak P } ^ 2 ( { \\rm i } ) { \\rm e } ^ { 2 \\pi { \\rm i } \\operatorname { T r } _ { K _ { \\mathfrak P } / \\Q _ 2 } ( { \\rm i } / \\pi _ { \\mathfrak P } ^ 4 ) } = - 2 = - N _ { K / \\Q } \\big ( \\mathfrak P ^ 2 \\big ) ^ { 1 / 2 } , \\end{gather*}"}
-{"id": "874.png", "formula": "\\begin{align*} f _ i g _ j = 0 i + j \\geq 2 i \\neq j + 1 . \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ M ) ( q \\ , | \\ , b ) = f ( 0 ) \\frac { d ^ M } { d t ^ M } \\Big \\vert _ { t = 0 } \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) + M ! \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j \\ , B ^ { ( f ) } _ { 1 } ( q + b _ 0 ) . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\widehat { y } _ N = \\omega _ { 2 N } ^ { - N } \\sum _ { l = 0 } ^ { 2 N - 1 } \\omega _ { 2 N } ^ { - N l } y _ l = - \\sum _ { l = 0 } ^ { 2 N - 1 } \\omega _ { 2 N } ^ { N l } y _ l = - \\widehat { y } _ N , \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\tau ^ { l ( x ) } ( h ( x ) ) = \\tau ^ { k ( x ) } ( h ( \\sigma ( x ) ) ) \\quad \\sigma ^ { l ' ( y ) } ( h ^ { - 1 } ( y ) ) = \\sigma ^ { k ' ( y ) } ( h ^ { - 1 } ( \\tau ( y ) ) ) \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} m = n ( n - 1 ) / 2 \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} D _ N ( s , \\alpha , f ) = \\sum _ { k = 0 } ^ { N - 1 } f ( \\{ s _ 1 + k \\alpha _ 1 \\} , \\dots , \\{ s _ d + k \\alpha _ d \\} ) - N \\int _ { [ 0 , 1 ] ^ d } f ( x ) \\ , \\textrm { d } x ( N \\in \\mathbb { N } ) , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} { \\bf P } \\bigl ( V _ N < s \\bigr ) = \\lim \\limits _ { \\beta \\rightarrow \\infty } { \\bf E } \\Bigl [ \\exp \\Bigl ( - e ^ { - \\beta s } \\ , Z _ { \\lambda _ 1 , \\lambda _ 2 , \\varepsilon } ( \\beta ) / C \\Bigr ) \\Bigr ] , \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} | \\mu _ A ( f ) | \\leq \\frac { 1 } { \\alpha } \\int _ \\Gamma | ( f \\circ \\iota ) | \\ , d \\lambda = \\frac { 1 } { \\alpha } \\| f \\| _ { L ^ 1 ( b \\Gamma ) } , \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\widetilde { c } _ { j i } ( - 4 n + 2 + r ) = - \\widetilde { c } _ { j i } ( r ) \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} C ^ * ( U _ i ) = \\overline { } \\{ U _ { i , m , n } \\ : \\ m , n \\in \\N \\} . \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} x \\in \\pi ^ { - 1 } \\iota ^ { - 1 } B & \\Leftrightarrow \\iota \\pi ( x ) \\in B \\\\ & \\Leftrightarrow \\pi ( x ) \\in B & & \\\\ & \\Rightarrow \\pi ( x ) + ( x - \\pi ( x ) ) \\in B + W & & \\\\ & \\Rightarrow x \\in B + W \\\\ & \\Leftrightarrow x \\in B & & , \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N ( c ^ u _ { i } + c ^ d _ { i } ) \\leq C _ { \\mathrm { T o t a l } } . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} G _ \\alpha ( x , y ) ~ = ~ \\exp ( \\langle \\alpha , y - x \\rangle ) G ( x , y ) , x , y \\in E , \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\dot { x } = F ( x ) . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ \\frac { Z _ { \\varepsilon } ( n \\beta ) } { Z ^ n _ { \\varepsilon } ( \\beta ) } \\Bigr ] \\approx N ^ { 1 + n ^ 2 \\beta ^ 2 - ( 1 + \\beta ^ 2 ) n } \\prod \\limits _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma ( 1 - n \\beta ^ 2 + ( j + 1 ) \\beta ^ 2 ) ^ 2 \\ , \\Gamma ( 1 - \\beta ^ 2 ) } { \\Gamma ( 1 - 2 n \\beta ^ 2 + ( j + 1 ) \\beta ^ 2 ) \\ , \\Gamma ( 1 + j \\beta ^ 2 ) } . \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} d S _ \\nu : = \\{ d f ^ n _ \\omega ( 0 ) \\ , | \\ , \\omega \\in s u p p ( \\nu ) ^ { \\mathbb N } , \\ , n \\in \\mathbb N \\} , \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} L _ { m _ 0 } ( Z _ K ) : = L _ { m _ 0 } ( X _ K ) \\otimes _ { R _ { K , m } } \\overline { R } _ { K , m } \\cong L _ { m _ 0 } ( X _ K ) / ( T _ 1 ^ { \\frac { 1 } { m _ 0 } } , \\ldots , T _ l ^ { \\frac { 1 } { m _ 0 } } ) \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\bar { \\bf C } _ { A S } ^ q ( M ) = 0 q > m . \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} h ( \\gamma z ) = \\tilde j ( \\gamma , z ) ^ { 2 k + 1 } \\chi ( d ) h ( z ) \\gamma = \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) \\in \\Gamma _ 0 ( 4 N ) \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\ & | \\Sigma _ 1 | + | \\Sigma _ 3 | \\leq \\sum _ { k \\leq k _ 0 - 2 \\atop k \\ge k _ 0 + 3 } \\ , | A _ k | + \\sum _ { k \\leq k _ 0 - 2 \\atop k \\ge k _ 0 + 2 } \\ , | D _ k | \\leq \\\\ \\ & \\leq C \\omega ( 2 ^ { - n } ) \\sum _ { \\nu = 1 } ^ { \\infty } \\frac { 1 } { \\nu ^ { 2 } } \\leq C \\omega ( 2 ^ { - n } ) . \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{gather*} - \\frac 1 4 \\mathcal G _ { 2 , ( 1 ; 3 ) } ^ \\ast ( \\tau _ 0 ) = \\frac 3 4 L ( G _ 2 ( q ) , 2 ) = L \\big ( \\eta ( 2 \\tau ) ^ 3 \\eta ( 6 \\tau ) ^ 3 , 2 \\big ) = \\frac 3 2 L ( f _ { 3 6 } , 1 ) ^ 2 , \\end{gather*}"}
-{"id": "3016.png", "formula": "\\begin{align*} G = G _ e \\times G _ { ( p _ 1 ) } \\times \\cdots \\times G _ { ( p _ m ) } \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\Phi - \\Psi = J _ h . \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} h \\alpha _ n = \\left ( \\begin{array} { c c } a p ^ n & b p ^ { - n } \\\\ c p ^ n & d p ^ { - n } \\end{array} \\right ) = \\left ( \\begin{array} { c c } d ^ { - 1 } p ^ n & \\ast \\\\ 0 & d p ^ { - n } \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & 0 \\\\ c d ^ { - 1 } p ^ { 2 n } & 1 \\end{array} \\right ) = : g _ 1 g _ 2 , \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} a _ f ( n ) = \\mu ^ 2 \\lambda _ f ( n ) , \\omega ( n ) = \\omega ' ( n ) = \\mu ^ 2 ( n ) | \\lambda _ f ( n ) | ^ 2 , \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} \\left \\{ B \\in \\mathcal B ( V \\oplus W ) \\vert B + W = B \\right \\} = \\sigma ( f \\circ \\pi : f \\in V ^ * ) . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k \\right ) = a _ { 1 } S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } - 1 , k - 1 \\right ) + \\left ( a _ { 1 } k + b _ { 1 } \\right ) S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } - 1 , k \\right ) . \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} T _ { i n t } = \\bigcup \\Big \\{ l _ t \\cap T : l _ t ~ ~ \\Gamma \\cap T \\Big \\} , \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} A _ 0 ( z ) = \\frac { z - 1 + \\cos z } { 1 - \\sin z } . \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\vert F ( z + h ) - F ( z ) - L ( h ) \\vert & = \\vert e \\vert \\\\ & \\leq \\frac 6 5 w _ x ( \\vert F ( z + h ) - F ( z ) \\vert ) \\cdot \\vert F ( z + h ) - f ( z ) \\vert \\\\ & \\leq 2 w _ x ( \\vert F ( z + h ) - F ( z ) \\vert ) \\cdot \\vert h \\vert \\\\ & = o ( \\vert h \\vert ) . \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} F ( q \\ , | \\ , \\beta , \\lambda _ 1 , \\lambda _ 2 ) = F ( q \\ , \\big | \\ , \\frac { 1 } { \\beta } , \\lambda _ 1 , \\lambda _ 2 ) , \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} R ( \\rho _ { w } ) = \\exp \\left ( - \\sum _ { j > 0 } v _ { j } ( w ) L _ { j } \\right ) v _ { 0 } ( w ) ^ { - L _ { 0 } } \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} B ' ( v , e ) = B ( v , e ) , ~ w ' ( e ) = \\bigg ( \\frac { w ( e ) ^ { p - r } } { m ^ { p ' - p } } \\bigg ) ^ { 1 / ( p ' - r ) } . \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} ~ ~ ^ { A B R } _ { a } D ^ \\alpha ( p ( t ) ~ ^ { A B R } D ^ \\alpha _ b x ( t ) ) + q ( t ) x ( t ) = \\lambda r ( t ) x ( t ) , t \\in ( a , b ) , \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} w ( e ) ^ { p - r } \\prod _ { v \\in e } B ( v , e ) = \\frac { w _ j ( e ) ^ { p - r } \\prod _ { v \\in e } B _ j ( v , e ) } { C ^ { p - r } \\alpha _ j } = \\frac { 1 } { C ^ { p - r } } . \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\phi ^ { \\prime \\prime } + { n - 1 \\over r } \\phi ^ { \\prime } + \\left ( p u _ R ^ { p - 1 } ( r ) - { \\lambda _ k \\over r ^ 2 } \\right ) \\phi = - { \\mu _ k } _ i \\phi \\ \\hbox { i n } \\ ( R , 1 ) , \\\\ \\phi ( R ) = \\phi ( 1 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} T _ H \\colon C _ c ( G ) \\to C _ c ( G / H ) , T _ H ( f ) ( g H ) = \\int _ H f ( g h ) \\ , d \\mu _ H ( h ) \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| g _ n ( s , t ) - g ( s , t ) \\| = 0 . \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} Q ( x ) = \\frac { 1 } { 2 } ( x , x ) , x \\in V . \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} [ y _ 1 ^ U ] _ \\beta ^ T \\alpha = \\alpha y _ 1 ^ T . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\chi _ { k } * ( b c ) = b ( \\chi _ { k } * c ) + ( \\chi _ { k } * b ) ( f _ { k } * c ) . \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\alpha D \\nu _ 0 + \\frac { 1 } { \\gamma } \\nu _ 0 - \\eta \\nu _ 0 ^ 3 = 0 , \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} ( T _ G D _ { { \\bf x } , t } ^ + ) ( D _ { { \\bf x } , t } ^ + { \\bf u } ) + T _ G { \\bf D } p = T _ G ( { \\bf f } ) . \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} c ( n ^ 2 | D | ) = c ( | D | ) \\sum _ { 0 < d \\mid n } \\mu ( d ) \\left ( \\frac { D } { d } \\right ) \\chi ( d ) d ^ { k - 1 } a _ { \\chi } ( n / d ) . \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} g & = ( q - 2 ) ( m - 1 ) + ( m - d ) \\quad \\\\ N & = \\begin{cases} ( q ^ 2 - 1 ) m + 2 d & ( q - 1 ) / d \\equiv 0 \\pmod { 3 } \\\\ ( q ^ 2 - 1 ) m & ( q - 1 ) / d \\not \\equiv 0 \\pmod { 3 } . \\end{cases} \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} \\mathcal { O } \\left ( \\sum _ { j = 0 } ^ { L - 1 } 2 ^ j + ( J - L ) 2 ^ L \\right ) = \\mathcal { O } \\left ( 2 ^ L + ( J - L ) 2 ^ L \\right ) = \\mathcal { O } \\left ( m \\log \\frac { 2 N } { m } \\right ) . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\sigma ^ K _ k = \\frac { 1 - \\frac { \\theta _ k } { \\theta _ { k - 1 } } \\frac { 1 - \\theta _ 0 } { 1 - \\theta _ k } } { 1 + \\frac { \\theta _ { k - 1 } } { \\theta _ 0 \\mu } } , k \\in \\{ 1 , \\dots , K - 1 \\} \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} \\psi _ N \\ ; = \\ ; \\sum _ { j = 0 } ^ { N _ 1 } \\sum _ { k = 0 } ^ { N _ 2 } \\ ; \\chi _ { j k } \\ , \\boxtimes \\ , \\big ( u _ 0 ^ { \\otimes ( N _ 1 - j ) } \\otimes v _ 0 ^ { \\otimes ( N _ 2 - k ) } \\big ) \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} \\sigma _ 0 = \\frac 1 2 \\pm \\frac 1 { \\log q } t _ 0 , \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} w ( e ) ^ { p - r } \\cdot \\prod _ { v \\in e } B ( v , e ) & = \\Bigg ( \\frac { r } { \\lambda ^ { ( p ) } ( G [ S ] ) } \\prod _ { u \\in e } x _ u \\Bigg ) ^ { p - r } \\cdot \\prod _ { v \\in e } \\frac { \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G [ S ] ) x _ v ^ p } \\\\ & = \\frac { r ^ { p - r } } { ( \\lambda ^ { ( p ) } ( G [ S ] ) ) ^ p } = \\alpha . \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} \\sum _ { i _ { 1 } + i _ { 2 } + \\cdots + i _ { r } = N } \\frac { a _ { 1 } ^ { i _ { 1 } } a _ { 2 } ^ { i _ { 2 } } \\cdots a _ { r } ^ { i _ { r } } ( - z _ 1 ; q ) _ { i _ { 1 } } ( - z _ 2 ; q ) _ { i _ { 2 } } \\cdots ( - z _ r ; q ) _ { i _ { r } } } { ( q ; q ) _ { i _ { 1 } } ( q ; q ) _ { i _ { 2 } } \\cdots ( q ; q ) _ { i _ { r } } } . \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} y _ \\xi : = t \\partial _ \\xi + \\sum _ { \\alpha \\in R _ + } c _ \\alpha \\langle \\alpha , \\xi \\rangle \\sigma _ { \\langle \\alpha ^ \\vee , \\lambda \\rangle } ( \\langle \\alpha , x \\rangle ) s _ \\alpha \\ , \\xi \\in V \\ . \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ 1 x _ { n } + \\sum _ { i = 2 } ^ { n - 1 } x _ i ^ 2 + x _ 1 x _ 2 ^ 2 , \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} | h ^ { ( j ) } ( x ) | \\ll N ^ { - j } , \\ , \\hbox { f o r } \\ , j = 0 , 1 , 2 , \\ldots \\ , \\int _ { - \\infty } ^ { \\infty } h ( x ) d x = N . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} R _ 1 ^ { A B } ( \\rho _ - , v _ - ) : \\left \\{ \\begin{array} { l l } \\xi = \\lambda _ 1 = v + \\beta t - \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } , \\\\ v - v _ - = - \\int _ { \\rho _ { - } } ^ { \\rho } \\frac { \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } } { \\rho } d \\rho , \\end{array} \\right . \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\left \\{ \\frac { 1 } { z _ 1 ^ { \\alpha _ { 1 } ^ 1 + \\cdots + \\alpha _ { k } ^ 1 } } ( d ^ { 1 } z ) ^ { \\alpha _ 1 } ( d ^ { 2 } z ) ^ { \\alpha _ 2 } \\cdots ( d ^ { k } z ) ^ { \\alpha _ k } \\right \\} _ { | \\alpha _ 1 | + 2 | \\alpha _ 2 | + \\cdots + k | \\alpha _ k | = N } , \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} \\theta _ u ^ { { \\rm { R F } } } \\left ( t \\right ) = 2 \\pi f _ u ^ { { \\rm { R F } } } t + \\varphi _ u ^ { { \\rm { R F } } } \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} p ( - t ) t ^ m = ( 1 - t ) ^ { n - d } \\left ( h ( t ) - f ( t ) ( 1 - t ) ^ d \\right ) . \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} I _ { 2 , 4 } = { \\Theta } \\frac { \\zeta ( 3 ) } { \\zeta ^ { 2 } ( 2 ) } x + O \\left ( ( \\log x ) ^ { 5 / 3 } ( \\log \\log x ) ^ { 4 / 3 } \\right ) . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} L _ 1 & : = t _ { \\alpha _ 1 } ^ { - 1 } t _ { \\delta _ 1 } ^ { - 1 } t _ { \\gamma _ 1 } ^ { - 1 } t _ { \\beta _ 1 } ^ { - 1 } t _ { x _ 1 } t _ { y _ 1 } t _ { z _ 1 } , \\\\ L _ 1 ^ \\prime & : = t _ { \\beta _ 1 } ^ { - 1 } t _ { \\gamma _ 1 } ^ { - 1 } t _ { \\delta ' _ 1 } ^ { - 1 } t _ { \\alpha _ 1 } ^ { - 1 } t _ { x ^ \\prime _ 1 } t _ { y ^ \\prime _ 1 } t _ { z ^ \\prime _ 1 } . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} h _ { i } ( x ) = \\frac { \\partial } { \\partial x _ { i } } h ( x ) = - 2 \\alpha ( x _ { i } - ( x _ { 0 } ) _ { i } ) e ^ { - \\alpha | x - x _ { 0 } | ^ { 2 } } , \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} \\frac { ( - z q ; q ) _ i } { ( q ; q ) _ i } = \\frac { 1 + z q ^ i } { 1 + z } \\frac { ( - z ; q ) _ i } { ( q ; q ) _ i } , \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta _ { \\omega ( t ) } ) t r _ { \\omega ( t ) } \\hat \\omega & = g ( t ) ^ { \\bar j i } g ( t ) ^ { \\bar l k } \\hat R _ { i \\bar j k \\bar l } - g ( t ) ^ { \\bar j i } g ( t ) ^ { \\bar l k } \\hat g ^ { \\bar b a } \\nabla ^ { g ( t ) } _ i \\hat g _ { k \\bar b } \\nabla ^ { g ( t ) } _ { \\bar j } \\hat g _ { a \\bar q } \\\\ & \\le g ( t ) ^ { \\bar j i } g ( t ) ^ { \\bar l k } \\hat R _ { i \\bar j k \\bar l } \\\\ & \\le A ( t r _ { \\omega ( t ) } \\hat \\omega ) ^ 2 , \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} ( i _ { Y _ { j _ 0 } } ( \\omega ) ) | _ { X _ { j _ 0 l _ 0 } } = ( \\sum _ { i \\ne j } 2 \\lambda _ { i j } \\hat { F } _ { i j } i _ { Y _ { j _ 0 } } ( d F _ i ) \\ , d F _ j ) | _ { X _ { j _ 0 l _ 0 } } = ( 2 \\lambda _ { j _ 0 l _ 0 } \\hat { F } _ { j _ 0 l _ 0 } d F _ { l _ 0 } ) | _ { X _ { j _ 0 l _ 0 } } . \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} & \\left \\langle \\rho , \\tau ^ { x _ { 3 } } \\alpha _ { 1 } , y \\alpha _ { 2 } \\alpha _ { 3 } ^ { a } \\right \\rangle \\cong M _ { 1 } \\\\ & \\ a , y _ { 3 } = 0 , . . . , p - 1 , \\ x _ { 3 } , y _ { 2 } = 1 , . . . , p - 1 \\ \\ a x _ { 3 } - y _ { 2 } - x _ { 3 } y _ { 2 } \\not \\equiv 0 \\ \\mathrm { m o d } \\ p . \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} \\bigcup _ { j \\in [ n ] } \\mathcal { L } _ j = \\mathcal { U } \\mbox { f o r } \\mathcal { L } _ j = \\{ ( u , v ) \\mid u _ j = 1 \\} \\enspace . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} \\log \\gamma \\leq - \\lim _ n \\frac { 1 } { n } \\log Q ( x , x \\star u ^ { \\star n } ) ~ = ~ 0 \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} t _ { x _ 1 } t _ { y _ 1 } t _ { \\alpha _ 1 } ^ { - 1 } ( z _ 1 ) & = t _ { \\beta _ 1 } t _ { \\gamma _ 1 } t _ { \\delta _ 1 } t _ { z _ 1 } ^ { - 1 } ( z _ 1 ) = z _ 1 , \\\\ t _ { y _ 1 ^ \\prime } t _ { z _ 1 ^ \\prime } t _ { \\gamma _ 1 } ^ { - 1 } ( x _ 1 ^ \\prime ) & = t _ { x _ 1 ^ \\prime } ^ { - 1 } t _ { \\alpha _ 1 } t _ { \\beta _ 1 } t _ { \\delta _ 1 ^ \\prime } ( x _ 1 ^ \\prime ) = x _ 1 ^ \\prime . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t \\omega ( t ) & = - R i c ( \\omega ( t ) ) \\\\ \\omega ( 0 ) & = \\hat \\omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\begin{psmallmatrix} 0 & \\cdots & 0 & ( \\omega ^ 0 _ i ) ^ { - 1 } & 0 & \\cdots & 0 \\end{psmallmatrix} \\circ \\omega ^ 0 = ( \\omega ^ 0 _ i ) ^ { - 1 } \\circ \\omega ^ 0 _ i = 1 _ { W ^ 0 } , \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} \\begin{aligned} I _ 1 ( u ) & : = \\int _ { - \\infty } ^ { \\infty } u ( t , x ) d x = I _ 1 ( u _ 0 ) , \\ ; \\ ; \\ ; I _ 2 ( u ) : = \\int _ { - \\infty } ^ { \\infty } u ^ 2 ( t , x ) d x = I _ 2 ( u _ 0 ) , \\\\ \\\\ I _ 3 ( u ) & : = \\int _ { - \\infty } ^ { \\infty } \\left ( \\frac 1 2 ( \\partial _ x u ) ^ 2 - G ( u ) \\right ) ( t , x ) d x = I _ 3 ( u _ 0 ) , \\end{aligned} \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s _ { 2 } } v + V _ { 2 } ( x ) v = f _ { 2 } ( v ) , x \\in \\mathbb { R } ^ { N } . \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\displaystyle M A E = \\displaystyle \\frac { 1 } { N } \\sum _ { i = 0 } ^ { N } | [ \\sum _ { \\substack { j \\in J \\setminus J _ { K ( n ) } } } \\frac { B _ { t _ { j + 1 } } + B _ { t _ { j } } } { 2 } ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) ] - \\frac { 1 } { 2 } B _ { T } ^ 2 | . \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} & I _ { \\chi } = \\delta _ { \\chi } ; \\\\ & I _ { q , 4 } = \\delta _ { p \\equiv _ 4 1 \\ ; \\forall p \\mid q } . \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} \\int _ { U _ - } P _ t u ( x ) - u ( x ) d x & = \\int _ { U _ - } \\int _ { \\R ^ d } u ( y ) p ( t , | x - y | ) d y d x - \\int _ { U _ - } u ( x ) d x \\\\ & = \\int _ { \\R ^ d } u ( y ) \\int _ { U _ - } p ( t , | x - y | ) d x d y - \\int _ { U _ - } u ( y ) d y \\\\ & = \\int _ { U _ - ^ c } u ( y ) \\int _ { U _ - } p ( t , | x - y | ) d x d y + \\int _ { U _ - } u ( y ) \\left ( \\int _ { U _ - } p ( t , | x - y | ) d x - 1 \\right ) d y \\\\ & \\ge \\int _ { U _ - } u ( y ) \\left ( \\int _ { U _ - } p ( t , | x - y | ) d x - 1 \\right ) d y . \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} \\varepsilon _ { i j } = \\# \\{ j \\rightarrow i \\} - \\# \\{ i \\rightarrow j \\} . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\lvert \\Phi \\rvert = m - \\frac { 4 \\pi } { \\emph { v o l } ( N ) } \\frac { k } { \\rho } + o ( \\rho ^ { - 1 } ) . \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\widetilde { H } _ { N } : = U _ { N } H _ { N } ^ { \\operatorname { M F } } U ^ * _ { N } - N e _ H . \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { M , M - 1 } ( a , b ) ^ q \\bigr ] = \\eta _ { M , M - 1 } ( q \\ , | a , \\ , b ) . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} M ( t , A ) : = A ^ { - s ( t ) / 2 } X ( t ) A ^ { - s ( t ) / 2 } \\ , . \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} \\omega = \\frac { \\partial T } { \\partial \\dot { \\alpha } } = \\rho ^ { 2 } ( \\dot { \\alpha } - \\frac { 1 } { 2 } \\cos \\varphi \\dot { \\theta } ) , \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} T _ p Z : = T _ p S _ \\alpha \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} [ \\beta ] = [ \\imath _ * \\imath ^ { ! * } \\beta ] + [ \\jmath _ { ! * } \\jmath ^ * \\beta ] . \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} ( \\mathbf { \\tau } ^ { \\ast } , \\mathbf { \\nu } ^ { \\ast } ) ^ { T } = \\left ( \\begin{array} [ c ] { c c } J _ { \\varphi } & J _ { \\theta } \\sin \\varphi \\\\ - J _ { \\theta } \\sin \\varphi & J _ { \\varphi } \\end{array} \\right ) ( \\frac { \\partial } { \\partial \\varphi } , \\frac { 1 } { \\sin \\varphi } \\frac { \\partial } { \\partial \\theta } ) ^ { T } , \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} P = \\bigoplus _ { \\lambda \\in \\Pi ( P ) } P _ { \\lambda } = \\bigoplus _ { \\lambda \\in \\Pi ^ { + } ( P ) } ( \\dim V ^ { \\lambda } ) L ^ { \\lambda } . \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} T ( v _ 1 \\cdot ( e , v ) ) = T ( t _ { V _ 1 } ( v _ 1 ) , \\Phi ( v _ 1 ) \\cdot v ) = ( t _ { V _ 1 } ( v _ 1 ) , h ( t _ { V _ 1 } ( v _ 1 ) ) \\cdot 0 ^ { V _ 2 } + \\Phi ( v _ 1 ) \\cdot v ) , \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} S = Z . \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} E | _ E = - l _ 1 - ( n - 1 ) l _ 2 , E ^ 3 = ( - l _ 1 - ( n - 1 ) l _ 2 ) ^ 2 = 2 n - 2 . \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} x ^ k = ( x ' , x _ n + \\lambda ^ k r ) y ^ k = ( y ' , y _ n + \\lambda ^ k r ) , \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} G _ { k , N } ^ \\chi ( Z _ 1 , Z _ 2 , r ; Q _ \\tau ) & = \\chi ( \\tau ) ^ { - 2 } \\sum _ { F \\in \\mathfrak { B } } C _ N ( \\pi _ F , \\chi , r ) \\frac { F ( Z _ 1 ) \\bar { F } ( Z _ 2 ) } { \\langle F , F \\rangle } \\\\ & = \\chi ( \\tau ) ^ { - 2 } \\sum _ { \\pi \\in \\Pi _ N ( \\mathbf { k } ) } C _ N ( \\pi , \\chi , r ) \\sum _ { \\substack { F \\in \\mathfrak { B } \\\\ F \\in V _ N ( \\pi ) } } \\frac { F ( Z _ 1 ) \\bar { F } ( Z _ 2 ) } { \\langle F , F \\rangle } . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} \\sup \\limits _ { t \\in [ 0 , T ] } \\sum \\limits _ { k = 1 } ^ { \\infty } \\langle T _ t h _ k , h _ k \\rangle < \\infty \\qquad \\lim \\limits _ { N \\rightarrow \\infty } \\sup \\limits _ { t \\in [ 0 , T ] } \\sum \\limits _ { k = N } ^ { \\infty } \\langle T _ t h _ k , h _ k \\rangle = 0 . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} x : = \\begin{bmatrix} \\widehat { K } _ 1 ( \\lambda _ 0 ) & 0 \\end{bmatrix} z \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} A ( y ^ { k + 1 } - y ^ k ) - D _ N H _ \\gamma ( p ^ k ) ( p ^ { k + 1 } - p ^ k ) = - A y ^ k + H _ \\gamma ( p ^ k ) . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { B ^ c _ { r _ { k k _ 0 } } } P _ { r _ { k k _ 0 } } \\tilde { g } d y = \\int _ { B _ { r _ { ( k - 1 ) k _ 0 } } \\backslash B _ { r _ { k k _ 0 } } } P _ { r _ { k k _ 0 } } \\tilde { g } d y + \\int _ { B ^ c _ { r _ { ( k - 1 ) k _ 0 } } } P _ { r _ { k k _ 0 } } \\tilde { g } d y . \\end{aligned} \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} h ( z ) = h ( z ; \\tau ) : = \\int _ { \\mathbb { R } } \\frac { \\cosh ( 2 \\pi z w ) } { \\cosh ( \\pi w ) } e ^ { \\pi i \\tau w ^ 2 } d w , \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ Z _ { \\varepsilon } ( q \\beta ) Z ^ { - n } _ { \\varepsilon } ( \\beta ) \\Bigr ] \\approx N ^ { 1 + q ^ 2 \\beta ^ 2 - ( 1 + \\beta ^ 2 ) n } \\prod \\limits _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma ( 1 - q \\beta ^ 2 + ( j + 1 ) \\beta ^ 2 ) ^ 2 \\ , \\Gamma ( 1 - \\beta ^ 2 ) } { \\Gamma ( 1 - 2 q \\beta ^ 2 + ( j + 1 ) \\beta ^ 2 ) \\ , \\Gamma ( 1 + j \\beta ^ 2 ) } . \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} & { \\rm E x p } _ { ( A _ r , B _ r , C _ r ) } ( \\xi , \\eta , \\zeta ) \\\\ : = & ( A _ r ^ { \\frac { 1 } { 2 } } \\exp ( A _ r ^ { - \\frac { 1 } { 2 } } \\xi A _ r ^ { - \\frac { 1 } { 2 } } ) A _ r ^ { \\frac { 1 } { 2 } } , B _ r + \\eta , C _ r + \\zeta ) \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\sum _ { k = k ( n ) } ^ { \\infty } \\omega ( 2 ^ { - k } ) \\leq C \\omega ( 2 ^ { - k ( n ) } ) \\leq C \\omega ( 2 ^ { - n } \\cdot \\| M _ 1 M _ 2 \\| ) . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\lim _ { s \\searrow 0 } \\frac { f ( \\alpha ( s ) ) - f ( \\alpha ( 0 ) ) } { s } = g _ { \\dot \\gamma ( f ( p ) ) } ( \\dot \\gamma ( f ( p ) ) , \\dot \\alpha ( 0 ) ) \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\ , x _ j ^ { \\uparrow } \\ge \\sum _ { j = 1 } ^ { k } \\ , y _ j ^ { \\uparrow } . \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} [ \\bar p _ { 1 } , \\bar p _ i ] = 0 , [ \\bar \\rho , \\bar p _ i ] = i \\bar p _ i . \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} g \\Bigl ( \\frac { x + z ^ q } { x + z } \\Bigr ) = ( 1 + a + b ) ^ { q - 1 } \\frac { y + z ^ q } { y + z } . \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} P _ m = \\prod _ { n \\geq m } ( 1 - p _ n ) = \\textrm { E x p } \\left ( \\sum _ { n \\ge m } \\log ( 1 - p _ n ) \\right ) . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\textbf { E } ^ \\rho _ { \\alpha , \\beta , \\omega , a ^ + } \\varphi ( x ) = \\int _ a ^ x ( x - t ) ^ { \\beta - 1 } E _ { \\alpha , \\beta } ^ \\rho ( \\omega ( x - t ) ^ \\alpha ) \\varphi ( t ) d t , ~ ~ x > a . \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} \\eta _ k ( t ) : = \\tilde \\omega _ k ( t - \\delta _ k ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\xi _ k ( t ) : = ( \\tilde v _ 3 ) _ k ( t - \\delta _ k ) , \\mbox { f o r } t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} ~ ~ \\mathbf { R } ^ s _ i ( X ) = \\int _ 0 ^ 1 ( 1 - t ) \\frac { d ^ 2 } { d t ^ 2 } \\mathbf { u } ^ s ( Y _ i ( t , X ) ) d t , ~ i \\in \\mathcal { I } ^ s , ~ \\forall X \\in T _ { \\ast } ^ s . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\sum _ { k \\leq x } \\frac { 1 } { k } \\sum _ { J = 1 } ^ { k } c _ { k } ( j ) \\log j & = \\left ( \\frac { \\log \\sqrt { 2 \\pi } } { \\zeta ( 2 ) } + \\frac { \\zeta ' ( 2 ) } { 2 \\zeta ^ 2 ( 2 ) } + \\frac { \\Theta } { \\zeta ( 3 ) } \\right ) x + O \\left ( ( \\log x ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} \\frac { | c ( | D | ) | ^ 2 } { \\langle h , h \\rangle } = 2 ^ { \\nu ( N ) } \\frac { ( k - 1 ) ! } { \\pi ^ k } | D | ^ { k - 1 / 2 } \\left ( \\prod _ { p \\mid M } \\frac { p } { p + 1 } \\right ) \\frac { L ( f , D , k ) } { \\langle f , f \\rangle } . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} - \\int _ { B _ R } \\int _ 0 ^ T u \\ , \\varphi _ t \\ , d t \\ , d \\mathcal { V } = \\int _ { B _ R } u _ 0 ( x ) \\ , \\varphi ( x , 0 ) \\ , d \\mathcal { V } ( x ) + \\int _ { B _ R } \\int _ 0 ^ T \\left ( u ^ m \\ , \\Delta \\varphi + u ^ p \\ , \\varphi \\right ) d t \\ , d \\mathcal { V } \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\int _ { \\Omega } U _ j ^ { 2 ^ { * } - 2 } Z _ { j , h } Z _ { 1 , l } = ( \\bar { c } + o ( 1 ) ) \\delta _ { h l } \\lambda ^ { 2 n _ l } , \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{gather*} W _ k ( z ; 0 ) = \\begin{cases} { \\rm i } ^ k \\sqrt { \\dfrac { \\pi } { 2 } } { \\rm e } ^ { { \\rm i } z } , & \\mbox { i f } \\Im ( z ) > 0 , \\\\ 0 , & \\mbox { i f } \\Im ( z ) < 0 . \\end{cases} \\end{gather*}"}
-{"id": "2914.png", "formula": "\\begin{align*} W ( E _ N ) = \\max _ { a , b , t } \\left | \\sum _ { j = 0 } ^ { t - 1 } e _ { a + j b } \\right | , \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l } g _ { i , 0 } \\ ! \\ ! \\ ! & = \\overline g _ i & \\qquad \\forall i = 1 , \\ldots , n , \\\\ g _ { i , t } ' + \\sum _ { j = 1 } ^ n H _ { j , i , t } \\ , g _ { j , t } \\ ! \\ ! \\ ! & = 0 & \\qquad \\forall i = 1 , \\ldots , n \\mbox { a n d f o r } \\mathcal { L } ^ 1 \\mbox { - a . e . \\ } t \\in [ 0 , 1 ] . \\end{array} \\right . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} \\tfrac { a } { c } , \\ ; a , c \\in \\Z , \\ ; c \\geq 1 , \\ ; c \\mid N \\gcd ( c , a ) = 1 . \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} s ( t ) = L _ { 1 } + \\int _ { t } ^ { T } F ( s ( r ) ) d r , \\ ; t \\in \\lbrack 0 , T ] . \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} \\Sigma _ N & = \\sum _ { ( n , v w ) = 1 } G \\left ( \\frac { n } { N } \\right ) e \\left ( \\frac { n \\overline { m \\ell _ 1 \\ell _ 3 v } } { w } \\right ) . \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\overline w ( t _ 2 ) - \\overline w ( t _ 1 ) = i \\int _ { t _ 1 } ^ { t _ 2 } e ^ { - i s ( \\Delta ^ 2 _ x - \\kappa \\Delta _ x ) } g ( w , p ) d s . \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} S _ { M } ( z \\ , | \\ , a ) = S _ { M - 1 } ( z \\ , | \\ , \\hat { a } _ i ) \\ , S _ M \\bigl ( z + a _ i \\ , | \\ , a \\bigr ) , \\ , i = 1 \\cdots M , \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\frac { 1 } { \\epsilon } L ( w ( R _ 1 ( \\epsilon ) , R _ 2 ( \\epsilon ) ) , x _ 0 ) = L ( w ( R _ 1 , R _ 2 ) , ( 0 , 0 ) ) , \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} { n \\brack \\delta } a ' _ 1 + \\sum _ { r = 1 } ^ { n - \\delta } { n - r \\brack \\delta } ( q ^ \\delta A ' _ r + B ' _ r ) = { n \\brack \\delta } ( 1 + q ^ \\delta ) ( q ^ { ( m + 1 ) ( n - \\delta ) } q ^ \\delta - \\abs { Y } ) . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\varphi ( u _ { * * } , y _ { * * } ) = \\frac { 4 y _ { * * } ^ 4 } { 2 8 y _ { * * } ^ 4 + 2 2 y _ { * * } ^ 2 - 1 } , \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} ( \\mu \\wedge d F _ { i _ 1 } \\wedge \\dots \\wedge d F _ { i _ j } ) | _ { \\C ( X _ I ) } = 0 , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} K [ u ] = \\inf _ { v \\in W ^ { 1 , 1 } ( \\Omega , \\R ^ n ) } J _ \\psi [ v ] \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} d x _ { t } ( \\zeta ) = \\overline { x } _ { t } ( \\zeta ) d t + \\sum _ { r = 1 } ^ { 3 } x _ { t } ^ { r } ( \\zeta ) d B _ { t } ^ { ( r ) } , \\ \\ x = e , h , f . \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} { \\tilde z } _ 1 = z _ 1 , \\ ; \\ ; \\ ; { \\tilde z } _ 2 = z _ 2 , \\ ; \\ ; \\ ; { \\tilde z } _ 3 = z _ 3 , \\ ; \\ ; \\ ; { \\tilde z } _ 4 = z _ 4 . \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} & \\| \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i + \\rho ( U ^ k ) ^ * \\rho ( S _ \\alpha S _ \\alpha ^ * ) \\rho ( U ) ^ k e _ i \\| ^ 2 = \\\\ & = \\| \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i + \\rho ( U ^ k ) ^ * \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i \\| ^ 2 = \\\\ & = \\| \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i \\| ^ 2 + \\| \\rho ( U ^ k ) ^ * \\rho ( S _ \\alpha S _ \\alpha ^ * ) e _ i \\| ^ 2 = 2 \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} N ( \\varphi ) = | U _ * + \\varphi | ^ { 2 ^ { * } - 2 } ( U _ * + \\varphi ) - | U _ * | ^ { 2 ^ { * } - 2 } U _ * - ( 2 ^ * - 1 ) | U _ * | ^ { 2 ^ { * } - 2 } \\varphi , \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} c _ 0 & + b _ 0 = \\frac { 1 } { \\theta _ 0 ^ 2 } \\\\ c _ 0 & = \\sum _ { l = 0 } ^ { K - 1 } \\frac { \\theta _ l } { 1 - \\theta _ l } \\frac { \\theta _ { - 1 } ^ 2 } { \\theta _ { l - 1 } ^ 2 } \\prod _ { j = 1 } ^ { l } \\frac { 1 } { 1 - \\sigma ^ K _ { j } } b _ 0 \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} P = A \\rho ^ { n } - \\frac { B } { \\rho ^ { \\alpha } } , \\ \\ \\ 1 \\leq n \\leq 3 , \\ \\ 0 < \\alpha \\leq 1 , \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} \\norm { P _ N ^ \\perp F f } _ { \\ell ^ 2 } ^ 2 & = \\sum _ { l = N + 1 } ^ { + \\infty } | F f ( l ) | ^ 2 \\\\ & \\le C ( A , d , \\rho ) M ^ 4 \\sum _ { l = N + 1 } ^ { + \\infty } \\frac { \\log ^ { 2 d - 2 } ( l + 1 ) } { l ^ 2 } \\\\ & \\le C ( A , d , \\rho ) M ^ 4 \\frac { \\log ^ { 2 d - 2 } ( N ) } { N } . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} & \\sum _ { a b c ^ 2 = n } \\lambda _ f ( a ) \\lambda _ f ( b ) \\mu ( b ) \\mu ( b c ) ^ 2 F _ { \\Upsilon , M } ( b c ) = \\sum _ { a b c ^ 2 = n } \\sum _ { d | ( a , b ) } \\lambda _ f \\left ( \\frac { a b } { d ^ 2 } \\right ) \\mu ( b ) \\mu ( b c ) ^ 2 F _ { \\Upsilon , M } ( b c ) , \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} | \\mathcal { S P F } _ n | = C _ { n - 1 } ( n - 1 ) ! \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\tau ( w ) = \\sum _ { 1 \\le | \\gamma | \\le N - 1 } c _ { \\gamma } ( \\tau ) w ^ { \\gamma } + \\sum _ { | \\gamma | = N } c _ { \\gamma } ( \\tau , w ) w ^ { \\gamma } \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c c } 0 & 0 & - 1 \\\\ 0 & Q _ 1 & 0 \\\\ - 1 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} ( \\ , \\check e _ i ) ^ { 1 - A _ { j i } } \\check e _ j = 0 , ( \\ , \\check f _ i ) ^ { 1 - A _ { j i } } \\check f _ j = 0 . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} \\abs { \\mathcal J ( t ) - \\mathcal J ( t _ n ) } = o ( 1 ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; t \\uparrow \\ , \\infty , \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u ( t , x ) = A u ( t , x ) , & x \\ge 0 , \\ t \\in [ 0 , T ] , \\\\ u ( 0 , x ) = f ( x ) , & x \\ge 0 , \\end{cases} \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} T = \\left ( \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) \\right ) _ { m , n \\in \\N ^ N } \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} T ( s ) = L ( f \\otimes f , s ) . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\alpha _ 1 \\kappa _ 1 + \\dots + \\alpha _ s \\kappa _ s & = \\log | \\det ( \\Lambda _ \\omega ) | \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log | \\det ( M ^ n _ \\omega ) | \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } \\log | \\det ( M _ { T ^ i \\omega } ) | . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} \\| f ( \\tau ) \\| _ { \\mathcal { H } ^ k ( \\Omega ) } : = \\left ( \\sum _ { | \\alpha | \\leq k } \\| \\partial ^ { \\alpha } f ( \\tau , \\cdot ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} f ^ { o , \\kappa , h } _ C ( \\rho , j ) \\coloneqq w _ { j , i _ { \\kappa } } \\sum \\limits _ { n = 0 } ^ { n _ { \\kappa } + 1 } \\biggl ( \\rho \\pi ^ { o , \\kappa , h , + } _ { i _ { \\kappa } , k } ( n ) + \\frac { 1 } { h } \\pi ^ { o , \\kappa , h } _ { \\rho } ( n ) \\biggr ) n \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} B _ { 1 } & = \\{ 4 , 6 , 7 , 9 , 1 0 , 1 1 \\} , & B _ { 5 } & = \\{ 2 , 3 , 4 , 8 , 1 0 , 1 1 \\} , & B _ { 9 } & = \\{ 1 , 3 , 4 , 6 , 7 , 8 \\} , \\\\ B _ { 2 } & = \\{ 1 , 5 , 7 , 8 , 1 0 , 1 1 \\} , & B _ { 6 } & = \\{ 1 , 3 , 4 , 5 , 9 , 1 1 \\} , & B _ { 1 0 } & = \\{ 2 , 4 , 5 , 7 , 8 , 9 \\} , \\\\ B _ { 3 } & = \\{ 1 , 2 , 6 , 8 , 9 , 1 1 \\} , & B _ { 7 } & = \\{ 1 , 2 , 4 , 5 , 6 , 1 0 \\} , & B _ { 1 1 } & = \\{ 3 , 5 , 6 , 8 , 9 , 1 0 \\} . \\\\ B _ { 4 } & = \\{ 1 , 2 , 3 , 7 , 9 , 1 0 \\} , & B _ { 8 } & = \\{ 2 , 3 , 5 , 6 , 7 , 1 1 \\} , \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} \\lambda ^ { - 1 } x = c \\cdot \\frac { x } { \\norm { x } } \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} L _ t ( m _ 1 ) L _ t ( m _ 2 ) = \\sum _ { m \\in \\mathbb { B } ^ { \\xi , \\flat } } c _ { m _ 1 , m _ 2 } ^ { m } L _ t ( m ) . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} \\nabla = d + \\Bigg ( p _ { - 1 } - \\frac { \\varphi } { h ^ \\vee } \\rho + v _ 0 \\delta + \\sum _ { j \\in E } v _ j p _ j \\Bigg ) d z , \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} e _ k ( x ) = \\sqrt { 2 } \\begin{cases} \\cos ( 2 \\pi k \\cdot x ) , & k \\in \\Z ^ 2 _ + , \\\\ \\sin ( 2 \\pi k \\cdot x ) , & k \\in \\Z ^ 2 _ - . \\end{cases} \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} Q _ { i + 1 } = \\sum _ { j = 1 } ^ { i } c _ { j } Q _ { j } . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} S ( x , y ) = Q ( x + y ) - Q ( x ) - Q ( y ) , \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} e ^ { - t \\Lambda _ { C _ \\infty } ( a , \\nabla a + b ) } : = s \\mbox { - } C _ \\infty \\mbox { - } \\lim _ n e ^ { - t \\Lambda _ { C _ \\infty } ( a _ n , \\nabla a _ n + b _ n ) } ( t \\geq 0 ) , \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} P _ i = \\begin{pmatrix} \\alpha _ i & \\beta _ i \\cr \\gamma _ i & \\delta _ i \\end{pmatrix} \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} Z ( s , f , \\phi ) ( g ) = & \\frac { L ^ S ( ( 2 n + 1 ) s + 1 / 2 , \\pi \\boxtimes \\chi , \\varrho _ { 2 n + 1 } ) } { L ^ S ( ( 2 n + 1 ) ( s + 1 / 2 ) , \\chi ) \\prod _ { j = 1 } ^ n L ^ S ( ( 2 n + 1 ) ( 2 s + 1 ) - 2 j , \\chi ^ 2 ) } \\\\ & \\times i ^ { n k } \\ , \\pi ^ { n ( n + 1 ) / 2 } c ( ( 2 n + 1 ) s - 1 / 2 ) \\phi ( g ) , \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\tau ^ l \\frac { \\Gamma \\bigl ( 2 - 2 l + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) \\bigr ) } { \\Gamma \\bigl ( 2 - l + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) \\bigr ) } = \\prod \\limits _ { j = 0 } ^ { l - 1 } \\frac { 1 } { 1 + \\lambda _ 1 + \\lambda _ 2 - ( l + j - 1 ) / \\tau } . \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} { \\left | { \\hat h _ { \\max } ^ { { \\rm { r o u g h } } } } \\right | ^ { \\rm { 2 } } } = { \\sum \\limits _ { { { \\bf { r } } _ n } \\in { I _ 1 } } ^ { } { \\max \\left ( { { { \\left | { r _ { n , 1 } ^ { } } \\right | } ^ { \\rm { 2 } } } , { { \\left | { r _ { n , 2 } ^ { } } \\right | } ^ { \\rm { 2 } } } } \\right ) } } / { \\left | { { I _ 1 } } \\right | } - { N _ 0 } \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} & \\Big ( A ^ { \\frac { 1 } { 2 } } \\big ( ( 1 - \\lambda ) A ^ { - 1 } + \\lambda B ^ { - 1 } \\big ) A ^ { \\frac { 1 } { 2 } } \\Big ) ^ { \\frac { 1 } { 2 } } A ^ { - \\frac { 1 } { 2 } } \\big ( ( 1 - \\lambda ) A + \\lambda B \\big ) A ^ { - \\frac { 1 } { 2 } } \\Big ( A ^ { \\frac { 1 } { 2 } } \\big ( ( 1 - \\lambda ) A ^ { - 1 } + \\lambda B ^ { - 1 } \\big ) A ^ { \\frac { 1 } { 2 } } \\Big ) ^ { \\frac { 1 } { 2 } } \\\\ & - I = \\lambda ( 1 - \\lambda ) A ^ { - \\frac { 1 } { 2 } } ( A - B ) B ^ { - 1 } ( A - B ) A ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} ( \\Phi _ d ( \\lambda ) ^ T \\otimes I _ n ) M _ \\phi ( \\lambda ) = \\kappa _ { d - 1 } P ( \\lambda ) , \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} i _ ! \\circ i ^ \\ast ( P _ ! ( \\theta ) ) = P _ ! ( \\theta ) u . \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} \\small \\left ( \\begin{array} { c } \\binom { s _ { i , 1 } } { 1 } + t _ { i , 2 } \\binom { s _ { i , 2 } } { 1 } + \\cdots + t _ { i , k _ { i } } \\binom { s _ { i , k _ { i } } } { 1 } , \\ldots , \\binom { s _ { i , 1 } } { n } + t _ { i , 2 } \\binom { s _ { i , 2 } } { n } + \\cdots + t _ { i , k _ { i } } \\binom { s _ { i , k _ { i } } } { n } \\end{array} \\right ) . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} \\mbox { I m } \\ , T = \\ker d _ 2 ^ { n - 2 , \\ , n } , \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} u _ { n } | | \\mathbb { P } _ { n } ^ { ( 0 ) } - P | | _ { \\mathcal { F } } < 1 + \\varepsilon , u _ { n } = \\sqrt { \\frac { n } { 2 \\sigma _ { \\mathcal { F } } ^ { 2 } L \\circ L ( n ) } } , \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} 1 _ { \\lambda } & = 0 & \\displaybreak [ 1 ] \\\\ 1 _ { \\lambda } 1 _ { \\mu } & = \\delta _ { \\lambda , \\mu } 1 _ { \\lambda } , \\sum _ { \\lambda \\in \\Z ^ n } 1 _ { \\lambda } = 1 \\displaybreak [ 1 ] \\\\ E _ i 1 _ { \\lambda } & = 1 _ { \\lambda + \\alpha _ i } E _ i , F _ i 1 _ { \\lambda } = 1 _ { \\lambda - \\alpha _ i } F _ i \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} y + P = x + z ( y ) + u _ x ( z ( y ) ) + P = x + u _ x ( z ( y ) ) + P . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} _ { q / 2 } \\Bigl ( \\frac { a b } { 1 + a ^ 2 + b ^ 2 } \\Bigr ) \\ , & = _ { q / 2 } \\Bigl ( \\frac { a b } { 1 + b } \\Bigr ) = _ { q / 2 } \\Bigl ( a + \\frac a { 1 + b } \\Bigr ) \\cr & = _ { q / 2 } \\Bigl ( \\frac { a ^ 2 } { ( 1 + b ) ^ 2 } \\Bigr ) = _ { q / 2 } \\Bigl ( \\frac b { 1 + b } \\Bigr ) = 0 . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} \\psi ( ( a b ^ i ) ^ p ) = \\big ( a ^ { i ( e _ 2 + \\dots + e _ { p - 1 } ) } b ^ i a ^ { i e _ 1 } , a ^ { i ( e _ 3 + \\dots + e _ { p - 1 } ) } b ^ i a ^ { i ( e _ 1 + e _ 2 ) } , \\dots , a ^ { i ( e _ 1 + \\dots + e _ { p - 1 } ) } b ^ i \\big ) . \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} C = G ( v _ n , c _ n ) \\to G ( v _ 0 , c _ 0 ' ) \\neq G ( v _ 0 , c _ 0 ) = C \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} S ( a , b ) = \\textrm { $ \\{ c \\in \\Q $ s u c h t h a t $ - a v _ \\ell ( q ) \\leq v _ \\ell ( c ) \\leq b v _ \\ell ( q ) , $ f o r e v e r y p r i m e $ \\ell \\} $ } . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} L _ Q ( \\lambda ) : = L _ \\mathcal { L } ( \\lambda ) \\begin{bmatrix} \\widehat { K } _ 2 ( \\lambda ) ^ T \\\\ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K P _ k ( m , n ) \\ll K . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} e ^ 2 _ { N , s } ( Y _ 1 z _ 1 , \\ldots , Y _ s z _ s ) & = - 1 + \\frac { 1 } { b ^ m } \\sum _ { k = 0 } ^ { b ^ m - 1 } q ( k ) . \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{gather*} \\int _ { B } g = \\int _ { r = 0 } ^ { R } e ^ { - r } \\sigma _ { 4 } r ^ { 4 } = 4 ! \\ , \\sigma _ { 4 } - e ^ { - R } \\sigma _ { 4 } ( R ^ { 4 } + 4 R ^ { 3 } + 1 2 R ^ { 2 } + 2 4 R + 2 4 ) . \\end{gather*}"}
-{"id": "4944.png", "formula": "\\begin{align*} ( f \\cup g ) ( \\gamma _ 1 , \\ldots , \\gamma _ { p + q } ) = f ( \\gamma _ 1 , \\ldots , \\gamma _ p ) g ( \\gamma _ { p + 1 } , \\ldots , \\gamma _ { p + q } ) , \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} \\beta _ V . D + \\sum _ { x } \\pm w _ e = 0 \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} r ^ j ( \\mathbf { S } , \\delta ) = \\pi _ 0 ^ j R ^ j _ 0 ( \\mathbf { S } , \\delta ) + \\pi _ 1 ^ j R ^ j _ 1 ( \\mathbf { S } , \\delta ) \\ ; , \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\zeta \\\\ \\varphi \\end{array} \\right ) = \\left ( \\begin{array} { c c } \\frac { 1 } { 2 } I & 0 \\\\ 0 & B \\\\ \\end{array} \\right ) \\left ( \\begin{array} { c } \\ell \\\\ \\psi \\end{array} \\right ) , \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} \\mathcal I _ 1 ( n ) = - ( 1 - p ^ { - 1 } ) + ( 1 - p ^ { - 1 } ) ^ 2 + ( 1 - p ^ { - 1 } ) ( p ^ { n - 1 } - 1 ) = p ^ { - 1 } ( 1 - p ^ { - 1 } ) ( p ^ n - p - 1 ) , \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} \\mathbf Y = \\begin{pmatrix} y _ { 1 , 1 } & \\cdots & y _ { 1 , n } \\\\ \\vdots & & \\vdots \\\\ y _ { m , 1 } & \\cdots & y _ { m , n } \\end{pmatrix} \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} f ( A ) : = \\int _ { \\sigma ( A ) } f ( \\lambda ) d P ^ { ( A ) } ( \\lambda ) \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} C \\langle y \\mid \\dd ( y ) = z _ { f } \\otimes 1 \\rangle \\otimes _ { R } M \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} \\R _ \\infty = \\R \\setminus [ - S _ \\infty , S _ \\infty ] , \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} u ( 0 ) ( A + \\alpha u ( 0 ) ) = \\beta u ( 0 ) \\dot { u } ( 0 ) \\leq 0 \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} r _ { 1 } & = - \\frac { 3 } { 4 } w _ { 3 } b _ { 4 } \\left ( b _ { 1 } ^ { - 1 } - 1 \\right ) - \\frac { 1 } { 2 } b _ { 4 } w _ { 1 } - \\frac { 1 } { 2 } u _ { 1 } b _ { 3 } , \\\\ r _ { 3 } & = \\frac { 1 } { 2 } u _ { 1 } b _ { 1 } , \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} P _ N U ( q _ 1 ) - P _ N U ( q _ 2 ) & = P _ N F ( q _ 1 - q _ 2 ) + P _ N ( B ( q _ 1 ) - B ( q _ 2 ) ) \\\\ & = F ( q _ 1 - q _ 2 ) - P _ N ^ \\perp F ( q _ 1 - q _ 2 ) + P _ N ( B ( q _ 1 ) - B ( q _ 2 ) ) , \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ M \\frac { 2 ^ { - k } r } { \\varphi ( 2 ^ { - k } r ) } V ( 6 \\cdot 2 ^ { - k } r ) & \\le \\sum _ { k = 1 } ^ M \\frac { 2 ^ { - k } r } { V ( 2 ^ { - k } r ) } = \\sum _ { k = 1 } ^ M \\int _ { 2 ^ { - k } r } ^ { 2 ^ { - k + 1 } r } \\frac { 1 } { V ( 2 ^ { - k } r ) } d s \\\\ & \\le \\int _ 0 ^ r \\frac { 1 } { V ( s ) } d s \\le c _ 5 \\frac { r } { V ( r ) } , \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} f _ 0 ( M _ 0 ) = - \\frac { 1 } { 4 \\pi } \\int \\limits _ { B _ { R _ 0 } ( \\mathbb { O } ) } \\frac { \\Delta f _ 0 ( M ) } { \\rho _ { M _ 0 } ( M ) } \\ , d m _ 3 ( M ) . \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} I _ { s , t } ^ { \\alpha } - I _ { 0 , t } ^ { \\alpha } = - \\alpha \\ , \\int _ 0 ^ s \\widehat I _ { u , t } ^ { \\ , \\alpha - 1 } \\ , { \\rm e } ^ { - \\alpha X _ u } \\ , d u . \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\displaystyle \\int _ { B ( x _ { 0 } , \\epsilon ) } | u _ { n } - u _ { m } | ^ { q } d x & = \\displaystyle \\int _ { B ( x _ { 0 } , \\epsilon ) } | a ( x ) | ^ { q } | a ( x ) | ^ { - q } | u _ { n } - u _ { m } | ^ { q } d x \\\\ & \\leq c \\| | a ( x ) | ^ { - q } \\chi _ { B ( x _ { 0 } , \\epsilon ) } \\| _ { ( \\frac { p ( x ) } { q } ) ' } \\| | a ( x ) | ^ { q } | u _ { n } - u _ { m } | ^ { q } \\| _ { \\frac { p ( x ) } { q } } , \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} v _ a \\mapsto \\omega ^ { a j } v _ j \\mapsto \\omega _ { k a } \\omega ^ { k j } v _ j = v _ a . \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{gather*} \\Phi _ m ( \\nu ) = \\gcd \\left ( [ m ] _ \\nu / [ m / p _ 1 ] _ \\nu , \\ldots , [ m ] _ \\nu / [ m / p _ t ] _ \\nu \\right ) , \\\\ \\Phi _ { p _ k } ( \\nu ^ { m / p _ k } ) = [ m ] _ { \\nu } / [ m / p _ k ] _ { \\nu } , k = 1 , \\ldots , t , \\\\ \\Phi _ n ( \\nu ) = \\Phi _ { m } ( \\nu ^ { n / m } ) . \\end{gather*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\inf _ { u \\in B V ( \\Omega , \\R ^ N ) } K [ u ] = \\inf _ { w \\in W ^ { 1 , 1 } ( \\Omega , \\R ^ N ) } J [ w ] \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} H _ { \\alpha \\alpha _ 2 } ( t ) = S _ { \\alpha \\alpha _ 2 } ( t ) + \\sum _ { l = 1 } ^ \\infty \\int _ 0 ^ t \\int _ 0 ^ { t _ 1 } \\cdots \\int _ 0 ^ { t _ { l - 1 } } \\Omega ^ { ( l ) } _ { \\alpha \\alpha _ 2 } ( t , t _ 1 , \\dots , t _ n ) d t _ l d t _ { l - 1 } \\cdots d t _ 1 . \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} A ( t ) & : = \\frac { c _ p } { c _ \\mu } \\left ( \\gamma - 2 ( \\alpha + 1 ) \\right ) \\rho ( x _ t ) ^ { \\gamma - \\alpha } \\\\ B ( t ) & : = - \\frac { 1 } { c _ \\mu } ( \\alpha + 1 ) \\rho ( x _ t ) ^ { - \\alpha } \\\\ C ( t ) & : = \\frac { c ^ 2 _ p } { c _ \\mu } \\left ( \\gamma - ( \\alpha + 1 ) \\right ) \\rho ( x _ t ) ^ { 2 \\gamma - \\alpha } . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} P _ a ( z , z ' ) = K _ k ( z , z ' ) - K _ k ( z , a ) - K _ k ( a , z ' ) + K _ k ( a , a ) = K _ k ( z , z ' ) - 1 \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} \\tilde { Z } ^ f _ { \\beta , N } = \\sum _ { t = 1 } ^ n \\tilde { Z } ^ c _ { \\beta , t } P ( N - t ) \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} P ^ { n - k } \\ 1 ( y ) ~ = ~ \\P _ y ( \\tau _ \\vartheta > n - k ) ~ \\geq ~ \\P _ y ( \\tau _ \\vartheta > n ) ~ = ~ P ^ n \\ 1 ( y ) \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} S _ { d , k } ( n ) = \\sum _ { i = d + 1 } ^ { k + 1 } S _ { d , k } ( n - i ) \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { n \\leq x } \\frac { ( \\mu * \\mu ) ( n ) } { n } \\Delta \\left ( \\frac { x } { n } \\right ) + O \\left ( ( \\log x ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} B _ \\lambda ( s ) & = 2 ^ n \\int \\limits _ A \\int \\limits _ N f _ k ( Q _ n \\cdot ( a n , 1 ) ) w _ \\lambda ( a n ) \\ , d n \\ , d a \\\\ & = 2 ^ n \\int \\limits _ A \\int \\limits _ N \\bigg ( \\prod _ { j = 1 } ^ n a _ j ^ { \\ell _ { n + 1 - j } + n + 1 - j } \\bigg ) f _ k ( Q _ n \\cdot ( a n , 1 ) ) \\ , d n \\ , d a . \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} | \\mu _ { w _ { \\langle 0 , n - 1 \\rangle } } ( P ) - \\nu ( P ) | = \\left | \\frac 1 n \\sum _ { i = 0 } ^ { n - 1 } 1 _ P ( \\sigma ^ i ( \\omega ) ) - \\nu ( P ) \\right | < \\eta . \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{gather*} \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } ( x ) \\bigr ) ( q \\ , | \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } ( x ) \\bigr ) ( 0 \\ , | \\ , b ) = M ! \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j \\Bigl ( B ^ { ( f ) } _ { 1 } ( q + b _ 0 ) - B ^ { ( f ) } _ { 1 } ( b _ 0 ) \\Bigr ) . \\end{gather*}"}
-{"id": "5073.png", "formula": "\\begin{align*} { \\bf { C o v } } \\left [ V _ { \\varepsilon } ( \\psi ) , \\ , V _ { \\varepsilon } ( \\xi ) \\right ] = & \\begin{cases} - 2 \\ , \\log | e ^ { 2 \\pi i \\psi } - e ^ { 2 \\pi i \\xi } | , \\ , | \\xi - \\psi | > \\varepsilon , \\\\ - 2 \\log \\varepsilon , \\psi = \\xi , \\end{cases} \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N I _ n ( x + k ) = \\dfrac { B _ { n + 1 } ( x + N + 1 ) - B _ { n + 1 } ( x ) } { n + 1 } , \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} e ^ { \\beta } . ( s \\otimes e ^ { \\alpha } ) : = \\epsilon ( \\beta , \\alpha ) s \\otimes e ^ { \\alpha + \\beta } \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} f _ k ( Q _ n \\cdot ( a n , 1 ) ) & = i ^ { n k } \\bigg ( \\prod _ { j = 1 } ^ n \\frac 1 { a _ j + a _ j ^ { - 1 } } \\bigg ) ^ { ( 2 n + 1 ) ( s + \\frac 1 2 ) } \\ ! f _ k ( \\left [ \\begin{smallmatrix} I _ n \\\\ & I _ n \\\\ & ^ t Z & I _ n \\\\ Z & & & I _ n \\end{smallmatrix} \\right ] \\ ! \\left [ \\begin{smallmatrix} C & & & S \\\\ & C & S \\\\ & - S & C \\\\ - S & & & C \\end{smallmatrix} \\right ] \\ ! \\left [ \\begin{smallmatrix} I _ n & & - X \\\\ & I _ n \\\\ & & I _ n \\\\ & & & I _ n \\end{smallmatrix} \\right ] ) . \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\mathcal { B } ( t , x ) = \\mathcal { C } ( 1 + t , x ) = \\frac { C ( ( 1 + t ) x ) } { 1 - x C ( ( 1 + t ) x ) } , \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} \\mathbb { V } _ { 1 } = V \\left ( \\mathbb { V } _ { 1 } s i \\right ) \\subseteq H S P I S \\mathcal { T } _ { i } \\left ( P \\left ( { \\mathbf { 2 } } \\right ) \\right ) \\subseteq V \\left ( \\mathcal { T } _ { i } \\left ( P \\left ( { \\mathbf { 2 } } \\right ) \\right ) \\right ) = V \\left ( \\left \\{ \\mathcal { T } _ { i } \\left ( \\mathbf { 2 } ^ { \\kappa } \\right ) : i \\in \\left \\{ 1 , 2 \\right \\} , \\kappa \\right \\} \\right ) . \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\log \\lambda ^ { ( p ) } ( G ) & \\leq \\bigg ( 1 - \\frac { r } { p } \\bigg ) \\log r - \\frac { 1 } { p } \\log \\alpha \\\\ & = \\mu \\log \\lambda ^ { ( p _ 1 ) } ( G ) + ( 1 - \\mu ) \\log \\lambda ^ { ( p _ 2 ) } ( G ) . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\psi _ { w _ 0 } y _ 1 ^ { c _ 1 } \\cdots y _ j ^ { c _ j } y _ { j + 1 } ^ { c _ { j + 1 } } \\cdots y _ n ^ { c _ n } \\psi _ { w _ { 0 } } y _ { \\min } = - \\psi _ { w _ 0 } y _ 1 ^ { c _ 1 } \\cdots y _ j ^ { c _ { j + 1 } } y _ { j + 1 } ^ { c _ { j } } \\cdots y _ n ^ { c _ n } \\psi _ { w _ { 0 } } y _ { \\min } . \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} 0 = 2 \\left ( h _ { d , j } ^ * \\right ) ^ { ( r ) } ( 1 ) + r \\ , \\left ( h _ { d , j } ^ * \\right ) ^ { ( r - 1 ) } ( 1 ) - \\left ( h _ { d , j - 1 } ^ * \\right ) ^ { ( r ) } ( 1 ) , r = 1 , \\dots , j , \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k ( R - \\rho ) ^ { \\beta _ j } f _ j ( \\rho ) \\leq C \\sum _ { \\ell = 1 } ^ m A _ \\ell ( R - \\rho ) ^ { \\gamma _ \\ell } . \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} g \\tau g ^ { - 1 } & = \\sigma \\left ( \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\cdot \\tau \\right ) \\sigma ^ { - 1 } = \\rho ^ { \\left ( a _ { 3 } - 1 \\right ) } \\tau \\in \\left \\langle \\rho , \\tau \\right \\rangle \\ \\\\ g \\rho g ^ { - 1 } & = \\sigma \\left ( \\alpha _ { 1 } ^ { a _ { 1 } } \\alpha _ { 2 } ^ { a _ { 2 } } \\alpha _ { 3 } ^ { a _ { 3 } } \\cdot \\rho \\right ) \\sigma ^ { - 1 } = \\rho \\in \\left \\langle \\rho , \\tau \\right \\rangle , \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} 1 = c _ 1 < c _ 2 < \\cdots < c _ { d - 1 } < c _ d = n \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\phi = \\frac { \\epsilon } { \\gamma ( 1 - \\gamma ) } \\nu _ 0 \\psi , \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { r } \\frac { ( - a _ { j } z _ j q ; q ) _ { N - 1 } } { ( a _ { j } q ; q ) _ { N - 1 } } . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} \\mathcal { W } = \\Phi ^ * \\left ( \\mathcal { W } _ q \\right ) . \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} \\left ( \\overline { M } ^ s \\right ) ^ T \\alpha _ i = \\alpha _ i , ~ ~ i = 1 , 2 . \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} A = ( A \\cap \\overline B ) \\cup ( A \\cap B ) \\subseteq \\overline B \\cup C . \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} S _ 2 & = \\left ( 2 P _ 1 ( 1 ) P _ 3 ( 1 ) + P _ 3 ( 1 ) ^ 2 + \\frac { 1 } { \\theta _ 3 } \\int _ 0 ^ 1 P _ 3 ' ( x ) ^ 2 d x + \\kappa + \\lambda + o ( 1 ) \\right ) \\sum _ { q } \\Psi \\left ( \\frac { q } { Q } \\right ) \\frac { q } { \\varphi ( q ) } \\varphi ^ + ( q ) , \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} u ^ 2 = v ^ 3 = 1 . \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\delta \\tau ( n g , h ) = & n g \\cdot \\tau ( h ) - \\tau ( n g h ) + \\tau ( n g ) \\\\ = & g \\cdot \\tau ( h ) - f ( n ) - \\tau ( g h ) + f ( n ) + \\tau ( g ) = \\delta \\tau ( g , h ) \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} X _ n = [ U _ n ] . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} S _ k = \\{ x \\in \\mathrm { E n d } ( V ) \\mid \\mathrm { r a n k } ( x ) = n - k \\} . \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\epsilon ( k , \\lambda _ n ) = \\frac { k ^ 2 \\beta } { \\lambda _ n ( 1 - \\beta ) } . \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\beta > 1 b = 0 \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} \\begin{cases} ( \\imath _ { k - 1 } , \\imath _ k , \\imath _ { k + 1 } ) = ( \\imath , \\jmath , \\imath ) , \\\\ ( \\imath ' _ { k - 1 } , \\imath ' _ k , \\imath ' _ { k + 1 } ) = ( \\jmath , \\imath , \\jmath ) , \\\\ \\imath _ s = \\imath ' _ s \\ \\ s \\ \\ s \\neq k , k \\pm 1 , \\end{cases} \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} - \\Delta u ^ m = - \\Delta v \\ge v = u ^ p . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N ( { \\mathbf { b } ^ { S } _ { i } } ) ^ T \\mathbf { w } _ i \\leq C _ { \\mathrm { T o t a l } } , \\allowdisplaybreaks \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\tau ( x ) : = \\int _ { 0 } ^ { 1 } \\exp [ s \\log x ] d s . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t \\psi + \\psi \\partial _ x \\psi = \\mu ^ { - 1 / 2 } ~ \\partial _ x ^ 2 \\psi , \\end{array} \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} & \\# ( \\widehat { \\mathcal { P } } \\cap R D ) = \\# \\left ( ( \\mathcal { P } _ * \\cap R D ) \\setminus \\bigcup _ { c \\in C } ( \\mathcal { P } _ * \\cap c \\mathcal { P } _ * \\cap R D ) \\right ) \\\\ & = \\# ( \\mathcal { P } _ * \\cap R D ) - \\# \\left ( \\bigcup _ { i = 1 } ^ n ( \\mathcal { P } _ * \\cap c _ i \\mathcal { P } _ * \\cap R D ) \\right ) , \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} \\hat { \\omega } ( ( g , \\epsilon ) , 1 ) \\hat { \\phi } ( x ; y ) = \\omega ( ( g , \\epsilon ) , 1 ) \\phi _ 1 ( x ) \\cdot \\phi _ 2 ( y g ) \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} j ^ * \\alpha ^ + = \\alpha ; j ^ * \\beta ^ + = \\beta ; j ^ * \\gamma ^ + = \\gamma . \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\sup _ { n \\in \\Z } \\lVert y _ { n + 1 } - A _ n y _ n \\rVert = \\sup _ { n \\in \\Z } \\lVert g _ n ( y _ n ) \\rVert \\le \\delta . \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} f _ { \\varepsilon , u } ( x ) \\triangleq ( \\chi _ u \\star \\phi _ \\varepsilon ) ( x ) = \\frac { 1 } { \\varepsilon } \\int \\chi _ u ( x - y ) \\phi ( y / \\varepsilon ) \\ , d y . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} \\mathcal { Q } _ { t } ( x _ { t - 1 } , \\xi _ { [ t - 1 ] } , 0 ) = 0 . \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\mathbb { H } _ { p o l } \\ni x = ( a , b , c ) \\mapsto \\tau ( x ) = \\left ( \\frac { a } { 2 } , \\frac { b } { 2 } , \\frac { c } { 2 } - \\frac { a b } { 1 2 } \\right ) . \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} { \\bf u } ^ { \\tt M } = \\sum _ { x = 0 } ^ \\infty \\binom { x + \\beta - 1 } { x } c ^ x \\ , \\delta _ x . \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\phi _ I = \\sum _ { i = 1 } ^ m f _ i ( x _ 1 ^ i , \\dotsc , x _ { ( f _ i ) } ^ i ) \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} { } \\rho _ { n } \\varphi _ { n + 1 } ( z ) = z \\varphi _ { n } ( z ) - \\bar { \\alpha } _ { n } \\varphi _ { n } ^ { * } ( z ) \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} ( \\nabla _ { U } \\varphi ) V = u ( V ) A _ { N } U + B ( U , V ) \\xi , \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} \\Omega _ p ( r _ 1 ) \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ 1 \\Gamma _ { 0 0 } ) = \\frac { p - G ( 1 , p ) } { p ( p - 1 ) } \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ 1 \\Gamma _ { 0 0 } ) , \\Omega _ p ( r _ u ) \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ u \\Gamma _ { 0 0 } ) = \\frac { p - G ( u , p ) } { p ( p - 1 ) } \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ u \\Gamma _ { 0 0 } ) . \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} C ( N , M , \\chi ) = | \\chi ( 2 ) | ^ { - 2 } M ^ { 3 - k } N ^ { - 1 } \\prod _ { p \\mid N } ( p + 1 ) ^ 2 \\prod _ { p \\mid M } ( p + 1 ) . \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} \\zeta ( \\alpha , \\beta , s ) = \\sum _ { n + \\alpha > 0 } \\frac { e ( n \\beta ) } { ( n + \\alpha ) ^ s } \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} p ^ * X _ { i , t } = \\prod _ { j \\in I } A _ { j , t } ^ { \\varepsilon _ { j i , t } } . \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( c _ 2 ^ { - 2 } - c _ 1 ^ { - 2 } ) \\varphi d y = 0 , \\ \\ \\hbox { f o r a l l h a r m o n i c f u n c t i o n s } \\varphi . \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} Q _ k ( E _ N ) = \\max _ { a , b , t , D } \\left | \\sum _ { j = 0 } ^ t e _ { a + j b + d _ 1 } e _ { a + j b + d _ 2 } \\dots e _ { a + j b + d _ { k } } \\right | \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} T _ 1 & = \\frac { 1 } { | \\mathcal { Z } _ { N , w _ d } | } \\sum _ { z \\in \\mathcal { Z } _ { N , w _ d } } \\gamma _ { \\{ d \\} } ^ { \\lambda } \\sum _ { h _ d \\in \\mathcal { D } _ { \\{ d \\} } ( Y _ d z ) } \\rho _ { \\alpha \\lambda } ( h _ d ) \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( t ) = & - \\frac { \\beta } { m - 1 } \\ , ( T - t ) ^ { - \\alpha - 1 } \\ , [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } - 1 } \\\\ & \\ , + \\frac { C ^ { m - 1 } m } { a ^ 2 ( m - 1 ) ^ 2 } \\ , ( T - t ) ^ { - \\alpha m } \\ , [ - \\log ( T - t ) ] ^ { \\frac { \\beta ( 2 - m ) } { m - 1 } } \\\\ \\geq & \\ , \\frac { C ^ { m - 1 } m } { 2 a ^ 2 ( m - 1 ) ^ 2 } \\ , ( T - t ) ^ { - \\alpha m } \\ , [ - \\log ( T - t ) ] ^ { \\frac { \\beta ( 2 - m ) } { m - 1 } } \\end{aligned} \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} = \\begin{bmatrix} \\widehat { K } _ 2 ( \\lambda ) ^ T \\\\ 0 \\end{bmatrix} Q ( \\lambda ) , \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} a _ k ^ b = a _ l ^ b . \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} \\left ( x \\alpha _ { 1 } \\right ) \\left ( y \\alpha _ { 2 } \\alpha _ { 3 } ^ { a } \\right ) & = \\rho ^ { y _ { 2 } } x y \\alpha _ { 1 } \\alpha _ { 2 } \\alpha _ { 3 } ^ { a } \\ \\\\ \\left ( y \\alpha _ { 2 } \\alpha _ { 3 } ^ { a } \\right ) \\left ( x \\alpha _ { 1 } \\right ) & = \\rho ^ { a x _ { 3 } - x _ { 3 } y _ { 2 } } x y \\alpha _ { 1 } \\alpha _ { 2 } \\alpha _ { 3 } ^ { a } , \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} u ( t ; t _ 0 , \\overline u _ 0 ) = \\frac { c _ 0 a } { c _ 0 b - e ^ { - a ( t - t _ 0 ) } } \\forall \\ , \\ , t \\geq t _ 0 , \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\Phi ( \\alpha , \\beta ; Z ) : = \\int _ { \\Delta ^ { ( n ) } } \\varphi ( \\alpha , \\beta ; Z ; u ) d u . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} k ( M / ( A _ i \\cup \\{ e \\} ) ) - v ( M / ( A _ i \\cup \\{ e \\} ) ) = k ( M / A _ i ) - v ( M / A _ i ) . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\gamma ( \\sigma \\times \\sigma ^ \\prime , \\psi ) & \\sum _ { g \\in U _ n ( \\mathfrak { f } ) \\backslash G _ n ( \\mathfrak { f } ) } W ( g ) W ^ \\prime ( g ) \\phi ( e _ n g ) \\\\ & = \\sum _ { g \\in U _ n ( \\mathfrak { f } ) \\backslash G _ n ( \\mathfrak { f } ) } W ( g ) W ^ \\prime ( g ) \\widehat { \\phi } ( e _ 1 \\ , ^ t g ^ { - 1 } ) , \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} \\sum _ { u \\pmod * { p ^ { b } } } \\psi ( x + p ^ { a } u ) = \\begin{cases} p ^ b \\psi ( x ) & s \\leq a , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) = Y ( 0 ) . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { \\lambda } = \\int _ { \\mathbb { R } } f ( x ) g ( x ) d \\psi ( x ) + \\lambda ( \\Delta f ) ( c ) ( \\Delta g ) ( c ) , \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\mathfrak { d } _ 1 \\dot { \\phi } ( n ) & = \\dot { \\phi } ( n ) - \\dot { \\phi } ( n + 1 ) , \\\\ \\mathfrak { d } _ 2 \\dot { \\phi } ( n ) & = \\dot { \\phi } ( n ) - \\dot { \\phi } ( n + 2 ) , \\quad \\forall n \\in \\N , \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} 0 \\le 1 - | K _ k ( x , z ) | ^ 2 - | K _ k ( y , z ) | ^ 2 = 1 - | K ( x , z ) | ^ { 2 \\beta _ k } - | K ( y , z ) | ^ { 2 \\beta _ k } . \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\pi ( h \\cdot _ \\sigma ( f _ i \\ast _ g a ) ) = \\binom { n + m } { n } ^ { - 1 } \\pi ( h ) \\cdot ( f \\ast _ g \\pi ( a ) ) , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{gather*} \\int \\limits _ { [ 0 , \\ , 1 ] ^ l } \\prod _ { i = 1 } ^ l s _ i ^ { \\lambda _ 1 } ( 1 - s _ i ) ^ { \\lambda _ 2 } \\ , \\prod \\limits _ { i < j } ^ l | s _ i - s _ j | ^ { - 2 / \\tau } d s _ 1 \\cdots d s _ l = \\\\ \\prod _ { k = 0 } ^ { l - 1 } \\frac { \\Gamma ( 1 - ( k + 1 ) / \\tau ) \\Gamma ( 1 + \\lambda _ 1 - k / \\tau ) \\Gamma ( 1 + \\lambda _ 2 - k / \\tau ) } { \\Gamma ( 1 - 1 / \\tau ) \\Gamma ( 2 + \\lambda _ 1 + \\lambda _ 2 - ( l + k - 1 ) / \\tau ) } , \\end{gather*}"}
-{"id": "5317.png", "formula": "\\begin{align*} X _ 2 = \\beta _ { 2 , 2 } ^ { - 1 } \\bigl ( 1 , b _ 0 = 2 , \\ , b _ 1 = 1 / 2 , \\ , b _ 2 = 1 / 2 \\bigr ) , \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} F _ { 8 , 4 } = - 2 2 5 \\ , \\nu ( \\xi _ 1 ) \\chi _ 5 , F _ { 8 , 1 0 } = - \\frac { 6 0 7 5 } { 5 1 2 } \\nu ( \\xi _ 2 ) \\chi _ 5 , F _ { 8 , 1 2 } = - \\frac { 6 8 3 4 3 7 5 } { 4 } \\nu ( \\xi _ 3 ) \\chi _ 5 , \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} \\langle \\pi _ { [ N ] } ( U ^ m ( U ^ * ) ^ n ) \\xi ^ { [ N ] } , \\eta ^ { [ N ] } \\rangle = \\sum _ { \\lambda \\in \\Lambda } \\left \\langle S _ { [ N ] } ^ m \\left ( S _ { [ N ] } ^ * \\right ) ^ n f _ \\lambda , g _ \\lambda \\right \\rangle = \\left ( S _ { [ N ] } ^ m \\left ( S _ { [ N ] } ^ * \\right ) ^ n T \\right ) , \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} & H ( v ) = - \\dfrac { 1 } { 2 \\sinh v } ( - 3 + \\cosh 2 v ) , E ( v ) = 1 / \\cosh ^ 2 v , F ( v ) = 0 , \\\\ & G ( v ) = \\sinh ^ 2 v / \\cosh ^ 2 v , \\lambda ( v ) = \\sinh v / \\cosh ^ 2 v \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} P ^ { x , y } : = P ( \\bullet | X _ 0 = x \\mbox { a n d } X _ 1 = y ) . \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\partial \\int E = \\left ( \\int \\partial + 1 \\right ) E = g _ q \\otimes c _ r \\otimes w _ v \\otimes z + g _ { q ' } \\otimes w _ r \\otimes c _ v \\otimes z + c _ Q \\otimes w _ r \\otimes w _ v \\otimes z . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{gather*} \\frac 3 2 L ( \\chi , 1 / 2 ) ^ 2 = L \\big ( \\chi ^ 2 , 1 \\big ) { \\frac 8 3 } L ( \\chi , 1 / 2 ) ^ 3 = L \\big ( \\chi ^ 3 , 3 / 2 \\big ) . \\end{gather*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\theta ( S ) = 2 \\theta _ 0 ( S ' ) \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} C _ { S L ^ \\epsilon _ n ( q ) } ( g \\alpha ) = D R \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} l _ 1 \\le l _ 2 \\implies \\quad \\prod _ { j = 1 } ^ d \\max ( | \\rho ( l _ 1 ) _ j | , 1 ) \\le \\prod _ { j = 1 } ^ d \\max ( | \\rho ( l _ 2 ) _ j | , 1 ) . \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} q & = x _ 1 \\\\ r & = \\frac { \\sqrt { - x _ { n + 1 } + \\bigl ( x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 \\bigr ) } } { \\sqrt { x _ 1 } } \\\\ t & = \\frac { x _ 1 x _ 2 } { - x _ { n + 1 } + \\bigl ( x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 \\bigr ) } \\\\ s _ j & = \\frac { x _ j \\sqrt { x _ 1 } } { \\sqrt { - x _ { n + 1 } + \\bigl ( x _ 1 x _ 2 + x _ 1 \\sum \\limits _ { j = 3 } ^ n x _ j ^ 2 \\bigr ) } } , \\ , j = 3 , \\ldots , n , \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} r ( p ) = \\begin{cases} \\displaystyle \\frac { L } { \\sqrt { p } \\log p } , & L ^ 2 \\leqslant p \\leqslant \\exp ( \\log ^ 2 L ) , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} \\partial _ { j } b _ { i l } + \\partial _ { l } b _ { i j } = 0 \\quad \\forall 1 \\le l , j \\le d \\ ; \\forall 1 \\le i \\le N . \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} & \\mathrm { w t } ( S _ { k + 1 } \\cdots S _ { n - l + k + 1 } T ^ { ( k ) } ) \\\\ & = ( \\underbrace { \\lambda _ { l - k } , \\ldots , \\lambda _ { l - 1 } } _ { k } , \\lambda _ { l - k - 1 } , \\lambda _ { l } , \\underbrace { 0 , \\ldots , 0 } _ { n - l } , \\lambda _ { l - k - 2 } , \\ldots , \\lambda _ { 1 } , 0 , \\ldots ) . \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} \\frac { d } { d t } Q _ { \\kappa ' \\kappa } ( t ) = ( A + B _ 1 + B _ 2 ) Q _ { \\kappa ' \\kappa } ( t ) = L ^ * Q _ { \\kappa ' \\kappa } ( t ) . \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\int _ { \\R ^ + _ 0 } n ( t , s ) \\d s = \\int _ { \\R ^ + _ 0 } n _ 0 ( s ) \\d s , \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} ( \\rho _ { w } \\ast \\mu _ { \\rho ( w ) } ) ( t ) & = \\mu _ { \\rho ( w ) } ( \\rho _ { w } ( t ) ) = \\mu ( \\rho ( w ) + \\rho _ { w } ( t ) ) - \\mu ( \\rho ( w ) ) \\\\ & = \\mu ( \\rho ( w ) + \\rho ( w + t ) - \\rho ( w ) ) - \\mu ( \\rho ( w ) ) \\\\ & = ( \\rho \\ast \\mu ) _ { w } ( t ) \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} F ^ * ( x ) = x + 2 x ^ 2 \\widetilde { F } ' ( x ) . \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\delta _ \\Gamma ( \\vec x ^ { k + 1 } , \\vec x ^ k ) : = \\Gamma ( \\vec x ^ { k + 1 } ) - \\Gamma ( \\vec x ^ k ) , \\ , \\ , \\ , \\Gamma \\in \\{ F , \\vec q , \\vec p \\} . \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\beta _ m \\nu _ { \\delta } ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ m ) p ^ { ( m - 1 ) / 2 } \\varepsilon ( 1 / 2 , \\underline { \\chi } _ p ) \\underline { \\chi } _ p ( 2 ) \\mathbf 1 _ { p ^ { m - 1 } \\Z _ p ^ { \\times } } ( x ) \\psi \\left ( \\frac { - \\delta x ^ 2 } { p ^ { 2 m - 1 } } \\right ) \\underline { \\chi } _ p ( x ) . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} D \\triangleq \\left \\{ t \\geq 0 \\colon P \\big ( X _ { t \\wedge \\tau _ { \\lambda _ { m } } } \\not = X _ { ( t \\wedge \\tau _ { \\lambda _ { m } } ) - } \\big ) = 0 \\right \\} . \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{gather*} \\mathcal G _ { 1 , ( 1 ; 3 ) } ^ \\ast ( \\tau ) = - 2 \\pi { \\rm i } \\phi ( \\tau ) / 6 . \\end{gather*}"}
-{"id": "3499.png", "formula": "\\begin{gather*} K _ a ( x ) = \\frac 1 2 \\int _ 0 ^ \\infty t ^ { a - 1 } { \\rm e } ^ { - \\frac x 2 ( t + 1 / t ) } { \\rm d } t \\ x > 0 , \\end{gather*}"}
-{"id": "3394.png", "formula": "\\begin{align*} S _ { a , b } ^ { a , b + 1 , q } \\left ( p , k \\right ) = \\sum _ { r = 0 } ^ { q } \\binom { q } { r } S _ { a , b } \\left ( p + r , k \\right ) . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} e ' = ( U _ { j - 1 } ^ * U ''' _ j L ) & < \\frac { e } { 2 q } \\sum _ { i = 1 } ^ { 2 q } \\frac { k ' _ i } { k _ i } + \\sum _ { i = 1 } ^ { 2 q } \\sigma _ i \\left ( k ' - ( k + 1 ) \\frac { k ' _ i } { k _ i } \\right ) \\\\ & = \\frac { e } { 2 q k _ { j - 1 } } + \\sum _ { i = 1 } ^ { 2 q } \\sigma _ i \\left ( 1 - ( k + 1 ) \\frac { k ' _ i } { k _ i } \\right ) \\\\ & = \\frac { e } { 2 q r _ j } + \\sigma _ { j - 1 } \\left ( 1 - \\frac { r } { r _ j } \\right ) + \\sum _ { i = 1 , i \\neq j - 1 } ^ { 2 q } \\sigma _ i \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} E _ t ( m ) = L _ t ( m ) + t ^ { - 1 } L _ t ( m ' ) . \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\omega ( n ) = | \\varpi ( n ) | ^ 2 , \\omega ' ( n ) = \\varpi ( n ) \\prod _ { p \\mid n } \\big ( \\lambda ^ { \\ast } _ f ( p ) + \\lambda ^ { \\ast } _ g ( p ) \\big ) \\geqslant 0 \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} \\| B \\| _ { F , \\Omega } ^ 2 = \\sum _ { i , j \\ , : \\ , ( i , j ) \\in \\Omega } B _ { i j } ^ 2 \\ , . \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\chi _ { ( p ) } ( - 1 ) = - 1 p \\mid M . \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} n = \\lfloor ( m + 1 ) / 2 \\rfloor \\quad c = q ^ { m ( m + 1 ) / ( 2 n ) } . \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} ( U _ 1 ( \\lambda ) ^ { - 1 } \\oplus I _ { m _ 1 } ) \\begin{bmatrix} 0 \\\\ I _ n \\\\ - X ( \\lambda ) \\end{bmatrix} = \\begin{bmatrix} \\widehat { N } _ 1 ( \\lambda ) ^ T & N _ 1 ( \\lambda ) ^ T & 0 \\\\ 0 & 0 & I _ { m _ 1 } \\end{bmatrix} \\begin{bmatrix} 0 \\\\ I _ n \\\\ - X ( \\lambda ) \\end{bmatrix} = \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - X ( \\lambda ) \\end{bmatrix} . \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} [ M ( m ) ] = [ L ( m ) ] + [ L ( m ' ) ] \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\prod _ { ( i , j ) \\in F _ k } X _ { i , j } = \\frac { L _ { 1 , b - k + 1 } } { L _ { 1 , b - k + 2 } } \\frac { L _ { 2 , b - k + 2 } } { L _ { 2 , b - k + 3 } } \\cdots \\frac { L _ { k - 1 , b - 1 } } { L _ { k - 1 , b } } L _ { k , b } = \\frac { \\prod \\limits _ { i = 1 } ^ k L _ { i , b - k + i } } { \\prod \\limits _ { i = 1 } ^ { k - 1 } L _ { i , b - k + i + 1 } } \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { a - 1 } \\prod _ { i = 0 } ^ { b - 1 } \\frac { m + j - i } { b + j - i } = \\prod _ { j = 0 } ^ { a - 1 } \\frac { ( m + j ) ! } { ( m - b + j ) ! } \\frac { j ! } { ( b + j ) ! } = \\prod _ { j = 0 } ^ { a - 1 } \\binom { m + j } { b } \\binom { b + j } { b } ^ { - 1 } . \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} \\mu = 2 \\beta ^ 2 = \\frac { 2 } { \\tau } . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} \\lambda ^ * = \\sigma \\sqrt { 2 \\log p } \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} W : = W ^ N = N , N - 1 , \\dots , 2 , 1 , 2 , \\dots N \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} - \\frac { 1 } { \\beta _ { v _ 2 } } | b | ^ 2 = r | b | ^ 2 = r | a + b - a | ^ 2 \\leq p | a + b | ^ 2 + q | a | ^ 2 = - \\frac { 1 } { \\beta _ { v _ 0 } } | a + b | ^ 2 + \\frac { 1 } { \\beta _ { v _ 1 } } | a | ^ 2 . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\sigma ^ { A B } ( t ) [ \\rho ] = [ \\rho ( v + \\beta t ) ] , \\\\ \\sigma ^ { A B } ( t ) [ \\rho v ] = [ \\rho v ( v + \\beta t ) + P ] , \\ \\ P = A \\rho ^ { n } - \\frac { B } { \\rho ^ { \\alpha } } , \\end{array} \\right . \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = k ' a } ^ { k a - 1 } \\frac { \\binom { m k / n + i } { b k / n } ^ { n / k } } { \\binom { b k / n + i } { b k / n } ^ { n / k } } - ( 2 \\sqrt { k } - 1 ) \\right ) \\left ( \\prod _ { i = 0 } ^ { k ' a - 1 } \\frac { \\binom { m k / n + i } { b k / n } ^ { n / k } } { \\binom { b k / n + i } { b k / n } ^ { n / k } } - \\prod _ { i = 0 } ^ { k ' a - 1 } \\frac { \\binom { m k ' / n + i } { b k ' / n } ^ { n / k ' } } { \\binom { b k ' / n + i } { b k ' / n } ^ { n / k ' } } \\right ) , \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} P _ x ( t _ 1 > n ) = \\P _ 0 ( \\tau _ \\vartheta > n ) . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} 1 _ { A } f \\in \\mathcal { F } , A \\in \\mathcal { A } _ { \\cup } ^ { ( N ) } = \\mathcal { A } ^ { ( 1 ) } \\cup . . . \\cup \\mathcal { A } ^ { ( N ) } , f \\in \\mathcal { F } . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} L ^ { \\Delta , \\sigma } = A ^ { \\Delta , \\sigma } + B ^ { \\Delta , \\sigma } = A _ \\upsilon ^ { \\Delta , \\sigma } + B _ \\upsilon ^ { \\Delta , \\sigma } , \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\langle r ^ { - } _ { \\psi } ( \\alpha _ n \\nu _ c ) \\mathbf h _ p , \\mathbf h _ p \\rangle = p ^ { ( 1 - n ) / 2 } \\chi _ { \\psi } ( p ^ n ) \\varepsilon ( 1 / 2 , \\underline { \\chi } _ p ) \\underline { \\chi } _ p ( 2 ) \\int _ { \\Z _ p ^ { \\times } } \\psi \\left ( \\frac { - c y ^ 2 } { p } \\right ) \\underline { \\chi } _ p ( y ) d y \\int _ { \\Z _ p ^ { \\times } } \\underline { \\chi } _ p ( x ) d x . \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\sum _ { n \\geq 0 } f _ n \\overline { g _ n } n ! , \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } \\frac { | T _ i | } { x ^ { i + 1 } } = 1 , \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} q ( k ) & : = \\prod _ { j = 1 } ^ { s } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\frac { k z _ j ^ 0 \\bmod b ^ m } { b ^ m } \\right ) \\right ) . \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} 1 \\geq \\sup _ { \\partial B _ { r m _ i } ( 0 ) } \\lvert \\Phi _ { m _ i } \\rvert = m _ i ^ { - 1 } \\sup _ { \\partial B _ { r } ( x ) } \\lvert \\Phi _ { i } \\rvert > \\delta _ i . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} F _ m = \\{ ( \\chi _ 1 , \\chi _ 2 ) \\ ; : \\ ; ( m , \\chi _ 1 , \\chi _ 2 ) \\in F \\} . \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} t _ a ^ \\Lambda = \\sup _ { \\mu \\in M ( f , \\Lambda ) } \\{ h _ \\mu : \\ , \\int \\varphi d \\mu = a \\} . \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} \\sum _ { \\{ i _ 0 , i _ 1 , i _ 2 \\} = \\{ 0 , 1 , 2 \\} } \\Delta _ { \\frac { 1 } { w _ { i _ 0 } } ( w _ { i _ 1 } , w _ { i _ 2 } ) } ( k + | w | ) = 1 + \\frac { k ( k + | w | ) } { 2 \\bar w } = : g _ { w , k } , \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} c _ \\pm = \\frac { 1 } { 2 } \\lim _ { n \\to \\infty } \\dot { \\phi } ( 2 n ) \\pm \\frac { 1 } { 2 } \\lim _ { n \\to \\infty } \\dot { \\phi } ( 2 n + 1 ) . \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} ( s ) \\int _ { a } ^ { b } f g d \\mu \\leq & ( s ) \\int _ { a } ^ { b } \\bigg ( m 2 ^ { 1 - s } f ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) ^ s [ f ( b ) - m f ( a ) ] \\bigg ) . \\\\ & \\qquad \\qquad \\bigg ( m 2 ^ { 1 - s } g ( a ) + \\bigg ( \\frac { x - m a } { b - m a } \\bigg ) ^ s [ g ( b ) - m g ( a ) ] \\bigg ) d \\mu \\\\ = & ( s ) \\int _ { a } ^ { b } p _ { 1 } ( x ) p _ { 2 } ( x ) d \\mu . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} u w _ { l _ 1 , l _ 2 , k , n } ( d , e ) = \\begin{pmatrix} d _ { l _ 1 , k } \\\\ P & e _ { l _ 2 , n - 2 k } \\\\ Q & R & d _ { l _ 1 , k } \\end{pmatrix} \\in U L . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} ( q _ 1 \\ast q _ 2 ) \\cdot ( q _ 2 \\cdot x ) = q _ 2 \\cdot ( q _ 1 \\cdot x ) . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ 0 } \\sum _ { \\xi \\subset \\eta } G ( \\xi , \\eta , \\eta \\setminus \\xi ) \\lambda ( d \\eta ) = \\int _ { \\Gamma _ 0 } \\int _ { \\Gamma _ 0 } G ( \\xi , \\eta \\cup \\xi , \\eta ) \\lambda ( d \\xi ) \\lambda ( d \\eta ) . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} \\mathbb { P } _ { a , b , r , p } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( z \\right ) = \\sum _ { k = 0 } ^ { r p + \\sigma } \\sum _ { i = 0 } ^ { r p + \\sigma } \\binom { r p + \\sigma - i } { r p + \\sigma - k } A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) \\left ( z - 1 \\right ) ^ { r p + \\sigma - k } . \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } - \\Delta P U _ { \\xi , \\lambda } & = U _ { \\xi , \\lambda } ^ { N + 2 \\over N - 2 } , & \\hbox { i n } \\ \\Omega , \\\\ P U _ { \\xi , \\lambda } & = 0 , & \\hbox { o n } \\ \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} x _ i = x _ j ^ g . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} ( - \\Delta _ { p , q } ) ^ { s } ( q ^ { - \\gamma } ( x ) ) = \\mu ( \\gamma ) \\frac { 1 } { q ^ { p s + \\gamma ( p - 1 ) } ( x ) } \\ , \\ , \\ ; \\ ; \\mathbb { G } \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} w ( e _ 1 ) ^ p = \\alpha \\prod _ { v \\in e _ 1 } \\sum _ { f : \\ , v \\in f } w ( f ) \\geq \\alpha ( \\delta \\cdot w ( e _ 1 ) ) ^ r , \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\left [ \\begin{array} { c | c } M ( \\lambda ) & L _ { \\eta } ( \\lambda ^ \\ell ) ^ { T } \\otimes I _ { m } \\\\ \\hline L _ { \\epsilon } ( \\lambda ^ \\ell ) \\otimes I _ { n } & \\phantom { \\Big { ( } } 0 \\phantom { \\Big { ( } } \\end{array} \\right ] \\ > , \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} c _ K = 0 = c _ 0 - \\sum _ { l = 0 } ^ { K - 1 } \\frac { \\theta _ l } { 1 - \\theta _ l } b _ l \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} I _ \\nu ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! \\Gamma ( k + \\nu + 1 ) } \\left ( \\frac { z } { 2 } \\right ) ^ { 2 k + \\nu } , \\ , \\ , z \\in \\R . \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} p _ n ^ 1 ( y ) = p _ n ^ 2 ( y ) , \\ \\ \\forall y \\in \\partial \\Omega \\ \\ \\hbox { a n d } \\ \\ \\forall n \\in \\N . \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} f ^ \\# : = ( R _ a ^ * \\otimes 1 _ { b _ 1 } \\otimes 1 _ { \\bar { a } } ) ( 1 _ { \\bar { a } } \\otimes f ^ * \\otimes 1 _ { \\bar { a } } ) ( 1 _ { \\bar { a } } \\otimes 1 _ { b _ 2 } \\otimes \\bar { R } _ a ) \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} x = j - r ( p - 1 ) y = - 2 j + r ( 2 p - 1 ) , \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} { \\rm O p } ^ \\tau ( \\sigma ) u ( x ) = \\int _ G \\ ! \\Big ( \\int _ { \\widehat { G } } \\ , { \\rm T r } _ \\xi \\Big [ \\xi ( y ^ { - 1 } x ) \\sigma \\big ( x \\tau ( y ^ { - 1 } x ) ^ { - 1 } \\ ! , \\xi \\big ) \\Big ] d \\mu ( \\xi ) \\Big ) u ( y ) d m ( y ) , \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\Gamma _ 0 ^ { ( 2 ) } ( N ) = \\left \\lbrace g = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\in \\Gamma _ 2 : C \\equiv 0 N \\right \\rbrace . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\nabla _ x b ( s , \\omega , X _ \\theta ( s ) ( \\omega ) ) \\cdot 0 _ d d s = 0 \\mbox { a n d } \\int _ 0 ^ t \\nabla _ x \\sigma ( s , \\omega , X _ \\theta ( s ) ( \\omega ) ) \\cdot 0 _ d d W ( s ) = 0 , \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} \\min \\Big \\{ \\widetilde { F } '' \\big ( | P | \\big ) , \\frac { 1 } { | P | } \\widetilde { F } ' \\big ( | P | \\big ) \\Big \\} | Q | ^ 2 \\leq D ^ 2 F ( P ) ( Q , Q ) \\\\ \\leq \\max \\Big \\{ \\widetilde { F } '' \\big ( | P | \\big ) , \\frac { 1 } { | P | } \\widetilde { F } ' \\big ( | P | \\big ) \\Big \\} | Q | ^ 2 , \\ ; \\ ; P , Q \\in \\R ^ { N \\times n } . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} ( 1 - \\frac { \\epsilon } { 3 } ) \\sum \\limits _ { \\substack { I ^ { c } _ { k + l } ( n ) } } & b ( i _ { 1 } ; n ) b ( i _ { 2 } ; n ) \\cdots b ( i _ { k + l } ; n ) \\\\ & \\leq \\sum \\limits _ { \\substack { I ^ { c } _ { k + l } ( n ) } } \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k + l } ; n ) \\\\ & \\leq ( 1 + \\frac { \\epsilon } { 3 } ) \\sum \\limits _ { \\substack { I ^ { c } _ { k + l } ( n ) } } b ( i _ { 1 } ; n ) b ( i _ { 2 } ; n ) \\cdots b ( i _ { k + l } ; n ) . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\sum _ { k \\equiv _ 2 0 \\ , ( \\ , 1 ) } \\dim S _ { j , k } ( \\Gamma _ 2 , \\epsilon ) \\ , t ^ k = \\frac { N _ j } { ( 1 - t ^ 4 ) ( 1 - t ^ 6 ) ( 1 - t ^ { 1 0 } ) ( 1 - t ^ { 1 2 } ) } \\ , , \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} ( \\log R ) ^ * ( w ) ~ = ~ \\sup _ { \\alpha \\in \\R ^ d } \\left ( \\langle \\alpha , w \\rangle - R ( \\alpha ) \\right ) , \\ ; w \\in \\R ^ d . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} \\langle x | f ( A ) x \\rangle = \\int _ { \\sigma ( A ) } f ( \\lambda ) d \\mu ^ { ( A ) } _ x \\quad \\mbox { a n d } \\| f ( A ) x \\| ^ 2 = \\int _ { \\sigma ( A ) } | f ( \\lambda ) | ^ 2 d \\mu ^ { ( A ) } _ x \\ : . \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} M ( \\omega ) = 4 ( 1 - \\gamma | \\omega | ) [ 1 - b \\gamma ^ 3 | \\omega | - a \\gamma ^ 2 ] - 1 \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} L \\left ( \\frac { 1 } { 2 } , \\overline { \\chi } \\right ) & = \\sum _ n \\frac { \\overline { \\chi } ( n ) } { n ^ { \\frac { 1 } { 2 } } } V _ 1 \\left ( \\frac { n } { q ^ { \\frac { 1 } { 2 } } } \\right ) + \\epsilon ( \\overline { \\chi } ) \\sum _ n \\frac { \\chi ( n ) } { n ^ { \\frac { 1 } { 2 } } } V _ 1 \\left ( \\frac { n } { q ^ { \\frac { 1 } { 2 } } } \\right ) , \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} \\begin{array} { l l } L _ { p } & = \\{ \\Phi : \\Phi ( \\omega , t ) , \\omega \\in \\Omega , t \\in [ 0 , T ] , \\mathbb { E } \\int _ { 0 } ^ { T } | \\Phi _ { t } | ^ { p } d t < \\infty \\} , p \\geq 1 . \\end{array} \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} e _ H = c _ 1 T ^ { ( 1 ) } _ { 0 0 } + c _ 2 T ^ { ( 2 ) } _ { 0 0 } + \\frac { c _ 1 ^ 2 } { 2 } V ^ { ( 1 ) } _ { 0 0 0 0 } + \\frac { c _ 2 ^ 2 } { 2 } V ^ { ( 2 ) } _ { 0 0 0 0 } + c _ 1 c _ 2 V ^ { ( 1 2 ) } _ { 0 0 0 0 } . \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} a \\cdot _ x b = b \\circ \\beta _ { \\bar { b } \\circ x } ( \\rho _ { \\bar { b } } ( a ) ) . \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} F _ { 8 , 1 4 } = \\frac { 1 0 2 5 1 5 6 2 5 } { 2 5 6 } \\nu ( \\xi _ 4 ) \\chi _ 5 , F _ { 8 , 1 6 } = 5 0 6 2 5 \\nu ( \\xi _ 5 ) \\chi _ 5 ^ 3 F _ { 8 , 1 8 } ^ { ( 1 ) } = \\frac { 1 5 1 8 7 5 } { 4 } \\nu ( \\xi _ 6 ^ { ( 1 ) } ) \\chi _ 5 ^ 3 , \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} f ( u ) = u ^ { j } + f _ { j } ( u ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; j = 5 , 6 , . . \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} \\begin{cases} Q _ { i + 1 , j + 1 } = ( \\min [ Q _ { i + 1 , j } , W _ { i + 1 , j } ] - \\min [ Q _ { i , j } , W _ { i , j } ] ) + Q _ { i , j } , \\\\ W _ { i , j } ^ + = ( \\min [ Q _ { i + 1 , j } , W _ { i + 1 , j } ] - \\min [ Q _ { i , j } , W _ { i , j } ] ) + W _ { i , j } , \\end{cases} \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} d _ { 2 } ( M ) = \\frac { 1 } { \\left | B _ { \\frac { 1 } { 8 } \\cdot 2 ^ { n - 1 } } ( M ) \\right | } \\int \\limits _ { B _ { \\frac { 1 } { 8 } \\cdot 2 ^ { n - 1 } } ( M ) } d _ { 1 } ( \\widetilde { M } ) \\ , d m _ { 3 } ( \\widetilde { M } ) , \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} L ( 1 , \\tau _ p , \\mathrm { a d } ) = \\frac { 1 } { 1 - p ^ { - 1 } } = \\frac { p } { p - 1 } . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} ( H S ) _ { i , j } = \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 2 ^ N \\dot { \\phi } ( i + j + 1 ) . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} \\xi ^ { h } _ { \\ell , t } = \\begin{cases} \\frac { 1 } { t ^ { h } _ { \\ell , s ^ * } - \\tau ( t ^ h _ { \\ell , s ^ * } ) } , & \\tau ( t ^ { h } _ { \\ell , s ^ * } ) \\leq t < t ^ { h } _ { \\ell , s ^ * } t ^ { h } _ { \\ell , s ^ * } = \\min \\limits _ { \\begin{subarray} ~ k '' \\in \\mathbb { N } _ + \\end{subarray} } \\{ t ^ { h } _ { \\ell , k '' } | t ^ { h } _ { \\ell , k '' } > t \\} , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} & \\sum _ { i _ { 1 } + i _ { 2 } + \\cdots + i _ { r } = N } \\frac { a _ { 1 } ^ { i _ { 1 } } a _ { 2 } ^ { i _ { 2 } } \\cdots a _ { r } ^ { i _ { r } } q ^ { \\binom { i _ 1 } { 2 } + \\binom { i _ 2 } { 2 } + \\cdots + \\binom { i _ r } { 2 } } } { ( q ; q ) _ { i _ { 1 } } ( q ; q ) _ { i _ { 2 } } \\cdots ( q ; q ) _ { i _ { r } } } \\\\ & \\quad \\quad \\times \\left ( 1 + \\sum _ { s = 1 } ^ { r } q ^ { i _ 1 + i _ 2 + \\cdots + i _ s } a _ s q ^ N \\prod _ { k = 1 } ^ { s - 1 } \\left ( 1 + a _ k q ^ N \\right ) \\right ) . \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} m : = \\inf _ { D ^ + _ { \\kappa ' r } } \\frac { u _ 1 } { V ( d _ D ) } \\geq 0 . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} ( a _ 1 , \\dots , a _ k ) _ b : = ( a _ 1 ) _ b \\cdots ( a _ k ) _ b , \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} 1 = f _ 0 - f _ 1 + \\cdots + ( - 1 ) ^ { t - 1 } f _ { t - 1 } + \\underbrace { ( - 1 ) ^ t f _ t + ( - 1 ) ^ { t + 1 } f _ { t + 1 } + \\cdots + ( - 1 ) ^ m f _ m } _ { ( - 1 ) ^ t r _ t } . \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} \\mathcal O ^ { * * } = \\sigma \\ ( R _ \\alpha , \\ , Z ; \\alpha \\in \\ker A \\ ) . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} \\Phi _ t = \\Phi ( t , x ) = - ( \\nabla \\cdot \\rho ( t , x ) \\nabla ) ^ { \\dd } \\frac { \\partial \\rho } { \\partial t } ( t , x ) . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} ( I - x K _ u ^ 2 ) ^ { - 1 } ( h ) - ( I - x H _ u ^ 2 ) ^ { - 1 } ( h ) & = ( I - x K _ u ^ 2 ) ^ { - 1 } \\left [ ( I - x H _ u ^ 2 ) - ( I - x K _ u ^ 2 ) \\right ] ( I - x H _ u ^ 2 ) ^ { - 1 } ( h ) \\\\ & = - x ( h \\vert ( I - x H _ u ^ 2 ) ^ { - 1 } ( u ) ) ( I - x K _ u ^ 2 ) ^ { - 1 } ( u ) . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} \\Gamma = \\R \\cup \\Gamma _ \\infty , \\Gamma _ \\infty = \\{ | z | = S _ \\infty \\} \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} s ( [ 5 , 2 ] ) & = s ( [ 1 3 , 2 ] ) = s _ 2 s _ 3 s _ 2 s _ 3 s _ 2 s _ 1 s _ 2 s _ 3 s _ 2 s _ 3 s _ 2 \\ \\\\ \\beta ( [ 5 , 2 ] ) & = \\beta ( [ 1 3 , 2 ] ) = \\alpha _ 1 + 3 \\sqrt 2 \\alpha _ 2 + 6 \\alpha _ 3 . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} d \\rho _ t = - L ( \\rho _ t ) ( d _ { \\rho _ t } \\mathcal { F } ( \\rho _ t ) + \\frac { \\beta } { 2 } d _ { \\rho _ t } \\log \\Pi ( \\rho _ t ) ) d t + \\sqrt { 2 \\beta } { L ( \\rho _ t ) } ^ { \\frac { 1 } { 2 } } d B _ t . \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} \\tau _ i ( T ) = S _ i T S _ i ^ * . \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} c _ { i , \\{ k \\} } T \\otimes c _ { j , \\{ k \\} } T ' = c _ { i j , \\{ k \\} } ( T \\otimes T ' ) \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\begin{aligned} g _ 1 = \\sum _ { \\mu = 0 , 1 } \\exp \\left ( \\sum _ { i = 1 } ^ N \\mu _ i ( \\xi _ i - \\phi _ i ) + \\sum _ { i < j } ^ N \\mu _ i \\mu _ j \\gamma _ { i j } \\right ) , \\\\ g _ 2 = \\sum _ { \\mu = 0 , 1 } \\exp \\left ( \\sum _ { i = 1 } ^ N \\mu _ i ( \\xi _ i + \\phi _ i ) + \\sum _ { i < j } ^ N \\mu _ i \\mu _ j \\gamma _ { i j } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} ( \\hat u _ { 1 , 0 } , \\hat \\theta _ 0 , \\hat h _ { 1 , 0 } ) ( x , \\eta ) : = ( u _ { 1 , 0 } , \\theta _ 0 , h _ { 1 , 0 } ) \\big ( x , y ( x , \\eta ) \\big ) . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} r ^ { 2 } R _ { 2 } - ( r ^ { 2 } - 2 r + 6 ) R = 0 . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} \\mu ( \\gamma A ) = N ( \\gamma ) ^ { - \\beta } \\mu ( A ) . \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\beta _ m \\nu _ { \\delta } ) \\mathbf h _ p ( x ) & = \\chi _ { \\psi } ( p ^ m ) p ^ { - m / 2 } \\int _ { p ^ { - m } \\Z _ p } \\psi ( 2 x z ) \\left ( \\int _ { \\Q _ p } \\psi ( - 2 z p ^ m y - \\delta p y ^ 2 ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) d y \\right ) d z = \\\\ & = \\chi _ { \\psi } ( p ^ m ) p ^ { m / 2 } \\int _ { \\Q _ p } \\psi ( - \\delta p y ^ 2 ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) \\left ( \\int _ { \\Z _ p } \\psi ( 2 ( x p ^ { - m } - y ) z ) d z \\right ) d y . \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} \\partial ^ \\alpha = \\partial ^ { \\alpha _ 0 } _ \\tau \\partial ^ { \\alpha _ 1 } _ \\xi \\partial ^ { \\alpha _ 2 } _ \\eta , ~ \\alpha = ( \\alpha _ 0 , \\alpha _ 1 , \\alpha _ 2 ) \\in \\mathbb { N } ^ 3 , \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} C ( 2 , i ) = \\sum _ { k } B _ { i - 1 , k } = \\sum _ { k } C _ { i - 1 , k } 2 ^ k , \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} \\mathcal Q ( \\mathbf h _ 1 , \\mathbf h _ 2 , \\mathbf g _ 1 , \\mathbf g _ 2 , \\pmb { \\phi } _ 1 , \\pmb { \\phi } _ 2 ) = \\frac { 1 } { 4 } \\frac { L ( 1 / 2 , \\pi \\times \\mathrm { a d } \\tau ) } { L ( 1 , \\pi , \\mathrm { a d } ) L ( 1 , \\tau , \\mathrm { a d } ) } \\prod _ v \\mathcal I _ v ( \\mathbf h _ { 1 , v } , \\mathbf h _ { 2 , v } , \\mathbf g _ { 1 , v } , \\mathbf g _ { 2 , v } , \\pmb { \\phi } _ { 1 , v } , \\pmb { \\phi } _ { 2 , v } ) , \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} p ^ { \\ast } ( x ^ { \\ast } ) = q ( x ^ { \\ast } ) . \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} \\int _ { \\Gamma } ^ { } u _ { \\Gamma } ( t ) d \\Gamma = \\int _ { \\Gamma } ^ { } u _ { 0 \\Gamma } d \\Gamma \\mbox { f o r a l l } t \\in [ 0 , T ] ; \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\begin{aligned} & - I \\geq \\frac { 1 } { 2 } \\int _ { B _ { R _ i } \\times \\left \\{ t \\right \\} } \\left ( u - l _ i \\right ) _ - ^ 2 \\eta _ i ^ 2 \\ , d x - \\frac { 2 ^ { 2 ( i + 1 ) } \\theta _ 0 ^ { \\alpha _ { 0 } } \\left ( 2 \\theta _ 0 \\right ) ^ 2 } { R _ i ^ 2 } \\int _ { - \\theta _ 0 ^ { - \\alpha _ { 0 } } R _ i ^ 2 } ^ { t } \\int _ { B _ { R _ i } } \\chi _ { \\left [ u \\leq l _ i \\right ] } \\ , d x d \\tau \\end{aligned} \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} x _ 1 z _ 2 & = x _ 2 z _ 1 \\\\ y _ 1 z _ 2 & = y _ 2 z _ 1 \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial \\ell } { \\partial \\zeta } \\bigg \\vert \\frac { \\partial \\ell } { \\partial \\varphi } \\right ) = - \\left ( D _ { \\ell } f \\right ) ^ { - 1 } \\left ( \\frac { \\partial f } { \\partial \\zeta } \\bigg \\vert \\frac { \\partial f } { \\partial \\varphi } \\right ) . \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} \\overline { V } ^ { ( l ) } ( p , g , \\pi _ n ) _ T = n ^ { p / 2 - 1 } \\sum _ { i : t _ { i , n } ^ { ( l ) } \\leq T } g ( \\Delta _ { i , n } ^ { ( l ) } X ^ { ( l ) } ) , ~ l = 1 , 2 , \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\dot { \\phi } ( n ) = \\frac { ( - 1 ) ^ n } { ( n + 1 ) ^ { \\alpha + 1 } } . \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} L _ M \\bigl ( w + a _ i \\ , | \\ , a \\bigr ) = L _ { M } ( w \\ , | \\ , a ) - L _ { M - 1 } ( w \\ , | \\ , \\hat { a } _ i ) , \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} ( W X W ^ T , Y ) = ( X , W ^ T Y W ) \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} m \\cdot g e n u s ( M ) + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } g e n u s ( \\Sigma ^ y _ \\ell ) \\leq g e n u s ( M _ k ) . \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} \\sup \\mathrm { ( O D P ) } _ { 0 _ { X } ^ { \\ast } } = - \\inf _ { S \\in U , { { { \\lambda } \\in } } \\mathbb { R } _ { + } ^ { S } } \\left \\{ { { { \\sum \\limits _ { s \\in S } { { \\lambda _ { s } b _ { s } : } \\sum \\limits _ { s \\in S } { { \\lambda _ { s } a _ { s } ^ { \\ast } = } } } } - c ^ { \\ast } } } \\right \\} . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\binom { n } { k } ^ 4 \\frac { 1 } { a ^ { 2 k } } \\leq \\mathbb { E } [ T _ { n , k } ^ 2 ] \\leq \\binom { n } { k } ^ 2 \\frac { 1 } { a ^ { k } } \\ , \\sum _ { j = 0 } ^ k \\binom { n - k } { j } ^ 2 \\binom { n - j } { k - j } ^ 2 \\frac { 1 } { a ^ { j } } . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} \\begin{aligned} \\| w \\| _ { L ^ { \\infty } ( B _ { r _ { k + 1 } } ) } & \\leq C M r _ { k } ^ { 2 - n / p } \\leq \\hat { C } M r _ { k + 1 } ^ { \\beta } \\cdot \\frac { C } { \\hat { C } \\tau _ 1 ^ { \\beta } } , \\end{aligned} \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} K ( z , z ' ) : = z ^ * z ' . \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} L _ { m _ 2 } = \\Delta + \\sum _ { \\alpha \\in R _ + } \\frac { 2 k _ \\alpha } { \\langle \\alpha , x \\rangle } \\partial _ \\alpha \\ , \\ , ( = \\Delta _ k | _ { } ) \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} B _ n \\varphi ( x ) ~ = ~ P ^ { n - 1 } A _ x \\varphi ( e ) ~ = ~ \\sum _ { y \\in E } b _ n ( x , y ) \\varphi ( y ) , x \\in E , \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} ( f _ i \\pm \\varphi ) ( \\mu _ i ) & = f _ i ( \\mu _ i ) = ( 1 - \\delta ) \\tilde { f _ i } ( \\mu _ i ) \\\\ & = ( 1 - \\delta ) g _ i ( \\mu _ i ) = 1 - \\delta \\\\ & > ( 1 - \\delta ) ^ 2 > 1 - \\alpha _ i . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} X _ \\theta ( t ) ( \\omega ) = \\theta + \\int _ 0 ^ t b \\Big ( s , \\omega , X ( s ) ( \\omega ) \\Big ) d s + \\int _ 0 ^ t \\sigma \\Big ( s , \\omega , X ( s ) ( \\omega ) \\Big ) d W ( s ) , \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { k } d _ j ( \\mathcal { C } ^ s ( A ) ) = \\sum \\limits _ { j = 1 } ^ { k _ 1 } d _ j ( B ) + \\sum \\limits _ { j = 1 } ^ { k _ 2 } d _ j ( C ) . \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} & C _ 1 ( 0 ) F ( 1 , 1 , 6 ; 1 , 3 , 4 ; p ) = F ( 2 , 1 , 5 ; 1 , 3 , 4 ; p ) , \\\\ & C _ 2 ( 0 ) F ( 2 , 1 , 5 ; 1 , 3 , 4 ; p ) = F ( 2 , 2 , 4 ; 1 , 3 , 4 ; p ) , C _ 2 ( 1 ) F ( 2 , 2 , 4 ; 1 , 3 , 4 ; p ) = F ( 2 , 3 , 3 ; 1 , 3 , 4 ; p ) . \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} ( d g ) g ^ { - 1 } & = \\check e _ i \\frac { d z } { ( z - x ) ^ 2 } , g u g ^ { - 1 } = u - \\frac { 1 } { z - x } [ \\check e _ i , u ] = u + \\frac { \\langle u , \\check \\alpha _ i \\rangle } { z - x } \\check e _ i , \\\\ g p _ { - 1 } g ^ { - 1 } & = p _ { - 1 } - \\frac { 1 } { z - x } [ \\check e _ i , p _ { - 1 } ] + \\frac { 1 } { 2 ( z - x ) ^ 2 } \\big [ \\check e _ i , [ \\check e _ i , p _ { - 1 } ] \\big ] = p _ { - 1 } - \\frac { \\alpha _ i } { z - x } - \\frac { \\check e _ i } { ( z - x ) ^ 2 } . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} h ( x _ i ) : = 0 h ' ( x _ i ) : = 0 . \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} H = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 2 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} \\theta ( \\mathbf Y _ M \\mathbf G , \\phi _ { \\breve { \\mathbf g } } ) ( x ) & = M ^ { - 1 } \\int _ { [ \\mathrm O _ { 2 , 2 } ] } \\left ( \\sum _ { v \\in V _ 4 ( \\Q ) } \\omega ( x , y ' y ) \\omega ( \\varpi _ M , h _ M ) \\phi _ { \\mathbf g ^ { \\sharp } } ( v ) \\right ) \\mathbf G ( y ' y h _ M ) d y = \\\\ & = M ^ { - 1 } \\int _ { [ \\mathrm O _ { 2 , 2 } ] } \\left ( \\sum _ { v \\in V _ 4 ( \\Q ) } \\omega ( x \\varpi _ M , y ' h _ M y ) \\phi _ { \\mathbf g ^ { \\sharp } } ( v ) \\right ) \\mathbf G ( y ' h _ M y ) d y . \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { 2 m } Z _ j \\le \\sum \\limits _ { j = 0 } ^ { m } j + \\sum \\limits _ { j = m + 1 } ^ { 2 m } ( 2 m + 1 - j ) = m ( m + 1 ) . \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\Delta _ M \\nu + \\nu \\big ( ( 1 - \\nu ^ 2 ) ( \\kappa - 4 \\tau ^ 2 ) + | \\sigma | ^ 2 + 2 \\tau ^ 2 \\big ) = 0 . \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} ( 1 - \\frac { \\epsilon } { 3 } ) ^ { 2 } \\leq \\frac { P ( \\tilde { V } _ { n } = k ) } { P ( V _ { n } = k ) } \\leq ( 1 + \\frac { \\epsilon } { 3 } ) ^ { 3 } . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} \\mathrm { d i m } ( H _ { J } ) = 2 g - \\mathrm { d i m } ( H _ { j } ) + \\mathrm { d i m } ( H _ { J ' } ) = 2 g - ( 2 g - 1 ) + \\mathrm { d i m } ( H _ { J ' } ) = 1 + \\mathrm { d i m } ( H _ { J ' } ) . \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} ( \\stackrel { \\stackrel { m } { \\smile } } { E ( \\beta ; \\cdot ) \\times \\cdots \\times E ( \\beta ; \\cdot ) } ) ( x ) = E ( m , \\beta ; x ) . \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} \\hat { u } ( x , k ) = k ^ 2 \\int _ { \\R ^ 3 } \\left ( c ^ { - 2 } ( y ) - 1 \\right ) \\hat { u } ( y , k ) \\Phi ( x - y ) d y - \\frac { i k } { 2 \\pi } \\int _ { \\R ^ 3 } \\frac { f ( y ) } { c ^ { 2 } ( y ) } \\Phi ( x - y ) d y , \\ \\ x \\in \\R ^ 3 . \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} w ( t ) = e ^ { i t ( \\Delta ^ 2 _ x - \\kappa \\Delta _ x ) } w _ 0 + \\int _ { 0 } ^ { t } e ^ { i ( t - \\tau ) ( \\Delta ^ 2 _ x - \\kappa \\Delta _ x ) } g ( u ( \\tau ) , v ( \\tau ) , p ) d \\tau \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} f ^ + ( x ) & : = \\ ! \\ ! \\int _ { \\underline { x } } ^ x \\ ! \\ ! | f ' ( t ) | \\ , \\ , t = \\ ! \\ ! \\int _ { \\underline { x } } ^ x \\ ! \\ ! \\ ! | f ' ( t ) | - f ' ( t ) + f ' ( t ) \\ , \\ , t \\\\ & = \\ ! \\ ! \\int _ { \\underline { x } } ^ x \\ ! \\ ! \\ ! | f ' ( t ) | - f ' ( t ) \\ , \\ , t + f ( x ) - f ( \\underline { x } ) \\\\ & = f ( x ) - f ( \\underline { x } ) + \\Delta ( x ) , \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } M \\Big ( [ u ] _ { s , A _ \\varepsilon } ^ 2 \\Big ) ( - \\Delta ) _ { A _ \\varepsilon } ^ s u + \\varepsilon ^ { - 2 s } V ( x ) u = \\varepsilon ^ { - 2 s } | u | ^ { 2 _ s ^ \\ast - 2 } u + \\varepsilon ^ { - 2 s } h ( x , | u | ^ 2 ) u , \\ , x \\in \\mathbb { R } ^ N , \\smallskip \\smallskip \\\\ u ( x ) \\rightarrow 0 , \\ \\indent a s \\ \\ | x | \\rightarrow \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} r < \\| \\vec v \\| _ 1 , \\ ; \\ ; \\ ; p = 1 . \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} \\overline { \\mathcal { M } ^ \\Delta _ { \\mathbb { P } ^ 1 , \\mathcal { O } ( t ) } } ( Q , \\mathbf { a } ) & \\cong \\mathbb { P } ^ q \\times \\prod _ { j = 1 } ^ { i ' } \\emph { G r } \\Big ( s _ j , d _ 3 - a _ j + t + 1 - \\sum _ { k = 1 } ^ { j - 1 } s _ { k } ( a _ k - a _ { j } + 1 ) \\Big ) \\\\ & \\qquad \\times \\prod _ { j = i ' + 1 } ^ { m } \\emph { G r } \\Big ( s _ j , a _ j - d _ 1 + t + 1 - \\sum _ { k = j } ^ { m - 1 } s _ { k } ( a _ k - a _ { j } + 1 ) \\Big ) \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\begin{cases} x ^ 0 , y ^ 0 \\in C , \\\\ x ^ { n + 1 } = \\underset { y \\in C } { \\textup { a r g m i n } } \\bigg \\{ \\lambda _ n f ( y ^ n , y ) + \\dfrac { 1 } { 2 } \\| y - x ^ n \\| ^ 2 \\bigg \\} , \\\\ y ^ { n + 1 } = \\underset { y \\in C } { \\textup { a r g m i n } } \\bigg \\{ \\lambda _ n f ( y ^ n , y ) + \\dfrac { 1 } { 2 } \\| y - x ^ { n + 1 } \\| ^ 2 \\bigg \\} , \\end{cases} \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\textnormal { l o g } \\mathcal { P } _ { X _ { A } } ( A ^ { + } ( n ) ) = \\left [ \\lvert \\alpha ^ { - } ( \\phi ) \\rvert - \\alpha ^ { - } ( \\phi ^ { - 1 } ) \\right ] h _ { t o p } ( \\sigma _ { A } ) . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} E _ i = e _ { i , i + 1 } - e _ { n + i + 1 , n + i } & F _ i = e _ { i + 1 , i } - e _ { n + i , n + i + 1 } \\displaybreak [ 1 ] \\\\ X _ { j } = e _ { j , n + j } & Y _ { j } = e _ { n + j , j } \\displaybreak [ 1 ] \\\\ Z _ { j } = e _ { j j } - e _ { n + j , n + j } & H _ i = Z _ { i } - Z _ { i + 1 } \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\Delta _ { \\bar g } \\rho + R _ { \\bar g } \\rho - \\delta _ { \\bar g } \\delta _ { \\bar g } ( e ^ { - 2 u } \\mathring { Q } _ m ( g , s ) ) = 0 . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} & \\frac { ( 2 - \\theta _ 0 ) } { k + ( 2 - \\theta _ 0 ) / \\theta _ 0 } \\leq \\theta _ k \\leq \\frac { 2 } { k + 2 / \\theta _ 0 } \\\\ & \\frac { 1 - \\theta _ { k + 1 } } { \\theta _ { k + 1 } ^ 2 } = \\frac { 1 } { \\theta _ k ^ 2 } , \\enspace \\forall k = 0 , 1 , \\dots \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} A _ { i , a } : = Y _ { i , a q _ i ^ { - 1 } } Y _ { i , a q _ i } \\left ( \\prod _ { j \\colon c _ { j i } = - 1 } Y _ { j , a } ^ { - 1 } \\right ) \\left ( \\prod _ { j \\colon c _ { j i } = - 2 } Y _ { j , a q ^ { - 1 } } ^ { - 1 } Y _ { j , a q } ^ { - 1 } \\right ) \\left ( \\prod _ { j \\colon c _ { j i } = - 3 } Y _ { j , a q ^ { - 2 } } ^ { - 1 } Y _ { j , a } ^ { - 1 } Y _ { j , a q ^ 2 } ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} e ^ { - \\infty } : = 0 \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} & \\widetilde { c } _ { j i } ( r - 2 ) + \\widetilde { c } _ { j i } ( r + 2 ) - \\sum _ { | j - k | = 1 } \\widetilde { c } _ { k i } ( r ) = 0 \\ \\ j \\in I \\setminus \\{ n \\} , \\\\ & \\widetilde { c } _ { n i } ( r - 1 ) + \\widetilde { c } _ { n i } ( r + 1 ) - \\widetilde { c } _ { n - 1 i } ( r - 1 ) - \\widetilde { c } _ { n - 1 i } ( r + 1 ) = 0 . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\nabla \\log \\rho = \\frac { 1 } { \\rho } \\nabla \\rho , \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} V _ r ( a _ j : u _ 0 \\leq j \\leq u _ K ) \\leq K ^ { 1 - 1 / r } \\Big ( \\sum _ { k = 1 } ^ K V _ r ( a _ j : u _ { k - 1 } \\leq j \\leq u _ k ) ^ r \\Big ) ^ \\frac { 1 } { r } . \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\varphi ( c ) ( \\beta _ { 1 } , \\ldots , \\beta _ { p } ) ( \\alpha _ { 1 } , \\ldots , \\alpha _ { q } ) = c _ { p } ( \\alpha _ { 1 } , \\ldots . \\alpha _ { q } , \\beta _ { 1 } , \\ldots , \\beta _ { p } ) \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\ , \\ , \\displaystyle { u _ { t t } - \\Delta u = | u | ^ p } & \\displaystyle { t > 0 , x \\in { \\Omega } , } \\\\ { } \\\\ \\displaystyle { u ( 0 , x ) = u _ 0 ( x ) , \\ ; \\ ; u _ t ( 0 , x ) = u _ 1 ( x ) \\qquad \\qquad } & \\displaystyle { x \\in { \\Omega } , } \\\\ { } \\\\ \\displaystyle { u = 0 , \\qquad \\qquad } & \\displaystyle { t > 0 , \\ ; x \\in { \\partial \\Omega } , } \\end{array} \\right . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} G ( s ) : = C ( s I _ n + A ) ^ { - 1 } B , s \\in { \\bf C } \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} C ^ * ( U _ i ) = \\overline { } \\{ U _ i ^ { m } ( U _ i ^ * ) ^ n \\ : \\ m , n \\in \\N \\} \\subset \\ell _ 2 ( X _ i ) . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} \\mathcal I _ 1 ( n ) = - p ^ { 2 n } \\frac { p - 1 } { p ( p + 1 ) } - p ^ { 2 n } \\frac { p - 1 } { p + 1 } = - p ^ { 2 n } \\frac { p - 1 } { p + 1 } ( 1 + p ^ { - 1 } ) = - p ^ { 2 n - 1 } ( p - 1 ) \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} T _ { n . k } = \\sum _ { 1 \\leq i _ 1 < \\cdots < i _ k \\leq n } \\sum _ { 1 \\leq j _ 1 < \\cdots < j _ k \\leq n } \\mathbf { 1 } ( X _ { i _ 1 } = Y _ { j _ 1 } , \\ldots , X _ { i _ k } = Y _ { j _ k } ) . \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} & \\psi _ 1 = ( 3 ) , \\psi _ 2 = ( 5 , 6 , 7 , 9 ) , \\psi _ 3 = ( 8 ) , \\psi _ 4 = ( 1 , 2 ) , \\\\ & \\psi _ 5 = ( 1 0 , 4 ) , \\sigma = ( 1 , 4 , 3 , 2 , 5 ) . \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} J _ { 1 , 4 } = { \\Theta } \\frac { \\zeta ( 3 + a ) } { ( 1 + a ) \\zeta ( 2 + a ) } x ^ { 1 + a } + O _ { a } \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} D _ \\psi = \\{ ( k , k ' ) \\in \\N ^ { 2 } \\ , ; \\ , 0 \\le k ' \\le \\psi ( k ) \\} . \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} | \\overline { h } | \\leq \\max \\bigl \\{ c \\| \\overline { f } \\| , c ^ 2 \\| d \\overline { f } \\| \\bigr \\} = c \\| \\overline { f } \\| \\leq \\tfrac { 1 } { r c } \\leq \\tfrac { 1 } { r } \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} q = q ( \\tau , \\xi , \\eta ) : = h _ 1 ^ 2 ( \\tau , \\xi , \\eta ) / 2 . \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} \\nu ' ( C ) = \\frac { 1 } { 4 } \\left ( ( d - 1 ) ( d - 3 ) - \\tau ( C ) + \\sum _ { j } \\nu _ j \\right ) . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} t ^ A v = \\big ( t ^ { \\abs { \\gamma } } v _ \\gamma : \\gamma \\in \\Gamma \\big ) . \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} y ^ { ( j ) } _ { k } = y _ { k } ^ { ( j + 1 ) } + y _ { k + 2 ^ { j } } ^ { ( j + 1 ) } , k \\in \\left \\{ 0 , \\ldots , 2 ^ { j } - 1 \\right \\} , \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} \\inf \\mathrm { ( R P ) } _ { 0 _ { X } ^ { \\ast } } = \\inf \\left \\{ \\left \\langle c ^ { \\ast } , x \\right \\rangle : { \\left \\langle a _ { s } ^ { \\ast } , x \\right \\rangle \\leq b _ { s } , \\ ; \\forall s \\in S } \\right \\} , \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { ( \\beta ) _ { n } n ^ m c ^ { n } } { ( n - 1 ) ! } = \\frac { 1 } { \\Gamma ( \\beta ) } \\lim _ { n \\rightarrow \\infty } n ^ { \\beta + m } c ^ { n } . \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} [ A ( n ) , Y ( B , w ) ] = \\sum _ { k \\ge 0 } \\binom { n } { k } w ^ { n - k } Y ( A ( k ) B , w ) . \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} \\int _ { \\Omega } ^ { } \\nabla \\bar { y } \\cdot \\nabla z d x + \\int _ { \\Gamma } ^ { } \\nabla _ \\Gamma \\bar { y } _ \\Gamma \\cdot \\nabla _ \\Gamma z _ \\Gamma d \\Gamma = \\langle \\bar { u } _ \\Gamma , z _ \\Gamma \\rangle _ { V ^ * _ \\Gamma , V _ \\Gamma } \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} & S \\times ( T \\times U ) : = \\big \\{ \\big ( s , ( t , u ) \\big ) \\ , \\big | \\ , s \\in S , t \\in T , u \\in U \\big \\} \\\\ = ^ ? & ( S \\times T ) \\times U : = \\big \\{ \\big ( ( s , t ) , u \\big ) \\ , \\big | \\ , s \\in S , t \\in T , u \\in U \\big \\} . \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} | v _ { m _ 1 } | > | v _ { m _ 1 + 1 } | = \\cdots = | v _ { m _ 0 } | > | v _ { m _ 0 + 1 } | . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} K ( z , \\lambda z ' ) = K ( z , \\rho \\lambda \\rho ^ { - 1 } z ' ) = K _ e ( \\rho ^ { - 1 } z , \\lambda \\rho ^ { - 1 } z ' ) = \\lambda ^ e K _ e ( \\rho ^ { - 1 } z , \\rho ^ { - 1 } z ' ) = \\lambda ^ e K ( z , z ' ) . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} e _ 1 = { } ^ t ( 1 , 0 , 0 , 0 ) , \\ , \\dots \\ , , e _ 4 = { } ^ t ( 0 , 0 , 0 , 1 ) \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} \\left | V _ r \\right | \\le \\sum _ { k = 0 } ^ { K } \\left | V _ { k , r } \\right | , \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\bigg | \\xi _ \\gamma - \\frac { a _ \\gamma ' } { q _ \\gamma ' } \\bigg | = \\bigg | \\xi _ \\gamma - \\frac { a _ \\gamma '' } { q '' } \\bigg | \\leq N ^ { - \\abs { \\gamma } } L ( \\log N ) ^ { \\beta ' / d } . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} d _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , a ^ + } x ( b ) + d _ 2 ~ ^ { A B R } _ { a } D ^ \\alpha x ( b ) = 0 , \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} m _ { \\alpha + 1 } = \\frac { 1 } { 4 } \\left ( ( d - 1 ) ( d - 3 ) - \\tau ( C ) + \\sum _ { j } \\nu _ j \\right ) . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} g _ { 0 } ( M ) = \\frac { 1 } { \\left | B _ { \\frac { 1 } { 8 } d _ { 0 } ( M ) } ( M ) \\right | } \\int \\limits _ { B _ { \\frac { 1 } { 8 } d _ { 0 } ( M ) } ( M ) } g _ { 2 } ( \\widetilde { M } ) \\ , d m _ { 3 } ( \\widetilde { M } ) , \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\omega _ n ( t ^ 0 , t ^ j ) = \\delta _ { n , j } \\ ; \\mbox { f o r a l l $ j , n \\in \\Z $ . } \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} v _ p ( f ( x ) ) & = v _ p \\left ( \\sum _ { n = l } ^ { + \\infty } a _ n ( x - x _ 0 ) ^ n \\right ) = v _ p \\left ( \\sum _ { n = l } ^ { + \\infty } a _ n ( x - x _ 0 ) ^ { n - l } \\right ) + v _ p \\left ( ( x - x _ 0 ) ^ l \\right ) \\\\ & = v _ p ( a _ l ) + l v _ p ( x - x _ 0 ) . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} D _ { a } ( p ) = \\sum \\nolimits _ { p = u a v } v u \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} & A _ r P ' + P ' A _ r + A _ r ' P + P A ' _ r - B _ r ' B _ r ^ T - B _ r B '^ { T } _ r = 0 , \\\\ & A X ' + X ' A _ r + X A _ r ' - B B _ r '^ T = 0 . \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} L ( 1 / 2 , \\pi _ p \\times \\mathrm { a d } \\tau _ p ) = \\frac { L ( 1 / 2 , \\pi _ p \\otimes \\tau _ p \\otimes \\tau _ p ) } { L ( 1 / 2 , \\pi _ p ) } = \\frac { p ^ 3 } { ( p ^ 2 + w _ p ) ( p + w _ p ) } . \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\gamma ( s , \\pi _ 1 \\times \\pi _ 2 , \\psi ) = \\frac { q ^ { n ( s - 1 ) } ( q ^ n - 1 ) } { \\lambda _ 1 \\lambda _ 2 - q ^ { n ( s - 1 ) } } + \\gamma ( \\sigma _ 1 \\times \\sigma _ 2 , \\psi ) . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} \\lambda ( t ) = \\displaystyle \\frac { \\beta } { 2 } \\left ( 1 + \\tanh ( \\frac { t - t _ L } { w } ) - \\big ( 1 + \\tanh ( \\frac { t - t _ R } { w } ) \\big ) \\right ) , \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} { \\bf E } [ M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ^ q ] = \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) , \\ ; \\Re ( q ) < \\tau . \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} G _ { t - s } = H _ { \\alpha ' \\alpha _ 2 } ( t - s ) G \\in \\mathcal { G } _ { \\alpha ' } , \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} W = T W + Q W , \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} Q \\left ( \\sum _ { i = 1 } ^ n \\lambda _ i e _ i \\right ) = \\sum _ { i = 1 } ^ \\frac { k b } { 2 } \\lambda _ i \\lambda _ { \\frac { k b } { 2 } + i } + Q ' \\left ( \\sum _ { i = k b + 1 } ^ n \\lambda _ i e _ i \\right ) \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t u ( t ) = ( \\lambda / \\mu ^ { 1 / 2 } ) u ( t ) - f ( t ) / \\mu ^ { 1 / 2 } . \\end{array} \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} S _ { 1 , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) = r ! \\sum _ { t = 0 } ^ { r } \\frac { 1 } { t ! } \\binom { b + k - t } { r - t } S _ { 1 , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p - 1 , k - t \\right ) . \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\gamma _ { \\infty } ( \\kappa _ 1 , \\kappa _ 2 ) & = \\begin{cases} \\beta , & f ^ U = f ^ L = 0 , \\\\ \\max \\{ 1 , \\beta \\} , & \\end{cases} \\\\ \\gamma _ 0 ( \\kappa _ 1 , \\kappa _ 2 ) & = \\begin{cases} \\alpha , & f ^ U = f ^ L = 0 , \\\\ \\min \\{ 1 , \\alpha \\} , & \\end{cases} \\\\ c _ 1 & : = g ^ L c _ 2 : = c _ 3 : = f ^ U + g ^ U \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} \\Sigma = \\bar { \\Omega } ( x ) \\Lambda ^ { ( 1 ) } - \\Omega _ 0 ( x ) + \\Omega _ 1 ( x ) \\Lambda - \\Omega _ 2 ( x ) \\Lambda ^ 2 \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} C _ { p } \\rtimes C _ { q ^ { m } } = \\langle s , t | \\ t ^ { p } = s ^ { q ^ { m } } = 1 , \\ s ^ { - 1 } t s = t ^ { k } \\rangle \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} i _ 4 ( M ) & < j _ 4 ( S _ t ) \\sum _ { \\lambda } \\sum _ { l _ 1 + 2 l _ 2 + 4 l _ 4 = t } i _ { 4 , \\lambda } ( C l _ m ( q ) ) ^ { l _ 1 } i _ { 2 , \\lambda } ( C l _ m ( q ) ) ^ { l _ 2 } | C l _ m ( q ) | ^ { 3 l _ 4 + l _ 2 } \\\\ & < | S _ t | ^ \\frac { 3 } { 4 } | C l _ m ( q ) | ^ { \\frac { 3 } { 4 } ( l _ 1 + 2 l _ 2 + 4 l _ 4 ) } q ^ { c '' n } \\\\ & < | M | ^ \\frac { 3 } { 4 } q ^ { c ''' n } . \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} & \\sigma _ 0 = ( - 1 ) ^ { m + 1 } { 2 m \\choose m } \\\\ & \\sigma _ { i + 1 } = \\sigma _ { i } + 2 ( - 1 ) ^ { i + 1 } { n \\choose i } . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} G ( x , y ) ~ \\leq ~ G _ S ( x , y ) ~ = ~ G _ S ( 0 , y - x ) , \\forall x , y \\in E . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} { \\rm d e g r e e } ( { \\mathcal N } ^ * _ { \\mathcal F } ) \\ , = \\ , { \\rm d e g r e e } ( D ) \\ , . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} \\sum _ { \\delta \\in \\{ 0 , 1 \\} } \\sum \\limits _ { k _ 1 \\pmod { 2 p } \\atop { k _ 2 \\pmod { 6 p } } } ( - 1 ) ^ { \\delta \\left ( k _ 1 + k _ 2 \\right ) } \\zeta _ { 2 p } ^ { \\ell _ 1 k _ 1 } \\zeta _ { 6 p } ^ { \\ell _ 2 k _ 2 } I _ { \\boldsymbol { k } } \\left ( - \\frac { 1 } { \\tau } \\right ) = 2 \\sum \\limits _ { { k _ 1 \\pmod { 2 p } \\atop { k _ 2 \\pmod { 6 p } } } \\atop { k _ 1 \\equiv k _ 2 \\pmod { 2 } } } \\zeta _ { 2 p } ^ { \\ell _ 1 k _ 1 } \\zeta _ { 6 p } ^ { \\ell _ 2 k _ 2 } I _ { \\boldsymbol { k } } \\left ( - \\frac { 1 } { \\tau } \\right ) . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ 1 \\Gamma _ { 0 0 } ) = \\mathrm { v o l } ( \\Gamma _ { 0 0 } r _ u \\Gamma _ { 0 0 } ) = \\frac { p ^ { - 3 } ( p - 1 ) ^ 2 } { 2 } . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\tau _ { K } ( \\chi \\otimes \\rho ) = \\rho ( ( \\mathfrak { f } ( \\chi ) \\mathfrak { D } _ { K } ) ^ { - 1 } ) \\tau _ { K } ( \\chi ) = \\rho ( c _ { \\chi } ^ { - 1 } ) \\tau _ K ( \\chi ) . \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} L ( x ) = \\exp \\Big ( \\int _ 1 ^ x \\frac { \\vartheta ( t ) } { t } { \\ : \\rm d } t \\Big ) \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\partial _ t f _ t ( x , v ) = - v \\ , \\partial _ x f _ t ( x , v ) + \\partial _ v ( v f _ t ( x , v ) ) + \\sigma \\ , \\partial _ { v v } f _ t ( x , v ) \\ , . \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} \\sum _ { i + j + k = n } m _ { i + 1 + k } \\left ( 1 ^ { \\otimes i } \\otimes m _ j \\otimes 1 ^ { \\otimes k } \\right ) = 0 . \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} Z : = \\bigcap _ { n \\geq 1 } \\overline { \\bigcup _ { i \\geq n } \\Phi _ i ^ { - 1 } ( 0 ) } . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} \\lim _ { \\| x \\| \\rightarrow \\infty } \\frac { \\Psi ( x ) } { \\| x \\| } = + \\infty . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} R = C _ { s } - \\log ( 1 + \\rho _ { e } ) . \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} \\frac { 1 } { \\zeta _ K ( s ) } = \\sum _ { I \\triangleleft \\mathcal { O } _ K } \\frac { \\mu ( I ) } { N ( I ) ^ s } . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } \\alpha ^ { j } \\gamma ^ { * ( n - j ) } \\otimes \\Delta ( u _ { j , k } ) . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} b ( 1 + a ^ { q + 1 } + b ^ { q + 1 } ) + a ^ { 2 q } = 0 \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} O r d _ { d } ( a b ^ { - 1 } ) = 2 k ~ ~ ~ \\Rightarrow ~ ~ ~ ( a b ^ { - 1 } ) ^ k \\equiv - 1 ( { \\rm m o d } ~ d ) , \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} ( y x \\cdot x ^ { - 1 } ) x \\cdot y ^ { \\alpha } & = y x \\cdot y ^ { \\alpha } & \\\\ y x \\cdot x ^ { - 1 } & = y \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} U ^ { \\ast } & = u _ { 0 } + \\grave { u } _ { 1 } t + \\grave { u } _ { 2 } t ^ { 2 } + . . . \\\\ U _ { \\varphi } ^ { \\ast } & = \\mu _ { 0 } + \\mu _ { 1 } t + \\mu _ { 2 } t ^ { 2 } + . . . \\\\ U _ { \\theta } ^ { \\ast } & = \\eta _ { 0 } + \\eta _ { 1 } t + \\eta _ { 2 } t ^ { 2 } + . . . . \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\zeta ( e _ n ) & = - f _ { n + 1 } , \\zeta ( x _ n ) = - y _ { n + \\frac { 1 } { 2 } } , \\zeta ( h _ { n } ) - h _ n + 2 \\delta _ { n , 0 } K , \\\\ \\zeta ( K ) & = K , \\zeta ( y _ n ) = x _ { n - \\frac { 1 } { 2 } } , \\zeta ( f _ n ) = - e _ { n - 1 } . \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} ( A , B ; C , D ) : = \\frac { [ A , C ] [ B , D ] } { [ A , D ] [ B , C ] } \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} \\begin{bmatrix} N _ 2 ( \\lambda ) & - N _ 2 ( \\lambda ) M ( \\lambda ) \\widehat { N } _ 1 ( \\lambda ) ^ T \\end{bmatrix} \\mathcal { L } ( \\lambda ) = Q ( \\lambda ) \\begin{bmatrix} \\widehat { K } _ 1 ( \\lambda ) & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} M _ { t } & : = \\exp \\left ( - \\int _ 0 ^ { t } V ( W _ s ) d W _ s - \\frac { 1 } { 2 } \\int _ 0 ^ { t } V ^ 2 ( W _ s ) d s \\right ) \\\\ & = \\exp \\left ( F ( W _ 0 ) - F ( W _ t ) + \\frac { 1 } { 2 } \\int _ 0 ^ t ( V ' ( W _ s ) - V ^ 2 ( W _ s ) ) d s \\right ) , \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} \\begin{cases} u _ t = u \\big ( a _ 0 ( t ) - a _ 1 ( t ) u - a _ 2 ( t ) v \\big ) \\cr v _ t = v \\big ( b _ 0 ( t ) - b _ 1 ( t ) u - b _ 2 ( t ) v \\big ) . \\end{cases} \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\mathbb { E } [ f ( \\Delta _ { i , n } ^ { ( 1 ) } X ^ { t o y , ( 1 ) } , \\Delta _ { j , n } ^ { ( 2 ) } X ^ { t o y , ( 2 ) } ) | \\mathcal { S } ] = ( \\sigma ^ { ( 1 ) } ) ^ { p _ 1 } ( \\sigma ^ { ( 2 ) } ) ^ { p _ 2 } \\mathbb { E } [ f ( Z , Z ' ) ] | \\mathcal { I } _ { i , n } ^ { ( 1 ) } | ^ { p _ 1 / 2 } | \\mathcal { I } _ { j , n } ^ { ( 2 ) } | ^ { p _ 2 / 2 } \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} g . ( \\phi _ 1 , v _ 1 , \\dots , \\phi _ n , v _ n ) : = \\big ( g \\phi _ 1 g ^ { - 1 } , g v _ 1 , \\dots , g \\phi _ n g ^ { - 1 } , g v _ n \\big ) . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + \\Delta u = 0 , x , t \\in \\R ^ + , \\\\ u ( x , 0 ) = u _ 0 ( x ) \\in H ^ { s _ 0 } ( \\R ^ + ) , u ( 0 , t ) = g ( t ) \\in H ^ { \\frac { 2 s + 1 } 4 } ( \\R ^ + ) , \\end{cases} \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} g _ { m , n } ( s , t ) : = \\begin{cases} \\displaystyle \\sum _ { j = 1 } ^ { m } \\left \\langle e _ j , g \\left ( s , \\tfrac { t ^ n _ k + t ^ n _ { k + 1 } } { 2 } \\right ) \\right \\rangle e _ j & t \\in ( t _ { k } ^ n , t _ { k + 1 } ^ n ) , k = 0 , \\ldots , N _ n - 1 , \\\\ \\displaystyle \\sum _ { j = 1 } ^ { m } \\langle e _ j , g ( s , t ^ n _ k ) \\rangle e _ j , & t = t _ { k } ^ n , k = 0 , \\ldots , N _ n , \\end{cases} \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\frac { d ( p , \\gamma _ i ( \\delta ) ) - d ( \\gamma _ i ( 0 ) , \\gamma _ i ( \\delta ) ) } { d _ i } = g _ { w _ \\infty } ( w _ \\infty , v ^ f ) . \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} F ^ n ( z , z ' ) : = F ( z , z ' ) ^ n ~ ~ ~ ( n = 1 , 2 , \\dots ) \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} \\dot { \\phi } ( n ) = \\frac { 1 } { ( n + 1 ) ^ \\alpha } , \\quad \\forall n \\in \\N . \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} A _ k \\ , x ^ Y = 0 \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\bar { u } ( x ; t ) = \\begin{cases} u _ n , \\bar { x } _ { n - 1 } < x \\leq \\bar { x } _ n \\ , ( n = 1 , 2 , \\ldots , N ) \\\\ u _ { N + 1 } , x > \\bar { x } _ N \\end{cases} \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { U } V = \\nabla _ { U } V + B ( U , V ) N , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} m = 1 + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( b ^ y _ { \\ell } - 1 ) . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\widetilde { P } ' ( x ) = \\sum \\limits _ { n \\geq 0 } n ! S _ n \\frac { x ^ n } { n ! } , \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} \\Vert y _ i - a _ i \\Vert & \\le \\Vert y _ i - b _ i \\Vert + \\Vert b _ i - a _ i \\Vert \\\\ & < 2 0 \\rho + 1 0 \\rho + 2 \\rho = 3 2 \\rho . \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\Omega _ { n , j } : = \\{ \\omega \\in \\Omega \\ , | \\ , \\forall \\alpha \\in \\Omega , \\ , \\exists k \\ , : \\ , \\Vert M ^ n _ { T ^ k \\omega } - M ^ n _ \\alpha \\Vert < \\varepsilon _ j \\} . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} B _ \\lambda ( s ) = i ^ { n k } \\ , \\frac { \\pi ^ { n ( n + 1 ) / 2 } } { \\prod _ { m = 1 } ^ { n } ( m - 1 ) ! } 2 ^ { - n ( 2 n + 1 ) s + 3 n / 2 } \\ , \\gamma _ n \\Big ( ( 2 n + 1 ) s - \\frac 1 2 + k \\Big ) \\prod _ { j = 1 } ^ n \\frac { \\beta ( \\ell _ j , s ) } { \\beta ( k - j , s ) } . \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } e ^ { i t x ^ 2 } { v } ( x ) d x = \\sqrt { 2 \\pi } \\pi ( { \\mathcal F } ^ \\ast N _ v ) ( t ) . \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\bar { c } = \\frac { \\int _ 0 ^ L c ( X , T ) \\ , \\mathrm { d } X } { \\int _ 0 ^ L \\mathrm { d } X } \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} \\Delta ( \\omega ) = \\omega \\otimes \\omega , S ( \\omega ) = \\omega ^ { - 1 } , \\epsilon ( \\omega ) = 1 . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\theta _ { \\emptyset } ^ { ( 0 ) } ( e ) = \\left | \\left \\{ ( a , b , c ) \\in ( [ 0 , r _ e - 1 ] \\cap \\N ) ^ 3 : \\ a + b + c \\equiv 0 \\mod n \\right \\} \\right | = \\left | A _ 0 \\right | + \\left | A _ 1 \\right | + \\left | A _ 2 \\right | , \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} v _ t ( x ) = E \\Big [ v _ T ( \\phi _ { t , T } ( x ) ) \\exp \\Big \\{ \\int _ t ^ T \\mathrm { d i v } \\beta _ j ( \\phi _ { t , s } ( x ) ) d \\tilde { \\Z } _ s ^ j \\Big \\} \\Big ] , \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} k \\cdot ( l , z ) = ( l k ^ { - 1 } , k z ) \\ k \\in K , l \\in L , z \\in Z . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} ( u \\otimes v ) ^ { \\natural } _ { s t } & : = - \\int _ s ^ t \\big [ \\dot { \\mu } _ { r } \\otimes \\delta v _ { s r } + \\delta u _ { s r } \\otimes \\dot { \\nu } _ r \\big ] d r + u _ { s t } ^ { \\natural } \\otimes v _ s + u _ s \\otimes v ^ { \\natural } _ { s t } + u _ { s t } ^ { \\flat } \\otimes v _ { s t } ^ { \\flat } \\\\ & + u _ { s t } ^ { \\flat } \\otimes B _ { s t } ^ 1 v _ s + A _ { s t } ^ 1 u _ s \\otimes v _ { s t } ^ { \\flat } . \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} F ( v ) : = \\begin{pmatrix} f _ { e _ 1 } ( 0 ) \\\\ \\vdots \\\\ f _ { e _ l } ( 0 ) \\\\ f _ { e _ { l + 1 } } ( L ( e _ { l + 1 } ) ) \\\\ \\vdots \\\\ f _ { e _ { m } } ( L ( e _ { m } ) ) \\end{pmatrix} F ' ( v ) : = \\begin{pmatrix} f ' _ { e _ 1 } ( 0 ) \\\\ \\vdots \\\\ f ' _ { e _ l } ( 0 ) \\\\ - f ' _ { e _ { l + 1 } } ( L ( e _ { l + 1 } ) ) \\\\ \\vdots \\\\ - f ' _ { e _ { m } } ( L ( e _ { m } ) ) \\end{pmatrix} . \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\sum _ { j _ N = 0 } ^ { \\infty } \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } \\left ( \\vec { d } ( x , y ) + 2 ( j _ 1 , . . . , j _ N ) + 2 \\chi ^ I \\right ) = \\sum _ { j = 0 } ^ { \\infty } a _ j - a _ { j + 1 } \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } & \\frac { ( \\phi _ { 1 + a } * \\phi ) ( n ) } { n } \\log \\frac { n } { e } = \\frac { \\zeta ( 1 - a ) } { \\zeta ^ { 2 } ( 2 ) } x \\log ^ { } x - \\frac { 2 \\zeta ( 1 - a ) } { \\zeta ^ { 2 } ( 2 ) } x \\\\ & + \\frac { \\zeta ( 1 + a ) } { ( 1 + a ) \\zeta ^ { 2 } ( 2 + a ) } x ^ { 1 + a } \\log x - \\frac { ( 2 + a ) \\zeta ( 1 + a ) } { ( 1 + a ) ^ { 2 } \\zeta ^ { 2 } ( 2 + a ) } x ^ { 1 + a } + \\widetilde { P } _ { a } ( x ) , \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} f ( m n ) = f ( m ) h _ f ( n ) + f ( n ) h _ f ( m ) \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} \\partial _ x ^ m \\ , \\mathbb { E } f \\bigl ( Y ^ n _ t ( x ) \\bigr ) & = \\partial _ x ^ { m - 1 } \\mathbb { E } f ' \\bigl ( Y ^ { n } _ { t } ( x ) \\bigr ) + \\frac { 1 } { 2 } \\sqrt { t } \\ , \\partial _ x ^ { m - 1 } \\mathbb { E } \\bigl ( \\xi g _ 1 \\bigl ( x , \\xi \\sqrt t , Y ^ { n - 1 } _ { t } \\bigr ) \\bigr ) \\\\ & = : P _ m ( t , x ) + R _ m ( t , x ) . \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\widetilde { d } _ 1 = \\overline { d _ 1 } , \\widetilde { d _ 2 } = \\overline { d _ 2 } \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} \\frac { \\mathrm d g ( t ) } { \\mathrm d t } & = \\lim _ { n \\to \\infty } \\frac { \\mathrm d g _ n ( t ) } { \\mathrm d t } \\quad \\ t \\in [ a , b ] . \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} ( x + t ) F ^ 2 - ( 2 t + 1 ) F + t + 1 = 0 , \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\alpha = ( \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 , \\alpha _ 4 , \\alpha _ { 1 ' } , \\alpha _ { 2 ' } , \\alpha _ { 3 ' } , \\alpha _ { 4 ' } ) : \\theta ( V ) \\rightarrow \\theta ( W ) \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} ( L v ) _ j = \\sum _ { \\gamma \\in \\Gamma } c _ { j , \\gamma } v _ \\gamma , \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} & \\partial _ t h + \\partial _ x ( u h ) = 0 , \\\\ & \\partial _ t ( h u ) + \\partial _ x ( h u ^ 2 ) + \\frac { g } { 2 } \\partial _ x h ^ 2 = 4 \\nu \\partial _ x ( h \\partial _ x u ) + h f \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} ( \\rho , u ) ( x , t ) = \\left \\{ \\begin{array} { l l } ( \\rho _ - , u _ - + \\beta t ) , \\ \\ \\ \\ - \\infty < x < u _ - t + \\frac { 1 } { 2 } \\beta t ^ { 2 } , \\\\ v a c u u m , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ u _ - t + \\frac { 1 } { 2 } \\beta t ^ { 2 } < x < u _ + t + \\frac { 1 } { 2 } \\beta t ^ { 2 } , \\\\ ( \\rho _ + , u _ + + \\beta t ) , \\ \\ \\ \\ u _ + t + \\frac { 1 } { 2 } \\beta t ^ { 2 } < x < \\infty . \\end{array} \\right . \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} \\langle f , \\phi \\rangle = \\langle - \\Delta u , | w | ^ { q - 2 } f \\rangle + ( q - 2 ) \\langle | w | ^ { q - 3 } w \\cdot \\nabla | w | , f \\rangle = : F _ 1 + F _ 2 . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\Vert ( a _ i ^ { \\Vert } , \\phi _ i ^ { \\Vert } ) \\Vert _ { L ^ { 2 , n } _ { \\nu + n } ( C ) } + \\Vert ( a _ i ^ { \\perp } , \\phi _ i ^ { \\perp } ) \\Vert _ { L ^ { 2 , n } ( C ) } = \\Vert ( a _ i , \\phi _ i ) \\Vert _ { H _ { n , \\nu + n } ( C ) } \\lesssim m _ i ^ { - 7 / 4 } , \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} Z _ { \\varepsilon } ( \\beta ) = \\sum \\limits _ { j = - N / 2 } ^ { N / 2 } e ^ { - \\beta V _ \\varepsilon ( \\psi _ j ) } . \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\lambda ^ { \\textrm { o p t } } = e ^ { j \\angle \\lambda _ { 1 } } \\sqrt [ 3 ] { | \\lambda _ { 1 } | } + e ^ { j \\angle \\lambda _ { 2 } } \\sqrt [ 3 ] { | \\lambda _ { 2 } | } - \\frac { a } { 3 } , \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} A ^ T \\Sigma ^ { - 1 } + \\Sigma ^ { - 1 } A + \\sum _ { k = 1 } ^ m N _ k ^ T \\Sigma ^ { - 1 } N _ k + \\sum _ { i , j = 1 } ^ v H _ i ^ T \\Sigma ^ { - 1 } H _ j k _ { i j } & \\leq - \\Sigma ^ { - 1 } B B ^ T \\Sigma ^ { - 1 } , \\\\ A ^ T \\Sigma + \\Sigma A + \\sum _ { k = 1 } ^ m N _ k ^ T \\Sigma N _ k + \\sum _ { i , j = 1 } ^ v H _ i ^ T \\Sigma H _ j k _ { i j } & \\leq - C ^ T C , \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} \\theta = \\inf _ { u \\ne 0 : \\| u _ { T ^ c } \\| _ 1 \\le \\sqrt { \\tilde k } \\| u \\| } \\frac { \\| \\bar \\Sigma ^ { 1 / 2 } u \\| } { \\| u \\| } \\tilde k = k \\ ; ( 2 4 \\log ( p / k ) ) ^ 2 . \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} 4 \\pi \\left ( ( 2 ^ { n + 1 } + 1 ) \\cdot \\Lambda _ { \\ ! n } ^ { 2 } + ( 2 ^ n + 1 ) \\cdot 4 \\Lambda _ { \\ ! n } ^ { 2 } \\right ) \\leq C \\cdot 2 ^ n \\cdot ( { 2 ^ { - n } } ) ^ 2 = C \\cdot 2 ^ { - n } \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} \\frac { d + 1 } { d } = \\frac { d + 1 } { d } \\norm { f } ^ { 2 } = \\sum _ { i = 1 } ^ { d + 1 } | \\langle \\vec { f } , \\vec { f } _ { i } \\rangle | ^ { 2 } . \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} P = \\bigoplus _ { \\lambda \\in \\Pi ^ { + } ( P ) } V ^ { \\lambda } \\otimes L ^ { \\lambda } \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} \\Omega _ 2 ( N , q ) = \\begin{cases} \\frac { 1 } { 2 } - \\frac { 3 } { 4 } e _ 2 & 0 \\leq e _ 2 \\leq 2 , \\\\ - 2 & e _ 2 \\geq 3 , s _ 2 = 0 , \\\\ - 2 s _ 2 & e _ 2 > s _ 2 \\geq 3 , \\\\ - e _ 2 & e _ 2 = s _ 2 \\geq 3 . \\end{cases} \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} B ( x , y ) = \\int \\limits _ 0 ^ \\infty \\frac { a ^ { x - 1 } } { ( a + 1 ) ^ { x + y } } \\ , d a = 2 \\int \\limits _ 0 ^ \\infty \\frac { a ^ { x - y } } { ( a + a ^ { - 1 } ) ^ { x + y } } \\ , d ^ \\times a \\qquad { \\rm R e } ( x ) , { \\rm R e } ( y ) > 0 . \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} \\mathbf { u } ^ s ( A _ i ) = \\mathbf { u } ^ s ( X ) + \\left ( ( A _ i - X ) ^ T \\otimes I _ 2 \\right ) \\emph { V e c } ( \\nabla \\mathbf { u } ^ s ( X ) ) + \\mathbf { R } ^ s _ { i } ( X ) , ~ i \\in \\mathcal { I } ^ s , ~ \\forall X \\in T _ { \\ast } ^ s , \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} \\lambda _ 1 ( \\widetilde H ) > 0 = \\lambda _ 1 ( H ) , \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} ( x , y , z , w ) = ( t ^ 5 , t ^ { 1 7 } + s t ^ { 1 8 } , t ^ { 2 8 } , t ^ { 4 6 } ) . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\theta ( \\beta , H ) : = \\mathbb { P } _ { \\mathbb { Z } ^ d , w , \\beta , H } ( 0 \\overset { \\mathbb { Z } ^ d } { \\longleftrightarrow } \\infty ) , \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\tilde h _ { i , k } = \\Im \\left \\{ \\frac { r _ { i , k } e ^ { - j \\hat \\theta ^ { \\mathrm f } _ { i , k - 1 } } } { s _ { i , k } } \\right \\} . \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} \\widetilde { P } ( x ) = \\sum \\limits _ { n \\geq 1 } ( n - 1 ) ! S _ { n - 1 } \\frac { x ^ n } { n ! } . \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} { \\underline { \\mathfrak { Q } } } _ t ^ k ( x _ { t - 1 } , D _ { t - 1 } , \\xi _ t , D _ { t } ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; D _ { t - 1 } f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + \\mathcal { Q } _ { t + 1 } ^ k ( x _ { t } , D _ t ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ { r _ { k _ 0 } } } G ( x , y ) | f | ( y ) d y \\leq C _ 1 M r _ { k _ 0 } ^ { s - n / p } \\end{aligned} \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} \\mathcal E _ 1 ( \\tau ) = J _ { ( 1 , 3 ) } ( \\tau ) . \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} \\lambda _ q ( n ) : = \\sum _ { q _ { 1 } ^ { 2 } q _ 2 q _ 3 = q } \\mu ( q _ 2 ) \\rho _ { q _ 3 } ( n ) . \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} W _ k : = \\mathrm { I n d } _ { H _ n ^ k } ^ { H _ n } ( \\Bbbk ) \\cong H _ n \\otimes _ { H _ n ^ k } \\Bbbk , \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} \\dd ( y ^ { ( i ) } ) = z y ^ { ( i - 1 ) } \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\Bigl \\Vert \\sum _ { i \\in A } z _ i \\Bigr \\Vert = \\vert A \\vert . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} L = \\begin{bmatrix} 1 & - a ^ { - 1 } u ^ T \\\\ 0 & I \\end{bmatrix} , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} \\varphi = q \\circ a . \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} E ^ a ( x , \\gamma \\setminus x ) : = \\sum _ { y \\in \\gamma \\setminus x } a ( x - y ) \\geq 0 . \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} A * B = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 0 & 0 \\end{array} \\right ) * \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) \\neq \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) * \\left ( \\begin{array} { c c } 0 & 1 \\\\ 0 & 0 \\end{array} \\right ) = B * A \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} \\begin{pmatrix} a ' - a & b _ 1 - b ' & c _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} & \\phi _ { ( j , N ) } f = \\left ( f - \\mathbb { E } \\left ( f \\ ; | \\ ; A _ { j } ^ { ( N ) } \\right ) \\right ) 1 _ { A _ { j } ^ { ( N ) } } , \\\\ & \\phi _ { ( N ) } f = \\sum _ { j = 1 } ^ { m _ { N } } \\phi _ { ( j , N ) } f = f - \\sum _ { j = 1 } ^ { m _ { N } } \\mathbb { E } ( f \\ ; | \\ ; A _ { j } ^ { ( N ) } ) 1 _ { A _ { j } ^ { ( N ) } } . \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} \\log \\hat { X } _ n = \\hat { x } _ n { \\partial \\over \\partial \\hat { x } _ n } F , \\log \\hat { Y } _ n = - \\hat { y } _ n { \\partial \\over \\partial \\hat { y } _ n } F \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} [ P ' ] : = \\left [ \\begin{pmatrix} x \\cdot H \\\\ y \\cdot H \\\\ 1 \\end{pmatrix} \\right ] . \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} | \\frac { 1 } { n } \\log P r ( ( X ^ n , Y ^ n ) = ( x ^ n , y ^ n ) ) + H ( X , Y ) | & < \\delta \\\\ | \\frac { 1 } { n } \\log P r ( X ^ n = x ^ n ) + H ( X ) | & < \\delta \\\\ | \\frac { 1 } { n } \\log P r ( Y ^ n = y ^ n ) + H ( Y ) | & < \\delta \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + \\ , { \\rm c o n s t } ) } \\ , { \\bf E } \\Bigl [ M ^ q _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\Bigr ] \\Gamma ( 1 - q ) , \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\sum _ { k \\in J } b _ { k s } \\lambda _ { k t } = 2 r _ { i _ s } \\delta _ { s t } \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} A _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , i \\right ) = \\sum _ { j = 0 } ^ { i } \\left ( - 1 \\right ) ^ { j } \\binom { r p + \\sigma + 1 } { j } \\binom { a \\left ( i - j \\right ) + b } { r } ^ { p } \\prod _ { s = 2 } ^ { L } \\binom { \\alpha _ { s } \\left ( i - j \\right ) + \\beta _ { s } } { r _ { s } } ^ { p _ { s } } . \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} r _ { i j } = \\begin{cases} 1 , & ( i , j ) = ( 0 , p ) , \\\\ \\frac { 1 } { p - | I | } , & i \\in I j = p , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} ( \\omega _ M \\otimes \\omega _ M - \\omega _ N \\otimes \\omega _ N ) ( x , y ) = \\sum _ { ( k , l ) \\in \\Lambda _ { M , N } } \\ < \\omega , \\tilde e _ k \\ > \\ < \\omega , \\tilde e _ l \\ > \\tilde e _ k ( x ) \\tilde e _ l ( y ) = : \\omega _ { M , N } ( x , y ) . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} u _ j = \\dfrac { 1 } { \\sigma _ { j } } C _ k ( F ) v _ j , j = 1 , \\dots , p \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} A ( z ) \\sum _ n \\frac { G ( t _ n ) \\bar c _ n } { A ' ( t _ n ) \\mu _ n ^ { 1 / 2 } ( z - t _ n ) } = G ( z ) S ( z ) \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} \\Psi _ { ( N _ 1 , N _ 2 ) } : \\mathbb { F } _ { \\ell } [ \\lambda ] ^ { ( m + m _ 2 ) \\times ( n + m _ 1 ) } & \\longrightarrow \\mathbb { F } _ { d } [ \\lambda ] ^ { m \\times n } \\\\ M ( \\lambda ) & \\longrightarrow N _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} f _ k ^ { ( l ) } ( x ) = \\sum _ { P \\in \\mathcal { A } ( x , k ) ^ { ( l ) } } \\delta _ P \\in \\ell _ 2 ( \\mathcal { X } ^ { ( l ) } ) \\subseteq \\bigoplus _ { j = 0 } ^ { N - 1 } \\ell _ 2 ( \\mathcal { X } ^ { ( j ) } ) . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} \\delta u _ { s t } ^ { \\otimes 2 } = \\int _ s ^ t 2 u _ r \\hat { \\otimes } \\Delta u _ r + 2 u _ r \\hat { \\otimes } \\mathrm { d i v } ( F ( u _ r ) ) d r + ( \\Gamma _ { s t } ^ 1 + \\Gamma _ { s t } ^ 2 ) u _ s ^ { \\otimes 2 } + u _ { s t } ^ { \\otimes 2 , \\natural } \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} \\gamma _ k : = \\sum _ { m \\geq 0 } \\alpha _ m \\left ( \\sum _ { \\substack { l ( J ) = m + 1 \\\\ \\vert J \\vert + m = k } } \\beta _ J \\right ) . \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{gather*} L \\big ( \\psi _ v ^ 2 , s - 1 \\big ) = \\frac { 1 } { 1 - N v ^ { 1 - s } } = \\frac { 1 } { 1 - p ^ 2 N v ^ { - s } } . \\end{gather*}"}
-{"id": "6233.png", "formula": "\\begin{align*} | B ' | = \\sum _ { x \\in B ' } 1 \\le \\sum _ { x \\in B ' } \\sum _ { y \\in N ( x ) } \\frac { 1 } { d ( y ) } \\le \\sum _ { y \\in N ( B ' ) } \\sum _ { x \\in N ( y ) } \\frac { 1 } { d ( y ) } = \\sum _ { y \\in N ( B ' ) } 1 = | N ( B ' ) | . \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} [ \\theta ] & = \\bigoplus _ { \\hat k \\in \\hat K } [ \\rho _ { \\hat k } ] = [ ( 0 , 0 ) ] \\oplus [ ( \\alpha , 0 ) ] \\oplus \\bigoplus _ { g \\in G ^ \\times } [ ( \\theta _ { ( g , - g ) } , \\bar g ) ] \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( a _ 1 , a _ 2 ; N ) } ( \\tau ) = \\sum _ { ( m , n ) \\in \\Z ^ 2 \\atop ( m , n ) \\equiv ( a _ 1 , a _ 2 ) \\mod N } \\frac 1 { ( m \\tau + n ) ^ k } . \\end{gather*}"}
-{"id": "337.png", "formula": "\\begin{align*} M _ 1 ( s ) = \\zeta ( 2 s ) \\varphi _ 0 + \\Sigma ( s ) - \\Sigma _ B ( s ) - \\frac { 2 \\zeta ( s ) } { \\pi } \\int _ { 0 } ^ { \\infty } \\frac { \\zeta ( s + 2 i t ) \\zeta ( s - 2 i t ) } { | \\zeta ( 1 + 2 i t ) | ^ 2 } \\hat { \\varphi } ( t ) d t . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} \\sum _ { x = ( x _ 1 , x _ 2 ) \\in \\Z ^ 2 } x _ 1 ^ 2 \\mu ( x ) = \\sum _ { x = ( x _ 1 , x _ 2 ) \\in \\Z ^ 2 } x _ 2 ^ 2 \\mu ( x ) = 1 \\sum _ { x = ( x _ 1 , x _ 2 ) \\in \\Z ^ 2 } x _ 1 x _ 2 \\mu ( x ) = \\rho \\in ] - 1 , 1 [ . \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} L = \\sum _ { j = 1 } ^ { N } r ^ { j - 1 } \\xi . \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\mathcal I _ 2 ( n ) = \\int _ { \\mathcal A _ 2 ^ + ( n ) } \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h + \\int _ { \\mathcal A _ 2 ^ - ( n ) } ( c , d ) _ p \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h . \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} r = 1 \\implies \\chi ^ 2 \\neq 1 \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} C \\langle y \\mid \\dd ( y ) = z _ { f } \\otimes 1 \\rangle \\otimes _ { R } M \\cong C \\langle y \\mid \\dd ( y ) = z _ { g } \\otimes 1 \\rangle \\otimes _ { R } M \\ , . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{gather*} A = u _ 1 \\prod _ { l \\ne 1 } ^ n a _ { 1 l } a _ { 1 l } ^ + \\ , , B = \\tau ^ { 2 n - 2 } \\tau _ n - A - \\sum _ { i \\ne 1 } ( C _ i + D _ i ) \\ , , \\\\ C _ i = u _ i b _ { 1 i } a _ { 1 i } ^ + \\prod _ { l \\ne 1 , i } a _ { i l } a _ { i l } ^ + \\ , , D _ i = u _ i ^ - b _ { 1 i } ^ + a _ { 1 i } \\prod _ { l \\ne 1 , i } a _ { l i } ^ - a _ { l i } \\ , . \\end{gather*}"}
-{"id": "9557.png", "formula": "\\begin{align*} D _ { n + 1 , t } ^ 0 D _ { n + 1 } ^ 2 - D _ { n + 2 , t } ^ 0 D _ n ^ 2 = 2 D _ { n + 2 } ^ { - 1 } ( 1 ; 2 ) D _ { n + 1 } ^ 1 . \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} f ( m ) \\ast _ R f ( n ) & = ( \\underbrace { 1 _ R + \\dots + 1 _ R } _ { m } ) \\ast _ R ( \\underbrace { 1 _ R + \\dots + 1 _ R } _ { n } ) \\\\ & = \\underbrace { 1 _ R \\ast ( \\underbrace { 1 _ R + \\dots + 1 _ R } _ { n } ) + \\dots + 1 _ R \\ast ( \\underbrace { 1 _ R + \\dots + 1 _ R } _ { n } ) } _ { m } \\\\ & = \\underbrace { 1 _ R + \\dots + 1 _ R } _ { m n } = f ( m \\ast n ) . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\cup _ { \\psi \\in \\Gamma } [ \\neg \\psi ] ^ { \\beta } \\cup \\cup _ { \\psi \\in \\Delta } [ \\psi ] ^ { \\beta } \\cup [ \\phi ( y ) ] ^ { \\beta } = \\Omega \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} n ( T ) & = n ^ + ( T ) - n ^ - ( T ) = 0 - 0 , \\\\ n ( C _ 2 ) & = n ^ + ( C _ 2 ) - n ^ - ( C _ 2 ) = n - 0 , \\\\ n ( L ) & = n ^ + ( L ) - n ^ - ( L ) = 1 2 n - n . \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} & ~ ( A \\sharp B ) \\Big [ ( A ! _ \\lambda B ) ^ { - 1 } - ( A \\nabla _ \\lambda B ) ^ { - 1 } \\Big ] ( A \\sharp B ) = A \\Big [ ( A ! _ \\lambda B ) ^ { - 1 } - ( A \\nabla _ \\lambda B ) ^ { - 1 } \\Big ] B \\\\ & = B \\Big [ ( A ! _ \\lambda B ) ^ { - 1 } - ( A \\nabla _ \\lambda B ) ^ { - 1 } \\Big ] A = \\lambda ( 1 - \\lambda ) ( B - A ) ( A \\nabla _ \\lambda B ) ^ { - 1 } ( B - A ) . \\qquad \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} L \\left ( \\frac { 1 } { 2 } , \\chi \\right ) = \\sum _ { n = 1 } ^ \\infty \\frac { \\chi ( n ) } { n ^ { \\frac { 1 } { 2 } } } , \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} \\int _ \\Omega \\delta \\nabla u _ 1 \\cdot \\nabla u _ 2 d x = 0 , \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} \\mathop { } M = \\{ c \\in C ~ | ~ M c = 0 \\} . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\vec z ^ k = \\frac { \\vec x ^ k } { \\norm { \\vec x ^ k } } , \\ , \\ , \\ , \\vec z ^ { k , i } = ( z ^ { k , i } _ 1 , \\ldots , z ^ { k , i } _ n ) = ( z _ 1 ^ { k + 1 } , \\ldots , z _ i ^ { k + 1 } , z _ { i + 1 } ^ k , \\ldots , z _ { n } ^ k ) . \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} f ^ { ( n ) } ( x _ 0 ) = \\sum _ { i = 1 } ^ k c _ i a _ i ^ { x _ 0 } ( \\log _ p a _ i ) ^ n , n \\in \\N . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\mu \\langle | w | ^ q \\rangle + I _ q ^ { } + c \\bar { I } _ { q } ^ { } + ( q - 2 ) ( J _ q + c \\bar { J } _ { q } ^ { } ) + \\langle [ \\nabla , A _ q ] _ - u ^ { } , w | w | ^ { q - 2 } \\rangle = \\langle f , \\phi \\rangle . \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\delta _ i \\delta _ j & = \\delta _ { i , j } \\delta _ i , & 1 & = \\sum _ { i = 0 } ^ { N - 1 } \\delta _ i , & \\Delta ( \\delta _ i ) & = \\sum _ { a + b = i } \\delta _ a \\otimes \\delta _ b , & \\epsilon ( \\delta _ i ) & = \\delta _ { i , 0 } , & S ( \\delta _ i ) = \\delta _ { - i } . \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\dot { q } _ t = L ^ { \\Delta , \\sigma } _ \\vartheta q _ t , q _ t | _ { t = 0 } = q _ 0 \\in \\mathcal { D } _ \\vartheta . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} A _ { u ^ { \\star n } } h = T _ { u ^ { \\star n } } P h - P T _ { u ^ { \\star n } } h . \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\partial _ t Y ^ { n , m } _ t + v \\partial _ x Y ^ { n , m } _ t + B ^ { n , m } _ t \\partial _ v Y ^ { n , m } _ t - \\partial _ v ( v Y ^ { n , m } _ t ) - \\sigma \\partial _ { v v } Y ^ { n , m } _ t = R ^ { n , m } _ t \\ , , \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} - a _ v E \\nu \\times \\nu _ v + a ( \\lambda _ v \\nu _ u + F _ v \\nu \\times \\nu _ u - E _ v \\nu \\times \\nu _ v - E \\nu \\times \\nu _ { v v } ) = 0 \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} \\begin{pmatrix} a ' & - b ' _ 1 & - c ' _ 2 \\\\ - a ' _ 2 & b ' & - c ' _ 1 \\\\ - a ' _ 1 & - b ' _ 2 & c ' \\end{pmatrix} \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} h ( M _ { i - 1 } ^ j ) > h _ { i - 1 } ^ j , ~ j = 1 , \\cdots , k . \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} \\eta _ { 2 , 2 } ( q \\ , | \\ , \\tau , b ) = \\exp \\Bigl ( - \\frac { b _ 1 b _ 2 } { \\tau } \\log ( q ) + O ( 1 ) \\Bigr ) , \\ ; q \\rightarrow \\infty , \\ ; | \\arg ( q ) | < \\pi . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} B ^ { \\Sigma } + B ^ { \\Sigma ' } = B ^ { \\Sigma } + I _ { G } B = B ^ { \\Sigma ' } + I _ { G } B \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\mu _ n \\vert _ { B } = \\sum _ { i = 1 } ^ k \\delta _ { x _ i ^ { ( n ) } } \\ ; \\ ; \\ ; \\ ; \\mu \\vert _ { B } = \\sum _ { i = 1 } ^ k \\delta _ { x _ i } \\ : , \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} ( \\pi ( f ) \\xi ) _ x = \\sum _ { g \\in G ^ x } D ( g ) ^ { - 1 / 2 } f ( g ) g \\xi _ { s ( g ) } , \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} \\displaystyle \\int _ { B ( x _ { 0 } , \\epsilon ) } | a ( x ) | ^ { \\frac { - q p ( x ) } { ( p ( x ) - q ) } } d x & \\leq \\displaystyle \\int _ { B ( 0 , \\epsilon ) } | x | ^ { \\frac { - s q p ^ { + } } { ( p ^ { - } - q ) } } d x \\\\ & = w _ { n } \\displaystyle \\int _ { 0 } ^ { \\epsilon } r ^ { N - 1 } r ^ { \\frac { - s q p ^ { + } } { ( p ^ { - } - q ) } } d r \\\\ & = w _ { n } \\frac { \\epsilon ^ { \\alpha } } { \\alpha } , \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} F ( z , z ' ) : = \\ < \\phi _ z , \\phi _ { z ' } \\ > \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} Q = \\bullet _ { 1 , 0 } \\longrightarrow \\bullet _ { 2 , 0 } \\longrightarrow \\bullet _ { 1 , 3 } \\longrightarrow \\bullet _ { 3 , - 2 } \\longrightarrow \\bullet _ { 1 , - 2 } \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} p : = \\mathop { \\sup } \\limits _ { u \\in U } { F _ { u } } ( \\cdot , { 0 _ { u } } ) \\ q : = \\mathop { \\inf } \\limits _ { \\left ( { u , y _ { u } ^ { \\ast } } \\right ) \\in \\Delta } \\ ; F _ { u } ^ { \\ast } ( \\cdot , y _ { u } ^ { \\ast } ) . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{gather*} \\bar { J } = \\xi _ 1 \\| u _ 1 \\| _ { L ^ 2 } ^ 2 + \\xi _ 2 \\| u _ 2 \\| _ { L ^ 2 } ^ 2 , \\\\ \\ell _ 1 = \\| u _ 1 \\| _ { L ^ 2 } ^ 2 ( 2 Q + \\sigma _ 1 ^ 2 - | \\xi _ 1 | ^ 2 ) , \\\\ \\ell _ 2 = \\| u _ 2 \\| _ { L ^ 2 } ^ 2 ( 2 Q + \\sigma _ 2 ^ 2 - | \\xi _ 2 | ^ 2 ) . \\end{gather*}"}
-{"id": "8160.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } - A _ { K W } U + \\nabla _ { U } ^ { s } K W + D ^ { l } ( U , K W ) \\\\ - A _ { L W } U + \\nabla _ { U } ^ { l } L W + D ^ { s } ( U , L W ) \\end{array} \\right ) = \\left ( \\begin{array} { c } K \\nabla _ { U } W + L \\nabla _ { U } W + \\breve { J } h ^ { l } ( U , W ) \\\\ + B h ^ { s } ( U , W ) + C h ^ { s } ( U , W ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} p : = \\gamma - 1 - \\ell > m \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\Theta _ { h + g } ^ \\Phi = \\Theta _ { g } ^ \\Psi + \\Theta _ { h } ^ \\Phi \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} [ \\rho ] [ \\bar \\rho ] & = \\bigoplus _ { g \\in G } [ \\alpha _ g ] & \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\int _ { A _ j } \\left ( \\frac { \\langle a _ j , n \\rangle } { 2 \\pi i | n | ^ 2 } e ^ { - 2 \\pi i \\langle n , x \\rangle } - f ( x ) e ^ { - 2 \\pi i \\langle n , x \\rangle } \\right ) \\ , \\textrm { d } x = \\int _ { \\partial A _ j } \\frac { \\langle n , \\nu ( x ) \\rangle } { 2 \\pi i | n | ^ 2 } f ( x ) e ^ { - 2 \\pi i \\langle n , x \\rangle } \\ , \\textrm { d } x . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} { \\sigma _ { \\tau } } _ M ( v , \\xi ) : = \\sum _ { | \\alpha | + | \\beta | < M } \\sum _ { | \\alpha | + | \\beta | \\le | \\delta | \\le N \\cdot ( | \\alpha | + | \\beta | ) } k _ \\delta ( \\tau , \\alpha , \\beta , x - y ) \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\partial _ \\xi ^ \\delta a ( v , v , \\xi ) . \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} I ( U _ * + \\varphi ) = & I ( U _ * ) + \\frac 1 2 \\int _ \\Omega | \\nabla \\varphi | ^ 2 + \\int _ \\Omega \\bigl ( u _ 0 ^ { 2 * - 1 } - \\sum _ { j = 1 } ^ k U _ j ^ { 2 ^ * - 1 } \\bigr ) \\varphi \\\\ & - \\frac 1 { 2 ^ * } \\int _ \\Omega \\Bigl ( | U _ * + \\varphi | ^ { 2 ^ * } - | U _ * | ^ { 2 ^ * } \\Bigr ) . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} \\bigcup _ { l \\in \\mathbb { Z } , \\ , \\ , l \\leq k } D _ { l } = A ^ { c } _ { k + 1 } \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} i ' = i _ { k ' \\leq l ' } ^ { 0 \\leq n } \\colon ( k ' , l ' ) \\to ( 0 , n ) \\qquad i = i _ { k \\leq l } ^ { 0 \\leq n } \\colon ( k , l ) \\to ( 0 , n ) , \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} & \\left \\lfloor \\frac { n _ 1 + t } { q ^ 3 + 1 } \\right \\rfloor + \\left \\lfloor \\frac { n _ 2 + t } { q ^ 3 + 1 } \\right \\rfloor + ( q ^ 2 - q ) \\left \\lfloor \\frac { t } { q ^ 2 - q + 1 } \\right \\rfloor + \\left \\lfloor \\frac { - t q ^ 3 } { q ^ 3 + 1 } \\right \\rfloor \\\\ = & \\begin{cases} q ^ 2 - 1 + ( q ^ 2 - q ) q - q ^ 3 = - 1 & t = q ^ 3 \\\\ q ^ 2 - 2 + ( q ^ 2 - q ) ( q - 1 ) - q ^ 3 + q ^ 2 - q + 1 = - 1 & t = ( q ^ 2 + 1 ) ( q - 1 ) . \\end{cases} \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} \\varSigma _ { \\beta } & = 4 \\cdot \\mathsf { C o v } ( \\zeta _ 1 ) , m ( \\varphi _ \\lambda ) = \\int _ 0 ^ { \\tau } \\varphi _ \\lambda ( u ) a ( u ) G _ C ( u ) d u , \\\\ \\sigma ^ 2 _ { \\varphi } & = 4 \\cdot \\mathsf { V a r } ~ \\bigl \\langle q ' ( Y , \\varDelta , W , \\lambda _ 0 , \\beta _ 0 ) , \\varphi \\bigr \\rangle = 4 \\cdot \\mathsf { V a r } ~ \\xi ( Y , \\varDelta , W ) , \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\phi ( \\eta ) ~ \\equiv ~ 0 \\mbox { f o r } \\eta \\in [ 0 , 1 ] , \\qquad \\phi ( \\eta ) ~ \\equiv ~ 1 \\quad \\mbox { f o r } \\quad \\eta \\geq 2 . \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ \\frac { \\bigl ( \\mathcal { S } _ { M - 1 } e ^ { - x t } \\bigr ) ( q \\ , | \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } e ^ { - x t } \\bigr ) ( 0 \\ , | \\ , b ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } + q e ^ { - t } \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j } { \\prod \\limits _ { i = 1 } ^ M a _ i } \\Bigr ] \\Bigl ) , \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } f _ M \\Phi _ N = \\Phi ^ { \\operatorname { g s } } , \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} E [ U _ { i j } ] = p _ { i j } \\frac { \\partial \\log Z } { \\partial p _ { i j } } . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} [ u ] _ { s , p , q } = \\left ( \\int _ { \\mathbb { G } } \\int _ { \\mathbb { G } } \\frac { | u ( x ) - u ( y ) | ^ { p } } { q ^ { Q + s p } ( y ^ { - 1 } \\circ x ) } d x d y \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} \\langle \\| x \\| x - y \\| y \\| , y - z \\rangle & = \\langle \\| x \\| ( x - y ) + y ( \\| x \\| - \\| y \\| ) , y - z \\rangle \\\\ & \\leq 2 \\big \\| x - y \\| \\big | \\big \\| y - z \\big \\| \\\\ & \\leq \\| x - y \\| ^ 2 + \\| y - z \\| ^ 2 . \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} X _ n ( T _ 0 ) = \\bigcup _ { i = 1 } ^ N X _ { n , i } ( T _ 0 ) . \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\lambda _ 1 = \\lambda _ 2 = \\lambda \\geq 0 . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} V _ { 1 1 2 1 } V _ { 1 1 1 } = 2 1 \\ , V _ { 1 1 1 1 1 1 1 1 } + 6 \\ , V _ { 1 1 2 1 1 1 1 } + V _ { 1 1 2 2 1 1 } + 3 \\ , V _ { 1 1 3 1 1 1 } + 1 0 \\ , V _ { 1 2 1 1 1 1 1 } + 1 5 \\ , V _ { 2 1 1 1 1 1 1 } . \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} P ( \\lambda ) = \\lambda ^ { k ( \\ell - 1 ) } B _ k ( \\lambda ) + \\lambda ^ { k ( \\ell - 2 ) } B _ { k - 1 } ( \\lambda ) + \\cdots + \\lambda ^ \\ell B _ 2 ( \\lambda ) + B _ 1 ( \\lambda ) . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} M _ t ( x , y ) = E \\Big [ M _ T \\big ( \\phi _ { t , T } ( x ) , \\phi _ { t , T } ( y ) \\big ) \\exp \\Big \\{ \\int _ t ^ T \\mathrm { d i v } \\beta _ j ( \\phi _ { t , s } ( x ) ) + \\mathrm { d i v } \\beta _ j ( \\phi _ { t , s } ( y ) ) d \\tilde { \\Z } _ s ^ j \\Big \\} \\Big ] , \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} \\dim ( C _ { S p _ \\frac { k a } { 2 } } { ( j _ { \\frac { k a } { 4 } , \\frac { k a } { 2 } } ) } ) = \\frac { 1 } { 1 6 } k ^ 2 a ^ 2 + \\frac { 1 } { 4 } k a , \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} \\lim _ { u \\to \\infty } F _ { s u p } ( u ) & = \\lim _ { u \\to \\infty } \\limsup _ { t \\to \\infty } \\P _ { u , x _ 0 } ( X _ { u + t } \\in B | \\tau _ { A } > u + t ) \\\\ & = \\limsup _ { t \\to \\infty } \\P _ { x _ 0 } ( X _ { t } \\in B | \\tau _ { A _ \\infty } > t ) \\\\ & = \\alpha _ \\infty ( B ) . \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { s _ n ( x , r ) } { n } = 0 . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\langle f , \\ , g \\rangle \\triangleq & \\Re \\int | w | \\hat { f } ( w ) \\overline { \\hat { g } ( w ) } \\ , d w , \\\\ = & - \\frac { 1 } { 2 \\pi ^ 2 } \\int f ' ( x ) g ' ( y ) \\log | x - y | d x \\ , d y \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t \\omega _ \\epsilon ( t ) & = - R i c ( \\omega _ \\epsilon ( t ) ) \\\\ \\omega _ \\epsilon ( 0 ) & = \\omega _ \\epsilon , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} \\operatorname { f p r } ( t , \\Omega ) = \\frac { | t ^ T \\cap S | } { | t ^ T | } . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} F _ n = \\Delta ^ { - 1 } ( c ^ { - 2 } F _ { n - 1 } ) , \\ \\ n \\geq 1 . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\left \\langle ( x - \\alpha - j ) _ { j + 1 } p ( x ) , q ( x ) \\right \\rangle _ { \\lambda , j , \\ell } = & \\left \\langle ( x - \\alpha - j ) _ { j + 1 } { \\bf u } ^ { M } , p ( x ) q ( x ) \\right \\rangle \\\\ [ 2 m m ] = & \\left \\langle p ( x ) , ( x - \\alpha - j ) _ { j + 1 } q ( x ) \\right \\rangle _ { \\lambda , j , \\ell } . \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } & \\mathbb { P } [ \\exists \\mathbf { x } \\in \\mathcal { F } ^ l ~ \\textrm { s . t . } ~ \\mathbf { D } \\mathbf { x } = \\mathbf { e } _ i ] \\ge \\lim _ { m \\to \\infty } \\dfrac { 1 } { 1 + \\frac { 1 } { \\mathbb { E } [ Y _ i ] } } = 1 . \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ^ q ( 1 , b _ 0 , b _ 1 , b _ 2 ) \\bigr ] = & \\frac { G ( b _ 0 ) } { G ( q + b _ 0 ) } \\frac { G ( q + b _ 0 + b _ 1 ) } { G ( b _ 0 + b _ 1 ) } \\frac { G ( q + b _ 0 + b _ 2 ) } { G ( b _ 0 + b _ 2 ) } \\times \\\\ & \\times \\frac { G ( b _ 0 + b _ 1 + b _ 2 ) } { G ( q + b _ 0 + b _ 1 + b _ 2 ) } . \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} h _ 1 = \\partial _ y \\psi , h _ 2 = - \\partial _ x \\psi , \\psi | _ { y = 0 } = 0 . \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} ( \\mathfrak { d } _ 1 ^ m a ) _ n & = \\sum _ { k = 0 } ^ m \\tbinom { m } { k } ( - 1 ) ^ k \\frac { ( - 1 ) ^ { n + k } } { ( n + k + 1 ) ^ \\alpha } \\\\ & = ( - 1 ) ^ n \\sum _ { k = 0 } ^ m \\tbinom { m } { k } \\frac { 1 } { ( n + k + 1 ) ^ \\alpha } \\\\ & \\sim \\frac { 1 } { ( n + 1 ) ^ \\alpha } . \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\tilde { G } _ t = \\Theta _ t ^ * \\Psi _ t ^ * G , \\ ; \\ , \\tilde { H } _ t = \\log \\frac { \\omega _ { \\C ^ m } ^ m \\wedge ( \\epsilon _ t ^ 2 \\Theta _ t ^ * \\Psi _ t ^ * \\omega _ F ) ^ n } { ( \\omega _ { \\C ^ m } + \\epsilon _ t ^ 2 \\Theta _ t ^ * \\Psi _ t ^ * \\omega _ F ) ^ { m + n } } . \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} U = \\sum _ { i < j } \\frac { m _ { i } m _ { j } } { r _ { i j } } , r _ { i j } = \\left \\vert \\mathbf { a } _ { i } - \\mathbf { a } _ { j } \\right \\vert \\mathbf { a } _ { i } \\in \\mathbb { R } ^ { 3 } , \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} \\lim _ { t \\searrow 0 } \\frac { f \\circ \\alpha ( t ) - f \\circ \\alpha ( 0 ) } { t } = g _ { \\dot \\gamma ( f ( p ) ) } ( \\dot \\gamma ( f ( p ) ) , \\dot \\alpha ( 0 ) ) \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} \\iota _ { \\leq p } \\colon A _ { \\leq p } = \\{ x \\in A \\mid f ( x ) \\leq p \\} \\to A . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} q ( n ) = e ^ { \\beta _ c } f _ n , \\ , n \\ge 1 , \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} a \\left ( u \\right ) = \\begin{pmatrix} 2 u _ { 1 } & 0 \\\\ 0 & D \\end{pmatrix} , F \\left ( u , u _ { x } \\right ) = \\begin{pmatrix} C u _ { 1 x } + A \\left ( 1 - u _ { 1 } \\right ) - u _ { 1 } u _ { 2 } ^ { 2 } \\\\ - B u _ { 2 } + u _ { 1 } u _ { 2 } ^ { 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} ( 2 p - 1 ) x + ( p - 1 ) y = j , \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} | G | _ { \\sigma , \\alpha } = \\int _ { \\Gamma _ 0 } | G ( \\eta ) | \\exp ( \\alpha | \\eta | ) e ( \\phi _ \\sigma ; \\eta ) \\lambda ( d \\eta ) . \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} h ( z ) = \\sum _ { \\substack { n \\geq 1 , \\\\ ( - 1 ) ^ k n \\equiv 0 , 1 \\ , ( 4 ) } } c ( n ) q ^ n . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { j = 0 } ^ { n - 1 } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) = \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J \\setminus J _ { K } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) . \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} ( M \\otimes _ { A } N ) ^ { k } = \\bigoplus _ { i + j = k } M ^ { i } \\otimes N ^ { j } \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} F | _ { k + 1 } [ \\gamma ] = \\chi ( \\gamma ) F \\gamma \\in \\Gamma _ 0 ^ { ( 2 ) } ( N ) . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} e ( t ) : = \\begin{bmatrix} v & 0 \\\\ 0 & I \\end{bmatrix} \\exp ( - t J ) \\begin{bmatrix} e & 0 \\\\ 0 & 0 \\end{bmatrix} \\exp ( t J ) \\begin{bmatrix} v ^ { - 1 } & 0 \\\\ 0 & I \\end{bmatrix} , t \\in [ 0 , \\frac \\pi 2 ] , \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} \\epsilon _ p = w _ p p \\mid N / M , \\epsilon _ p = - w _ p p \\mid M . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} t _ { a } t _ { c } t _ { b } t _ { a } t _ { c } = t _ { a } t _ { c } t _ { b } t _ { c } t _ { a } = t _ { a } t _ { b } t _ { c } t _ { b } t _ { a } . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} ( n + 1 ) ^ { m } \\mathfrak { d } _ 2 \\dot { \\phi } ( n ) = ( n + 1 ) ^ { m - N - \\frac { 1 } { 2 } } , \\quad \\forall n \\in \\N . \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} & i p = e = F ( i ' ) F ( p ' ) , \\ , p i = 1 _ x , \\ , F ( p ' ) F ( i ' ) = F ( p ' i ' ) = F ( 1 _ { x ' } ) = 1 _ { F ( x ' ) } , & & \\\\ & F ( p ' ) i \\circ p F ( i ' ) = F ( p ' ) F ( i ' ) F ( p ' ) F ( i ' ) = 1 _ { F ( x ' ) } & & \\\\ & p F ( i ' ) \\circ F ( p ' ) i = p i p i = 1 _ x . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} \\sum _ { j \\geq - n } \\int _ { \\mathcal L _ { 2 j + 1 } } \\mathbf h _ p ( h ) d h = - p \\sum _ { j \\geq - n } \\mathrm { v o l } ( \\mathcal L _ { 2 j + 1 } ) = - p ( 1 - p ^ { - 1 } ) ^ 2 \\sum _ { j \\geq - n } p ^ { - 2 j - 1 } = - p ^ { 2 n } \\frac { ( 1 - p ^ { - 1 } ) ^ 2 } { 1 - p ^ { - 2 } } = - p ^ { 2 n } \\frac { p - 1 } { p + 1 } . \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} \\mu _ { p } ( \\Delta _ { r } ) = \\mu _ { p } ( \\Gamma _ { r } ) = \\sum _ { \\{ i \\in [ 1 , 2 ^ { p } ] \\ , ; \\ , i \\equiv r \\ ! \\ ! \\ ! \\ ! \\mod 2 ^ { q } \\} } \\mu _ { p } ( \\{ \\lambda _ { i } \\} ) \\le ( 1 - \\varepsilon ) ^ { q } \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} g = \\left ( \\delta _ { i j } + \\frac { 1 } { 3 } _ { i k l j } x ^ k x ^ l + O ( \\vert x \\vert ^ 2 ) \\right ) d x ^ i \\otimes d x ^ j . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} \\Gamma _ 2 ^ { - 1 } ( z \\ , | \\ , a _ 1 , a _ 2 ) = ( 2 \\pi ) ^ { - z / 2 a _ 1 } a _ 2 ^ { 1 + { } _ 2 S _ 0 ( z \\ , | \\ , a _ 1 , a _ 2 ) } G \\Bigl ( \\frac { z } { a _ 1 } \\ , \\Big | \\ , \\frac { a _ 2 } { a _ 1 } \\Bigr ) . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} C = \\mathbf { V } ( e ) ^ { S } / / \\mathrm { P G L ( \\mathbf { 1 } ) } \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} { \\rm d e g r e e } ( { \\mathcal N } ^ * _ { \\mathcal F } ) - { \\rm d e g r e e } ( D ) \\ , = \\ , { \\rm d e g r e e } ( { \\rm a d } ( E _ H ) ) \\ , . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} w _ i ^ k = w _ j ^ g \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , q _ { 1 } + q _ { 2 } } \\left ( p _ { 1 } + p _ { 2 } , l \\right ) = \\sum _ { m = 0 } ^ { p _ { 2 } + q _ { 2 } } S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , q _ { 2 } } \\left ( p _ { 2 } , m \\right ) S _ { a _ { 1 } , a _ { 1 } m + b _ { 1 } } ^ { a _ { 2 } , a _ { 2 } m + b _ { 2 } , q _ { 1 } } \\left ( p _ { 1 } , l - m \\right ) . \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} n _ 1 ^ p ( t ) = \\sqrt { { { \\rm { 2 } } \\mathord { \\left / { \\vphantom { { \\rm { 2 } } T } } \\right . \\kern - \\nulldelimiterspace } T } } w _ 1 ^ i \\left ( t \\right ) \\cos \\left ( { 2 \\pi f ' t + { { \\varphi ' } _ n } } \\right ) - \\sqrt { { { \\rm { 2 } } \\mathord { \\left / { \\vphantom { { \\rm { 2 } } T } } \\right . \\kern - \\nulldelimiterspace } T } } w _ 1 ^ q \\left ( t \\right ) \\sin \\left ( { 2 \\pi f ' t + { { \\varphi ' } _ n } } \\right ) \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ C _ 2 ^ q \\bigr ] = \\frac { ( 2 ^ { 2 q + 2 } - 1 ) } { q + 1 } \\bigl ( \\frac { 2 } { \\pi } \\bigr ) ^ { q + 1 } \\xi ( 2 q + 2 ) , \\ ; q \\in \\mathbb { C } , \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} P \\Big ( \\frac { 2 } { k + 1 + 2 / \\theta _ 0 } \\Big ) & \\geq \\frac { 4 } { ( k + 1 + 2 / \\theta _ 0 ) ^ 2 } + \\Big ( \\frac { 2 } { k + 1 + 2 / \\theta _ 0 } - 1 \\Big ) \\frac { 4 } { ( k + 2 / \\theta _ 0 ) ^ 2 } \\\\ & = \\frac { 4 } { ( k + 1 + 2 / \\theta _ 0 ) ^ 2 ( k + 2 / \\theta _ 0 ) ^ 2 } \\geq 0 . \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} \\nabla \\left ( c ( x + \\beta ) { \\bf u } ^ { \\tt M } \\right ) = \\left ( x ( c - 1 ) + \\beta c \\right ) { \\bf u } ^ { \\tt M } . \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} l _ ! \\circ l ^ \\ast ( \\theta _ { \\tilde Z } ) = \\theta _ { \\tilde Z } \\cdot u \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} \\exp \\left ( \\int \\limits _ 0 ^ \\infty \\frac { d x } { x } \\frac { e ^ { - b x } ( 1 - e ^ { - c x } ) ( 1 - e ^ { - d x } ) } { ( 1 - e ^ { - x } ) ( 1 - e ^ { - x \\tau } ) } ( e ^ { x q } - 1 ) \\right ) = & \\frac { G ( b \\ , | \\ , \\tau ) } { G ( b - q \\ , | \\ , \\tau ) } \\frac { G ( b - q + c \\ , | \\ , \\tau ) } { G ( b + c \\ , | \\ , \\tau ) } \\times \\\\ & \\times \\frac { G ( b - q + d \\ , | \\ , \\tau ) } { G ( b + d \\ , | \\ , \\tau ) } \\frac { G ( b + c + d \\ , | \\ , \\tau ) } { G ( b - q + c + d \\ , | \\ , \\tau ) } . \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} s _ { l + x } \\Pi ^ + _ { k - 1 } = \\Pi ^ + _ { k - 1 } s _ { \\sigma _ { h - 1 , k } + x } , \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} { \\nabla } _ { \\mathfrak { U } ^ 1 } ^ p ( s _ { U ^ 1 } ) = \\theta _ p \\theta _ { p - 1 } \\cdots \\theta _ 1 g { \\nabla } _ { \\mathfrak { U } ^ 2 } ^ p ( s _ { U ^ 2 } ) + \\sum _ { j = 0 } ^ { p - 1 } P _ j ^ p { \\nabla } _ { \\mathfrak { U } ^ 2 } ^ j ( s _ { U ^ 2 } ) . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} E _ { r } ^ { p , q } = Z _ { r } ^ { p , q } / ( Z _ { r - 1 } ^ { p - 1 , q + 1 } + \\delta Z _ { r - 1 } ^ { p + 1 - r , q + r - 2 } ) \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} \\lambda ^ { ( p ) } ( G _ 1 \\times G _ 2 ) = \\frac { r ^ { 1 - r / p } } { \\alpha ^ { 1 / p } } = ( r - 1 ) ! \\lambda ^ { ( p ) } ( G _ 1 ) \\lambda ^ { ( p ) } ( G _ 2 ) . \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} v \\ , t _ 0 ^ { - \\beta } = 1 . \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } - \\Delta u & = | u | ^ { 4 \\over N - 2 ' } u , & \\hbox { i n } \\Omega , \\\\ u & = 0 , & \\hbox { o n } \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} \\Delta ^ m \\langle x \\rangle _ n = \\dfrac { n ! } { ( n - m ) ! } \\langle x + m \\rangle _ { n - m } , m = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} \\mathcal { \\dot M } ( t ) \\leq 2 \\sum _ { \\mu , \\iota = 1 } ^ N \\int _ { \\R ^ d \\times \\R ^ d } \\Delta _ x \\psi ( x , y ) \\widetilde K ( t , x , y ) \\ , d x d y & \\\\ - \\frac { 4 p } { p + 1 } \\sum _ { \\substack { \\mu , \\nu , \\iota = 1 } } ^ N \\gamma _ { \\mu \\nu } \\int _ { \\R ^ d \\times \\R ^ d } | u _ \\mu ( x ) | ^ { p + 1 } | u _ \\nu ( x ) | ^ { p + 1 } m _ { u _ \\iota } ( y ) \\Delta _ x \\psi ( x , y ) \\ , d x d y , & \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} r ^ { i j } r ^ { k l } f _ { i k } ^ { \\ ; \\ ; \\ ; m } + r ^ { m i } r ^ { k l } f _ { i k } ^ { \\ ; \\ ; \\ ; j } + r ^ { m i } r ^ { j k } f _ { i k } ^ { \\ ; \\ ; \\ ; l } = 0 . \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} f ^ L ( \\tilde { \\vec x } ) & = \\int _ 0 ^ { \\| \\tilde { \\vec x } \\| _ \\infty } f ( V _ t ^ + ( \\tilde { \\vec x } ) , V _ t ^ - ( \\tilde { \\vec x } ) ) d t \\\\ \\ & = \\int _ 0 ^ \\alpha f ( V ^ + _ 0 , V ^ - _ 0 ) d t + \\int _ \\alpha ^ { \\| x \\| _ \\infty + \\alpha } f ( V _ t ^ + ( \\tilde { \\vec x } ) , V _ t ^ - ( \\tilde { \\vec x } ) ) d t \\\\ & = \\alpha f ( V ^ + _ 0 , V ^ - _ 0 ) + \\int _ 0 ^ { \\| x \\| _ \\infty } f ( V _ t ^ + ( { \\vec x } ) , V _ t ^ - ( { \\vec x } ) ) d t \\\\ & = \\alpha f ( V ^ + _ 0 , V ^ - _ 0 ) + f ^ L ( \\vec x ) . \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} \\P _ { s , x } ( X _ { s + u _ { 0 } + \\gamma } \\in \\cdot | \\tau _ { h } > s + u _ { 0 } + \\gamma ) & = \\phi _ { s + u _ 0 + \\gamma , s } ( \\delta _ x ) \\\\ & = \\phi _ { s + u _ 0 + \\gamma , s + g ( s ) } \\circ \\phi _ { s + g ( s ) , s } ( \\delta _ x ) \\\\ & = \\P _ { s + g ( s ) , \\phi _ { s + g ( s ) , s } ( \\delta _ x ) } ( X _ { s + u _ 0 + \\gamma } \\in \\cdot | \\tau _ h > s + u _ 0 + \\gamma ) . \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} p ^ * \\left ( \\frac { \\prod \\limits _ { i = 1 } ^ k L _ { i , b - k + i } } { \\prod \\limits _ { i = 1 } ^ { k - 1 } L _ { i , b - k + i + 1 } } \\right ) = \\frac { \\Delta _ { \\{ k - a , \\dots , k - 1 \\} } } { \\Delta _ { \\{ k - a + 1 , \\dots , k \\} } } \\prod _ { i = k - a } ^ { k - 1 } \\lambda _ i = p ^ * ( M _ k ) , \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} U ^ { \\ast } ( \\varphi , \\theta ) = \\sum _ { i = 1 } ^ { 3 } \\frac { 1 } { \\sqrt { ( 1 - \\sin \\varphi \\cos ( \\theta - \\theta _ { i } ) } } \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { I ^ { c } _ { k + l } ( n ) } } & \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k + l } ; n ) \\\\ & = \\sum \\limits _ { \\substack { I ^ { c } _ { k + l } ( n ) } } \\left ( \\frac { \\tilde { b } ( i _ { 1 } , i _ { 2 } , . . . , i _ { k + l } ; n ) } { b ( i _ { 1 } ; n ) b ( i _ { 2 } ; n ) \\cdots b ( i _ { k + l } ; n ) } \\cdot b ( i _ { 1 } ; n ) b ( i _ { 2 } ; n ) \\cdots b ( i _ { k + l } ; n ) \\right ) , \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} \\mathcal { C } _ 1 : = \\left \\{ w _ 1 \\ ; \\Big | \\ ; \\Re ( w _ 1 ) = - 2 \\delta - \\frac { c } { \\log ^ { 3 / 4 } \\big ( 2 + | \\Im ( w _ 1 ) | \\big ) } \\right \\} , \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} \\mathrm { d i m } ( H _ { 1 , 2 } ) = 2 g - 2 . \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} ( i _ { \\ell + 1 } , j _ { \\ell + 1 } ) = ( i _ { * } , j _ { * } ) \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} u ^ { 1 - \\alpha } ( t ) = u ^ { 1 - \\alpha } ( 0 ) + ( 1 - \\alpha ) \\widetilde { q } t . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} B _ 2 = T _ 1 - T _ 1 ( A _ 2 + T _ 1 ) ^ { - 1 } T _ 1 . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} \\sum _ { e \\ , \\cup f : \\ , v \\in e \\ , \\cup f } B ( v , e \\cup f ) & = \\sum _ { e \\ , \\cup f : \\ , v \\in e \\ , \\cup f } B _ 1 ( v , e ) w _ 2 ( f ) \\\\ & = \\sum _ { e \\in E ( G _ 1 ) : \\ , v \\in e } \\sum _ { f \\in E ( G _ 2 ) } B _ 1 ( v , e ) w _ 2 ( f ) \\\\ & = \\Bigg ( \\sum _ { e \\in E ( G _ 1 ) : \\ , v \\in e } B _ 1 ( v , e ) \\Bigg ) \\Bigg ( \\sum _ { f \\in E ( G _ 2 ) } w _ 2 ( f ) \\Bigg ) \\\\ & = 1 . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} H _ N ^ { \\rm G P } = \\sum _ { i = 1 } ^ N ( - \\Delta _ { z _ i } + U _ i ( z _ i ) ) + \\sum _ { i < j } ^ N V _ { i j } ( z _ i - z _ j ) . \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} x ^ B ( t ) = \\sigma _ { 0 } ^ B t + \\frac { 1 } { 2 } \\beta t ^ 2 . \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} u \\in L ^ q ( 0 , T ; L ^ r ( \\R ^ d ) ) , \\ \\ \\frac { 2 } { q } + \\frac { d } { r } = 1 , \\ \\ 3 \\leq d \\leq r \\leq \\infty . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} ( \\mathcal { S } _ N f ) ( q \\ , | \\ , b _ 0 + x ) = ( \\mathcal { S } _ N f ) ( q + x \\ , | \\ , b ) , \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} u ( \\varphi U ) = p u ( U ) - u ( U ) v ( E ) , \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} r _ t = ( - 1 ) ^ { t + 1 } ( f _ 0 - f _ 1 + \\cdots ( - 1 ) ^ { t - 1 } f _ { t - 1 } ) + ( - 1 ) ^ t = \\sum _ { j = 0 } ^ { t - 1 } ( - 1 ) ^ { t + j + 1 } f _ j + ( - 1 ) ^ t . \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} b _ k = \\prod _ { l = 0 } ^ { k - 1 } \\frac { 1 } { ( 1 - \\theta _ l ) ( 1 - \\sigma ^ K _ { l + 1 } ) } b _ 0 = \\frac { \\theta _ { - 1 } ^ 2 } { \\theta _ { k - 1 } ^ 2 } \\prod _ { l = 1 } ^ { k } \\frac { 1 } { 1 - \\sigma ^ K _ { l } } b _ 0 , \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} T : = \\{ i \\in [ r ] \\mid n _ i \\} \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\Delta _ 2 ' ( h ( \\gamma ) ) & = \\sum _ { a \\geqslant 1 } \\sum _ { j \\leqslant r + 1 } ( \\gamma ^ a ) _ { ( j ) } \\otimes ( \\gamma ^ a ) ^ { ( j ) } \\\\ & = \\sum _ { a \\geqslant 1 } \\sum _ { j \\leqslant i _ a + 1 } ( \\gamma ^ a ) _ { ( j ) } \\otimes ( \\gamma ^ a ) ^ { ( j ) } + \\sum _ { a \\geqslant 1 } \\sum _ { i _ a < j - 1 \\leqslant r } ( \\gamma ^ a ) _ { ( j ) } \\otimes ( \\gamma ^ a ) ^ { ( j ) } \\\\ \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} S _ { j k } & = \\sum _ { x = 0 } ^ N \\frac { \\theta ( a q ^ { 2 x } ) } { \\theta ( a ) } \\frac { ( a , b , c , d , e , f , g , q ^ { - N } ) _ { x } } { ( q , a q / b , a q / c , a q / d , a q / e , a q / f , a q / g , a q ^ { N + 1 } ) _ { x } } \\ , q ^ { x ( 2 n - 1 ) } \\\\ & \\quad \\times \\frac { ( b q ^ x , b q ^ { - x } / a ) _ { j - 1 } ( c q ^ x , c q ^ { - x } / a ) _ { n - j } ( e q ^ x , e q ^ { - x } / a ) _ { k - 1 } ( f q ^ x , f q ^ { - x } / a ) _ { n - k } } { ( b , b / a ) _ { j - 1 } ( c , c / a ) _ { n - j } ( e , e / a ) _ { k - 1 } ( f , f / a ) _ { n - k } } . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} \\begin{cases} ( 1 ) & \\mathfrak { m } _ \\lambda ^ { ( k ) } \\neq 0 , 0 \\leq k \\leq m + n \\\\ ( 2 ) & E _ { i , i + 1 } ^ { ( M ) } \\ , \\mathfrak { m } _ \\lambda ^ { ( k ) } = 0 , 1 \\leq i \\leq m + n - 1 , \\ ; i \\neq m , \\\\ ( 3 ) & E _ { - \\beta _ i } \\ , \\mathfrak { m } _ \\lambda ^ { ( k ) } = 0 = E _ { \\beta _ j } \\ , \\mathfrak { m } _ \\lambda ^ { ( k ) } , 1 \\leq i \\leq k < j \\leq m n . \\end{cases} \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} E = \\{ f \\in \\ell _ 2 ( \\N ^ N ) \\ : \\ \\exists \\ \\dot { f } : \\N \\to \\C , \\ f ( m ) = \\dot { f } ( | m | ) \\} , \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} T ( \\Gamma ; x , y ) : = \\sum _ { A \\subseteq E } ( x - 1 ) ^ { r ( \\Gamma ) - r ( \\Gamma \\backslash A ^ { c } ) } ( y - 1 ) ^ { n ( \\Gamma \\backslash A ^ { c } ) } , \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} \\Sigma _ x ^ { X _ 1 ( N ) } = \\bigcup _ { i = 1 } ^ m T ( x _ i ) . \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} S \\bigg ( \\sum _ { i = 1 } ^ m x _ i \\beta _ i , \\sum _ { j = 1 } ^ m y _ j \\beta _ j \\bigg ) = \\sum _ { i , j = 1 } ^ m B _ { i j } x _ i y _ j , \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} \\Phi _ { k } ^ { ( N ) } ( f ) - \\Phi _ { k } ^ { ( N ) } [ \\mathcal { A } ^ { ( N + 1 ) } ] \\cdot \\mathbb { E } [ f \\mid \\mathcal { A } ^ { ( N + 1 ) } ] = \\Phi _ { k } ^ { ( N + 1 ) } ( f ) . \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\mathcal W _ { \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) \\otimes \\underline { \\chi } , B } ( g _ { \\infty } ) = \\mathcal W _ { \\theta ( \\mathbf h , \\phi _ { \\mathbf h } ) , B } ( g _ { \\infty } ) . \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} - \\int _ M \\int _ 0 ^ T u \\ , \\varphi _ t \\ , d t \\ , d \\mathcal { V } = \\int _ M u _ 0 ( x ) \\ , \\varphi ( x , 0 ) \\ , d \\mathcal { V } ( x ) + \\int _ M \\int _ 0 ^ T \\left ( u ^ m \\ , \\Delta \\varphi + u ^ p \\ , \\varphi \\right ) d t \\ , d \\mathcal { V } \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| u ( \\cdot , t + t _ 0 ; t _ 0 , u _ 0 , v _ 0 ) - u ^ { * * } ( \\cdot , t + t _ 0 ) \\| _ { L ^ 2 ( \\Omega ) } = \\lim _ { t \\to \\infty } \\| \\psi ( t + t _ 0 ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } = 0 , \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\max _ { \\lambda } & ~ U _ { B S } = \\lambda \\theta P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } - \\theta ( A P _ { B S } ^ { 2 } + B P _ { B S } ) , \\\\ s . t . & ~ \\lambda \\geq 0 . \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} \\hat { \\varphi } ( s ) = \\mathbb { E } \\left [ \\left . \\Gamma _ { 1 } ( T ) \\Gamma _ { 2 } ( T ) \\varepsilon _ { 8 } ( T ) + \\int _ { s } ^ { T } \\Gamma _ { 1 } ( r ) \\Gamma _ { 2 } ( r ) I ( r ) d r \\right \\vert \\mathcal { F } _ { s } ^ { t } \\right ] . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} g ( \\phi X , Y ) = - g ( X , \\phi Y ) . \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} \\mathop { \\inf } \\limits _ { x \\in X } \\left \\{ { F \\left ( { x , { 0 _ { Y } } } \\right ) - \\left \\langle { { x ^ { \\ast } } , x } \\right \\rangle } \\right \\} = \\mathop { \\sup } \\limits _ { { y ^ { \\ast } } \\in { Y ^ { \\ast } } } - { F ^ { \\ast } } \\left ( { { x ^ { \\ast } } , { y ^ { \\ast } } } \\right ) , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} ( 2 p - 1 ) x + ( p - 1 ) y + z = m , \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ k \\frac 1 { | \\lambda \\xi _ j - \\lambda \\xi _ 1 | ^ \\tau } \\le \\frac { C k } { \\lambda ^ \\tau } \\le C . \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} G ( \\hat { \\vec z } ) = \\frac { r - ( \\hat { \\vec z } , | \\vec v | ) } { \\| \\hat { \\vec z } \\| _ p } \\le \\frac { r - ( \\vec z ^ * , | \\vec v | ) } { \\| \\hat { \\vec z } \\| _ p } < \\frac { r - ( \\vec z ^ * , | \\vec v | ) } { \\| \\vec z ^ * \\| _ p } = G ( \\vec z ^ * ) , \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\iota _ { 1 \\leq 3 } ^ { ( 3 ) } [ 6 ] = ( S ^ { ( 3 ) } _ { 0 , 3 } ) ^ { 3 } \\iota _ { 1 \\leq 3 } ^ { ( 3 ) } ( S ^ { ( 3 ) } _ { 0 , 1 } ) ^ { - 3 } = \\Sigma ^ 3 \\iota _ { 1 \\leq 3 } ^ { ( 3 ) } \\Sigma ^ { - 2 } = \\Sigma \\iota _ { 1 \\leq 3 } ^ { ( 3 ) } . \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} \\Sigma _ B ( k , s ) = 2 \\pi ( - 1 ) ^ k \\zeta ( 2 s ) \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { s } } \\sum _ { q = 1 } ^ { \\infty } \\frac { S ( 1 , n ^ 2 ; q ) } { q } J _ { 2 k - 1 } \\left ( \\frac { 4 \\pi n } { q } \\right ) . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} \\mathrm { E M } _ p ( s , z , z ' , u , v , w ) = O \\Bigl ( \\frac { p ^ { 2 \\theta } } { p ^ { 2 ( 1 - \\theta ) } } \\Bigr ) = O \\Bigl ( \\frac { 1 } { p ^ { 2 - 4 \\theta } } \\Bigr ) . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } u & = & x _ 2 x _ 1 x _ 2 ^ { - 1 } x _ 1 ^ { - 1 } \\\\ v & = & x _ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} g _ d ( q ) = \\sum _ { \\substack { 0 \\leq j < n \\\\ \\gcd ( j , n ) = d } } q ^ j , \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} h ( z ) = \\sum _ { n \\geq 1 } c ( n ) q ^ n . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\int _ { \\R } \\int _ { \\R } | \\sum _ { \\mu = 1 } ^ N m _ { u _ \\mu } ( t , x ) | ^ { 2 } \\ , d t \\ , d x + \\int _ { \\R } \\int _ { \\R } | \\sum _ { \\mu = 1 } ^ N \\nabla _ x m _ { u _ \\mu } ( t , x ) | ^ { 2 } \\ , d t \\ , d x & \\\\ + \\sum _ { \\mu = 1 } ^ N \\gamma _ { \\mu \\mu } \\int _ { \\R } \\int _ { \\R } | u _ \\mu ( t , x ) | ^ { 2 p + 4 } \\ , d t \\ , d x \\leq C \\sum _ { \\mu = 1 } ^ N \\| u _ { \\mu , 0 } \\| ^ 4 _ { H ^ 2 _ x } ; & \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} w _ 0 ^ B = \\big \\{ \\rho _ + \\rho _ - \\big ( ( u _ + - u _ - ) ^ 2 - ( \\frac { 1 } { \\rho _ + } - \\frac { 1 } { \\rho _ - } ) ( \\frac { B } { \\rho _ + ^ \\alpha } - \\frac { B } { \\rho _ - ^ \\alpha } ) \\big ) \\big \\} ^ \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\frac { 1 } { l + 1 } 2 ^ { l H ( \\alpha ) } \\le \\binom { l } { \\alpha l } \\le 2 ^ { l H ( \\alpha ) } , \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} \\sigma ( \\beta \\alpha ) _ 4 = \\beta _ 4 \\alpha _ 4 = \\sigma ( \\beta ) _ 4 \\tau ( \\alpha ) _ 4 = ( \\sigma ( \\beta ) \\tau ( \\alpha ) ) _ 4 . \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} f ^ I ( A ) = \\langle \\pi _ I ( A ) \\xi ^ I , \\eta ^ I \\rangle , \\quad \\forall A \\in \\mathcal { A } . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} \\norm { x _ { k + 1 } - x _ * } _ v ^ 2 = ( 1 - \\theta _ k ) \\norm { x _ k - x _ * } _ v ^ 2 + \\frac { \\theta _ k } { \\theta _ 0 } \\norm { z _ { k + 1 } - x _ * } _ v ^ 2 - \\frac { \\theta _ k } { \\theta _ 0 } ( 1 - \\theta _ 0 ) \\norm { z _ k - x _ * } _ v ^ 2 \\\\ + \\frac { ( 1 - \\theta _ k ) \\theta _ k / \\theta _ 0 ( 1 - \\theta _ 0 ) } { 1 - \\theta _ k / \\theta _ 0 } \\norm { x _ k - z _ k } _ v ^ 2 - \\frac { \\theta _ k / \\theta _ 0 } { 1 - \\theta _ k / \\theta _ 0 } \\norm { x _ { k + 1 } - z _ { k + 1 } } _ v ^ 2 \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { D _ \\varepsilon } \\theta _ k ^ 2 > C _ { \\max } ( \\alpha , \\beta ) \\varepsilon ^ 2 \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\varphi _ 0 ( F _ 0 , t ) = \\frac { \\sigma ^ { \\frac { p - 2 + m } { p - 1 } } } { \\gamma ^ { \\frac { m - 1 } { p - 1 } } } \\left [ \\left ( \\frac { m - 1 } { p - 2 + m } \\right ) ^ { \\frac { m - 1 } { p - 1 } } - \\left ( \\frac { m - 1 } { p - 2 + m } \\right ) ^ { \\frac { p - 2 + m } { p - 1 } } \\right ] - \\delta \\ , . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\langle P ( x ) , Q ( y ) \\rangle = \\sum _ { \\substack { n _ 1 = d ( x _ 1 , y _ 1 ) \\\\ \\vdots \\\\ n _ N = d ( x _ N , y _ N ) } } ^ \\infty \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( ( n _ 1 , . . . , n _ N ) + \\chi ^ I ) \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\sigma ^ { \\frac { l } { 2 } } & = \\lim _ { m \\to \\infty } \\left ( 1 - 2 \\frac { c _ 1 } { m } \\right ) ^ { \\frac { 1 } { 2 } m ( 1 + \\epsilon ) } = e ^ { - c _ 1 ( 1 + \\epsilon ) } \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( a ; N ) } ^ \\ast ( \\tau ) = \\frac { ( - 2 \\pi { \\rm i } ) ^ k } { ( k - 1 ) ! } \\sum _ { n \\ge 1 } n ^ { k - 1 } \\ ( \\frac { q ^ { n a } + ( - 1 ) ^ k q ^ { n ( N - a ) } } { 1 - q ^ { n N } } \\ ) . \\end{gather*}"}
-{"id": "2722.png", "formula": "\\begin{align*} L ( p _ 1 , p _ 2 ) = \\{ ( k , l , m ) \\in ( 2 \\mathbb { N } _ 0 ) ^ 3 : k \\leq p _ 1 , l + m \\leq p _ 2 , p _ 1 + p _ 2 - ( k + l + m ) \\in 2 \\mathbb { N } _ 0 \\} . \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\beta = \\frac { N - 2 } 2 - 2 \\tau - \\frac 1 2 ( N - 2 - 2 \\tau - \\frac { N + 2 } 2 ) = \\frac { N + 2 } 4 - \\tau , \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} u \\colon \\R \\to \\R & , \\ u ( z ) : = \\sum _ { k = 1 } ^ \\infty \\frac { \\cos ( 2 ^ k z ) } { 2 ^ k \\sqrt k } \\\\ \\intertext { a n d } U \\colon \\R \\to \\R ^ 2 & , \\ U ( z ) : = \\binom z { u ( z ) } . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} - a \\cdot \\nabla ^ 2 + b \\cdot \\nabla \\equiv - \\sum _ { i , j = 1 } ^ d a _ { i j } ( x ) \\nabla _ i \\nabla _ j + \\sum _ { j = 1 } ^ d b _ j ( x ) \\nabla _ j . \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} T ^ { { A } _ { ( U ) } } _ { i } = \\sum _ { j = 1 } ^ M ( ( T ^ { t } _ { i j } + T ^ { r } _ { i j } ) ( x ^ a _ { i j } + x ^ c _ { i j } ) + T ^ { a } _ { i j } x ^ a _ { i j } ) , \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\mathfrak D _ 1 \\mathfrak D _ 1 ^ \\dag + \\cdots + \\mathfrak D _ n \\mathfrak D _ n ^ \\dag = 0 \\ \\ \\mathfrak D _ 1 = \\cdots = \\mathfrak D _ n = 0 . \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} \\mathcal L _ c ( \\psi ) = - \\frac { d } { d \\ , x } \\left [ ( 1 - x ^ 2 ) \\frac { d \\psi } { d \\ , x } \\right ] + c ^ 2 x ^ 2 \\psi . \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} | \\nabla _ x p _ D ( s , x , y ) | & \\le \\frac { c _ { 1 2 } } { r } p _ D ( s , x , y ) \\le \\frac { c _ { 1 3 } } { r } \\left ( 1 \\land { \\frac { V ( d _ D ( x ) ) } { s ^ { 1 / 2 } } } \\right ) \\left ( 1 \\land { \\frac { V ( d _ D ( y ) ) } { s ^ { 1 / 2 } } } \\right ) p ( s , | x - y | / 4 ) \\\\ & \\le \\frac { c _ { 1 4 } } { r } \\frac { V ( r ) } { \\sqrt s } p ( s , | x - y | / 4 ) . \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} m ( s ) & = p ( s ) h ( s ) ; \\\\ n ( s ) & = \\left ( 1 - p ( s ) \\sigma _ { z } ( s ) \\right ) ^ { - 1 } [ b _ { z } ( s ) p ( s ) ^ { 2 } + p ( s ) g _ { z } ( s ) + q ( s ) ] h ( s ) . \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} A \\cdot E = \\frac { 1 } { q } , B \\cdot E = \\frac { 1 } { p } , ( A \\cdot A ) _ { S _ { P , \\omega } } = ( A \\cdot A ) _ S - \\frac { p } { q } , ( B \\cdot B ) _ { S _ { P , \\omega } } = ( B \\cdot B ) _ S - \\frac { q } { p } . \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} \\gamma = 1 . \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} \\mathbb { E } [ T _ n ] = \\sum _ { k = 1 } ^ { n } \\binom { n } { k } ^ 2 \\frac { 1 } { a _ n ^ k } \\sim \\binom { n } { k ^ * } ^ 2 \\frac { 1 } { a _ n ^ { k ^ * } } \\sqrt { \\frac { \\pi n } { A _ n } } . \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} M _ 2 & = - \\frac { P _ 3 ( 1 ) } { L } \\sum _ { w \\mid q } \\varphi ( w ) \\mu ( q / w ) \\mathop { \\sum \\sum } _ { \\substack { b \\leq y _ 2 \\\\ b \\equiv \\pm n ( w ) \\\\ ( b n , q ) = 1 } } \\frac { \\Lambda ( b ) P _ 2 [ b ] } { ( b n ) ^ { \\frac { 1 } { 2 } } } F ( n ) . \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} \\begin{dcases} \\frac { w _ 1 ^ p } { 2 w _ 1 ^ 2 ( 2 w _ 1 + 2 w _ 2 ) } = \\alpha \\\\ \\frac { w _ 2 ^ p } { w _ 2 ^ 2 ( 2 w _ 1 + 2 w _ 2 ) } = \\alpha \\\\ 2 w _ 1 + 2 w _ 2 = 1 . \\end{dcases} \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\begin{array} { c c c c } & \\quad \\ : \\ , q _ 1 & & \\quad \\ ; \\ ; \\ : q _ s \\\\ C _ k = & \\multicolumn { 3 } { c } { t \\begin{bmatrix} C _ k ( f _ 1 ) | & \\cdots & | C _ k ( f _ s ) \\end{bmatrix} , } \\end{array} \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} x _ v ^ * = \\left ( \\frac { w ( e ) } { r B ( v , e ) } \\right ) ^ { 1 / p } , \\ v \\in e . \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} h \\Lambda = \\epsilon ( h ) \\Lambda . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} \\left \\| f ( x ) - \\sum _ { n = 0 } ^ { N - 1 } a _ n x ^ n \\right \\| \\leq C _ N ( S ' ) | x | ^ N , S ' . \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\bigg | \\frac { \\underline { \\vec z _ { k - 2 } } } { | \\underline { \\vec z _ { k - 2 } } | } - \\frac { \\underline { \\vec z _ { k } } } { | \\underline { \\vec z _ { k } } | } \\bigg | ^ 2 = 2 \\bigg ( 1 - \\frac { \\langle \\underline { \\vec z _ { k - 2 } } , \\underline { \\vec z _ { k } } \\rangle } { | \\underline { \\vec z _ { k - 2 } } | \\cdot | \\underline { \\vec z _ { k } } | } \\bigg ) , \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} \\ , \\ , \\ , a \\leq b \\ , \\ , \\ , \\ , \\ , a ^ { \\prime } \\wedge b = 0 , \\ , \\ , \\ , \\ , a = b . \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} \\langle \\sigma \\frac { \\partial u } { \\partial \\nu } , \\psi \\rangle _ { H ^ { - 1 / 2 } ( \\partial \\Omega ) , H ^ { 1 / 2 } ( \\partial \\Omega ) } = \\int _ { \\Omega } \\sigma \\nabla u \\cdot \\nabla \\psi \\ , d x , \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{gather*} a ( \\tau ) : = \\sum _ { ( n , m ) \\in \\Z ^ 2 } q ^ { n ^ 2 + n m + m ^ 2 } = \\frac { 3 \\eta ( 3 \\tau ) ^ 3 + \\eta ( \\tau / 3 ) ^ 3 } { \\eta ( \\tau ) } , \\\\ b ( \\tau ) : = \\sum _ { ( n , m ) \\in \\Z ^ 2 } \\zeta _ 3 ^ { m - n } q ^ { n ^ 2 + n m + m ^ 2 } = \\frac { \\eta ( \\tau ) ^ 3 } { \\eta ( 3 \\tau ) } , \\\\ c ( \\tau ) : = \\sum _ { ( n , m ) \\in \\Z ^ 2 } q ^ { \\ ( n + 1 / 3 \\ ) ^ 2 + \\ ( n + 1 / 3 \\ ) \\ ( m + 1 / 3 \\ ) + \\ ( m + 1 / 3 \\ ) ^ 2 } = 3 \\frac { \\eta ( 3 \\tau ) ^ 3 } { \\eta ( \\tau ) } , \\end{gather*}"}
-{"id": "3110.png", "formula": "\\begin{align*} n ( n + 1 ) = \\chi _ n ( 0 ) \\leq \\chi _ n ( c ) \\leq n ( n + 1 ) + c ^ 2 , n \\geq 0 . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\begin{pmatrix} j _ { l _ 1 , k } \\\\ P & j _ { l _ 2 , n - 2 k } \\\\ Q & R & j _ { l _ 1 , k } \\end{pmatrix} \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} T _ u P \\varphi = P T _ u \\varphi + A _ u \\varphi . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} - \\Delta u = u ^ q , \\textrm { o n } \\ \\mathbb { H } _ h ^ n \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\tilde { \\bar { R } } ( X , Y , Z , W ) & = \\tilde { R } ( X , Y , Z , W ) - g \\big ( \\tilde { h } ( X , W ) , \\tilde { h } ( Y , Z ) \\big ) \\\\ & + g \\big ( \\tilde { h } ( X , Z ) , \\tilde { h } ( Y , W ) \\big ) , \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} d _ { k } = a _ { k } - \\sum _ { l \\in \\mathbb { Z } , \\ , \\ , l \\geq k + 1 } d _ { l } . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} k ( M / A ) + k ( M \\backslash A ^ c ) = k ( M ) + f ( M \\backslash A ^ c ) , \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} \\sup _ { \\mathcal { F } } \\sup _ { \\mathcal { X } } \\left \\vert \\phi _ { ( N ) } f \\right \\vert = \\sup _ { \\mathcal { F } } \\max _ { 1 \\leqslant j \\leqslant m _ { N } } \\sup _ { \\mathcal { X } } \\left \\vert \\phi _ { ( j , N ) } f \\right \\vert \\leqslant M \\left ( 1 + \\frac { 1 } { p _ { N } } \\right ) \\leqslant \\frac { 2 M } { p _ { N } } , \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\rightarrow \\infty } \\alpha \\Psi ( x _ { \\alpha , \\varepsilon } , y _ { \\alpha , \\varepsilon } ) = 0 . \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} ^ { A B } _ { a } I ^ \\alpha ~ ^ { A B R } _ { a } D ^ \\alpha f ( t ) = f ( t ) , \\quad ~ ~ ~ ^ { A B } I _ b ^ \\alpha ~ ^ { A B R } D _ b ^ \\alpha f ( t ) = f ( t ) . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ { x ^ i } U ( x ^ i , \\xi ^ i ) ~ u _ { \\xi ^ i } = K ( x ^ i ) U ( x ^ i , \\xi ^ i ) ~ u _ { \\xi ^ i } \\end{array} \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\Phi \\Delta \\Phi ^ { - 1 } = \\bigoplus _ { N = 0 } ^ { \\infty } \\bigoplus _ { k = 1 } ^ { \\beta _ N } J _ N ( k ) , \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} & \\mu \\log \\bigg ( \\frac { ( t _ { 1 } - 1 ) \\theta + 1 } { ( t _ { 2 } - 1 ) \\theta + 1 } \\bigg ) \\\\ & ~ ~ ~ = \\mu ( 1 - \\theta ) \\bigg [ \\frac { t _ { 1 } - 1 } { ( t _ { 1 } - 1 ) \\theta + 1 } - \\frac { t _ { 2 } - 1 } { ( t _ { 2 } - 1 ) \\theta + 1 } \\bigg ] - t _ { 3 } . \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} \\sigma ( X ) = - n ( T ) - 7 n ( C _ 2 ) + n ( L ) . \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} \\left \\Vert \\mathbb { G } ^ { ( 2 m ) } - \\mathbb { G } _ { } ^ { ( \\infty ) } \\right \\Vert _ { \\mathcal { F } } & = \\left \\Vert ( S _ { 1 , } ^ { ( m - 1 ) } - S _ { 1 , } ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { A } ] + ( S _ { 2 , } ^ { ( m - 2 ) } - S _ { 2 , } ) ^ { t } \\cdot \\mathbb { G } [ \\mathcal { B } ] \\right \\Vert _ { \\mathcal { F } } \\\\ & \\leqslant c _ { 0 } \\max ( \\lambda _ { 1 } , \\lambda _ { 2 } ) ^ { m - 2 } Z , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} T _ d ( z ) & = \\sum _ { k = 0 } ^ { N - 1 } \\omega \\left ( \\frac { k b ^ { w _ d } z \\bmod N } { N } \\right ) q _ d ( k ) . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} \\frac { R } { Z ( R ) } = \\langle a + Z , b + Z : 4 ( a + Z ) = 4 ( b + Z ) = Z \\rangle \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} u ( 0 , x ) = w _ 0 ( x ) , u ' ( 0 , x ) = w _ 1 ( x ) . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} N _ { \\mathrm { d } m } = ( 2 m ) ! . \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} & \\dot C _ 1 = G ( C _ 1 ) - C _ 1 \\ , , \\\\ & \\dot C _ 2 = - 2 C _ 2 + 2 \\sigma \\ , . \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} | v x _ 0 - x _ 0 | & \\geqslant n \\ ; | { u } _ j x _ 0 - { u } _ { j + 1 } x _ 0 | - 4 8 n \\delta . \\\\ & = 5 \\nu \\ ; | { u } _ j x _ 0 - { u } _ { j + 1 } x _ 0 | - 2 4 0 \\nu \\delta > 3 \\nu \\ , | { u } _ j x _ 0 - { u } _ { j + 1 } x _ 0 | + A \\delta + 1 0 ^ 7 \\delta . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { p _ { 2 } - 1 } \\sum _ { j = 0 } ^ { p _ { 1 } } \\binom { p _ { 1 } } { j } m ^ { p _ { 1 } - j } S \\left ( j , l - m \\right ) S \\left ( p _ { 2 } , m \\right ) = \\sum _ { k = 1 } ^ { p _ { 2 } - 1 } \\left ( - 1 \\right ) ^ { p _ { 2 } + k + 1 } s \\left ( p _ { 2 } , k \\right ) S \\left ( p _ { 1 } + k , l \\right ) . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} \\frac { d ( \\mu _ { \\rho } ) _ y } { d \\mu _ { \\rho } } ( x H ) = \\frac { \\rho ( y x ) } { \\rho ( x ) } \\forall x , y \\in G . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} u _ { \\epsilon , M } & \\to u \\mbox { i n $ L ^ 2 _ { l o c } \\cap L ^ { 1 + m } _ { l o c } $ } \\\\ U ^ { \\frac { m - 1 } { 2 } } \\left | \\nabla u _ { \\epsilon , M } \\right | & \\to U ^ { \\frac { m - 1 } { 2 } } \\left | \\nabla u \\right | \\mbox { i n $ L ^ 2 _ { l o c } $ } \\\\ \\epsilon \\left | \\nabla u _ { \\epsilon , M } \\right | & \\to 0 \\mbox { i n $ L ^ 2 _ { l o c } $ } \\end{aligned} \\end{cases} \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{gather*} \\mathcal G _ { k , ( a ; N ) } ^ \\ast ( \\tau ) : = \\sum _ { i = 0 } ^ { N - 1 } \\mathcal G _ { k , ( a , i ; N ) } ^ \\ast ( \\tau ) = \\sum _ { m , n \\in \\Z } \\frac { 1 } { \\ ( ( N m + a ) \\tau + n \\ ) ^ k } \\end{gather*}"}
-{"id": "2576.png", "formula": "\\begin{align*} T _ { u ^ { \\star n } } h ~ = ~ h _ n + G \\varphi _ { n } \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta \\phi _ 0 ( x ) = 0 , \\ ; \\mbox { i n } \\ ; \\Omega , \\ ; \\ ; n = 2 , \\\\ \\phi _ 0 | _ { \\partial \\Omega } = 0 , \\\\ | x | \\rightarrow \\infty , \\phi _ 0 ( x ) \\rightarrow + \\infty , \\ ; \\ ; \\mbox { a n d } \\ ; \\phi _ 0 ( x ) \\ ; \\mbox { i n c r e a s e a t t h e r a t e o f } \\ ; \\ln ( | x | ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\begin{pmatrix} X \\\\ P & Y \\\\ Q & R & X \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} J _ \\nu ( r ) = \\sum _ { j = 0 } ^ \\infty \\frac { ( - 1 ) ^ j ( r / 2 ) ^ { \\nu + 2 j } } { j ! \\ , \\Gamma ( \\nu + j + 1 ) } . \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} ( { \\cal L _ \\gamma } ) ' ( 0 ) = g _ { \\dot \\gamma ( b ) } ( { \\dot \\gamma ( b ) } , U ( b ) ) - g _ { \\dot \\gamma ( a ) } ( { \\dot \\gamma ( a ) } , U ( a ) ) \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} I ^ F = ( ( x _ 1 , \\ldots , x _ t ) R _ P ) ^ F = ( x _ 1 , \\ldots , x _ t ) ^ F R _ P = ( x _ 1 , \\ldots , x _ t ) ^ * R _ P = I ^ * , \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\log \\eta _ { M , N } ( \\kappa \\ , q \\ , | \\ , \\kappa \\ , a , \\kappa \\ , b ) & = \\bigl ( \\mathcal { S } _ N L _ M \\bigr ) ( \\kappa \\ , q \\ , | \\ , \\kappa \\ , a , \\kappa \\ , b ) - \\bigl ( \\mathcal { S } _ N L _ M \\bigr ) ( 0 \\ , | \\ , \\kappa \\ , a , \\kappa \\ , b ) , \\\\ & = \\log \\eta _ { M , N } ( q \\ , | \\ , a , \\ , b ) . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} F ' = e ^ F ( 1 + x F ' ) ^ 2 . \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} q _ 1 ( t ) = 2 a _ { 1 , \\inf } ( t ) \\underbar r _ 1 + a _ { 2 , \\inf } ( t ) \\underbar r _ 2 + \\frac { \\chi _ 1 \\left ( k \\underbar r _ 1 + l \\underbar r _ 2 \\right ) } { 2 d _ 3 } , \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\mathcal { N } _ \\ell ( P ) & : = \\{ y ( \\lambda ) ^ T \\in \\mathbb { F } ( \\lambda ) ^ { 1 \\times m } \\mbox { s u c h t h a t } y ( \\lambda ) ^ T P ( \\lambda ) = 0 \\} , \\\\ \\mathcal { N } _ r ( P ) & : = \\{ x ( \\lambda ) \\in \\mathbb { F } ( \\lambda ) ^ { n \\times 1 } \\mbox { s u c h t h a t } P ( \\lambda ) x ( \\lambda ) = 0 \\} , \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} Y _ { 1 } & = 8 9 . 0 1 7 4 4 6 6 8 9 9 1 : \\rho _ { 0 } = 0 . 0 2 0 7 6 , \\rho _ { 1 } = 0 , v _ { 0 } = 1 3 7 4 1 . 7 9 8 \\\\ Y _ { 2 } & = 8 . 0 8 3 1 6 1 3 3 5 2 5 8 : \\rho _ { 0 } = 2 . 5 1 4 7 5 , \\rho _ { 1 } = 0 , v _ { 0 } = 1 0 . 3 0 1 8 4 \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} u _ { t } - \\nabla \\cdot \\left ( \\left | \\nabla u \\right | ^ { p - 2 } \\nabla u \\right ) = F \\left ( u \\right ) p \\ge 2 , \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} V ' _ { 2 2 } = V ' _ 2 V ' _ 2 - 2 V ' _ 1 V ' _ 3 + 3 V ' _ 4 . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\tilde { f } ( t , x ) = \\Bigg \\{ \\begin{array} { c l } f ( t , x ) , & t \\in T , \\\\ \\delta _ { B } ( x ) , & t = \\omega _ 0 . \\end{array} \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} A ^ \\circ = \\{ w \\in W : { \\uparrow } w \\subseteq A \\} ; \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} H _ { 2 , \\boldsymbol { \\alpha } } ( \\tau ) = - 2 i \\int _ { \\mathbb R ^ 2 } w _ 2 \\left ( \\mathcal G _ { \\alpha _ 1 } ( w _ 1 ) \\mathcal F _ { \\alpha _ 2 } ( w _ 2 ) + \\mathcal F _ { \\alpha _ 1 } ( w _ 1 ) \\mathcal G _ { \\alpha _ 2 } ( w _ 2 ) \\right ) e ^ { 2 \\pi i \\tau Q ( \\boldsymbol { w } ) } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} z = \\frac { 1 } { 2 } \\left ( \\max _ i z _ i + \\min _ i z _ i \\right ) . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} B _ n ( x ) = \\sum _ { j = 0 } ^ \\infty j ^ n \\dfrac { x ^ j } { j ! } e ^ { - x } = \\sum _ { j = 0 } ^ n S ( n , j ) x ^ j , \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} K _ u ( u _ k ^ K ) = \\sigma _ k \\Psi _ k ^ K u _ k ^ K . \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} I _ { l _ 1 , l _ 2 , k , n } & < c q ^ { \\frac { 1 } { 4 } ( n - 2 k - 2 l _ 2 ) ^ 2 + ( k - 2 l _ 1 ) ( n - 2 k - 2 l _ 2 ) + 2 l _ 1 ( n - 2 k - 2 l _ 2 ) + 2 l _ 2 ( k - 2 l _ 1 ) + 4 l _ 1 l _ 2 + 2 l _ 1 ^ 2 + 2 l _ 1 ( k - 2 l _ 1 ) + ( k - 2 l _ 1 ) ^ 2 } \\\\ & = c q ^ { \\frac { 1 } { 4 } n ^ 2 - l _ 2 ( n - 2 k - l _ 2 ) - 2 l _ 1 ( k - l _ 1 ) } . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} u = \\sum _ { k \\geq 1 } u _ k ^ K + u _ \\infty ^ K = \\sum _ { \\substack { k \\geq 1 \\\\ \\sigma _ k \\in \\Sigma _ u ^ K } } u _ k ^ K + u _ \\infty ^ K , \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\P _ { x + y } ( S ( n ) \\in { \\cal C } , \\ ; \\forall n \\geq 0 ) & = P _ x ( S ( n ) + y \\in { \\cal C } , \\ ; \\forall n \\geq 0 ) \\\\ & \\geq P _ x ( S ( n ) + N m \\in { \\cal C } , \\ ; \\forall n \\geq 0 ) > 0 \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} f _ { i j } ^ { \\ ; \\ ; \\ ; l } \\omega _ { l k } + f _ { i k } ^ { \\ ; \\ ; \\ ; l } \\omega _ { l j } + f _ { j k } ^ { \\ ; \\ ; \\ ; l } \\omega _ { l i } = 0 , \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} \\sum _ { n \\le z } \\tau _ \\nu ( n ) = O \\ ( z ( \\log z ) ^ { \\nu - 1 } \\ ) \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} & { \\rm m i n i m i z e } \\hat { m } _ { ( ( A _ r ) _ k , ( B _ r ) _ k , ( C _ r ) _ k ) } ( \\xi , \\eta , \\zeta ) \\\\ & { \\rm s u b j e c t \\ , t o } | | ( \\xi , \\eta , \\zeta ) | | _ { ( ( A _ r ) _ k , ( B _ r ) _ k , ( C _ r ) _ k ) } \\leq \\Delta _ k , \\\\ & { \\rm w h e r e } \\hat { m } _ k ( \\xi , \\eta , \\zeta ) : = \\hat { m } _ { ( ( A _ r ) _ k , ( B _ r ) _ k , ( C _ r ) _ k ) } ( \\xi , \\eta , \\zeta ) , \\\\ & \\quad \\quad ( \\xi , \\eta , \\zeta ) \\in T _ { ( ( A _ r ) _ k , ( B _ r ) _ k , ( C _ r ) _ k ) } M . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} f ( x , y ) : = \\left \\{ \\begin{aligned} & \\frac { ( \\sqrt { x } - \\sqrt { y } ) ^ 2 } { ( x - y ) ^ 2 } ~ ~ ~ ~ x \\ne y \\\\ & \\frac { 1 } { 4 \\ , x } ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{aligned} \\right . , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ g ( x , y ) : = \\left \\{ \\begin{aligned} & \\frac { \\log ( x ) - \\log ( y ) } { x - y } ~ ~ ~ ~ x \\ne y \\\\ & \\frac { 1 } { x } ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{aligned} \\right . \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} M _ { } & = ( 1 + o ( 1 ) ) P _ 1 ( 1 ) P _ 3 ( 1 ) \\varphi ^ + ( q ) , \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} F _ n = \\sum \\limits _ { r \\geq 0 } \\frac { 1 } { r ! } \\sum \\limits _ { \\sum \\limits _ { i = 0 } ^ r k _ i = n } P _ { k _ { 0 } } F _ { k _ { 1 } } \\cdots F _ { k _ { r } } { n \\choose k _ 0 , k _ 1 , \\hdots , k _ r } ^ 2 ( k _ 0 ) ^ r . \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\lambda ^ { ( p + 1 ) } ( H ) - \\lambda ^ { ( p + 1 ) } ( H ' ) & \\geq r \\sum _ { e \\in E ( H ) } \\prod _ { v \\in e } y _ v - r \\sum _ { e \\in E ( H ' ) } \\prod _ { v \\in e } x _ v \\\\ & = r ( y _ { v _ 0 } - x _ { u _ 0 } ) \\prod _ { v \\in e _ 0 \\backslash \\{ v _ 0 \\} } x _ v + r ( y _ { v _ 0 } - x _ { v _ 0 } ) \\sum _ { \\substack { e \\in E ( H ) \\backslash e _ 0 , \\\\ v _ 0 \\in e } } \\prod _ { v \\in e \\backslash \\{ v _ 0 \\} } x _ v \\\\ & > 0 , \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} \\frac { C ( x ) } { ( x C ( x ) ) ' } = \\frac { 1 - 2 x C ( x ) } { 1 - x C ( x ) } . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} 2 d ( \\lambda C - D ) ^ { 2 } + 2 ( 1 - d D ) ( \\lambda C - D ) - a d C = 0 . \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} & \\begin{pmatrix} 1 & 1 & 0 \\\\ 1 & 2 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} T _ \\theta = T '' _ \\theta - d L \\times \\{ e \\} \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} \\tilde { J } N = \\xi + v ( E ) N , \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\mu ^ \\Lambda ( d \\eta ) = R ^ \\Lambda _ \\mu ( \\eta ) \\lambda ( d \\eta ) , \\eta \\in \\Gamma _ \\Lambda . \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} G ^ { ( 1 ) , n } _ p ( t ) = G ^ { ( 2 ) , n } _ p ( t ) \\overset { \\mathbb { P } } { \\longrightarrow } G _ p ( t ) , ~ ~ ~ t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} | a _ { n _ 0 } - a _ { n _ 0 + k } - k a | \\leq \\sum _ { n = 0 } ^ { k - 1 } | a _ { n _ 0 + n } - a _ { n _ 0 + n + 1 } - a | \\leq k \\frac { | a | } { 2 } , \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\big ( \\tilde \\partial ( \\lambda _ 1 \\ast \\lambda _ 2 ) \\big ) ( q ) & = ( \\lambda _ 1 \\ast \\lambda _ 2 ) \\big ( ( 1 , q ) \\big ) \\\\ & = \\lambda _ 1 \\big ( ( 1 , q ) \\big ) \\ast \\lambda _ 2 \\big ( ( 1 , q ) \\big ) \\\\ & = \\tilde \\partial ( \\lambda _ 1 ) \\ast \\tilde \\partial ( \\lambda _ 2 ) ( q ) , \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} \\mathcal { F } _ c P _ k ( x ) = \\int _ { - 1 } ^ 1 e ^ { i c x y } P _ k ( y ) \\ , d y = i ^ k \\sqrt { \\frac { 2 \\pi } { c x } } J _ { k + \\frac { 1 } { 2 } } ( c x ) , x \\in I , \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\psi _ 0 ( z ) = A z + B , \\quad \\psi _ 1 ( z ) = C \\ne 0 , \\ , \\ , \\forall z \\in \\C , \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} H = ( \\mathfrak { d } _ 1 \\dot { \\phi } ( i + j ) ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} L ( \\chi _ { E } ) = d \\chi _ { \\Phi ( E ) } \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} ( \\nabla ( \\Phi _ 4 ) ^ k ) ^ * ( \\Phi _ { 2 1 } ) = \\Phi _ { 1 4 \\cdot 3 ^ { k } } \\cdots \\Phi _ { 4 2 } \\Phi _ { 2 1 } . \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} g = g ^ { ( 0 ) } + g ^ { ( 2 ) } \\epsilon ^ 2 + g ^ { ( 4 ) } \\epsilon ^ 4 + \\cdots , g ^ { ( m ) } \\geq 0 \\ , ( m = 0 , 2 , \\ldots ) . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} y _ k = b _ k \\theta _ k + \\varepsilon \\xi _ k , k \\in J , \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} { n \\brack k } = q ^ { 2 k } { n - 1 \\brack k } + { n - 1 \\brack k - 1 } = { n - 1 \\brack k } + q ^ { 2 ( n - k ) } { n - 1 \\brack k - 1 } . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} \\lambda _ { 1 } = \\inf _ { u \\in W _ { 0 } ^ { s , p , q } ( \\Omega ) , \\ , \\ , u \\neq 0 } \\frac { [ u ] ^ { p } _ { s , p , q } } { \\| u \\| ^ { p } _ { L ^ { p } ( \\mathbb { G } ) } } . \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{gather*} \\theta _ 3 ^ 4 ( \\tau ) = \\frac { 4 E _ 2 ( 2 \\tau ) - E _ 2 ( \\tau / 2 ) } { 3 } , \\theta _ 3 ^ 4 ( \\tau ) + \\theta _ 2 ^ 4 ( \\tau ) = 2 E _ 2 ( \\tau ) - E _ 2 ( \\tau / 2 ) . \\end{gather*}"}
-{"id": "8866.png", "formula": "\\begin{align*} \\big \\langle \\Lambda _ v P _ { v , \\rm R } F ( v ) , P _ { v , \\rm R } F ( v ) \\big \\rangle = 0 . \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} \\phi ( t ) = \\det ( \\gamma ' ( t ) , \\eta ( t ) ) . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} & i + 1 \\leq k + j ( = X ) \\leq 2 n - i - 1 \\ \\ 1 - i \\leq k - j ( = Y ) \\leq i - 1 \\\\ & \\ ( k + j ) - ( i + 1 ) , ( k - j ) - ( 1 - i ) \\in 2 \\mathbb { Z } _ { \\geq 0 } , \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} u _ { * } v _ { * } = \\frac { 1 - 4 y _ { * } ^ 4 } { - y _ { * } ^ 3 \\sqrt { 1 + y _ { * } ^ 2 } } . \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} S _ \\nu : = \\left \\{ f ^ n _ \\omega \\ , | \\ , \\omega \\in \\Omega , \\ , n \\in \\mathbb N \\right \\} . \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} B ( u , u ) ( t ) = \\int _ 0 ^ t \\frac { 1 } { ( t - \\tau ) ^ { \\frac { d + 1 } { 2 } } } G \\Big ( \\frac { x } { \\sqrt { t - \\tau } } \\Big ) * ( u \\otimes u ) ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} \\begin{aligned} v _ { 2 } ( x ) & \\leq M + C _ 1 M r _ { k _ 0 } ^ { s - n / p } \\\\ & \\leq M + C _ 1 M r _ { k _ 0 } ^ { \\beta } \\\\ & \\leq \\hat { C } M r _ { 2 k _ 0 } ^ { \\beta } \\left ( \\frac { 1 } { \\hat { C } \\tau _ 1 ^ { 2 k _ 0 \\beta } } + \\frac { C _ 1 } { \\hat { C } \\tau _ 1 ^ { k _ 0 \\beta } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} d \\sigma ^ { 2 } = \\frac { 1 } { 4 } ( d \\varphi ^ { 2 } + \\sin ^ { 2 } ( \\varphi ) d \\theta ^ { 2 } ) , \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} \\hat { u } _ { n + 1 } = - { \\gamma _ 1 \\over \\alpha _ 1 } \\hat { x } _ n \\ ; . \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} R _ N = \\sum _ { | k | \\leq N } C _ { k , l } C _ { k , m } \\big ( \\ < \\omega , e _ k e _ l \\ > \\ < \\omega , e _ k e _ m \\ > - \\delta _ { l , m } \\big ) . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} d h ^ { i } _ { t } ( \\zeta ) = \\frac { 1 } { \\rho _ { t } ( \\zeta ) } d B _ { t } ^ { ( i ) } \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} w ^ * = \\left ( \\sum _ { i = d + 1 } ^ { k + 1 } i \\rho ^ { i } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\Delta U _ I ( X ) = \\sum _ { J K } a _ { \\overline { I } ^ \\sim } ^ { \\overline { J } ^ \\sim \\overline { K } ^ \\sim } U _ J ( X ) \\otimes U _ K ( X ) , \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\rho = \\rho _ 1 \\cdots \\rho _ r , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} \\overline { \\phi ( v _ { i - a + 1 } ) } = \\vartheta _ i \\overline { v _ { i - a + 1 } } . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} F ( z , z ' ) : = \\log K ( z , z ' ) , \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} \\mathcal { C } ^ * ( \\chi ) = \\Big \\{ \\big ( \\chi ( I , 1 ) , \\chi ( I , 2 ) , \\ldots , \\chi ( I , n ) \\big ) \\mid I \\in \\{ 1 , \\ldots , n \\} ^ { d - 1 } \\Big \\} . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\mathcal C _ x : = \\{ ( q , i ) \\ ; | \\ ; q \\in Q , \\ , q \\le x _ i \\} \\subseteq Q \\times \\{ 1 , \\dotsc , d \\} \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} \\dot { x } = A ( t ) x , \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} d ( \\Z ^ n , f ) = \\operatorname * { g c d } \\lbrace f ( r _ 1 , r _ 2 , \\ldots , r _ n ) : 0 \\leq r _ i \\leq m _ i \\rbrace . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} ( f \\cdot b ) \\ast _ g a = ( f \\otimes b ) \\ast _ g \\Delta ( a ) \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} \\mathbf { i } _ 0 = ( 1 , \\dots , n - 1 , 1 , \\dots , n - 2 , \\dots , 1 , 2 , 1 ) . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\sigma \\left ( \\frac { L ^ N ( t , \\psi ) } { ( 2 \\pi i ) ^ t G ( \\psi ) } \\right ) = \\frac { L ^ N ( t , { } ^ \\sigma \\ ! \\psi ) } { ( 2 \\pi i ) ^ t G ( { } ^ \\sigma \\ ! \\psi ) } . \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} h _ { 1 , i } & : = x _ { m + i - 1 } , \\\\ h _ { 1 , n + j } & : = x _ j , \\\\ h _ { i _ 1 , i _ 2 } & : = x _ { m + i _ 1 - 1 } - x _ { m + i _ 2 - 1 } , \\\\ h _ { i , n + j } & : = x _ j - x _ { m + i - 1 } , \\\\ h _ { n + j _ 1 , n + j _ 2 } & : = x _ { j _ 1 } - x _ { j _ 2 } . \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} \\xi _ 1 & = A ^ { - 1 } \\phi _ 1 , \\\\ \\xi _ n & = A ^ { - 1 } \\Big ( \\phi _ n + \\chi _ n - \\sum _ { \\stackrel { 1 \\le k , m \\le n - 1 , } { \\mu _ k + \\mu _ m = \\mu _ n } } B ( \\xi _ k , \\xi _ m ) \\Big ) \\quad 2 \\le n \\le N _ * , \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\mathcal { W } _ q = q \\frac { \\Delta _ { \\{ b + 1 , \\dots , n - 1 , 1 \\} } } { \\Delta _ { \\{ b + 1 , \\dots , n \\} } } + \\sum _ { i = 1 } ^ { n - 1 } \\frac { \\Delta _ { \\{ i - a + 1 , \\dots , i - 1 , i + 1 \\} } } { \\Delta _ { \\{ i - a + 1 , \\dots , i \\} } } . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} B _ { n , k } = \\sum _ { s = k } ^ n \\binom { s } { k } C _ { n , s } . \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\sum _ { i = k - m _ D } ^ { k } { k \\choose i } { d - k \\choose k - i } / { d \\choose k } \\ge \\frac { 1 } { J + 1 } \\ , . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} \\mathbf { c } = & \\overline { K } ^ { - 1 } \\mathbf { b } - \\overline { K } ^ { - 1 } \\overline { L } ( I _ 2 + \\overline { \\Psi } ^ { - T } \\overline { K } ^ { - 1 } \\overline { L } ) ^ { - 1 } \\overline { \\Psi } ^ { - T } \\overline { K } ^ { - 1 } \\mathbf { b } \\\\ = & \\overline { K } ^ { - 1 } \\mathbf { b } - \\overline { K } ^ { - 1 } \\overline { L } K Q ^ T \\Xi ^ { - 1 } Q \\overline { \\Psi } ^ { - T } \\overline { K } ^ { - 1 } \\mathbf { b } . \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} \\lambda _ { A \\Gamma } ( n ) = \\Lambda ( n ) . \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { P ( \\tilde { V } _ { n } = k ) } { P ( V _ n = k ) } = 1 \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{gather*} \\mathcal G _ { 1 , ( 1 : 3 ) } ^ \\ast \\ ( s , \\frac { - 1 } { \\sqrt { - 3 } } \\ ) = \\ ( \\frac { - 1 } { \\sqrt { - 3 } } \\ ) ^ { - 1 - s } \\sum _ { m , n \\in \\Z } \\frac { 1 } { \\big ( ( 3 m + 1 ) + n \\sqrt { - 3 } \\big ) \\big | ( 3 m + 1 ) + n \\sqrt { - 3 } \\big | ^ { 2 s } } \\\\ \\hphantom { \\mathcal G _ { 1 , ( 1 : 3 ) } ^ \\ast \\ ( s , \\frac { - 1 } { \\sqrt { - 3 } } \\ ) } { } = \\ ( \\frac { - 1 } { \\sqrt { - 3 } } \\ ) ^ { - 1 - s } L ( f _ { 3 6 } , s + 1 ) . \\end{gather*}"}
-{"id": "5463.png", "formula": "\\begin{align*} n _ { \\mathrm { M } } : = \\begin{cases} n - 1 & \\ ( n - 1 , r _ { \\mathrm { M } } ) \\in \\widehat { I } _ { \\xi } , \\\\ n + 1 & \\ ( n , r _ { \\mathrm { M } } ) \\in \\widehat { I } _ { \\xi } . \\end{cases} & & & \\left ( n _ { \\mathrm { m } } : = \\begin{cases} n - 1 & \\ ( n - 1 , r _ { \\mathrm { m } } ) \\in \\widehat { I } _ { \\xi } , \\\\ n + 1 & \\ ( n , r _ { \\mathrm { m } } ) \\in \\widehat { I } _ { \\xi } . \\end{cases} \\right ) \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\Omega _ { n } \\stackrel { \\rm d e f } = \\widetilde { \\Omega } _ { n } \\setminus \\overline { \\widetilde { \\Omega } } _ { n + 1 } . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} \\theta _ { \\emptyset } ^ { ( 0 ) } ( e ) = \\left | \\left \\{ ( a , b ) \\in ( [ 0 , r _ q - 1 ] \\cap \\N ) ^ 2 : \\ a + b \\equiv 0 \\mod n \\right \\} \\right | = \\max \\{ 1 , 2 r _ e - n + 1 \\} . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\dfrac { x ^ a - \\left ( \\frac { x } { \\sqrt [ a ] { 1 + c x ^ a } } \\right ) ^ a } { x ^ a \\left ( \\frac { x } { \\sqrt [ a ] { 1 + c x ^ a } } \\right ) ^ a } = c > 0 . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} A = \\bigcup _ { i = 0 } ^ { \\min \\{ k , N - 1 \\} } \\bigcup _ { z \\in B _ i ( x , y ) } A ( z , i ) , \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} \\Delta ( \\beta ^ { - 1 } ) = [ U : \\beta ( U ) ] = p . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} \\gamma _ n ( z ) = \\prod _ { m = 1 } ^ { n } ( m - 1 ) ! \\prod _ { j = 1 } ^ m \\frac { 1 } { z - m - 1 + 2 j } . \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} M ^ 1 H ^ 3 ( X ) = N ^ 1 H ^ 3 ( X ) . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & I _ { \\widehat { n } } \\\\ I _ { \\widehat { m } } & 0 \\end{bmatrix} \\left [ \\begin{array} { c } \\widehat { L } ( \\lambda ) \\\\ \\hline \\phantom { \\Big { ( } } \\widetilde { L } ( \\lambda ) \\phantom { \\Big { ( } } \\end{array} \\right ] = \\left [ \\begin{array} { c } \\widetilde { L } ( \\lambda ) \\\\ \\hline \\phantom { \\Big { ( } } \\widehat { L } ( \\lambda ) \\phantom { \\Big { ( } } \\end{array} \\right ] \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} \\eta _ { i t } ( m n ) = \\sum _ { d | ( m , n ) } \\mu ( d ) \\eta _ { i t } \\left ( \\frac { m } { d } \\right ) \\eta _ { i t } \\left ( \\frac { n } { d } \\right ) . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\lim \\limits _ { A , B \\rightarrow 0 } u _ * ^ { A B } = \\lim \\limits _ { A , B \\rightarrow 0 } \\sigma _ 1 ^ { A B } = \\lim \\limits _ { A , B \\rightarrow 0 } \\sigma _ 2 ^ { A B } = \\sigma _ { 0 } + \\beta t . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} h _ m ( y ) = h _ m ( y _ m ) : = x _ m . \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} q ^ { 2 r } { n \\brack r } \\prod _ { j = 0 } ^ { r - 1 } ( c - q ^ { 2 j } ) . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\left . \\frac { \\partial } { \\partial x } ( W _ { s } ( s , x ) + G ( s , x , W ( s , x ) , W _ { x } ( s , x ) , W _ { x x } ( s , x ) , \\bar { u } ( s ) ) ) \\right \\vert _ { x = \\bar { X } ^ { t , x ; \\bar { u } } ( s ) } = 0 . \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ M \\frac { D _ { { \\textrm { o u t } } } ( i j ) x _ { i j } } { \\eta ^ d _ i c ^ d _ i } \\leq D ^ d _ { i } , \\ \\forall i , \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} Q _ { \\alpha \\alpha _ 1 } ( t ; \\mathbf { B } ) = \\Sigma _ { \\alpha \\alpha _ 1 } ( t ) + \\sum _ { l = 1 } ^ { \\infty } \\int _ { 0 } ^ { t } \\int _ 0 ^ { t _ 1 } . . . \\int _ 0 ^ { t _ { l - 1 } } \\Pi ^ { ( l ) } _ { \\alpha \\alpha _ 1 } ( t , t _ 1 , t _ 2 , . . . , t _ l ; \\mathbf { B } ) d t _ l . . . d t _ 1 . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} ( \\phi ( \\hat J ) ) ^ \\bot = \\Phi ( J ^ \\bot ) \\ , . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} \\left ( u _ { \\lambda } , u _ { \\lambda } \\right ) _ { \\lambda } ^ { \\varphi } & = 1 & ( x . u _ { 1 } , u _ { 2 } ) _ { \\lambda } ^ { \\varphi } & = ( u _ { 1 } , \\varphi ( x ) . u _ { 2 } ) _ { \\lambda } ^ { \\varphi } \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\frac { 1 } { | G | } \\sum _ { g \\in G _ c } \\overline { \\chi _ { \\alpha } ( g ) } \\prod _ { i = 1 } ^ d \\sum _ { a _ i = 0 } ^ { p ^ e - 1 } ( \\xi _ { g , e , i } ) ^ { a _ i } = \\varphi _ { c } ^ { ( \\alpha ) } p ^ { c e } , \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} f ( \\underline { x } ) = \\sum \\limits _ { \\substack { \\mathbf { 0 } \\leq \\mathbf { i } \\leq \\mathbf { m } \\\\ \\vert \\mathbf { i } \\vert \\le k } } b ( \\mathbf { i } ) B _ { \\mathbf { i } } ( \\underline { x } ) . \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} \\frac { \\Phi _ { \\pmb { \\phi } _ p } ( \\alpha _ n ) } { | | \\pmb { \\phi } _ p | | ^ 2 } = \\chi _ { \\overline { \\psi } _ p } ( p ^ n ) p ^ { - | n | / 2 } , \\frac { \\Phi _ { \\pmb { \\phi } _ p } ( \\beta _ m ) } { | | \\pmb { \\phi } _ p | | ^ 2 } = \\chi _ { \\overline { \\psi } _ p } ( p ^ m ) p ^ { - | m | / 2 } . \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} ( \\omega ^ \\bullet _ t ) ^ { m + n } = c _ t e ^ { - n t + G } \\omega _ \\infty ^ m \\wedge \\omega _ F ^ n \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} \\langle { \\rm g r a d } f ( S ) , \\xi \\rangle _ S & = { \\rm D } f ( S ) [ \\xi ] \\\\ & = { \\rm D } \\bar { f } ( S ) [ \\xi ] \\\\ & = { \\rm t r } \\ , ( \\xi ^ T { \\rm s y m } ( \\nabla \\bar { f } ( S ) ) ) . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} ( \\Lambda _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m ) \\Sigma _ { P } ^ { ( \\epsilon , \\eta ) } ( \\lambda ) ( \\Lambda _ { \\epsilon } ( \\lambda ^ \\ell ) \\otimes I _ n ) = P ( \\lambda ) . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} \\limsup _ { s \\to \\infty } Y _ s = + \\infty { \\rm a . s . } \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} \\widetilde { D } _ { w \\lambda , w ' \\lambda } : = v ^ { - ( w \\lambda - w ' \\lambda , w \\lambda - w ' \\lambda ) / 4 + ( w \\lambda - w ' \\lambda , \\rho ) / 2 } D _ { w \\lambda , w ' \\lambda } . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} W : = 1 , u ^ 1 , u ^ 2 , \\dots , u ^ { 2 ^ n - 1 } \\quad \\left \\{ \\begin{array} { l l } u ^ k : = 2 , 3 , \\dots , n & \\\\ u ^ k : = n - 1 , n - 2 , \\dots , 1 & . \\end{array} \\right . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} Q = S _ i { r } ^ { i j } { X } _ j , \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} \\langle \\varphi ( y ^ { k } - x ^ { k } ) , y ^ { k + 1 } - y ^ { k } \\rangle & = \\varphi \\big [ \\| y ^ { k + 1 } - x ^ k \\| ^ 2 - \\| x ^ k - y ^ { k } \\| ^ 2 - \\| y ^ { k + 1 } - y ^ k \\| ^ 2 \\big ] . \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} \\alpha _ n ^ { - 1 } B _ \\lambda ( s ) & = i ^ { n k } \\ , 2 ^ { - n ( 2 n + 1 ) ( s + \\frac 1 2 ) + n } \\\\ & \\times \\int \\limits _ { \\mathfrak { a } ^ + } \\prod _ { 1 \\leq i < j \\leq n } ( \\cosh ( a _ i ) ^ 2 - \\cosh ( a _ j ) ^ 2 ) \\prod _ { j = 1 } ^ n \\cosh ( a _ j ) ^ { - ( 2 n + 1 ) s + \\frac 1 2 - n - k } \\prod _ { j = 1 } ^ n \\sinh ( a _ j ) \\ , d H . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} M ( m ) \\geq p ^ { \\frac { v _ p ( q \\cdot m ! ) } { r } } = p ^ { \\frac { v _ p ( q ) + \\frac { m - s _ p ( m ) } { p - 1 } } { r } } > p ^ { \\frac { v _ p ( q ) + \\frac { m } { p - 1 } - 1 - \\log _ p m } { r } } . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} \\textrm { H e s s } _ W \\mathcal { F } ( \\rho ) ( \\sigma _ 1 , \\sigma _ 2 ) = & g _ W ( \\sigma _ 1 , \\nabla _ { \\sigma _ 2 } \\nabla _ W \\mathcal { F } ( \\rho ) ) \\\\ = & \\sigma _ 2 \\big ( g _ W ( \\sigma _ 1 , \\nabla _ W \\mathcal { F } ( \\rho ) ) \\big ) - g _ W ( \\nabla _ { \\sigma _ 2 } \\sigma _ 1 , \\nabla _ W \\mathcal { F } ( \\rho ) ) . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} \\begin{array} { r l l } H _ { 1 , n + j } : & x _ j = 0 & j = 1 , \\ldots , m , \\\\ H _ { i , n + j } : & x _ j - x _ { m + i - 1 } = 0 & i = 2 , \\ldots , n , j = 1 , \\ldots , m . \\end{array} \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\begin{gathered} d r _ t = ( k - a r _ t ) d t + \\sigma \\sqrt { r _ t } d W _ t , a , k , \\sigma > 0 . \\end{gathered} \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\Theta _ h ^ \\Phi ( \\pi _ 1 , \\pi _ 2 ) : = h ( [ \\pi _ 1 , \\pi _ 2 ] ) - [ h ( \\pi _ 1 ) , h ( \\pi _ 2 ) ] - \\Phi _ 1 ( \\pi _ 1 ) \\cdot h ( \\pi _ 2 ) + ( - 1 ) ^ l h ( \\pi _ 1 ) \\cdot \\Phi _ 1 ( \\pi _ 2 ) . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} M _ 0 = \\{ m _ { s , r + 2 } ^ { ( i - 1 ) } , m _ { s , r + 2 } ^ { ( i + 1 ) } \\} , \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} \\left ( - 2 L _ { - 2 } + \\frac { \\kappa } { 2 } L _ { - 1 } ^ { 2 } \\right ) v _ { \\lambda } = \\left ( - ( 1 - 2 \\lambda ^ { 2 } ) H _ { 1 } ( - 1 ) ^ { 2 } - \\sum _ { i = 2 } ^ { \\ell } H _ { i } ( - 1 ) ^ { 2 } \\right ) v _ { \\Lambda } , \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} s _ p ( h ) = \\begin{cases} ( c , d ) _ p & c d \\neq 0 , \\mathrm { o r d } _ p ( c ) , \\\\ 1 & \\end{cases} x ( h ) = \\begin{cases} c & c \\neq 0 , \\\\ d & c = 0 , \\end{cases} x ( h \\alpha _ n ) = \\begin{cases} c p ^ n & c \\neq 0 , \\\\ d p ^ { - n } & c = 0 . \\end{cases} \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} q _ j ( z ) : = ( z + 1 ) h _ { d j } ^ * ( z ) - h _ { d , j - 1 } ^ * ( z ) - \\sum _ { k = 1 } ^ { j - 1 } w _ { j , j - k } \\ , ( z - 1 ) ^ k h _ { d , j - 1 - k } ^ * ( z ) . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} { \\overline { \\mathcal { F } } } _ t = \\sigma ( \\xi _ j , D _ j , j \\leq t ) \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} ( V ) \\ , : = \\ , ( c _ 1 ( V ) \\cup \\omega ^ { d - 1 } ) \\cap [ X ] \\ , \\in \\ , { \\mathbb R } \\ , . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} H = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 1 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} { \\big ( I \\odot \\bar { I } \\big ) _ o } ^ { A } { } _ { B } = & \\ , \\ , { \\mathbb Y } _ { r } { } ^ A { } _ B \\big ( v ^ s ( \\nabla _ s \\bar { v } ^ r ) + \\bar { v } ^ s ( \\nabla _ s { v } ^ r ) - \\tfrac { 2 } { 3 } v ^ r ( \\nabla _ t \\bar { v } ^ t ) - \\tfrac { 2 } { 3 } \\bar { v } ^ r ( \\nabla _ t { v } ^ t ) \\big ) \\\\ & + \\mathrm { \" l o w e r \\ , \\ , a d j o i n t \\ , \\ , t r a c t o r \\ , \\ , s l o t s \" } . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} b = \\frac { 1 } { q } N _ G b = \\frac { 1 } { q } N _ G b _ 1 + \\frac { 1 } { q } N _ G b _ 2 , \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} \\Phi _ { \\breve { \\mathbf g } _ p } ( \\alpha _ n ) = p ^ { - 2 | n | } , \\Phi _ { \\mathbf h _ p } ( \\alpha _ n ) = p ^ { - 3 | n | / 2 } ( - 1 ) ^ n \\chi _ { \\psi } ( p ^ n ) . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{gather*} W _ k ( z ; s ) = \\frac { \\partial W _ { k - 1 } ( z ; s ) } { \\partial z } . \\end{gather*}"}
-{"id": "3264.png", "formula": "\\begin{align*} ( \\nabla _ { U } u ) V = B ( U , V ) v ( E ) - B ( U , \\varphi V ) - \\tau ( U ) u ( V ) , \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = { \\bf E } \\bigl [ M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] , \\ ; \\Re ( q ) < \\tau , \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} \\phi + _ r \\psi = \\mu _ X \\bigl ( \\eta _ X ( \\phi ) + _ r \\eta _ X ( \\psi ) \\bigr ) \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\begin{array} { l l } A ( t ) ~ u _ s = ( I + \\kappa U ( s , t ) ) ~ \\partial _ { t } { \\rm L o g } ~ ( U ( t , s ) + \\kappa I ) ~ u _ s , \\end{array} \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\mathcal { S } _ 1 = 1 + \\frac { M ^ { - 2 ( 1 - \\Upsilon ) \\delta } - M ^ { - 2 \\delta } } { ( 2 \\delta \\Upsilon \\log M ) ^ 2 } + \\mathcal { O } \\left ( \\frac { \\log \\log q } { \\log q } M ^ { - 2 ( 1 - \\Upsilon ) \\delta } \\right ) . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} _ { r } F _ { s } \\left ( \\begin{array} { c } a _ 1 , \\dots , a _ r \\\\ b _ 1 , \\dots , b _ s \\end{array} ; z \\right ) : = \\sum _ { k = 0 } ^ \\infty \\frac { ( a _ 1 , \\dots , a _ r ) _ k } { ( b _ 1 , \\dots , b _ s ) _ k } \\dfrac { z ^ k } { k ! } . \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\begin{aligned} \\prod _ { t = s ' } ^ { { s } } { ( s - s ' ) + \\lambda _ { i _ t } - ( t - s ' ) \\brack 1 } _ q \\prod _ { t = 1 } ^ { { s ' - 1 } } { \\lambda _ { h } - \\lambda _ { i _ t } - a _ { t - 1 } \\cdots - a _ 1 \\brack a _ t } _ q \\neq 0 . \\end{aligned} \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} h \\alpha _ n = \\left ( \\begin{array} { c c } a p ^ n & b p ^ { - n } \\\\ c p ^ n & d p ^ { - n } \\end{array} \\right ) = \\left ( \\begin{array} { c c } c ^ { - 1 } p ^ { - n } & \\ast \\\\ 0 & c p ^ { n } \\end{array} \\right ) \\left ( \\begin{array} { c c } 0 & - 1 \\\\ 1 & d c ^ { - 1 } p ^ { - 2 n } \\end{array} \\right ) = : g _ 1 g _ 2 , \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} \\left ( \\lambda _ { G / H _ 1 } ( k ) \\oplus \\lambda _ { G / H _ 2 } ( k ) \\right ) \\left ( \\varphi ( \\delta _ { g ( H _ 1 \\cap H _ 2 ) } ) \\right ) & = \\left ( \\lambda _ { G / H _ 1 } ( k ) \\oplus \\lambda _ { G / H _ 2 } ( k ) \\right ) \\left ( \\delta _ { ( g H _ 1 , g H _ 2 ) } \\right ) \\\\ & = \\delta _ { k g H _ 1 , k g H _ 2 } \\\\ & = \\varphi ( \\delta _ { k g ( H _ 1 \\cap H _ 2 ) } ) \\\\ & = \\varphi \\left ( \\lambda _ { G / ( H _ 1 \\cap H _ 2 ) } ( k ) ( \\delta _ { g ( H _ 1 \\cap H _ 2 ) } ) \\right ) . \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { ( a \\delta _ j X - \\gamma _ j Y ) } = \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { ( a \\delta _ j X - a \\gamma _ j Y ) } = a ^ { \\ell } \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { ( \\delta _ j X - \\gamma _ j Y ) } = \\prod \\limits _ { j = 0 } ^ { \\ell - 1 } { ( \\delta _ j X - \\gamma _ j Y ) } \\cdot \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} E ( V ) = V e V ^ { - 1 } = \\begin{bmatrix} S _ 0 ^ 2 & S _ 0 ( 1 + S _ 0 ) Q \\\\ S _ 1 D & 1 - S _ 1 ^ 2 \\end{bmatrix} . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} Q = \\rho _ 1 ^ 2 - \\sigma _ 1 ^ 2 + \\rho _ 2 ^ 2 - \\sigma _ 2 ^ 2 . \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} ( a ) z ^ 2 _ { m i n } > 3 ( X _ u - x _ i ) ^ 2 - ( Y _ u - y _ i ) ^ 2 , \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\\\ ( b ) z ^ 2 _ { m i n } > 3 ( Y _ u - y _ i ) ^ 2 - ( X _ u - x _ i ) ^ 2 , \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\\\ ( c ) z ^ 2 _ { m i n } > 3 ( X _ u - x _ i ) ^ 2 + 3 ( Y _ u - y _ i ) ^ 2 , \\forall i \\in I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} \\begin{dcases} \\sum _ { e \\in E ( G ) } w _ i ( e ) = 1 , \\\\ \\sum _ { e : \\ , v \\in e } B _ i ( v , e ) = 1 , \\ \\ v \\in V ( G ) , \\\\ w _ i ( e ) ^ { p _ i - r } \\prod _ { v \\in e } B _ i ( v , e ) = \\alpha _ i , \\ \\ e \\in E ( G ) . \\end{dcases} \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} \\nu ' ( C ) = \\dim G r _ F ^ 1 H ^ 2 ( F , \\C ) _ { - 1 } . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\langle b \\nabla G _ { \\beta } ^ { a } u , b \\nabla u \\rangle = \\langle G _ { \\beta } ^ { a } b \\nabla u , b \\nabla u \\rangle + \\langle G _ { \\beta } ^ { a } R G _ { \\beta } ^ { a } u , b \\nabla u \\rangle . \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} \\mu _ { j , t } ( \\Psi _ t ( \\hat { x } ) ) = d ^ { \\hat { g } _ t } ( \\hat { x } , \\partial ( B _ { \\lambda _ t } \\times Y ) ) ^ j | ( \\nabla ^ { j , \\hat { g } _ t } \\hat { g } ^ \\bullet _ t ) ( \\hat { x } ) | _ { \\hat { g } _ t ( \\hat { x } _ t ) } . \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} P \\Big ( \\frac { 2 } { k + 1 + 2 / \\theta _ 0 } \\Big ) = \\frac { 4 } { ( k + 1 + 2 / \\theta _ 0 ) ^ 2 } + \\frac { 2 } { k + 1 + 2 / \\theta _ 0 } \\theta _ k ^ 2 - \\theta _ k ^ 2 \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} \\min _ { \\mu = 1 , \\dots , N } & \\ , \\gamma _ { \\mu \\mu } \\sum _ { \\mu = 1 } ^ N \\int _ { \\R } \\int _ { \\R ^ 3 } | u _ \\mu ( t , x ) | ^ { 2 p + 4 } \\ , d t \\ , d x \\\\ & \\gtrsim \\sum _ { \\mu = 1 } ^ N \\sum _ { n } \\int _ { t _ n } ^ { t _ n + \\bar t } \\int _ { \\widetilde Q _ { x _ n } } | u _ \\mu ( t , x ) | ^ { 2 p + 4 } \\ , d t \\ , d x = \\infty , \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\| f \\| _ { \\alpha , T } : = \\sup _ { t \\in [ 0 , T ] } \\sup _ { x \\neq y , | x - y | \\leq 1 } \\frac { | f ( t , x ) - f ( t , y ) | } { | x - y | ^ \\theta } < \\infty \\ , . \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} & \\left \\langle \\rho , \\tau ^ { x _ { 3 } } \\alpha _ { 1 } , \\sigma ^ { y _ { 2 } } \\tau ^ { y _ { 3 } } \\alpha _ { 2 } \\alpha _ { 3 } ^ { \\left ( 1 + x _ { 3 } \\right ) y _ { 2 } x _ { 3 } ^ { - 1 } } \\right \\rangle \\cong C _ { p } ^ { 3 } \\\\ & \\ y _ { 3 } = 0 , . . . , p - 1 , \\ y _ { 2 } , x _ { 3 } = 1 , . . . , p - 1 , \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\| \\pi ( \\sigma _ { \\check F } ( f ) ) \\xi \\| ^ 2 & = \\int _ { r ( W ) } D ( r ^ { - 1 } ( x ) ) ^ { - 1 } | f ( r ^ { - 1 } ( x ) ) | ^ 2 \\| \\xi _ { s ( r ^ { - 1 } ( x ) ) } \\| ^ 2 d \\nu ( x ) \\\\ & = \\int _ { s ( W ) } | f ( s ^ { - 1 } ( x ) ) | ^ 2 \\| \\xi _ x \\| ^ 2 d \\nu ( x ) , \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} u _ { t } + u \\cdot \\nabla u - \\nu \\Delta u + \\nabla p = f \\ ; \\ ; i n \\ ; \\Omega , \\\\ \\nabla \\cdot u = 0 \\ ; \\ ; i n \\ ; \\Omega , \\\\ u = 0 \\ ; \\ ; o n \\ ; \\partial \\Omega . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} 2 . 4 3 8 4 = ( s ) \\int _ { 1 } ^ { 4 } x ^ 2 d \\mu \\leq m i n \\{ 2 . 5 3 0 2 , 4 - 1 \\} = 2 . 5 3 0 2 . \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & \\epsilon \\end{pmatrix} . \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} H _ { [ s _ 0 : s _ 1 ] } \\ , = \\ , \\{ \\ , s _ 0 ( X - Y ) \\ , + \\ , s _ 1 ( Z - W ) \\ , = \\ , 0 \\ , \\} , [ s _ 0 : s _ 1 ] \\in \\P ^ 1 _ k \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} h _ 3 ( \\tau / 2 ) & = { { \\eta ( \\tau ) ^ 3 \\eta ( 3 \\tau ) ^ 3 } } = \\ ( \\frac { b c } 3 \\ ) ^ { 3 / 2 } = 3 ^ { - 3 / 2 } \\big ( s \\big ( 1 - s ^ 3 \\big ) ^ { 1 / 3 } a ^ 2 \\big ) ^ { 3 / 2 } \\\\ & = \\big ( s \\big ( 1 - s ^ 3 \\big ) a ^ 2 / 3 \\big ) \\big ( s ^ { 1 / 2 } \\big ( 1 - s ^ 3 \\big ) ^ { - 1 / 2 } 3 ^ { - 1 / 2 } a \\big ) . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} \\rho \\frac { \\partial ^ 2 u } { \\partial t ^ 2 } - \\left ( \\frac { p _ 0 } { \\lambda } + \\frac { E } { 2 L } \\int _ 0 ^ L \\left | \\frac { \\partial u } { \\partial x } \\right | ^ 2 d x \\right ) \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } = 0 \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} \\widetilde b _ { d j } ^ * ( z ) = ( z ^ { - 1 } - 1 ) ^ { d - j } \\ , h _ { d j } ^ * ( z ^ { - 1 } ) , \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\| T ' \\| _ { S _ 1 } \\leq \\left [ \\prod _ { i = 1 } ^ N \\frac { q _ i + 1 } { q _ i - 1 } \\right ] \\left ( \\| \\phi \\| _ { c b } - | c _ + | - | c _ - | \\right ) . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} B ^ { ( f ) } _ { m } ( x ) \\triangleq \\frac { d ^ m } { d t ^ m } \\Big \\vert _ { t = 0 } \\bigl [ f ( t ) e ^ { - x t } \\bigr ] . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} a _ i \\leq \\left ( 1 - \\frac { i } { p } \\right ) b _ i + \\frac { i } { p } b _ { i + 1 } \\leq \\left ( 1 - \\frac { i } { p + 1 } \\right ) b _ i + \\frac { i } { p + 1 } b _ { i + 1 } \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} \\mathcal { H } _ u & = \\bigcap _ { n \\in \\N } V ^ n \\mathcal { H } , & \\mathcal { H } _ s & = \\bigoplus _ { n \\in \\N } V ^ n \\left ( V ^ * \\right ) . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{gather*} c _ p ( n ) = \\frac { 1 } { p 2 ^ p } \\Bigl [ \\bigl ( \\zeta ( p , \\ , 1 + 2 \\lambda ) - 2 \\zeta ( p , 1 + \\lambda ) \\bigr ) \\frac { B _ { p + 1 } ( n ) - B _ { p + 1 } } { p + 1 } + \\zeta ( p ) \\frac { B _ { p + 1 } ( n + 1 ) - B _ { p + 1 } } { p + 1 } - n \\zeta ( p ) \\Bigr ] . \\end{gather*}"}
-{"id": "1402.png", "formula": "\\begin{align*} d ( g x _ 0 , x _ 0 ) = d ( a x _ 0 , x _ 0 ) = \\sqrt { ( \\log a _ 1 ) ^ 2 + \\ldots + ( \\log a _ d ) ^ 2 } , \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} f ^ { ( i ) } _ { \\varepsilon , u } ( x ) \\triangleq ( \\chi ^ { ( i ) } _ u \\star \\phi _ \\varepsilon ) ( x ) = \\frac { 1 } { \\varepsilon } \\int \\chi ^ { ( i ) } _ u ( x - y ) \\phi ( y / \\varepsilon ) \\ , d y . \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} M = ( M _ 1 , \\dots , M _ n ) \\colon \\ { \\rm C o n f } _ n ^ \\times ( a ) \\longrightarrow T ^ \\vee . \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{gather*} \\tilde \\psi ( \\mathfrak I ) = \\tilde \\psi ( ( a + b { \\rm i } ) ) = ( - 1 ) ^ { b / 2 } ( a - b { \\rm i } ) \\end{gather*}"}
-{"id": "1910.png", "formula": "\\begin{align*} I _ X ( x , y ) \\coloneqq \\begin{cases} I & x = y \\\\ 0 & x \\neq y . \\end{cases} \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} \\dot \\rho = - L ( \\rho ) d _ \\rho \\mathcal { F } ( \\rho ) = \\textrm { d i v } ( \\rho \\nabla _ G d _ \\rho \\mathcal { F } ( \\rho ) ) . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} ( f \\cdot _ \\sigma b ^ \\tau ) \\ast _ g a ^ \\rho = ( f \\ast _ g ( \\alpha _ { \\rho ( i _ 1 ) } \\otimes \\cdots \\otimes \\alpha _ { \\rho ( i _ n ) } ) ) \\cdot _ \\sigma ( ( \\beta _ { \\tau ( 1 ) } \\otimes \\cdots \\otimes \\beta _ { \\tau ( m ) } ) \\ast _ g ( \\alpha _ { \\rho ( i _ 1 ) } \\otimes \\cdots \\otimes \\alpha _ { \\rho ( i _ m ) } ) ) , \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} \\bar { \\nabla } ^ { ' } _ { X } Y = \\bar { \\nabla } _ X Y + \\eta ( Y ) X . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\int _ { \\Omega _ 1 \\setminus S } u _ 0 ^ { 2 ^ * } = \\int _ { \\Omega _ 1 } u _ 0 ^ { 2 ^ * } + O ( \\lambda ^ { - \\frac { N } { 2 } } ) . \\end{array} \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} q ( k ) & = \\left [ \\prod _ { j = 1 } ^ { d } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\left \\{ \\frac { k Y _ j z _ j } { N } \\right \\} \\right ) \\right ) \\right ] \\left [ \\prod _ { j = d + 1 } ^ { s } \\left ( 1 + \\gamma _ j \\ , \\omega \\left ( \\left \\{ \\frac { k z _ j ^ 0 } { N } \\right \\} \\right ) \\right ) \\right ] . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x _ 1 x _ 2 - x _ 2 x _ 1 & = & \\alpha ( x _ 3 x _ 4 + x _ 4 x _ 3 ) , \\\\ x _ 1 x _ 2 + x _ 2 x _ 1 & = & x _ 3 x _ 4 - x _ 4 x _ 3 , \\\\ x _ 1 x _ 3 - x _ 3 x _ 1 & = & \\beta ( x _ 4 x _ 2 + x _ 2 x _ 4 ) , \\\\ x _ 1 x _ 3 + x _ 3 x _ 1 & = & x _ 4 x _ 2 - x _ 2 x _ 4 , \\\\ x _ 1 x _ 4 - x _ 4 x _ 1 & = & \\gamma ( x _ 2 x _ 3 + x _ 3 x _ 2 ) , \\\\ x _ 1 x _ 4 + x _ 4 x _ 1 & = & x _ 2 x _ 3 - x _ 3 x _ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sum _ { i = 0 } ^ { n _ k - 1 } f \\circ T ^ i x = 0 \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} B _ n ( x ) = \\sum _ { k = 0 } ^ n \\dfrac { ( - 1 ) ^ k } { k + 1 } \\Delta ^ k I _ n ( x ) , E _ n ( x ) = \\sum _ { k = 0 } ^ n \\dfrac { ( - 1 ) ^ k } { 2 ^ k } \\Delta ^ k I _ n ( x ) . \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} X _ t ^ { t o y } = \\sigma W _ t \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} g _ m = g _ E + m ^ { - 2 } \\gamma + O ( m ^ { - 3 } ) . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} P ^ T E Y + X ^ T F R & = 0 , \\\\ Q ^ T F X + P ^ T E P + X ^ T F Q & = 0 . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} m \\left ( \\Omega _ { n } \\cap \\Omega _ { 0 } \\right ) & = m \\left ( \\Omega _ { 0 } \\backslash \\left ( \\Omega _ { n } ^ { c } \\cap \\Omega _ { 0 } \\right ) \\right ) \\\\ & \\geq m \\left ( \\Omega _ { 0 } \\right ) - m \\left ( \\Omega _ { n } ^ { c } \\cap \\Omega _ { 0 } \\right ) \\\\ & \\geq \\gamma _ { 0 } - \\frac { 1 } { n } > \\frac { \\gamma _ { 0 } } { 2 } \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} a _ m \\ ; : = \\ ; a ( u _ m ) , a ^ * _ m \\ ; : = \\ ; a ^ * ( u _ m ) , b _ n \\ ; : = \\ ; b ( v _ n ) , b ^ * _ n \\ ; : = \\ ; b ^ * ( v _ n ) \\ , . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} f ' ( g , s ) = \\int \\limits _ { K } \\int \\limits _ { K } | f ( g ( k _ 2 , k _ 1 ^ { - 1 } ) , s ) | \\ , d k _ 1 \\ , d k _ 2 , g \\in G _ { 4 n } ( F ) . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ N L _ M \\bigr ) ( \\kappa \\ , q \\ , | \\ , \\kappa \\ , a , \\kappa \\ , b ) = \\bigl ( \\mathcal { S } _ N L _ M \\bigr ) ( q \\ , | \\ , a , b ) - \\frac { \\log \\kappa } { M ! } \\bigl ( \\mathcal { S } _ N B _ { M , M } \\bigr ) ( q \\ , | \\ , b ) \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} W ^ { ( 1 ) } - W ^ { ( 3 ) } = & b _ 0 y + ( 1 6 2 b _ 0 + b _ 1 ) y ^ 5 + ( 1 1 1 8 6 b _ 0 + 1 1 6 b _ 1 + b _ 2 ) y ^ 9 \\\\ & + ( 3 9 4 4 9 8 b _ 0 + 5 0 8 1 b _ 1 + 7 0 b _ 2 + b _ 3 ) y ^ { 1 3 } \\\\ & + ( 4 6 2 8 8 2 6 b _ 0 + 6 5 9 3 6 b _ 1 + 1 0 9 2 b _ 2 + 2 4 b _ 3 + b _ 4 ) y ^ { 1 7 } \\\\ & + ( - 2 2 6 3 9 7 7 1 0 b _ 0 - 2 9 8 3 5 1 9 b _ 1 - 4 3 7 5 8 b _ 2 - 7 8 1 b _ 3 - 2 2 b _ 4 ) y ^ { 2 1 } \\\\ & + \\cdots . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q + k a _ i \\ , | \\ , a , b ) = \\eta _ { M , N } ( q \\ , | \\ , a , b ) \\exp \\Bigl ( - \\sum _ { l = 0 } ^ { k - 1 } \\bigl ( \\mathcal { S } _ N L _ { M - 1 } \\bigr ) ( q + l a _ i \\ , | \\ , \\hat { a } _ i , b ) \\Bigr ) . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} h _ 1 = \\partial _ y \\psi > 0 , \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} v _ { 0 } & = \\sqrt { \\varphi _ { 1 } ^ { 2 } + g _ { 0 } \\theta _ { 1 } ^ { 2 } } \\\\ v _ { 1 } & = \\frac { 1 } { v _ { 0 } } [ 2 \\varphi _ { 1 } \\varphi _ { 2 } + \\frac { f _ { 0 } } { 2 } \\varphi _ { 1 } \\theta _ { 1 } ^ { 2 } + 2 g _ { 0 } \\theta _ { 1 } \\theta _ { 2 } ] , \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} f ( t ) = \\frac { \\ln ( 1 + t ) } { t } = 1 - \\frac { t } { 2 } + \\cdots . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} p _ { s } ^ { \\textrm { m a x } } = \\frac { E _ { s } } { 1 - \\theta } = \\frac { \\xi \\theta P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } } { 1 - \\theta } . \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\mathcal { Q } _ t ^ k ( x _ { t - 1 } , 1 ) = \\displaystyle \\max _ { 0 \\leq j \\leq k } \\ ; \\theta _ t ^ j + \\langle \\beta _ t ^ j , x _ { t - 1 } \\rangle . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} & L _ j : = { } _ { r _ k ^ { j - 1 } } ( L _ 1 ) , & & L _ j ^ \\prime : = { } _ { r _ k ^ { j - 1 } } ( L _ 1 ^ \\prime ) , & \\\\ & \\delta _ j : = r _ k ^ { j - 1 } ( \\delta _ 1 ) , & & \\delta _ j ^ \\prime : = r _ k ^ { j - 1 } ( \\delta _ 1 ^ \\prime ) , \\\\ & X _ { i , j } : = { } _ { r _ k ^ { j - 1 } } ( X _ i ) , & & Y _ { i , j } : = { } _ { r _ k ^ { j - 1 } } ( Y _ i ) , & \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\gamma _ 1 - 1 - i \\ell = n + ( a - 1 - i ) \\ell , i = 0 , \\dots , \\left \\lfloor \\frac { m } { \\ell } \\right \\rfloor , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} & \\pi _ 1 ( S _ 2 ) e _ { 2 k } = e _ { 4 k } \\\\ & \\pi _ 1 ( S _ 2 ) e _ { 2 k + 1 } = e _ { 4 k + 1 } \\\\ & \\pi _ 1 ( S _ 1 ) e _ { 2 k } = e _ { 4 k + 2 } \\\\ & \\pi _ 1 ( S _ 1 ) e _ { 2 k + 1 } = e _ { 4 k + 3 } \\ ; . \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} \\Delta u + e ^ u = 0 . \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} \\widehat { d } ( G ( w , A _ 1 , \\ldots , A _ m ) ) \\prec _ { \\log } \\prod _ { j = 1 } ^ { m } \\widehat { d \\ , \\ , } ^ { w _ { j } } ( A _ j ) . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\int _ A f \\ , d \\lambda = \\lambda ( \\chi _ A f ) , \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} T ( X , X ' ) = \\sum _ { X < k \\leq X ' } e ^ { 2 \\pi i F ( k ) } . \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} \\Psi ( z _ 1 , . . . , z _ N ) = \\prod _ { i = 1 } ^ { N } u _ i ( z _ i ) \\prod _ { j < k } ^ { N } f _ { j k } ( z _ j - z _ k ) . \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} { \\rm m u l t } _ \\lambda ( \\mathbf { m } ) & = \\sum _ { \\sigma \\in S _ n } \\varepsilon ( \\sigma ) Q \\Big ( \\sigma \\Big ( ( \\ell _ 1 + 1 + m ) \\sum _ { j = 1 } ^ n e _ j + \\frac 1 2 \\sum _ { j = 1 } ^ n ( n + 1 - 2 j ) e _ j \\Big ) - \\lambda - \\frac { n + 1 } 2 \\sum _ { j = 1 } ^ n e _ j \\Big ) \\\\ & = \\sum _ { \\sigma \\in S _ n } \\varepsilon ( \\sigma ) Q \\Big ( \\sum _ { j = 1 } ^ n ( \\ell _ 1 + 1 + m - \\ell _ j ) e _ j - \\sigma \\Big ( \\sum _ { j = 1 } ^ n j e _ j \\Big ) \\Big ) . \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} T '' _ \\theta = T ' _ \\theta + d K \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} \\hat { g } _ t = g _ { \\C ^ m } + \\lambda _ t ^ { 2 } e ^ { - t } g _ { Y , z _ t } , \\ ; \\ , \\hat { \\omega } _ t ^ \\natural = \\omega _ { \\C ^ m } + \\lambda _ t ^ { 2 } e ^ { - t } \\Psi _ t ^ * \\Phi ^ * \\omega _ F , \\\\ C ^ { - 1 } \\hat { g } _ t \\leq \\hat { g } ^ \\bullet _ t \\leq C \\hat { g } _ t . \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} B ( v , e ) & = \\begin{cases} 1 / d _ v , & \\ v \\in e , \\\\ 0 , & , \\end{cases} \\\\ w ( e ) & = \\frac { 1 } { C } \\prod _ { v \\in e } d _ v ^ { 1 / ( p - r ) } . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} & \\sum _ { d } \\frac { \\eta _ { \\nu } ( d ^ 2 \\ell _ 1 \\ell _ 2 ) } { d ^ s } = \\frac { \\zeta ( s ) \\zeta ( s + 2 \\nu ) \\zeta ( s - 2 \\nu ) } { \\zeta ( 2 s ) } \\prod _ { p \\big | \\frac { \\ell _ 1 \\ell _ 2 } { ( \\ell _ 1 , \\ell _ 2 ) ^ 2 } } \\frac { \\eta _ { \\nu } ( p ) } { 1 + p ^ { - s } } \\prod _ { p | ( \\ell _ 1 , \\ell _ 2 ) } \\frac { \\eta _ \\nu ( p ^ 2 ) - p ^ { - s } } { 1 + p ^ { - s } } . \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\| \\mathbf { u } \\| _ { k , p , \\widetilde { \\Omega } } = \\| u _ 1 \\| _ { k , p , \\widetilde { \\Omega } } + \\| u _ 2 \\| _ { k , p , \\widetilde { \\Omega } } ~ ~ ~ \\textrm { a n d } ~ ~ ~ | \\mathbf { u } | _ { k , p , \\widetilde { \\Omega } } = \\| D ^ { \\alpha } u _ 1 \\| _ { 0 , p , \\widetilde { \\Omega } } + \\| D ^ { \\alpha } u _ 2 \\| _ { 0 , p , \\widetilde { \\Omega } } , ~ ~ | \\alpha | = k . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} H _ K ( \\beta ) \\begin{cases} \\infty , & \\beta \\in [ 0 , \\beta _ P ( 2 ) ] , \\\\ \\in ( 0 , \\infty ) , & \\beta \\in ( \\beta _ P ( 2 ) , \\beta _ c ( 2 ) ) , \\\\ 0 , & \\beta \\geq \\beta _ c ( 2 ) . \\end{cases} \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} \\tau _ g ( b ( P , \\sigma _ - ) ) = \\frac { 1 } { 2 i \\sin \\varphi } . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} R ( r ) = A e ^ { \\eta { r } } r ^ { ( \\frac { 1 + \\tau } { 2 } ) } \\Big [ e ^ { - 2 \\eta { r } } r ^ { - ( 1 + \\tau + \\mathrm { a } ) } \\Big ] _ { - ( 1 + \\mathrm { a } E ^ { - 1 } ) } . \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} B _ \\lambda ( s ) = \\frac { \\pi ^ { n ( n + 1 ) / 2 } } { \\prod _ { m = 1 } ^ n ( m - 1 ) ! } \\ , i ^ { n k } \\ , 2 ^ { - n ( 2 n + 1 ) s + 3 n / 2 } \\ , \\gamma _ n \\Big ( ( 2 n + 1 ) s - \\frac 1 2 + k \\Big ) , \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} \\gamma _ n = \\frac { c n ( n + \\beta - 1 ) } { ( 1 - c ) ^ 2 } , \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{gather*} \\frac { \\eta \\big ( \\sqrt { - 3 } \\big ) ^ { 2 4 } } { \\eta ( { \\zeta _ 3 } ) ^ { 2 4 } } = \\frac { \\eta ( 2 { \\zeta _ 3 } ) ^ { 2 4 } } { \\eta ( { \\zeta _ 3 } ) ^ { 2 4 } } = - \\frac 1 { 2 5 6 } . \\end{gather*}"}
-{"id": "1328.png", "formula": "\\begin{align*} \\underset { t \\rightarrow \\infty } { } \\ , ( x _ { i } - x _ { j } ) = 0 , \\ , \\ , \\ , \\ , i , j = 1 , \\ldots , N . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\mathcal O _ { \\mathcal P \\left ( { \\bf w } \\right ) } ( k ) \\otimes \\mathcal O _ { \\mathcal P \\left ( { \\bf w } \\right ) } ( l ) = \\mathcal O _ { \\mathcal P \\left ( { \\bf w } \\right ) } ( k + l ) , ~ k , l \\in \\mathbb Z , \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} \\P _ x ( X _ \\alpha ( t ) = y ) ~ = ~ \\exp ( \\langle \\alpha , y - x \\rangle ) \\P _ x ( X ( t ) = y ) \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\sphericalangle \\big ( P ( y , r _ { x , i } ) , P _ y \\big ) & \\leq \\sphericalangle \\big ( P ( y , r _ { x , i } ) , P ( y , r _ { x , k } ) \\big ) + \\sphericalangle \\big ( P ( y , r _ { x , k } ) , P _ y \\big ) \\\\ & \\leq \\sum \\limits _ { l = 0 } ^ { k - i - 1 } \\sphericalangle \\big ( P ( y , r _ { x , i + l } ) , P ( y , r _ { x , i + l + 1 } ) \\big ) + \\varepsilon \\\\ & \\leq \\sum \\limits _ { l = 0 } ^ { \\infty } \\sphericalangle \\big ( P ( y , r _ { x , i + l } ) , P ( y , r _ { x , i + l + 1 } ) \\big ) + \\varepsilon . \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} \\Phi _ { \\omega } ( \\eta ) : = \\sum _ { x \\in \\eta } \\sum _ { y \\in \\eta \\setminus x } \\left [ a ( x - y ) - \\omega \\int _ { \\mathbb { R } ^ d } b ( z | x , y ) d z \\right ] \\ge - \\upsilon | \\eta | . \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\kappa _ t ( u ) = & \\ , \\frac { - a ^ 3 \\lambda _ v E R ( 2 \\lambda _ v Q + 2 F _ v D + E \\det ( \\nu , \\nu _ v , \\nu _ { v v } ) ) } { a ^ 4 \\lambda _ v ^ 2 E ^ 2 R ^ 2 } ( u , 0 ) \\\\ & + \\frac { ( a ^ 3 \\lambda _ v E ^ 2 R D ) ( a ^ 2 E F _ v R ) } { | a ^ 2 E ^ 2 R | \\ , | a ^ 4 \\lambda _ v ^ 2 E ^ 2 R ^ 2 | } ( u , 0 ) \\\\ = & \\ , \\frac { - 2 \\lambda _ v Q - F _ v D - E \\det ( \\nu , \\nu _ v , \\nu _ { v v } ) } { a \\lambda _ v E R } ( u , 0 ) . \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} A _ { n _ 1 , \\tilde { n } } = e ^ { 2 \\pi i \\sum _ { \\ell < \\abs { \\gamma } \\leq d } \\theta _ \\gamma ( n _ 1 , \\tilde { n } ) ^ \\gamma } , \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} x _ i ( \\tau , \\lambda _ i ) \\triangleq 1 + \\tau ( 1 + \\lambda _ i ) , \\ , i = 1 , 2 , \\ , a _ 1 = 1 , \\ , a _ 2 = \\tau , \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} \\psi _ 3 ( h _ 0 ) = \\big ( a ^ { ( i - j ) e _ 1 } v _ 1 , a ^ { ( i - j ) ( e _ 1 + e _ 2 ) } v _ 2 , \\dots , v _ { p - 1 } , v _ p \\big ) , \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} c _ k = c _ 0 - \\sum _ { l = 0 } ^ { k - 1 } \\frac { \\theta _ l } { 1 - \\theta _ l } b _ l \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} ( b / x ) ( b \\backslash z ^ { \\alpha } x ^ { \\alpha } \\cdot x ) = b ( z x \\backslash \\left [ z ( b \\backslash z ^ { \\alpha } x ^ { \\alpha } \\cdot x ) \\right ] ) \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} \\omega _ { \\infty } ( \\tilde k _ { \\theta } , k ' ) \\phi _ { \\mathbf h , \\infty } = e ^ { - \\sqrt { - 1 } ( k + 1 / 2 ) \\theta } \\det ( \\mathbf k ) ^ { k + 1 } \\phi _ { \\mathbf h , \\infty } \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} E ( V ) = V e V ^ { - 1 } = \\begin{bmatrix} S _ 0 ^ 2 & S _ 0 ( 1 + S _ 0 ) Q \\\\ S _ 1 D & 1 - S _ 1 ^ 2 \\end{bmatrix} . \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} \\langle \\xi _ x ^ { ( n ) } , \\xi _ x ^ { ( n ) } \\rangle = r ^ 2 - r t \\phi ( p _ x ^ { ( n ) } ) - r t \\phi ( p _ x ^ { ( n ) } ) + t ^ 2 \\phi ( p _ x ^ { ( n ) } ) = r ( t - r ) . \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} b ( U , n ) = \\sum ( \\dim \\alpha ) ^ { 2 } , \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} \\ensuremath { { \\dot { x } _ { i } } ( t ) = { u _ { i } } ( t ) } , \\quad \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} C ^ { - 1 } & = \\frac { 1 } { 4 } \\begin{pmatrix} 3 & 2 & 1 \\\\ 2 & 4 & 2 \\\\ 1 & 2 & 3 \\end{pmatrix} & C ( z ) & = \\begin{pmatrix} [ 2 ] _ z & - 1 & 0 \\\\ - 1 & [ 2 ] _ z & - 1 \\\\ 0 & - 1 & [ 2 ] _ z \\end{pmatrix} & \\widetilde { C } ( z ) & = \\frac { 1 } { [ 4 ] _ z } \\begin{pmatrix} [ 3 ] _ z & [ 2 ] _ z & 1 \\\\ [ 2 ] _ z & ( [ 2 ] _ z ) ^ 2 & [ 2 ] _ z \\\\ 1 & [ 2 ] _ z & [ 3 ] _ z \\end{pmatrix} . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} S _ Y ( n , m ; x ) = 0 , m = n + 1 , n + 2 , \\ldots \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} U ^ T A U & = \\begin{pmatrix} 1 . 9 6 1 3 & 0 . 0 5 0 7 & 0 . 7 5 1 0 \\\\ 0 . 0 5 0 7 & 2 . 8 5 6 6 & 1 . 6 6 6 6 \\\\ 0 . 7 5 1 0 & 1 . 6 6 6 6 & 3 . 1 4 8 6 \\end{pmatrix} . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{gather*} \\frac { 1 } { 2 \\pi { \\rm i } } \\frac { { \\rm d } } { { \\rm d } \\tau } \\log \\Delta ( \\tau ) = E _ 2 ( \\tau ) , \\end{gather*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\partial _ t \\begin{pmatrix} \\zeta \\\\ { \\bf v } _ \\beta \\end{pmatrix} + J \\ ; H ( \\zeta , { \\bf v } _ \\beta ) = 0 , \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} q ( B ^ G ) \\cap N _ D B = q ( B ^ D ) . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} g _ { t } ( z ) = z + \\sum _ { n \\le 0 } g _ { n } ( t ) z ^ { n } \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} C = [ C _ { s } - \\max _ { k } C _ { e , k } ] ^ { + } . \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\langle x \\rangle \\cap \\langle y ^ g \\rangle = 1 \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} q a - \\beta - 1 - ( - | w | ) = q ( a + 1 ) + ( d - \\beta ) > 0 \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} \\Theta _ { \\ell , n } : = \\bigl \\{ ( a _ 1 , \\cdots , a _ { n } ) \\bigm | 0 \\leq a _ 1 \\leq \\cdots \\leq a _ { n } \\leq \\ell - n , a _ i \\in \\Z , \\forall \\ , i \\bigr \\} . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} u ^ m _ { r r } ( r , t ) = \\frac { C ^ m m } { a ^ 2 ( m - 1 ) ^ 2 } \\zeta ^ m ( t ) \\eta ^ 2 ( t ) F ^ { \\frac 1 { m - 1 } - 1 } \\ , . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t U ( t , s ) ~ u _ s = A ( t ) U ( t , s ) ~ u _ s \\end{array} \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} \\varphi _ k ( x ) : = \\begin{cases} - \\sum _ { i = 1 } ^ { j - 1 } f _ k ( T ^ { - i } x ) & \\textrm { i f } x \\in I _ { j , k } \\textrm { f o r } j = 2 , \\dots , 2 a _ k ; \\\\ 0 & \\textrm { e l s e w h e r e } . \\end{cases} \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} C = \\left ( \\begin{array} { c c } A & - B \\\\ B & A \\end{array} \\right ) . \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} y ^ k - x ^ { k - 1 } = \\frac { 1 + \\varphi } { \\varphi } ( y ^ k - x ^ k ) = \\varphi ( y ^ k - x ^ k ) \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} E ( D ) \\ , = \\ , \\begin{bmatrix} I - e ^ { - D D ^ * } & - D e ^ { - \\frac 1 2 D ^ * D } \\\\ \\\\ - \\frac { I - e ^ { - D ^ * D } } { D ^ * D } e ^ { - \\frac 1 2 D ^ * D } D ^ * & e ^ { - D ^ * D } \\end{bmatrix} . \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{gather*} L ( f _ { 3 6 } , 1 ) = \\ ( \\frac { - 1 } { \\sqrt { - 3 } } \\ ) \\mathcal G _ { 1 , ( 1 ; 3 ) } ^ \\ast \\big ( { - } 1 / \\sqrt { - 3 } \\big ) , \\end{gather*}"}
-{"id": "864.png", "formula": "\\begin{align*} a _ m * a _ n : = ( m + 1 ) a _ m a _ n . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} 2 m & = \\sum _ { v \\in V ' } d ( v ) + \\sum _ { v \\in V '' } d ( v ) \\\\ & = d ( w ) + \\sum _ { v \\in V ' \\setminus \\{ w \\} } d ( v ) + \\sum _ { v \\in V '' } d ( v ) \\\\ & \\geq 1 + 3 | V ' \\setminus \\{ w \\} | + 4 | V '' | \\\\ & = 1 + 3 \\big ( | V ' | - 1 \\big ) + 4 \\big ( n - | V ' | \\big ) \\\\ & = 4 n - | V ' | - 2 . \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N \\tbinom { N } { k } ( - 1 ) ^ k \\dot { \\varphi } _ n ( i + j + k ) = \\begin{cases} \\tbinom { N } { n - i - j } ( - 1 ) ^ { n - i - j } & n - N \\leq i + j \\leq n \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\mathcal { D } _ \\alpha ^ \\Delta = \\{ k \\in \\mathcal { K } _ \\alpha : \\Psi k \\in \\mathcal { K } _ \\alpha \\} . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} & c _ H ( \\psi _ g ( x _ 1 ) , \\ldots , \\psi _ g ( x _ n ) ) = A ( \\hat { g } ) ( c _ K ( x _ 1 , \\ldots , x _ n ) ) . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} \\epsilon _ p = \\left ( \\frac { D } { \\pi _ p } \\right ) = \\left ( \\frac { D } { p } \\right ) = \\begin{cases} w _ p & p \\mid N / M , \\\\ - w _ p & p \\mid M . \\end{cases} \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} \\begin{bmatrix} \\ell & 0 & - n & 0 \\\\ 0 & \\ell & 0 & - m \\\\ 0 & 0 & 1 & 1 \\\\ \\end{bmatrix} \\begin{bmatrix} m _ 1 \\\\ m _ 2 \\\\ \\epsilon \\\\ \\eta \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\\\ d - \\ell \\end{bmatrix} \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} t _ M \\to 0 \\mbox { a s $ M \\to \\infty $ } \\mbox { a n d } U _ M ( x , t ) = U ( x , t ) \\forall x \\in \\R ^ n , \\ , \\ , t \\geq t _ M . \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} T ( K ) = \\sum _ { 1 \\leq k \\leq K } \\exp \\Big ( 2 \\pi i \\big ( \\xi P ( j k ) + m ( \\varphi _ 1 ( j k ) - \\tau \\psi ( j k ) ) \\big ) \\Big ) . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} 0 < \\alpha < \\frac 1 { m - 1 } \\ , , \\beta = \\frac { \\alpha ( m - 1 ) + 1 } 2 \\ , . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} u ^ { \\varepsilon } = 0 \\ ; \\partial \\Omega \\times \\left ( 0 , T \\right ) , u ^ { \\varepsilon } \\left ( x , T \\right ) = u _ { f } ^ { \\varepsilon } \\left ( x \\right ) \\ ; \\Omega . \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} \\begin{aligned} & A ( z ) \\sum _ n \\frac { c _ n \\mu _ n ^ { 1 / 2 } } { z - t _ n } = G _ 1 ( z ) S _ 1 ( z ) , \\\\ & A ( z ) \\sum _ n \\frac { G ( t _ n ) \\bar c _ n } { A ' ( t _ n ) \\mu _ n ^ { 1 / 2 } ( z - t _ n ) } = G _ 2 ( z ) S _ 2 ( z ) . \\end{aligned} \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} D _ { i j } ^ { 1 } = \\frac { 1 } { 2 } M _ { k k i j } ^ { ( 2 ) } - \\frac { 1 } { 4 } M _ { k k l l } ^ { ( 2 ) } \\delta _ { i j } D _ { i j } ^ { 2 } = \\frac { 1 } { 2 } M _ { i j k k } ^ { ( 2 ) } - \\frac { 1 } { 4 } M _ { k k l l } ^ { ( 2 ) } \\delta _ { i j } \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} Q _ { \\alpha _ 2 \\alpha _ 1 } ( t ) = Q _ { \\alpha _ 2 \\alpha _ 1 } ( t ; B ^ \\Delta _ \\upsilon ) , t < T ( \\alpha _ 2 , \\alpha _ 1 ; B ^ \\Delta _ \\upsilon ) = T ( \\alpha _ 2 , \\alpha _ 1 ) \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} \\int _ { Q ( r _ { k - 1 } ) } \\left | | u | ^ 2 - [ | u | ^ 2 ] _ { B ( r _ { k - 1 } ) } \\right | | u | d x d t \\leq C r ^ { 1 + d + 2 / d } _ { k - 1 } \\epsilon _ 1 ^ { 1 + 1 / d } , \\ \\ \\ k = 2 , \\ , 3 . \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\forall X , Y \\in \\mathfrak { g } , \\ , \\ , \\lambda > 0 , \\ , \\ , [ D _ { \\lambda } X , D _ { \\lambda } Y ] = D _ { \\lambda } [ X , Y ] . \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} S _ 1 ( w \\ , | \\ , a ) = 2 \\sin ( \\pi w / a ) . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} { \\rm D } h ( x ) [ \\xi ] = \\langle { \\rm g r a d } \\ , h ( x ) , \\xi \\rangle _ x \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} S ( n , m ; x ) : = \\dfrac { \\Delta ^ m I _ n ( x ) } { m ! } , m \\leq n , \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} Z = \\bigcap _ { n \\geq 1 } \\overline { \\bigcup _ { i \\geq n } \\Phi _ i ^ { - 1 } ( 0 ) } , \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { 4 \\pi } \\int \\limits _ { \\partial \\sum _ { n } } \\left ( f _ 0 ( M ) \\right ) ' _ { \\bar { n } ( M ) } \\ , \\frac { 1 } { \\rho _ { M _ 0 } ( M ) } \\ , d S ( M ) \\right | \\leq C \\alpha _ { n } \\cdot 2 ^ { n } \\cdot 2 ^ { - n } = C \\alpha _ { n } \\\\ \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} F _ { \\chi } ( Z ) = \\sum _ { B = \\left ( \\begin{smallmatrix} n & r / 2 \\\\ r / 2 & m \\end{smallmatrix} \\right ) } \\left ( \\sum _ { \\substack { a \\mid \\gcd ( n , r , m ) , \\\\ \\gcd ( a , N ) = 1 } } a ^ k \\chi ( a ) c \\left ( \\frac { 4 n m - r ^ 2 } { a ^ 2 } \\right ) \\right ) e ^ { 2 \\pi \\sqrt { - 1 } \\mathrm { T r } ( B Z ) } . \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} G ( z = 1 \\ , | \\ , \\tau ) = & 1 , \\\\ G ( z + 1 \\ , | \\ , \\tau ) = & \\Gamma \\Big ( \\frac { z } { \\tau } \\Bigr ) \\ , G ( z \\ , | \\ , \\tau ) , \\\\ G ( z + \\tau \\ , | \\ , \\tau ) = & ( 2 \\pi ) ^ { \\frac { \\tau - 1 } { 2 } } \\ , \\tau ^ { - z + \\frac { 1 } { 2 } } \\ , \\Gamma ( z ) \\ , G ( z \\ , | \\ , \\tau ) . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} \\psi _ 3 ( x _ i ) = \\big ( b ^ { a ^ { i e _ 1 } } , b ^ { a ^ { i ( e _ 1 + e _ 2 ) } } , \\dots , b , b \\big ) \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} ( j - i + 1 ) f _ i g _ j = j f _ 1 g _ { i + j - 1 } . \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} 0 = \\delta ^ d \\alpha ^ d \\eta = ( - 1 ) ^ d \\Sigma ^ d ( \\nu ) \\gamma ^ { d + 1 } \\xi ^ d \\eta = ( - 1 ) ^ d \\Sigma ^ d ( \\nu ) \\gamma ^ { d + 1 } \\phi . \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} X _ n = \\{ x \\in \\mathrm { E n d } ( V ) \\mid \\mathrm { r a n k } ( x ) < n \\} = S _ 1 \\cup \\cdots \\cup S _ n \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} \\gamma _ { N ' } : = \\sqrt { \\frac { \\ln N ' } { N ' } } . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} n _ i = | V ' _ i | \\geq \\log ^ 6 { n } \\geq w ^ 3 \\enspace \\enspace m _ i = | X ' _ i | \\leq \\log ^ 2 { n } . \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { D _ \\varepsilon } \\theta _ k ^ 2 > C _ { \\max } ( \\alpha , \\beta ) \\sqrt { \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 4 } } \\Rightarrow \\mathbb { P } _ \\theta ( \\Delta _ { \\alpha , \\varepsilon } = 0 ) \\leq \\beta . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\displaystyle w _ { j j } & = \\Big ( \\frac { 2 h _ i h _ { i j } } { ( \\beta - h ) ^ 2 } \\Big ) _ j + \\Big ( \\frac { 2 h ^ 2 _ i h _ j } { ( \\beta - h ) ^ 3 } \\Big ) _ j \\\\ \\displaystyle & = \\frac { 2 h ^ 2 _ { i j } } { ( \\beta - h ) ^ 2 } + \\frac { 2 h _ i h _ { i j j } } { ( \\beta - h ) ^ 2 } + \\frac { 8 h _ i h _ j h _ { i j } } { ( \\beta - h ) ^ 3 } + \\frac { 2 h ^ 2 _ i h _ { j j } } { ( \\beta - h ) ^ 3 } + \\frac { 6 h ^ 2 _ i h ^ 2 _ j } { ( \\beta - h ) ^ 4 } . \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} S _ { d , k } ( n ) = \\sum _ W \\sum _ { \\ell } \\binom { W } { \\ell } S _ { d + 1 , k } \\big ( n - \\ell ( d + 1 ) , W - \\ell \\big ) . \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} c = \\lim _ { | n | \\to \\infty } \\tilde { \\phi } ( n ) \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} \\mathcal A _ 1 ^ + ( n ) : = \\{ h \\in \\mathcal A _ 1 ( n ) : c \\neq 0 \\mathrm { o r d } _ p ( c ) \\} , \\mathcal A _ 1 ^ - ( n ) : = \\{ h \\in \\mathcal A _ 1 ( n ) : c = 0 \\mathrm { o r d } _ p ( c ) \\} , \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\tilde { \\phi } ( m + n ) = c _ + + ( - 1 ) ^ { | m | + | n | } c _ - + f _ \\phi ^ { [ N ] } ( U ( m , n ) ) , \\quad \\forall m , n \\in \\N ^ N . \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\varphi _ q ^ { ( p ) } \\circ \\Delta _ t \\circ { \\rm g w } _ t ( x ) = 0 , \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} r _ A : = \\sum _ { i \\in N _ A } t _ i & & r _ B : = 1 - r _ A = \\sum _ { i \\in N _ B } t _ i \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{gather*} L \\left ( \\psi _ { v ' } , s - \\frac 1 2 \\right ) = \\frac { 1 } { 1 - ( - 1 ) ^ { b / 2 } ( a + b { \\rm i } ) p ^ { - s } } . \\end{gather*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j _ 1 \\leq \\cdots \\leq j _ { t + 1 } \\leq n } { d _ { j _ 1 } + \\cdots + d _ { j _ { t + 1 } } - 1 \\choose n - 1 } = \\begin{cases} \\operatorname { r k } F _ { n - t - 1 } = f _ { n - t - 1 } & t \\geq n - m - 1 , \\\\ 0 & t < n - m - 1 . \\end{cases} \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} E q 1 & : \\rho _ { 0 } ^ { 3 } v _ { 0 } ^ { 2 } + 2 \\omega \\frac { \\rho _ { 0 } v _ { 0 } } { K _ { 0 } } = 4 \\mathfrak { S } _ { 0 } \\\\ E q 2 & : \\frac { \\rho _ { 1 } } { \\rho _ { 0 } v _ { 0 } } - \\omega \\lbrack J _ { 3 } \\frac { 1 } { \\rho _ { 0 } ^ { 2 } v _ { 0 } } + J _ { 4 } \\rho _ { 0 } v _ { 0 } ] \\ = J _ { 2 } \\ \\\\ E q 3 & : \\rho _ { 0 } ( \\rho _ { 1 } ^ { 2 } - 2 h ) + \\frac { \\omega ^ { 2 } } { \\rho _ { 0 } } - \\frac { \\omega } { 2 K _ { 0 } } \\rho _ { 0 } v _ { 0 } = J _ { 1 } \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} I ( \\vec x ) \\geq I ( \\vec y ) + ( \\vec x - \\vec y , \\vec s ) = ( \\vec x , \\vec s ) , \\ , \\ , \\ , \\forall \\ , \\vec s \\in \\partial I ( \\vec y ) , \\ , \\ , \\ , \\forall \\ , \\vec x , \\vec y \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} \\chi _ 1 \\ ! \\left ( - \\frac { c } { \\gcd ( m , c ) } \\right ) \\chi _ 2 \\ ! \\left ( \\frac { m } { \\gcd ( m , c ) } \\right ) = 0 , \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { f ( x + \\varepsilon h ) - f ( x ) } { \\varepsilon } = \\frac { d } { d \\varepsilon } f ( x + \\varepsilon h ) , \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} ( x _ 1 ^ { n _ 1 } , \\ldots , x _ d ^ { n _ d } ) ^ * = ( x _ 1 ^ { n _ 1 } , \\ldots , x _ d ^ { n _ d } ) ^ F \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} \\kappa _ n ( \\delta ) = \\frac { 1 } { 8 } \\sum \\limits _ { m = 0 } ^ \\infty \\frac { \\bigl ( - ( \\delta - 1 / 2 ) \\bigr ) ^ m } { m ! } \\ , 3 2 ^ { ( m + n ) / 2 } \\ , \\Gamma \\bigl ( \\frac { m + n } { 2 } \\bigr ) \\ , { \\bf E } \\bigl [ C _ 2 ^ { - ( m + n ) / 2 } \\bigr ] . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} A _ 1 ' ( z ) = A _ 1 ( z ) \\cdot E ( z ) + z E ( z ) - \\frac { z ^ 2 } { 2 ! } , \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} E _ { v _ 0 , \\rm i } : = E _ { v _ 1 , \\rm i } \\cup E _ { v _ 2 , \\rm i } E _ { v _ 0 , \\rm t } : = E _ { v _ 1 , \\rm t } \\cup E _ { v _ 2 , \\rm t } \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} f ( \\beta X ) = \\beta X ( 1 + a \\beta ^ { 1 - q } X ^ { q ( 1 - q ) } + \\beta ^ { 2 ( q + 1 ) } X ^ { 2 ( q - 1 ) } ) , \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} ( i + j + 1 ) \\mathfrak { d } _ 1 ^ { N + 1 } \\dot { \\phi } ( i + j ) = & \\mathfrak { d } _ 1 ^ { N } \\dot { \\psi } ( i + j ) + N \\mathfrak { d } _ 1 ^ { N } \\dot { \\phi } ( i + j + 1 ) , \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} d _ B ( E ( \\pi _ 1 , s _ 1 ) , E ( \\pi _ 2 , s _ 2 ) ) = d _ B ( \\pi _ 1 , \\pi _ 2 ) . \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 1 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\mathcal { Q } _ { t + 1 } ( x _ { t } , 1 ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ { t j } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} \\begin{cases} 0 < \\underbar s _ 1 ^ { n - 1 } \\leq \\underbar s _ 1 ^ n \\leq \\bar s _ 1 ^ n \\leq \\bar s ^ { n - 1 } _ 1 \\le \\bar B _ 1 \\cr 0 < \\underbar s _ 2 ^ { n - 1 } \\leq \\underbar s _ 2 ^ n \\leq \\bar s _ 2 ^ n \\leq \\bar s _ 2 ^ { n - 1 } \\le B _ 2 \\end{cases} \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} { \\psi _ { \\rm { 1 } } } \\left ( { { v _ n } , { v _ { n + 1 } } } \\right ) = \\delta \\left ( { { { \\tilde \\theta } _ { n + 1 } } - { { [ { { \\tilde \\theta } _ n } + \\vartheta ] } _ { 2 \\pi } } } \\right ) \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} I _ 0 : { \\mathcal C } ^ \\infty ( { \\mathcal X } ) [ 1 ] \\to { \\mathcal C } ^ \\infty ( { \\mathcal M } ) [ 1 ] , \\ \\ \\ I _ 0 = \\sum _ { i \\in S } \\chi _ i \\ , \\sigma _ i ^ * \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\alpha _ n \\nu _ c ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ n ) p ^ { - n / 2 } \\mathbf 1 _ { \\Z _ p } ( x p ^ n ) \\int _ { \\Q _ p } \\psi ( - 2 x p ^ n y - c p y ^ 2 ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) d y . \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{gather*} X \\beta S ( Y ) \\alpha Z = 1 , S ( P ) \\alpha Q \\beta R = 1 . \\end{gather*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\gamma _ i = s _ { i _ 1 } \\cdots s _ { i _ { k - 1 } } ( \\alpha _ { i _ k } ) \\ \\ i _ k = i . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} \\O _ S = \\O _ { X , \\iota ( 0 ) } \\subset \\O _ { Y , \\iota ( 0 ) } . \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} F _ k = f _ k ^ { - 1 } ( \\{ \\delta \\} ) \\cap Z _ k \\cap B \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\delta ^ s ( B _ 1 ) + \\delta ^ s ( B _ 2 ) + \\delta ^ s ( B _ 3 ) + \\delta ^ s ( B _ 4 ) = \\delta ^ s ( B _ 5 ) + \\delta ^ s ( B _ 6 ) + \\delta ^ s ( B _ 7 ) + \\delta ^ s ( C ) . \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} \\varrho : = \\varrho \\left ( \\beta \\right ) = \\sqrt { \\frac { \\underline { M } } { T } \\log \\left ( \\log ^ { \\kappa } \\left ( \\gamma \\left ( T , \\beta \\right ) \\right ) \\right ) } > 0 , \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} \\mathbf L _ { k j } f ( \\vect y ) & = \\mathbf L _ { k j } ( f \\circ T \\circ T ^ { - 1 } ) ( \\vect y ) \\\\ & = \\sum _ { q = 1 } ^ n \\left [ y _ k \\frac { \\partial \\theta _ q } { \\partial y _ j } - y _ j \\frac { \\partial \\theta _ q } { \\partial y _ k } \\right ] \\frac { \\partial f \\circ T } { \\partial \\theta _ q } ( T ^ { - 1 } ( \\vect y ) ) . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} \\left [ \\prod _ { i = 1 } ^ N \\frac { q _ i + 1 } { q _ i - 1 } \\right ] \\left ( \\| \\phi \\| _ { c b } - | c _ + | - | c _ - | \\right ) , \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} v _ { 1 2 } \\ & = \\ ( l _ { 1 2 } , 0 , d _ { 1 1 2 } , d _ { 2 1 2 } ) , \\\\ v _ { 1 1 } \\ & = \\ v _ { 1 2 } \\ + \\ ( 0 , 0 , l _ { 2 1 } + l _ { 2 2 } , 0 ) . \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} E _ { 1 } ^ { ( b ) } F _ { 1 } ^ { ( a ) } 1 _ { ( k , l ) } ( T ) = \\sum _ { R \\subseteq T ^ { 1 } , | R | = a } \\sum _ { S \\subseteq T ^ { 2 } \\cup R , | S | = b } c _ { 1 , S } c _ { 2 , R } T \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} e ^ { \\alpha } e ^ { \\beta } = \\epsilon ( \\alpha , \\beta ) e ^ { \\alpha + \\beta } \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} L _ { \\mathbb { T } } \\colon \\mathbb { R } ^ { n + m } \\to \\mathbb { R } ^ { n + m } , \\ , L _ { \\mathbb { T } } ( w ) = ( m _ i ^ { ( k ) } ( w ) ) _ { i , k } . \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} f = g h , \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\sum _ { \\substack { n m \\leqslant N \\\\ ( n , m ) = 1 } } \\frac { r ( n ) ^ 2 r ( m ) \\omega ( n ) \\omega ' ( m ) } { \\sqrt { m } } = \\big ( 1 + o ^ { \\star } ( 1 ) \\big ) \\prod _ p \\left ( 1 + r ( p ) ^ 2 \\omega ( p ) + \\frac { r ( p ) } { \\sqrt { p } } \\omega ' ( p ) \\right ) . \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\langle \\nabla \\varrho , \\nabla \\varphi \\rangle \\ , d x = \\int _ { \\mathbb { R } ^ d } \\langle \\varrho b , \\nabla \\varphi \\rangle \\ , d x \\forall \\varphi \\in C _ 0 ^ \\infty ( \\mathbb { R } ^ d ) . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} \\bigl \\| | \\nabla \\varrho / \\varrho | \\bigr \\| _ { L ^ 2 ( \\mu ) } ^ 2 = \\int _ { \\mathbb { R } ^ d } \\Bigl | \\frac { \\nabla \\varrho } { \\varrho } \\Bigr | ^ 2 \\varrho \\ , d x \\le \\int _ { \\mathbb { R } ^ d } | b | ^ 2 \\varrho \\ , d x = \\bigl \\| | b | \\bigr \\| _ { L ^ 2 ( \\mu ) } ^ 2 . \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} h \\left ( D _ 1 \\cdots D _ n \\right ) = \\sum _ { i = 1 } ^ n h ( D _ i ) = \\sum _ { i = 1 } ^ n h _ i \\iota \\left ( D _ 1 \\cdots D _ n \\right ) = \\sum _ { i = 1 } ^ n \\iota ( D _ i ) + \\sum _ { 1 \\leq j < k \\leq n } m \\left ( h _ k , \\partial h _ j \\right ) , \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} \\delta _ { \\Phi } ^ * ( \\eta _ 1 , & \\ldots , \\eta _ { n - 1 } ) = \\\\ & ( \\phi _ 1 ^ * \\eta _ 1 ^ * , \\mbox { } \\phi _ 2 ^ * \\eta _ 2 ^ * - \\eta _ 1 ^ * \\phi _ 1 ^ * , \\ldots , \\phi _ { n - 1 } ^ * \\eta _ { n - 1 } ^ * - \\eta _ { n - 2 } ^ * \\phi _ { n - 2 } ^ * , - \\eta _ { n - 1 } ^ * \\phi _ { n - 1 } ^ * ) \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m } x _ j ^ { \\downarrow } = \\prod _ { j = 1 } ^ { m } y _ j ^ { \\downarrow } . \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} 2 \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } x \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } v \\ , \\sqrt { 1 + v ^ { 2 m } } \\phi \\ , v \\ , \\partial _ v \\phi & = - \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } x \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } v \\ , \\sqrt { 1 + v ^ { 2 m } } \\phi ^ 2 \\\\ & - \\int _ { \\mathbb { T } } \\ ! \\mathrm { d } x \\int _ { \\mathbb { R } } \\ ! \\mathrm { d } v \\ , \\frac { m v ^ { 2 m } } { \\sqrt { 1 + v ^ { 2 m } } } \\phi ^ 2 \\ , , \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} T _ w T _ s = \\left \\{ \\begin{aligned} & T _ { w s } , & \\mbox { i f } \\ell ( w s ) > \\ell ( w ) ; \\\\ & ( q - 1 ) T _ w + q T _ { w s } , & \\mbox { o t h e r w i s e } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} \\vec { d } ( x , y ) = ( d _ 1 ( x _ 1 , y _ 1 ) , . . . , d _ N ( x _ N , y _ N ) ) \\in \\N ^ N . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} \\mathcal { Q } _ t ( x _ { t - 1 } ) = \\mathbb { E } _ { \\xi _ t } [ \\mathfrak { Q } _ t \\left ( x _ { t - 1 } , \\xi _ t \\right ) ] , \\ ; t = 2 , \\ldots , T _ { \\max } . \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} & ( i ) I \\nabla _ \\lambda T - I ! _ \\lambda T = \\lambda ( 1 - \\lambda ) ( I - T ) ( T \\nabla _ \\lambda I ) ^ { - 1 } ( I - T ) . \\\\ & ( i i ) T ^ { \\frac { 1 } { 2 } } \\big ( I \\nabla _ \\lambda T ^ { - 1 } - I ! _ \\lambda T ^ { - 1 } \\big ) T ^ { \\frac { 1 } { 2 } } = \\lambda ( 1 - \\lambda ) ( T - I ) ( I \\nabla _ \\lambda T ) ^ { - 1 } ( T - I ) . \\\\ & ( i i i ) \\quad ( I ! _ \\lambda T ) ^ { - \\frac { 1 } { 2 } } ( I \\nabla _ \\lambda T ) ( I ! _ \\lambda T ) ^ { - \\frac { 1 } { 2 } } - I = \\lambda ( 1 - \\lambda ) ( I - T ) T ^ { - 1 } ( I - T ) \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} ( { \\bf Q } D _ { { \\bf x } , t } ^ + { \\bf u } - { \\bf Q } F _ { \\Gamma } D _ { { \\bf x } , t } ^ + { \\bf u } ) + { \\bf Q } T _ G { \\bf D } p = { \\bf Q } T _ G ( { \\bf f } ) . \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} \\ ; \\ ; \\mathop { \\inf } \\limits _ { x \\in X } \\mathop { \\sup } \\limits _ { u \\in U } \\left \\{ { { F _ { u } } \\left ( { x , { 0 _ { u } } } \\right ) - \\left \\langle { { x ^ { \\ast } } , x } \\right \\rangle } \\right \\} = \\mathop { \\sup } \\limits _ { \\left ( { u , y _ { ^ { u } } ^ { \\ast } } \\right ) \\in \\Delta } - { F _ { u } ^ { \\ast } } \\left ( { { x ^ { \\ast } } , { y _ { u } ^ { \\ast } } } \\right ) . \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} 1 - e ^ { - 2 \\beta _ P ( d ) } = p _ c ^ b ( d ) \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\delta ( [ X , Y ] ) = [ \\delta ( X ) , 1 \\otimes Y + Y \\otimes 1 ] + [ 1 \\otimes X + X \\otimes 1 , \\delta ( Y ) ] , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\forall X , Y \\in \\bf g , \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} P ( m | n ) = \\{ A = ( A _ \\alpha ) _ { \\alpha \\in \\Phi } \\ , | \\ , A _ \\alpha \\in \\mathbb N \\mbox { i f } \\bar { \\alpha } = \\bar 0 \\mbox { a n d } A _ \\alpha \\in \\{ 0 , 1 \\} \\mbox { i f } \\bar { \\alpha } = \\bar 1 \\} . \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} e ( G [ A ] ) + e ( G [ B ] ) \\le 3 t = 6 6 \\delta n ^ 2 ~ \\Rightarrow ~ e ( G _ 1 [ A , B ] ) \\ge e ( G _ 1 ) - 6 6 \\delta n ^ 2 \\ge \\frac { n ^ 2 } { 4 } - 8 8 \\delta n ^ 2 , \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} e ^ { - 2 t \\zeta _ j ^ { ( k ) } } P _ { I , J } \\left ( \\zeta , \\varphi \\right ) = \\frac { \\vert \\Delta _ { I , J } \\vert ^ 2 } { \\vert \\Delta _ j ^ { ( k ) } \\vert ^ 2 } < C e ^ { - t \\delta } , \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} \\binom { m + 1 } { m - s + 1 } = \\binom { m + 1 } { s } = \\binom { m + 1 } { m + 1 - ( m - s + 1 ) } \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\Xi _ { \\imath } = \\begin{cases} \\xi ( \\imath ) & \\ 1 \\leq \\imath \\leq n - 1 , \\\\ \\frac { 1 } { 2 } ( \\xi ( n - 1 ) + \\xi ( n ) ) + 1 & \\ \\imath = n \\ \\ \\flat = > , \\\\ \\frac { 1 } { 2 } ( \\xi ( n - 1 ) + \\xi ( n ) ) & \\ \\imath = n \\ \\ \\flat = < , \\\\ \\xi ( \\imath - 1 ) & \\ n + 1 \\leq \\imath \\leq 2 n - 1 . \\end{cases} \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} ( \\rho , u ) ( x , t ) = \\left \\{ \\begin{array} { l l } ( \\rho _ - , u _ - + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x < x _ { 1 } ^ { A B } ( t ) , \\\\ ( \\rho _ * , v _ * + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x _ { 1 } ^ { A B } ( t ) < x < x _ { 2 } ^ - ( t ) , \\\\ R _ { 2 } ^ { A B } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x _ { 2 } ^ - ( t ) < x < x _ { 2 } ^ + ( t ) , \\\\ ( \\rho _ + , u _ + + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x > x _ { 2 } ^ + ( t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} \\lambda = \\frac { b k ( k - 1 ) } { ( d + 1 ) d } \\quad r = \\frac { d \\lambda } { k - 1 } . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} f _ \\phi = \\sum _ { I \\subset [ N ] } f _ \\phi ^ I \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} \\underbar r _ 2 = \\frac { b _ { 0 , \\inf } - b _ { 1 , \\sup } \\bar r _ 1 - l \\frac { \\chi _ 2 } { d _ 3 } \\bar r _ 2 } { b _ { 2 , \\sup } - l \\frac { \\chi _ 2 } { d _ 3 } } , \\bar r _ 2 = \\frac { b _ { 0 , \\sup } - b _ { 1 , \\inf } \\underbar r _ 1 - l \\frac { \\chi _ 2 } { d _ 3 } \\underbar r _ 2 } { b _ { 2 , \\inf } - l \\frac { \\chi _ 2 } { d _ 3 } } . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} F _ t ( m _ { k ( i , r ) , r } ^ { ( i ) } ) ^ T = [ \\underline { W _ { k ( i , r ) , r } ^ { ( i ) } } ] ^ T = \\underline { m _ { k ( i , r ) , r } ^ { ( i ) } } \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} g ( p ) = \\frac { 1 } { p } \\sum _ { \\ell = 1 } ^ { p } \\frac { 1 } { s + ( r + \\ell ) / p } \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} \\Delta _ { y _ 1 , \\ldots , y _ m } ^ m f ( x ) = ( - 1 ) ^ m f ( x ) + \\sum _ { k = 1 } ^ m ( - 1 ) ^ { m - k } \\sum _ { I _ m ( k ) } f ( x + y _ { i _ 1 } + \\cdots + y _ { i _ k } ) . \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} \\rho ( \\int _ { T } x ^ * ( t ) d \\mu ( t ) - x ^ * ) & \\leq \\rho ( \\int _ { T } \\tilde { x } _ n ^ * ( t ) d \\mu ( t ) - x ^ * ) + \\int _ { A _ n } \\| \\tilde { x } _ n ^ * ( t ) \\| d \\mu ( t ) \\\\ & + \\int _ { A _ n } \\| y ^ * ( t ) \\| d \\mu ( t ) \\leq \\varepsilon / 3 + \\varepsilon / 3 + \\varepsilon / 3 = \\varepsilon , \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} \\sigma ^ { \\beta } & = \\sigma , \\ \\tau ^ { \\beta } = \\rho \\tau \\ \\\\ \\sigma ^ { \\gamma } & = \\rho \\sigma , \\ \\tau ^ { \\gamma } = \\tau . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} Y _ { L _ { \\infty } } : = \\Delta ( G _ K ) _ { G _ { L _ { \\infty } } } = \\Z _ p \\widehat \\otimes _ { \\Lambda ( G _ { L _ { \\infty } } ) } \\Delta ( G _ K ) \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} m = Y _ { n , r } Y _ { n , r + 2 p } \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( 1 \\right ) = \\frac { z } { z - 1 } . \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} S _ { 1 } = c \\big \\langle ( \\nabla _ r \\mathsf { f } ) \\cdot ( \\nabla _ r w ) ( \\mathsf { f } \\cdot w ) | w | ^ { q - 2 } \\big \\rangle & + c ( q - 2 ) \\big \\langle ( \\nabla _ r \\mathsf { f } ) \\cdot ( \\nabla | w | ) ( \\mathsf { f } \\cdot w ) w _ r | w | ^ { q - 3 } \\big \\rangle , \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\zeta ^ 2 ( x , t ) & \\leq \\int _ { \\mathbb R } | \\zeta \\zeta _ x | d x = \\frac { 1 } { \\sqrt { \\mu | c | } } \\int _ { \\mathbb R } \\sqrt { \\mu | c | } | \\zeta \\zeta _ x | d x \\leq \\frac { 1 } { 2 \\sqrt { \\mu | c | } } \\int _ { \\mathbb R } ( \\zeta ^ 2 + \\mu | c | \\zeta _ x ^ 2 ) d x \\\\ & \\leq \\frac { 1 } { ( 1 - \\gamma ) \\sqrt { \\mu | c | } } H ( ( \\zeta ( \\cdot , t ) , v ( \\cdot , t ) ) \\leq \\frac { 1 } { ( 1 - \\gamma ) \\sqrt { \\mu | c | } } H ( \\zeta _ 0 , v _ 0 ) \\equiv \\alpha ^ 2 \\end{array} \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} r ( x , h + y ) : = \\frac { 1 } { ( p - 1 ) ! } f _ y ^ { ( p ) } ( 0 , 0 ) [ h ] ^ { p - 1 } [ h + y ] - f ( x , h + y ) . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\varphi = g - P g . \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} U _ { S W } ( \\theta , P _ { B S } ) & ~ = \\mu ( 1 - \\theta ) \\bigg [ \\log \\bigg ( 1 + \\frac { \\theta \\xi P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } | h _ { s } | ^ { 2 } } { ( 1 - \\theta ) \\sigma _ { s } ^ { 2 } } \\bigg ) \\\\ & ~ ~ ~ ~ - \\log ( 1 \\ ! + \\ ! \\rho _ { e } ^ { \\textrm { o p t } } ) \\bigg ] \\ ! - \\ ! \\theta ( A P _ { B S } ^ { 2 } \\ ! + \\ ! B P _ { B S } ) . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} ( \\ell ' , m ' , n ' ) = ( b c , a c , a b ) . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\varphi ( u _ { * * } , y _ { * * } ) = \\frac { 4 y _ { * * } ^ 4 } { 1 6 y _ { * * } ^ 6 + 1 8 y _ { * * } ^ 4 + 1 1 y _ { * * } ^ 2 - 1 } . \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} \\mathbb { P } _ { a , b , r , p } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( z \\right ) = \\frac { 1 } { \\left ( r ! \\right ) ^ { p } \\prod _ { s = 2 } ^ { L } \\left ( r _ { s } ! \\right ) ^ { p _ { s } } } \\sum _ { k = 0 } ^ { r p + \\sigma } k ! S _ { a , b , r } ^ { \\alpha _ { s } , \\beta _ { s } , r _ { s } , p _ { s } } \\left ( p , k \\right ) \\left ( z - 1 \\right ) ^ { r p + \\sigma - k } . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , k } ( x | a ) \\bigr ) ( q \\ , | \\ , b ) & - \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , k } ( x | a ) \\bigr ) ( 0 \\ , | \\ , b ) = 0 . \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} H _ x = \\{ y \\in Y : T ( V ^ A _ { x , \\{ u \\} } ) \\subseteq V ^ B _ { y , \\{ v \\} } \\ ; { \\rm f o r \\ ; s o m e \\ ; } u \\in S ( E ) \\ ; { \\rm a n d } \\ ; v \\in S ( F ) \\} . \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} a _ n = u _ n + [ m _ { n + 1 } , p _ { - 1 } ] + w _ n . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} \\frac { \\partial U _ { S W } } { \\partial P _ { B S } } = \\frac { a d } { 1 + d P _ { B S } } - 2 \\theta A P _ { B S } - \\theta B = 0 , \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} R _ { \\{ 1 \\} } ( \\Delta ) = R _ { \\{ 2 \\} } ( \\Delta ) = \\frac { 1 } { 2 } \\log \\frac { \\sigma ^ 2 ( 1 + r _ { \\min } ^ 2 ) } { { \\Delta - \\sigma ^ 2 ( 1 - r _ { \\min } ^ 2 ) } } , \\sigma ^ 2 ( 1 - r _ { \\min } ^ 2 ) \\leq \\Delta \\leq 2 \\sigma ^ 2 . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} M _ { n } ( x ; \\beta , c ) = \\frac { c ^ n \\ , n ! } { ( 1 - c ) ^ n } \\sum _ { j = 0 } ^ n c ^ { - j } \\binom { x } { j } \\binom { - x - \\beta } { n - j } . \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} e _ { i , j } = \\begin{cases} w _ j ^ { ( i ) } , & j \\in \\{ 0 , 1 , \\cdots , l - 1 \\} \\\\ d _ i , & j = l . \\end{cases} \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\widetilde { M } _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\overset { { \\rm i n \\ , l a w } } { = } L \\ , N , \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} & \\dot C _ n = \\dot M _ n - \\sum _ { j = 1 } ^ { n - 1 } \\binom { n - 1 } { j } \\big ( \\dot M _ j C _ { n - j } + M _ j \\dot C _ { n - j } \\big ) \\\\ & \\ , \\ , \\ , = \\dot M _ n - G ( M _ 1 ) M _ { n - 1 } - 2 \\sigma ( n - 1 ) M _ { n - 2 } - \\sum _ { j = 1 } ^ { n - 1 } \\binom { n - 1 } { j } \\big ( \\dot M _ j - ( n - j ) M _ j \\big ) C _ { n - j } \\ , , \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} \\gamma _ n : = C _ 1 ^ { n + 1 } C _ 2 ^ { ( \\beta _ 1 - M ) n } \\prod _ { j = 1 } ^ n B \\left ( { M } / { 2 } , { j M } / { 2 } \\right ) = C _ 1 \\Gamma ( M / 2 ) \\frac { \\left ( C _ 1 C _ 2 ^ { \\beta _ 1 - M } \\Gamma ( M / 2 ) \\right ) ^ n } { \\Gamma \\left ( ( n + 1 ) M / 2 \\right ) } . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle | _ { H \\times X } = ( \\cdot , \\cdot ) . \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 _ n & 0 & X \\\\ X ^ t I _ { p , q } & 1 & \\frac 1 2 \\sum _ { i = 1 } ^ p x _ i ^ 2 - \\frac 1 2 \\sum _ { i = p + 1 } ^ n x _ i ^ 2 \\\\ 0 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\Theta = \\Big \\{ ( x _ 1 , \\tilde { x } , x _ 1 ' , \\tilde { x } ' ) \\in \\Omega \\times \\Omega : ( x _ 1 , \\tilde { x } ' ) , ( x _ 1 ' , \\tilde { x } ) \\in \\Omega \\Big \\} , \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} ( f , b ) = & \\begin{cases} ( m + s + 1 , m - s ) , & , \\\\ ( m - s , m + s + 1 ) , & \\end{cases} \\\\ \\intertext { a n d } w = & \\begin{cases} t , & , \\\\ - t , & . \\end{cases} \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} \\left ( { 1 + x } \\right ) _ q ^ \\alpha = \\frac { \\left ( { 1 + x } \\right ) _ q ^ { \\alpha + \\beta } } { \\left ( { 1 + q ^ \\alpha x } \\right ) _ q ^ \\beta } \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} \\Gamma ^ { * } = \\left ( \\Gamma \\Delta _ m \\right ) ^ { \\intercal } \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} & \\tilde { J } = \\tilde { J } ( \\{ y _ i \\} _ { i = 1 } ^ n ) : = \\sum _ { \\beta \\in Q _ + } J _ \\beta \\prod _ { i = 1 } ^ n y _ i ^ { ( \\beta - \\lambda , \\omega _ i ^ \\vee ) } \\in N _ \\lambda , \\\\ & J = J ( \\{ y _ i \\} _ { i = 1 } ^ n ) : = \\sum _ { \\beta \\in Q _ + } J _ \\beta \\prod _ { i = 1 } ^ n y _ i ^ { ( \\beta - \\lambda - \\rho , \\omega _ i ^ \\vee ) } \\in N _ { \\lambda + \\rho } . \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} Q ( t ) = \\int _ 0 ^ t D _ s X ( t ) D _ s X ( t ) ^ T d s = J ( t ) \\Bigg ( \\int _ 0 ^ t K ( s ) A ( s , t ) A ( s , t ) ^ T K ( s ) ^ T d s \\Bigg ) J ( t ) ^ T . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} K ( z , z ' ) = \\min ( z , z ' ) , \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} f ( m , n ) & = ( 2 m + n , m ) , \\\\ f ( n , m ) & = ( 2 n + m , n ) , \\\\ f ( m , - n ) & = ( 2 m - n , m ) \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} \\sup _ { D _ r } \\frac { u } { V ( d _ D ) } - \\inf _ { D _ r } \\frac { u } { V ( d _ D ) } \\leq \\sup _ { D _ { r _ k } } \\frac { u } { V ( d _ D ) } - \\inf _ { D _ { r _ k } } \\frac { u } { V ( d _ D ) } \\leq c _ 2 ( M _ k - m _ k ) = c _ 2 V ( r _ { k + 1 } / 2 ) ^ \\gamma \\leq c _ 2 V ( r ) ^ \\gamma . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} n = \\left \\lfloor m / 2 \\right \\rfloor , \\quad c = q ^ { m ( m - 1 ) / ( 2 n ) } , \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} & - \\\\ & \\phantom { } = 2 \\pi w ( P ) + j \\left ( \\varepsilon + ( u - 2 ) \\delta \\right ) \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} B _ { 2 , 2 } ( w \\ , | \\ , a ) = \\frac { w ^ 2 } { a _ 1 a _ 2 } - \\frac { w ( a _ 1 + a _ 2 ) } { a _ 1 a _ 2 } + \\frac { a _ 1 ^ 2 + 3 a _ 1 a _ 2 + a _ 2 ^ 2 } { 6 a _ 1 a _ 2 } . \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} | q _ 2 ' - q _ 2 | & = | q _ 1 - p _ 1 | - | q _ 2 ' - p _ 2 | - | p _ 1 - p _ 2 | \\\\ & \\leqslant | u _ 1 v x _ 0 - u _ 1 x _ 0 | - | u _ 2 v x _ 0 - u _ 2 x _ 0 | + 7 2 \\delta \\leqslant 7 2 \\delta , \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} d X _ t = b ( X _ { t - } ) \\ , d t + \\sigma ( X _ { t - } ) \\ , d L _ t , X _ 0 \\sim \\mu , \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\bar { P } ^ { i n } _ { \\mathbb { Z } ^ d , \\vec { H } } ( T ^ { \\vec { H } } ( x ) = 1 ) < p _ c ^ s ( d ) / 2 x \\in \\mathbb { Z } ^ d . \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 0 } ^ { 2 n + 1 - m } a _ i t ^ i \\right ) = \\left ( \\sum _ { i = 0 } ^ { 2 n + 1 - m } ( - 1 ) ^ i \\beta _ i t ^ { 2 i } \\right ) \\left ( \\sum _ { i \\geq 0 } \\delta _ i t ^ i \\right ) . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\psi _ p ( u ) = \\sum _ { j = 0 } ^ \\kappa d _ { j , p } ( a u + b ) ^ j , \\forall p \\in \\{ 0 , . . . , \\kappa \\} , \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} \\Psi _ p ( \\xi ; \\alpha _ p ) = p ^ { e _ p ( 3 / 2 - k ) } a ( p ^ { e _ p } ) W _ { \\tilde { \\varphi } _ p , \\xi } ( 1 ) . \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\psi ^ { k , t } ( x ) = \\psi ( x , \\zeta _ 1 ^ { k , t } ) = e ^ { \\zeta _ 1 ^ { k , t } \\cdot x } ( 1 + r ^ { k , t } ( x ) ) , x \\in \\R ^ d , \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\{ x \\in F _ { p ^ m } ( \\ ! ( t ) \\ ! ) \\colon p ^ k x = 0 \\} = p ^ { m - k } F _ { p ^ m } ( \\ ! ( t ) \\ ! ) . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} f _ k ( \\omega _ { n _ 1 } ^ { j _ 1 } , \\dots , \\omega _ { n _ { m } } ^ { j _ { m } } ) = f _ { ( n _ { m + 1 } , k ) } ( \\omega _ { n _ 1 } ^ { j _ 1 } , \\dots , \\omega _ { n _ { m } } ^ { j _ { m } } ) , \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} A _ { x \\star u } = A _ x T _ u + T _ x A _ u \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} \\ell + \\deg ( X ( \\lambda ) h ( \\lambda ) ) & = \\deg ( K _ 2 ( \\lambda ) ^ T X ( \\lambda ) h ( \\lambda ) ) \\leq \\deg ( M ( \\lambda ) ) + \\deg ( N _ 1 ( \\lambda ) ^ T h ( \\lambda ) ) \\\\ & \\leq \\ell + \\deg ( N _ 1 ( \\lambda ) ^ T h ( \\lambda ) ) , \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} \\phi \\left ( \\left [ \\frac { a t ^ n } { f ^ l t ^ { d l } } \\right ] \\right ) = \\left [ \\frac { a } { f ^ l } \\right ] \\cdot t ^ { n - d l } . \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} P ( m , n ) = \\frac { \\lambda ( \\mathcal { E } _ m \\cap \\mathcal { E } _ n ) } { \\lambda ( \\mathcal { E } _ m ) \\lambda ( \\mathcal { E } _ n ) } . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} P _ s ( x ) = \\frac { p _ s ^ { j _ s } } { q } P \\bigg ( \\frac { q } { p _ s ^ { j _ s } } x \\bigg ) . \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = m } \\left ( \\sum _ { i = 1 } ^ n \\frac { \\partial c _ \\alpha ( z ) } { \\partial z _ i } d ^ 1 z _ i ( d ^ { 1 } z ) ^ { \\alpha _ 1 } \\cdots ( d ^ { k } z ) ^ { \\alpha _ k } + \\sum _ { j = 1 } ^ k \\sum _ { i = 1 } ^ n c _ \\alpha ( z ) \\alpha _ j ^ i d ^ { j + 1 } z _ i ( d ^ { 1 } z ) ^ { \\alpha _ 1 } \\cdots ( d ^ j z ) ^ { \\alpha ' _ { j , i } } \\cdots ( d ^ { k } z ) ^ { \\alpha _ k } \\right ) , \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} T = \\{ ( t _ 1 , \\ldots , t _ n ) \\in \\R ^ n \\ : : \\ : t _ 1 > \\ldots > t _ n > 1 \\} . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( \\lambda , w ) = \\frac { \\sinh \\beta } { \\pi } \\frac { \\Gamma ( w + \\lambda ) } { c ^ { w + \\lambda } } . \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} ( \\hat u _ 1 , \\hat \\theta , \\hat h _ 1 ) | _ { \\tau = 0 } = ( u _ { 1 , 0 } , \\theta _ 0 , h _ { 1 , 0 } ) \\big ( \\xi , y ( \\xi , \\eta ) \\big ) = : ( \\hat u _ { 1 , 0 } , \\hat \\theta _ 0 , \\hat h _ { 1 , 0 } ) ( \\xi , \\eta ) , \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} \\log | f _ { \\omega , 1 } ^ n ( z , w ) | \\le \\log | z | + \\sum _ { i = 1 } ^ n \\log \\left ( 1 + \\frac { | w | } { 2 ^ { \\alpha ( i ) } } \\right ) . \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\{ \\beta \\to \\beta ' \\mid b _ { \\beta , \\beta ' } = 1 \\ \\ b _ { \\beta ' , \\beta } = - 1 \\} . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\bigcup _ { l \\in \\mathbb { Z } , \\ , \\ , l \\geq k } D _ { l } = A _ { k } . \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} { k \\over N } = { 1 - \\beta ^ 2 \\sum _ { i = 1 } ^ N { k \\over N } \\over 1 - \\beta ^ 2 } \\implies \\beta = \\sqrt { { N - k \\over k N - k } } . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} \\gamma _ 1 = ( \\ell - 1 ) ( m - 1 ) = 2 a ( b + 1 ) = n + ( a - 1 ) \\ell + 1 \\ge \\gamma . \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} \\hat { g } _ { \\hat { z } , t } = \\lambda _ t ^ { 2 } \\Psi _ t ^ * g _ { z , t } = g _ { \\C ^ m } + \\lambda _ t ^ { 2 } e ^ { - t } g _ { Y , z } \\ ; \\ , ( z = \\lambda _ t ^ { - 1 } \\hat { z } ) \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} \\chi _ j : = \\begin{cases} 1 & S _ j : = \\sum _ { i = 1 } ^ { N _ j } X _ j ( i ) > 0 , \\\\ - 1 & . \\end{cases} \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} x ^ n = n x \\quad \\mbox { f o r a l l $ \\ , n \\in \\Z $ . } \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} ( H ^ \\omega - z ) ^ { - 1 } & = ( \\tilde { H } ^ \\omega - z ) ^ { - 1 } + ( \\tilde { H } ^ \\omega - z ) ^ { - 1 } K ( \\tilde { H } ^ \\omega - z ) ^ { - 1 } \\\\ & + ( H ^ \\omega - z ) ^ { - 1 } K ( \\tilde { H } ^ \\omega - z ) ^ { - 1 } K ( \\tilde { H } ^ \\omega - z ) ^ { - 1 } . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\dbinom { n } { r } \\right ) = \\frac { z } { \\left ( z - 1 \\right ) ^ { r + 1 } } . \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} M ( s , f ) : = \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { x _ \\ell ( s ) } { \\ell ^ { 1 / 2 } } \\lambda _ f ( \\ell ) , \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} \\int _ { \\mathcal B _ 1 ^ - ( m ) } ( c , d ) _ p \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h = \\sum _ { j > - m } \\mathrm { v o l } ( \\mathcal R _ { 2 j } ) = ( 1 - p ^ { - 1 } ) ^ 2 \\sum _ { j > - m } p ^ { - 2 j } = p ^ { 2 m - 2 } \\frac { 1 - p ^ { - 1 } } { 1 + p ^ { - 1 } } . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} 1 - c \\cdot \\frac { 2 t + 1 } { N } \\leq & R _ { B , o p t } \\left ( N , t \\right ) \\leq 1 - \\frac { t } { N } , \\\\ 1 - c \\cdot \\frac { 8 t + 1 } { N } \\leq & R _ { G , o p t } \\left ( N , t \\right ) \\leq 1 - \\frac { t } { N } , \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} p _ { - 1 } \\coloneqq \\sum _ { i = 0 } ^ \\ell \\check f _ i . \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) = 2 ^ { k + 1 - \\nu ( M ) } C ( N , M , \\chi ) \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } \\frac { | \\langle ( \\mathrm { i d } \\otimes U _ M ) F _ { \\chi | \\mathcal H \\times \\mathcal H } , g \\times g \\rangle | ^ 2 } { \\langle g , g \\rangle ^ 2 } , \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\vert \\pi _ { P _ { F ( w ) } } ( D F ( z ) a - D F ( w ) a ) \\vert & = \\vert \\pi _ { P _ { F ( w ) } } ( D f ( z ) a - D f ( w ) a ) \\vert \\\\ & = \\vert ( \\pi _ { P _ { F ( w ) } } - \\pi _ { P _ x } ) ( D f ( z ) a - D f ( w ) a ) \\vert \\\\ & \\leq C _ 2 ( m ) \\tilde \\delta _ { x , k } \\vert D f ( z ) a - D f ( w ) a \\vert . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} H _ { \\tilde { u } } ( e _ 1 ) & = \\frac { | p | ^ 2 } { ( 1 - | p | ^ 2 ) ^ 2 } e _ 1 + \\frac { 1 } { 1 - | p | ^ 2 } e _ 2 , \\\\ H _ { \\tilde { u } } ( e _ 2 ) & = \\frac { 2 | p | ^ 2 } { ( 1 - | p | ^ 2 ) ^ 3 } e _ 1 + \\frac { 1 } { ( 1 - | p | ^ 2 ) ^ 2 } e _ 2 . \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} \\langle \\mathbf h , \\mathbf h \\rangle = 2 ^ { - 1 } \\zeta _ { \\Q } ( 2 ) ^ { - 1 } \\langle h , h \\rangle , \\langle \\breve { \\mathbf g } , \\breve { \\mathbf g } \\rangle = \\langle \\mathbf g , \\mathbf g \\rangle = \\zeta _ { \\Q } ( 2 ) ^ { - 1 } \\langle g , g \\rangle , \\langle \\pmb { \\phi } , \\pmb { \\phi } \\rangle = | | \\pmb { \\phi } _ { \\infty } | | ^ 2 = 2 ^ { - 1 } . \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { ( \\beta ) _ { n } n ^ m c ^ { n } } { ( n - 1 ) ! } = 0 , \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} f ( z ) = \\sum _ n \\frac { c _ n } { t _ n - z } , \\sum _ n \\frac { | c _ n | } { | t _ n | } < \\infty . \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} & ( b - m a ) ^ 2 - ( b - m a ) ^ 2 \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } g ( a ) } { g ( b ) - m g ( a ) } \\bigg ) ^ { \\frac { 1 } { s } } - ( b - m a ) ^ 2 \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } f ( a ) } { f ( b ) - m f ( a ) } \\bigg ) ^ { \\frac { 1 } { s } } \\\\ & + ( b - m a ) ^ 2 \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } f ( a ) } { f ( b ) - m f ( a ) } \\bigg ) ^ { \\frac { 1 } { s } } . \\bigg ( \\frac { \\beta - m 2 ^ { 1 - s } g ( a ) } { g ( b ) - m g ( a ) } \\bigg ) ^ { \\frac { 1 } { s } } = \\beta . \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} E _ 1 ( x , \\zeta ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! } \\left ( i ( x _ 1 \\zeta _ 1 + x _ 2 \\zeta _ 2 - x _ 0 ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) ) \\right ) ^ k , \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} \\mathcal { P } _ 1 ( r ) = \\omega _ n ^ { - 1 } \\int _ { B ( 0 , r ) ^ c } \\Big ( 1 \\land \\frac { | x | ^ 2 } { r ^ 2 } \\Big ) J ( | x | ) d x \\le \\omega _ n ^ { - 1 } \\mathcal { P } ( r ) \\le c _ 4 V ( r ) ^ { - 2 } , r > 0 . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} \\frac { \\partial \\mathbb { P } ( t , \\rho ) } { \\partial t } = \\Pi ( \\rho ) ^ { \\frac { 1 } { 2 } } \\nabla _ \\rho \\cdot ( L ( \\rho ) \\big ( \\mathbb { P } ( t , \\rho ) d _ \\rho \\mathcal { F } ( \\rho ) + \\beta d _ \\rho \\mathbb { P } ( t , \\rho ) \\big ) \\Pi ( \\rho ) ^ { - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\frac { \\theta } { \\pi } | x ^ 2 J _ c ^ { - 1 } f ( x ) | & = | \\int _ { \\mathbb R } e ^ { - \\theta | x - y | } x ^ 2 f ( y ) d y | \\leqq \\int _ { \\mathbb R } e ^ { - \\theta | x - y | } | x - y | ^ 2 | f ( y ) | d y + \\int _ { \\mathbb R } e ^ { - \\theta | x - y | } y ^ 2 | f ( y ) | d y \\\\ & \\leqq | f | _ { \\infty } \\int _ { \\mathbb R } e ^ { - \\theta | p | } p ^ 2 d p + | p ^ 2 f | _ { \\infty } \\int _ { \\mathbb R } e ^ { - \\theta | p | } d p < \\infty . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} B : = C _ { k + 1 } Q _ 2 = U \\Sigma V ^ H . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { x } _ { 1 } & = & g _ { 1 } ( t , x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } ) \\\\ \\dot { x } _ { 2 } & = & g _ { 2 } ( t , x _ { 2 } , \\ldots , x _ { n } ) \\\\ & \\vdots & \\\\ \\dot { x } _ { n } & = & g _ { n } ( t , x _ { n } ) , \\end{array} \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} & T ^ { \\frac 3 2 - 3 b } \\| u _ 0 \\| _ { L ^ 2 } \\lesssim 1 , \\\\ & T ^ { \\frac 3 2 - 3 b - \\epsilon } \\| u _ 0 \\| _ { L ^ 2 } \\lesssim 1 , \\\\ & T ^ { \\frac 3 2 - 3 b } ( \\| n _ 0 \\| _ { H ^ { s _ 1 } } + \\| n _ 1 \\| _ { { H } ^ { s _ 1 - 1 } } + \\| h \\| _ { H ^ { s _ 1 } } ) \\lesssim 1 , \\\\ & T ^ { \\frac 3 2 - 3 b - \\epsilon } \\| u _ 0 \\| _ { L ^ 2 } ^ 2 \\lesssim ( \\| n _ 0 \\| _ { H ^ { s _ 1 } } + \\| n _ 1 \\| _ { { H } ^ { s _ 1 - 1 } } ) . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} \\P _ { x _ { j - 1 } } ( X ( 1 ) = x _ j ) ~ = ~ \\tilde \\mu ( x _ j - x _ { j - 1 } ) \\geq \\delta > 0 , \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} B ( u ) \\zeta : = \\sum _ { i = 1 } ^ { N } \\langle b _ { i } , \\nabla u \\rangle \\zeta ^ { i } , u \\in S , \\ ; \\zeta \\in U . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} J _ 2 = \\int _ { I _ q } \\lambda _ q ^ \\frac { 5 } { 2 } \\sum _ { r \\geq q - 2 } \\| u _ r \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} K _ { \\widehat { X } } = f ^ * K _ Y + a E \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\sum _ { n = n _ 0 } ^ \\infty \\sum _ { i = 1 } ^ \\infty r _ n t ^ { n + i } x _ i y _ { i + 2 n } = \\sum _ { n = n _ 0 } ^ \\infty \\sum _ { i = \\max \\{ 1 , 2 ( m - n ) \\} } ^ \\infty r _ n t ^ { n + i } x _ i y _ { i + 2 n } . \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} W _ { 0 , j } : = \\frac { 1 } { 2 } + \\frac { j d / 2 } { \\log q } , H _ j : = \\frac { 3 ( j + 1 ) d / 2 } { \\log q } . \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} W ^ t ( q ) = \\min _ { i \\in I ^ 0 } \\big \\{ \\vartheta _ { p _ i } ^ t ( q ) \\big \\} . \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } x ^ a ( f ' ( 0 ) + \\varepsilon ( x , 0 ) / x ) ^ a = 0 , \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} | v _ t | ^ 2 _ 0 & + \\int _ 0 ^ t | \\partial _ x v _ r | ^ 2 _ 0 d r \\\\ & \\lesssim ( v _ 0 ^ 2 , m _ 0 ) - 2 \\int _ 0 ^ t ( u _ r ^ { ( 1 ) } \\partial _ x v _ r + v _ r \\partial _ x u ^ { ( 2 ) } _ r , v _ r m _ r ) d r \\\\ & = ( u _ 0 ^ 2 , m _ 0 ) + \\int _ 0 ^ t ( ( v _ r ) ^ 2 , \\partial _ x ( u _ r ^ { ( 1 ) } m _ r ) ) d t + 2 \\int _ 0 ^ t ( ( v _ r ) ^ 2 \\partial _ x u ^ { ( 2 ) } _ r , m _ r ) d r . \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} h ( f ( m ) ) = \\sum _ h h _ 3 f ( S ^ { - 1 } ( h _ 2 ) h _ 1 m ) = \\sum _ h ( h _ 2 \\cdot f ) ( h _ 1 m ) = \\sum _ h \\epsilon ( h _ 2 ) f ( h _ 1 m ) = f ( h m ) . \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} h ( z ) = \\sum _ { \\xi \\in \\Z , \\xi > 0 } c ( \\xi ) q ^ { \\xi } , \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} B ^ { k , l } : = \\sum _ { a , b \\geq 0 , a - b = k - l } ( - 1 ) ^ { k - a } E _ { 1 } ^ { ( b ) } F _ { 1 } ^ { ( a ) } 1 _ { ( k , l ) } : M ^ { k , l } \\rightarrow M ^ { l , k } \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{gather*} \\zeta ( x , s ) : = \\sum _ { { n = 0 } } ^ { \\infty } { \\frac { 1 } { ( n + x ) ^ { { s } } } } , \\operatorname { R e } ( s ) > 0 , \\operatorname { R e } ( x ) > 0 . \\end{gather*}"}
-{"id": "8068.png", "formula": "\\begin{align*} K ( [ z ] , \\lambda [ z ' ] ) = K ( [ z ] , [ \\lambda z ' ] ) = K ( z , \\lambda z ' ) = \\lambda ^ e K ( z , z ' ) = \\lambda ^ e K ( [ z ] , [ z ' ] ) , \\end{align*}"}
-{"id": "9012.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": "5573.png", "formula": "\\begin{align*} J _ { \\mathrm { A } } = \\{ ( \\imath , - \\imath + 1 - 2 k ) \\mid k = 0 , 1 , \\dots , 2 n - \\imath - 1 \\ \\ \\imath \\in I _ { \\mathrm { A } } \\} . \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} d ( \\ker ( G ^ { A , S } \\twoheadrightarrow G ) ) = d ( R / R ( S ) ) = d ( R ) = \\vert G \\vert ( \\vert A \\vert - 1 ) + 1 \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} \\Psi _ n = \\sum _ { I \\vDash n } ( - 1 ) ^ { \\ell ( I ) - 1 } V _ I . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} d ( \\underline { S } , f ) = ( b ( { \\mathbf { 0 } } ) \\mathbf { 0 } ! _ { \\underline { S } } , \\ldots , b ( { \\mathbf { i } } ) \\mathbf { i } ! _ { \\underline { S } } , \\ldots , b ( { \\mathbf { m } } ) \\mathbf { m } ! _ { \\underline { S } } ) . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} M _ 0 = \\begin{pmatrix} z & x & y \\\\ y ^ b & z & x ^ a \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{gather*} y ^ 4 = 1 - x ^ 2 . \\end{gather*}"}
-{"id": "5969.png", "formula": "\\begin{align*} p ^ * \\left ( \\prod _ { f \\in I } X _ f \\right ) = \\prod _ { i = 1 } ^ n A _ { i ' } = \\prod _ { i = 1 } ^ n \\frac { \\lambda _ { i - a } \\Delta _ { \\{ i - a , i - a + 1 , \\dots , i - 1 \\} } } { \\Delta _ { \\{ i - a + 1 , i - a + 2 , \\dots , i \\} } } = \\prod _ { i = 1 } ^ n \\lambda _ i . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell \\binom { \\ell } { j } \\psi _ 0 ^ { ( \\ell - j ) } ( w ) f ^ { ( j ) } ( w ) + \\sum _ { j = n } ^ { \\ell + n } \\binom { \\ell } { j - n } \\psi _ n ^ { ( \\ell + n - j ) } ( w ) f ^ { ( j ) } ( w ) = \\lambda f ^ { ( \\ell ) } ( w ) . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} & ~ w ( f ) ^ { p - r } \\prod _ { ( u , v ) \\in f } B ( ( u , v ) , f ) \\\\ = & ~ \\frac { [ w _ 1 ( \\pi _ 1 ( f ) ) w _ 2 ( \\pi _ 2 ( f ) ) ] ^ { p - r } } { ( r ! ) ^ { p - r } } \\cdot \\prod _ { ( u , v ) \\in f } \\frac { B _ 1 ( u , \\pi _ 1 ( f ) ) B _ 2 ( v , \\pi _ 2 ( f ) ) } { ( r - 1 ) ! } \\\\ = & ~ \\frac { r ^ r \\alpha _ 1 \\alpha _ 2 } { ( r ! ) ^ p } . \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = { \\bf E } \\bigl [ M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] , \\ ; \\Re ( q ) < \\tau , \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} S K _ 1 ( \\Lambda ^ { \\mathcal { O } } ( \\mathcal { G } ) ) : = \\varprojlim S K _ 1 ( \\mathcal { O } [ \\mathcal { G } / \\mathcal { N } ] ) , \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } e ^ { \\sigma | x | } \\nu ( x ) = C , \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} ( a b ) ^ { k / 2 } = ( b a ) ^ { k / 2 } . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} \\int \\limits _ T \\varphi ( h \\ , ^ t h ) \\ , d h = \\int \\limits _ { P _ + } \\varphi ( Y ) \\det ( Y ) ^ { - \\frac { n + 1 } 2 } \\ , d Y . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta _ { \\omega ( t ) } ) \\partial _ t \\varphi & = - t r _ { \\omega ( t ) } R i c ( \\hat \\omega ) \\\\ & \\le B t r _ { \\omega ( t ) } \\hat \\omega \\\\ & \\le B C _ 1 , \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dfrac { n \\epsilon } { \\ell } \\\\ \\dfrac { m d } { \\ell } - \\frac { m \\epsilon } { \\ell } - m \\\\ \\epsilon \\\\ d - \\ell - \\epsilon \\end{bmatrix} \\mbox { w i t h } \\epsilon \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} \\frac { H _ t ( - \\delta ) } { H _ t ( \\delta ) } = \\frac { \\Gamma ( - \\delta + 1 + i t ) \\Gamma ( - \\delta + 1 - i t ) } { \\Gamma ( \\delta + 1 + i t ) \\Gamma ( \\delta + 1 - i t ) } = 1 + \\mathcal { O } \\big ( | \\delta | \\big ) . \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} M = \\theta ( y _ 0 ) \\lesssim s _ 0 ^ 3 e _ 0 \\lesssim \\varepsilon ; \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} \\min _ { j = 1 , . . . , p : \\beta ^ * _ j \\ne 0 } | \\beta _ j ^ * | \\ge \\lambda / \\sqrt n \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{gather*} E _ u ( s ) : = \\ker ( H _ u ^ 2 - s ^ 2 I ) , \\\\ F _ u ( s ) : = \\ker ( K _ u ^ 2 - s ^ 2 I ) . \\end{gather*}"}
-{"id": "438.png", "formula": "\\begin{align*} \\begin{cases} \\underbar s _ 1 ^ 1 - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 1 ^ 1 + \\epsilon \\cr \\underbar s _ 2 ^ 1 - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 2 ^ 1 + \\epsilon \\end{cases} \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} f _ o ^ { L } ( \\vec x ) = \\int _ { \\min _ { 1 \\le i \\le n } x _ i } ^ { \\max _ { 1 \\le i \\le n } x _ i } f ( V _ t ) d t + f ( V ) \\min _ { 1 \\le i \\le n } x _ i , \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} Q _ W ( x _ 0 , x _ 1 , x _ 2 , u _ 0 , u _ 1 ) = k ( u _ 0 - x ) ^ 2 + ( u _ 1 - u _ 0 ) ^ 2 \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} | G | _ \\alpha = \\int _ { \\Gamma _ 0 } e ^ { \\alpha | \\eta | } \\left ( G ^ { + } ( \\eta ) + G ^ { - } ( \\eta ) \\right ) \\lambda ( d \\eta ) = | G ^ { + } | _ \\alpha + | G ^ { - } | _ \\alpha . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\big ( U _ N \\psi _ N \\big ) _ { j k } = \\Big ( \\big ( Q ^ { ( 1 ) } \\big ) ^ { \\otimes j } \\otimes \\big ( Q ^ { ( 2 ) } \\big ) ^ { \\otimes k } \\frac { a ^ { N _ 1 - j } _ 0 \\ , \\ , b ^ { N _ 2 - k } _ 0 } { \\sqrt { ( N _ 1 - j ) ! ( N _ 2 - k ) ! } } \\Psi _ N \\Big ) _ { j k } , \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} N = \\underbrace { 1 + 1 + \\cdots + 1 } _ { N - 1 \\ ; \\mathrm { n o n \\ ; e m p t y \\ ; b o x e s } } + \\underbrace { 1 } _ { \\ ; 1 \\ ; \\mathrm { o b j e c t } } . \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} & ( A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } ) ^ { \\frac { 1 } { 2 } } A ^ { \\frac { 1 } { 2 } } \\Big [ \\big ( ( 1 - \\lambda ) A ^ { - 1 } + \\lambda B ^ { - 1 } \\big ) - \\big ( ( 1 - \\lambda ) A + \\lambda B \\big ) ^ { - 1 } \\Big ] A ^ { \\frac { 1 } { 2 } } ( A ^ { - \\frac { 1 } { 2 } } B A ^ { - \\frac { 1 } { 2 } } ) ^ { \\frac { 1 } { 2 } } \\\\ & = \\lambda ( 1 - \\lambda ) A ^ { - \\frac { 1 } { 2 } } ( B - A ) \\big ( ( 1 - \\lambda ) A + \\lambda B \\big ) ^ { - 1 } ( B - A ) A ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} \\mathbb { E } _ 1 = \\frac { \\sqrt { 8 } } { 8 } ( e _ 1 \\otimes e _ 1 \\otimes e _ 1 \\otimes e _ 1 + e _ 2 \\otimes e _ 2 \\otimes e _ 2 \\otimes e _ 2 - e _ 1 \\otimes e _ 1 \\otimes e _ 2 \\otimes e _ 2 - e _ 1 \\otimes e _ 2 \\otimes e _ 1 \\otimes e _ 2 \\\\ - e _ 2 \\otimes e _ 1 \\otimes e _ 1 \\otimes e _ 2 - e _ 2 \\otimes e _ 1 \\otimes e _ 2 \\otimes e _ 1 - e _ 1 \\otimes e _ 2 \\otimes e _ 2 \\otimes e _ 1 - e _ 2 \\otimes e _ 2 \\otimes e _ 1 \\otimes e _ 1 ) \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} ( \\mathfrak { d } _ 1 ^ 0 a ) _ n & = a _ n , \\\\ ( \\mathfrak { d } _ 1 ^ { m + 1 } a ) _ n & = ( \\mathfrak { d } _ 1 ^ m a ) _ n - ( \\mathfrak { d } _ 1 ^ m a ) _ { n + 1 } , \\quad \\forall m , n \\in \\N . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} u _ \\Gamma ( 0 ) = u _ { 0 \\Gamma } \\mbox { o n } \\Gamma , u ( 0 ) = u _ 0 \\mbox { i n } \\Omega , \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} \\dot { k } _ t = L ^ \\Delta _ { \\alpha _ 2 } k _ t , k _ t | _ { t = 0 } = k _ 0 \\in \\mathcal { K } _ { \\alpha _ 1 } . \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\lVert \\delta _ { A } ^ { W ^ { - } ( n , \\phi ^ { - 1 } ) } \\rVert _ { 1 } = - \\alpha ^ { - } ( \\phi ^ { - 1 } ) \\log \\rho ( \\delta _ { A } ^ { - 1 } ) = \\lvert \\alpha ^ { - } ( \\phi ^ { - 1 } ) \\rvert \\log \\rho ( \\delta _ { A } ^ { - 1 } ) . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} \\frac { x } { ( 1 - x \\rho _ j ^ 2 ) ^ 2 ( 1 - y \\rho _ j ^ 2 ) } - \\frac { y } { ( 1 - x \\rho _ j ^ 2 ) ( 1 - y \\rho _ j ^ 2 ) ^ 2 } & = \\frac { x - y } { ( 1 - x \\rho _ j ^ 2 ) ^ 2 ( 1 - y \\rho _ j ^ 2 ) ^ 2 } , \\\\ \\frac { 1 } { ( 1 - x \\rho _ l ^ 2 ) ( 1 - y \\rho _ j ^ 2 ) } - \\frac { 1 } { ( 1 - x \\rho _ j ^ 2 ) ( 1 - y \\rho _ l ^ 2 ) } & = \\frac { ( \\rho _ l ^ 2 - \\rho _ j ^ 2 ) ( x - y ) } { ( 1 - x \\rho _ j ^ 2 ) ( 1 - x \\rho _ l ^ 2 ) ( 1 - y \\rho _ j ^ 2 ) ( 1 - y \\rho _ l ^ 2 ) } , \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} u _ 1 = & \\ \\frac { 1 } { l _ { 0 1 } + 1 } ( d _ { 1 0 1 } - l _ { 0 1 } , d _ { 2 0 1 } ) , & u _ 2 = & \\ u _ 1 \\ + \\ \\left ( \\frac { l _ { 0 1 } } { l _ { 0 1 } + 1 } , 0 \\right ) , \\\\ u _ 3 = & \\ \\frac { 1 } { 2 } ( - 1 , 0 ) , & u _ 4 = & \\ \\frac { 1 } { 2 l _ { 0 1 } + 1 } ( l _ { 0 1 } ^ 2 + d _ { 1 0 1 } + l _ { 0 1 } , d _ { 2 0 1 } ) \\\\ u _ 5 = & \\ \\frac { 1 } { l _ { 0 1 } + 2 } ( - d _ { 1 0 1 } , - d _ { 2 0 1 } ) , & u _ 6 = & \\ u _ 5 \\ + \\ \\left ( \\frac { l _ { 0 1 } + 1 } { l _ { 0 1 } + 2 } , 0 \\right ) . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\langle { \\bf u } ^ { \\tt M } , f \\rangle : = \\sum _ { x = 0 } ^ \\infty \\binom { x + \\beta - 1 } { x } c ^ x \\ , f ( x ) , \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} w = - p ( \\rho ) + \\mu ( \\rho ) \\partial _ x u . \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} T \\acute { N } ^ { \\perp } = \\cup _ { x \\in \\acute { N } } \\left \\{ u \\in T _ { x } \\breve { N } \\mid \\breve { g } \\left ( u , v \\right ) = 0 , \\forall v \\in T _ { x } \\acute { N } \\right \\} . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} F ( z , z ' ) : = \\sum _ { \\ell \\in L } w _ \\ell F _ \\ell ( z , z ' ) \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} t ^ k \\omega ( x , y ) = \\omega ( t ^ k x , t ^ k y ) \\ ; \\ , \\mbox { f o r a l l $ x , y \\in A $ a n d $ k \\in \\Z $ . } \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} \\dd Y ^ m _ t = b ^ { n , m } ( Y ^ m _ { t - } ) \\dd t + \\sigma ^ { n , m } ( Y ^ m _ { t - } ) \\dd W _ t + \\int _ E v ^ { n , m } ( y , Y ^ m _ { t - } ) ( p - q ) ( \\dd y , \\dd t ) , Y ^ m _ 0 = \\zeta . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} A u ( x ) & = \\lim _ { t \\downarrow 0 } \\int _ { \\R ^ n } \\left ( \\frac { u ( x + y ) + u ( x - y ) } { 2 } - u ( x ) \\right ) \\frac { p ( t , | y | ) } { t } d y \\\\ & = \\int _ { \\R ^ n } \\left ( \\frac { u ( x + y ) + u ( x - y ) } { 2 } - u ( x ) \\right ) J ( | y | ) d y = L u ( x ) . \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} u ^ i ( x , t ) \\leq U ( x , t ) \\forall ( x , t ) \\in \\R ^ n \\times \\left [ 0 , \\infty \\right ) , \\ , \\ , i = 1 , \\cdots , k . \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} H & = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } a _ { i + j } \\right ) _ { i , j \\in \\N } , & T & = \\left ( a _ { | m | + | n | } \\right ) _ { m , n \\in \\N ^ N } . \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} e ^ { 4 \\gamma - 2 \\sum _ { j = 1 } ^ 4 ( u _ j - \\mu _ j ) } = 1 \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} T \\acute { N } = S ( T \\acute { N } ) \\bot R a d T \\acute { N } , \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} S ( X , x ) = \\sum _ { X < n \\leq x } \\exp \\Big ( 2 \\pi i \\big ( \\xi P ( n ) - m \\varphi _ 1 ( n ) \\big ) \\Big ) \\Lambda ( n ) . \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} ( \\rho , u ) ( x , t ) = \\left \\{ \\begin{array} { l l } ( \\rho _ - , u _ - + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x < x ^ B ( t ) , \\\\ ( w ^ B ( t ) \\delta ( x - x ^ B ( t ) ) , u _ \\delta ^ { B } ( t ) ) , \\ \\ \\ \\ \\ x = x ^ B ( t ) , \\\\ ( \\rho _ + , u _ + + \\beta t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x > x ^ B ( t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} x _ 1 = 2 y _ 4 \\ ; , \\ ; \\ ; x _ 2 = y _ 2 \\ ; , \\ ; \\ ; x _ 3 = y _ 3 \\ ; , \\ ; \\ ; x _ 4 = \\frac { 4 y _ 1 - e ^ { 2 y _ 4 } ( 4 y _ 1 + y _ 2 ^ 2 + 4 y _ 2 y _ 3 - y _ 3 ^ 2 ) + 4 y _ 2 y _ 3 } { 2 ( e ^ { 2 y _ 4 } - 1 ) } , \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} m _ X k = m _ X . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\psi ' ( z ) = \\int _ 0 ^ { \\infty } \\frac { t e ^ { - ( z - 1 ) t } } { e ^ t - 1 } d t , z > 0 , \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\phi _ { \\mathbf h , p } ( z ) = \\pmb { \\phi } _ p ( x _ 3 ) \\phi _ { \\breve { \\mathbf g } , p } ( X ) = \\left ( \\mathbf 1 _ { \\Z _ p } ( x _ 1 ) - p ^ { - 1 } \\mathbf 1 _ { p ^ { - 1 } \\Z _ p } ( x _ 1 ) \\right ) \\mathbf 1 _ { \\Z _ p } ( x _ 2 ) \\mathbf 1 _ { \\Z _ p } ( x _ 3 ) \\mathbf 1 _ { \\Z _ p } ( x _ 4 ) \\mathbf 1 _ { p \\Z _ p } ( x _ 5 ) . \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} \\dot \\sigma _ i ( 0 ) = \\frac { 1 } { F ( \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) ) } \\exp ^ { - 1 } _ p ( \\gamma _ i ( 0 ) ) . \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} \\chi ^ I _ i = \\begin{cases} 1 & i \\in I \\\\ 0 & i \\notin I . \\end{cases} \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} y _ k = b _ k \\theta _ k + \\varepsilon \\xi _ k , k \\in \\mathbb { N } ^ { * } . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} d ( \\Z ^ n , f ) = \\operatorname * { g c d } \\lbrace f ( r _ 1 , r _ 2 , \\ldots , r _ n ) : 0 \\leq r _ i \\leq m _ i , r _ 1 + r _ 2 + \\ldots + r _ n \\leq k \\rbrace . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\mu _ { p , s } ( \\{ \\lambda _ { s } \\} ) = \\mu _ { p , s - 1 } ( \\{ \\lambda _ { s } \\} ) \\ , ( 1 - \\varepsilon ) = \\mu _ { p } ( \\{ \\lambda _ { s } \\} ) \\ , ( 1 - \\varepsilon ) \\le ( 1 - \\varepsilon ) ^ { p + 1 } \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} a \\cdot _ x ( b \\cdot c ) = a \\cdot b \\cdot a ^ { - 1 } \\cdot ( a \\cdot _ x c ) . \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} r - \\lambda = \\frac { b k ( d + 1 - k ) } { d ( d + 1 ) } \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} ( y x \\cdot b z ) x \\cdot b y ^ { \\alpha } = y x \\cdot b ( z x \\cdot b y ^ { \\alpha } ) . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} G \\cong \\prod _ { i = 1 } ^ { r } \\Z _ { n _ i } , \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} \\pi _ { i , k } ^ { o , \\kappa , h } ( n ) = \\pi _ { i , k } ^ { o , \\kappa , h } ( 0 ) \\prod _ { n ' = 1 } ^ n \\frac { \\alpha ^ { \\varphi ( o , \\kappa ) } _ { i , k } ( n ' - 1 ) h \\lambda _ { \\ell ( i ) } } { n ' \\mu _ i } , \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} \\delta u ^ { \\natural } _ { s \\theta t } & = A _ { \\theta t } ^ 2 \\delta u _ { s \\theta } + A ^ 1 _ { \\theta t } ( \\delta u _ { s \\theta } - A _ { s \\theta } ^ 1 u _ s ) \\\\ & = ( A _ { \\theta t } ^ 2 + A _ { \\theta t } ^ 1 ) \\delta \\mu _ { s \\theta } + ( A _ { \\theta t } ^ 2 A _ { s \\theta } ^ 1 + A _ { \\theta t } ^ 2 A _ { s \\theta } ^ 2 + A _ { \\theta t } ^ 1 A _ { s \\theta } ^ 2 ) u _ s + ( A _ { \\theta t } ^ 1 + A _ { \\theta t } ^ 2 ) u _ { s \\theta } ^ { \\natural } \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} \\lambda ^ { ( p + 1 ) } ( G ^ { r + 1 } ) = \\Big ( \\Big ( \\frac { r + 1 } { r } \\Big ) ^ { p - r } \\big ( \\lambda ^ { ( p ) } ( G ) \\big ) ^ p \\Big ) ^ { 1 / ( p + 1 ) } . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} \\sigma _ { 2 k - 2 \\dim { Y } - 1 } ( P ( t ) ) & = \\sigma _ { k + ( k - 2 \\dim { Y } ) - 1 } ( P ( t ) ) \\\\ & \\le M t ^ { 1 - 2 \\alpha } \\exp { - \\eta \\sqrt { k - 2 \\dim { Y } } } + \\hat { M } \\exp { - 2 \\tilde { \\eta } ( t ) \\sqrt { k - 2 \\dim { Y } } } \\\\ & \\le 2 \\max ( M t ^ { 1 - 2 \\alpha } , \\hat { M } ) \\exp { - \\min ( \\eta , 2 \\tilde { \\eta } ( t ) ) \\sqrt { k - 2 \\dim { Y } } } , \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , \\Big | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\lvert B _ 3 \\rvert = \\lvert B _ 3 \\setminus B _ 2 \\rvert + \\lvert B _ 3 \\cap B _ 2 \\rvert \\leq \\lvert B _ 1 \\rvert + \\lvert B _ 2 \\rvert . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\mathbb { P } [ \\sum _ { j = 1 } ^ k d ^ { e _ j } _ { l ' } = 0 ] = \\frac { 1 } { 2 } ( 1 + \\sigma ^ k ) . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} ~ ^ { A B C } \\nabla _ b ^ { \\alpha } f ( t ) = ~ ^ { A B R } \\nabla _ b ^ { \\alpha } f ( t ) - f ( b ) \\frac { B ( \\alpha ) } { 1 - \\alpha } E _ { \\overline { \\alpha } } \\left ( \\frac { - \\alpha } { 1 - \\alpha } , b - t \\right ) . \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} b _ N ^ { ( n ) } ( \\xi ) = b _ N ( \\xi ) , G _ N ^ { k , ( n ) } ( \\xi ) = G _ N ^ k ( \\xi ) , \\xi \\in B _ n ( H _ N ) , \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\Phi _ { p ^ { - 1 } , \\nu } = ( \\Phi _ { p , \\nu } ) ^ { - 1 } . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\{ X _ f , X _ g \\} = 2 \\varepsilon _ { f g } X _ f X _ g . \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} E _ t ( m ) = L _ t ( m ) + t ^ { - 1 } P ' _ { m , m ' } ( t ) L _ t ( m ' ) \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} \\hat { \\phi } _ { \\mathbf h , p } ( x ; y ) = \\mathbf 1 _ { \\Z _ p } ( x _ 1 ) \\mathbf 1 _ { \\Z _ p } ( x _ 2 ) \\mathbf 1 _ { \\Z _ p } ( x _ 3 ) \\mathbf 1 _ { p \\Z _ p } ( y _ 1 ) \\int _ { \\Q _ p } \\left ( \\mathbf 1 _ { \\Z _ p } ( z ) - p ^ { - 1 } \\mathbf 1 _ { p ^ { - 1 } \\Z _ p } ( z ) \\right ) \\psi _ p ( - y _ 2 z ) d z . \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\lbrace S ^ i ( x ^ 1 , . . . , x ^ { 2 n } ) , S ^ j ( x ^ 1 , . . . , x ^ { 2 n } ) \\rbrace = \\tilde { f } ^ { i j } _ { \\ ; \\ ; \\ ; k } S ^ k ( x ^ 1 , . . . , x ^ { 2 n } ) . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} M _ e ^ \\ell = ( R _ e ^ \\ell ) ^ T M ^ S _ e R _ e ^ \\ell , M _ e ^ { \\ell - 1 } = ( R _ e ^ { \\ell - 1 } ) ^ T M ^ S _ e R _ e ^ { \\ell - 1 } , M _ e ^ { \\ell , \\ell - 1 } = ( R _ e ^ \\ell ) ^ T M ^ S _ e R _ e ^ { \\ell - 1 } . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} p _ { n + 2 } ( x ) & = \\sum _ { \\substack { m = 0 \\\\ \\textnormal { m e v e n } } } ^ { n - 1 } \\frac { i ^ { m + 1 } } { 8 \\pi ^ 2 m ! } \\int _ { \\Omega } ( 1 - c ^ { - 2 } ) ( y ) L ^ { \\frac { n - m + 1 } { 2 } } ( f _ 2 - f _ 1 ) ( y ) | x - y | ^ { m - 1 } d y \\\\ & \\ \\ \\ \\ - \\frac { i ^ { n + 2 } } { 8 \\pi ^ 2 ( n + 1 ) } \\int _ { \\Omega } c ^ { - 2 } ( y ) ( f _ 2 - f _ 1 ) ( y ) | x - y | ^ n d y . \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} | F \\chi _ { R _ i } ( l ) | \\le ( A \\pi ) ^ { - d } M \\frac { 1 } { \\prod _ { j = 1 } ^ d \\max ( | \\rho ( l ) _ j | , 1 ) } \\le C ( A , d , \\rho ) M \\frac { \\log ^ { d - 1 } ( l + 1 ) } { l } . \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{gather*} E _ 2 ^ \\ast \\big ( \\sqrt { - 3 } \\big ) = 2 ^ { - 8 / 3 } \\frac { B ( 1 / 3 , 1 / 3 ) ^ 2 } { \\pi ^ 2 } , E _ 2 \\big ( \\sqrt { - 3 } \\big ) = \\frac { \\sqrt { 3 } } { \\pi } + 2 ^ { - 8 / 3 } \\frac { B ( 1 / 3 , 1 / 3 ) ^ 2 } { \\pi ^ 2 } . \\end{gather*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\left [ \\prod _ { i = 1 } ^ N \\frac { d _ i } { d _ i - 2 } \\right ] \\left ( \\| \\phi \\| _ { c b } - | c _ + | - | c _ - | \\right ) , \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} p _ H ( x \\star u , y \\star u ) ~ = ~ G A _ { x \\star u } \\ 1 _ { \\{ y \\star u \\} } ( e ) ~ = ~ G A _ x T _ u \\ 1 _ { \\{ y \\star u \\} } ( e ) + G T _ x A _ u \\ 1 _ { \\{ y \\star u \\} } ( e ) \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} \\lambda \\alpha = \\alpha ( y _ 2 + 1 ) ^ T , \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\mathbb { E } _ { n } ^ { ( N - 1 ) } ( f | A ) - \\mathbb { E } ( f | A ) \\right ) = \\mathbb { E } \\left ( \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( f 1 _ { A } ) \\left ( \\frac { 1 } { \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A ) } - \\frac { 1 } { P ( A ) } \\right ) \\right ) \\neq 0 , \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} \\mathcal { K } f ( x ) \\triangleq \\sum _ { i = 1 } ^ n ( & \\langle x , A ^ * y ^ * _ i \\rangle + \\langle b ( x ) , y ^ * _ i \\rangle ) \\partial _ i f ( x ) + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\langle a ( x ) y ^ * _ i , y ^ * _ j \\rangle \\partial ^ 2 _ { i j } f ( x ) \\\\ & + \\int _ { \\mathbb { B } } \\Big ( f ( x + y ) - f ( x ) - \\sum _ { i = 1 } ^ n \\langle h ( y ) , y ^ * _ i \\rangle \\partial _ i f ( x ) \\Big ) K ( x , \\dd y ) . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} G ( ( A \\times C ) ( x ) , z ) = G ( A ( 0 ) , z ) G ( C ( 0 ) , z ) e ^ { x z } . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} X _ 1 & : = \\{ x \\in X : f ( n , x ) > 0 f ( - n , x ) < 0 n > N \\} \\quad \\\\ X _ 2 & : = \\{ x \\in X : f ( n , x ) < 0 f ( - n , x ) > 0 n > N \\} \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} Q ( x , y ) ~ = ~ \\P _ { x } \\bigl ( X ( t ) = y \\ ; \\ ; 0 < t < + \\infty \\bigr ) , x , y \\in E . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} ( 1 + \\lambda | y - \\xi _ { j } | ) ^ { \\frac { N + 2 } { 2 } + \\tau - 4 } \\le C = C \\lambda ^ { 2 - N } \\lambda ^ { N - 2 } . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} m \\leq 1 + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( b ^ y _ \\ell - 1 ) \\leq 1 + \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } \\left ( \\frac { 3 } { 4 } i n d e x ( \\Sigma ^ y _ l ) + \\frac { 1 } { 4 } \\right ) \\leq 1 + \\frac { 3 } { 4 } j + \\frac { 1 } { 4 } \\sum _ { y \\in \\mathcal { Y } } J _ y \\leq 1 + j . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\sup _ { | y | \\leq R } \\sup _ { | \\xi | \\leq R ^ { - 1 } } | q ( y , \\xi ) | = 0 . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} d \\rho _ { t } ( z ) = \\frac { 2 } { \\rho _ { t } ( z ) } d t - d B _ { t } \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} C = \\left ( \\begin{matrix} 0 & 0 & 0 & 1 / 4 \\cr 0 & 0 & 1 & 0 \\cr 0 & 1 & 0 & 0 \\cr 1 & 0 & 0 & 0 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\eta _ r ( x , y ) = \\eta _ r ( x _ 0 t ^ 0 , y _ { 2 m } t ^ { 2 m } ) + \\eta _ r ( x _ 0 t ^ 0 , y ' ) + \\eta _ r ( x ' , y ) \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k _ 1 , . . . , k _ N = 0 } ^ \\infty \\eta _ { k _ 1 } ^ + ( x _ 1 ) \\otimes \\cdots \\otimes \\eta _ { k _ N } ^ + ( x _ N ) \\otimes B e _ { ( k _ 1 , . . . , k _ N ) } \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\limsup _ { t \\to 0 } | \\Delta ( t ) | = \\lim _ { t \\to 0 } | \\Delta ( t ) | = 0 \\ , . \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} N = \\int _ 0 ^ \\infty p ( N , s ) n ( s ) \\d s . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} ( \\nabla _ { U } u ) V = - B ( U , \\varphi V ) - u ( V ) \\tau ( U ) , \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} 0 < c < c ^ * _ n : = \\frac { \\pi n } { 2 } - \\log ( n ) - 6 . \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} \\| y ^ { k + 1 } - z \\| ^ 2 & = ( 1 + \\varphi ) \\| x ^ { k + 1 } - z \\| ^ 2 - \\varphi \\| x ^ { k } - z \\| ^ 2 + \\varphi ( 1 + \\varphi ) \\| x ^ { k + 1 } - x ^ k \\| ^ 2 \\\\ & = ( 1 + \\varphi ) \\| x ^ { k + 1 } - z \\| ^ 2 - \\varphi \\| x ^ { k } - z \\| ^ 2 + \\frac { 1 } { \\varphi } \\| y ^ { k + 1 } - x ^ k \\| ^ 2 . \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} P _ n = \\big \\{ Q \\cdot w : Q \\mid Q _ 0 w \\in \\Pi \\cup \\{ 1 \\} \\big \\} . \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} t ^ { \\overline { n } } = t ( t + 1 ) ( t + 2 ) . . . ( t + n - 1 ) \\quad ( n \\in \\mathbf { N } ) , \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} \\dim ( D ^ \\circ ) = \\frac { n ^ 2 } { 8 } + \\frac { 3 s ^ 2 } { 8 } - \\frac { s } { 2 } . \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} \\begin{array} { l l } 2 I = \\int _ { \\mathbb R } [ ( 1 - \\gamma ) - | \\omega | ] \\xi ^ 2 ( x ) - c \\mu ( \\xi ' ( x ) ) ^ 2 ( ( 1 - \\gamma ) + \\frac { b } { c } | \\omega | ) d x \\\\ \\geqq [ ( 1 - \\gamma ) - | \\omega | ] \\int _ { \\mathbb R } \\xi ^ 2 - c \\mu ( \\xi ' ) ^ 2 d x = [ ( 1 - \\gamma ) - | \\omega | ] \\| J _ c ^ { 1 / 2 } \\xi \\| ^ 2 . \\end{array} \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} ( A \\psi ) ^ * \\phi = \\psi ^ * ( A ^ * \\phi ) , \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} \\Sigma _ q = \\left ( \\begin{matrix} 1 - \\frac { r } { r _ 2 } & 1 & 1 & 1 & \\cdots & 1 \\\\ 1 & 1 & 1 - \\frac { r } { r _ 4 } & 1 & & \\vdots \\\\ \\vdots & & & & & \\\\ 2 - \\frac { r } { r _ 2 } & 2 - r & 2 & 2 & \\cdots & \\\\ 2 & 2 & 2 - \\frac { r } { r _ 4 } & 2 - r & & \\\\ \\vdots & & & & & \\\\ 2 & 2 & 2 & 2 & \\cdots & 2 - r \\end{matrix} \\right ) \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} \\left . \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } ( W _ { s } ( s , x ) + G ( s , x , W ( s , x ) , W _ { x } ( s , x ) , W _ { x x } ( s , x ) , \\bar { u } ( s ) ) ) \\right \\vert _ { x = \\bar { X } ^ { t , x ; \\bar { u } } ( s ) } \\geq 0 . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} \\Pi = \\bigcup _ { k = 1 } ^ D \\Pi _ k , \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} \\begin{aligned} a _ { \\omega } & = n _ { 4 , 2 } + 2 n _ { 3 , 3 } + n _ { 2 , 4 } + 4 \\tilde { \\delta } , \\\\ a ' _ { \\omega ' } & = n _ { 3 , 1 } + 2 n _ { 2 , 2 } + n _ { 1 , 3 } - 4 \\tilde { \\delta } , \\end{aligned} \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} M _ t ^ \\nu = \\exp ( \\nu S _ t ) - \\nu \\exp ( \\nu S _ t ) ( S _ t - X _ t ) . \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} \\kappa _ { N } = \\prod _ { k = 1 } ^ { N } \\left ( 1 + M m _ { N } \\right ) \\leqslant \\prod _ { k = 1 } ^ { N } \\left ( 1 + \\frac { M } { p _ { k } } \\right ) \\leqslant \\left ( 1 + \\frac { M } { p _ { ( N _ { 0 } ) } } \\right ) ^ { N _ { 0 } } , \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} ( 1 - \\rho ) b = - \\bigl ( ( 1 + \\sigma ) \\rho \\bigr ) b + ( 1 + \\sigma \\rho ) b . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} \\nu ( \\Gamma _ { \\mathbf { m } , k } ( \\underline { S } ) ) = \\max _ { \\substack { 0 \\leq r < l _ { \\mathbf { m } , k } \\\\ 0 \\leq s \\leq r } } \\nu \\left ( \\frac { \\Delta _ { \\mathbf { m } } ( \\underline { a } _ 0 , \\underline { a } _ 1 , \\ldots , \\underline { a } _ r ) } { \\Delta _ { \\mathbf { m } } ( s ; \\underline { a } _ 0 , \\underline { a } _ 1 , \\ldots , \\underline { a } _ { r - 1 } ) } \\right ) . \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} a ^ { \\prime } = \\left \\{ \\begin{array} [ c ] { l } f \\left ( a \\right ) a \\in L - \\left \\{ 0 \\right \\} \\\\ 0 a = 0 \\\\ f ^ { - 1 } \\left ( a \\right ) a \\in M \\end{array} \\right . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} \\sum _ { M ^ h \\in M ^ G } \\frac { | x ^ G \\cap M ^ h | } { | x ^ G | } \\frac { | \\{ K ^ g : g \\in G , K ^ g \\le M ^ h \\} | } { | K ^ G | } & = | G : M | \\frac { | x ^ G \\cap M | } { | x ^ G | } \\frac { | \\{ K ^ g : g \\in G , K ^ g \\le M \\} | } { | K ^ G | } \\\\ & = \\operatorname { f p r } ( x , M ^ G ) \\operatorname { f i x } ( K , M ^ G ) . \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} \\lambda _ f ( m ) \\lambda _ f ( n ) = \\sum _ { d | ( m , n ) } \\varepsilon _ q ( d ) \\lambda _ f \\left ( \\frac { m n } { d ^ 2 } \\right ) , \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\alpha _ m ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ m ) p ^ { - m / 2 } \\mathbf 1 _ { \\Z _ p ^ { \\times } } ( p ^ m x ) \\underline { \\chi } _ p ^ { - 1 } ( p ^ m x ) = \\chi _ { \\psi } ( p ^ m ) p ^ { - m / 2 } \\mathbf 1 _ { p ^ { - m } \\Z _ p ^ { \\times } } ( x ) \\underline { \\chi } _ p ^ { - 1 } ( x ) , \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} \\psi ( 0 , x , y ) = \\eta ( x , y ) . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} k = \\Bigl ( \\frac { 1 + b } { a _ 1 } \\Bigr ) ^ 2 + \\frac { 1 + b } { a _ 1 } , \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} \\det \\big [ M ( \\lambda ; t \\mid \\hat { u } _ n , \\hat { U } _ n ) \\big ] = ( \\lambda - t ) \\hat { U } _ n , \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} \\varphi _ i ^ { ( k ) } = \\psi _ i ^ { ( k - 1 ) } + \\mbox { l i n e a r c o m b i n a t i o n o f h i g h e r $ \\psi $ i n t h e o r d e r } . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\big | [ \\Phi ^ { - 1 } u ^ { N , k , r } ] ( v ) \\big | & = | u ^ { N , k , r } _ { m - N } \\varphi _ { N _ i , k , m - N } ( v ) | \\leq \\frac { C ( \\varepsilon ) e ^ { ( L ( E ) - \\varepsilon ) N } e ^ { - ( L ( E ) - \\varepsilon ) m } } { 2 ^ { ( m - N ) / 2 } } \\\\ & \\lesssim _ N C ( \\varepsilon ) e ^ { - \\big ( L ( E ) + \\frac { \\log 2 } { 2 } - \\varepsilon \\big ) m } . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} A ( x , k ) = \\left \\{ y \\in X \\ : \\ \\begin{matrix} \\exists \\ , \\omega _ x \\\\ \\omega _ x ( 0 ) = x , \\ , \\omega _ x ( k ) = y , \\ , | \\omega _ x \\Delta \\omega _ 0 | < \\infty \\end{matrix} \\right \\} . \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\chi _ { \\psi , V } ( a ) = ( a , ( - 1 ) ^ { m / 2 } \\det ( V ) ) _ F , a \\in F ^ { \\times } . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} W \\downarrow { g \\alpha } = W ( 2 ) ^ b + W ( 4 ) ^ c + V ( 2 ) ^ d \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\binom { k b + i } { k a } \\binom { k a + i } { k a } ^ { - 1 } & = \\frac { ( k b + i ) ( k b + i - 1 ) \\cdots ( k b + i - k a + 1 ) } { ( k a + i ) ( k a + i - 1 ) \\cdots ( i + 1 ) } \\\\ & \\geq \\left ( \\frac { k b + i } { k a + i } \\right ) ^ { k a } \\\\ & \\geq \\left ( \\frac { b + a } { 2 a } \\right ) ^ { k a } \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} x _ 1 z _ 1 = x _ 0 z _ 0 , x _ 2 y _ 2 = x _ 0 y _ 0 , \\\\ x _ 1 ^ 3 - y _ 1 ^ 3 + z _ 1 ^ 3 = - ( x _ 0 ^ 3 - y _ 0 ^ 3 + z _ 0 ^ 3 ) , \\\\ x _ 2 ^ 3 + y _ 2 ^ 3 - z _ 2 ^ 3 = - ( x _ 0 ^ 3 + y _ 0 ^ 3 - z _ 0 ^ 3 ) . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} w ( e ) ^ { p - r } \\cdot \\prod _ { v \\in e } B ( v , e ) & = \\Bigg ( \\frac { r } { \\lambda ^ { ( p ) } ( G ) } \\prod _ { u \\in e } x _ u \\Bigg ) ^ { p - r } \\cdot \\prod _ { v \\in e } \\frac { \\prod _ { u \\in e } x _ u } { \\lambda ^ { ( p ) } ( G ) x _ v ^ p } \\\\ & = \\frac { r ^ { p - r } } { ( \\lambda ^ { ( p ) } ( G ) ) ^ p } = \\alpha . \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { i + 1 } a _ { j } = m + 1 - s . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{gather*} G _ k ( q ) : = \\sum _ { m , n \\in \\Z } \\big ( 3 m + 1 - n \\sqrt { - 3 } \\big ) ^ k q ^ { ( 3 m + 1 ) ^ 2 + 3 n ^ 2 } , \\end{gather*}"}
-{"id": "6416.png", "formula": "\\begin{align*} q ( \\phi , \\eta ^ 0 ) = \\psi ( \\varphi ^ 0 \\eta ^ 0 \\phi ) = \\psi ( \\varphi ^ 0 \\Sigma ^ { - d } ( \\omega ^ { d + 1 } ) ) = \\psi ( \\Sigma ^ { - d } \\xi ^ { d + 1 } ) \\neq 0 , \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} \\frac { p _ 1 } { d _ 1 } + \\frac { q _ 2 } { d _ 2 } - r = 1 . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} s ' & = ( \\alpha _ i ( 1 + b ) ^ { i + 1 } ) \\circ ( 1 + c ) - \\alpha _ i ( 1 + c + b \\circ ( 1 + c ) ) ^ { i + 1 } \\\\ & = \\alpha _ i \\left ( ( 1 + b ) ^ { i + 1 } \\circ ( 1 + c ) - ( 1 + c + b \\circ ( 1 + c ) ) ^ { i + 1 } \\right ) . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} F _ { 2 , 1 6 } \\wedge F _ { 2 , 2 2 } \\wedge F _ { 2 , 2 4 } = - 2 8 8 0 \\ , ( u ^ { 3 } + u - u ^ { - 1 } - u ^ { - 3 } ) \\ , X ^ 7 Y ^ { 1 1 } + \\ldots \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\bigg ( w ( e ) ^ { p - r } \\prod _ { v \\in e } B ( v , e ) \\bigg ) ^ { 1 / p } & \\ ! \\geq \\ ! \\bigg ( w _ 1 ( e ) ^ { \\xi ( p - r ) } w _ 2 ( e ) ^ { ( 1 - \\xi ) ( p - r ) } \\prod _ { v \\in e } ( B _ 1 ( v , e ) ) ^ { \\eta } ( B _ 2 ( v , e ) ) ^ { 1 - \\eta } \\bigg ) ^ { 1 / p } \\\\ & \\ ! = \\ ! \\bigg ( \\ ! w _ 1 ( e ) ^ { p _ 1 - r } \\prod _ { v \\in e } B _ 1 ( v , e ) \\ ! \\bigg ) ^ { \\mu / p _ 1 } \\ ! \\bigg ( \\ ! w _ 2 ( e ) ^ { p _ 2 - r } \\prod _ { v \\in e } B _ 2 ( v , e ) \\ ! \\bigg ) ^ { ( 1 - \\mu ) / p _ 2 } \\\\ & = \\alpha _ 1 ^ { \\mu / p _ 1 } \\alpha _ 2 ^ { ( 1 - \\mu ) / p _ 2 } . \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\# \\{ \\sum _ { \\substack { j \\in J \\setminus J _ { k } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) | \\forall J _ { k } = \\{ i _ 1 , \\cdots , i _ k \\} \\subset J , 1 \\leq k \\leq K \\} \\\\ = C _ { n } ^ { K } ( 2 ^ { K } - 1 ) . \\end{array} \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\lbrace z _ 1 , z _ 3 \\rbrace = 1 ~ , ~ ~ ~ ~ \\lbrace z _ 2 , z _ 4 \\rbrace = 1 . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} j _ { \\frac { k a } { 2 } , n } , \\begin{pmatrix} j _ { \\frac { k a } { 4 } , \\frac { k a } { 2 } } \\\\ & I _ s \\\\ & & j _ { \\frac { k a } { 4 } , \\frac { k a } { 2 } } \\end{pmatrix} \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N I _ n ( x + k ) = \\sum _ { m = 0 } ^ { n \\wedge N } \\binom { N + 1 } { m + 1 } \\Delta ^ m I _ n ( x ) , \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} \\big ( \\alpha _ F ( r ) , h _ 2 \\big ( r \\otimes \\frac { r r _ 2 ' } { r _ 1 ' } \\big ) - n w ' \\big ) & = \\big ( \\alpha _ F ( r ) , h _ 2 \\big ( r \\otimes \\frac { r r _ 2 s _ 2 } { r _ 1 s _ 1 } \\big ) - n h _ 2 \\big ( 1 \\otimes \\frac { s _ 2 } { s _ 1 } \\big ) - n w \\big ) \\\\ & = \\big ( \\alpha _ F ( r ) , h _ 2 \\big ( r \\otimes \\frac { r r _ 2 s _ 2 } { r _ 1 s _ 1 } + r \\otimes \\frac { s _ 1 } { s _ 2 } \\big ) - n w \\big ) \\\\ & = \\big ( \\alpha _ F ( r ) , h _ 2 \\big ( r \\otimes \\frac { r r _ 2 } { r _ 1 } \\big ) - n w \\big ) . \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} \\P _ x ( X ( m ) = y ) ~ \\geq ~ \\delta ^ { \\kappa | y - x | } \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\nu _ c ) \\mathbf h _ p ( x ) = \\int _ { \\Q _ p } \\psi ( - 2 x y - c p y ^ 2 ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) d y = \\mathbf 1 _ { \\Z _ p } ( x ) \\int _ { \\Q _ p } \\psi ( - 2 x y - c p y ^ 2 ) \\mathfrak G ( 2 y , \\underline { \\chi } _ p ^ { - 1 } ) d y , \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} \\max f ^ { - 1 } ( 1 ) & = \\max \\{ i \\in [ n ] \\mid \\forall j \\leq i , f ( i ) \\leq f ( j ) \\} \\\\ & = \\max \\{ i \\in [ n ] \\mid \\forall j \\leq i , g ( i ) \\leq g ( j ) \\} \\\\ & = \\max g ^ { - 1 } ( 1 ) \\\\ & = \\max ( g ^ { - 1 } ( \\sigma ( 1 ) ) , \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\mathcal T = \\coth ( \\sqrt { \\mu _ 2 } | D | ) - \\frac { 1 } { \\sqrt { \\mu _ 2 } | D | } , \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\hat x ( t ) = f _ t ( y ( 0 : t - 1 ) ) . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} R _ i = \\{ ( Q , Q ' ) \\in \\Q \\times \\Q : Q - Q ' \\in \\Q _ i \\} . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} c _ + + ( - 1 ) ^ { | m | + | n | } c _ - = \\lim _ { k \\to \\infty } \\tilde { \\phi } ( m + n + 2 k \\chi ^ J ) = f _ \\phi ^ \\varnothing ( U ( m , n ) ) + f _ \\phi ^ I ( U ( m , n ) ) , \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} ( { \\bf Y \\star X } ) ^ \\tau : = { \\bf ( Y \\otimes X ) } ( \\Delta \\tau ) = \\sum _ { s \\in \\textsf { S u b } ( \\tau ) } Y ^ { \\tau \\backslash s } X ^ s , \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} n ( M \\backslash e ) = n ( \\Gamma \\backslash e ) . \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} \\mathbf g _ 2 ( x t ( 2 ^ { - 1 } ) ) \\overline { \\mathbf g _ 2 ( x t ( 2 ^ { - 1 } ) ) } = \\begin{cases} 2 ^ { - 2 } & x \\in L _ 0 , \\\\ 1 & x \\in L _ 1 , \\\\ 2 ^ { 2 } & x \\in L _ 2 . \\end{cases} \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} \\beta ^ { \\kappa } _ { M , M - 1 } ( \\kappa \\ , a , \\kappa \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } \\kappa ^ { \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j / \\prod \\limits _ { i = 1 } ^ M a _ i } \\ ; \\beta _ { M , M - 1 } ( a , \\ , b ) . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} \\Gamma _ 1 & = ( \\zeta ^ 4 + \\zeta ^ 3 ) x ^ 2 + ( \\zeta ^ 5 + \\zeta ^ 2 ) ( \\zeta ^ 5 + \\zeta ^ 2 + 1 ) x y + ( \\zeta ^ 5 + \\zeta ^ 2 ) y ^ 2 \\\\ & + ( - \\zeta ^ 5 - \\zeta ^ 2 + 1 ) x z + ( - \\zeta ^ 4 - \\zeta ^ 3 + 1 ) y z + ( \\zeta ^ 5 + \\zeta ^ 2 ) ( \\zeta ^ 4 + \\zeta ^ 3 - 1 ) z ^ 2 , \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} f ^ - ( x ) & : = f ( x ) - f ^ + ( x ) = f ( \\underline { x } ) - \\Delta ( x ) \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} \\mu _ { k , t } ( \\Psi _ t ( \\hat { x } ) ) = d ^ { \\hat { g } _ t } ( \\hat { x } , \\partial ( B _ { \\lambda _ t } \\times Y ) ) ^ k | ( \\nabla ^ { k , \\hat { g } _ t } \\hat { g } ^ \\bullet _ t ) ( \\hat { x } ) | _ { \\hat { g } _ t ( \\hat { x } ) } . \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} T ^ * ( t ) e _ k = e ^ { - \\lambda _ k t } e _ k t \\in [ 0 , T ] , \\ , k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} ( f _ 1 \\sqcup f _ 2 ) _ * ( g _ 1 \\sqcup g _ 2 ) ^ * = ( f _ 1 ) _ * ( g _ 1 ) ^ * + ( f _ 2 ) _ * ( g _ 2 ) ^ * . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} G ( A ( x ) , z ) = G ( A ( 0 ) , z ) e ^ { x z } . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} \\varphi _ B ( x ) = \\sum _ { l \\equiv 1 ( 2 ) } 2 l \\check { \\varphi } ( l ) J _ l ( x ) . \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} V ( T ^ * ) & : = V ( F ) \\cup \\{ x _ 1 , \\dots , x _ { 2 ( k + 1 ) \\eta n } \\} \\\\ E ( T ^ * ) & : = E ( F ) \\cup \\{ s _ i t _ i : i \\in [ 2 \\eta n ] \\} \\cup \\{ s _ i q : q \\in X ' _ i , i \\in [ { 2 \\eta \\log { n } } / { p ' } ] \\} . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} \\ u _ { 0 } & = 1 , u _ { 1 } = - 0 . 0 1 , w _ { 0 } = 0 . 2 , w _ { 1 } = 0 , K _ { 0 } = 0 . 1 , K _ { 1 } = 0 . 0 5 6 7 \\\\ P ( Y ) & = 0 . 0 0 1 9 9 - 0 . 0 2 4 1 \\cdot Y + 0 . 0 8 6 4 \\cdot Y ^ { 2 } - 0 . 1 2 8 \\cdot Y ^ { 3 } + 0 . 0 8 0 4 \\cdot Y ^ { 4 } = 0 \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} \\mathcal L _ { + \\infty } = \\frac 1 \\gamma - \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } | D | + \\frac { \\mu } { \\gamma } \\big ( a - \\frac { 1 } { \\gamma ^ 2 } \\Big ) \\partial _ x ^ 2 . \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\frac { ( 2 c ) ^ k } { k ! } \\leq \\frac { ( 2 c ) ^ n } { n ! } \\sum _ { k = 0 } ^ \\infty \\left ( \\frac { n } { 2 c } \\right ) ^ k \\leq 1 . 2 3 \\frac { ( 2 c ) ^ n } { n ! } . \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} ( U ( q _ 1 ) ) _ l & = \\langle e ^ { \\zeta _ 2 ^ { k _ l , t _ { k _ l } } \\cdot x } , ( \\Lambda _ { q _ 1 } - \\Lambda _ { 0 } ) \\psi _ 1 ^ l \\rangle _ { { H ^ { \\frac 1 2 } ( \\partial \\Omega ) \\times H ^ { - \\frac 1 2 } ( \\partial \\Omega ) } } \\\\ & = \\langle e ^ { \\zeta _ 2 ^ { k _ l , t _ { k _ l } } \\cdot x } , ( \\Lambda _ { q _ 2 } - \\Lambda _ { 0 } ) \\psi _ 2 ^ l \\rangle _ { { H ^ { \\frac 1 2 } ( \\partial \\Omega ) \\times H ^ { - \\frac 1 2 } ( \\partial \\Omega ) } } \\\\ & = ( U ( q _ 2 ) ) _ l , \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\psi ^ * \\phi : = \\phi ( \\psi ) , \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} a _ 0 = ( 1 - \\theta _ 0 ) ( b _ 0 + c _ 0 ) = \\frac { 1 - \\theta _ 0 } { \\theta _ 0 ^ 2 } \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} L _ { \\boldsymbol { u } } ( A \\times C ) ( x ) = ( A \\times L _ { \\boldsymbol { u } } C ) ( x ) = ( L _ { \\boldsymbol { u } } A \\times C ) ( x ) , A ( x ) , C ( x ) \\in \\mathcal { A } . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} r ^ { 2 } R _ { 2 } - ( 5 r ^ { 2 } - 2 r + 2 ) R = 0 , \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} A \\tilde { w } _ r ( x ) = \\frac { c _ 2 } { C _ 3 } A V ( \\Psi ) ( x ) + A \\eta _ r ( x ) \\ge - \\frac { c _ 2 } { C _ 3 } | L V ( \\Psi ) ( x ) | + A \\eta _ r ( x ) \\ge - \\frac { c _ 2 } { V ( 4 r ) } + \\frac { c _ 2 } { V ( 4 r ) } = 0 \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\Psi \\left ( H , \\left ( u - k \\right ) _ + , c \\right ) = 0 \\mbox { i f $ u \\leq k = \\frac { \\omega } { 2 } $ } . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} i ( T _ { 0 0 } ) & = 0 . 6 5 5 4 0 1 0 3 8 6 . . . = \\frac { 1 } { 6 } ( 3 . 9 3 2 4 0 6 2 3 1 9 . . . ) \\\\ & = \\frac { \\pi } { 1 8 \\sqrt { 3 } } { \\displaystyle \\int \\limits _ { 0 } ^ { \\pi } } \\frac { \\xi ( x ) \\eta ( x ) } { \\sqrt { b ( x ) \\left ( a ( x ) - c ( x ) \\right ) } } \\sin \\left ( \\frac { x } { 2 } \\right ) d x \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} \\mathcal { E } ( u , v ) : = \\int _ { \\mathbb { T } ^ { d } } \\langle \\nabla u , \\nabla v \\rangle \\ , d \\xi , u , v \\in D ( \\mathcal { E } ) , \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} \\nabla _ { U } \\omega V = \\nabla _ { U } ^ { \\ast } \\omega V + C ( U , \\omega V ) E , \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} \\widehat { S } _ i : = \\varprojlim _ { r , s } \\bigl ( S _ { i , r } / ( \\ker \\theta _ { \\log } ) ^ s \\bigr ) , \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\rho _ { e } ( p _ { s } ) = \\frac { ( N _ { T } - 1 ) \\{ [ 1 - ( 1 - \\varepsilon ) ^ { \\frac { 1 } { K } } ] ^ { \\frac { 1 } { 1 - N _ { T } } } - 1 \\} \\delta _ { e } ^ { 2 } } { P _ { B S } \\gamma _ { e } ^ { 2 } } p _ { s } . \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} & x ^ k = \\dfrac { ( \\varphi - 1 ) y ^ k + x ^ { k - 1 } } { \\varphi } , \\\\ & y ^ { k + 1 } = { \\rm a r g m i n } \\Big \\{ \\lambda _ k f ( y ^ k , y ) + \\dfrac { 1 } { 2 } \\| y - x ^ k \\| ^ 2 : y \\in C \\Big \\} . \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} \\mathcal F _ 0 ( w _ 1 ) = \\left ( \\mathcal F _ 0 ( w _ 1 ) - \\frac { 1 } { \\pi w _ 1 } \\right ) + \\frac { 1 } { \\pi w _ 1 } . \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\mathsf { P } ( \\eta = 0 ~ | ~ C > 0 ) = 0 . \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} E _ N = ( e _ 1 , \\dots , e _ N ) \\in \\{ - 1 , + 1 \\} ^ N . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} \\abs { Z } \\le \\begin{cases} q ^ { m ( m - d + 2 ) / 2 } & \\\\ q ^ { ( m + 1 ) ( m - d + 1 ) / 2 } & . \\end{cases} \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} \\left | \\tilde { f } _ { x _ 0 } ( x ) - d _ { x _ 0 } ( x ) \\right | & = \\int _ G \\mathfrak { c } ( g ^ { - 1 } x ) \\left | f _ { x _ 0 } ( g ^ { - 1 } x ) - d _ { x _ 0 } ( g ^ { - 1 } x ) \\right | d g \\leq \\delta _ U \\leq \\epsilon . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} Z _ { f t } ( \\tilde { K } ) & = \\int D \\tilde { \\Psi } \\ , e ^ { - L + \\tilde { \\Psi } Q \\tilde { K } } \\ , , & Q & = \\begin{pmatrix} 0 & - 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , & \\tilde { K } & = \\begin{pmatrix} \\tilde { k } \\\\ \\tilde { \\bar { k } } \\\\ \\tilde { j } \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{gather*} \\Gamma _ \\C ( 1 ) L ( g , 1 ) = \\epsilon \\big ( \\psi ^ 2 , 0 \\big ) \\Gamma _ \\C ( 2 ) L ( g , 2 ) , \\end{gather*}"}
-{"id": "1637.png", "formula": "\\begin{align*} T _ { d } ^ + ( \\alpha ^ + ) : = \\left \\{ ( A , \\Sigma ^ + , B ) \\in T _ d ^ + \\mid \\Sigma ^ + \\in X _ { \\alpha ^ + } ^ + \\right \\} . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} \\begin{gathered} X _ t = X _ 0 + \\int _ 0 ^ t ( k - a X _ s ) d s + \\sigma \\int _ 0 ^ t \\sqrt { X _ s } \\circ d B _ s ^ H , \\end{gathered} \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} \\langle \\mathbf h _ 2 ^ { ( 2 ) } , \\mathbf h _ 2 ^ { ( 2 ) } \\rangle = \\langle \\mathbf h _ 2 , \\mathbf h _ 2 \\rangle , \\langle \\breve { \\mathbf g } _ 2 , \\breve { \\mathbf g } _ 2 \\rangle = \\langle \\mathbf g _ 2 , \\mathbf g _ 2 \\rangle , \\langle \\pmb { \\phi } _ 2 ^ { ( 2 ) } , \\pmb { \\phi } _ 2 ^ { ( 2 ) } \\rangle = 2 \\langle \\pmb { \\phi } _ 2 , \\pmb { \\phi } _ 2 \\rangle . \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} \\delta = \\min \\left ( \\frac { \\psi ( m ) } { m } , \\frac { \\psi ( n ) } { n } \\right ) , \\Delta = \\max \\left ( \\frac { \\psi ( m ) } { m } , \\frac { \\psi ( n ) } { n } \\right ) . \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\phi _ t ( \\alpha ( x ) ) = \\alpha ( \\phi _ t ( x ) ) . \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} ( 1 , 1 ) _ { L } = 1 & , & & ( f _ { i } x , y ) _ { L } = \\frac { 1 } { 1 - v _ { i } ^ { 2 } } ( x , e ' _ { i } ( y ) ) _ { L } , & & ( x f _ { i } , y ) _ { L } = \\frac { 1 } { 1 - v _ { i } ^ { 2 } } ( x , { _ { i } e ' } ( y ) ) _ { L } . \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} \\pi ( A P ( f ) ) = A P ( g ) ~ ~ \\pi ( R e c ( f ) ) = R e c ( g ) . \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\chi ( m ) \\overline { \\chi } ( n ) = \\frac { 1 } { 2 } \\mathop { \\sum \\sum } _ { \\substack { v w = q \\\\ w \\mid m \\pm n } } \\mu ( v ) \\varphi ( w ) . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} x _ 2 ^ 3 = 1 . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nu _ 4 ( 1 + t ) ^ { - \\mu } \\leq \\min \\left \\{ \\frac { \\psi ' ( t ) } { t } , \\psi '' ( t ) \\right \\} , \\\\ & \\max \\left \\{ \\frac { \\psi ' ( t ) } { t } , \\psi '' ( t ) \\right \\} \\leq \\nu _ 5 \\frac { 1 } { 1 + t } \\end{aligned} \\right . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } - \\gamma _ 2 + 2 \\epsilon = ( 1 - \\gamma _ 2 ) - \\frac { 1 } { 2 } + 2 \\epsilon < 0 , \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} v _ \\mu ( x ) : = \\psi ( x ) - \\sum _ { j = 1 } ^ N a _ j \\mathcal { G } _ { \\mu } ( x - y _ j ) = \\sum _ { j = 1 } ^ N a _ j \\left ( - \\frac { 1 } { 2 \\pi } \\log | x - y _ j | - \\mathcal { G } _ { \\mu } ( x - y _ j ) \\right ) . \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} & \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} , & \\begin{pmatrix} 1 & - 1 & 0 \\\\ - 1 & 1 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} S ( f _ t | f _ \\infty ) \\le \\mathrm { e } ^ { - c t } S ( f _ 0 | f _ { \\infty } ) + \\begin{cases} K \\gamma ^ { - 1 } | c - \\gamma | ^ { - 1 } \\ , \\mathrm { e } ^ { - ( c \\wedge \\gamma ) t } & \\gamma \\ne c \\ , , \\\\ K c ^ { - 1 } t \\ , \\mathrm { e } ^ { - c t } & \\gamma = c \\ , . \\end{cases} \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} p ^ { m _ i } _ i ( s ' , q ) = \\left \\{ \\begin{array} { l l } 1 & q \\geq \\epsilon ^ { 1 / 2 } \\\\ 0 & q < \\epsilon ^ { 1 / 2 } . \\end{array} \\right . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} { \\rm E x p } _ { ( A _ r , B _ r ) } ( \\xi , \\eta ) = ( A _ r ^ { \\frac { 1 } { 2 } } \\exp ( A _ r ^ { - \\frac { 1 } { 2 } } \\xi A _ r ^ { - \\frac { 1 } { 2 } } ) A _ r ^ { \\frac { 1 } { 2 } } , B _ r + \\eta ) . \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{gather*} \\Lambda ( \\xi , s ) : = L _ \\infty ( \\xi _ \\infty , s ) L ( \\xi , s ) \\end{gather*}"}
-{"id": "7778.png", "formula": "\\begin{align*} \\mathrm { E L } _ p ( s , z , z ' , u , v , w ) = O \\Bigl ( \\ , \\frac { p ^ { 2 \\theta } } { p ^ { 2 ( 1 - \\theta ) } } \\ , \\Bigr ) = O \\Bigl ( \\ , \\frac { 1 } { p ^ { 2 - 4 \\theta } } \\ , \\Bigr ) \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} A : = \\begin{bmatrix} L _ 1 & \\cdots & L _ n \\end{bmatrix} \\ , N _ k \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} v _ t ( x ) = E \\Big [ v _ T ( \\phi _ { t , T } ( x ) ) \\exp \\Big \\{ \\int _ t ^ T \\mathrm { d i v } \\beta _ j ( \\phi _ { t , s } ( x ) ) \\dot { Z } _ s ^ j d s \\Big \\} \\Big ] . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\mathcal { M } ^ * _ { 1 , s } \\left ( \\tau \\right ) - \\frac { 1 } { \\sqrt { - i \\tau } } \\sqrt { \\frac { 2 } { p } } \\sum _ { k = 1 } ^ { p - 1 } \\sin \\left ( \\frac { \\pi k ( p - s ) } { p } \\right ) \\mathcal { M } ^ * _ { 1 , k } \\left ( - \\frac { 1 } { \\tau } \\right ) = i \\sqrt { 2 p } \\cdot r _ { f _ { p - s , p } } ( \\tau ) , \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\ell = b + 2 , m = 2 a + 1 , n = a b + b + 1 ( = ( a + 1 ) \\ell - m ) , \\quad \\gcd ( \\ell , m ) = 1 . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\vec { h } } _ { + } ( \\Lambda ) : = \\{ \\{ z , g ^ { + } \\} : z \\in \\Lambda h _ z > 0 \\} , \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( ( M ^ n _ \\omega ) ^ * M ^ n _ \\omega \\right ) ^ { \\frac { 1 } { 2 n } } = \\Lambda _ \\omega . \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} v ^ * = p + \\frac r { | | ( \\hat A Q ^ { - \\frac 1 2 } ) ^ + \\hat b | | } ( \\hat A Q ^ { - \\frac 1 2 } ) ^ + \\hat b \\ ; , \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} & \\Gamma = \\{ - n , - n + 2 , . . . , n - 2 , n \\} \\ , , \\qquad | \\Gamma | = n \\ , , \\\\ & | \\gamma | = \\begin{pmatrix} n \\\\ k \\end{pmatrix} \\ , , \\gamma = n - 2 k \\in \\Gamma \\ , . \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} M _ n = \\frac { \\sum _ { i = 1 } ^ { E _ n } s _ i } { E _ n } . \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} Q ^ { ( m ) } _ k ( 2 s + 1 ) = Q ^ { ( m ) } _ k ( 2 s , 1 ) - q ^ { m - s } \\ , Q ^ { ( m - 1 ) } _ { k - 1 } ( 2 s , 1 ) \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{gather*} e ^ { - \\beta ^ 2 ( \\kappa - \\log \\varepsilon ) } \\int _ \\phi ^ \\psi e ^ { \\beta V _ { \\varepsilon } ( \\theta ) } \\ , d \\theta \\longrightarrow M _ { \\beta } ( \\phi , \\psi ) , \\\\ { \\bf { E } } [ M _ { \\beta } ( \\phi , \\psi ) ] = | \\psi - \\phi | . \\end{gather*}"}
-{"id": "8369.png", "formula": "\\begin{align*} k _ { u , v } = \\frac { u - 3 v } { 2 v } . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} a ^ 2 + b ^ 2 + c ^ { - 2 } + a ^ 2 b ^ 2 c ^ 2 d ^ 2 - a b ( 1 + c ) ( 1 + d ) & = ( a - b ) ^ 2 + c ^ { - 2 } + a b ( 1 - c ) ( 1 - d ) + \\\\ & + ( a b c d - 1 ) ^ 2 - 1 \\geq c ^ { - 2 } - 1 > 0 . \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} B = \\left ( \\begin{array} { c c } n & r / 2 \\\\ r / 2 & m \\end{array} \\right ) , Z = \\left ( \\begin{array} { c c } \\tau & z \\\\ z & \\tau ' \\end{array} \\right ) , \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N + 1 } \\epsilon ^ { n ( n - 1 ) } h _ n g ^ { N - n + 1 } = o ( \\epsilon ^ { N ( N + 1 ) } ) , \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\mathcal J _ 1 ( m ) = \\int _ { \\mathcal B _ 1 ^ + ( m ) } \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h + \\int _ { \\mathcal B _ 1 ^ - ( m ) } ( c , d ) _ p \\chi _ { \\psi } ( c ) \\chi _ { \\delta } ( c ) | c | ^ { - 3 / 2 } _ p \\mathbf h _ p ( h ) d h . \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} Y \\overset { { \\rm i n \\ , l a w } } { = } Y ' = \\beta _ { 1 , 0 } ^ { - 1 } \\bigl ( \\tau = 1 , b _ 0 = 1 \\bigr ) . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} \\int | \\nabla _ { z _ i } u _ i | ^ 2 \\frac { | \\Psi | ^ 2 } { | u _ i | ^ 2 } \\leqslant \\int | \\nabla _ { z _ i } u _ i | ^ 2 | \\Psi _ i | = \\| \\nabla u _ i \\| _ { L ^ 2 } ^ 2 \\| \\Psi _ i \\| _ { L ^ 2 } ^ 2 \\leqslant \\frac { 1 } { 1 - C \\ell ^ 2 } \\| \\nabla u _ i \\| _ { L ^ 2 } ^ 2 \\| \\Psi \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\mathcal A _ 2 ^ + ( n ) : = \\{ h \\in \\mathcal A _ 2 ( n ) : d = 0 \\mathrm { o r d } _ p ( c ) \\} , \\mathcal A _ 2 ^ - ( n ) : = \\{ h \\in \\mathcal A _ 2 ( n ) : d \\neq 0 \\mathrm { o r d } _ p ( c ) \\} , \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} \\frac { L ( 1 , \\pi _ { \\infty } , \\mathrm { A d } ) L ( 1 , \\tau _ { \\infty } , \\mathrm { A d } ) } { L ( 1 / 2 , \\pi _ { \\infty } \\times \\mathrm { A d } ( g ) ) } = \\frac { 2 ( 2 \\pi ) ^ { - k - 1 } \\Gamma ( k + 1 ) \\pi ^ { - 1 } \\Gamma ( 1 ) \\cdot 2 ( 2 \\pi ) ^ { - 2 k } \\Gamma ( 2 k ) \\pi ^ { - 1 } \\Gamma ( 1 ) } { 2 ^ 2 ( 2 \\pi ) ^ { - 2 k - 1 } \\Gamma ( 2 k ) \\Gamma ( k - k + 1 ) \\cdot 2 ( 2 \\pi ) ^ { - k } \\Gamma ( k ) } = \\frac { k } { 2 \\pi ^ 2 } , \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} - \\Xi ' _ { \\imath } + \\Xi ' _ { \\jmath } \\begin{cases} = r ^ { ( \\jmath ) } & \\ \\imath \\in { { } ^ { \\jmath } B } , \\\\ \\in r ^ { ( \\jmath ) } + 4 \\mathbb { Z } _ { > 0 } & \\ \\imath \\notin { { } ^ { \\jmath } B } \\ \\ \\jmath \\neq n , \\\\ \\in r ^ { ( n ) } + 2 \\mathbb { Z } _ { > 0 } & \\ \\imath \\notin { { } ^ { \\jmath } B } \\ \\ \\jmath = n . \\end{cases} \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} \\nabla _ { ( a } \\nabla _ b v _ { c ) } = 0 \\quad \\quad \\nabla ^ { [ a } v ^ { b ] _ 0 } = 0 , \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\nabla _ { U } V = \\overset { \\mu _ { 0 } } { \\nabla } _ { U } V + \\frac { 1 } { q } \\alpha _ { 1 } ( U , V ) \\tilde { J } E + \\frac { 1 } { q } \\alpha _ { 2 } ( U , V ) \\tilde { J } N + \\alpha _ { 3 } ( U , V ) E . \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} C = \\left ( \\begin{matrix} 0 & 0 & 0 & - 1 \\cr 1 & 0 & - 1 & 0 \\cr 0 & 1 & 0 & 0 \\cr a & 0 & 0 & 0 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} | | T | | _ 1 = | | \\ : | T | \\ : | | _ 1 = t r ( | T | ) \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} p ^ * ( Y _ { j , k } ) = \\frac { \\Delta _ { I ( j , k ) } \\Delta _ { J ( j + 1 , k ) } } { \\Delta _ { I ( j , k - 1 ) } \\Delta _ { I ( j + 1 , k + 1 ) } } . \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\mathbf h _ 2 ^ { ( 2 ) } : = \\tilde { \\pi } _ 2 ( t ( 2 ) ) \\mathbf h _ 2 , \\pmb { \\phi } _ 2 ^ { ( 2 ) } : = 2 ^ { 1 / 2 } \\omega _ 2 ( t ( 2 ) ) \\pmb { \\phi } _ 2 . \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} \\Phi ( u ^ k , u ^ l ) = \\begin{cases} l , & \\mbox { i f } k + l = 0 \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} \\left ( a ^ { \\sim } \\vee \\lozenge a \\right ) \\wedge p = \\left ( b ^ { \\sim } \\vee \\lozenge b \\right ) \\wedge p \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} \\nabla _ { a } ^ { - \\nu } U ( t ) = \\sum _ { s = a } ^ { t } \\frac { [ t - \\varphi ( s ) ] ^ { \\overline { \\nu - 1 } } } { \\Gamma ( \\nu ) } U ( s ) , \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} I = \\sum m _ { i } \\left \\vert \\mathbf { a } _ { i } \\right \\vert ^ { 2 } T = \\frac { 1 } { 2 } \\sum m _ { i } \\left \\vert \\mathbf { \\dot { a } } _ { i } \\right \\vert ^ { 2 } \\mathbf { \\Omega } = \\sum m _ { i } ( \\mathbf { a } _ { i } \\times \\mathbf { \\dot { a } } _ { i } ) \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} \\nabla \\log R _ k ( \\alpha ) = \\frac { \\nabla R _ k ( \\alpha ) } { R _ k ( \\alpha ) ) } \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ { n - 1 } ( v ^ 2 - v ^ { - 2 } ) ^ 2 \\left ( v ^ { 4 n + 2 - 4 i } F _ i K _ { \\varpi _ { i + 1 } } E _ i K _ { \\varpi _ i } + v ^ { - 4 n + 2 + 4 i } F _ i K _ { - \\varpi _ i } E _ i K _ { - \\varpi _ { i + 1 } } \\right ) + \\\\ & ( v - v ^ { - 1 } ) ^ 2 ( F _ n K _ { - \\varpi _ n } E _ n + v ^ 2 F _ n E _ n K _ { \\varpi _ n } ) + c ^ 2 v ^ 2 F _ n ^ 2 K _ { - \\varpi _ n } E _ n ^ 2 K _ { \\varpi _ n } . \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} \\widetilde { c } _ { j i } ( - r _ j ) & = \\delta _ { j i } & \\widetilde { c } _ { j i } ( r ) & = 0 \\ \\ r > - r _ j . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} g ( \\gamma , x ) = ( \\gamma r _ K ( g ) ^ { - 1 } , g x ) \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} F _ u ( \\sigma _ k ) & = \\left \\lbrace \\frac { f } { D } u _ k ^ K \\ ; \\middle | \\ ; f \\in \\mathbb { C } _ { m ' - 1 } [ z ] \\right \\rbrace , \\\\ E _ u ( \\sigma _ k ) & = \\left \\lbrace \\frac { z g } { D } u _ k ^ K \\ ; \\middle | \\ ; g \\in \\mathbb { C } _ { m ' - 2 } [ z ] \\right \\rbrace , \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i = 1 } ^ n \\frac { u ^ i } { \\lambda _ i } e _ i \\Big \\| _ \\mathbb { K } \\leq \\sum _ { i = 1 } ^ n \\frac { \\| \\iota \\| _ o m + \\epsilon _ n } { \\lambda _ i } \\triangleq C = C ( n , m ) . \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} \\chi ( M ) = 2 k ( M ) - s ( M ) . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} \\gamma ( U _ 1 ^ { m _ 1 } ( U _ 1 ^ * ) ^ { n _ 1 } \\otimes \\cdots \\otimes U _ N ^ { m _ N } ( U _ N ^ * ) ^ { n _ N } ) = 1 , \\qquad \\forall \\ , m _ 1 , n _ 1 , . . . , m _ N , n _ N \\in \\N . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{gather*} \\dot { u } _ k ^ K = B _ u u _ k ^ K - i J w _ k ^ K , \\\\ \\dot { w } _ k ^ K = B _ u w _ k ^ K + i \\bar { J } ( 2 Q + \\sigma _ k ^ 2 ) u _ k ^ K . \\end{gather*}"}
-{"id": "1480.png", "formula": "\\begin{align*} \\widehat I _ { s , t } = { \\rm e } ^ { X _ s } \\ , I _ { s , t } = { \\rm e } ^ { X _ s } \\ , \\int _ s ^ t { \\rm e } ^ { - X _ u } \\ , d u , \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} A & : = i ( 0 , \\beta ( X ) ) \\cap \\cap _ { n \\in L ^ 0 ( \\mathbb { N } ) } ( ( [ i ( x , X ) ] ^ { \\beta [ \\frac { n } { x } ] } ) ^ c \\cup [ i ( x + 1 , X ) ] ^ { \\beta [ \\frac { n } { x } ] } ) \\\\ B & : = \\cap _ { n \\in L ^ 0 ( \\mathbb { N } ) } [ i ( x , X ) ] ^ { \\beta [ \\frac { n } { x } ] } \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} \\sup _ { \\tilde { B } } \\abs { S _ x ( u - P _ { \\tilde { x } } ( \\cdot - \\tilde { x } ) ) } = \\sup _ { \\hat { B } _ { c _ \\beta \\rho ( x ) } ( x ) } \\abs { u - P _ { \\tilde { x } } ( \\cdot - \\tilde { x } ) } \\leq C \\Lambda \\rho ( x ) ^ q . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} { \\theta _ { 2 , u } } \\left ( t \\right ) = 2 \\pi { f _ { 2 , u } } t - 2 \\pi \\Delta f n T + \\varphi _ { n , u } ^ { { \\rm { C F S K } } } + { \\varphi _ u } \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} \\begin{array} { l l } 2 I = \\int _ { \\mathbb R } [ ( 1 - \\gamma ) - | \\omega | ] \\xi ^ 2 ( x ) + \\mu b ( \\xi ' ( x ) ) ^ 2 ( \\frac { | c | } { b } ( 1 - \\gamma ) - | \\omega | ) d x \\\\ \\geqq \\Big [ \\frac { | c | } { b } ( 1 - \\gamma ) - | \\omega | \\Big ] \\int _ { \\mathbb R } \\xi ^ 2 + \\mu b ( \\xi ' ) ^ 2 d x = \\Big [ \\frac { | c | } { b } ( 1 - \\gamma ) - | \\omega | \\Big ] \\| J _ b ^ { 1 / 2 } \\xi \\| ^ 2 . \\end{array} \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\underline { m } ^ * \\circ \\mathsf { c o f } ^ { \\underline { 1 } } & = \\infty ^ * \\circ ( 0 _ { \\underline { m } ^ { \\vee } } ) ^ * \\circ \\mathsf { c o f } ^ { \\underline { 1 } } \\\\ & \\cong \\infty ^ * \\circ \\mathsf { c o f } ^ { \\underline { 1 } } \\circ ( 1 _ { \\underline { m } ^ { \\vee } } ) ^ * \\\\ & = C ^ { | \\underline { m } | } \\circ ( 1 _ { \\underline { m } ^ { \\vee } } ) ^ * . \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} Q ( \\lambda ) : = N _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T , \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} A = \\bigcup _ { p \\in P } A _ p \\qquad p \\neq q A _ p \\cap A _ q = \\varnothing . \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\rho ( t , 0 ) = ( ( \\xi _ 1 ( t ) , \\dots , \\xi _ n ( t ) ) , 0 ) , \\xi _ k ( t ) = \\lim _ { s \\to 0 } x _ k ( s t ) / s ^ { \\ell _ k } = \\sigma ( x _ k ) ( t ) . \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} n _ 1 : = & ( \\beta + 1 ) q ^ { n - 3 } ( q ^ 2 - q + 1 ) + \\alpha ( q ^ n + 1 ) \\\\ n _ 2 : = & ( q ^ 2 - 3 ) ( q ^ n + 1 ) + 3 q ^ { n - 3 } ( q ^ 2 - q + 1 ) - ( \\beta + 1 ) q ^ { n - 3 } ( q ^ 2 - q + 1 ) - \\alpha ( q ^ n + 1 ) , \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} \\sqcap _ k N _ k | A _ k : = V | B \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} { \\bf P } \\bigl [ \\log \\beta _ { M , N } ( b ) = 0 \\bigr ] = \\exp \\Bigl ( - \\int \\limits _ 0 ^ \\infty d K ^ { ( f ) } _ { M , N } ( t \\ , | \\ , b ) / t \\Bigr ) . \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} M _ \\phi ( U ( m , n ) ) = \\tilde { \\phi } ( m + n ) U ( m , n ) . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{gather*} W _ 0 ( z ; s ) = | { \\Im } ( z ) | ^ { 1 / 2 - s } { \\rm e } ^ { { \\rm i } \\operatorname { R e } ( z ) } K _ { s - 1 / 2 } ( | \\Im ( z ) | ) , \\end{gather*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } S ( n ) / n = m \\in { \\cal C } . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} \\varepsilon _ p ( f ) = - 1 p \\mid M . \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\frac { D _ r ( n ) } { n ! } = \\frac { 1 } { e } \\sum _ { k = 0 } ^ { r } A ( r , k ) \\binom { n + r - k } { n } + O ( \\varepsilon ^ n ) . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\sum ^ N _ { \\mu = 1 } \\| ( - \\Delta ) ^ { \\frac { 1 } { 2 } } | u _ \\mu ( t , x ) | ^ 2 \\| ^ 2 _ { L ^ 2 ( ( T _ 1 , T _ 2 ) ; L _ x ^ 2 ) } \\lesssim \\sup _ { t \\in [ T _ 1 , T _ 2 ] } | \\mathcal { M } ( t ) | ; \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} \\int \\limits _ T \\phi ( h ) \\ , d h = 2 ^ n \\int \\limits _ A \\int \\limits _ { N _ 1 } \\phi ( a n ) \\ , d n \\ , d a , \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} P _ n ^ * = n ( n - 1 ) \\widetilde { P } _ { n - 1 } , \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} \\mathfrak v _ i = \\sum _ { j = 0 } ^ { m _ i } \\partial _ x ^ j v _ { j i } ( x ) \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} T _ x P ^ n \\ 1 = P ^ n T _ x \\ 1 + \\sum _ { k = 1 } ^ n P ^ { k - 1 } A _ x P ^ { n - k } \\ 1 = P ^ n \\ 1 + \\sum _ { k = 1 } ^ n P ^ { k - 1 } A _ x P ^ { n - k } \\ 1 , \\forall n \\geq 1 . \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\varepsilon } ( x ^ { \\ast } ) = \\bigcap \\limits _ { \\eta > 0 } \\bigcup \\limits _ { y ^ { \\ast } \\in Y ^ { \\ast } } \\Big \\{ x \\in X \\ , : \\ , \\left ( x , 0 _ { Y } \\right ) \\in \\left ( M ^ { \\varepsilon + \\eta } F \\right ) ( x ^ { \\ast } , y ^ { \\ast } ) \\Big \\} . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 0 } ^ { 2 n + 1 - m } a _ i t ^ i \\right ) \\left ( \\sum _ { i = 0 } ^ { 2 n + 1 - m } ( - 1 ) ^ i a _ i t ^ i \\right ) = \\left ( \\sum _ { i = 0 } ^ { 2 n + 1 - m } ( - 1 ) ^ i \\beta _ i t ^ { 2 i } \\right ) \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} \\dd { x } { t } = Y I _ E \\ , v _ k ( x ) = Y A _ k \\ , x ^ Y , \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} M _ { ( n ) } ( k ) = \\left \\{ \\begin{array} { c c c } 1 , & k \\in \\{ - n , - n + 2 , . . . , n - 2 , n \\} \\\\ 0 , & \\end{array} \\right \\} . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} \\psi _ N = c ( \\omega , u ^ 1 , \\cdots , u ^ N ) , \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} & 1 \\leq p < p ^ * ( d ) , \\ \\ \\ \\ p d > 4 , \\ \\ \\ \\ p ^ * ( d ) = \\begin{cases} + \\infty \\ \\ \\ \\ \\ \\ , & \\ \\ \\ d = 3 , 4 , \\\\ \\frac 4 { d - 4 } \\ \\ \\ \\ \\ \\ & \\ \\ \\ 5 \\leq d \\leq 8 . \\end{cases} \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} \\alpha : = \\left ( \\begin{array} { c c } p & 0 \\\\ 0 & p ^ { - 1 } \\end{array} \\right ) , \\beta : = s \\alpha = \\left ( \\begin{array} { c c } 0 & p ^ { - 1 } \\\\ - p & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\Theta _ { t } = e ^ { H _ { 1 } \\otimes h ^ { 1 } _ { t } ( \\zeta ) } \\cdots e ^ { H _ { \\ell } \\otimes h ^ { \\ell } _ { t } ( \\zeta ) } , \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\rho _ t + ( \\rho u ) _ x = 0 , \\\\ ( \\rho u ) _ t + ( \\rho u ^ 2 ) _ x = \\beta \\rho , \\end{array} \\right . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} E _ { b , c } ^ { ( M ) } { K _ b \\brack \\lambda _ b } = { K _ b ; - M \\brack \\lambda _ b } E _ { b , c } ^ { ( M ) } ; E _ { b , c } ^ { ( M ) } { K _ c \\brack \\lambda _ c } = { K _ c ; M \\brack \\lambda _ c } E _ { b , c } ^ { ( M ) } . \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\lambda ' \\wedge \\lambda = 0 . \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} H M H ^ { - 1 } \\ , = \\ , \\pm M ^ { - 1 } , \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} ( a \\cdot b ) \\cdot _ x c = a \\cdot _ x ( b \\cdot _ x c ) , e _ A \\cdot _ x a = a , \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} N ( t ) : = \\int _ 0 ^ \\infty p ( s ) n ( t , s ) \\d s , t \\in [ 0 , T ) . \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} h \\varpi _ p = \\left ( \\begin{array} { c c } x p ^ { - 1 } & y \\\\ z p ^ { - 1 } & t \\end{array} \\right ) = \\left ( \\begin{array} { c c } p ^ { - 1 } t ^ { - 1 } \\det ( h ) & y \\\\ 0 & t \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & 0 \\\\ t ^ { - 1 } p ^ { - 1 } z & 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} ( D _ { w \\lambda , w ' \\lambda } , x ) _ { L } = ( u _ { w \\lambda } , x . u _ { w ' \\lambda } ) _ { \\lambda } ^ { \\varphi } \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\mathbb { E } [ \\sum _ { \\substack { j \\in J _ { K } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) ] = 0 . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} U = \\begin{pmatrix} 0 . 8 9 0 6 & 0 . 1 1 8 9 & - 0 . 1 0 2 5 \\\\ - 0 . 1 1 1 7 & 0 . 7 2 1 6 & 0 . 0 3 7 3 \\\\ - 0 . 0 6 5 0 & - 0 . 1 5 5 8 & 0 . 8 9 9 4 \\\\ - 0 . 2 1 4 4 & 0 . 6 1 3 8 & 0 . 0 3 0 2 \\\\ 0 . 3 7 9 8 & 0 . 2 5 3 2 & 0 . 4 2 2 3 \\end{pmatrix} , \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} S _ d \\omega _ { 1 } & = \\pm d S _ d \\\\ S _ d \\omega _ 2 \\pm S _ { d - 1 } \\omega _ 1 & = \\pm d S _ { d - 1 } \\\\ & \\vdots \\\\ S _ d \\omega _ { r + 1 } + ( \\omega _ r , \\dots , \\omega _ { r - d } ) & = 0 \\ ( r \\ge d ) \\\\ & \\vdots \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} \\Theta ( f ) \\Theta ( \\mu - \\delta _ e ) T = 0 , \\ \\ \\ f \\in L ^ 1 ( G ) , \\ \\mu \\in \\Lambda . \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} { \\bf P } \\bigl [ \\beta _ { M , N } ( a , b ) = 1 \\bigr ] & = \\exp \\Bigl ( - \\int \\limits _ 0 ^ \\infty e ^ { - b _ 0 t } \\frac { \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } \\frac { d t } { t } \\Bigr ) , \\\\ & = \\exp \\bigl ( - ( \\mathcal { S } _ N \\log \\Gamma _ M ) ( 0 \\ , | a , \\ , b ) \\bigr ) . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} \\Gamma _ 2 ( z \\ , | \\ , 1 , \\tau ) = ( 2 \\pi ) ^ { \\frac { z } { 2 } } \\tau ^ { \\frac { ( z - z ^ 2 ) } { 2 \\tau } - \\frac { z } { 2 } } e ^ { \\gamma \\frac { ( z ^ 2 - z ) } { 2 \\tau } } \\Gamma ( z ) \\prod \\limits _ { m = 1 } ^ \\infty ( m \\tau ) ^ { 1 - z } e ^ { \\frac { ( z - z ^ 2 ) } { 2 m \\tau } } \\frac { \\Gamma ( z + m \\tau ) } { \\Gamma ( 1 + m \\tau ) } . \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} L _ 2 \\leqq \\| \\nu _ { n _ k } \\| \\| [ \\mathcal H , \\varphi _ k ] \\nu ' _ { n _ k } \\| \\leqq C | \\varphi ' _ k | _ { \\infty } \\| \\nu _ { n _ k } \\| ^ 2 = O ( \\epsilon ) . \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} B _ \\lambda ( s ) = \\frac { \\pi ^ { n ( n + 1 ) / 2 } } { \\prod _ { m = 1 } ^ n ( m - 1 ) ! } \\ , i ^ { n k } \\ , 2 ^ { - n ( 2 n + 1 ) s + 3 n / 2 } \\ , \\gamma _ n \\Big ( ( 2 n + 1 ) s - \\frac 1 2 + k \\Big ) , \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} & \\left \\lfloor \\frac { 1 + q ^ 3 } { q ^ 3 + 1 } \\right \\rfloor + \\left \\lfloor \\frac { q ^ 5 - q ^ 3 + q ^ 2 - 2 } { q ^ 3 + 1 } \\right \\rfloor + ( q ^ 2 - q ) \\left \\lfloor \\frac { q ^ 3 } { q ^ 2 - q + 1 } \\right \\rfloor + \\left \\lfloor \\frac { - q ^ 6 } { q ^ 3 + 1 } \\right \\rfloor \\\\ = \\quad & 1 + q ^ 2 - 2 + ( q ^ 2 - q ) q - q ^ 3 = - 1 < 0 . \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} F ^ { ( m ) } _ r ( s ) = \\sum _ { j = 0 } ^ r ( - 1 ) ^ { r - j } q ^ { ( r - j ) ( r - j - 1 ) } { n - j \\brack n - r } { n - s \\brack j } \\ , c ^ j , \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} Q ^ { - } ( u ) = - \\operatorname { d i v } \\left ( \\frac { \\nabla u } { \\sqrt { 1 - { | \\nabla u | ^ 2 } } } \\right ) . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{gather*} \\phi ( \\tau ) = \\ ( \\frac { \\eta ( \\tau / 3 ) ^ 3 } { \\eta ( 3 \\tau ) ^ 3 } + 3 \\ ) \\phi _ 1 ( \\tau ) . \\end{gather*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\varTheta _ \\lambda ^ \\epsilon : = \\bigl \\{ & f : [ 0 , \\tau ] \\to \\mathbb { R } ~ | ~ f ( t ) \\geq \\epsilon , \\ ; \\forall t \\in [ 0 , \\tau ] \\ \\\\ & \\big | f ( t ) - f ( s ) \\big | \\leq ( L - \\epsilon ) | t - s | , \\forall t , s \\in [ 0 , \\tau ] \\bigr \\} . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\mathrm { d i m } _ \\nu ( V _ k ) & = \\dfrac { 1 - \\nu ^ n } { 1 - \\nu ^ { n _ k } } , & \\mathrm { d i m } _ \\nu ( W _ k ) & = \\prod _ { l \\neq k } \\dfrac { 1 - \\nu ^ n } { 1 - \\nu ^ { n _ l } } . \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} \\begin{cases} \\underbar s _ 1 ^ k - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 1 ^ k + \\epsilon \\cr \\underbar s _ 2 ^ k - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar s _ 2 ^ k + \\epsilon \\end{cases} \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { \\infty } \\gamma _ { i , \\pm r } ^ { \\pm } z ^ { \\pm r } = q _ i ^ { \\deg ( Q _ i ) - \\deg ( R _ i ) } \\frac { Q _ i ( z q _ i ^ { - 1 } ) R _ i ( z q _ i ) } { Q _ i ( z q _ i ) R _ i ( z q _ i ^ { - 1 } ) } \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\mu _ i : = m _ i ^ { - 1 } e ( A _ i , \\Phi _ i ) \\mathcal { H } ^ 3 . \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} \\sum _ { m \\in \\Z } \\Omega _ p ( \\beta _ m ) \\mathrm { v o l } ( \\Gamma _ 0 \\beta _ m \\Gamma _ 0 ) = ( 1 - p ^ { - 1 } ) \\frac { 1 - w _ p } { p ^ 2 + w _ p } . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} \\tilde { F } = \\pm \\left ( \\frac { 2 } { 2 \\sigma _ { p , q } - p } \\tilde { J } - \\frac { p } { 2 \\sigma _ { p , q } - p } I \\right ) . \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} \\R \\langle A z _ 0 , z _ 0 \\rangle = - \\int _ { \\Omega _ + } | \\nabla w ( x , y ) | ^ 2 \\ , \\dd ( x , y ) \\le 0 , \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\mathcal W _ { B , 2 } ( 1 ) = \\begin{cases} 2 ^ { - 7 / 2 } \\sum _ { n = 0 } ^ { \\mathrm { m i n } ( \\mathrm { o r d } _ 2 ( b _ i ) ) } 2 ^ { n / 2 } \\Psi _ 2 ( 2 ^ { - 2 n + 2 } \\xi ; \\alpha _ 2 ) & b _ 1 , b _ 2 , b _ 3 \\in \\Z _ 2 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} ( \\tilde { \\omega } _ t ^ \\natural + \\tilde { \\eta } _ t ) ^ { m + n } = c _ t e ^ { \\tilde { G } _ t + \\tilde { H } _ t } ( \\tilde { \\omega } _ t ^ \\natural ) ^ { m + n } , \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\begin{aligned} \\abs { \\Sigma _ { 2 2 } } + \\abs { \\Sigma _ 3 } \\lesssim X ^ { 1 + \\epsilon } \\Big ( u ^ { - \\frac { 1 } { 2 } } + \\abs { m } ^ { \\frac { 1 } { 4 ( 2 ^ d - 1 ) } } X ^ { - \\frac { d - 1 } { 8 ( 2 ^ d - 1 ) } } + X ^ { - \\frac { 1 } { 2 ^ { d + 2 } } } + \\big ( \\varphi _ 1 ( X ) \\sigma _ 1 ( X ) \\big ) ^ { - \\frac { 1 } { 2 ^ { d + 1 } } } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\mathcal I _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { L ( 1 , \\pi _ v , \\mathrm { a d } ) L ( 1 , \\tau _ v , \\mathrm { a d } ) } { L ( 1 / 2 , \\pi _ v \\times \\mathrm { a d } \\tau _ v ) } \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) , \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} \\phi X = T X + F X , \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} \\begin{pmatrix} a ' - a & b _ 1 - b ' _ 1 & c _ 2 - c ' _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { L ^ { p } \\left ( 0 , T ; X \\right ) } = \\left ( \\int _ { 0 } ^ { T } \\left \\Vert u \\left ( t \\right ) \\right \\Vert _ { X } ^ { p } d t \\right ) ^ { 1 / p } < \\infty \\quad 1 \\le p < \\infty , \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} \\inf \\mathrm { ( R P ) } _ { 0 _ { X } ^ { \\ast } } = \\lim _ { r } \\inf \\left \\{ f ( x ) : { g _ { t } } ( x ) \\leq 0 , \\ \\forall t \\in S _ { r } \\right \\} , \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} K = K ( h ) = \\lfloor \\varepsilon h \\log 4 \\rfloor . \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\sum _ { d = 1 } ^ { \\infty } \\gamma _ d ^ { \\frac { 1 } { \\alpha - 2 \\delta } } b ^ { w _ d } < \\infty . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} 0 & = 1 - \\sum _ { i = 1 } ^ n \\dfrac { x _ i ^ 2 } { t _ i } = 1 - \\lambda \\sum _ { i = 1 } ^ n t _ i = 1 - \\lambda = ( 1 + \\mu ) ( 1 - \\mu ) \\\\ & = ( 1 + \\sum _ { i = 1 } ^ n \\epsilon _ i x _ i ) ( 1 - \\sum _ { i = 1 } ^ n \\epsilon _ i x _ i ) . \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} I ( q \\ , | \\ , a , \\tau ) = \\log \\frac { G ( 1 + a + \\tau \\ , | \\ , \\tau ) } { G ( 1 - q + a + \\tau \\ , | \\ , \\tau ) } - q \\log \\Bigl [ \\Gamma \\bigl ( 1 + \\frac { a } { \\tau } \\bigr ) \\Bigr ] + \\frac { ( q ^ 2 - q ) } { 2 \\tau } \\psi \\bigl ( 1 + \\frac { a } { \\tau } \\bigr ) . \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{gather*} \\tau \\partial _ t u - \\Delta u + \\xi + \\pi ( u ) = f \\mbox { a . e . \\ i n } Q , \\\\ \\mu _ \\Gamma = \\partial _ { \\boldsymbol { \\nu } } u - \\Delta _ \\Gamma u _ \\Gamma + \\xi _ \\Gamma + \\pi _ \\Gamma ( u _ \\Gamma ) - f _ \\Gamma \\mbox { a . e . \\ o n } \\Sigma , \\\\ u ( 0 ) = u _ 0 \\mbox { a . e . \\ i n } \\Omega , u _ \\Gamma ( 0 ) = u _ { 0 \\Gamma } \\mbox { a . e . \\ o n } \\Gamma . \\end{gather*}"}
-{"id": "1807.png", "formula": "\\begin{align*} \\langle f , h \\rangle & = \\langle \\pi ( z ) ^ * f , \\pi ( z ) ^ * h \\rangle = \\sum _ { \\lambda \\in \\Lambda } \\langle \\pi ( z ) ^ * f , \\pi ( \\lambda ) g \\rangle \\langle \\pi ( \\lambda ) \\gamma , \\pi ( z ) ^ * h \\rangle \\\\ & = \\sum _ { \\lambda \\in \\Lambda } V _ g f ( z + \\lambda ) \\overline { V _ { \\gamma } h ( z + \\lambda ) } = \\Phi ( z ) \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} { \\rm E x p } _ S ( \\xi ) : = \\Gamma _ { ( S , \\xi ) } ( 1 ) = S ^ { \\frac { 1 } { 2 } } \\exp ( S ^ { - \\frac { 1 } { 2 } } \\xi S ^ { - \\frac { 1 } { 2 } } ) S ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} \\mathrm { e v } _ { v = 1 } \\colon \\mathcal { A } _ v [ N _ - ] \\to \\mathbb { C } \\otimes _ { \\mathbb { Z } [ v ^ { \\pm 1 / 2 } ] } \\mathcal { A } _ v [ N _ - ] \\simeq \\mathbb { C } [ N _ - ] , \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} \\mathfrak U = [ \\vec { \\mathfrak u _ 1 } \\ \\vec { \\mathfrak u _ 2 } \\ \\dots \\ \\vec { \\mathfrak u _ N } ] ^ T . \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\dot { \\phi } ( n ) = \\frac { ( - 1 ) ^ n } { ( n + 1 ) ^ { \\alpha + 1 } } , \\quad \\forall n \\in \\N , \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} K _ \\phi ( \\lambda ) ^ T \\otimes I _ n = \\begin{bmatrix} - \\alpha _ { d - 2 } & & & \\\\ \\lambda - \\beta _ { d - 2 } & - \\alpha _ { d - 3 } \\\\ - \\gamma _ { d - 2 } & \\lambda - \\beta _ { d - 3 } & \\ddots \\\\ & \\ddots & \\ddots & - \\alpha _ 0 \\\\ & & - \\gamma _ 1 & \\lambda - \\beta _ 0 \\end{bmatrix} \\otimes I _ n \\in \\mathbb { F } [ \\lambda ] ^ { d n \\times ( d - 1 ) n } . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} \\mu _ M ^ H & = \\frac { 3 } { 2 } + \\cos ^ 2 \\biggl ( \\frac { 5 \\pi } { 8 } \\biggr ) + \\cos \\biggl ( \\frac { 5 \\pi } { 8 } \\biggr ) \\cdot \\sin \\biggl ( \\frac { 5 \\pi } { 8 } \\biggr ) \\\\ & = 2 - \\frac { 1 } { \\sqrt { 2 } } \\\\ & \\cong 1 , 2 9 . \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} m _ A \\circ r = m _ A . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( c _ 2 ^ { - 2 } - c _ 1 ^ { - 2 } ) \\varphi d y = 0 \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} T \\tilde { N } = \\{ \\tilde { J } ( R a d T \\acute { N } ) \\oplus \\tilde { J } ( l t r ( T \\acute { N } ) ) \\} \\bot \\mu _ { 0 } \\bot \\{ R a d ( T \\acute { N } ) \\oplus l t r ( T \\acute { N } ) \\} . \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} I _ { s , t } : = \\int _ s ^ t \\exp ( - X _ u ) d u , 0 \\leq s < t \\leq \\infty , \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} ( \\sigma ^ { a _ { 1 } } \\tau ^ { a _ { 2 } } ) ^ { n } = \\rho ^ { \\frac { 1 } { 2 } a _ { 1 } a _ { 2 } n ( n - 1 ) } \\sigma ^ { n a _ { 1 } } \\tau ^ { n a _ { 2 } } . \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} E ^ { n } U ( t ) = U ( t + n ) , \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} A \\otimes B = \\left [ \\begin{array} { c c c } a _ { 1 1 } B & \\cdots & a _ { 1 n } B \\\\ \\vdots & \\ddots & \\vdots \\\\ a _ { m 1 } B & \\cdots & a _ { m n } B \\end{array} \\right ] . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} { } _ 0 ^ C \\mathbb { D } _ x ^ { \\varrho ( x ) } u ( x ) = \\int _ 0 ^ x { \\frac { 1 } { { \\Gamma ( n - \\varrho ( x - r ) ) } } \\frac { { u ^ { ( n ) } ( r ) d r } } { { ( x - r ) ^ { \\varrho ( x - r ) - n + 1 } } } } , \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} \\widehat { I } : = \\{ ( i , r ) \\in I \\times \\mathbb { Z } \\mid r - \\xi ( i ) \\in 2 \\mathbb { Z } \\} . \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\mathcal { B } ( t , x ) = \\mathcal { C } ( 1 + t , x ) = \\frac { C ( ( 1 + t ) x ) } { 1 - x C ( ( 1 + t ) x ) } , \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} L M ( s , f ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda _ f ( n ) } { n ^ { s } } c ( n ) , \\textrm { w h e r e } c ( n ) = \\sum _ { d | n } \\mu ( d ) F _ { \\Upsilon , M } ( d ) , \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} \\sigma ^ { A B } ( t ) = \\frac { \\rho ( v + \\beta t ) - \\rho _ { - } v _ { - } } { \\rho - \\rho _ { - } } = v + \\frac { \\rho _ { - } ( v - v _ { - } ) } { \\rho - \\rho _ { - } } + \\beta t = v _ { - } + \\frac { \\rho ( v - v _ { - } ) } { \\rho - \\rho _ { - } } + \\beta t . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} F ( u , n ) & = \\eta _ T ( t ) n u q ( t ) = \\eta _ T ( t ) \\Bigl [ \\int _ 0 ^ t e ^ { i ( t - t ^ \\prime ) \\Delta } F ( u , n ) \\d t ' \\Bigr ] _ { x = 0 } \\\\ n _ \\pm & = \\mp i \\int _ 0 ^ t e ^ { \\pm i ( t - t ^ \\prime ) D } G ( u ) \\d t ' \\\\ G ( u ) & = \\eta _ T ( t ) D ^ { - 1 } | u | ^ 2 , z ( t ) = \\eta _ T ( t ) [ n _ + + n _ - ] _ { x = 0 } \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} m _ X ( ( a , b ) \\circledcirc ( c , x ) ) = a \\circ b \\circ c \\circ x , \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} 1 = \\liminf _ { n \\to \\infty } C ^ { a _ n } ( 0 ) \\le \\liminf _ { n \\to \\infty } \\sqrt { 2 \\pi } n ^ { 1 / 2 } P ^ { a _ n } _ { x _ n } ( n ) \\le \\limsup _ { n \\to \\infty } \\sqrt { 2 \\pi } n ^ { 1 / 2 } P ( n ) = 1 \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} 2 \\int _ { \\R ^ d \\times \\R ^ d } \\Delta ^ 2 _ x \\psi ( x , y ) m _ { \\nabla u _ \\mu } ( t , x ) m _ { u _ \\iota } ( t , y ) \\ , d x d y & \\\\ = - 2 \\int _ { \\R ^ d \\times \\R ^ d } \\nabla _ x \\cdot \\nabla _ y \\Delta _ x \\psi ( x , y ) m _ { \\nabla u _ \\mu } ( t , x ) m _ { u _ \\iota } ( t , y ) \\ , d x d y & \\\\ = - 2 \\int _ { \\R ^ d \\times \\R ^ d } \\Delta _ x \\psi ( x , y ) \\nabla _ x m _ { \\nabla u _ \\mu } ( t , x ) \\cdot \\nabla _ y m _ { u _ \\iota } ( t , y ) \\ , d x d y . & \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} \\varphi = \\begin{bmatrix} x _ 1 ^ { d _ 1 } & x _ 1 ^ { d _ 2 } & \\cdots & x _ 1 ^ { d _ n } \\\\ x _ 2 ^ { d _ 1 } & x _ 2 ^ { d _ 2 } & \\cdots & x _ 2 ^ { d _ n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ x _ n ^ { d _ 1 } & x _ n ^ { d _ n } & \\cdots & x _ n ^ { d _ n } \\\\ g _ 1 & \\cdots & \\cdots & g _ n \\end{bmatrix} , \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\Sigma ( a ) = \\alpha ^ { * } , & \\Sigma ( a ^ { * } ) = \\alpha , & \\Sigma ( c ) = - q \\gamma , \\\\ & \\Sigma ( c ^ { * } ) = - q ^ { - 1 } \\gamma ^ { * } . \\end{array} \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ x \\log \\left [ \\frac { \\partial _ t u ( t , x ) - \\mu ^ { - 1 / 2 } \\partial _ x ^ 2 u ( t , x ) } { u ( t , x ) } \\right ] = 0 \\end{array} \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} \\begin{aligned} E _ { - \\beta _ t } \\ , \\mathfrak { m } _ \\lambda ^ { ( n ^ 2 ) } = 0 , \\ 1 \\leq t \\leq n ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} _ { q / 2 } \\Bigl ( 1 + \\frac { ( c _ 2 ^ 2 + c _ 1 c _ 3 ) ^ 3 } { c _ 3 ^ 2 ( c _ 1 c _ 2 + c _ 0 c _ 3 ) ^ 2 } \\Bigr ) = 1 . \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} 0 \\le q : = \\sum _ { n = 1 } ^ N \\frac { p _ n } { 1 - p _ 0 } q _ n \\le \\sum _ { n = 1 } ^ N \\frac { p _ n } { 1 - p _ 0 } = 1 \\ : . \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} D _ { 0 } = \\frac { 2 D _ { 1 } ^ { \\prime } \\sigma _ { \\mathcal { F } } ^ { 2 } } { M } , D _ { 1 } = N _ { 0 } 2 ^ { N _ { 0 } } M _ { N _ { 0 } } , D _ { 2 } = \\frac { D _ { 2 } ^ { \\prime \\prime } P _ { N _ { 0 } } ^ { 2 } } { ( 1 + M + D _ { 0 } ) ^ { 2 N _ { 0 } } } , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} \\frac { n } { 2 q _ 1 } + \\frac { 1 } { q _ 2 } = 1 - \\kappa _ 1 \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{gather*} a _ { k + 1 } ( p ) = \\big ( a + b \\sqrt { - 3 } \\big ) ^ k + \\big ( a - b \\sqrt { - 3 } \\big ) ^ k , a \\equiv 1 \\mod 3 , \\end{gather*}"}
-{"id": "5275.png", "formula": "\\begin{align*} B ^ { ( f ) } _ { 1 } ( x ) - B ^ { ( f ) } _ { 1 } ( x + y ) = f ( 0 ) \\ , y , \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\nu = K _ 3 \\ast G _ 4 ( \\nu ) \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} l ( c ) : = \\sup \\{ \\sum _ { i = 1 } ^ { k } \\ ; d ( c ( t _ { i - 1 } ) , c ( t _ i ) ) \\ : | \\ : a = : t _ 0 < t _ 1 < \\dots < t _ { k - 1 } < t _ k : = b \\} . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle \\int _ { S } | U _ * | ^ { 2 ^ * } \\\\ = \\displaystyle \\int _ S V ^ { 2 ^ * } - \\displaystyle 2 ^ * \\int _ S V ^ { 2 ^ * - 1 } u _ 0 + O ( \\int _ S v ^ { 2 ^ * - 1 - \\delta } u _ 0 ^ \\delta ) \\\\ = \\displaystyle \\int _ S V ^ { 2 ^ * } - \\displaystyle 2 ^ * \\int _ S V ^ { 2 ^ * - 1 } u _ 0 + O ( \\lambda ^ { - \\frac { ( 1 + \\delta ) ( N - 2 ) } { 2 } } ) , \\\\ \\end{array} \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} f ( s , m ) = - \\binom { m + 1 } { m - s + 1 } & N _ { \\mathrm { d } \\ , m - s + 1 } - \\binom { m + 1 } { m - s } N _ { \\mathrm { d } \\ , m - s } \\mathcal { C } _ { s } ^ { s + 1 } - \\cdots - \\binom { m + 1 } { m + 1 - ( s + r ) } N _ { \\mathrm { d } \\ , m + 1 - ( s + r ) } \\mathcal { C } _ { s } ^ { s + r } - \\cdots \\\\ & - \\binom { m + 1 } { 2 } N _ { \\mathrm { d } 2 } \\mathcal { C } _ { s } ^ { m - 1 } - \\binom { m + 1 } { 1 } N _ { \\mathrm { d } 1 } \\mathcal { C } _ { s } ^ { m } , \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} - \\Delta u - ( 2 ^ { * } - 1 ) U _ { 0 , \\Lambda } ^ { 2 ^ { * } - 2 } u = 0 , \\ , \\ , \\ , \\ , \\ , \\ , \\R ^ { N } , \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\tilde { g } _ { \\tilde { z } , t } = g _ { \\C ^ m } + \\epsilon _ t ^ 2 g _ { Y , z } \\ ; \\ , ( \\epsilon _ t = d _ t ^ { - 1 } \\delta _ t , z = d _ t \\lambda _ t ^ { - 1 } \\tilde { z } ) , \\ ; \\ , \\tilde { \\omega } _ t ^ \\natural = \\omega _ { \\C ^ m } + \\epsilon _ t ^ 2 \\Theta _ t ^ * \\Psi _ t ^ * \\omega _ F . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} d _ r : = \\det ( \\tau _ { 1 + j - i } ' ) _ { 1 \\leq i , j \\leq r } \\textrm { a n d } b _ r : = ( \\tau _ r ' ) ^ 2 + 2 \\sum _ { i \\geq 1 } ( - 1 ) ^ i \\tau _ { r + i } ' \\tau _ { r - i } ' , \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} H _ { t , n } ( \\chi ) = \\begin{cases} \\chi \\bigl ( \\frac { t + p ^ s \\omega } { 2 } \\bigr ) \\bigl ( 2 + \\alpha \\frac { p ^ e + p ^ { e - 1 } - 2 } { p - 1 } \\bigr ) & r \\geq 2 e , \\\\ \\chi \\bigl ( \\frac { t + \\sqrt { d } \\ell } { 2 } \\bigr ) + \\chi \\bigl ( \\frac { t - \\sqrt { d } \\ell } { 2 } \\bigr ) & r < 2 e \\psi _ d ( p ) = 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} \\Omega _ { \\Phi ( n ) - k - 2 } = \\Omega _ { \\Phi ( n ) - k - 1 } + 2 ( - 1 ) ^ { k + 1 } { n \\choose k + 1 } . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} R _ \\mathcal { L } ( \\lambda ) = \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ - \\widehat { N } _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T \\end{bmatrix} R _ Q ( \\lambda ) , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} \\lim \\limits _ { A \\rightarrow 0 } u _ * ^ { A } = \\lim \\limits _ { A \\rightarrow 0 } \\sigma _ 1 ^ { A } = \\lim \\limits _ { A \\rightarrow 0 } \\sigma _ 2 ^ { A } = \\widehat { \\sigma _ { 0 } ^ { B } } + \\beta t , \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} R ( \\alpha ) ~ = ~ \\sum _ { x \\in \\Z ^ d } e ^ { \\alpha \\cdot x } \\mu ( x ) \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} a _ { m } ^ * = [ a ^ * _ { m , 1 } \\cdots a ^ * _ { m , { f _ { m - 1 } \\choose r _ { m } } } ] a _ { m - 1 } = \\begin{bmatrix} a _ { m - 1 , 1 } \\\\ \\vdots \\\\ a _ { m - 1 , { f _ { m - 2 } \\choose r _ { m - 1 } } } \\end{bmatrix} . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} r B ( v , e ) ( x _ v ^ * ) ^ p = w ( e ) , \\ v \\in e . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} E ^ { ( r ) } _ i 1 _ { \\lambda } & : T \\mapsto \\sum _ { R \\subseteq T ^ { i + 1 } , | R | = r } c _ { i , R } T \\\\ F ^ { ( r ) } _ i 1 _ { \\lambda } & : T \\mapsto \\sum _ { R \\subseteq T ^ { i } , | R | = r } c _ { i + 1 , R } T \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} V \\left ( \\mathbf { D } _ { 3 } \\right ) = M o d \\left ( E q \\left ( \\mathbb { A O L } \\right ) \\cup \\left \\{ \\right \\} \\right ) . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} h _ \\theta ( x ) = \\sqrt { 2 \\pi } \\sum _ { m = 1 } ^ \\infty \\frac { t a n \\ ; \\eta _ m } { \\eta _ m t a n \\ ; \\eta _ m - \\theta - 1 } e ^ { - \\eta _ m | x | } \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} V ^ { \\otimes d } = \\bigoplus _ { \\lambda \\in \\Lambda ^ { + } ( n , d ) } V ^ { \\lambda } \\otimes L ^ { \\lambda } \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\sum _ { j | k } \\mu ( k / j ) \\left ( \\prod _ { i = 0 } ^ { j a - 1 } \\binom { m j / n + i } { b j / n } \\binom { b j / n + i } { b j / n } ^ { - 1 } \\right ) ^ { n / j } , \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} N _ { w , G } ^ { \\chi } = \\langle N _ { w , G } , \\chi \\rangle = \\frac { 1 } { | G | } \\sum _ { g \\in G } N _ { w , G } ( g ) \\overline { \\chi } ( g ) = \\frac { 1 } { | G | } \\sum _ { ( g _ { 1 } , \\cdots , g _ { r } ) \\in G ^ { r } } \\overline { \\chi } ( w ( g _ { 1 } , \\cdots , g _ { r } ) ) , \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( 1 + q ) } \\frac { S _ 2 ( \\tau - q \\ , | \\ , \\tau ) } { S _ 2 ( \\tau \\ , | \\ , \\tau ) } = & \\frac { \\Gamma _ 2 ( 1 + q + \\tau \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( 1 - q + \\tau \\ , | \\ , \\tau ) } \\ , \\frac { \\tau ^ { q / \\tau } } { \\Gamma ( 1 - q / \\tau ) } , \\\\ \\tau ^ { q / \\tau } \\frac { \\Gamma _ 2 ( \\tau - q \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( \\tau \\ , | \\ , \\tau ) } = & \\Gamma ( 1 - \\frac { q } { \\tau } ) \\ , \\frac { \\Gamma _ 2 ( 1 - q + \\tau \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( 1 + \\tau \\ , | \\ , \\tau ) } . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} 0 < \\underline { M } \\left | \\xi \\right | ^ { 2 } \\le \\sum _ { i , j = 1 } ^ { N } A _ { i j } \\left ( x , t ; \\mathbf { p } ; \\mathbf { q } \\right ) \\xi _ { i } \\xi _ { j } \\le \\overline { M } \\left | \\xi \\right | ^ { 2 } , \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} \\mathcal B ( \\Theta _ z ) & = \\sigma \\ ( A \\iota _ z , \\ , R _ \\alpha \\iota _ z : \\alpha \\in \\mathbf R ^ { J ( z ) } \\cap \\ker A \\ ) \\subset \\iota _ z ^ { - 1 } \\sigma \\ ( A , \\ , R _ \\alpha : \\alpha \\in \\mathbf R ^ { J ( z ) } \\cap \\ker A \\ ) \\\\ & \\subset \\iota _ z ^ { - 1 } \\sigma \\ ( A , \\ , R _ \\alpha , \\ , Z : \\alpha \\in \\mathbf \\ker A \\ ) \\subset \\sigma \\ ( A , \\ , R _ \\alpha , \\ , Z : \\alpha \\in \\mathbf \\ker A \\ ) . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} ( { \\rm g w } _ t ) _ * \\pi _ { \\mathfrak { k } ^ * } = ( h ^ { - 1 } \\circ \\gamma ) _ * ( t \\pi _ { \\mathfrak { k } ^ * } ) = t \\pi _ { K ^ * } . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} \\omega _ f ( x ) : = \\sum _ { d l ^ 2 \\leq x } \\frac { \\lambda _ f ( d ^ 2 ) } { d l ^ 2 } . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} \\dot { x } = g ( t , x ) , \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} h _ t ( v ) : = f _ t \\big ( v - \\int _ 0 ^ t \\ ! \\mathrm { d } s \\ , G ( M _ 1 ( s ) ) \\big ) \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} E ^ m ( g , \\phi ) = L ^ m ( \\phi ) , \\phi \\in \\Phi \\ , , \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\sqrt { J } = \\sqrt { I _ { r } ( \\delta ) } \\ , . \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} { } ^ g ( h , a ) = ( g h , a ) \\mbox { f o r a l l } g \\in G \\mbox { a n d } ( h , a ) \\in E . \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { \\substack { j \\in J _ { K } } } \\Phi ( B _ { t _ { j } } , t _ { j } ) ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) = 0 \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} Z ( s , f ^ { ( Q _ \\tau ) } , \\phi ) ( g ) = & \\frac { L ^ N ( ( 2 n + 1 ) s + 1 / 2 , \\pi \\boxtimes \\chi , \\varrho _ { 2 n + 1 } ) } { L ^ N ( ( 2 n + 1 ) ( s + 1 / 2 ) , \\chi ) \\prod _ { j = 1 } ^ n L ^ N ( ( 2 n + 1 ) ( 2 s + 1 ) - 2 j , \\chi ^ 2 ) } \\\\ & \\times i ^ { n k } \\ , \\chi ( \\tau ) ^ { - n } \\ , \\pi ^ { n ( n + 1 ) / 2 } \\bigg ( \\prod \\limits _ { p | N } { \\rm v o l } ( \\Gamma _ { 2 n } ( p ^ { m _ p } ) ) \\bigg ) A _ { \\mathbf { k } } ( ( 2 n + 1 ) s - 1 / 2 ) \\phi ( g ) , \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{gather*} h _ 3 ( \\tau ) = \\eta ( 2 \\tau ) ^ 3 \\eta ( 6 \\tau ) ^ 3 = \\sum _ { k \\ge 0 , m , n \\in \\Z \\atop { m \\equiv n \\mod 2 } } ( - 3 ) ^ k \\big ( ( 3 m + 1 ) ^ 2 - 3 n ^ 2 \\big ) q ^ { 3 ^ k ( ( 3 m + 1 ) ^ 2 + 3 n ^ 2 ) } . \\end{gather*}"}
-{"id": "9572.png", "formula": "\\begin{align*} K _ 0 ( C ^ * _ r G ) = \\bigoplus _ { j \\in \\Z } \\Z \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} \\mathcal W _ { B , p } ( 1 ) = \\begin{cases} \\sum _ { n = 0 } ^ { \\mathrm { m i n } ( \\mathrm { o r d } _ p ( b _ i ) ) } p ^ { n / 2 } \\Psi _ p ( p ^ { - 2 n } \\xi ; \\alpha _ p ) & b _ 1 , b _ 2 , b _ 3 \\in \\Z _ p , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\mathbf { \\Gamma } _ k = k { \\pi } _ { 2 , k } ^ * \\Gamma _ 2 + ( k + k - 1 ) { \\pi } _ { 3 , k } ^ * \\Gamma _ 3 + \\cdots + ( k + \\cdots + 3 ) { \\pi } ^ * _ { k - 1 , k } \\Gamma _ { k - 1 } + ( k + \\cdots + 2 ) \\Gamma _ k \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} F ( z , z ' ) - F ( z , a ) - F ( a , z ' ) + F ( a , a ) = q _ z ^ * q _ { z ' } . \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\zeta _ j ^ { k , t } \\cdot \\zeta _ j ^ { k , t } = 0 j = 1 , 2 , \\zeta _ 1 ^ { k , t } + \\zeta _ 2 ^ { k , t } = - 2 \\pi i k . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} Z ( i _ { k ' \\leq l ' } ^ { k \\leq l } ) = z _ { k ' \\leq l ' } ^ { k \\leq l } , L ( i _ { k ' \\leq l ' } ^ { k \\leq l } ) = l _ { k ' \\leq l ' } ^ { k \\leq l } , \\quad R ( i _ { k ' \\leq l ' } ^ { k \\leq l } ) = r _ { k ' \\leq l ' } ^ { k \\leq l } . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} ( \\sigma \\star f ) ( \\gamma _ 1 , \\ldots , \\gamma _ { p + q } ) = \\sigma ( \\gamma _ 1 , \\ldots , \\gamma _ p ) f ( \\gamma _ { p + 1 } , \\ldots , \\gamma _ { p + q } ) , \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} v _ { i j } = \\begin{cases} C , & ( i , j ) = ( 0 , p ) , \\\\ \\frac { C } { p - | I | } , & i \\in I j = p , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\nu \\biggl ( \\bigcap _ { i = 1 } ^ r T ^ { \\mathbf { n } _ i } B _ i \\biggr ) \\longrightarrow \\prod _ { i = 1 } ^ r \\nu ( B _ i ) \\enspace \\textup { a s } \\enspace \\| \\mathbf { n } _ i - \\mathbf { n } _ j \\| \\to \\infty \\enspace \\textup { f o r } \\enspace 1 \\le i < j \\le r . \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} \\Phi _ { s , t } \\circ \\Phi _ { r , s } = \\Phi _ { r , t } 0 \\le r \\le s \\le t \\le T . \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} \\mathcal { F C } _ P : = \\big \\{ F : H ^ { - 1 - } \\to \\R \\ | \\ , \\exists \\ , \\Lambda \\Subset \\Z ^ 2 _ 0 \\mbox { a n d } f \\in C _ P ^ \\infty ( H _ \\Lambda ) \\mbox { s . t . } F = f \\circ \\Pi _ \\Lambda \\big \\} . \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k \\geq 0 } P _ k ( x ) \\otimes \\tilde { B } e _ k , Q ( x ) = \\sum _ { k \\geq 0 } Q _ k ( x ) \\otimes \\tilde { A } e _ k , \\forall x \\in X , \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} \\sum _ i x _ i x _ i ^ * = { N \\over k } I _ k , \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\sum _ { n \\in \\Z } \\Omega _ p ( \\alpha _ n ) \\mathrm { v o l } ( \\Gamma _ 0 \\alpha _ n \\Gamma _ 0 ) + \\sum _ { m \\in \\Z } \\Omega _ p ( \\beta _ m ) \\mathrm { v o l } ( \\Gamma _ 0 \\beta _ m \\Gamma _ 0 ) . \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} { \\mathbb D } ^ { B } { } _ { A } { \\mathbb D } ^ { D } { } _ { C } - \\ , { \\mathbb D } ^ { D } { } _ { C } { \\mathbb D } ^ { B } { } _ { A } = \\tfrac { 2 } { 3 } \\delta _ { [ A } { } ^ { ( D } [ { \\mathbb D } , { \\mathbb D } ] _ { C ] } { } ^ { B ) } - \\tfrac { 2 } { 3 } \\delta _ { ( C } { } ^ { [ B } [ { \\mathbb D } , { \\mathbb D } ] _ { A ) } { } ^ { D ] } . \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} \\omega _ 1 & : = d x _ 1 \\wedge ( d x _ 2 - d x _ 3 ) + ( d x _ 3 + d x _ 2 ) \\wedge d x _ 4 , \\\\ \\omega _ 2 & : = d x _ 1 \\wedge ( d x _ 2 + d x _ 3 ) + ( d x _ 3 - d x _ 2 ) \\wedge d x _ 4 . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} \\varphi ( \\alpha , \\beta ; Z ; u ) = \\left ( \\displaystyle \\prod _ { i = 1 } ^ n u _ i ^ { \\alpha _ i - 1 } \\right ) \\displaystyle \\prod _ { j = 1 } ^ l \\left ( ( Z \\cdot u ) _ j ^ { - \\beta _ j } \\right ) \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\gamma \\phi ^ 1 \\omega ^ 0 = \\gamma \\eta ^ 0 \\phi ^ 0 = 1 _ { V ^ 0 } \\circ \\phi ^ 0 = \\phi ^ 0 , \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} { \\bf P } \\bigl [ \\beta _ { M , N } ( a , b ) = 1 \\bigr ] = \\prod \\limits _ { n _ 1 , \\cdots , n _ M = 0 } ^ \\infty \\Bigl [ \\frac { b _ 0 + \\Omega } { \\prod \\limits _ { j _ 1 = 1 } ^ N b _ 0 + b _ { j _ 1 } + \\Omega } \\frac { \\prod \\limits _ { j _ 1 < j _ 2 } ^ N b _ 0 + b _ { j _ 1 } + b _ { j _ 2 } + \\Omega } { \\prod \\limits _ { j _ 1 < j _ 2 < j _ 3 } ^ N b _ 0 + b _ { j _ 1 } + b _ { j _ 2 } + b _ { j _ 3 } + \\Omega } \\cdots \\Bigr ] . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} I ( i , j ) : = \\{ \\underbrace { b - j + 1 , \\dots , b - j + i } _ , \\underbrace { b + i + 1 , \\dots , n } _ \\} . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\frac { d x } { d t } = u , \\ \\ \\frac { d u } { d t } = \\beta . \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } D _ S ( f _ t ) = - 2 D _ S ( f _ t ) + \\frac 2 \\sigma G ' ( M _ 1 ) ( G ( M _ 1 ) - M _ 1 ) ^ 2 + R \\ , , \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} R ^ D f ( x ) : = \\int _ 0 ^ \\infty \\int _ { D } p _ D ( t , x , y ) f ( y ) d y d t . \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} y _ { k } ^ { ( J - 1 ) } = y _ { k } + y _ { k + 2 ^ { J - 1 } } = y _ { 2 ^ { J } - 1 - k } + y _ { 2 ^ { J } - 1 - ( k + 2 ^ { J - 1 } ) } = y _ { 2 ^ { J - 1 } - 1 - k } ^ { ( J - 1 ) } \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { \\lim } \\underset { \\alpha \\in _ { n } \\Omega } { \\sup } ( f _ { \\alpha } ( A _ { \\mathcal { F } } , . . . , A _ { \\mathcal { F } } ) ) = 0 \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} \\Pi ( r , s ) : = \\{ [ \\mu , \\nu ] ~ | ~ \\mu \\vdash r , ~ \\nu \\vdash s , ~ l ( \\mu ) + l ( \\nu ) \\leq n \\} \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} N _ D - N _ G - \\sum _ { \\begin{subarray} { l } \\widetilde { \\Sigma } < D \\\\ \\vert \\widetilde { \\Sigma } \\rvert = 2 \\end{subarray} } N _ { \\widetilde { \\Sigma } } = - q \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} \\bar { P } ^ { i n } _ { \\mathbb { Z } ^ d , \\vec { H } } ( \\cdot ) = \\int _ { \\mathbb { R } ^ { \\mathbb { Z } ^ d } } P ^ { i n } _ { \\mathbb { Z } ^ d , \\vec { h } } ( \\cdot ) P r o b ( d \\vec { h } ) . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} [ { \\bf a } , Y ( B , w ) ] = Y ( { \\bf a } _ { w } B , w ) , \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} \\varphi \\left ( a \\right ) = \\left \\{ \\begin{array} [ c ] { l } a a \\in P \\left ( { \\mathbf { L } } \\right ) \\\\ f \\left ( a ^ { \\prime { \\mathbf { L } } } \\right ) \\end{array} \\right . \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} \\phi ( a b ) = \\phi ( b \\sigma _ { i \\beta } ( a ) ) , \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} c _ { ( i , r ) } = u _ { i , r } ( m ) \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\left | L \\left ( \\frac { 1 } { 2 } , \\chi \\right ) \\right | ^ 2 | \\psi _ { } ( \\chi ) + \\psi _ { } ( \\chi ) | ^ 2 = \\lambda \\varphi ^ + ( q ) + O \\left ( q L ^ { - 1 + \\epsilon } \\right ) , \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} ( \\mathfrak { d } _ 1 ^ m a ) _ n & = \\sum _ { j = 0 } ^ { m } \\tbinom { m } { j } ( - 1 ) ^ j a _ { n + j } . \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\chi _ { \\alpha } ^ { \\lambda } = \\sum _ { T \\in ( \\lambda , \\alpha ) } ( - 1 ) ^ { ( T ) } , \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} T ( \\Gamma ; x + 1 , y + 1 ) = \\widetilde { \\mathcal Q } ( M ; x , y , a x ^ 2 , b y ^ 2 ) . \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} ( X _ 0 , 0 ) = ( \\{ f _ { 1 , 1 } = \\cdots = f _ { m , n } = 0 \\} , 0 ) \\subset ( Z , 0 ) \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} C ( x ) = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n + 1 } \\binom { 2 n } { n } x ^ n = \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 x } . \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} \\int _ X \\lvert _ A \\Phi \\rvert ^ 2 = \\lim _ { r \\to \\infty } \\int _ { \\rho ^ { - 1 } ( 0 , r ] } \\lvert _ A \\Phi \\rvert ^ 2 . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} & E _ 1 ( x , \\zeta ) = e ^ { i ( x _ 1 \\zeta _ 1 + x _ 2 \\zeta _ 2 ) } \\left ( \\cosh ( x _ 0 | \\zeta | _ { \\C } ) - \\frac { i ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) } { | \\zeta | _ { \\C } } \\sinh ( x _ 0 | \\zeta | _ { \\C } ) \\right ) . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} N _ k ( E _ N ) \\leq \\frac { 1 } { 2 ^ k } \\sum _ { t = 1 } ^ k \\binom { k } { t } C _ t ( E _ N ) \\leq \\max _ { 1 \\leq t \\leq k } C _ t ( E _ N ) . \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} [ g , h ] ^ { n + 1 } & = [ g , h ] [ g , h ] ^ n = g h g ^ { - 1 } h ^ { - 1 } [ g , h ] ^ n = g h g ^ { - 1 } [ g , h ] ^ n h ^ { - 1 } = \\\\ & = [ g , h ^ 2 ] h [ g , h ] ^ { n - 1 } h ^ { - 1 } = [ g , h ^ 2 ] [ h g h ^ { - 1 } , h ] . \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } & a \\\\ b \\end{array} \\right ) ^ { - 1 } \\left ( \\begin{array} { c c } x \\\\ & x ^ { - 1 } \\end{array} \\right ) \\left ( \\begin{array} { c c } & a \\\\ b \\end{array} \\right ) = \\left ( \\begin{array} { c c } x ^ { - 1 } \\\\ & x \\end{array} \\right ) \\neq g ^ { s } . \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} \\max _ { \\mathclap { 1 \\leq j \\leq d + 1 } } | \\langle \\vec { g } , \\vec { f } _ { j } \\rangle | = \\frac { 1 } { \\sqrt { d } } \\quad \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\Re ( { \\bf Q } T _ G { \\bf D } p ) = \\Re ( { \\bf Q } T _ G { \\bf f } ) . \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} S _ { [ a , b ] } = \\frac { i } { 2 } ( { \\mathcal M } _ a H { \\mathcal M } _ { - a } - { \\mathcal M } _ b H { \\mathcal M } _ { - b } ) , \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} W ^ \\Lambda : = W _ 1 ^ { \\Lambda _ 1 } \\cdots W _ n ^ { \\Lambda _ n } , \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ m \\widehat { J } _ i = I + \\mathcal { O } \\left ( | \\lambda - a | ^ k \\right ) \\forall a \\in \\gamma _ 0 ~ . \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { p } a _ i \\leq \\frac { 1 } { p + 1 } \\sum _ { i = 1 } ^ { p } ( p + 1 - i ) b _ i + \\frac { 1 } { p + 1 } \\sum _ { i = 1 } ^ { p } i b _ { i + 1 } . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} f _ n = \\frac { 1 } { \\sqrt { 2 \\pi } } n ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} x _ \\ell ( s ) : = \\frac { \\mu ( \\ell ) } { \\ell ^ { s - 1 / 2 } } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\mu ^ 2 ( \\ell n ) F _ { \\Upsilon , M } ( \\ell n ) } { n ^ { 2 s } } . \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} H _ 0 : \\theta = 0 _ { l _ 2 ( \\mathbb { N } ^ * ) } \\mathrm { a g a i n s t } H _ 1 : \\theta \\in \\Theta , \\ : \\left \\| \\theta \\right \\| ^ 2 \\geq r _ \\varepsilon ^ 2 , \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} \\frac { \\partial G ( \\vec z ^ * ) } { \\partial z _ i } = - \\frac { | v _ i | \\| \\vec z ^ * \\| _ p + ( r - ( \\vec z ^ * , | \\vec v | ) ) \\| \\vec z ^ * \\| _ p ^ { 1 - p } ( z _ i ^ * ) ^ { p - 1 } } { \\| \\vec z ^ * \\| _ p ^ 2 } . \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} i \\partial _ t u = \\Pi ( | u | ^ 2 u ) . \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\int _ { \\R ^ d \\times \\R ^ d } \\nabla _ x m _ { \\nabla u _ \\mu } ( t , x ) \\cdot \\nabla _ y m _ { u _ \\iota } ( t , y ) \\Delta _ x \\psi ( x , y ) \\ , d x d y & & \\\\ = ( \\nabla m _ { \\nabla u _ { \\mu } } ( t , \\cdot ) , ( - \\Delta ) ^ { \\frac { 1 - d } 2 } \\nabla m _ { u _ \\iota } ( t , \\cdot ) ) \\geq 0 , \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} a \\cdot b = a \\circ ( \\bar { a } \\rhd b ) , a , b \\in A , \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} Z \\subseteq \\bigcap _ { n \\geq 1 } \\bigcup _ { j = 1 } ^ k \\overline { B _ { c m _ n ^ { - 1 / 2 } } ( x _ j ) } = \\bigcup _ { j = 1 } ^ k \\bigcap _ { n \\geq 1 } \\overline { B _ { c m _ n ^ { - 1 / 2 } } ( x _ j ) } = \\bigcup _ { j = 1 } ^ k \\lbrace x _ j \\rbrace . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} E _ { s } = \\xi \\theta P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } , \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\Phi _ i = \\begin{pmatrix} 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\\\ * & * & \\cdots & * \\end{pmatrix} . \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\left ( \\frac { \\Lambda ( f \\otimes \\mathrm { A d } ( g ) , k ) } { \\langle g , g \\rangle ^ 2 c ^ + ( f ) } \\right ) ^ { \\sigma } = \\frac { \\Lambda ( f ^ { \\sigma } \\otimes \\mathrm { A d } ( g ^ { \\sigma } ) , k ) } { \\langle g ^ { \\sigma } , g ^ { \\sigma } \\rangle ^ 2 c ^ + ( f ^ { \\sigma } ) } . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} \\begin{cases} \\mathrm { d } x _ t = v _ t \\mathrm { d } t \\ , , \\\\ \\mathrm { d } v _ t = - v _ t \\mathrm { d } t + \\sqrt { 2 \\sigma } \\mathrm { d } W _ t \\ , , \\end{cases} \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} & \\int \\limits _ { B _ { 3 \\cdot 2 ^ { - n } \\| M _ 1 M _ 2 \\| } ( M _ 2 ) } \\frac { \\omega ( d ( M ) ) } { d ^ 2 ( M ) } \\ , d m _ 3 ( M ) = \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\int \\limits _ { \\sigma _ { n k } } \\frac { \\omega ( d ( M ) ) } { d ^ 2 ( M ) } \\ , d m _ 3 ( M ) = \\\\ & = \\sum _ { k = k ( n ) } ^ { \\infty } \\int \\limits _ { \\sigma _ { k n } } \\frac { \\omega ( d ( M ) ) } { d ^ 2 ( M ) } \\ , d m _ 3 ( M ) . \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} R [ I t ] = R \\oplus I t \\oplus I ^ 2 t ^ 2 \\oplus \\cdots \\subset R [ t ] , \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\Pr \\left ( { \\left . { { s _ n } , { { \\tilde \\theta } _ n } , \\vartheta } \\right | { { \\bf { R } } _ { \\rm { 1 } } } } \\right ) = { m _ { \\uparrow { v _ n } } } { m _ { \\searrow { v _ n } } } { m _ { \\swarrow { v _ n } } } \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} i _ { 2 \\times 2 } ( H ^ F ) & < \\sum _ { a _ { k , n } } | a _ { k , n } ^ { H ^ F } | i _ { 2 \\times 2 } ( C _ { H ^ F } ( a _ { k , n } ) ) + \\sum _ { b _ { k , n } } | b _ { k , n } ^ { H ^ F } | i _ { 2 \\times 2 } ( C _ { H ^ F } ( b _ { k , n } ) ) + \\sum _ { c _ { k , n } } | c _ { k , n } ^ { H ^ F } | i _ { 2 \\times 2 } ( C _ { H ^ F } ( c _ { k , n } ) ) \\\\ & = \\Sigma _ a + \\Sigma _ b + \\Sigma _ c . \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{gather*} \\omega = \\frac { { \\rm d } x } { y ^ 2 } = \\frac 1 { y ^ 2 } \\frac { { \\rm d } x } { { \\rm d } \\tau } { \\rm d } \\tau = \\frac 1 { y ^ 2 } \\frac { { \\rm d } x } { { \\rm d } q } \\frac { { \\rm d } q } { { \\rm d } \\tau } { \\rm d } \\tau = 2 \\pi { \\rm i } \\ ( q \\frac 1 { y ^ 2 } \\frac { { \\rm d } x } { { \\rm d } q } \\ ) { \\rm d } \\tau = \\ ( \\frac 1 { y ^ 2 } \\frac { { \\rm d } x } { { \\rm d } q } \\ ) { \\rm d } q . \\end{gather*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\Gamma _ 0 \\alpha _ n \\Gamma _ 0 = \\bigcup _ { \\gamma , \\delta \\in \\Z _ p / p \\Z _ p } \\Gamma _ { 0 0 } \\nu _ { \\gamma } \\alpha _ n \\nu _ { \\delta } \\Gamma _ { 0 0 } , \\Gamma _ 0 \\beta _ m \\Gamma _ 0 = \\bigcup _ { \\gamma , \\delta \\in \\Z _ p / p \\Z _ p } \\Gamma _ { 0 0 } \\nu _ { \\gamma } \\beta _ m \\nu _ { \\delta } \\Gamma _ { 0 0 } . \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} \\dot h - L ( h ) L ( \\rho ) ^ { \\dagger } \\dot \\rho = 0 , h ( 0 ) = h ( 1 ) = 0 . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} h ( f \\ast _ g n ) = [ g ^ { - 1 } h g f ] \\ast _ { g } ( g ' ) ^ { - 1 } h g ' n . \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} v _ n ( x , t ) & = \\lambda _ n u ( X _ 0 + \\lambda _ n x , T _ * + \\lambda ^ 2 _ n t ) , \\\\ q _ n ( x , t ) & = \\lambda _ n ^ 2 q ( X _ 0 + \\lambda _ n x , T _ * + \\lambda ^ 2 _ n t ) . \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{gather*} \\tau > 1 , \\ ; \\lambda _ 1 , \\lambda _ 2 < 0 , \\\\ 1 + \\tau ( 1 + \\lambda _ i ) > 0 , \\ , i = 1 , 2 , \\\\ 1 + \\tau ( \\lambda _ 1 + \\lambda _ 2 ) < 0 . \\end{gather*}"}
-{"id": "6152.png", "formula": "\\begin{align*} \\| \\alpha ( t \\wedge \\tau _ m ( \\lambda ) ) - \\alpha ( s \\wedge \\tau _ m ( \\lambda ) ) \\| = \\begin{cases} \\| \\alpha ( \\tau _ m ( \\lambda ) ) - \\alpha ( s ) \\| , & t \\geq \\tau _ m ( \\lambda ) \\geq s , \\\\ \\| \\alpha ( t ) - \\alpha ( \\tau _ m ( \\lambda ) ) \\| , & t \\leq \\tau _ m ( \\lambda ) \\leq s , \\\\ \\| \\alpha ( t ) - \\alpha ( s ) \\| , & t , s \\leq \\tau _ m ( \\alpha ) , \\\\ 0 , & t , s \\geq \\tau _ m ( \\alpha ) . \\end{cases} \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\rho _ { a A } \\otimes \\frac { I _ B } { B } = \\Bigl ( 1 - \\dim ( B ) ^ { - 2 } \\Bigr ) \\mu _ { a A B } + \\dim ( B ) ^ { - 2 } \\rho _ { a A B } . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} \\begin{pmatrix} I _ l \\\\ & & I _ l \\\\ & I _ { m - 2 l } \\end{pmatrix} \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} ( i , r ) ^ + = ( i , r - 2 r _ i ) \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} \\eta _ { 2 , 2 } ( q + 1 \\ , | \\ , \\tau , b ) & = \\eta _ { 2 , 2 } ( q \\ , | \\ , \\tau , b ) \\ , \\frac { \\Gamma \\bigl ( ( q + b _ 0 + b _ 1 ) / \\tau \\bigr ) \\Gamma \\bigl ( ( q + b _ 0 + b _ 2 ) / \\tau \\bigr ) } { \\Gamma \\bigl ( ( q + b _ 0 ) / \\tau \\bigr ) \\Gamma \\bigl ( ( q + b _ 0 + b _ 1 + b _ 2 ) / \\tau \\bigr ) } , \\\\ \\eta _ { 2 , 2 } ( q + \\tau \\ , | \\ , \\tau , b ) & = \\eta _ { 2 , 2 } ( q \\ , | \\ , \\tau , b ) \\ , \\frac { \\Gamma ( q + b _ 0 + b _ 1 ) \\Gamma ( q + b _ 0 + b _ 2 ) } { \\Gamma ( q + b _ 0 ) \\Gamma \\bigl ( q + b _ 0 + b _ 1 + b _ 2 ) } . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } i n d e x ( \\Sigma ^ y _ { \\ell } ) \\leq \\liminf _ { k \\to \\infty } i n d e x ( M _ k ) . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} \\frac { \\log \\left ( t ^ { \\varepsilon } \\right ) } { t ^ { \\varepsilon } } = - C _ { 1 } \\log \\left ( \\gamma \\left ( T , \\beta \\right ) \\right ) . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} r _ \\alpha ( h ) = \\begin{cases} 1 & \\mbox { i f $ h = 0 $ , } \\\\ b ^ { - \\alpha \\psi _ b ( h ) } & \\mbox { i f $ h \\neq 0 $ , } \\end{cases} \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} - 2 d _ { j - 1 } + \\frac { d } { r } + d _ { j + 1 } + 3 t = \\frac { e } { 2 q } \\left ( \\frac { 1 } { r _ j } + \\frac { 1 } { r _ { j + 1 } } \\right ) + \\sigma _ { j - 1 } \\left ( 2 - \\frac { r } { r _ j } \\right ) + \\sigma _ { j } \\left ( 2 - \\frac { r } { r _ { j + 1 } } \\right ) + \\sum _ { i = 1 , i \\neq j - 1 , j } ^ { 2 q } 2 \\sigma _ i \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\Phi _ i ^ { - 1 } ( 0 ) \\subset \\left ( \\bigcup _ { j = 1 } ^ l B _ { c m _ i ^ { - 1 / 2 } } ( x _ j ) \\right ) \\cup \\left ( \\bigcup _ { j = l + 1 } ^ k B _ { c m _ i ^ { - 1 / 2 } } ( m _ i x _ j ) \\right ) , \\ \\ \\mathrm { d e g } ( \\Phi _ i \\Big \\vert _ { \\partial B _ { c m _ i ^ { - 1 / 2 } } ( x _ j ) } ) = 1 . \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\frac { d y } { d x } = f ( x , y ) , y ( 0 ) = 0 , \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} \\mu _ 0 ( v ) = \\frac { 1 } { \\sqrt { 2 \\pi \\sigma } } \\ , \\exp \\left [ - \\frac { v ^ 2 } { 2 \\sigma } \\right ] \\ , , \\mu _ \\pm ( v ) = \\frac { 1 } { \\sqrt { 2 \\pi \\sigma } } \\ , \\exp \\left [ - \\frac { ( v \\mp 1 ) ^ 2 } { 2 \\sigma } \\right ] \\ , . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\widehat { u ^ 1 } _ { 2 k + 1 } = - \\widehat { u ^ 0 } _ { 2 k + 1 } , k \\in \\left \\{ 0 , \\dotsc , 2 ^ j - 1 \\right \\} , \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} v _ { 0 } = 2 \\sqrt { \\frac { \\mathfrak { S } _ { 0 } } { \\rho _ { 0 } ^ { 3 } } } \\rho _ { 1 } = 2 J _ { 2 } \\sqrt { \\frac { \\mathfrak { S } _ { 0 } } { \\rho _ { 0 } } } \\rho _ { 0 } = \\frac { 1 } { 2 h } ( 4 J _ { 2 } ^ { 2 } \\mathfrak { S } _ { 0 } - J _ { 1 } ) h \\neq 0 . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\left ( u \\alpha _ { 1 } \\right ) \\left ( v \\alpha _ { 3 } \\right ) & = \\rho ^ { v _ { 2 } } u v \\alpha _ { 1 } \\alpha _ { 3 } \\ \\\\ \\left ( v \\alpha _ { 3 } \\right ) \\left ( u \\alpha _ { 1 } \\right ) & = \\rho ^ { u _ { 3 } } v u \\alpha _ { 1 } \\alpha _ { 3 } = \\rho ^ { u _ { 3 } + u _ { 2 } v _ { 3 } - u _ { 3 } v _ { 2 } } u v \\alpha _ { 1 } \\alpha _ { 3 } , \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} \\epsilon ( s , \\pi _ 1 \\times \\pi _ 2 , \\psi ) & = \\frac { \\gamma ( s , \\pi _ 1 \\times \\pi _ 2 , \\psi ) L ( s , \\pi _ 1 \\times \\pi _ 2 ) } { L ( 1 - s , \\tilde { \\pi } _ 1 \\times \\tilde { \\pi } _ 2 ) } \\\\ & = \\frac { ( \\lambda _ 1 \\lambda _ 2 ) ^ { - 1 } q ^ { - n } [ q ^ n - 1 - \\gamma ( \\sigma _ 1 \\times \\sigma _ 2 , \\psi ) ] q ^ { n s } + \\gamma ( \\sigma _ 1 \\times \\sigma _ 2 , \\psi ) } { q ^ { n s } - \\lambda _ 1 \\lambda _ 2 } \\cdot q ^ { n s } . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} P _ I : = B W _ I B , \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} M ^ f \\triangleq f ( X ) - f ( X _ 0 ) - \\int _ 0 ^ { \\cdot } \\mathcal { K } f ( X _ { s - } ) \\dd s \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} R _ { l , m } ( N ) = \\sum _ { k \\in \\Gamma _ { N } } C _ { k , l } C _ { k , m } \\big ( \\ < \\omega , e _ k e _ { - l } \\ > \\ < \\omega , e _ k e _ { - m } \\ > - \\delta _ { l , m } \\big ) \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} \\mathcal I _ v ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) ^ { - 1 } = \\begin{cases} p & v = p , \\ , p \\mid N / M , \\\\ \\frac { p ( p + 1 ) } { 2 } & v = p , \\ , p \\mid M , \\\\ 1 & v = 2 v = \\infty . \\\\ \\end{cases} \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} f ( 0 ) = P , f ( 1 ) = Q , f ( \\infty ) = R . \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} \\mathcal I _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\frac { 2 } { p ( p + 1 ) } . \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} \\omega _ { \\psi , V } ( g , h ) \\phi = \\omega _ { \\psi , V } \\left ( g \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & \\det ( g ) ^ { - 1 } \\end{array} \\right ) , 1 \\right ) L ( h ) \\phi ( g , h ) \\in R ( F ) \\phi \\in \\mathcal S ( V ) , \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\begin{cases} x ^ \\ell - \\tilde \\alpha _ i = 0 \\\\ y ^ \\ell - \\tilde \\beta _ i = 0 , \\end{cases} \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} | \\zeta | _ { \\C } ^ 2 = \\zeta _ 1 ^ 2 + \\zeta _ 2 ^ 2 = \\| \\xi \\| ^ 2 - \\| \\eta \\| ^ 2 + 2 i \\xi \\cdot \\eta , \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} \\begin{cases} \\lim _ { n \\to \\infty } \\underbar r _ 1 ^ n = \\underbar r _ 1 , \\lim _ { n \\to \\infty } \\bar r _ 1 ^ n = \\bar r _ 1 , \\cr \\lim _ { n \\to \\infty } \\underbar r _ 2 ^ n = \\underbar r _ 2 , \\lim _ { n \\to \\infty } \\bar r _ 2 ^ n = \\bar r _ 2 . \\end{cases} \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} \\overline { \\mathcal { L } _ j } = \\bigcup _ { i \\in [ d ] } \\overline { \\mathcal { L } _ j ^ { ( i ) } } \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} \\kappa \\int _ { \\R ^ d \\times \\R ^ d } \\Delta ^ 2 _ x \\psi ( x , y ) m _ { u _ \\mu } ( t , x ) m _ { u _ \\iota } ( t , y ) \\ , d x d y & \\\\ = - \\kappa \\int _ { \\R ^ d \\times \\R ^ d } D ^ 2 _ { x y } D ^ 2 _ x \\psi ( x , y ) m _ { u _ \\mu } ( t , x ) m _ { u _ \\iota } ( t , y ) \\ , d x d y & \\\\ = - \\kappa \\int _ { \\R ^ d \\times \\R ^ d } \\nabla _ x m _ { u _ \\mu } ( t , x ) D ^ 2 _ x \\psi ( x , y ) \\nabla _ y m _ { u _ \\iota } ( t , y ) \\ , d x d y & . \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} f ( u ) = u ^ { 2 k } + f _ { 2 k } ( u ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; k \\in \\Z ^ + . \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} & g ( ( A \\cup C ) \\setminus ( B \\cup D ) , ( B \\cup D ) \\setminus ( A \\cup C ) ) + g ( A \\cap C , B \\cap D ) \\\\ = \\ , & \\sqrt { \\abs { A ' \\cup C ' } + \\abs { B ' \\cup D ' } } + \\sqrt { \\abs { A ' \\cap C ' } + \\abs { B ' \\cap D ' } } \\\\ \\le \\ , & \\sqrt { \\abs { A ' } + \\abs { B ' } } + \\sqrt { \\abs { C ' } + \\abs { D ' } } \\\\ \\le \\ , & \\sqrt { \\abs { A } + \\abs { B } } + \\sqrt { \\abs { C } + \\abs { D } } = g ( A , B ) + g ( C , D ) , \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} [ ( v _ 1 \\wedge v _ 2 ) \\otimes ( v _ 2 \\wedge v _ 3 ) \\otimes ( v _ 1 \\wedge v _ 4 ) ] \\ast _ f ( v _ 1 \\otimes v _ 1 \\otimes v _ 2 ) = ( v _ 2 \\wedge v _ 3 \\wedge v _ 1 ) \\otimes ( v _ 3 \\wedge v _ 4 \\wedge v _ 1 ) \\otimes ( v _ 2 \\wedge v _ 5 \\wedge v _ 6 ) \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} & g _ 1 = 1 + \\sum _ { n = 1 } ^ N \\epsilon ^ { n ( n + 1 ) } g ^ { - n } \\sum _ { 1 \\leq i _ 1 < \\cdots < i _ n \\leq N } ( \\lambda _ { i _ 1 } \\cdots \\lambda _ { i _ n } ) ^ 2 \\Delta _ n ( i _ 1 , \\cdots , i _ n ) z _ { i _ 1 } \\cdots z _ { i _ n } + o ( \\epsilon ^ { N ( N + 1 ) } ) , \\\\ & g _ 2 = 1 + \\sum _ { n = 1 } ^ N \\epsilon ^ { n ( n - 1 ) } g ^ { - n } \\sum _ { 1 \\leq i _ 1 < \\cdots < i _ n \\leq N } \\Delta _ n ( i _ 1 , \\cdots , i _ n ) z _ { i _ 1 } \\cdots z _ { i _ n } + o ( \\epsilon ^ { N ( N - 1 ) } ) . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} \\theta _ { 0 , N } & : = 1 , \\\\ \\theta _ { i + 1 , N } & : = \\frac { 1 + \\sqrt { 4 \\theta _ { i , N } ^ 2 + 1 } } { 2 } , i = 0 , 1 , \\dots , N - 1 , \\\\ \\theta _ { N , N } & : = \\frac { 1 + \\sqrt { 8 \\theta _ { N - 1 , N } ^ 2 + 1 } } { 2 } . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\sigma _ { 0 } ^ { B } = \\frac { \\rho _ + u _ + - \\rho _ - u _ - + w _ 0 ^ B } { \\rho _ + - \\rho _ - } , \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} K \\ni g \\longmapsto j ( g , I ) ^ { - k } , \\qquad I = i I _ n , \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} \\chi _ 2 ( g ) \\chi _ 1 ( e ) = 0 \\tfrac { a } { c } \\notin C _ { \\Gamma , \\chi } . \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} \\exp ( \\pi | r | ) \\Psi _ k ( s , x ) \\ll ( 1 - x ) ^ { 1 / 4 } \\frac { ( 1 + | r | ) ^ { 2 b } } { k ^ { 2 b - 1 / 2 } } \\left ( \\frac { x } { 1 - x } \\right ) ^ { b } , \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\| D ^ { ( l ) } \\| _ { S _ 1 } = \\left ( \\left ( \\left ( D ^ { ( l ) } \\right ) ^ * D ^ { ( l ) } \\right ) ^ { \\frac { 1 } { 2 } } \\right ) = \\left ( \\left ( D ^ { ( l ) } \\right ) ^ * D ^ { ( l ) } \\right ) = l + 1 . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\begin{pmatrix} - \\vartheta & \\overline \\mu & 0 & \\overline \\gamma \\\\ [ 0 . 1 c m ] \\mu & \\overline \\vartheta & \\overline { \\gamma } & 0 \\\\ [ 0 . 1 c m ] 0 & - \\gamma & \\vartheta & \\mu \\\\ [ 0 . 1 c m ] - \\gamma & 0 & \\overline \\mu & - \\overline { \\vartheta } \\end{pmatrix} , \\mbox { w i t h $ \\mu , \\vartheta , \\gamma \\in \\mathbb C $ . } \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} n ! S _ n = \\sum _ I c _ I \\Psi ^ I = \\sum _ I \\frac { n ! } { i _ 1 ( i _ 1 + i _ 2 ) \\cdots ( i _ 1 + i _ 2 + \\cdots i _ r ) } \\Psi ^ I \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} C ^ { \\varepsilon } ( x ) = \\bigcap \\limits _ { \\eta > 0 } D ^ { \\varepsilon + \\eta } ( x ) , \\ \\forall ( \\varepsilon , x ) \\in \\mathbb { R } _ { + } \\times X , \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { m } ( - 1 ) ^ i d _ i \\right ) \\cdot \\left ( \\sum _ { i = 1 } ^ { m + 1 } ( - 1 ) ^ i d _ i \\right ) > 0 . \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} d _ 2 p _ 1 + d _ 1 q _ 2 = d _ 1 d _ 2 + r d _ 1 d _ 2 \\geq d _ 1 d _ 2 . \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} \\big ( \\big ( n , \\big ( \\frac { r _ 1 } { r _ 2 } , w \\big ) \\big ) , b \\big ) & = \\big ( \\big ( b n , \\big ( \\frac { \\tilde { b } r _ 1 } { r _ 2 } , w \\big ) \\big ) , 1 \\big ) = \\big ( \\big ( b n , \\big ( \\frac { \\tilde { b } r _ 1 } { r _ 2 } , 0 \\big ) \\big ) , 1 \\big ) \\\\ & = \\big ( ( b n , ( r ' , 0 ) ) , 1 \\big ) \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\varpi ( \\alpha ; B ^ \\Delta _ { \\upsilon } ) = 2 \\langle b \\rangle + \\upsilon + \\langle a \\rangle e ^ { \\alpha } , \\varpi ( \\alpha ; B ^ \\Delta _ { 2 , \\upsilon } ) = 2 \\langle b \\rangle + \\upsilon . \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ { r / 2 } ( x _ { 2 i - 1 } y _ { 2 i } + x _ { 2 i } y _ { 2 i - 1 } ) , \\\\ \\sum _ { i = 1 } ^ { r / 2 - 1 } ( x _ { 2 i - 1 } y _ { 2 i } + x _ { 2 i } y _ { 2 i - 1 } ) + S _ 0 , \\end{gather*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\Phi _ i ^ { - 1 } ( 0 ) \\subset \\bigcup _ { j = 1 } ^ k B _ { 1 0 m _ i ^ { - 1 / 2 } } ( x _ j ) \\quad \\Phi _ i ^ { - 1 } ( 0 ) \\cap B _ { 1 0 m _ i ^ { - 1 / 2 } } ( x _ j ) \\neq \\emptyset , \\quad \\forall j \\in \\{ 1 , \\ldots , k \\} . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} P _ { B S } ^ { \\textrm { o p t } } = \\left [ \\frac { \\lambda \\| \\mathbf { h } \\| ^ { 2 } - B } { 2 A } \\right ] ^ { + } . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} R _ { G , o p t } ( N , t ) & \\leq 1 - \\frac { \\log \\left [ \\prod \\limits _ { k = 1 } ^ { t } ( N - k ) \\right ] } { \\log N ! } \\\\ & \\leq 1 - \\frac { t } { N } . \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} d c x ^ { p ^ e } \\in ( I ^ { [ p ^ e ] } , x _ { s + 1 } ^ { p ^ e } , \\ldots , x _ { s + t } ^ { p ^ e } ) = ( I , x _ { s + 1 } , \\ldots , x _ { s + t } ) ^ { [ p ^ e ] } \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} \\Delta _ { \\alpha , \\varepsilon } ^ { I P } = \\mathbf { 1 } _ { \\lbrace T _ { D _ \\varepsilon } > t _ { \\alpha , \\varepsilon } \\rbrace } , T _ { D _ \\varepsilon } = \\sum _ { k = 1 } ^ { D _ \\varepsilon } b _ k ^ { - 2 } ( y _ k ^ 2 - \\varepsilon ^ 2 ) \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} S _ { d , k } ( n ) = \\sum _ { W = 0 } ^ n S _ { d , k } ( n , W ) . \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\lim _ { | n | \\to \\infty } \\varphi ( n ) = 0 . \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} P _ { m , m ' } ( t ) \\begin{cases} = 1 & m ' = m , \\\\ \\in t ^ { - 1 } \\mathbb { Z } [ t ^ { - 1 } ] & m ' < m , \\\\ = 0 & \\end{cases} \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} \\left ( \\frac { \\dot { x } } { x } \\right ) ^ 2 = Q ^ 2 P \\left ( \\frac { \\sigma _ 1 ^ 2 - \\sigma _ 2 ^ 2 } { Q } x \\right ) , \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} \\omega ^ 0 = \\begin{psmallmatrix} \\omega ^ 0 _ 1 \\\\ \\vdots \\\\ \\omega ^ 0 _ t \\end{psmallmatrix} : W ^ 0 \\rightarrow W _ 1 \\oplus \\cdots \\oplus W _ t . \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} & \\pi _ 1 ( S _ 2 ) e _ { 2 k } = e _ { 4 k + 3 } \\\\ & \\pi _ 1 ( S _ 2 ) e _ { 2 k + 1 } = e _ { 4 k + 2 } \\\\ & \\pi _ 1 ( S _ 1 ) e _ { 2 k } = e _ { 4 k + 1 } \\\\ & \\pi _ 1 ( S _ 1 ) e _ { 2 k + 1 } = e _ { 4 k } \\ ; . \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\tau _ { i r } ( n ^ 2 ) } { n ^ s } = \\frac { \\zeta ( s ) \\zeta ( s + 2 i r ) \\zeta ( s - 2 i r ) } { \\zeta ( 2 s ) } , \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} A _ i = \\begin{cases} D _ { i + 1 } - D _ i & \\\\ - K _ S - ( D _ n - D _ 1 ) & \\end{cases} \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} ( \\phi _ { ( j , N ) } f ) ( \\phi _ { ( j ^ { \\prime } , N ) } g ) = 0 , \\quad 1 \\leqslant j \\neq j ^ { \\prime } \\leqslant m _ { N } . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} \\Phi _ N : \\mathbb { F } _ { b } [ \\lambda ] ^ { ( s + t ) \\times r } & \\longrightarrow \\mathbb { F } _ { n + b } [ \\lambda ] ^ { t \\times r } \\\\ B ( \\lambda ) & \\longrightarrow N ( \\lambda ) B ( \\lambda ) , \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} u \\left ( x , T \\right ) = u _ { f } \\left ( x \\right ) \\quad \\Omega , \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} \\psi _ t & = d _ 1 \\Delta \\psi - \\chi _ 1 \\nabla \\cdot ( \\psi \\nabla w ) - \\chi _ 1 \\nabla \\cdot ( u ^ { * * } \\nabla ( w - w ^ { * * } ) ) \\\\ & + \\psi \\Big ( a _ 0 ( t , x ) - a _ 1 ( t , x ) ( u + u ^ { * * } ) - a _ 2 ( t , x ) v \\Big ) - a _ 2 ( t , x ) u ^ { * * } \\phi , \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\nabla \\xi ( x ) = \\sum _ { k \\in \\Lambda _ N } \\xi _ k \\nabla \\tilde e _ k ( x ) = 2 \\pi { \\rm i } \\sum _ { k \\in \\Lambda _ N } \\xi _ k \\tilde e _ k ( x ) k . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} D _ n ^ m = \\sum _ { 1 \\leq i _ 1 < \\cdots < i _ n \\leq N } \\Delta _ n ( i _ 1 , \\cdots , i _ n ) ( \\lambda _ { i _ 1 } \\cdots \\lambda _ { i _ n } ) ^ m E _ { i _ 1 } \\cdots E _ { i _ n } , \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} \\alpha & = \\sum _ { i \\ne j \\ne k } \\lambda _ { i j } \\hat { F } _ { i j k } F _ k ' d F _ i \\wedge d F _ j + \\sum _ { i \\ne j } 2 \\lambda _ { i j } \\hat { F } _ { i j } d F _ i ' \\wedge d F _ j + \\beta \\\\ & = d \\rho ( \\lambda , \\underline { F } ) ( 0 , ( F _ i ' ) _ { i = 1 } ^ m ) + \\beta . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} d _ r : = \\det ( \\tau _ { 1 + j - i } ' ) _ { 1 \\leq i , j \\leq r } \\textrm { a n d } b _ r : = ( \\tau _ r ' ) ^ 2 + 2 \\sum _ { i \\geq 1 } ( - 1 ) ^ i \\tau _ { r + i } ' \\tau _ { r - i } ' , \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} \\big ( \\ker \\rho ^ \\vee | _ m \\cap \\ker ( \\rho ^ T _ * ) ^ \\vee | _ m \\big ) = \\big ( \\ker \\rho ^ \\vee | _ m \\cap \\ker \\rho _ * ^ \\vee | _ m \\big ) , \\ \\ \\ \\forall m \\in M . \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} \\rho ( \\mathfrak { f } ( \\chi ) ) = \\prod _ U \\rho _ U ( \\mathfrak { f } ( \\lambda _ U ) ) ^ { z _ U } . \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) & = \\sum _ { I \\subset [ N ] } \\sum _ { q \\geq \\chi ^ I } ( - 1 ) ^ { | I | } T _ { n + q , m + q } ' \\\\ & = \\sum _ { q \\in \\N ^ N } T _ { n + q , m + q } ' \\sum _ { \\substack { I \\subset [ N ] \\\\ q \\geq \\chi ^ I } } ( - 1 ) ^ { | I | } . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} a \\left ( u \\right ) = \\begin{pmatrix} a _ { 1 0 } + 2 a _ { 1 1 } u _ { 1 } + a _ { 1 2 } u _ { 2 } & a _ { 1 2 } u _ { 1 } \\\\ a _ { 2 1 } u _ { 2 } & a _ { 2 0 } + 2 a _ { 2 2 } u _ { 2 } + a _ { 2 1 } u _ { 1 } \\end{pmatrix} , \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} b _ 1 + b _ 2 = b _ 1 + \\rho ^ t ( b ' ) = b _ 1 + b ' - ( 1 - \\rho ^ t ) b ' \\in B ^ { \\Sigma } + I _ { G } B \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} { \\Omega _ { \\min } } = \\left [ { \\max \\left ( { \\left | { \\hat h _ { \\min } ^ { { \\rm { r o u g h } } } } \\right | - \\beta , 0 } \\right ) , \\left | { \\hat h _ { \\min } ^ { { \\rm { r o u g h } } } } \\right | + \\beta } \\right ] \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} \\Pr ( R ) = \\frac { | Z | } { 1 6 | Z | } + \\frac { ( 2 \\times 2 ) | Z | ^ 2 + ( 1 1 \\times 8 ) | Z | ^ 2 + ( 2 \\times 4 ) | Z | ^ 2 } { { 1 6 } ^ 2 | Z | ^ 2 } = \\frac { 2 9 } { 6 4 } . \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} m - s + 1 = \\underbrace { n _ { 1 } + \\cdots + n _ { 1 } } _ { M _ { 1 } \\ ; \\mathrm { t i m e s } } + \\underbrace { n _ { 2 } + \\cdots + n _ { 2 } } _ { M _ { 2 } \\ ; \\mathrm { t i m e s } } + \\cdots + \\underbrace { n _ { \\ell } + \\cdots + n _ { \\ell } } _ { M _ { \\ell } \\ ; \\mathrm { t i m e s } } \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} L ( 1 / 2 , \\pi _ p \\otimes \\tau _ p \\otimes \\tau _ p ^ { \\vee } ) = L ( 1 / 2 , \\pi _ p ) ^ 2 = \\frac { p ^ 2 } { ( p - 1 ) ^ 2 } . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} d e _ { t } ( \\zeta ) & = \\overline { e } _ { t } ( \\zeta ) d t + \\sum _ { r = 1 } ^ { 3 } e _ { t } ^ { r } ( \\zeta ) d B _ { t } ^ { ( r ) } , \\\\ d h _ { t } ( \\zeta ) & = \\overline { h } _ { t } ( \\zeta ) d t + \\sum _ { r = 1 } ^ { 3 } h _ { t } ^ { r } ( \\zeta ) d B _ { t } ^ { ( r ) } , \\\\ d f _ { t } ( \\zeta ) & = \\overline { f } _ { t } ( \\zeta ) d t + \\sum _ { r = 1 } ^ { 3 } f _ { t } ^ { r } ( \\zeta ) d B _ { t } ^ { ( r ) } . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} A _ i & : = \\{ x \\in D _ { T } ( x _ i ) \\colon | T ( x ) | > ( n p ' ) ^ { - 1 } | T ( x _ i ) | \\} , & B _ i & : = \\left \\{ \\begin{array} { l l } \\{ y _ { i ' } \\colon 1 \\leq i ' \\leq 2 n p ' \\} & s > 5 n p ' \\\\ \\{ y _ { i ' } \\colon 1 \\leq i ' \\leq s \\} & s \\leq 5 n p ' , \\\\ \\end{array} \\right . \\\\ B ' _ i & : = \\{ z _ { i ' } \\colon 1 \\leq i ' \\leq \\min \\{ s ' , n p ' \\} \\} . & & \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} h ^ 0 ( V \\otimes L ( p ) ) = k _ 0 . \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} \\tilde { J } ( A _ r , B _ r ) : = | | B ^ T ( s I _ n + A ) ^ { - 1 } B - B _ r ^ T ( s I _ r + A _ r ) B _ r | | ^ 2 _ { H ^ 2 } \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} ( \\imath , r ) \\mapsto ( \\imath , - \\imath + 1 - 2 k ) \\ \\ k = \\# \\{ r ' \\mid r ' > r , ( \\imath , r ' ) \\in \\widehat { J } _ { \\mathrm { A } } \\} . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} \\begin{pmatrix} S _ 1 & S _ 2 \\\\ S _ 3 & S _ 4 \\end{pmatrix} \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} A _ r = U ^ T A U , \\ B _ r = U ^ T B , \\ C _ r = C U , \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} { \\bf { C o v } } \\left [ V _ { \\varepsilon } ( u ) , \\ , V _ { \\varepsilon } ( v ) \\right ] = & \\begin{cases} - 2 \\ , \\log | u - v | , \\ , \\varepsilon < | u - v | \\leq 1 , \\\\ - 2 \\log \\varepsilon , \\ , u = v , \\end{cases} \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} d ( g , h ) : = \\log | | g ^ { - 1 } h | | , \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} F S ( e ) = \\varphi _ d p ^ { d e } + \\varphi _ { d - 1 } p ^ { ( d - 1 ) e } + \\cdots + \\varphi _ { 1 } p ^ e + \\varphi _ { 0 } . \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} Z ^ * = \\big \\{ S | _ W : S \\in Z \\big \\} , \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\exp \\left ( \\bar p _ 1 \\frac { \\mu '' ( s ) } { 2 \\mu ' ( s ) } \\right ) \\mu ' ( s ) ^ { \\bar \\rho } \\ , \\nabla \\ , \\mu ' ( s ) ^ { - \\bar \\rho } \\exp \\left ( \\ ! - \\bar p _ 1 \\frac { \\mu '' ( s ) } { 2 \\mu ' ( s ) } \\right ) \\\\ = d + \\bar p _ { - 1 } d s + \\sum _ { i \\in \\bar E } \\tilde { \\bar v } _ i ( s ) \\bar p _ i d s \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} [ ( \\phi \\wedge \\psi ) ( x ) ] ^ { \\beta [ \\frac { n } { x } ] } & = [ \\phi ( x ) ] ^ { \\beta [ \\frac { n } { x } ] } \\cap [ \\psi ( x ) ] ^ { \\beta [ \\frac { n } { x } ] } \\\\ & = i ( n , N _ \\phi | A _ \\phi ) \\cap i ( n , N _ \\psi | A _ \\psi ) \\\\ & = i ( n , N _ \\phi | A _ \\phi \\sqcap N _ \\psi | A _ \\psi ) \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\tilde U \\tilde S _ 2 & = U ^ { - 2 } S _ 2 S _ 1 S _ 1 ^ * + U ^ { - 2 } S _ 1 ^ 2 S _ 2 ^ * \\\\ & = U ^ { - 2 } S _ 2 U S _ 2 S _ 1 ^ * + U ^ { - 2 } U S _ 2 U S _ 2 S _ 2 ^ * \\\\ & = U ^ { - 2 } U ^ 2 S _ 2 S _ 2 S _ 1 ^ * + U ^ { - 2 } U U ^ 2 S _ 2 S _ 2 S _ 2 ^ * \\\\ & = S _ 2 ^ 2 S _ 1 ^ * + U S _ 2 ^ 2 S _ 2 ^ * \\\\ & = S _ 2 ^ 2 S _ 1 ^ * + S _ 1 S _ 2 S _ 2 ^ * = \\tilde S _ 1 \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & ( - \\Delta ) ^ { s / 2 } v _ k = | f | & & ~ ~ \\mbox { i n } ~ ~ B _ { r _ { ( k - 1 ) k _ 0 } } ; \\\\ & - v _ k \\leq u \\leq v _ k & & ~ ~ \\mbox { i n } ~ ~ R ^ { n } ; \\\\ & v _ k \\leq \\hat { C } M r _ { k k _ 0 } ^ { \\beta } & & ~ ~ \\mbox { i n } ~ ~ B _ { r _ { k k _ 0 } } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} \\theta ( g , h ; \\phi ) : = \\sum _ { x \\in V ( F ) } \\omega ( g , h ) \\phi ( x ) . \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } [ \\zeta _ k \\nu _ { n _ k } + \\varphi _ k \\nu _ { n _ k } ] D ( b _ k ) d x \\leqq 2 \\| D ( b _ k ) \\| \\| \\nu _ { n _ k } \\| \\leqq C \\| { \\bf { w } } _ { k } \\| _ { 1 \\times 1 } \\| { \\bf { h } } _ { n _ k } \\| _ { 1 \\times 1 } = O ( \\epsilon ) . \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} f ^ { \\star ' } _ 0 ( x ) = \\frac { \\omega ( x ) } { x } , x \\in ( 0 , \\ , 1 ] . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ { 1 } & a _ { 2 } \\\\ a _ { 3 } & a _ { 4 } \\end{pmatrix} \\cdot \\alpha _ { 1 } = \\alpha _ { 1 } ^ { a _ { 4 } } \\alpha _ { 3 } ^ { - a _ { 2 } } , \\ \\begin{pmatrix} a _ { 1 } & a _ { 2 } \\\\ a _ { 3 } & a _ { 4 } \\end{pmatrix} \\cdot \\alpha _ { 3 } = \\alpha _ { 1 } ^ { - a _ { 3 } } \\alpha _ { 3 } ^ { a _ { 1 } } . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} \\sigma ^ { \\alpha } & = \\rho ^ { r _ { 1 } } \\sigma \\\\ \\tau ^ { \\alpha } & = \\rho ^ { r _ { 2 } } \\tau \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} T _ { n , k } = _ d & \\sum _ { \\mathcal { S } _ 1 } h ( \\sigma ( i _ 1 ) , \\ldots , \\sigma ( i _ k ) ; \\gamma ( j _ 1 ) , \\ldots , \\gamma ( j _ k ) ) \\\\ = & \\sum _ { \\mathcal { S } _ 2 } h ( \\sigma ( i _ 1 ) , \\ldots , \\sigma ( i _ k ) ; \\gamma ( j _ 1 ) , \\ldots , \\gamma ( j _ k ) ) \\mathbf { 1 } ( 1 \\leq i _ 1 < \\cdots < i _ k \\leq n ) \\mathbf { 1 } ( 1 \\leq j _ 1 < \\cdots < j _ k \\leq n ) , \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} T ( \\frac { d \\Gamma } { d t } ) = \\frac { 1 } { 2 } \\left \\vert \\frac { d \\Gamma } { d t } \\right \\vert ^ { 2 } = \\frac { 1 } { 2 } ( \\frac { d u } { d t } ) ^ { 2 } \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { \\epsilon _ a - \\epsilon _ b \\in \\Phi ^ + _ 0 } A _ { b , a } ( \\epsilon _ b - \\epsilon _ a ) = \\sum _ { j = k + 1 } ^ { m n } { \\sigma _ j } { \\beta _ j } - \\sum _ { i = 1 } ^ { k } { \\sigma _ i } { \\beta _ i } + { \\beta _ k } = : ( \\nu ^ { ( 0 ) } , \\nu ^ { ( 1 ) } ) \\in \\mathbb Z ^ { m + n } . \\end{aligned} \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\log \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = q \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } B ( t ) + \\frac { 1 } { 2 } q ^ 2 \\sigma ^ 2 ( \\mu ) + \\int \\limits _ { \\mathbb { R } \\setminus \\{ 0 \\} } \\bigl ( e ^ { u q } - 1 - u q \\bigr ) \\ , d \\mathcal { M } _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ( u ) . \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} t ^ { \\overline { \\nu } } = \\frac { \\Gamma ( t + \\nu ) } { \\Gamma ( t ) } , \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} F ( p ) : = \\Phi _ \\mu \\big ( | p | \\big ) , \\ ; \\Phi _ \\mu ( t ) : = \\intop _ 0 ^ t \\intop _ 0 ^ s ( 1 + r ) ^ { - \\mu } \\ , d r \\ , d s , \\ ; p \\in \\R ^ { n \\times N } , \\ ; t \\geq 0 . \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} S ( x , r ) : = ( T x , r + f ( x ) ) . \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} \\| u _ i \\pm v \\| & = \\| T x _ i - u _ i \\| + \\| T ( x _ i \\pm z ) \\| + \\| T z - w \\| \\\\ & \\le \\frac { \\delta } { 2 } + ( 1 + \\frac { \\delta } { 2 } ) \\| x _ i \\pm z \\| + \\frac { \\delta } { 2 } \\le ( 1 + \\delta ) ^ 2 + \\delta \\le 1 + \\varepsilon . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\gamma ( \\Pi _ { \\imath } , \\Pi _ { \\jmath } ) = \\delta ( \\imath \\in { { } ^ { \\jmath } B } ) . \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} \\sum _ { \\substack { B \\subset \\{ 1 , 2 , \\cdots , n \\} \\\\ | B | = k } } \\prod _ { i \\in B } \\frac { b ( i ; n ) } { 1 - b ( i ; n ) } \\geq \\frac { \\beta _ n ^ k } { k ! } \\ , \\left ( 1 - \\ , \\frac { k \\ , \\beta } { \\lambda _ n \\ , ( 1 - \\beta ) } \\right ) ^ k . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Z ^ d } x \\mu ( x ) ~ = ~ 0 , \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\mathbf { D } = [ \\mathbf { d } _ 1 , \\mathbf { d } _ 2 , \\dots , \\mathbf { d } _ l ] . \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} \\mu _ { | _ \\Sigma } = \\mu _ \\Gamma , u _ { | _ \\Sigma } = u _ \\Gamma \\mbox { o n } \\Sigma , \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} B _ n ( X ) = \\bigoplus _ { u \\in X ^ + } B _ n ( X ; u ) . \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} g ( 2 ) = 4 g ( 3 ) = 6 g ( 4 ) = 8 g ( 2 ) = 5 g ( 3 ) = 6 g ( 4 ) = 8 . \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} \\left ( u \\alpha _ { 1 } \\right ) \\left ( v \\alpha _ { 2 } \\right ) & = \\rho ^ { v _ { 2 } } u v \\alpha _ { 1 } \\alpha _ { 2 } \\ \\\\ \\left ( v \\alpha _ { 2 } \\right ) \\left ( u \\alpha _ { 1 } \\right ) & = \\rho ^ { \\frac { 1 } { 2 } u _ { 2 } \\left ( u _ { 2 } - 1 \\right ) + v _ { 3 } u _ { 2 } - u _ { 3 } v _ { 2 } - u _ { 2 } ^ { 2 } - u _ { 2 } v _ { 2 } } \\tau ^ { u _ { 2 } } u v \\alpha _ { 1 } \\alpha _ { 2 } , \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} & { \\rm D } \\bar { J } ( A _ r , B _ r , C _ r ) [ ( A ' _ r , B ' _ r , C ' _ r ) ] \\\\ = & { \\rm t r } ( A _ r '^ T \\nabla _ { A _ r } \\bar { J } ( A _ r , B _ r , C _ r ) ) + { \\rm t r } ( B _ r '^ T \\nabla _ { B _ r } \\bar { J } ( A _ r , B _ r , C _ r ) ) \\\\ & + { \\rm t r } ( C _ r '^ T \\nabla _ { C _ r } \\bar { J } ( A _ r , B _ r , C _ r ) ) , \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} \\mathrm { v o l } ( \\mathcal L _ j ) = \\mathrm { v o l } ( \\mathcal R _ j ) = p ^ { - j } ( 1 - p ^ { - 1 } ) ^ 2 j > 0 . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} Y ^ { + } _ t ( x ) & : = ( \\sqrt { x } + \\xi \\sqrt { t } ) ^ 2 , Y ^ { - } _ t ( x ) : = ( \\sqrt { x } - \\xi \\sqrt { t } ) ^ 2 . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} J _ { 2 , 4 } = { \\Theta } \\frac { \\zeta ( 3 + a ) } { ( 1 + a ) \\zeta ^ { 2 } ( 2 + a ) } x ^ { 1 + a } + O _ { a } \\left ( \\log x \\right ) . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} g _ { i } = \\frac { \\sum _ { j \\in \\Lambda _ i } f _ { j } } { \\norm { \\sum _ { j \\in \\Lambda _ i } f _ { j } } } . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} c \\vee a = c \\neq a = \\left ( \\left ( c \\vee a \\right ) \\wedge a ^ { \\sim } \\right ) \\vee \\left ( \\left ( c \\vee a \\right ) \\wedge \\lozenge a \\right ) \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} X ^ { N } _ y ( t ) = \\frac { 1 } { N ^ { 1 / 2 } } X _ { \\lfloor N y \\rfloor } ( t ) + \\frac { 1 } { N ^ { 1 / 2 } } ( N y - \\lfloor N y \\rfloor ) ( X _ { \\lfloor N y \\rfloor + 1 } ( t ) - X _ { \\lfloor N y \\rfloor } ( t ) ) , \\ , t \\ge 0 , \\ , y \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\dot { R } _ t = L ^ \\dagger R _ t , R _ t | _ { t = 0 } = R _ { \\mu _ 0 } \\in \\mathcal { D } ^ \\dagger . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} e ^ { - \\frac { ( k - n p ) ^ 2 } { 2 n p ( 1 - p ) } } \\left | \\frac { P \\left ( V _ n = k \\right ) } { \\frac { 1 } { \\sqrt { 2 \\ , \\pi n p ( 1 - p ) } } e ^ { - ( k - n p ) ^ 2 / ( 2 n p ( 1 - p ) ) } } - 1 \\right | < \\frac { C ' } { \\sqrt { n p ( 1 - p ) } } , \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} \\lambda = \\left \\{ \\begin{array} { c c } \\lambda ^ - & \\ ; X \\in \\Omega ^ - , \\\\ \\lambda ^ + & \\ ; X \\in \\Omega ^ + , \\end{array} \\right . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mu = \\left \\{ \\begin{array} { c c } \\mu ^ - & \\ ; X \\in \\Omega ^ - , \\\\ \\mu ^ + & \\ ; X \\in \\Omega ^ + . \\end{array} \\right . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} \\widetilde { c } _ { j i } ( - 2 k - 1 ) = 0 \\ \\ i \\in I \\setminus \\{ n \\} , & & & \\widetilde { c } _ { j n } ( - 2 k - 2 ) = 0 . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} \\Phi ^ * \\left ( q \\frac { \\Delta _ { \\{ b + 1 , \\dots , n - 1 , 1 \\} } } { \\Delta _ { \\{ b + 1 , \\dots , n \\} } } \\right ) = \\vartheta _ b . \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} | \\ < q _ z , \\psi \\ > - \\psi _ j ( z ) | = | \\ < q _ z , \\psi \\ > - \\ < q _ z , \\psi _ j \\ > | = | \\ < q _ z , \\psi - \\psi _ j \\ > | \\le \\| q _ z \\| \\ , \\| \\psi - \\psi _ j \\| \\to 0 , \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} \\big \\langle \\widetilde \\Lambda _ { v _ 2 } \\widetilde P _ { v _ 2 , \\rm R } \\widetilde F ( v _ 2 ) , \\widetilde P _ { v _ 2 , \\rm R } \\widetilde F ( v _ 2 ) \\big \\rangle = \\alpha _ { v _ 2 } | \\widetilde f ( v _ 2 ) | ^ 2 \\leq 0 . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} Q _ { m + 1 } ^ { [ r ] } : = \\begin{cases} Q _ { m , n } ^ { [ 1 ] } \\sqcup \\coprod \\limits _ { i = r + 1 } ^ { n - 1 } Q _ { m , i } & \\\\ Q _ { m , 1 } ^ { [ 1 ] } \\sqcup \\coprod \\limits _ { i = 2 } ^ { n - r } Q _ { m , i } & \\end{cases} \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} Z = \\lbrace x _ 1 , \\ldots , x _ l \\rbrace . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} M _ { a , b , c } ( q ) : = \\sum _ { \\pi \\in P ( a , b , c ) } q ^ { | \\pi | } = \\prod _ { i = 1 } ^ a \\prod _ { j = 1 } ^ b \\prod _ { k = 1 } ^ c \\frac { \\left [ i + j + k - 1 \\right ] _ q } { \\left [ i + j + k - 2 \\right ] _ q } . \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\partial _ t h _ 1 + ( u _ 1 \\partial _ x + u _ 2 \\partial _ y ) h _ 1 + h _ 1 ( \\partial _ x u _ 1 + \\partial _ y u _ 2 ) = \\nu \\partial _ { y } ^ 2 h _ 1 + ( h _ 1 \\partial _ x + h _ 2 \\partial _ y ) u _ 1 . \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\begin{aligned} & \\ , 2 K _ 1 ^ { \\frac { p - 1 } { p - 2 + m } } \\frac { C ^ { m - 1 } m ( N - 1 ) k \\coth ( k ) } { a ( m - 1 ) } ( T - t ) ^ { - \\alpha m } [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } } \\\\ \\leq \\ , & C ^ { \\frac { ( p - 1 ) ( m - 1 ) } { p - 2 + m } } \\left ( \\frac { C ^ { m - 1 } m } { 2 a ^ 2 ( m - 1 ) ^ 2 } \\right ) ^ { \\frac { p - 1 } { p - 2 + m } } ( T - t ) ^ { \\frac { - \\alpha p ( m - 1 ) - \\alpha m ( p - 1 ) } { p - 2 + m } } [ - \\log ( T - t ) ] ^ { \\frac { \\beta } { m - 1 } } \\forall t \\in ( 0 , T ) \\ , . \\end{aligned} \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} \\psi _ 3 ( ( h ^ { - 1 } ) ^ a h ) = ( h _ 3 ^ { - 1 } h _ 1 , h _ 1 ^ { - 1 } h _ 2 , h _ 2 ^ { - 1 } h _ 3 ) = ( [ a , b ] , 1 , 1 ) . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} ( p + 1 ) ( a _ 1 + \\dotsb + a _ p ) & \\leq \\sum _ { i = 1 } ^ { p } ( p + 1 - i ) b _ i + \\sum _ { i = 2 } ^ { p + 1 } ( i - 1 ) b _ { i } \\\\ & \\leq p \\sum _ { i = 1 } ^ { p } b _ i + \\sum _ { i = 1 } ^ { p } ( 1 - i ) b _ i + p b _ { p + 1 } + \\sum _ { i = 2 } ^ { p } ( i - 1 ) b _ { i } \\\\ & \\leq p ( b _ 1 + \\dotsb + b _ { p + 1 } ) . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\mathcal { W } = \\frac { L _ { 0 , 0 } } { L _ { 1 , 1 } } + L _ { a , b } + \\sum _ { i = 1 } ^ { a - 1 } \\sum _ { j = 1 } ^ b \\frac { L _ { i , j } } { L _ { i + 1 , j } } + \\sum _ { j = 1 } ^ { b - 1 } \\sum _ { i = 1 } ^ a \\frac { L _ { i , j } } { L _ { i , j + 1 } } . \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} s _ i \\in ( \\partial I ( \\vec x ) ) _ i = [ p _ i - q _ i , p _ i + q _ i ] , \\forall \\ , \\vec s = ( s _ 1 , \\ldots , s _ n ) \\in \\partial I ( \\vec x ) . \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} \\sum _ { i \\geq 0 } C ( 2 , i ) x ^ i = \\frac { 1 + 2 x C ( 2 x ) } { 1 + x } = \\frac { 1 } { 1 - x C ( 2 x ) } = \\frac { 4 } { 3 + \\sqrt { 1 - 8 x } } . \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} L ( x ) = \\exp \\Big ( \\int _ 1 ^ x \\frac { \\vartheta ( t ) } { t } { \\ : \\rm d } t \\Big ) \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { 1 - \\theta } { m } + \\frac { \\theta } { 2 } , \\ \\ , s _ p = ( s _ m + \\delta ) ( 1 - \\theta ) . \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\Pr ( R ) = \\frac { | Z | } { p ^ 2 | Z | } + \\frac { p ( p - 1 ) ^ 2 | Z | ^ 2 + 2 p ( p - 1 ) | Z | ^ 2 } { p ^ 4 | Z | ^ 2 } = \\frac { p ^ 2 + p - 1 } { p ^ 3 } . \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\sum _ { a \\in Q _ 1 } [ a , D _ a ( w ) ] & = \\sum _ { a \\in Q _ 1 } \\sum _ { \\{ s | b _ s = a \\} } \\big ( a b _ { s + 1 } \\ldots b _ r \\ldots b _ { s - 1 } - b _ { s + 1 } \\ldots b _ r \\ldots b _ { s - 1 } a \\big ) \\\\ & = \\sum _ { s = 1 } ^ r b _ s b _ { s + 1 } \\ldots b _ r \\ldots b _ { s - 1 } - \\sum _ { s = 1 } ^ r b _ { s + 1 } \\ldots b _ r \\ldots b _ { s - 1 } b _ s \\\\ & = 0 . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} [ d _ h , \\ , \\overline { d } _ { - h } ^ \\star ] = - [ \\tau _ h , \\ , \\overline { d } _ { - h } ^ \\star ] = - [ d _ h , \\ , \\overline { \\tau } _ { - h } ^ \\star ] . \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} \\left \\{ \\Sigma ^ + \\in X ^ + \\mid A \\subset \\Sigma ^ + \\subset B \\right \\} \\cap X _ { \\alpha ^ + } ^ + = \\{ P \\} = \\left \\{ \\Sigma \\in X \\mid A \\subset \\Sigma \\subset B \\right \\} , \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\epsilon ( \\chi ) \\chi ( m \\ell _ 1 \\ell _ 3 ) \\overline { \\chi } ( n ) & = \\frac { 1 } { q ^ { 1 / 2 } } \\mathop { \\sum \\sum } _ { \\substack { v w = q \\\\ ( v , w ) = 1 } } \\mu ^ 2 ( v ) \\varphi ( w ) \\cos \\left ( \\frac { 2 \\pi n \\overline { m \\ell _ 1 \\ell _ 3 v } } { w } \\right ) . \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} - K _ S - ( D _ n - D _ 1 ) = H - E _ 1 - \\ldots - E _ k \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} H _ { 2 , \\boldsymbol { \\alpha } } ( \\tau ) = - 2 i \\int _ { \\mathbb R ^ 2 } \\mathcal F _ { 0 } ( w _ 1 ) \\mathcal G _ { \\alpha _ 2 } ^ * ( w _ 2 ) e ^ { 2 \\pi i \\tau Q ( \\boldsymbol { w } ) } \\boldsymbol { d w } . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} W _ { \\sigma } [ \\xi _ { \\sigma } , c _ { \\sigma } ] = ( \\pi _ { \\sigma } ^ { - 1 } \\circ \\xi _ { \\sigma } ^ { - 1 } ) ( c _ { \\sigma } ) , \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u '' ( t ) + A ( t , u ' ( t ) ) + B u ( t ) \\in F ( t , u ( t ) , u ' ( t ) ) \\ \\mbox { f o r a l m o s t a l l } \\ t \\in T , \\\\ u ( 0 ) = u _ { 0 } , \\ u ' ( 0 ) = u _ 1 . \\end{array} \\right \\} \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\mathfrak { d } _ 2 \\dot { \\phi } ( n ) = f ( n ) - f ( n + 2 ) = - \\int _ 0 ^ 2 f ' ( n + t ) \\ , d t . \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} & u _ { * } [ v _ { * } ^ 2 ( y _ { * } ^ 4 + y _ { * } ^ 2 ) + 1 - 4 y _ { * } ^ 4 ] = - y _ { * } ^ 3 \\sqrt { 1 + y _ { * } ^ 2 } v _ { * } ( u _ { * } ^ 2 - 4 ) , \\\\ & v _ { * } [ u _ { * } ^ 2 ( y _ { * } ^ 4 + y _ { * } ^ 2 ) + 1 - 4 y _ { * } ^ 4 ] = - y _ { * } ^ 3 \\sqrt { 1 + y _ { * } ^ 2 } u _ { * } ( v _ { * } ^ 2 - 4 ) . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} \\{ ( y , z ) \\mid x \\in R ^ m y = S ( g ) ( x ) S ( h ) ( x ) = z \\} . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } ( \\mathbf { P } _ { \\mathcal { B } | \\mathcal { A } } \\mathbf { P } _ { \\mathcal { A } | \\mathcal { B } } ) ^ { k } \\cdot \\mathbb { E } [ f | \\mathcal { A } ] = P _ { 1 } [ f ] , \\lim _ { k \\rightarrow + \\infty } ( \\mathbf { P } _ { \\mathcal { A } | \\mathcal { B } } \\mathbf { P } _ { \\mathcal { B } | \\mathcal { A } } ) ^ { k } \\cdot \\mathbb { E } [ f | \\mathcal { B } ] = P _ { 2 } [ f ] , \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\sum _ { | I | = i } e ^ { t \\sum _ { j \\in I } \\lambda _ j ( A ) } = \\sum _ { | I | = | J | = i } | \\Delta _ { I , J } ( b _ t ) | ^ 2 . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} s ( [ n , n + 1 ] ) & = s _ 1 ( s _ 3 s _ 1 ) ^ { n - 1 } , & s ( [ n + 1 , 1 ] ) & = s _ 3 ( s _ 1 s _ 3 ) ^ { n - 1 } , & s ( [ 1 , 1 ] ) & = s _ 2 , \\\\ \\beta ( [ n , n + 1 ] ) & = n \\alpha _ 1 + ( n - 1 ) \\alpha _ 3 , & \\beta ( [ n + 1 , 1 ] ) & = ( n - 1 ) \\alpha _ 1 + n \\alpha _ 3 , & \\beta ( [ 1 , 1 ] ) & = \\alpha _ 2 . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} u _ 1 = & \\ \\frac { 1 } { 2 } ( d _ { 1 1 1 } - 1 , d _ { 2 1 1 } ) , & u _ 2 = & \\ u _ 1 \\ + \\ \\left ( \\frac { 1 } { 2 } , 0 \\right ) , \\\\ u _ 3 = & \\ \\frac { 1 } { 2 } ( - 1 , 0 ) , & u _ 4 = & \\ u _ 3 \\ + \\ \\left ( \\frac { 5 } { 6 } , 0 \\right ) , \\\\ u _ 5 = & \\ \\frac { 1 } { 3 } ( - 2 d _ { 1 1 1 } - 1 , - 2 d _ { 2 1 1 } ) , & u _ 6 = & \\ u _ 5 \\ + \\ \\left ( \\frac { 2 } { 3 } , 0 \\right ) . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m _ { N + 1 } } \\mathbb { E } ( f \\mid A _ { j } ^ { ( N + 1 ) } ) \\Phi _ { k } ^ { ( N ) } ( A _ { j } ^ { ( N + 1 ) } ) ^ { t } = \\left [ \\Phi _ { k } ^ { ( N ) } [ \\mathcal { A } ^ { ( N + 1 ) } ] \\cdot \\mathbb { E } [ f \\mid \\mathcal { A } ^ { ( N + 1 ) } ] \\right ] ^ { t } . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} a \\ell - b _ 1 m - c _ 2 n & = 0 , \\\\ - a _ 2 \\ell + b m - c _ 1 n & = 0 , \\\\ - a _ 1 \\ell - b _ 2 m + c n & = 0 . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} f _ j \\left ( - \\frac 1 \\tau \\right ) = ( - i \\tau ) ^ { \\kappa _ 1 } \\sum _ { 1 \\le k \\le N } \\chi _ { j , k } \\ : f _ k ( \\tau ) , g _ \\ell \\left ( - \\frac 1 \\tau \\right ) = ( - i \\tau ) ^ { \\kappa _ 2 } \\sum _ { 1 \\le m \\le M } \\psi _ { \\ell , m } \\ : g _ m ( \\tau ) . \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q \\ , | \\ , b ) & = \\exp \\Bigl ( - \\int \\limits _ 0 ^ \\infty ( 1 - e ^ { - t q } ) d K ^ { ( f ) } _ { M , N } ( t \\ , | \\ , b ) / t \\Bigr ) , \\\\ d K ^ { ( f ) } _ { M , N } ( t \\ , | \\ , b ) & \\triangleq e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) f ( t ) \\ , d t / t ^ M . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\theta ( \\zeta ) = \\theta ( \\zeta ) _ { - } + \\theta ( \\zeta ) _ { + } , \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{gather*} L ( g , 2 ) = ( 1 6 ) ^ { - 1 / 2 } 2 \\pi B ( 1 / 4 , 1 / 4 ) ^ 2 / ( 3 2 \\pi ) = B ( 1 / 4 , 1 / 4 ) ^ 2 / 6 4 \\end{gather*}"}
-{"id": "3266.png", "formula": "\\begin{align*} U ( v ( E ) ) = - B ( U , \\xi ) - u ( A _ { N } U ) . \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} \\mathcal { T S } _ 0 ^ { o p } & = \\{ B _ 1 , B _ 2 , \\ldots , B _ { n - 1 } , B _ n \\} \\\\ & = \\{ A _ { n - 1 } , A _ { n - 2 } , \\ldots , A _ 2 , A _ 1 , A _ n \\} \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} t _ { s _ { 1 , 1 } } ^ { 2 m } = \\prod _ { i = 1 } ^ { m } { } _ { t _ { c _ 1 } ^ { - 2 i } } ( t _ { t _ { c _ 1 } ( z _ 1 ) } ^ { - 1 } t _ { a _ 1 } t _ { z _ 1 } ^ { - 1 } t _ { a _ 1 } ) \\cdot t _ { c _ 1 } ^ { - 2 m } t _ { a _ 1 } ^ { 2 m } \\cdot t _ { d _ { g - 1 } } ^ { 2 m } t _ { d _ 1 } ^ { 2 m } \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} \\phi _ N ( t ) : = \\frac { 1 } { Z ^ f _ { \\beta _ c , N } } \\frac { P ( N ( ( 1 - t ) ) } { Q ( N ( 1 - t ) ) } . \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\frac { \\ , a b \\ , } { c d } \\ , , \\begin{array} { l } a , b , c \\in \\N , \\\\ a + b + c = d . \\end{array} \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} \\frac { F ( X , Y ) } { Q ( Y ) } = X ^ 2 + X + k + 1 + \\frac { ( C _ 1 ^ 2 + C _ 0 C _ 2 ) ( C _ 2 ^ 2 + C _ 1 C _ 3 ) } { ( C _ 1 C _ 2 + C _ 0 C _ 3 ) ^ 2 } \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} f ' ( Q _ n \\cdot ( \\exp ( H ) , 1 ) , s ) = \\Big ( e ^ { d + ( 2 n + 1 ) ( { \\rm R e } ( s ) + 1 / 2 ) } \\Big ) ^ { a _ 1 + \\cdots + a _ n } f ' ( \\left [ \\begin{smallmatrix} I _ n \\\\ & I _ n \\\\ & A & I _ n \\\\ A & & & I _ n \\end{smallmatrix} \\right ] \\left [ \\begin{smallmatrix} I _ n \\\\ & & & I _ n \\\\ & & I _ n \\\\ & - I _ n \\end{smallmatrix} \\right ] , s ) \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} \\partial ( q _ 1 \\ast q _ 2 ) = \\partial ( q _ 1 ) \\ast \\big ( \\phi ( e _ { q _ 1 } ) \\cdot \\partial ( q _ 2 ) \\big ) ~ \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} \\Pr \\left ( { \\left . { { s _ n } , { { \\tilde \\theta } _ n } , \\vartheta } \\right | { { \\bf { R } } _ q } } \\right ) = \\Pr \\left ( { \\left . { { s _ n } , { { \\tilde \\theta } _ n } , \\vartheta } \\right | { { \\bf { R } } _ q } , \\left | { { h _ A } } \\right | { \\rm { , } } \\left | { { h _ B } } \\right | } \\right ) \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} J = \\begin{pmatrix} & I _ { \\frac { k a } { 8 } } \\\\ I _ { \\frac { k a } { 8 } } \\end{pmatrix} , \\ J ' = \\begin{pmatrix} & I _ { \\frac { s } { 2 } } \\\\ I _ { \\frac { s } { 2 } } \\end{pmatrix} , \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} \\psi _ \\zeta ^ i ( x ) = e ^ { i A ( 0 ) x } \\phi _ \\zeta ^ i ( x ) \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { U } \\tilde { J } V = \\nabla _ { U } \\tilde { J } V + B ( U , \\tilde { J } V ) N , \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} \\mathcal A _ r : = \\{ \\omega \\in \\Omega \\ , | \\ , \\mathbb B _ r \\Subset U _ \\omega \\} . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} \\frac { 1 } { p _ s } = \\frac { s } { p } + \\frac { 1 - s } { 2 } \\ , . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} \\| T ' \\| _ { S _ 1 } & = \\left [ \\prod _ { i = 1 } ^ N \\left ( 1 - \\frac { 1 } { q _ i } \\right ) ^ { - 1 } \\right ] \\left \\| \\left [ \\prod _ { i = 1 } ^ N \\left ( I - \\frac { \\tau _ i } { q _ i } \\right ) \\right ] T \\right \\| _ { S _ 1 } \\\\ & \\leq \\left [ \\prod _ { i = 1 } ^ N \\left ( 1 - \\frac { 1 } { q _ i } \\right ) ^ { - 1 } \\right ] \\left [ \\prod _ { i = 1 } ^ N \\left ( 1 + \\frac { 1 } { q _ i } \\right ) \\right ] \\| T \\| _ { S _ 1 } \\\\ & = \\left [ \\prod _ { i = 1 } ^ N \\frac { q _ i + 1 } { q _ i - 1 } \\right ] \\| T \\| _ { S _ 1 } . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} H ^ { l 1 } _ { k , l } = H ^ { r 1 } _ { n - l , n - k } H ^ { l 2 } _ { k , l } = H ^ { r 2 } _ { n - l , n - k } . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} A \\| g \\| _ G \\le \\| \\{ \\langle f _ n , g \\rangle \\} _ { n = 1 } ^ { \\infty } \\| _ Y \\le B \\| g \\| _ G . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} F _ 3 = \\ , & 1 + a _ 1 ^ 2 + a _ 1 ^ 3 + a _ 1 ^ 4 + b + a _ 1 b + a _ 1 ^ 4 b + b ^ 2 + a _ 1 b ^ 2 + a _ 1 ^ 3 b ^ 2 + b ^ 3 + b ^ 4 + a _ 1 ^ 2 b ^ 4 \\\\ & + b ^ 5 + a _ 1 b ^ 5 + b ^ 6 + a _ 1 b ^ 6 + b ^ 7 + ( a _ 1 ^ 2 + a _ 1 ^ 4 + a _ 1 ^ 5 + a _ 1 ^ 2 b + a _ 1 ^ 4 b ^ 2 + a _ 1 ^ 2 b ^ 4 + a _ 1 ^ 2 b ^ 5 ) k \\cr & + ( a _ 1 ^ 4 + a _ 1 ^ 4 b + a _ 1 ^ 5 b + a _ 1 ^ 4 b ^ 2 + a _ 1 ^ 5 b ^ 2 + a _ 1 ^ 4 b ^ 3 ) k ^ 2 + ( a _ 1 ^ 6 + a _ 1 ^ 6 b ) k ^ 3 , \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 1 ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; f _ t ( x _ t , x _ { t - 1 } , \\xi _ { t j } ) + \\mathcal { Q } _ { t + 1 } ( x _ { t } , 1 ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ { t j } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} q ^ { 0 , h } ( \\iota , \\iota ' ) = \\begin{cases} n _ { \\iota } \\mu _ { i } , & i = i ' , \\ n _ { \\iota ' } = n _ { \\iota } - 1 , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\Theta ) ( e ) = \\displaystyle \\sum _ { e ' : ( e , e ' ) \\in K } \\Theta ( e , e ' ) - \\displaystyle \\sum _ { e ' : ( e ' , e ) \\in K } \\Theta ( e ' , e ) , \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} | \\lambda - \\lambda ' | \\ge | \\lambda _ { r } - \\lambda _ { r ' } | - 3 \\eta _ { q } \\ge 4 \\eta _ { q } - 3 \\eta _ { q } = \\eta _ { q } > 0 . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\Delta = \\Delta ( h , f ) = \\Phi ( \\Gamma ( h , f ) ) \\subset D \\times D ' , \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} \\big \\langle \\Lambda _ v P _ { v , \\rm R } F ( v ) , P _ { v , \\rm R } F ( v ) \\big \\rangle = \\alpha _ v | f ( v ) | ^ 2 . \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} \\frac { 2 p } { 2 - p } \\log ( \\frac { \\left \\| \\sqrt { n _ { \\pi } } a _ { \\pi } \\right \\| _ p } { \\left \\| \\sqrt { n _ { \\pi } } \\widehat { a _ { \\pi } } \\right \\| _ { p ' } } ) & = \\frac { 2 p } { 2 - p } \\log ( \\frac { \\left \\| \\sqrt { n _ { \\pi } } a _ { \\pi } \\right \\| _ p } { n _ { \\pi } ^ { \\frac { 1 } { 2 } - \\frac { 1 } { p } } } ) \\\\ & \\leq \\frac { 2 p } { 2 - p } \\log ( B _ 1 ^ { \\frac { 2 - p } { p } } ) = \\log ( B _ 1 ^ 2 ) , \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} U _ j & = \\sum _ { k = 1 } ^ d u _ k ( \\gamma ^ { - 1 } ) _ { k j } , j = 1 , \\dots , d , \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} \\left ( { 1 + x } \\right ) _ q ^ \\infty = \\sum \\limits _ { j = 0 } ^ \\infty { q ^ { j \\left ( { j - 1 } \\right ) / 2 } \\frac { { x ^ j } } { { \\left ( { 1 - q } \\right ) \\left ( { 1 - q ^ 2 } \\right ) \\ldots \\left ( { 1 - q ^ j } \\right ) } } } , \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} I ^ * = \\{ x \\mid c x ^ { p ^ e } \\in I ^ { [ p ^ e ] } c \\in R ^ { \\circ } e \\gg 0 \\} . \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac \\pi 2 } ( \\sin t ) ^ { 2 q + 1 } ( \\cos t ) ^ { 2 q + 1 } \\ , d t \\ , = \\ , \\frac 1 2 B ( q + 1 , q + 1 ) \\ , = \\ , \\frac 1 2 \\frac { ( q ! ) ^ 2 } { ( 2 q + 1 ) ! } . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} b = d > 0 , \\ ; a < 0 , \\ ; c < 0 . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} ( \\mathsf { e } _ t \\ f ) \\star ( \\mathsf { e } _ t \\ g ) = \\mathsf { e } _ t \\ ( f \\star g ) , \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\mathbb { W } _ { { X } _ k ( D ) } : = { \\rm S p a n } \\{ \\omega _ { \\ ! D } ( s _ 1 , \\ldots , s _ k ) \\mid s _ 1 , \\ldots , s _ k \\in H ^ 0 ( X , L ) \\} \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} M _ 1 = & M _ { ( a , x ) } \\ ; \\ ; a = ( 1 , \\tau ) , \\ ; x = \\bigl ( 1 + \\tau ( 1 + \\lambda _ 1 ) , \\ , 1 + \\tau ( 1 + \\lambda _ 2 ) \\bigr ) , \\\\ M _ 2 = & M _ { ( a , x ) } \\ ; \\ ; a = ( 1 , \\tau ) , \\ ; x = ( - \\tau \\lambda _ 1 , \\ , - \\tau \\lambda _ 2 ) . \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} ( { \\cal L _ \\gamma } ) ' ( 0 ) = g _ { \\dot \\gamma ( b ) } ( \\dot \\gamma , U ) | _ a ^ b - \\int _ a ^ b g _ { \\dot \\gamma } ( D _ { \\dot \\gamma } { \\dot \\gamma } , U ) d t , \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} & A \\Big ( A ^ { - 1 } \\sharp ( A ! _ \\lambda B ) ^ { - 1 } \\Big ) ( A \\nabla _ \\lambda B ) \\Big ( A ^ { - 1 } \\sharp ( A ! _ \\lambda B ) ^ { - 1 } \\Big ) A = A ( A ! _ \\lambda B ) ^ { - 1 } ( A \\nabla _ \\lambda B ) \\qquad \\qquad \\qquad \\\\ & = ( A \\nabla _ \\lambda B ) ( A ! _ \\lambda B ) ^ { - 1 } A = \\lambda ( 1 - \\lambda ) ( A - B ) B ^ { - 1 } ( A - B ) + A . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\mathcal K _ { \\boldsymbol { k } } ( \\tau ) : = 2 \\mathcal J _ { \\boldsymbol { k } } ( \\tau ) + \\mathcal J _ { \\left ( \\frac { k _ 1 + k _ 2 } { 2 } , \\frac { k _ 2 - 3 k _ 1 } { 2 } \\right ) } ( \\tau ) , \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} G ( z ) = A ( z ) \\bigg ( 1 + \\sum _ n a _ n \\bar b _ n t _ n \\nu _ n + \\sum _ n \\frac { a _ n \\bar b _ n t ^ 2 _ n \\nu _ n } { z - t _ n } \\bigg ) . \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} w ( x ) : = \\int _ { \\R ^ 3 } g ( y ) \\Phi _ 0 ( x - y ) d y = \\int _ { \\Omega } g ( y ) \\Phi _ 0 ( x - y ) d y . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} S _ Y ( n , m ) = \\sum _ { r = m } ^ n S ( n , r ) S ( r , m ) \\lambda ^ r , m \\leq n . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { l = 0 } ^ \\infty u _ l ( x ) \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } \\log \\left ( X ^ { 1 / 2 } A _ j ^ { - 1 } X ^ { 1 / 2 } \\right ) = 0 . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} u = \\abs { m } ^ { - \\frac { 2 } { 3 d ( d + 1 ) } } \\big ( \\varphi _ 1 ( X ) \\sigma _ 1 ( X ) \\big ) ^ { \\frac { 4 } { 3 d ( d + 1 ) ^ 2 } } , \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} \\begin{cases} a _ { 0 , \\sup } + \\frac { a _ { 2 , \\sup } } { 2 } \\bar r _ 1 + \\frac { b _ { 1 , \\sup } } { 2 } \\bar r _ 2 < 2 a _ { 1 , \\inf } \\underbar r _ 1 + a _ { 2 , \\inf } \\underbar r _ 2 \\cr b _ { 0 , \\sup } + \\frac { b _ { 1 , \\sup } } { 2 } \\bar r _ 2 + \\frac { a _ { 2 , \\sup } } { 2 } \\bar r _ 1 < 2 b _ { 2 , \\inf } \\underbar r _ 2 + b _ { 1 , \\inf } \\underbar r _ 1 . \\end{cases} \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} S ^ { o , h } _ A ( i _ { \\kappa } , \\kappa ) = f ^ { o , \\kappa , h } _ A ( \\rho ^ h _ { \\kappa } ) \\leq \\min \\{ 1 - \\Delta ^ { \\kappa } _ A ( h ) , 1 \\} \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} & \\langle ( \\xi _ 1 , \\eta _ 1 , \\zeta _ 1 ) , ( \\xi _ 2 , \\eta _ 2 , \\zeta _ 2 ) \\rangle _ { ( A _ r , B _ r , C _ r ) } \\\\ : = & { \\rm t r } ( A _ r ^ { - 1 } \\xi _ 1 A _ r ^ { - 1 } \\xi _ 2 ) + { \\rm t r } ( \\eta _ 1 ^ T \\eta _ 2 ) + { \\rm t r } ( \\zeta _ 1 ^ T \\zeta _ 2 ) \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } \\bigr ) ( q \\ , | \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } \\bigr ) ( 0 \\ , | \\ , b ) = - q f ( 0 ) \\ , M ! \\prod _ { j = 1 } ^ { M - 1 } b _ j . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} F ( k ) = F ( a b ^ 2 ) = f ( b ) g ( b ) H ( a ) , \\textrm { w h e r e } H ( a ) = \\sum _ { a = r m n } h ( r ) f ( m ) g ( n ) . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} \\mu _ i = \\sum _ { j = 0 } ^ k | ( \\nabla ^ { j , g _ P } \\eta _ i ' ) ( { x } _ i ) | _ { g _ P ( x _ i ) } \\to \\infty . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} a _ 1 v _ { \\sigma ( 1 ) } a _ 2 \\cdots a _ { t - 1 } v _ { \\sigma ( t ) } a _ { t + 1 } = 0 \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} \\overline { \\mathrm { L } } ( y ) = \\sup _ { x \\in [ 0 , y ] } \\mathcal { L } [ f - f _ 1 ] ( x ) \\mathrm { \\ a n d \\ } \\underline { \\mathrm { L } } ( y ) = \\inf _ { x \\in [ 0 , y ] } \\mathcal { L } [ f - f _ 1 ] ( x ) . \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} P _ \\mu ( A ) = \\frac 1 { n ! } \\ , \\langle \\Pi _ \\mu ( A ) \\mathbf { 1 } , \\mathbf { 1 } \\rangle , \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} & \\mathrm { C o v } ( \\mathbb { G } ^ { ( N ) } ( f ) , \\mathbb { G } ^ { ( N ) } ( g ) ) - \\mathrm { C o v } ( \\mathbb { G } ( f ) , \\mathbb { G } ( g ) ) \\\\ & = \\frac { 1 } { 2 } \\left ( \\Upsilon _ { N } ( f , g ) - 2 \\Gamma _ { N } ( f , g ) \\right ) + \\frac { 1 } { 2 } \\left ( \\Upsilon _ { N } ( g , f ) - 2 \\Gamma _ { N } ( g , f ) \\right ) , \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} w = \\frac { M _ 0 } { M } \\mathcal { B } _ { M } \\mbox { a . e . i n $ \\R ^ n \\times ( 0 , \\infty ) $ } . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} V ( x ) ~ = ~ \\lim _ { n \\to \\infty } { \\P _ x ( \\tau _ \\vartheta > n ) } / { \\P _ 0 ( \\tau _ \\vartheta > n ) } ~ = ~ { \\cal V } ( x ) ; \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} \\mathrm { d } { x } _ t ^ { i , N } & = v _ t ^ { i , N } \\ , \\mathrm { d } t \\ , , \\\\ \\ \\mathrm { d } { v } _ t ^ { i , N } & = - v _ t ^ { i , N } \\mathrm { d } t + G \\left ( \\frac { \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x _ t ^ { i , N } - x _ t ^ { j , N } ) v _ t ^ { j , N } } { \\frac 1 N \\sum _ { j = 1 } ^ N \\varphi ( x _ t ^ { i , N } - x _ t ^ { j , N } ) } \\right ) \\mathrm { d } t + \\sqrt { 2 \\sigma } \\mathrm { d } W _ t ^ { i } \\ , , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} \\omega _ n ( t ^ { i _ 0 } , t ^ { j _ 0 } ) = \\sum _ { i = i _ 0 } ^ \\infty \\delta _ { i _ 0 , i } \\delta _ { j _ 0 , i + n } t ^ i = \\delta _ { j _ 0 , i _ 0 + n } t ^ { i _ 0 } \\ ; \\ ; \\mbox { f o r a l l $ i _ 0 , j _ 0 \\in \\Z $ } \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} P _ \\mu ( A ) = \\sum _ { j = 1 } ^ n a _ { i j } P _ \\mu ( A _ { i j } ( \\mu ) ) = \\sum _ { i = 1 } ^ n a _ { i j } P _ \\mu ( A _ { i j } ( \\mu ) ) \\ , . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} w ^ { ( i ) } w ^ { ( j ) } = \\frac { ( i + j ) ! } { i ! j ! } \\ , w ^ { ( i + j ) } \\quad \\dd ( w ^ { ( i ) } ) = \\dd ( w ) w ^ { ( i - 1 ) } \\quad i , j \\ge 0 \\ , . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\big \\| \\rho ^ N _ 0 \\circ \\Pi _ N - \\rho _ 0 \\big \\| _ { L ^ 2 ( \\mu ) } = 0 . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{gather*} L ( f _ { 3 6 } , 1 ) = - 2 \\pi { \\rm i } \\frac { \\rm i } { \\sqrt 3 } \\frac { \\phi ( \\tau _ 0 ) } { 6 } = \\frac 1 { 2 ^ { 4 / 3 } \\cdot 3 } B ( 1 / 3 , 1 / 3 ) , \\end{gather*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\delta \\cdot \\delta ^ { \\prime } = \\sum m _ { i } \\mathbf { a } _ { i } \\cdot \\mathbf { b } _ { i } , \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} M & = \\frac { 2 P _ 3 ( 1 ) } { L } \\sideset { } { ^ + } \\sum _ { \\chi ( q ) } \\epsilon ( \\chi ) \\sum _ { b \\leq y _ 2 } \\frac { \\Lambda ( b ) \\overline { \\chi } ( b ) P _ 2 [ b ] } { b ^ { \\frac { 1 } { 2 } } } \\sum _ n \\frac { \\overline { \\chi } ( n ) } { n ^ { \\frac { 1 } { 2 } } } V \\left ( \\frac { n } { q } \\right ) . \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} u ( U ) = 0 . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} \\sum _ i \\ell _ i \\lambda _ i + \\sum _ j m _ j \\mu _ j = 0 \\\\ \\sum _ i \\ell _ i + \\sum _ j m _ j = 0 , \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} V ( \\bullet _ a ^ * ) : = V _ a , V \\big ( ( [ \\tau _ 1 \\dots \\tau _ n ] _ a ) ^ * \\big ) : = \\frac { 1 } { \\sigma \\big ( { [ \\tau _ 1 \\dots \\tau _ n ] _ a } \\big ) } ( D ^ n V _ a ) \\big ( V ( \\tau _ 1 ^ * ) , \\dots , V ( \\tau _ n ^ * ) \\big ) ; \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} K _ 1 : = \\left ( \\frac { m - 1 } { p - 2 + m } \\right ) ^ { \\frac { m - 1 } { p - 1 } } - \\left ( \\frac { m - 1 } { p - 2 + m } \\right ) ^ { \\frac { p - 2 + m } { p - 1 } } > 0 \\ , . \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} f ( A ) = \\sum _ { I \\subset [ N ] } \\langle \\pi _ I ( A ) \\xi ^ I , \\eta ^ I \\rangle , \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} L _ { X , i j } Q = T ^ { j - i } Q + Q T ^ { i - j } , \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} [ [ x , y ] , \\alpha ( z ) ] = [ [ x , z ] , \\alpha ( y ) ] + [ \\alpha ( x ) , [ y , z ] ] . \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} L ( r ^ k , \\vec s ^ k ) = r ^ k \\| \\vec x ^ { k + 1 } \\| _ \\infty - ( \\vec x ^ { k + 1 } , \\vec s ^ k ) < 0 . \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , M } ( x \\ , | \\ , a ) \\ , \\log ( x ) \\bigr ) ( q | b ) = - M ! \\bigl ( \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j / \\prod \\limits _ { i = 1 } ^ M a _ i \\bigr ) q \\log ( q ) + O ( q ) . \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\mathcal J _ { \\boldsymbol { k } } ( \\tau ) : = \\sum _ { \\delta \\in \\{ 0 , 1 \\} } \\mathcal I _ { ( k _ 1 + p \\delta , k _ 2 + 3 p \\delta ) } ( \\tau ) , \\quad \\mathcal I _ { \\boldsymbol { k } } ( \\tau ) : = - \\frac { \\sqrt { 3 } } { 8 \\pi } I _ { \\Theta _ 1 ( 2 p , k _ 1 , 2 p ; \\cdot ) , \\Theta _ 0 ( 6 p , k _ 2 , 6 p ; \\cdot ) } ( \\tau ) . \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } ( \\beta _ m ) \\mathbf h _ p ( x ) = \\chi _ { \\psi } ( p ^ m ) p ^ { - m / 2 } \\int _ { p ^ { - m } \\Z _ p ^ { \\times } } \\psi ( 2 x y ) \\underline { \\chi } _ p ^ { - 1 } ( y ) d y = \\chi _ { \\psi } ( p ^ m ) p ^ { m / 2 } \\mathfrak G ( 2 x p ^ { - m } , \\underline { \\chi } _ p ^ { - 1 } ) . \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} C _ { n , k } = \\frac { n - k + 1 } { n + 1 } \\binom { n + k } { n } . \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ { + } } \\ell ^ { \\varepsilon } = \\infty , \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} \\psi ( t ) = \\det ( \\gamma ' ( t ) , \\eta ( t ) ) . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} p ^ * \\left ( \\frac { \\prod \\limits _ { i = 1 } ^ a L _ { i , b - k + i } } { \\prod \\limits _ { i = 1 } ^ { a } L _ { i , b - k + i + 1 } } \\right ) = \\frac { \\Delta _ { \\{ k - a , \\dots , k - 1 \\} } } { \\Delta _ { \\{ k - a + 1 , \\dots , k \\} } } \\prod _ { i = k - a } ^ { k - 1 } \\lambda _ i = p ^ * ( M _ k ) . \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} \\frac { d } { d t } u ( x , t ) = L u ( x , t ) + \\lambda u ( x , t ) - \\lambda ( u ( x , t ) ) ^ { 1 + \\beta } , u ( x , 0 , \\theta ) = 1 - e ^ { i \\theta g _ k ( x ) } . \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { N = N _ n } ^ { N _ { n + 1 } - 1 } \\big | Y _ { N + 1 } f - Y _ N f \\big | \\Big \\| _ { \\ell ^ p } \\leq C _ { p , \\rho } n ^ { \\rho - 1 } \\| f \\| _ { \\ell ^ p } , \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} E ^ 2 = - \\frac { d } { p q } A \\cdot E = \\frac { 1 } { q } , B \\cdot E = \\frac { 1 } { p } , ( A \\cdot A ) _ { S _ { P , \\omega } } = ( A \\cdot A ) _ S - \\frac { p } { d q } , ( B \\cdot B ) _ { S _ { P , \\omega } } = ( B \\cdot B ) _ S - \\frac { q } { d p } . \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } D \\imath \\\\ D ^ 2 q D \\beta ^ { - 1 } \\end{array} \\right ) \\beta \\left ( \\begin{array} { l l } \\imath & \\beta ^ { - 1 } D q \\end{array} \\right ) = \\left ( \\begin{array} { l l } \\alpha & 0 \\\\ 0 & \\gamma \\end{array} \\right ) . \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} \\overrightarrow a ( x ) \\cdot \\nabla p ( x ) = 0 , \\ \\ \\mbox { f o r a l l } \\ x \\in \\Omega . \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} \\zeta _ S ( \\beta ) : = \\sum _ { s \\in S } N ( s ) ^ { - \\beta } \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} U = X + i Y , V = Z + i W , \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} W _ { \\lambda } & \\cong \\bigoplus _ { ( r , s ) \\vDash d } \\bigoplus _ { \\substack { \\mu \\in \\Lambda ( n , r ) \\\\ \\nu \\in \\Lambda ( n , s ) \\\\ \\lambda = \\mu - \\nu } } \\left ( M ^ { \\mu } \\boxtimes M ^ { \\nu } \\right ) \\otimes _ { \\C [ S _ r \\times S _ s ] } \\C [ S _ d ] \\\\ & = \\bigoplus _ { ( r , s ) \\vDash d } \\bigoplus _ { \\substack { \\mu \\in \\Lambda ( n , r ) \\\\ \\nu \\in \\Lambda ( n , s ) \\\\ \\lambda = \\mu - \\nu } } M ^ { ( \\mu , \\nu ) } \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} & \\left \\langle \\rho , u \\alpha _ { 1 } , v \\alpha _ { 3 } \\right \\rangle \\cong C _ { p } ^ { 3 } \\\\ & \\ A = \\begin{pmatrix} u _ { 2 } & v _ { 2 } \\\\ u _ { 3 } & v _ { 3 } \\end{pmatrix} \\in \\mathrm { G L } _ { 2 } ( \\mathbb { F } _ { p } ) \\ \\ v _ { 2 } = u _ { 3 } + \\mathrm { d e t } ( A ) . \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\vec { g } _ { 1 } = \\frac { \\vec { f } _ { 1 } + \\vec { f } _ { 2 } } { \\| \\vec { f } _ { 1 } + \\vec { f } _ { 2 } \\| } , \\ \\vec { g } _ { 2 } = \\frac { \\vec { f } _ { 1 } + \\vec { f } _ { 3 } } { \\| \\vec { f } _ { 1 } + \\vec { f } _ { 3 } \\| } , \\ \\vec { g } _ { 3 } = \\frac { \\vec { f } _ { 1 } + \\vec { f } _ { 4 } } { \\| \\vec { f } _ { 1 } + \\vec { f } _ { 4 } \\| } . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} ( { \\mathbb D } \\odot { \\mathbb D } ) ^ { B } { } _ { A } { } ^ { D } { } _ { C } = { \\mathbb D } ^ { B } { } _ { A } { \\mathbb D } ^ { D } { } _ { C } + \\ , { \\mathbb D } ^ { D } { } _ { C } { \\mathbb D } ^ { B } { } _ { A } \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} e = \\sum _ { i = 1 } ^ { 2 q + 1 } d _ i + \\sum _ { i = 1 , i } ^ { 2 q - 1 } \\left ( ( r _ { i + 1 } + 2 ) t - ( r _ { i + 1 } + 1 ) d _ i \\right ) . \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} \\begin{cases} u '' ( x ) + \\omega ( x ) u ( x ) = 0 , \\ , \\ , x \\in ( a , b ) , \\\\ u ( a ) = u ( b ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} d _ X ( x _ 1 , x _ 2 ) = \\inf _ { m \\in \\mathbb Z } | x _ 1 + m - x _ 2 | . \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } ^ k , 1 , \\xi _ { t j } , 0 ) = \\left \\{ \\begin{array} { l } \\sup _ { \\lambda } \\ ; \\lambda ^ T ( b _ { t j } - B _ { t j } x _ { t - 1 } ^ k ) \\\\ A _ { t j } ^ T \\lambda \\leq c _ { t j } . \\end{array} \\right . \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} \\left [ n \\right ] _ q = \\frac { { q ^ n - 1 } } { { q - 1 } } = 1 + q + q ^ 2 + \\cdots q ^ { n - 1 } . \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} \\{ \\lambda _ { a , u } ( t ) ~ | ~ 2 \\leq | u | \\leq | w | , ~ s ( u ) = i , ~ t ( u ) = j \\} \\bigcup \\{ \\gamma _ { b , v } ( t ) ~ | ~ b \\in Q _ 1 , ~ 1 \\leq | v | < | w | \\} . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} \\min _ { v \\in E } | u - v | = | u - [ u ] | , \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} y _ { n + 1 } = G _ n ( y _ n ) n \\in \\Z , \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } ( - \\Delta ) ^ { s _ { 1 } } u + V _ { 1 } ( x ) u = f _ { 1 } ( u ) + \\lambda ( x ) v , & x \\in \\mathbb { R } ^ { N } , \\\\ ( - \\Delta ) ^ { s _ { 2 } } v + V _ { 2 } ( x ) v = f _ { 2 } ( v ) + \\lambda ( x ) u , & x \\in \\mathbb { R } ^ { N } , \\end{array} \\right . \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} f \\circ \\alpha = \\alpha _ A \\circ f . \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} { ( { \\mathbb D } ^ 2 ) } ^ A { } _ { B } = & \\ , \\ , { \\mathbb D } ^ { A } { } _ { R } { \\mathbb D } ^ { R } { } _ { B } + { \\mathbb D } ^ { R } { } _ { B } { \\mathbb D } ^ { A } { } _ { R } = ( { \\mathbb D } ^ { A } { } _ { R } { \\mathbb D } ^ { R } { } _ { B } - { \\mathbb D } ^ { R } { } _ { B } { \\mathbb D } ^ { A } { } _ { R } ) + 2 \\ , { \\mathbb D } ^ { R } { } _ { B } { \\mathbb D } ^ { A } { } _ { R } \\\\ = & \\ , \\ , 3 { \\mathbb D } ^ { A } { } _ { B } + 2 \\ , { \\mathbb D } ^ { R } { } _ { B } { \\mathbb D } ^ { A } { } _ { R } , \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\sum _ { e \\ , \\cup f \\in E ( G _ 1 * G _ 2 ) } w ( e \\cup f ) = \\Bigg ( \\sum _ { e \\in E ( G _ 1 ) } w _ 1 ( e ) \\Bigg ) \\Bigg ( \\sum _ { f \\in E ( G _ 2 ) } w _ 2 ( f ) \\Bigg ) = 1 . \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} \\varepsilon ^ \\prime _ { 2 k + 3 , r } = ( - 1 ) ^ { k + 1 } \\varepsilon _ { 2 k + 3 , r } = ( - 1 ) ^ { k + 1 } ( - 1 ) ^ { k + 4 } = - 1 . \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\tilde e _ k ( x ) \\tilde e _ l ( x ) = \\tilde e _ { k + l } ( x ) , \\overline { \\tilde e _ k ( x ) } = \\tilde e _ { - k } ( x ) , ( \\tilde e _ k \\ast \\tilde e _ l ) ( x ) = \\delta _ { k , l } \\ , \\tilde e _ k ( x ) , \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} | N | ^ { r } N _ { w , H } ( g N ) = \\sum _ { y \\in g N } N _ { w , G } ( y ) \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{gather*} j = \\frac { ( 1 + 2 5 6 u ) ^ 3 } { u } . \\end{gather*}"}
-{"id": "1236.png", "formula": "\\begin{align*} w _ { \\beta } ^ { \\varepsilon } \\left ( x , T \\right ) = u _ { \\beta f } ^ { \\varepsilon } \\left ( x \\right ) - u _ { f } \\left ( x \\right ) \\quad x \\in \\Omega . \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\Lambda ^ { + } ( X ) = \\sum _ { i \\in \\mathcal { I } ^ - } ( A _ i - \\overline { X } ) ^ T \\otimes \\Phi ^ + _ i ( X ) \\overline { M } ^ { + } + \\sum _ { i \\in \\mathcal { I } ^ + } ( A _ i - \\overline { X } ) ^ T \\otimes \\Phi ^ + _ i ( X ) - ( X - \\overline { X } ) ^ T \\otimes I _ 2 , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} \\alpha ( x ) \\ , = \\ , U _ \\alpha \\ , ( 1 \\otimes x ) \\ , U _ \\alpha ^ * \\ \\ \\ ( x \\in N ) \\ , . \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\begin{bmatrix} \\lambda ^ 4 & 0 \\\\ 0 & 0 \\end{bmatrix} = \\begin{bmatrix} \\lambda & 1 & 0 \\\\ 0 & \\lambda & 1 \\end{bmatrix} M ( \\lambda ) \\begin{bmatrix} \\lambda & 0 \\\\ 1 & \\lambda \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} f _ 1 g _ { 2 j } = 0 j \\geq 1 . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} { \\fam 2 O } = v ^ a \\nabla _ a + \\sum \\limits _ { j = 0 } ^ \\infty \\gamma ^ { a _ 1 } \\ldots \\gamma ^ { a _ { 2 j } } w ^ j _ { a _ 1 \\ldots a _ { 2 j } } , w ^ j _ { a _ 1 \\ldots a _ { 2 j } } \\in { \\mathcal E } _ { ( a _ 1 \\ldots a _ { 2 j } ) } \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} \\nu ' ( C ) = \\dim { \\rm G r } _ F ^ 1 H ^ 2 ( F , \\C ) _ { \\eta } . \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} e ( G _ 2 [ Z ] ) \\le \\frac { | X ' | \\cdot | Z | } { 2 } = \\frac { \\alpha \\beta n ^ 2 } { 2 } . \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} \\langle \\ ! \\langle G , q _ 0 ^ { \\Lambda , N } \\rangle \\ ! \\rangle = \\langle \\ ! \\langle K G , R _ 0 ^ { \\Lambda , N } \\rangle \\ ! \\rangle \\geq 0 . \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\nabla ^ { j , g _ P } \\hat { g } ^ \\bullet _ \\infty = 0 \\ ; \\ , ( 0 < j < k ) , \\ ; \\ , \\sup \\nolimits _ { \\mathbb { C } ^ m \\times Y } | \\nabla ^ { k , g _ P } \\hat { g } ^ \\bullet _ \\infty | _ { g _ P } = 1 . \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} \\tilde { R } _ k ( \\alpha ) = \\sum _ { x \\in \\Z ^ d } \\exp ( \\langle \\alpha , x \\rangle ) \\tilde \\mu _ k ( x ) \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} \\tilde { P } ( m ) = P ( Q m + r ) = \\frac { a } { q } Q ^ d m ^ d + . \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\frac { 1 } { z - 1 } - \\sum _ { i = 1 } ^ d \\frac { 1 } { z ^ { 1 / v _ i } - 1 } > 0 , \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { c l } d \\varphi ( r ) = & - \\left [ A ( r ) \\varphi ( r ) + C ( r ) \\nu ( r ) + p ( r ) \\varepsilon _ { 1 } ( r ) + q ( r ) \\varepsilon _ { 2 } ( r ) + \\varepsilon _ { 3 } ( r ) \\right . \\\\ & \\left . + H _ { z } ( r ) ( 1 - p ( r ) \\sigma _ { z } ( r ) ) ^ { - 1 } p ( r ) \\varepsilon _ { 2 } ( r ) \\right ] d r + \\nu ( r ) d B ( r ) , \\\\ \\varphi ( T ) = & \\varepsilon _ { 4 } ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} & L w ( x ) : = \\frac { 1 } { 2 } \\int _ { \\R ^ n } \\left ( w ( x + y ) + w ( x - y ) - 2 w ( x ) \\right ) J ( y ) \\ , d y \\\\ & = \\frac { 1 } { 2 } \\int _ { B _ r } \\left ( w ( x + y ) + w ( x - y ) - 2 w ( x ) \\right ) J ( y ) \\ , d y + \\frac { 1 } { 2 } \\int _ { B _ r ^ c } \\left ( w ( x + y ) + w ( x - y ) - 2 w ( x ) \\right ) J ( y ) \\ , d y \\\\ & \\ge \\frac { 1 } { r ^ 3 } \\int _ { B _ r } | y | ^ 2 J ( y ) d y \\ , = : \\delta ( r ) > 0 \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\lim \\limits _ { y \\rightarrow + \\infty } ( \\rho , u _ 1 , \\theta , h _ 1 ) ( t , x , y ) = ( \\rho ^ e , u _ 1 ^ e , \\theta ^ e , h _ 1 ^ e ) ( t , x , 0 ) . \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} \\nabla _ { k , l } : & M ^ { k , l } \\rightarrow \\C = M ^ { k + l } ~ ; ~ \\sum _ { g \\in G } k _ g g \\mapsto \\sum _ { g \\in G } k _ g & \\\\ \\Delta _ { k , l } : & \\C = M ^ { k + l } \\rightarrow M ^ { k , l } ~ ; ~ 1 \\mapsto \\sum _ { g \\in S _ { k , l } / S _ { k { + } l } } g = \\frac { 1 } { k ! l ! } \\sum _ { g \\in S _ { k { + } l } } S _ { k , l } g \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} & \\lim _ { N \\to \\infty } \\langle u \\otimes v , P _ x \\otimes P _ y U _ R ( x - y ) P _ x \\otimes P _ y u \\otimes v \\rangle \\\\ & = \\lim _ { N \\to \\infty } \\langle | P u | ^ 2 , U _ R * | P v | ^ 2 \\rangle = 4 \\pi \\int _ { \\R ^ 3 } | P u ( x ) | ^ 2 | P v ( x ) | ^ 2 \\d x . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} v = \\sum _ { n \\in \\N } a _ n v _ n \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\| \\vec x \\| & = \\sum _ { i = 1 } ^ n d _ i | x _ i | , \\\\ I ^ + ( \\vec x ) & = \\sum _ { i < j } w _ { i j } | x _ i + x _ j | . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} h _ { d j } ^ * ( 1 ) = 2 ^ { d - j } , j = 0 , \\dots , d , \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} I _ { - \\alpha - \\beta } \\varphi ( z ) = \\sum _ { n \\geq 0 } ( 1 + n ) ^ { \\alpha + \\beta } a _ n z ^ n \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} 2 ( v - v _ { - } ) v _ { \\rho } = \\frac { 1 } { \\rho ^ { 2 } } \\Big ( A ( \\rho ^ { n } - \\rho _ { - } ^ { n } ) - B ( \\frac { 1 } { \\rho ^ { \\alpha } } - \\frac { 1 } { \\rho _ { - } ^ { \\alpha } } ) \\Big ) + \\frac { \\rho - \\rho _ { - } } { \\rho \\rho _ { - } } ( A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } ) > 0 , \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} V _ { \\delta , t } ( y ) & = \\zeta _ q ( 1 + 2 \\delta ) y ^ { - \\delta } + \\frac { H _ { t } ( - \\delta ) } { H _ { t } ( \\delta ) } \\zeta _ q ( 1 - 2 \\delta ) y ^ { \\delta } + \\mathcal { O } _ { N } \\left ( y ^ N ( 1 + | t | ) ^ { B } \\right ) , \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} y _ \\xi ^ { c } : = p _ \\xi + \\sum _ { \\alpha \\in R _ + } c _ \\alpha \\langle \\alpha , \\xi \\rangle \\sigma _ { \\langle \\alpha ^ \\vee , \\lambda \\rangle } ( \\langle \\alpha , x \\rangle ) s _ \\alpha \\ , \\xi \\in V \\ . \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} \\check { \\phi } _ { \\mathbf g , p } \\left ( \\left ( \\begin{smallmatrix} x _ 1 & x _ 2 \\\\ x _ 3 & x _ 4 \\end{smallmatrix} \\right ) \\right ) = \\mathbf 1 _ { \\Z _ p } ( x _ 1 ) \\mathbf 1 _ { \\Z _ p } ( x _ 2 ) \\mathbf 1 _ { p \\Z _ p } ( x _ 3 ) \\mathbf 1 _ { \\Z _ p ^ { \\times } } ( x _ 4 ) \\underline { \\chi } _ p ^ { - 1 } ( x _ 4 ) . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\frac { a } { c } \\in C _ { \\Gamma } , \\\\ q \\nmid \\frac { N } { M } } } \\log \\left | 1 - \\chi ( T _ { a / c } ) \\right | = \\sum _ { c \\mid N , q \\nmid \\frac { N } { M } } \\sum _ { a \\in ( \\Z / M \\Z ) ^ \\times } \\log \\left | 1 - \\chi \\ ! \\left ( 1 - a \\tfrac { N } { M } \\right ) \\right | . \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} \\begin{aligned} r ( X ) & \\geq H ( X | V \\setminus X ) , \\forall X \\subsetneq V , \\\\ r ( V ) & = H ( V ) . \\end{aligned} \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} \\rho ( \\theta ) = e ^ { - \\theta } , \\theta \\geq 0 . \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} { } M _ { n } ^ { E } = \\begin{pmatrix} \\det ( E - H _ { v , [ 1 , n ] } ) & - \\det ( E - H _ { v , [ 2 , n ] } ) \\\\ \\det ( E - H _ { v , [ 1 , n - 1 ] } ) & - \\det ( E - H _ { v , [ 2 , n - 1 ] } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\sigma _ z : = \\left ( { \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\\\ \\end{array} } \\right ) . \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} \\Lambda _ 1 & = x + ( \\zeta ^ 4 + \\zeta ^ 3 ) y - ( \\zeta ^ 4 + \\zeta ^ 3 + 1 ) z , \\\\ \\Lambda _ 2 & = - ( \\zeta ^ 4 + \\zeta ^ 3 + 1 ) x + y + ( \\zeta ^ 4 + \\zeta ^ 3 ) z , \\\\ \\Lambda _ 3 & = ( \\zeta ^ 4 + \\zeta ^ 3 ) x - ( \\zeta ^ 4 + \\zeta ^ 3 + 1 ) y + z , \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} V _ N = 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\log X + \\log Y + \\log Y ' + o ( 1 ) . \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} \\bigg | \\sum _ n \\frac { d _ n } { z - t _ n } \\bigg | = o ( 1 ) , \\bigg | \\sum _ n \\frac { t _ n d _ n } { z - t _ n } \\bigg | = o ( | z | ) , | z | \\to \\infty , \\ \\ { \\rm d i s t } \\ , ( z , T ) \\ge 1 . \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} S _ { 1 } ( N , y ) = \\# \\Phi ( N , y ) \\ll \\frac { N } { \\log y } , \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\kappa \\psi _ j ( u ) \\overline { z } ^ j = \\sum _ { j = 0 } ^ \\kappa \\psi _ j ( \\overline { a z + b } ) ( a u + b ) ^ j , \\quad \\forall z , u \\in \\C . \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} J _ h ( v _ \\gamma ) = 0 _ \\gamma \\cdot h ( s _ { V _ 1 } ( v _ \\gamma ) ) ^ { - 1 } + h ( t _ { V _ 1 } ( v _ \\gamma ) ) \\cdot 0 _ \\gamma \\ ; . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} ( - i \\varepsilon \\nabla + A ) ^ { 2 } u + V ( x ) u = f ( x , | u | ) u , \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} i = i ^ { k \\leq l } _ { k ' \\leq l ' } \\colon ( k ' , l ' ) \\to ( k , l ) . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\tau ^ { - \\frac { q } { \\tau } } \\frac { \\Gamma _ 2 ( 1 - q \\ , | \\tau ) \\Gamma _ 2 ( \\tau \\ , | \\tau ) } { \\Gamma _ 2 ( 1 \\ , | \\tau ) \\Gamma _ 2 ( \\tau - q \\ , | \\tau ) } = \\frac { \\Gamma ( 1 - q ) } { \\Gamma ( 1 - \\frac { q } { \\tau } ) } , \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} E ( \\lambda ) M ( \\lambda ) = N ( \\lambda ) H ( \\lambda ) , \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\widetilde { Y } _ { i , r } \\widetilde { Y } _ { j , s } = t ^ { \\gamma ( i , r ; j , s ) } \\widetilde { Y } _ { j , s } \\widetilde { Y } _ { i , r } , \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} { \\tilde \\theta _ n } = \\tilde \\theta \\left ( { n T } \\right ) = 2 n \\pi \\tilde f T + \\tilde \\varphi + \\varphi _ { n , B } ^ { { \\rm { C F S K } } } - \\varphi _ { n , A } ^ { { \\rm { C F S K } } } \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} A _ f ( \\chi ) = \\sum _ { \\ell \\leqslant L } \\frac { \\lambda _ f ( \\ell ) } { \\sqrt { \\ell } } \\mu ^ 2 ( \\ell ) \\chi ( \\ell ) . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} \\begin{cases} u _ t \\ , = \\ , \\Delta ( u ^ m ) + u ^ p & \\textrm { i n } \\ ; \\ ; M \\times ( 0 , T ) \\ , , \\\\ u \\ , = \\ , u _ 0 & \\textrm { i n } \\ ; \\ ; M \\times \\{ 0 \\} \\ , , \\end{cases} \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} \\| A T _ t f \\| _ p = \\| T _ { t - 1 } A T _ 1 f \\| _ p \\leq \\| T _ { t - 1 } \\| \\| A T _ 1 f \\| _ p \\leq M e ^ { - ( t - 1 ) } c _ 1 \\| f \\| _ p \\leq c _ 2 t ^ { - 1 } \\| f \\| _ p . \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\begin{aligned} w _ { i , j } = \\ , & ( s _ { \\sigma _ { i - 1 , j } } s _ { \\sigma _ { i - 1 , j } - 1 } \\cdots s _ { \\widetilde { m } _ { i - 1 , j } + 1 } ) \\\\ & ( s _ { \\sigma _ { i - 1 , j } + 1 } s _ { \\sigma _ { i - 1 , j } } \\cdots s _ { \\widetilde { m } _ { i - 1 , j } + 2 } ) \\cdot \\cdots \\cdot \\\\ & ( s _ { \\sigma _ { i - 1 , j } + m _ { i , j } - 1 } s _ { \\sigma _ { i - 1 , j } + m _ { i , j } - 2 } \\cdots s _ { \\widetilde { m } _ { i , j } } ) \\end{aligned} \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} \\mu _ A ( f ) = \\int _ { b \\Gamma } f ( k ) \\overline { \\rho _ A ( k ) } \\ , d m _ { b \\Gamma } ( k ) , \\textrm { f o r a l l $ f \\in C ( b \\Gamma ) $ } . \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} x \\backslash ( x y ) = y = x ( x \\backslash y ) , ( x y ) / y = x = ( x / y ) y x \\backslash x = y / y \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\pi } ( g ) = \\omega _ { \\pi } ( a ) | U _ n ( \\mathfrak { f } ) | ^ { - 1 } \\sum _ { h \\in U _ n ( \\mathfrak { f } ) } \\psi ( h ^ { - 1 } ) \\mathrm { t r } _ \\sigma ( \\overline { k } h ) , \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} \\rho \\circ r _ K = r _ { \\Q } \\circ N _ { K \\vert \\Q } , \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} { n \\choose j + 1 } = \\sum _ { k = j } ^ { n - 1 } { k \\choose j } , 0 \\le j \\le n - 1 , \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} X ( i ) = \\begin{cases} 1 & u , \\\\ - 1 & 1 - u . \\end{cases} \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} \\log \\prod _ p \\big ( 1 + r ( p ) ^ 2 \\omega ( p ) \\big ) & \\leqslant \\sum _ p r ( p ) ^ 2 \\omega ( p ) = A ^ 2 \\sum _ { A _ 0 ^ 2 \\leqslant p \\leqslant N ^ { c / A _ 0 ^ 2 } } \\frac { \\omega ( p ) } p \\\\ & \\leqslant A ^ 2 b _ { \\omega 2 } \\log \\frac { c \\log N } { 2 A _ 0 ^ 2 \\log A _ 0 } + O _ { \\omega } \\bigg ( \\frac { A ^ 2 } { \\log A _ 0 } \\bigg ) . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & d W _ { x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) \\\\ & = \\left \\{ W _ { s x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) + W _ { x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) b ( s ) + \\frac { 1 } { 2 } W _ { x x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) ( \\sigma ( s ) ) ^ { 2 } \\right \\} d s \\\\ & \\ \\ + W _ { x x } ( s , \\bar { X } ^ { t , x ; \\bar { u } } ( s ) ) \\sigma ( s ) d B ( s ) . \\end{array} \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} p _ { s } ^ { \\textrm { o p t } } = \\frac { \\theta \\xi P _ { B S } \\| \\mathbf { h } \\| ^ { 2 } } { 1 - \\theta } . \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} L ( s , \\pi \\boxtimes \\chi , \\varrho _ { 2 n + 1 } ) = \\frac 1 { 1 - \\chi ( \\varpi ) q ^ { - s } } \\prod _ { i = 1 } ^ n \\frac 1 { ( 1 - \\chi ( \\varpi ) \\alpha _ i q ^ { - s } ) ( 1 - \\chi ( \\varpi ) \\alpha _ i ^ { - 1 } q ^ { - s } ) } . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} \\Psi _ { \\Gamma } ( X ) = X + O ( X ^ { 3 / 4 } \\log { X } ) . \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} g Z = ( A Z + B ) ( C Z + D ) ^ { - 1 } g = \\left ( \\begin{array} { c c } A & B \\\\ C & D \\end{array} \\right ) . \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} a \\rhd b = \\lambda _ a ( b ) = a ^ { - 1 } \\cdot ( a \\circ b ) , a \\lhd b = \\overline { \\rho _ { \\bar { b } } ( \\bar { a } ) } = \\overline { ( \\bar { b } \\circ \\bar { a } ) \\cdot \\bar { b } ^ { - 1 } } = \\overline { \\bar { a } \\cdot b } \\circ b . \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} p ( { \\bf M ^ { ( 2 ) } } ) = p ( Q ( \\theta ) * { \\bf M ^ { ( 2 ) } } ) p ( { \\bf M ^ { ( 2 ) } } ) = p ( \\widetilde { Q } * { \\bf M ^ { ( 2 ) } } ) , \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} p ^ * ( Y _ { 1 , k } ) = 1 + p ^ * ( X _ { 1 , k } ) = 1 + \\frac { \\Delta _ { I ( 1 , k + 1 ) } \\Delta _ { I ( 2 , k ) } } { \\Delta _ { I ( 1 , k - 1 ) } \\Delta _ { I ( 2 , k + 1 ) } } = \\frac { \\Delta _ { I ( 1 , k ) } \\Delta _ { J ( 2 , k ) } } { \\Delta _ { I ( 1 , k - 1 ) } \\Delta _ { I ( 2 , k + 1 ) } } . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} c ( n ^ 2 | D | ) = c ( | D | ) \\sum _ { 0 < d \\mid n } \\mu ( d ) \\left ( \\frac { D } { d } \\right ) \\chi ( d ) d ^ { k - 1 } a _ { \\chi } ( n / d ) . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} S _ 1 & : = \\{ \\bar { J } \\in S _ { A ' } | \\beta ( s + 1 ) = 0 \\mbox { f o r a l l } \\beta \\in \\bar { J } \\} , \\\\ S _ 2 & : = \\{ \\bar { J } \\in S _ { A ' } | \\beta ( s + 1 ) > 0 \\mbox { f o r s o m e } \\beta \\in \\bar { J } \\} . \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} \\mathcal { K } ^ \\star _ \\alpha \\subset \\mathcal { K } _ { \\alpha } ^ { + } : = \\{ k \\in \\mathcal { K } _ \\alpha : k ( \\eta ) \\geq 0 \\} . \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} \\begin{array} { r l } V _ N ^ { ( \\alpha ) , \\operatorname { M F } } ( x ) \\ ; : = & \\ ! \\ ! V ^ { ( \\alpha ) } ( x ) , \\\\ V _ N ^ { ( \\alpha ) , \\operatorname { G P } } ( x ) \\ ; : = & \\ ! \\ ! N ^ 3 V ^ { ( \\alpha ) } ( N x ) , \\end{array} \\qquad \\alpha \\in \\{ 1 , 2 , 1 2 \\} \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} G ( x , u ( x ) , \\nabla u ( x ) , \\nabla ^ 2 u ( x ) , u ( \\cdot ) ) = 0 , \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} w ^ { ( i ) } : = \\sum _ { \\substack { { 1 \\le a _ j < b _ j \\le n } \\\\ { 1 \\le j \\le i } } } ( r _ { a _ 1 , b _ 1 } \\cdots r _ { a _ i , b _ i } ) \\ , x _ { a _ 1 } x _ { b _ 1 } \\cdots x _ { a _ i } x _ { b _ i } \\quad i \\ge 1 \\ , . \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} F _ 1 ( x ) = \\begin{bmatrix} x _ 1 ^ 3 \\\\ x _ 2 ^ 3 - 1 \\end{bmatrix} , \\ F _ 2 ( x ) = \\begin{bmatrix} x _ 1 ^ 3 - 1 \\\\ x _ 2 ^ 3 \\end{bmatrix} . \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( \\chi ( \\Sigma ^ y _ { \\ell } ) - b ^ y _ { \\ell } ) & = \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( 2 - 2 g e n u s ( \\Sigma ^ y _ { \\ell } ) - 2 b ^ y _ { \\ell } ) \\\\ & \\geq \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } ( 2 b ^ y _ { \\ell } - 3 ) - 3 \\sum _ { y \\in \\mathcal { Y } } \\sum _ { \\ell = 1 } ^ { J _ y } i n d e x ( \\Sigma ^ y _ { \\ell } ) \\geq 3 - 3 j . \\end{aligned} \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ \\omega ( \\ddot { u } _ 3 ( t , x ' ) - \\ddot { v } _ 3 ( t , x ' ) ) ( \\dot { u } _ 3 ( t , x ' ) - \\dot { v } _ 3 ( t , x ' ) ) \\ , d x ' \\ , d t = \\frac { 1 } { 2 } \\| \\dot { u } _ 3 ( T ) - \\dot { v } _ 3 ( T ) \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} \\Delta ( \\alpha ^ { - 1 } ) = [ V : \\alpha ( V ) ] . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} \\vec x = \\vec 1 _ S - \\vec 1 _ { V \\setminus S } , \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } f ^ n ( x _ 0 ) , ( f ^ n = \\underbrace { f \\circ f \\circ \\cdots \\circ f } _ { \\textrm { $ n $ t i m e s } } , f ^ 0 = \\operatorname { I d } ) , \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} P _ 1 ( \\lambda ) \\left ( L _ \\epsilon ( \\lambda ^ \\ell ) \\otimes I _ n \\right ) + \\left ( L _ \\eta ( \\lambda ^ \\ell ) ^ T \\otimes I _ m \\right ) P _ 2 ( \\lambda ) = 0 \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} i \\partial _ t u = 2 J \\Pi ( | u | ^ 2 ) + \\bar { J } u ^ 2 , \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} W _ { 1 3 5 } = - 2 ^ { 3 2 } \\cdot 3 ^ { 8 } \\cdot 5 ^ 8 \\cdot 7 ^ 2 \\cdot 1 3 \\cdot 2 3 \\ , \\chi _ 5 ^ 3 \\chi _ { 3 0 } ^ 4 \\ , . \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} V _ 0 ^ t ( \\phi _ \\pm , h ) = \\frac 1 2 \\Bigl [ e ^ { i t D } \\phi _ + + e ^ { - i t D } \\phi _ - \\Bigr ] + V _ 0 ^ t ( 0 , h - r ) ( x ) , \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - c \\mathcal W \\xi + \\mathcal Z \\nu = \\frac { \\epsilon } { \\gamma } \\xi \\nu , \\\\ \\xi = \\frac { 1 } { 1 - \\gamma } \\Big ( c \\nu + \\frac { \\epsilon } { 2 \\gamma } \\nu ^ 2 \\Big ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\mathbb { P } _ { n } ^ { ( N ) } ( \\left \\{ X _ { i } \\right \\} ) = \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( \\left \\{ X _ { i } \\right \\} ) \\frac { P ( A _ { j } ^ { ( N ) } ) } { \\mathbb { P } _ { n } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) } , \\quad X _ { i } \\in A _ { j } ^ { ( N ) } \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} W _ { l _ 3 } = \\langle W _ 0 , w _ { 4 l _ 4 + 1 } , \\dots , w _ { 4 l _ 3 + l _ 3 } , w _ { 4 l _ 4 + 1 } \\theta , \\dots , w _ { 4 l _ 4 + l _ 3 } \\theta \\rangle . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} w ^ B ( t ) u _ \\delta ^ B ( t ) = ( \\sigma _ { 0 } ^ { B } + \\beta t ) w _ { 0 } ^ { B } t , \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} N = \\left [ \\begin{array} { c c c c } A _ 0 & 0 & \\cdots & 0 \\\\ A _ 1 & A _ 0 & \\cdots & 0 \\\\ A _ 2 & A _ 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & A _ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\dot { x } = g ( t , x ) \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} w _ j = \\begin{cases} \\sqrt { N _ j } & , \\\\ N _ j & , \\end{cases} \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} w _ { \\leq k } & : = s _ { i _ 1 } \\cdots s _ { i _ k } , \\ w _ { \\leq 0 } : = e , \\\\ k ^ { + } & : = \\min ( \\{ \\ell + 1 \\} \\cup \\{ k + 1 \\leq j \\leq \\ell \\mid i _ { j } = i _ { k } \\} ) , \\\\ k ^ { - } & : = \\max ( \\{ 0 \\} \\cup \\{ 1 \\leq j \\leq k - 1 \\mid i _ { j } = i _ { k } \\} ) , \\\\ k ^ - ( i ) & : = \\max ( \\{ 0 \\} \\cup \\{ 1 \\leq j \\leq k - 1 \\mid i _ { j } = i , c _ { i i _ k } \\neq 0 \\} ) . \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} \\omega _ 2 ( t ( 2 ) ) \\pmb { \\phi } _ 2 = \\omega _ 2 ( t ( 2 ) ) \\mathbf 1 _ { \\Z _ 2 } = \\chi _ { \\psi } ( 2 ) | 2 | _ 2 ^ { 1 / 2 } \\mathbf 1 _ { \\frac { 1 } { 2 } \\Z _ 2 } . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\mathfrak { S } = \\frac { U _ { \\nu } ^ { \\ast } } { K ^ { \\ast } } \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} _ { q / 2 } \\Bigl ( \\frac { F _ 4 } { E _ 2 ^ 2 } \\Bigr ) = 0 . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{gather*} \\mathcal G _ { 1 , ( a ; N ) } ^ \\ast ( \\tau ) = - { \\rm i } \\pi \\ ( 1 - \\frac { 2 a } { N } \\ ) - 2 \\pi { \\rm i } \\sum _ { n > 0 } \\frac { q ^ { n a } - q ^ { n ( N - a ) } } { 1 - q ^ { N n } } , \\\\ \\mathcal G _ { 2 , ( a ; N ) } ^ \\ast ( \\tau ) = - \\frac { \\pi } { N \\Im ( \\tau ) } - ( 2 \\pi ) ^ 2 \\sum _ { n > 0 } n \\frac { q ^ { n a } + q ^ { n ( N - a ) } } { 1 - q ^ { N n } } , \\end{gather*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N I _ n ( x + k ) = \\sum _ { m = 0 } ^ { n \\wedge N } \\binom { N + 1 } { m + 1 } m ! S ( n , m ; x ) = \\sum _ { k = 0 } ^ { n \\wedge N } c _ { n , N } ( k ) I _ n ( x + k ) , \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} ( \\lceil j / 2 \\rceil - 1 ) ( j - \\lceil j / 2 \\rceil ) = \\frac { 1 } { 4 } \\tau ( C , p ) . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\binom { \\underline { x } } { \\mathbf { i } } _ { \\underline { S } } = \\binom { x _ 1 } { i _ 1 } _ { S _ 1 } \\binom { x _ 2 } { i _ 2 } _ { S _ 2 } \\cdots \\binom { x _ n } { i _ n } _ { S _ n } . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} U _ r = \\frac { 1 } { r } V _ r + O ( N / x ) , \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} \\sum _ { a \\in A } c _ a b _ a v _ L = 0 , \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} \\left | \\frac { \\P _ { s , x } ( \\tau _ { A } > t ) } { \\P _ { s , y } ( \\tau _ { A } > t ) } - \\frac { \\P _ { s , x } ( \\tau _ { A } > u ) } { \\P _ { s , y } ( \\tau _ { A } > u ) } \\right | \\leq C _ { s , y } \\inf _ { v \\in [ s + t _ 0 , t ] } \\frac { 1 } { d ' _ v } \\prod _ { k = 0 } ^ { \\left \\lfloor \\frac { v - s } { t _ 0 } \\right \\rfloor - 1 } ( 1 - d _ { v - k } ) . \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} D _ r ( n ) = \\sum _ { j = r } ^ n { j - 1 \\choose r - 1 } \\frac { n ! } { ( n - j ) ! } D ( n - j ) , \\mbox { } n \\geq r . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\psi } ( n + 2 \\chi ^ I ) = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( n + 2 \\chi ^ I ) , \\quad \\forall n \\in \\N ^ N . \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} \\widetilde h [ f ] - h [ f ] = - \\frac { 1 } { \\beta _ { v _ 1 } } | a | ^ 2 - \\frac { 1 } { \\beta _ { v _ 2 } } | b | ^ 2 = - \\Big ( \\frac { 1 } { \\beta _ { v _ 1 } } + \\frac { 1 } { \\beta _ { v _ 2 } } \\Big ) | a | ^ 2 = 0 \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\alpha > \\frac 1 { m - 1 } \\ , , \\ \\beta > 0 \\alpha = \\frac 1 { m - 1 } \\ , , \\ 0 < \\beta \\leq 1 \\ , . \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} c ( \\xi ) = c ( \\mathfrak d _ { \\xi } ) \\chi ( \\mathfrak f _ { \\xi } ) a ( \\mathfrak f _ { \\xi , N } ) \\prod _ { p \\mid \\mathfrak f _ { \\xi , 0 } } ( a ( p ^ { e _ p } ) - p ^ { k - 1 } \\chi _ { - \\xi } ( p ) a ( p ^ { e _ p - 1 } ) ) = c ( \\mathfrak d _ { \\xi } ) \\chi ( \\mathfrak f _ { \\xi } ) a ( \\mathfrak f _ { \\xi , N } ) \\mathfrak f _ { \\xi , 0 } ^ { k - 1 / 2 } \\prod _ { p \\mid \\mathfrak f _ { \\xi , 0 } } \\Psi _ p ( \\xi ; \\alpha _ p ) . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} | \\nabla \\eta _ j | \\leq \\frac { 2 } { R _ j - R _ { j + 1 } } = c ( R _ 0 , n ) \\Big ( \\frac { n } { n - 1 } \\Big ) ^ j . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} ( \\hat u _ 1 , \\partial _ \\eta \\hat h _ 1 ) | _ { \\eta = 0 } = \\mathbf { 0 } , \\hat \\theta | _ { \\eta = 0 } = \\theta ^ \\ast ( \\tau , \\xi ) , \\lim _ { \\eta \\rightarrow + \\infty } ( \\hat u _ 1 , \\hat \\theta , \\hat h _ 1 ) ( \\tau , \\xi , \\eta ) = ( U , \\Theta , H ) ( \\tau , \\xi ) . \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} { \\dim } V _ { c \\omega _ a } ( \\mu ) = \\# P ( a , b , c ) ( \\mu ) . \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\mathbb { G } ^ { ( N ) } ( f ) = \\mathbb { G } ^ { ( N - 1 ) } ( f ) - \\sum _ { j = 1 } ^ { m _ { N } } \\mathbb { E ( } f | A _ { j } ^ { ( N ) } ) \\mathbb { G } ^ { ( N - 1 ) } ( A _ { j } ^ { ( N ) } ) . \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} \\tau _ { t + 1 , n , m + 1 } \\tau _ { t - 1 , n + 1 , m } = \\tau _ { t , n + 1 , m } ^ { k _ 1 } \\tau _ { t , n , m + 1 } ^ { k _ 2 } + \\tau _ { t , n , m } ^ { l _ 1 } \\tau _ { t , n + 1 , m + 1 } ^ { l _ 2 } ( k _ 1 , l _ 1 , k _ 2 , l _ 2 \\in \\Z _ + ) . \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} \\eta _ { 2 , 1 } ( q | a , b ) = \\frac { \\Gamma _ 2 ( q + b _ 0 \\ , | \\ , a ) } { \\Gamma _ 2 ( b _ 0 \\ , | \\ , a ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 1 \\ , | \\ , a ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , a ) } . \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} T _ { n , m } = \\sum _ { I \\subset [ N ] } ( - 1 ) ^ { | I | } \\tilde { \\phi } ( m + n + 2 \\chi ^ I ) , \\quad \\forall m , n \\in \\N ^ N , \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\| T \\| _ { S _ 1 } \\leq \\sum _ { \\lambda \\in \\Lambda } \\| f _ \\lambda \\| \\| g _ \\lambda \\| \\leq \\left ( \\sum _ { \\lambda \\in \\Lambda } \\| f _ \\lambda \\| ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\left ( \\sum _ { \\lambda \\in \\Lambda } \\| g _ \\lambda \\| ^ 2 \\right ) ^ \\frac { 1 } { 2 } = \\| \\xi ^ { [ N ] } \\| \\| \\eta ^ { [ N ] } \\| . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\Lambda ^ { - } ( X ) = \\sum _ { i \\in \\mathcal { I } ^ - } ( A _ i - \\overline { X } ) ^ T \\otimes \\Phi ^ - _ i ( X ) \\overline { M } ^ { + } + \\sum _ { i \\in \\mathcal { I } ^ + } ( A _ i - \\overline { X } ) ^ T \\otimes \\Phi ^ - _ i ( X ) - \\left ( ( X - \\overline { X } ) ^ T \\otimes I _ 2 \\right ) \\overline { M } ^ + . \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} & d \\omega = \\star \\sigma d u , \\\\ & d \\star \\left ( \\frac { 1 } { \\sigma } d \\omega \\right ) = 0 . \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} X ^ x _ t = x + \\int _ 0 ^ t \\theta \\bigl ( \\kappa - X ^ x _ s \\bigr ) \\ , d s + \\int _ 0 ^ t \\sigma \\sqrt { X ^ x _ s } \\ , d B _ s , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ n } | w ' ( t , x ) | ^ 2 \\ , d x + \\mathcal { W } ( w ( t , \\cdot ) ) , \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} f + b = n . \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} \\partial _ t M _ t = - ( \\nabla _ x + \\nabla _ y ) ^ 2 M _ t + [ I \\otimes \\mathrm { d i v } _ y ( \\beta _ j \\cdot ) + \\mathrm { d i v } _ x ( \\beta _ j \\cdot ) \\otimes I ] M _ t \\dot { Z } _ t ^ j , \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} Y ^ \\circ = \\{ S \\in \\S : \\langle Q , S \\rangle = 1 \\ ; \\} \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} r ( \\delta ) = \\pi _ 0 R _ 0 ( \\delta ) + \\pi _ 1 R _ 1 ( \\delta ) \\ ; , \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} \\Sigma ( x _ i , y _ i ) = \\bigcup _ { g \\in G } \\ , \\Big ( \\langle x _ i \\rangle ^ g \\cup \\langle y _ i \\rangle ^ g \\cup \\langle x _ i y _ i \\rangle ^ g \\Big ) , \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} N _ { T } = \\left [ \\begin{array} { c c c c c } A _ 0 T & 0 & \\cdots & 0 & - A _ 2 \\ , T _ 2 / ( \\gamma T _ 0 ) \\\\ A _ 1 & A _ 0 & \\cdots & 0 & 0 \\\\ A _ 2 & A _ 1 & \\ddots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & A _ 1 & A _ 0 \\\\ 0 & 0 & \\cdots & A _ 2 & A _ 1 - A _ 2 T _ 1 / \\gamma T _ 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} M ^ n _ k : = \\sup _ { s \\in [ t ^ n _ k , t ^ n _ { k + 1 } ] } L ( s ) u , \\qquad m ^ n _ k : = \\inf _ { s \\in [ t ^ n _ k , t ^ n _ { k + 1 } ] } L ( s ) u . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\mbox { s u p p } u _ i \\subset B ( R ) , i = 0 , 1 . \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} l _ { k } = \\Bigl [ | U _ * | ^ { 2 ^ { * } - 2 } U _ * - u _ 0 ^ { 2 ^ * - 1 } + \\sum _ { j = 1 } ^ { k } P _ j ^ { 2 ^ { * } - 1 } \\Bigr ] + \\sum _ { j = 1 } ^ { k } \\bigl ( U _ j ^ { 2 ^ * - 1 } - P _ j ^ { 2 ^ { * } - 1 } \\bigr ) . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} d _ B ( \\pi _ 1 , \\pi _ 2 ) & = \\lvert A ( \\pi _ 1 ) \\setminus A ( \\pi _ 2 ) \\rvert , \\\\ d _ B ( \\sigma _ 1 , \\sigma _ 2 ) & = \\lvert A ( \\sigma _ 1 ) \\setminus A ( \\sigma _ 2 ) \\rvert . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} n ( f ) _ { k } = n ( f ) _ { T - k } , \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} A ^ { \\lambda , \\delta } ( x ) : = - \\operatorname { d i v } ( a ^ { \\ast } \\phi ^ { \\lambda } ( a \\nabla x ) ) - \\frac { 1 } { 2 } J _ { \\delta } ^ { a } L ^ { b } J _ { \\delta } ^ { a } x , x \\in S . \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} \\lim _ { t \\to \\pm \\infty } \\left \\| u ( t , \\cdot ) - e ^ { i t ( \\Delta ^ 2 - \\kappa \\Delta ) } u _ { 0 } ^ \\pm ( \\cdot ) \\right \\| _ { H ^ 2 } = 0 . \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\Omega } u _ { s s } \\psi _ 1 \\ , d x \\ , d s - \\int _ 0 ^ t \\int _ { \\Omega } \\Delta u \\psi _ 1 \\ , d x \\ , d s - \\int _ 0 ^ t \\int _ { \\Omega } \\Delta u _ s \\psi _ 1 \\ , d x \\ , d s = \\int _ 0 ^ t \\int _ { \\Omega } | u | ^ p \\psi _ 1 \\ , d x \\ , d s . \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} \\eta _ s ( x , y ) = \\sum _ { n = n _ 0 } ^ \\infty s _ n t ^ n \\omega _ { 2 n } ( x , y ) , \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} \\langle \\tau _ p ( \\nu _ { \\gamma } ) \\breve { \\mathbf g } _ p , \\breve { \\mathbf g } _ p \\rangle = \\int _ { K _ { 0 0 } } \\mathbf g _ p ( h \\nu _ { \\gamma } \\varpi _ p ) \\overline { \\mathbf g _ p ( h \\varpi _ p ) } d h . \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} \\left | 2 ^ n \\Delta f _ n ^ { ( k ) } ( \\gamma ) - f _ n ^ { ( k + 1 ) } ( \\gamma ) \\right | \\le \\frac { \\varepsilon _ n } { 2 ^ { n ( d - k ) } } + \\sum _ { \\ell = 2 } ^ { d - k } \\frac { | t _ { k , k + \\ell } | } { 2 ^ { n \\ell } } \\left | f _ n ^ { ( k + \\ell ) } ( \\gamma ) \\right | \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} d _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , b ^ - } \\overline { x } ( b ) + d _ 2 ~ ^ { A B R } D _ b ^ \\alpha \\overline { x } ( b ) = 0 , \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} \\frac { p ^ { - 3 } ( p - 1 ) } { 2 p } \\left ( 2 p - G ( 1 , p ) - G ( u , p ) \\right ) = p ^ { - 3 } ( p - 1 ) , \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} [ M ( I ) ^ { - 1 } \\Gamma _ 0 M ( I ) , M ( J ) ^ { - 1 } \\Gamma _ 0 M ( J ) ] = 1 \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} x ^ l _ { i j } + x ^ a _ { i j } + x ^ c _ { i j } = 1 . \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} R _ 2 ^ { A B } ( \\rho _ - , v _ - ) : \\left \\{ \\begin{array} { l l } \\xi = \\lambda _ 2 = v + \\beta t + \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } , \\\\ v - v _ - = \\int _ { \\rho _ { - } } ^ { \\rho } \\frac { \\sqrt { A n \\rho ^ { n - 1 } + \\frac { \\alpha B } { \\rho ^ { \\alpha + 1 } } } } { \\rho } d \\rho , \\ \\ \\rho > \\rho _ { - } . \\end{array} \\right . \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ 1 x _ { n } + \\sum _ { i = 2 } ^ { n - 1 } x _ i ^ 2 + x _ { 1 } ^ 2 x _ 2 + \\alpha x _ { 1 } ^ 4 ; \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} F ( t ) = \\xi P ( t ) + m ( \\varphi _ 1 ( t ) - \\tau \\psi ( t ) ) . \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( \\beta ) } \\int _ 0 ^ \\infty e ^ { - t v } v ^ { \\beta - 1 } d v = t ^ { - \\beta } . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} 1 - h ^ + ( G ) = \\min \\limits _ { \\vec x \\ne \\vec 0 } \\frac { I ^ + ( \\vec x ) } { \\| \\vec x \\| } , \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} n ^ { 1 / ( k + 1 ) } \\leq \\Delta < \\min \\Big \\{ n ^ { 1 / k } , \\frac { n } { M \\log { n } } \\Big \\} \\enspace \\enspace p : = \\max \\big \\{ n ^ { - k / ( k + 1 ) } , \\Delta ^ { k + 1 } n ^ { - 2 } \\big \\} . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\vert \\tau _ { j + k } ( y ) - \\tau _ j ( y ) \\vert & \\leq \\sum _ { i = 1 } ^ k \\vert \\tau _ { j + i } ( y ) - \\tau _ { j + i - 1 } ( y ) \\vert \\\\ & \\leq \\frac 5 4 \\left ( 3 6 C _ 1 ( m ) + 2 4 \\right ) \\delta \\sum _ { i = 1 } ^ k r _ { j + i - 1 } \\\\ & = \\frac 5 4 \\left ( 3 6 C _ 1 ( m ) + 2 4 \\right ) \\delta r _ j \\sum _ { i = 0 } ^ { k - 1 } 4 ^ { - i } \\\\ & \\xrightarrow [ j \\to \\infty ] { } 0 . \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} C _ k ( s ) = \\int \\limits _ { \\tilde N _ 1 } \\int \\limits _ { \\tilde N _ 2 } f _ k ( \\left [ \\begin{smallmatrix} I _ n \\\\ & I _ n \\\\ & ^ t V & I _ n \\\\ V & X & & I _ n \\end{smallmatrix} \\right ] , s ) \\ , d X \\ , d V \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} x ^ { a _ 1 } f _ 2 = y ^ { b _ 1 } f _ 3 - z ^ { c _ 1 } f _ 1 , z ^ { c _ 2 } f _ 2 = x ^ { a _ 2 } f _ 3 - y ^ { b _ 2 } f _ 1 . \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} \\begin{array} { l l } & \\displaystyle \\int _ { 0 } ^ { T } \\Phi _ { t } d B _ { t } = \\sum _ { k = 1 } ^ { \\infty } \\int _ { \\tau _ { k - 1 } \\wedge T } ^ { \\tau _ { k } \\wedge T } \\Phi ^ { ( k ) } d B _ { t } , \\ \\ ( \\tau _ { 0 } = 0 ) \\\\ & = \\displaystyle \\sum _ { k = 1 } ^ { \\infty } \\sum _ { \\substack { j \\in [ \\tau _ { k - 1 } \\wedge T , \\tau _ { k } \\wedge T ] \\bigcap \\{ J \\setminus J _ { K } \\} } } \\Phi ^ { ( k ) } ( B _ { t _ { j + 1 } } - B _ { t _ { j } } ) \\end{array} \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} \\abs { S _ { r , N ' } } & = \\Big | \\sum _ { n = 1 } ^ { N ' } ( \\tilde { S } _ { r , n } - \\tilde { S } _ { r , n - 1 } ) e ^ { 2 \\pi i \\theta n ^ d } \\Big | \\leq \\abs { \\tilde { S } _ { r , N ' } } + \\abs { \\tilde { S } _ { r , 0 } } + \\sum _ { n = 1 } ^ { N ' - 1 } \\abs { \\tilde { S } _ { r , n } } \\Big | e ^ { 2 \\pi i \\theta n ^ d } - e ^ { 2 \\pi i \\theta ( n + 1 ) ^ d } \\Big | . \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} \\left ( \\frac { | D | ^ { k - 1 / 2 } \\Lambda ( f , D , k ) } { c ^ + ( f ) c _ h ( | D | ) ^ 2 } \\right ) ^ { \\sigma } = \\frac { | D | ^ { k - 1 / 2 } \\Lambda ( f ^ { \\sigma } , D , k ) } { c ^ + ( f ^ { \\sigma } ) c _ { h ^ { \\sigma } } ( | D | ) ^ 2 } . \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\mathfrak R ( G ) = \\prod _ { \\mathfrak s \\in \\mathfrak B ( G ) } \\mathfrak R ^ { \\mathfrak s } ( G ) . \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} H ^ { r 1 } = H ^ { r 1 } _ { k , l } = ( H ^ { r 1 } _ { k , l } ) ^ { ( n ) } , \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} \\psi = \\sup _ { u \\in \\R ^ p : \\| u \\| = 1 , \\| u \\| _ 0 \\le C k \\log ^ 2 ( p / k ) } \\| \\bar \\Sigma ^ { 1 / 2 } u \\| \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} \\Re ( { \\bf Q } T _ G { \\bf D } p ) = \\Re \\Big [ { \\bf Q } T _ G \\left ( { \\bf f } - \\Re ( { \\bf u } { \\bf D } ) { \\bf u } \\right ) \\Big ] . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\mathbb { H } \\ni x = ( a , b , c ) \\mapsto \\tau ( x ) = \\left ( \\frac { a } { 2 } , \\frac { b } { 2 } , \\frac { c } { 2 } + \\frac { a b } { 6 } \\right ) . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ x \\left ( \\begin{array} { l l } { \\tilde u } \\\\ { \\tilde v } \\end{array} \\right ) - \\tilde { \\mathcal A } \\left ( \\begin{array} { l l } { \\tilde u } \\\\ { \\tilde v } \\end{array} \\right ) = 0 \\end{array} \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} \\begin{cases} 1 - \\sum _ { k = 1 } ^ n t _ k = 0 , \\\\ \\lambda - \\dfrac { z _ i } { t _ i ^ 2 } = 0 , \\\\ \\lambda ( 1 - \\sum _ { k = 1 } ^ n t _ k ) + ( 1 - \\sum _ { k = 1 } ^ n \\dfrac { z _ i } { t _ k } ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\omega ' _ { \\log } ( b _ 1 , \\ldots , b _ { p - 1 } , a _ I \\cdot \\tau ^ { ( r + k ) I } , b _ { p + 1 } , \\ldots , b _ k ) = ( p \\circ { \\pi } _ { 0 , k } ) ^ * \\tau ^ { r I } \\cdot \\omega ' _ { \\log , p , I } ( b _ 1 , \\ldots , b _ { p - 1 } , a _ I , b _ { p + 1 } , \\ldots , b _ k ) . \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} x ^ s _ { i j } ( x ^ s _ { i j } - 1 ) = 0 , \\forall i , j , \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} A = { S } _ 1 \\otimes \\omega _ 1 + a ( r ) ( { S } _ 2 \\otimes \\omega ^ 2 + { S } _ 3 \\otimes \\omega ^ 3 ) , \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} \\tau _ y = \\inf \\{ t \\geq 0 : ~ X ( t ) = y \\} . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} g ( 1 ) = i _ 1 , g ( 2 ) = i _ 5 , g ( 3 ) = i _ 2 , g ( 4 ) = i _ 6 . \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\rho ( \\pi ) = \\rho ( \\tau ( A _ i ) ) \\sqcup \\rho ( \\tau ( \\pi - A _ i ) ) \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} { T } = \\{ \\vec t \\in \\mathbb { R } ^ n : \\ , \\vec z ^ * + \\epsilon \\vec t \\in B _ \\infty \\epsilon > 0 \\} \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\mathrm d \\rho _ { \\infty } Y _ + v _ { m , \\ell } & = v _ { m + 1 , \\ell } , \\mathrm d \\rho _ { \\infty } X _ + v _ { m , \\ell } = - \\frac { 1 } { 2 \\pi N } v _ { m + 2 , \\ell } , \\\\ \\mathrm d \\rho _ { \\infty } Y _ - v _ { m , \\ell } & = - 2 \\pi N m v _ { m - 1 , \\ell } , \\mathrm d \\rho _ { \\infty } X _ - v _ { m , \\ell } = \\pi N m ( m - 1 ) v _ { m - 2 , \\ell } - \\frac { \\ell } { 4 } ( 2 k + \\ell - 1 ) v _ { m , \\ell - 2 } . \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\Phi _ i ^ { - 1 } ( 0 ) \\subset \\bigcup _ { j = 1 } ^ k B _ { c m _ i ^ { - 1 / 2 } } ( x _ j ) , \\ \\ \\mathrm { d e g } ( \\Phi _ i \\Big \\vert _ { \\partial B _ { c m _ i ^ { - 1 / 2 } } ( x _ j ) } ) = 1 . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} \\begin{aligned} \\| w \\| _ { L ^ { \\infty } ( B _ { r _ { k + 1 } } ) } \\leq C M r _ { k } \\leq \\hat { C } M r _ { k + 1 } ^ { \\beta } \\cdot \\frac { C } { \\hat { C } \\tau _ 1 ^ { \\beta } } , \\end{aligned} \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} W ^ { ( 1 ) } - W ^ { ( 3 ) } = & b _ 0 y + ( 3 6 b _ 0 + b _ 1 ) y ^ 5 + ( - 4 1 5 b _ 0 - 1 0 b _ 1 ) y ^ 9 \\\\ & + ( - 3 9 0 5 6 b _ 0 - 7 2 4 b _ 1 ) y ^ { 1 3 } + ( - 7 4 2 1 3 1 b _ 0 - 3 6 9 4 b _ 1 ) y ^ { 1 7 } + \\cdots . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} T _ 0 = \\min \\{ 1 , \\tilde { T } _ 0 \\} \\quad \\mbox { w i t h } \\quad \\tilde { T } _ 0 = \\frac { \\delta } { C ( 1 + T ^ { k - 1 } ) ( \\sqrt { M _ 0 } + \\sqrt { M _ e } ) } , \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} \\sup _ { n \\in \\Z } \\lVert y _ { n + 1 } - A _ n y _ n \\rVert = \\sup _ { n \\in \\Z } \\lVert g _ n ( y _ n ) \\rVert \\le \\delta . \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} u _ { w \\lambda } = f _ { i _ { 1 } } ^ { ( \\langle h _ { i _ { 1 } } , s _ { i _ { 2 } } \\cdots s _ { i _ { \\ell } } \\lambda \\rangle ) } \\cdots f _ { i _ { \\ell - 1 } } ^ { ( \\langle h _ { i _ { \\ell - 1 } } , s _ { i _ { \\ell } } \\lambda \\rangle ) } f _ { i _ { \\ell } } ^ { ( \\langle h _ { i _ { \\ell } } , \\lambda \\rangle ) } . u _ { \\lambda } \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} \\Phi & \\coloneqq \\sum _ { \\substack { i , j = 1 \\\\ i < j } } ^ N ( \\lambda _ i | \\lambda _ j ) \\log ( z _ i - z _ j ) - \\sum _ { i = 1 } ^ N \\sum _ { j = 1 } ^ m ( \\lambda _ i | \\alpha _ { c ( j ) } ) \\log ( z _ i - w _ j ) \\\\ & \\qquad \\qquad \\qquad \\qquad + \\sum _ { \\substack { i , j = 1 \\\\ i < j } } ^ m ( \\alpha _ { c ( i ) } | \\alpha _ { c ( j ) } ) \\log ( w _ i - w _ j ) . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} \\widetilde { A } _ { i , r } ^ { - 1 } \\widetilde { A } _ { j , s } ^ { - 1 } = t ^ { \\alpha ( i , r ; j , s ) } \\widetilde { A } _ { j , s } ^ { - 1 } \\widetilde { A } _ { i , r } ^ { - 1 } , \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} \\Re \\int _ { \\R ^ d \\times \\R ^ d } F ( x ) | u _ { \\iota } ( y ) | ^ 2 \\Delta _ y \\psi ( x , y ) \\ , d x d y & \\\\ = 2 \\Re \\int _ { \\R ^ d \\times \\R ^ d } F ( x ) \\overline u _ { \\iota } ( y ) \\nabla _ y u _ { \\iota } ( y ) \\cdot \\nabla _ y \\psi ( x , y ) \\ , d x d y . & \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\lambda ( \\sum _ i f _ 0 ^ i \\otimes f _ 1 ^ i \\otimes \\ldots \\otimes f _ q ^ i ) : = \\sum _ i f _ 0 ^ i d f _ 1 ^ i \\ldots d f _ q ^ i \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} E ( x , \\zeta ) = E _ 1 ( x , \\zeta ) - E _ 2 ( x , \\zeta ) = \\left ( 1 + \\frac { i ( \\zeta _ 1 e _ 1 + \\zeta _ 2 e _ 2 ) } { | \\zeta | _ { \\C } } \\right ) E _ 1 ( x , \\zeta ) . \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} \\mathcal { Q } _ t ( x _ { t - 1 } , 1 ) = ( 1 - q _ t ) \\sum _ { j = 1 } ^ { M _ { t } } p _ { t j } \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 1 ) + q _ t \\sum _ { j = 1 } ^ { M _ t } p _ { t j } \\mathfrak { Q } _ t ( x _ { t - 1 } , 1 , \\xi _ { t j } , 0 ) , \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} | P _ 1 ^ { 2 ^ * - 2 } \\bigl ( u _ 0 + \\sum _ { j = 2 } ^ k P _ j ) | \\le C U _ 1 ^ { 2 ^ * - 2 } \\le \\lambda ^ { - \\frac { N + 2 } 4 + \\frac \\tau 2 } \\frac { \\lambda ^ { \\frac { N + 2 } 2 } } { ( 1 + \\lambda | y - \\xi _ 1 | ) ^ { \\frac { N + 2 } 2 + \\tau } } , y \\in S . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} g ( y ) : = \\left \\{ \\mathring { f } ( f ( y ) + f ( 1 ) - f ( p ) ) \\right \\} ^ { - 1 / 2 } y ^ { 1 / 2 } ( f ( 1 ) - f ( y ) ) . \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} V _ 5 : = \\{ x \\in \\tilde V : ( x , x _ 0 ) = 0 \\} . \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ a X ^ { k _ i } Y ^ { l _ i } + \\prod _ { i = a + 1 } ^ { a + b } X ^ { k _ i } Y ^ { l _ i } \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} ( 1 - x ) F ' _ r ( x ) = \\frac { r x ^ { r - 1 } e ^ { - x } } { ( 1 - x ) ^ r } + ( x + r ) F _ r ( x ) \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} v _ \\mu ( y _ k ) & = \\sum _ { j = 1 } ^ N a _ j \\left ( \\hat { \\delta } _ { j k } ( - ( 2 \\pi ) ^ { - 1 } \\log | y _ k - y _ j | - \\mathcal { G } _ { \\mu } ( y _ k - y _ j ) ) + \\frac { \\delta _ { j k } } { 2 \\pi } ( \\log ( \\mu / 2 i ) + \\gamma ) \\right ) \\\\ & = \\sum _ { j = 1 } ^ N [ \\Gamma _ { \\alpha , Y } ( \\mu ) ] _ { k j } a _ j , \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} F ( x ) : = f ( x ) + N _ C ( x ) \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} R : = k [ | t _ { 1 } , \\dots , t _ { d } | ] / ( t _ { 1 } ^ { u _ { 1 } } , \\dots , t _ { c } ^ { u _ { c } } ) \\ , , \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} [ \\tau ( w ) ] ^ { \\alpha } [ w - \\tau ( w ) ] ^ { \\beta } = \\sum _ { | \\alpha | + | \\beta | \\le | \\delta | \\le N \\cdot { ( | \\alpha | + | \\beta | ) } } E _ \\delta ( \\tau , w ) \\prod _ { i = 1 } ^ { n } w _ i ^ { \\delta _ i } \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle E | \\sum _ { \\substack { j \\in J _ { K } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) | ^ 2 \\\\ = \\displaystyle \\| \\sum _ { \\substack { j \\in J _ { K } } } X ( u _ { j } ) ( t _ { j + 1 } - t _ { j } ) \\| ^ 2 \\\\ \\leq \\displaystyle [ \\sum _ { \\substack { j \\in J _ { K } } } \\| X ( u _ { j } ) \\| ( t _ { j + 1 } - t _ { j } ) ] ^ 2 \\\\ \\end{array} \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\mathcal D \\partial _ t \\zeta + \\mathcal B \\partial _ x v - \\frac { \\epsilon } { \\gamma } \\partial _ x ( \\zeta v ) = 0 \\\\ \\partial _ t v + ( 1 - \\gamma ) \\partial _ x \\zeta - \\frac { \\epsilon } { 2 \\gamma } \\partial _ x ( v ^ 2 ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} \\qquad \\partial _ t v ^ \\gamma _ t + \\nabla _ { \\gamma ' _ t } v ^ \\gamma _ t = 0 a . e . \\ t . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ \\Bigl ( \\int _ \\mathcal { D } \\varphi ( x ) \\ , M _ \\beta ( d x ) \\Bigr ) ^ q \\Bigr ] = \\bar { \\varphi } ^ q \\exp \\Bigl ( \\sum _ { p = 1 } ^ \\infty \\mu ^ { p } \\ , c _ p ( q ) \\Bigr ) , \\ ; \\Re ( q ) < \\tau . \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} P _ n = ( 2 n - 2 ) ! \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} W ^ t _ 0 ( u _ 0 ^ e , g ) = e ^ { i t \\Delta } u _ 0 ^ e + W ^ t _ 0 ( 0 , g - p ) , p ( t ) = \\eta ( t ) [ e ^ { i t \\Delta } u _ 0 ^ e ] _ { x = 0 } . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} \\log \\Gamma \\bigl ( 1 + \\frac { z - q } { \\tau } \\bigr ) - \\log \\Gamma \\bigl ( 1 + \\frac { z } { \\tau } \\bigr ) = \\int \\limits _ { 0 } ^ \\infty \\frac { d t } { t } \\Bigl [ e ^ { - t z } \\bigl ( \\frac { e ^ { t q } - 1 } { e ^ { t \\tau } - 1 } \\bigr ) - \\frac { q } { \\tau } e ^ { - t \\tau } \\Bigr ] . \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} \\frac { \\binom { k b + i } { k a } ^ { 1 / k } } { \\binom { k a + i } { k a } ^ { 1 / k } } \\geq \\frac { \\binom { j b + i } { j a } ^ { 1 / j } } { \\binom { j a + i } { j a } ^ { 1 / j } } . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\mathcal { Q } _ { t } ( x _ { t - 1 } , D _ { t - 1 } ) = \\mathbb { E } _ { \\xi _ t , D _ t } \\Big [ \\mathfrak { Q } _ t ( x _ { t - 1 } , D _ { t - 1 } , \\xi _ t , D _ t ) | D _ { t - 1 } \\Big ] \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} ( \\omega _ t ^ \\bullet ) ^ { m + n } = c _ t e ^ { - n t } e ^ { G } \\omega _ { \\infty } ^ m \\wedge \\omega _ F ^ n , \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\tilde j ( \\gamma , z ) = \\left ( \\frac { c } { d } \\right ) \\epsilon ( d ) ( c z + d ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} \\mathcal M _ i : = \\{ \\vec { \\mathfrak u } \\in \\Omega ( x ) ^ { \\oplus N } : \\vec { \\mathfrak u } ^ T \\mathfrak V _ j = \\vec 0 ^ T , \\ \\forall i \\neq j \\} , \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} \\sum _ { \\substack { I \\subset [ N ] \\\\ q \\geq \\chi ^ I } } ( - 1 ) ^ { | I | } = \\sum _ { k = 0 } ^ r \\tbinom { r } { k } ( - 1 ) ^ k , \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} e ' < \\frac { e } { 2 q } \\sum _ { i = 1 } ^ { 2 q } \\frac { k ' _ i } { k _ i } + \\sum _ { i = 1 } ^ { 2 q } \\sigma _ i \\left ( r ' - ( k + 1 ) \\frac { k ' _ i } { k _ i } \\right ) . \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\Gamma ^ { n , \\ , n - 2 } _ { \\omega } = - \\Delta _ \\omega ^ { '' - 1 } \\bar \\partial ^ \\star _ \\omega \\ , ( \\partial \\omega ^ { n - 1 } ) . \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} \\int \\limits _ G F ( g ) \\ , d g & = 2 ^ { n ( n + 3 ) / 2 } \\ , \\pi ^ { n ( n + 1 ) / 2 } \\prod _ { m = 1 } ^ n \\frac { \\Gamma ( k - ( n + m ) / 2 ) } { \\Gamma ( k - ( n - m ) / 2 ) } \\\\ & = 2 ^ { n ( n + 2 ) } \\ , \\pi ^ { n ( n + 1 ) / 2 } \\prod _ { m = 1 } ^ n \\ , \\prod _ { j = 1 } ^ m \\frac { 1 } { 2 k - n - m - 2 + 2 j } . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} f ( a k ) = \\frac { 1 } { k a } \\sum _ { \\ell = 1 } ^ { a k } \\frac { 1 } { s + ( i + \\ell ) / ( a k ) } = \\sum _ { \\ell = 1 } ^ { k a } \\frac { 1 } { k t - k a + i + \\ell } & = \\sum _ { \\ell = 0 } ^ { k a - 1 } \\frac { 1 } { k t + i - \\ell } . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\zeta \\circ \\Delta _ t \\circ { \\rm g w } _ t ( A ) = \\lim _ { t \\to \\infty } \\frac { 1 } { t } \\ln \\left ( p ( e ^ { f ( t A ) } ) \\right ) , \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} \\Phi _ { n } = \\sum _ { k = 0 } ^ { 2 ^ { n - 2 } } \\ , \\phi _ { k n } ( M ) . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} \\mathfrak { Q } _ t ( x _ { t - 1 } , D _ { t - 1 } , \\xi _ t , D _ { t } ) = \\left \\{ \\begin{array} { l } \\inf _ { x _ t } \\ ; D _ { t - 1 } f _ t ( x _ t , x _ { t - 1 } , \\xi _ t ) + \\mathcal { Q } _ { t + 1 } ( x _ { t } , D _ t ) \\\\ x _ t \\in X _ t ( x _ { t - 1 } , \\xi _ t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\mathcal { P } ( t , x ) = \\mathcal { A } ( 1 + t , x ) \\mathcal { P } ( t , x ) = x \\mathcal { B } ( t x , x ) . \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\left \\{ \\zeta _ i , \\zeta _ j \\right \\} _ { \\infty } = 0 , \\ , \\left \\{ \\varphi _ i , \\varphi _ j \\right \\} _ { \\infty } = 0 , \\ , \\left \\{ \\zeta _ i , \\varphi _ j \\right \\} _ { \\infty } = \\pi _ { i , j } \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\| u ( t ) \\| _ { H ^ 1 ( x > v t ) } = 0 , \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} \\left \\{ [ n - ( k - l + 1 ) + 1 , n - ( k - l + 1 ) + j ] \\cup G : j \\in [ k - l + 1 ] , G \\in \\binom { [ n - ( k - l + 1 ) ] } { i _ { l - 1 } } \\right \\} \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } \\Phi _ { t } \\circ d W _ { t } = \\displaystyle \\mathop { l . i . m } \\limits _ { \\| \\bigtriangleup _ { n } \\| \\longrightarrow 0 } \\sum _ { i = 0 } ^ { n - 1 } \\Phi ( \\frac { B _ { t _ { i } } + B _ { t _ { i + 1 } } } { 2 } , t _ { i } ) ( B _ { t _ { i + 1 } } - B _ { t _ { i } } ) \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} T _ { u ^ { \\star n } } h - ( 1 - \\varepsilon ) ^ n h = \\gamma _ n ' h + G \\varphi _ n ' \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} \\lambda _ { n } ( c ) \\leq \\frac { 2 c } { \\pi } \\sum _ { k = n } ^ { \\infty } ( k + 1 / 2 ) \\norm { \\mathcal { F } _ c P _ k } ^ 2 . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} & \\mu _ H \\circ ( ( a _ H \\otimes b _ H ) \\otimes c _ H ) \\circ ( \\rho _ { p , q } ( G / H ) \\otimes I d _ r ) \\circ \\rho _ { p + q , r } ( G / H ) \\\\ & = \\mu _ H \\circ ( ( a _ H \\otimes b _ H ) \\otimes c _ H ) \\circ ( I d _ p \\otimes \\rho _ { q , r } ( G / H ) ) \\circ \\rho _ { p , q + r } ( G / H ) \\\\ & + ( - 1 ) ^ { r q } \\mu _ H \\circ ( ( a _ H \\otimes b _ H ) \\otimes c _ H ) \\circ ( I d _ p \\otimes \\tau _ { r , q } ( G / H ) \\circ \\rho _ { r , q } ( G / H ) ) \\circ \\rho _ { p , q + r } ( G / H ) . \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\theta ( \\mathbf G , \\phi _ { \\mathbf g } ) ( x ) = \\int _ { [ \\mathrm O _ { 2 , 2 } ] } \\Theta ( x , y ' y ; \\phi _ { \\mathbf g } ) \\mathbf G ( y ' y ) d y = \\int _ { [ \\mathrm O _ { 2 , 2 } ] } \\left ( \\sum _ { v \\in V _ 4 ( \\Q ) } \\omega ( x , y ' y ) \\phi _ { \\mathbf g } ( v ) \\right ) \\mathbf G ( y ' y ) d y , \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} [ h \\beta _ m , \\epsilon ] = \\left [ \\left ( \\begin{array} { c c } c ^ { - 1 } p ^ m & \\ast \\\\ 0 & c p ^ { - m } \\end{array} \\right ) , e \\right ] \\left [ \\left ( \\begin{array} { c c } 1 & 0 \\\\ \\frac { - d } { c } p ^ { 2 m } & 1 \\end{array} \\right ) , 1 \\right ] , \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\frac { R } { Z ( R ) } = \\langle a + Z , b + Z : 2 ( a + Z ) = 4 ( b + Z ) = Z \\rangle \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} X _ { i } \\hat { X } _ { j } & = v ^ { - d _ i \\delta _ { i j } } \\hat { X } _ { j } \\hat { X } _ { i } \\ ( i \\in J , j \\in J _ e ) & \\hat { X } _ { i } \\hat { X } _ { j } & = v ^ { d _ i b _ { i j } } \\hat { X } _ { j } \\hat { X } _ { i } \\ ( i , j \\in J _ e ) . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} u _ j = \\left ( \\begin{array} { c c } \\mathrm { I d } _ 2 & E _ j \\\\ 0 & \\mathrm { I d } _ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} | \\phi _ j ^ * \\psi _ j - \\phi _ k ^ * \\psi _ k | = \\Big | ( \\phi _ j - \\phi _ k ) ^ * \\psi _ j + \\phi _ k ^ * ( \\psi _ j - \\psi _ k ) \\Big | \\le \\| \\phi _ j - \\phi _ k \\| \\ , \\| \\psi _ j \\| + \\| \\phi _ k \\| \\ , \\| \\psi _ j - \\psi _ k \\| \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} 2 \\sum _ { g , \\sigma } f _ 1 \\ast _ g ( x _ { i _ \\sigma ( 1 ) i _ \\sigma ( 2 ) } x _ { i _ \\sigma ( 3 ) i _ \\sigma ( 4 ) } ) = 1 8 f _ 2 . \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} D : = \\{ ( \\omega , z ) \\in \\Omega \\times \\mathbb C ^ m \\ , | \\ , z \\in D _ \\omega \\} . \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} A u ( x ) = \\lim _ { t \\downarrow 0 } \\frac { P _ t u ( x ) - u ( x ) } { t } = \\lim _ { t \\downarrow 0 } \\frac { P _ t ^ U u ( x ) - u ( x ) } { t } . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} \\Delta _ { i , n } ^ { ( l ) } X = X _ { t _ { i , n } ^ { ( l ) } } - X _ { t _ { i - 1 , n } ^ { ( l ) } } , i \\geq 1 , l = 1 , 2 , \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} \\partial _ t \\rho ^ N _ t = \\L _ N ^ \\ast \\rho ^ N _ t , \\rho ^ N | _ { t = 0 } = \\rho ^ N _ 0 \\in C _ P ^ \\infty ( H _ N ) , \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} G ( u _ n , v _ n ) ~ \\geq ~ \\P _ { u _ n } ( X ( m _ n ) = v _ n ) ~ \\geq ~ \\delta ^ { \\kappa | v _ n - u _ n | } \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} e ( A , \\Phi ) : = \\frac { 1 } { 2 } \\left ( \\lvert F _ A \\rvert ^ 2 + \\lvert _ A \\Phi \\rvert ^ 2 \\right ) \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} b ( i _ { 1 } , i _ { 2 } , . . . , i _ { k } ; n ) = P ( X _ { i _ { 1 } , n } = X _ { i _ { 2 } , n } = \\cdots = X _ { i _ { k } , n } = 1 ) . \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\alpha _ p ^ { \\sharp } ( \\mathbf h , \\breve { \\mathbf g } , \\pmb { \\phi } ) = \\sum _ { r \\in \\mathcal R } \\Omega _ p ( r ) \\mathrm { v o l } ( \\Gamma _ { 0 0 } r \\Gamma _ { 0 0 } ) , \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} I _ m ( \\alpha ) & : = \\int _ 0 ^ { 2 \\pi \\alpha } \\cos ( x ) ( \\cos ( m x ) - 1 ) \\ , d x , \\\\ J _ m ( \\alpha ) & : = \\int _ { 0 } ^ { 2 \\pi \\alpha } \\left ( \\cos ( ( m + 1 ) x ) + \\cos ( m x ) - \\cos ( x ) - 1 \\right ) \\ , d x . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} ( M * N ) ( i , k ) = \\sum _ { j } M ( i , j ) * N ( j , k ) . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} w : = \\phi ( n ^ * n ) ^ { - 1 / 2 } \\phi ( m ^ * m ) ^ { - 1 / 2 } d n ^ * m d . \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} \\Delta w _ i = ( \\| A \\| ^ 2 - 4 H ^ 2 ) w _ i + 2 H \\langle A E _ i , \\xi \\rangle - 2 g _ i \\langle A , \\nabla \\xi \\rangle + \\langle E _ i , \\Delta \\xi \\rangle , \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} c ( P _ 2 , \\mathcal F ) = \\sum _ { C \\in \\mathcal K } \\abs { E ( G _ C ) } \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} P ( y ) & = \\beta _ { 0 } + \\beta _ { 1 } \\omega y + \\ \\beta _ { 2 } \\omega ^ { 2 } y ^ { 2 } + \\ \\beta _ { 3 } \\omega ^ { 3 } y ^ { 3 } + \\ \\beta _ { 4 } \\omega ^ { 4 } y ^ { 4 } = 0 , \\\\ \\beta _ { 0 } & = a ^ { 2 } - J _ { 1 } d = 4 \\mathfrak { S } _ { 0 } ( 4 \\mathfrak { S } _ { 0 } J _ { 2 } ^ { 2 } - J _ { 1 } ) , \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} \\omega ( k _ { \\theta } , ( k _ { \\theta _ 1 } , k _ { \\theta _ 2 } ) ) \\phi _ { \\mathbf g , \\infty } = \\mathrm { e x p } ( \\sqrt { - 1 } ( k + 1 ) ( - \\theta + \\theta _ 1 + \\theta _ 2 ) ) \\phi _ { \\mathbf g , \\infty } . \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} b _ m ( \\omega ) : = \\omega ( t ^ 0 , t ^ m ) \\ ; \\ , \\mbox { f o r $ m \\in \\Z $ . } \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n X _ i ' X _ i ^ { - 1 } = \\sum _ { i = 1 } ^ n \\Omega _ i + \\frac 1 2 \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\gamma _ { i j } \\left ( f ( X _ j X _ i ^ { - 1 } ) - f ( X _ i X _ j ^ { - 1 } ) \\right ) . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} N _ { n } = \\frac { n ! } { 2 } \\frac { N _ { \\mathrm { d } \\ , n + 1 } } { ( n + 1 ) ! } ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; N _ { \\mathrm { d } n } = \\frac { 1 } { 2 } N _ { \\mathrm { d 1 } } N _ { \\mathrm { d } n } \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\wedge ^ { r _ { m - 1 } } \\alpha _ { m - 1 } = a _ { m - 1 } \\circ a _ m ^ * . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} J _ { \\sigma , \\psi } ( g ) = | U _ n ( \\mathfrak { f } ) | ^ { - 1 } \\sum _ { h \\in U _ n ( \\mathfrak { f } ) } \\psi ( h ^ { - 1 } ) \\mathrm { t r } _ \\sigma ( g h ) . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( r ) & = \\frac { 1 } { 2 } P ( r ) ( \\hat { X } ( r ) ) ^ { 2 } ; \\\\ \\tilde { \\nu } ( r ) & = P ( r ) \\hat { X } ( r ) ( \\hat { \\Theta } ( r ) D \\sigma ( r ) + \\varepsilon _ { 2 } ( r ) ) + \\frac { 1 } { 2 } Q ( r ) \\left ( \\hat { X } ( r ) \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} \\log | x - y _ j | = \\log | x | - \\frac { x \\cdot y _ j } { | x | ^ 2 } + \\mathcal { O } ( | x | ^ { - 2 } ) \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{gather*} \\frac { a ( 3 \\tau ) } { c ( 3 \\tau ) } = \\frac 1 3 \\frac { { \\eta ( \\tau ) ^ 3 } } { \\eta ( 9 \\tau ) ^ 3 } + 1 = \\frac 1 3 \\big ( q ^ { - 1 } + 5 q ^ 2 - 7 q ^ 5 + 3 q ^ 8 + 1 5 q ^ { 1 1 } + \\cdots \\big ) \\end{gather*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow + \\infty } \\displaystyle \\int _ { \\Omega } D ( x ) ( | u _ { n } ( x ) | ^ { p ( x ) - 1 } u _ { n } ( x ) - | u _ { 0 } ( x ) | ^ { p ( x ) - 1 } u _ { 0 } ( x ) ) ( u _ { n } ( x ) - u _ { 0 } ( x ) ) d x = 0 . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} \\Lambda ( f , D , k ) = 2 ^ { 1 - k - \\nu ( N ) } | D | ^ { 1 / 2 - k } | c ( | D | ) | ^ 2 \\left ( \\prod _ { p \\mid M } \\frac { p + 1 } { p } \\right ) \\frac { \\langle f , f \\rangle } { \\langle h , h \\rangle } . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} \\Delta h & = \\partial _ r ^ 2 h + \\frac { d - 1 } { r } \\partial _ r h + \\frac { 1 } { r ^ 2 } \\Delta _ { 0 } h , h \\in C ^ 2 ( B ) , \\\\ \\sum _ { i , j = 1 } ^ d x _ i x _ j \\partial _ i \\partial _ j h ( x ) \\Big | _ { x = r \\xi } & = r ^ 2 \\partial _ r ^ 2 h ( r \\xi ) , h \\in C ^ 2 ( B ) , 0 < r \\le 1 , \\xi \\in S ^ { d - 1 } , \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} { \\displaystyle \\lim _ { \\varepsilon \\to 0 ^ { + } } \\gamma ^ { C _ { 1 } T } \\left ( T , \\beta \\right ) \\varepsilon = K \\in \\left ( 0 , \\infty \\right ) } . \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\tau ( \\alpha _ 2 , \\alpha _ 1 ) : = \\frac { 1 } { 3 } T ( \\alpha _ 2 , \\alpha _ 1 ) < \\min \\{ T ( \\alpha _ 2 , \\alpha ' ) ; T ( \\alpha , \\alpha _ 1 ) \\} . \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} B = \\begin{bmatrix} E & U ^ T \\\\ U & B '' \\end{bmatrix} \\quad E = \\begin{bmatrix} a & b \\\\ b & c \\end{bmatrix} \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} \\omega _ { Q } : = \\{ x \\in \\mathbb { G } : \\ , q ( x ) = 1 \\} , \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\mathfrak { E } ( M ^ { 2 n } , P ^ { k } ) = \\{ ( x , y ) \\in C ( M ^ { 2 n } , P ^ { k } ) \\times M ^ { 2 n } : x \\in P _ { \\sigma } ^ { ' } , y \\in W _ { \\sigma } , \\widehat { \\pi } ( x ) = \\mu ( y ) \\} . \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\nu = - \\frac { 1 } { a \\sigma \\mu } K _ 1 \\star G _ 2 ( \\nu ) , \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} \\mathcal L _ { \\mu _ 2 } = \\frac 1 \\gamma - \\frac { \\sqrt { \\mu } } { \\gamma ^ 2 } | D | \\coth ( \\sqrt { \\mu _ 2 } | D | ) + \\frac { \\mu } { \\gamma } \\big ( a - \\frac { 1 } { \\gamma ^ 2 } \\coth ^ 2 ( \\sqrt { \\mu _ 2 } | D | ) \\Big ) \\partial _ x ^ 2 , \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} - \\frac { g _ t } { g - g _ x } = \\frac { ( \\sum _ { i = 1 } ^ N c _ i ) g ^ { N + 1 } + \\sum _ { n = 1 } ^ N \\epsilon ^ { ( n - 1 ) n } g ^ { N - n + 1 } d _ n e ^ { ( n - 1 ) x } ( e ^ x D _ { N - n , t } ^ 2 - 2 D _ { N - n + 1 , t } ^ 0 ) + o ( \\epsilon ^ { N ( N + 1 ) } ) } { g ^ { N + 1 } + \\sum _ { n = 1 } ^ N \\epsilon ^ { ( n - 1 ) n } g ^ { N - n + 1 } d _ n e ^ { n x } D _ { N - n } ^ 2 + o ( \\epsilon ^ { N ( N + 1 ) } ) } . \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} b ( f g ) = b ( f ) g + f b ( g ) , \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\{ \\imath _ 1 ^ { ( s ) } , \\dots , \\imath _ { k _ s } ^ { ( s ) } \\} = \\{ \\imath \\mid \\delta ( \\imath , s ) = 1 \\} . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} \\lim _ { M > N \\to \\infty } S _ 2 = 0 . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} P ( x ) = \\sum \\limits _ { n \\geq 1 } \\frac { C _ { n - 1 } } { n } x ^ n = \\sum \\limits _ { n \\geq 1 } ( 2 n - 2 ) ! \\frac { x ^ n } { ( n ! ) ^ 2 } . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} u _ { i } = - \\alpha \\nabla _ { x _ { i } } L _ { i } + h _ { i } . \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} y & = x + \\pi _ P ( y - x ) + \\pi _ P ^ \\perp ( y - x ) \\\\ & = x + z ( y ) + u _ x \\left ( z ( y ) \\right ) . \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\sum _ { j \\leq i } s _ { i j } ^ { a _ i } \\frac { \\prod _ { k \\leq i - 1 } ( 1 - s _ { i j } / s _ { i - 1 , k } ) } { \\prod _ { k \\leq i } ^ { k \\ne j } ( 1 - s _ { i j } / s _ { i k } ) } = ( t _ i ^ 2 v ^ { 2 d _ { i - 1 } - 2 d _ i } ) ^ { a _ i } . \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\lim \\limits _ { A , B \\rightarrow 0 } \\int _ { x _ 1 ^ { A B } } ^ { x _ 2 ^ { A B } } \\rho _ * ^ { A B } d x = \\sqrt { \\rho _ { + } \\rho _ { - } } ( u _ - - u _ + ) t , \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} s _ { ( a ^ b ) } ( 1 ^ m ) = \\prod _ { ( i , j ) \\in ( a ^ b ) } \\frac { m + j - i } { ( a - j ) + ( b - i ) + 1 } , \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\tilde { p } _ H \\bigl ( ( y \\star u , y ) , ( z \\star u , z ) \\bigr ) ~ = ~ p _ H ( y , z ) , \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} c _ 1 \\textbf { E } ^ 1 _ { \\alpha , 1 , \\frac { - \\alpha } { 1 - \\alpha } , a ^ + } x ( a ) + c _ 2 ~ ^ { A B R } _ { a } D ^ \\alpha x ( a ) = 0 , \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{gather*} 2 L ( \\psi , { 1 / 2 } ) ^ 2 = L \\big ( \\psi ^ 2 , { 1 } \\big ) . \\end{gather*}"}
-{"id": "7496.png", "formula": "\\begin{align*} f ( t , \\omega + \\varepsilon h , x ) - f ( t , \\omega , x ) = \\int _ 0 ^ \\varepsilon D ^ h f ( t , \\omega + r h , x ) d r . \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\beta = \\frac { 1 } { 2 } \\log X + \\frac { i } { 2 T } . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} \\mathfrak { a } ^ + = \\{ { \\rm d i a g } ( a _ 1 , \\ldots , a _ n , - a _ 1 , \\ldots , - a _ n ) : a _ 1 < \\ldots < a _ n < 0 \\} . \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} { { d } _ { k } } ( 1 - 6 { x _ { k } ^ { * } } ^ 2 + { x _ { k } ^ { * } } ^ 4 ) - { \\omega _ { k } } ( x _ { k } ^ { * } - { x _ { k } ^ { * } } ^ 3 ) = 0 . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} ( q , r ) : = \\left ( \\frac { 8 ( p + 1 ) } { n p } , 2 p + 2 \\right ) . \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\phi _ { \\mathbf g , \\infty } \\left ( \\left ( \\begin{smallmatrix} x _ 1 & x _ 2 \\\\ x _ 3 & x _ 4 \\end{smallmatrix} \\right ) \\right ) = ( x _ 1 + \\sqrt { - 1 } x _ 2 + \\sqrt { - 1 } x _ 3 - x _ 4 ) ^ { k + 1 } \\mathrm { e x p } ( - \\pi \\mathrm { t r } ( x ^ t x ) ) . \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} \\lambda _ n = \\lambda _ { k , j } = ( 1 - \\tfrac 1 k ) e ^ { \\frac { 2 \\pi i j } { k } } , j = 0 , \\dots , k - 1 , k = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} ( \\hat { u } _ 1 , \\hat { \\theta } , \\hat { h } _ 1 ) ( \\tau , \\xi , \\eta ) ~ : = ~ ( u _ 1 , \\theta , h _ 1 ) ( t , x , y ) . \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} \\hat { \\tau } = \\frac { 1 } { m ( m + 1 ) } \\sum \\limits _ { i , j = 1 } ^ { m + 1 } \\hat { R } ( E _ i , E _ j , E _ j , E _ i ) . \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} F _ { 1 } = \\sqrt { \\frac { d } { k ( d + 1 - k ) } } F K \\quad G _ { 1 } = F _ { 1 } ^ { T } F _ { 1 } = \\frac { d } { k ( d + 1 - k ) } K ^ { T } G K . \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\zeta ( t , \\cdot ) = \\sum _ { \\mu = 1 } ^ N m _ { u _ \\mu } ( t , \\cdot ) . \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} ( Y + x _ 0 ^ 3 + y _ 0 ^ 3 - z _ 0 ^ 3 ) = 3 D _ 1 , \\\\ ( Y + x _ 0 ^ 3 - y _ 0 ^ 3 + z _ 0 ^ 3 ) = 3 D _ 2 , \\\\ ( Y + x _ 1 ^ 3 + y _ 1 ^ 3 - z _ 1 ^ 3 ) = 3 D _ 3 , \\\\ ( Y - x _ 2 ^ 3 + y _ 2 ^ 3 + z _ 2 ^ 3 ) = 3 D _ 4 . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} & g ' _ 0 ( x , a , b ) = a \\bigl ( 2 F ^ { 0 , 2 } + 4 a ^ 2 F ^ { 1 , 3 } \\bigr ) \\quad \\mbox { a n d } \\\\ & g '' _ 0 ( x , a , b ) = a \\bigl ( 2 F ^ { 0 , 3 } + 8 a ^ 2 F ^ { 1 , 4 } + 8 a ^ 4 F ^ { 2 , 5 } \\bigr ) , \\qquad \\qquad \\qquad \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} N ( m ) = \\frac { 1 } { 2 ^ { m + 1 } } \\sum _ { i = 0 } ^ { m } ( - 1 ) ^ { i } \\sum _ { a _ { 1 } , \\cdots , a _ { i + 1 } = 1 } ^ { \\infty } \\delta _ { a _ { 1 } + \\cdots + a _ { i + 1 } , m + 1 } \\prod _ { j = 1 } ^ { i + 1 } \\frac { ( 2 a _ { j } ) ! } { a _ { j } ! } . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} g _ { d _ i } ^ { ( i ) } ( q _ i ) = \\sum _ { \\substack { 0 \\leq j < n _ i \\\\ ( j , n _ i ) = d _ i } } q _ i ^ j , \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} \\lambda _ { 2 1 } \\begin{pmatrix} A _ 1 & A _ 2 \\end{pmatrix} & = 0 , \\\\ \\lambda _ { 3 1 } \\begin{pmatrix} A _ 1 & A _ 2 \\end{pmatrix} & = 0 , \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} { \\tilde \\theta _ n } = 2 n \\pi \\tilde f T + \\tilde \\varphi + 2 \\pi { k _ n } \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} \\Omega ( t ) = \\int _ { - \\infty } ^ { \\infty } x u ( t , x ) d x \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\sum _ { \\mu , \\nu = 1 } ^ N \\Big \\| \\| u _ \\mu \\| _ { W ^ { 2 , r } _ x } \\| u _ \\nu \\| _ { L _ x ^ { r } } ^ { 2 p } \\Big \\| _ { L ^ { q ' } _ { t > T } } & \\\\ \\lesssim \\sum ^ N _ { \\mu , \\nu = 1 } \\Big \\| \\| u _ \\mu \\| _ { W ^ { 2 , r } _ x } \\Big ( \\| u _ \\nu \\| _ { L _ x ^ { r } } ^ { 2 p ( 1 - \\theta ) } \\| u _ \\nu \\| _ { L _ x ^ { r } } ^ { 2 p \\theta } \\Big ) \\| _ { L ^ { q ' } _ { t > T } } . & \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} \\mathfrak { M } ( - q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = \\exp \\left ( \\frac { 2 q ^ 2 \\log 2 } { \\tau } + \\frac { q \\log q } { \\tau } + O ( q ) \\right ) , \\ , \\ , q \\rightarrow \\infty , \\ , \\ , | \\arg ( q ) | < \\pi . \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} \\begin{dcases} \\sum _ { e \\in E ( G ) } w ( e ) = 1 , \\\\ \\sum _ { e : \\ , v \\in e } B ( v , e ) = 1 , \\ \\ v \\in V ( G ) , \\\\ w ( e ) ^ { p - r } \\prod _ { v \\in e } B ( v , e ) = \\alpha , \\ \\ e \\in E ( G ) . \\end{dcases} \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { \\alpha _ 1 } \\left \\| U \\left ( \\cdot , t \\right ) - \\mathcal { B } _ { M } \\left ( \\cdot , t \\right ) \\right \\| _ { L ^ { \\infty } } = 0 \\mbox { u n i f o r m l y $ x \\in \\R ^ n $ } . \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} \\tilde { p } _ H \\bigl ( ( y \\star u , y ) , ( z \\star u , \\vartheta ) \\bigr ) ~ = ~ p _ H ( y \\star u , z \\star u ) - p _ H ( y , z ) ~ = ~ G T _ y A _ u \\ 1 _ { \\{ z \\star u \\} } ( e ) \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} { \\rm G r } _ F ^ p \\tilde H ^ j ( F , \\C ) = \\frac { F ^ p \\tilde H ^ j ( F , \\C ) } { F ^ { p + 1 } \\tilde H ^ j ( F , \\C ) } . \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} I _ { \\alpha } \\varphi ( z ) = \\sum _ { n \\geq 0 } ( 1 + n ) ^ { - \\alpha } \\hat { \\varphi } ( n ) z ^ n . \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} \\tilde { \\Psi } = \\Phi ^ { \\vee } \\qquad \\tilde { \\Phi } = \\Psi ^ { \\vee } . \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} B _ { ( u , y _ { u } ^ { \\ast } ) } ( x ^ { \\ast } ) : = B _ { ( u , y _ { u } ^ { \\ast } ) } ^ { 0 } ( x ^ { \\ast } ) = \\Big \\{ x \\in J ( u ) \\ , : \\ , ( x , 0 _ { u } ) \\in ( M F _ { u } ) ( x ^ { \\ast } , y _ { u } ^ { \\ast } ) \\Big \\} . \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} \\tilde { \\eta } = ( \\eta | \\mathcal { I } _ { i , n } ^ { ( l ) } | ) ^ { p _ l / ( 2 p _ l \\wedge 1 ) } , a = \\mathbb { E } \\big [ \\big ( \\int _ { t _ { i - 1 , n } ^ { ( l ) } } ^ { t _ { i , n } ^ { ( l ) } } \\| \\tilde { \\sigma } _ s - \\tilde { \\sigma } _ s ( r ) \\| ^ 2 d s \\big ) ^ { p _ l \\vee \\frac { 1 } { 2 } } \\big | \\mathcal { S } \\big ] \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} \\int _ { \\Omega } g ( y ) | x - y | ^ n d y = - n ( n + 1 ) \\int _ { \\Omega } \\Delta ^ { - 1 } ( g ) ( y ) | x - y | ^ { n - 2 } d y \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} P ( \\tau _ g \\otimes \\alpha _ g ) = \\alpha _ g \\circ P , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , g \\in G . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\left \\vert \\left ( \\pi _ { P _ x \\mid \\tilde \\Sigma _ 1 } \\right ) ^ { - 1 } ( z ) - \\left ( \\pi _ { P _ x \\mid \\tilde \\Sigma _ 1 } \\right ) ^ { - 1 } ( z ' ) \\right \\vert & = \\vert y - y ' \\vert \\\\ & \\leq \\vert \\pi _ { P _ x } ( y - y ' ) \\vert + \\vert \\pi _ { P _ x } ^ \\perp ( y - y ' ) \\vert \\\\ & \\leq \\vert z - z ' \\vert + \\frac 1 2 \\vert y - y ' \\vert . \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} \\begin{cases} \\underbar r _ 1 ^ n - \\epsilon \\le u ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 1 ^ n + \\epsilon \\cr \\underbar r _ 2 ^ n - \\epsilon \\le v ( x , t ; t _ 0 , u _ 0 , v _ 0 ) \\le \\bar r _ 2 ^ n + \\epsilon , \\end{cases} \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} = \\mu ( \\gamma ) \\int _ { \\mathbb { G } } \\frac { | u ( x ) | ^ { p } } { q ^ { p s } ( x ) } d x . \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} m _ { 3 / 2 } = \\frac { 1 } { 4 } \\left ( ( d - 2 ) ^ 2 - \\tau ( C ) + \\sum _ { j } \\nu _ j \\right ) . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} \\gamma _ i ( U _ i ^ m ( U _ i ^ * ) ^ n ) = 1 , \\qquad \\forall \\ , m , n \\in \\N . \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\Phi _ { k } ^ { ( N ) } ( f ) = \\mathbb { E } \\left [ f \\mid \\mathcal { A } ^ { ( k ) } \\right ] + \\sum _ { \\substack { 1 \\leqslant L ^ { \\prime } \\leqslant N + 1 - k \\\\ k < l _ { 1 } < . . . < l _ { L ^ { \\prime } } < N + 1 } } ( - 1 ) ^ { L } \\mathbf { P } _ { \\mathcal { A } ^ { ( l _ { 1 } ) } | \\mathcal { A } ^ { ( k ) } } \\dots \\mathbf { P } _ { \\mathcal { A } ^ { ( l _ { L ^ { \\prime } } ) } | \\mathcal { A } ^ { ( l _ { L ^ { \\prime } - 1 } ) } } \\cdot \\mathbb { E } \\left [ f | \\mathcal { A } ^ { ( l _ { L ^ { \\prime } } ) } \\right ] . \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} \\frac { G ( z ) } { A ( z ) } \\sum _ n \\frac { | c _ n | ^ 2 } { z - t _ n } = \\bigg ( \\sum _ n \\frac { c _ n \\mu _ n ^ { 1 / 2 } } { z - t _ n } \\bigg ) \\cdot \\bigg ( \\sum _ n \\frac { G ( t _ n ) \\bar c _ n } { A ' ( t _ n ) \\mu _ n ^ { 1 / 2 } ( z - t _ n ) } \\bigg ) . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\{ f _ i = \\frac { \\sqrt { M } } { \\sqrt { a _ 1 ^ i \\cdots a _ d ^ i } } \\chi _ { R _ i } : i = 1 , \\dots , M \\} \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\begin{dcases} \\sum _ { e \\in E ( G ) } w _ i ( e ) = 1 , \\\\ \\sum _ { e : \\ , v \\in e } B _ i ( v , e ) = 1 , \\ \\ v \\in V ( G ) , \\\\ w _ i ( e ) ^ { 1 - r / p _ i } \\prod _ { v \\in e } ( B _ i ( v , e ) ) ^ { 1 / p _ i } = \\alpha _ i ^ { 1 / p _ i } , \\ \\ e \\in E ( G ) . \\end{dcases} \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\prod _ { 0 \\leq \\ell \\leq q } ( x - \\theta ^ { \\ell } ) = x ^ { q + 1 } - 1 \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\dd Y _ t = b ( Y _ { t - } ) \\dd t + \\sigma ( Y _ { t - } ) \\dd W _ t + \\int v ( x , Y _ { t - } ) ( p - q ) ( \\dd x , \\dd t ) , \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} \\sum _ { i \\geq 0 } C ( 2 , i ) x ^ i = \\frac { 1 + 2 x C ( 2 x ) } { 1 + x } = \\frac { 1 } { 1 - x C ( 2 x ) } = \\frac { 4 } { 3 + \\sqrt { 1 - 8 x } } . \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} d ( \\gamma _ i ( - \\delta ) , \\gamma _ i ( 0 ) ) - d ( \\gamma _ i ( - \\delta ) , p ) = \\int _ 0 ^ { d _ i } h ' ( t ) d t = \\int _ 0 ^ { d _ i } g _ { w _ i ( t ) } ( w _ i ( t ) , \\dot \\sigma _ i ( t ) ) d t \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} D ' _ { A ^ \\delta } : = \\{ a \\in A ^ \\delta : a d = d a d \\in D \\} \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\ < \\sigma _ k \\cdot \\nabla \\omega , D F \\ > ^ 2 = 2 \\pi ^ 2 \\sum _ { l , m \\in \\Lambda } C _ { k , l } C _ { k , m } \\ , ( \\partial _ l f ) ( \\Pi _ \\Lambda \\omega ) ( \\partial _ m f ) ( \\Pi _ \\Lambda \\omega ) \\ , \\ < \\omega , e _ k e _ { - l } \\ > \\ < \\omega , e _ k e _ { - m } \\ > . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} P ^ { \\vec { h } } _ { \\Lambda , \\eta , \\beta , H } ( \\sigma ^ { \\eta } _ x = s g n ( h _ x ) ) \\geq a ( \\beta , H , | h _ x | ) : = \\frac { e ^ { - 2 d \\beta + H | h _ x | } } { e ^ { - 2 d \\beta + H | h _ x | } + e ^ { 2 d \\beta - H | h _ x | } } , \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} S _ { a _ { 1 } , a _ { 1 } + b _ { 1 } } ^ { a _ { 2 } , a _ { 2 } + b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k \\right ) = S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k \\right ) + \\left ( k + 1 \\right ) S _ { a _ { 1 } , b _ { 1 } } ^ { a _ { 2 } , b _ { 2 } , p _ { 2 } } \\left ( p _ { 1 } , k + 1 \\right ) . \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} M _ { h } & = \\left \\{ p \\in M ; h + U ( p ) \\geq 0 \\right \\} \\\\ d s _ { h } ^ { 2 } & = ( h + U ) d s ^ { 2 } \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\mathrm { w t } ( T ^ { ( k + 1 ) } ) & = \\mathrm { w t } ( S _ { 1 } \\cdots S _ { n - l + k + 1 } T ^ { ( k ) } ) \\\\ & = ( \\underbrace { \\lambda _ { l - k - 1 } , \\ldots , \\lambda _ { l } } _ { k + 2 } , \\underbrace { 0 , \\ldots , 0 } _ { n - l } , \\lambda _ { l - k - 2 } , \\ldots , \\lambda _ { 1 } , 0 , \\ldots ) . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } \\Phi ( B ( t ) , t ) \\circ d B ( t ) = \\int _ { 0 } ^ { T } \\Phi ( B ( t ) , t ) d B ( t ) + \\displaystyle \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } \\frac { \\partial \\Phi ( B ( t ) , t ) } { \\partial B ( t ) } d t \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} [ { m _ \\epsilon } ^ { - 1 } \\gamma _ { 0 } { m _ \\epsilon } , { m _ { \\epsilon ' } } ^ { - 1 } \\gamma _ { 0 } { m _ { \\epsilon ' } } ] = 1 . \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} E _ G \\ , = \\ , E _ H \\times ^ H G \\ , \\longrightarrow \\ , X \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} { \\tilde \\theta _ n } = 2 n \\pi \\tilde f T + \\tilde \\varphi = 2 n \\pi \\tilde f T + \\varphi _ B ^ { { \\rm { R F } } } - \\varphi _ A ^ { { \\rm { R F } } } + { \\varphi _ { { h _ B } } } - { \\varphi _ { { h _ A } } } \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} H = \\left ( \\tbinom { N + i - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\tbinom { N + j - 1 } { N - 1 } ^ { \\frac { 1 } { 2 } } \\mathfrak { d } _ 1 ^ N \\dot { \\phi } ( i + j ) \\right ) _ { i , j \\in \\N } \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} \\langle P ( x ) , Q ( y ) \\rangle = \\tilde { \\phi } \\left ( d _ 1 ( x _ 1 , y _ 1 ) , . . . , d _ N ( x _ N , y _ N ) \\right ) . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} 0 = \\Delta _ { ( \\phi ( v _ { b + 1 } ) - p ^ * ( \\vartheta _ n ) v _ { b + 1 } ) \\wedge v _ { b + 2 } \\wedge \\dots \\wedge v _ n } = \\lambda _ { b } \\Delta _ { \\{ b , b + 2 , \\dots , n \\} } - p ^ * ( \\vartheta _ n ) \\Delta _ { \\{ b + 1 , \\dots , n \\} } . \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} & [ \\Gamma _ i , \\Gamma _ j ] = 1 , ( 1 \\leq i , j \\leq n ) , \\\\ & ( \\Gamma _ 0 \\Gamma _ i ) ^ 2 = ( \\Gamma _ i \\Gamma _ 0 ) ^ 2 ( 1 \\leq i \\leq n ) , \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} U _ { L } ( \\theta , \\lambda , P _ { B S } ) = \\mu T _ { s } - E , \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} v _ t = \\int _ 0 ^ t V _ { \\vartheta '' } ( t - s ) C ^ { \\Delta , \\sigma } _ { \\vartheta '' \\vartheta } v _ s d s \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} \\int _ { \\Gamma } ^ { } u _ { \\lambda } ( t ) d \\Gamma = \\int _ { \\Gamma } ^ { } u _ { 0 \\Gamma } d \\Gamma \\mbox { f o r a l l ~ } t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} \\varphi _ \\psi ( f ) = \\psi ( f | _ { G _ Z } ) \\ \\ \\ \\ f \\in C _ c ( G ) . \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } n ^ a = \\frac { x ^ { 1 + a } } { 1 + a } + \\zeta ( - a ) + O _ { a } \\left ( x ^ a \\right ) , \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ p \\lambda _ i \\vec 1 _ { V _ i ^ + , V _ i ^ - } = \\vec x \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle ( z _ i ^ * ) ^ { p - 1 } \\le a _ i , \\ , \\ , \\ , \\ , i = 1 , 2 \\ldots , m _ 0 , \\\\ \\displaystyle ( z _ i ^ * ) ^ { p - 1 } = a _ i , \\ , \\ , \\ , \\ , i = m _ 0 + 1 , \\ldots , n , \\end{cases} \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} f ( m ) = ( \\Lambda \\cdot g ) ( m ) = \\sum _ \\Lambda \\Lambda _ { 2 } g ( S ^ { - 1 } ( \\Lambda _ { 1 } ) m ) \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\sup _ { k \\in \\mathbb { N } } \\| \\sigma _ { 1 - 1 / k } g \\| _ k = \\sup _ { k \\in \\mathbb { N } } \\biggl ( \\frac 1 k \\sum _ { j = 0 } ^ { k - 1 } | g ( \\lambda _ { k , j } ) | ^ 2 \\biggr ) ^ { 1 / 2 } \\asymp \\| \\{ \\langle \\widehat { K } _ { \\lambda _ n } , g \\rangle \\} _ { n = 1 } ^ { \\infty } \\| _ { \\infty , 2 } . \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} ( A , B , C , H _ i , N _ k ) \\mapsto ( \\tilde A , \\tilde B , \\tilde C , \\tilde H _ i , \\tilde N _ k ) : = ( S A S ^ { - 1 } , S B , C S ^ { - 1 } , S H _ i S ^ { - 1 } , S N _ k S ^ { - 1 } ) , \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} \\Pr ( R ) = \\frac { | Z ( R ) | } { | R | } + \\frac { 1 } { | R | ^ 2 } \\underset { r \\in R \\setminus Z ( R ) } { \\sum } | C _ R ( r ) | \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} r ^ { - } _ { \\psi } \\left [ \\left ( \\begin{array} { c c } a \\\\ & a ^ { - 1 } \\end{array} \\right ) , 1 \\right ] \\varphi ( x ) & = | a | _ p ^ { 1 / 2 } \\chi _ { \\psi } ( a ) \\varphi ( a x ) , \\\\ r ^ { - } _ { \\psi } \\left [ \\left ( \\begin{array} { c c } 1 & b \\\\ & 1 \\end{array} \\right ) , 1 \\right ] \\varphi ( x ) & = \\psi ( b x ^ 2 ) \\varphi ( x ) , \\\\ r ^ { - } _ { \\psi } \\left [ s , 1 \\right ] \\varphi ( x ) & = \\gamma ( \\psi ) \\int _ { \\Q _ p } \\varphi ( y ) \\psi ( 2 x y ) d y . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\frac { \\partial ^ 2 u } { \\partial t ^ 2 } - { \\rm d i v } \\ , ( a ( t , z ) | D u _ t | ^ { p - 2 } D u _ t ) + \\beta ( z ) u _ t - \\Delta u \\in \\left [ f _ l ( t , z , u ) , f _ u ( t , z , u ) \\right ] \\mbox { i n } \\ T \\times \\Omega , \\\\ u | _ { T \\times \\partial \\Omega } = 0 , \\ u ( 0 , z ) = u _ 0 ( z ) , \\ u _ t ( 0 , z ) = u _ 1 ( z ) . \\end{array} \\right \\} \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} \\Phi ^ N _ t ( \\xi ) = \\Phi ^ { ( m ) } _ t ( \\xi ) \\mbox { a . s . f o r a l l } t > 0 . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} \\partial _ t \\psi + ( u _ 1 \\partial _ x + u _ 2 \\partial _ y ) \\psi = \\nu \\partial _ { y } ^ 2 \\psi . \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} ( a ) _ b : = \\dfrac { \\Gamma ( a + b ) } { \\Gamma ( a ) } , \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} y _ k \\coloneqq \\begin{cases} x _ k , & k \\in \\{ 0 , \\dotsc , N - 1 \\} , \\\\ x _ { N - 1 - k } , & k \\in \\{ N , \\dotsc , 2 N - 1 \\} . \\end{cases} \\end{align*}"}